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Monotone Iterative Methods
Chapter V
Monotone Iterative Methods 1
An Abstract Formulation
Let Z be a B a n a c h space. An order cone K C Z is a closed set such t h a t for all u and v E K , u + v E K , for a l l t E R + a n d u E K , t u E K , if u E K and - u E K t h e n u - 0. Such a cone K induces an order on Z" u<_v
if and only if
v-uEK.
We write equivalently u < v or v >_ u. T h e cone is said to be normal if t h e r e exists c > 0 such t h a t 0 < u < v implies [[ul[ <_ cl{v[I. T h e following t h e o r e m gives conditions for an increasing sequence ( a n ) n to converge to a fixed point of T. T h e o r e m 1.1. Let X C Z be continuously included Banach spaces so that Z has a normal order cone. Let a and ~ E X , a < ~, E = {u E X l a < u < Z}
and let T 9 s ---, X be completely continuous in X . sequence (an)n defined by ao = a,
an = T a n - 1
(1.1)
Assume that the
(1.2)
is bounded in X and for all n E N a n ~__ C~n+l ~--- Z.
Then the sequence (~n)n converges monotonically in X to a fixed point u of T such that
a
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Proof. Claim - The sequence (an)n converges in X . The sequence (an)n is increasing and included in C. As the set A = {an ] n E N} is bounded in X , T ( A ) is relatively compact in Z . Hence, any sequence (ank)k C (an)n has a converging subsequence in X and therefore in Z. As the order cone is normal and the sequence is monotone, the sequence itself converges in Z, i.e. there exists u E Z so t h a t Z
a _<_u <_/~ and an --* u. It follows t h a t all such subsequence converging in X have the same limit u, which implies X
an----~U.
T.
Next, we deduce from the continuity of T that u is a fixed point of [::] A similar result holds to prove the convergence of the sequence (/~n)n.
T h e o r e m 1.2. Let X C Z be continuously included Banach spaces so that Z has a normal order cone. Let a and ~ E X , a < ~, C be defined by (1.1) and T " E ~ X be completely continuous in X . Assume the sequence (13n)n defined by ~0=~, ~n=T~n_l, (1.3)
is bounded in X and for all n E N ~>~.+1>~. Then the sequence (~n)n converges monotonically in X to a fixed point v of T such that ~
a < Ta
and
T ~ < ~.
Then, the sequence (an)n and (~n)n defined by (1.2) and (1.3) converge monotonically in X to fixed points Umin and Umax of T such that a <_ Umi~ < Um~ < ~.
Further, any fixed point u E E of T verifies Umin ~_ U ~ Umax.
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Proof. Claim 1 - The sequence (an)n converges in X to a fixed point Umin of T such that a < Umin <_ ~. As T is monotone increasing, we prove by recurrence that for any n C N, an ~_ an+l <_ ~. Hence, (an)n C s and since T(s is relatively compact in X the sequence (an)n is bounded in X. The claim follows now from Theorem 1.1. Claim 2 - The sequence (~n)n converges in X to a fixed point Umax of T such that Umin <__Uma~ <_ ~. As for Claim 1 we prove existence of a fixed point u , ~ such t h a t a _< U m ~ <_ ~. Next, as T is monotone increasing and a <__ ~ we deduce by recurrence t h a t an <__ ~n. At last, going to the limit we obtain Umin <_ U m ~ . Claim 3 - A n y fixed point u c g of T verifies Umin <_ u <_ Umax. Since a _< u < /3, we deduce by recurrence an = T a n - 1 <_ T u = u <_ T ~ n - I ~n. The claim follows now going to the limit. [::] -
-
R e m a r k 1.1. In Theorem 1.1, we cannot prove the fixed point u is minimal since we only have some control on T ( u ) for u C {an [ n C N}. Consider for example any continuous function T : [ - 1 / 2 , 1/2] ~ R such that 1 T ( _ 2~+1) = 1 T ( - ~ )1 = 2(n+1), 2n+1" The assumptions of the theorem are satisfied for a = - 1 / 2 and ~ - 1/2. 1 The sequence (an)n - (--~-~)n converges to the fixed point u - 0. However 1 1 E [ - ~1 ' ~] are fixed points u - 0 is not minimal since the points Un 2n+1 ofT.
2 2.1
Well-ordered The
Lower
Periodic
and
Upper
Solutions
Problem
Consider the periodic b o u n d a r y value problem u"= f(t,u), u(a) = u(b), u' (a) = u'(b),
(2.1)
where f is a continuous function. Our aim is to build an approximation scheme, easy to compute, that converges to solutions of (2.1). To this end, given continuous functions a and ~, and M > 0, we consider the sequences (an)n and (~n)n defined by OZ0 :
C~,
Ol nt, - - M o~n - - f ( t , Oln -- 1 ) - - M O~n - 1 , / / : : n(b)
(2.2)
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and
0=Z, ~ - M~n = f(t, ~ n - s ) - Mfln-1, ~n (a) = ~,~ (b), ~" (a) = ~" (b).
(2.3)
The approximations an and/~n are "easy to compute", in the sense that for every n, the problems (2.2) and (2.3) are linear and have unique solutions which read explicitely
an(t) = n(t)
L
G(t,s)(f(S, an_l(S)) - M a n _ l ( S ) ) d s ,
=
-
where G(t, s) is the Green function of the problem
u " - M u = f(t), u(a) = u(b), u'(a) = u'(b).
(2.4)
Clearly this method does not avoid numerical difficulties such as those related to stiff systems. The next theorem proves the convergence of the a~ and/3n. T h e o r e m 2.1. Let a and/3 C C2([a, b]), a < ~ and E := {(t, u) e [a, b] x R I ~(t) _ u _ fl(t)}.
(2.5)
Assume f : E ~ ]R is a continuous function, there exists M > 0 such that aZl
(t, ux),
e E,
Ul ~ U2 implies f(t, u 2 ) - f(t, UX) ~ M ( u 2 - Ul) and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , a(a) = a(b), a'(a) >_ a'(b), ~"(t) <_ f ( t , ~ ( t ) ) , ~(a) = ~(b), ~'(a) <_ ~'(b). Then the sequences (an),~ and (fln)n defined by (2.2) and (2.3) converge monotonically in Cl([a,b]) to solutions Umin and Umax of (2.1)such that a ~__ Umi n ~ Urea x ~ ~.
Further, any solution u of (2.1) with graph in E verifies Umi n ~_~U ~_ Uma x.
Proof. Let X = CS([a,b]), Z = C([a,b]), K = {u E Z l u ( t ) >_ 0 on [a,b]} be the ordered cone in Z and $ be defined from (1.1). Define the operator T'$---+ X by Tu(t) = Sa b G(t, s)(f(s, u(s)) - Mu(s)) ds,
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where G(t, s) is the Green function of (2.4). This operator is continuous in X and monotone increasing. Further, T(E) is relatively compact in X, a < T~ and ~ >_ T~. The proof follows now from Theorem 1.3. K] R e m a r k Notice that the assumption a and ~ E C2([a, b]) is not restrictive. If these functions are only C2-10wer and upper solutions, the first iterates al and ~1 satisfy the assumptions of the theorem and are such that a <_
al <~1 <~. R e m a r k Recall that existence of the minimal and maximal solutions and Umax follows from Theorem I-2.4.
Umi n
Next, we consider a derivative dependent problem (2.6)
u" - f (t, u, u'), u(a) - u(b), u'(a) = u'(b).
As above, given a, 13 C C~([a, b]) and L > O, we consider the approximation schemes Ol0
--
Ol~
, 1) - L a n _ 1, a n'' -- L a n -- f (t, an-- 1, an-Ol n ( a )
- - oL n ( b ) ,
! oL n ( a )
(2.r)
! - - oL n ( b )
and ~0 : ~
3 ~ - L3n = f(t, Z n - 1 , 3 ~ _ 1 ) - L3n-1, ~ (a) = Zn (b), /3" (a) = Z~ (b).
(2.8)
Such problems lead to a major difficulty. A straightforward application of the previous ideas would be to assume that for any ul, u2, Vl and v2, ul _ u2 implies f ( t , u2, v2) - f ( t , ul, vl) <_ L(u2 - ul). This would mean that f does not depend on derivatives. The following result turns out the difficulty. T h e o r e m 2.2. Let a and Z C
C2([a, b]),
a < 13 and
E "= {(t, u, v) e [a, b] x R2 I a(t) _< u <__Z(t)}.
(2.9)
A s s u m e f " E -o R is a continuous function, there exists M >_ 0 such that f o r all (t, ul, v), (t, u2, v) C E, ul <_ u2
implies
f ( t , u2, v) - f ( t , ul, v) ~ M ( u 2 - Ul);
(2.10)
there exists N >_ 0 such that f o r all (t, u, vl), (t, u, v2) c E,
If(t, u, v2) - f(t, u, Vl)l ~
Nlv2 -
Vl];
(2.11)
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and for all t E [a, b]
a"(t) >_f(t,a(t),a'(t)), Z"(t) <_ f(t, Z(t), Z'(t)),
a(a) = a(b), a'(a) >_ a'(b), ~(a) - ~(b), ~'(a) <_ ~'(b).
At last, let L > 0 be such that N 2 + -~ N v/N2 Jr 4M L >_ M + --~-
(2.12)
and for all t E [a, b] f ( t , a ( t ) , a ' ( t ) ) - f ( t , ~ ( t ) , ~ ' ( t ) ) + L(~(t) - a ( t ) ) >_ O.
(2.13)
Then, the sequences (an)n and (13n)n defined by (2.7) and (2.8) converge monotonically in Cl([a, b]) to solutions u and v of (2.6) such that
a
(a) The function w - ~ -
~ _ 0 satisfies
- w " + Niw' I + (M + 1)w = h(t) > O,
w(a) = w(b), w'(b) > w'(a). Thus, using the m a x i m u m principle (Theorem A-5.3 with p(t) - - N s i g n w ' and q(t) - - ( M + 1)) we can prove that if a ~=/3, our assumptions imply a < /3 on [a,b]. Also i f u i s a solution of (2.6) such t h a t a ~
Proof of Theorem 2.2 9 The proof uses Theorems 1.1 and 1.2 with X = Cl([a, b]), Z = C([a, b]) and g = {u E Z Iu(t) > 0 on [a, b]} as the ordered cone in Z. Let E be defined from (1.1). The operator T " E ~ X, defined by Tu(t) =
G(t, s)(f(s, u(s), u'(s)) - Lu(s)) ds,
where G(t, s) is the Green function of (2.4) with M - L, is completely continuous in X. With these notations, the approximation schemes (2.7) and (2.8) are equivalent to (1.2) and (1.3).
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A : Claim - Let L > 0 satisfy (2.12). Then the functions an defined recursively by (2.7) are such that for all n E N, ( a ) a n is a lower solution, i.e. It
I
an(t) >_ f ( t , a n ( t ) , a n ( t ) ) , ! ! an (a) = an (b), a n (a) >_ a n (b),
(2.14)
(b) a n + l ~ a n . The proof is by recurrence.
Initial step 9 n = O. The condition (2.14) for n - 0 is an assumption. Next, w = a l - a 0 is a solution of - w " + L w = a~(t) - f ( t , do(t), a'o(t)) >_ O, w(a) = w(b), w'(a) <_ w'(b). Hence, we deduce (b) from the m a x i m u m principle (Theorem A-5.3).
Recurrence step - 1st part : assume (a) and (b) hold for some n and let us prove that 1!
a n + l ( t ) >_ f ( t , a n + , ( t ) , a n!+ 1 ( t ) ) , a n + l ( a ) = an+l(b) , a n' + l ( a ) _> an+ l(b). Let w = a n + l - an. We have I!
!
-an+l + f(t, an+l,an+l) - f ( t , ~ + 1 , ~'n+l) - f ( t , an, an) - L ( a n + l - a n ) -<_ M ( a n + l - an) + NI an+ ' 1 - anl - L ( a n + l - an) =(M-L)w+g[w'[. On the other hand, w satisfies
- w " + L w = h(t),
w(a) = w(b), w'(b) - w'(a) = A,
(2.15)
with h ( t ) " = a nI ! ( t ) - f ( t , a n ( t ) , a nI ( t ) ) >_ 0 and A >_ 0. Its solution w reads
w(t) = k
h(s) cosh v/L(k~ -~ + s -
+
h(s) c o s h , / - s
b-a
t)ds + t - s)
+ A cosh v
(t -
where k = ( 2 v ~ sinh v ~ - a )
-1 .
Hence, to prove an+ 1 is a lower solution, we only have to verify
~t [(M - L)cosh v ~ ( - ~ + s - t ) + N v / L I sinh x / T ( L ~ + s - t)l ] h ( s ) d s < O,
]
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ITERATIVE METHODS
b
ft
[(M - L)cosh v / L ( L ~ + t - s) + N v ~ I sinh v / - L ( ~ + t - s)l ] h(s) ds < 0
and (M - L) cosh V~(t - ~2-q~)+ NvrLI sinh v ~ ( t - ~ ) 1
-< 0.
Since h is nonpositive and (M-
L ) c o s h x + N v ~ I sinh x l _ ( M -
L + N v ~ ) I sinhxl
for all x C R, we obtain ( M - L ) w + N I w ' ! <_ 0 if M - L + N V ~ < 0, which follows from (2.12). Recurrence step - 2d part 9 assume (a) and (b) hold for some n and let us prove that an+2 _> an+l. The function w = a n + 2 - an+l satisfies (2.15),
where I!
h(t) ": a n + l ( t ) - f ( t , a n + l ( t ) , an+l(t)) '
and
A = 0.
From the previous step h(t) >_ 0 and the claim follows from the maximum principle (Theorem A-5.3). B : Claim - Let L > 0 satisfy (2.12). Then the functions 13n defined recursively by (2.8) are such that for all n c N, (a) ~n is an upper solution, i.e.
Z"(t) _< f(t, Zn (a) = (b),
Z'(t)), (a) _< Z" (b),
(b) Z +I The proof of this claim parallels the proof of Claim A. C : Claim - a n
< 3n. Define, for all i E N, wi = 3i - a i
and
hi(t) := f ( t , a i ( t ) , a:(t)) - f ( t , ~i(t), ~ ( t ) ) + L(~i(t) - a i ( t ) ) .
The proof of the claim is by recurrence. Initial step 9 a l _ 0 and A -- 0. Using the maximum principle (Theorem A-5.3), we deduce that Wl _> 0, i.e. a l __ 2. I f hn-2 >_ 0 and a n - 1 <_ ~n-1, then hn-1 >_~0 and an <_ ~n. First, let us prove that for all t C [a, b], the function hn_l is
nonnegative. Indeed, we have t
h n - i - f ( ' , a n - l , a n _ 1) - f(', ~n-1,/3tn_i)+ L ( ~ n - i - a n - i ) ! >_ - i ( ~ n - i -an-I)NI/3tn_i _ _ a n--l[ + L(/3n-1 - - a n - I ) -- ( L - M ) w n - i - N[w~n_ i [.
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Notice that wn-1 is a solution of (2.15) with h(t) - hn-2(t) _ 0 and A = 0. Hence, we can proceed as in the proof of Claim A to show that hn-1 _> 0. It follows then from the maximum principle (Theorem A-5.3) that Wn is nonnegative, i.e. an _ ~n. D " Claim - There exists R > 0 such that any solution u of u" >_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with ~ <_ u <_ 13 satisfies [lu'lloo < R. We deduce from the assumptions that u" - f (t, u, u') + h(t), where h (t) _> 0, f ( t , u, u') + h(t) >_ - max If(t, u, 0)1 - Nlu' I F
(2.16)
and F = {(t, u) l t C [a, b], a(t) <_ u <_/3(t)}. The proof follows now using Proposition I-4.5. E 9 Claim - There exists R > 0 such that any solution u of u" <_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with a <__u <_ 13 satisfies I[u'lloo < R. The proof repeats the argument of Claim D. F 9 Conclusion - We deduce from Theorems 1.1 and 1.2 that the sequences (an)n and (~n)n converge monotonically in Cl([a, b]) to functions u and v which are solutions of (2.6) such that a _< u __ fl and a _< v <_/3. Further, as an _< ~n for any n, we have u _ v. K] R e m a r k 2.1. In Claim D or E, the a priori bound on ilu'll~ has to be independent of h. We cannot deduce it from an usual Nagumo condition, the control we have on f ( t , u, u') + h(t) being basically one-sided (see (2.16)), which is why we used a one-sided Nagumo condition. Convergence to minimal and maximal solutions are easy to prove in case L is large enough. C o r o l l a r y 2.3. Let the assumptions of Theorem 2.2 be satisfied. Then, for L large enough, the sequences ((~n)n and (~n)n defined by (2.7) and (2.8)converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.6) such that < Umi~ <_ U m ~ <_ ~. Further, any solution u of (2.6), such that a < u < Z, verifies U m i n ~ U ~__ U m a x .
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Proof. Existence of extremal solutions umin and Umax follows from Theorem I-5.6. If a is not a solution, we deduce from the above Remark (a) t h a t umin > c~. Hence, choosing L > 0 large enough so that (2.12) is satisfied and f ( t , a ( t ) , a ' ( t ) ) - f(t, Umin(t),U~min(t)) + L(umin(t) - a ( t ) )
>_ 0
on [a, b], we can apply Theorem 2.2 with fl - Umin and obtain t h a t u = lim an is such t h a t a __ u <_ umin, whence u = Umin. n---~oo
Similarly, we prove lim
~ n --" U m a x .
[-1
n---,cx~
Remark
Observe t h a t the problem u" - u + (u') 2 - sin t, -
u'(0)
-
cannot be worked out from Theorem 2.2 as it does not satisfy (2.11). However, we can proceed as follows. Notice first that such a problem satisfies a Nagumo condition. Next, we know that lower and upper solutions, c~ and E [-1, 1], of problems t h a t satisfy this Nagumo condition have a pr/o r / b o u n d e d derivatives: Ila'll~ and I1~'11~r - R. We can modify then the equation for lull > R so t h a t the same a prior/bound on the derivatives can be obtained for the modified problem together with (2.11). It follows then t h a t the approximations defined from (2.7) and (2.8) are the corresponding approximations for the modified problem so that convergence follows from Theorem 2.2. The following theorem uses an approximation scheme which, from a computational point of view, is more involved since it uses piecewise linear equations. On the other hand, the analysis is simplified since the coefficients in the approximation scheme are the Lipschitz constants on the nonlinearity. Notice also t h a t in this theorem we use different approximation schemes for the c~ and for the ~n. 2.4. Let ~ and ~ E C2([a, b]), r < ~ and E be defined by (2.9). Assume f 9E ~ R is a continuous function, there exists M > 0 such that for all (t, ul, v), (t, u2, v) E E, Theorem
Ul ~ U2
implies
f(t, u2, v) -- f(t, ul, v) < M(u2 - ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, v2) E E, If(t, u, v2) - f(t, u, Vl)l <_ g[v2 - vii; and for all t E [a, b] a"(t) >_ f(t, a(t), a'(t)), ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) <_ ~'(b).
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Let c~o = ~ and ~o = 13. Then the problems I
gl~
c~,~" -
!
I
- c~n_l[ - Mc~n = f ( t , a ~ - l , ~ ! ! a n ( a ) = C~n(b), C~n(a) - an(b)
1)
--
MOln-1,
-
M~n-1,
and
~
+ NI~'~ - ~'~-~1- M~n
=
f(t,~n-l,~-l)
~n(a) = ~n(b), ~ ' ( a ) - ~ ' ( b ) , define sequences (C~n)n and (13n)n that converge monotonically in Cl([a, bl) to solutions Umin and Umax of (2.6) such that OL < U m i n < Uma x ~_< /3.
Furthermore, any solution u of (2.6), such that c~ < u
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u C Z l u ( t ) > 0 on [a,b]} be the ordered cone in Z and g be defined from (1.1). A " Definition of a completely continuous operator T. u C X the problem v"
-
NIv' -
u'(t)l-
Claim - For any
M y = f ( t , u(t), u'(t)) - M u ( t ) , v(a) = v(b), v'(a) = v'(b),
(2.17)
has a unique solution v. Notice t h a t if k > 0 is large enough, - k and k are well-ordered lower and upper solutions of (2.17). Existence of a solution of this problem follows then from Theorem I-5.3 with ~(s) = K ( s + 1) and K > 0 large enough. Let Vl and v2 be two solutions of (2.17). Define w = v 2 - Vl and assume t h a t for some to E [a, b] w(to) = m a x w(t) > O. tE[a,b]
We compute w'(to) = v ~ ( t o ) - v~(to) - 0 and obtain the contradiction w ' ( t o ) = M w ( t o ) + g ( I v ~ ( t o ) - u'(to)[ - I v y ( t 0 ) - u'(to)[) > 0. Hence w <_ 0. Similarly we prove w _ 0, which implies w = 0, and the solution of (2.17) is unique. Now, we can define the operator T'E~X,
uHTu,
where s is defined from (1.1) and T u is the solution of (2.17). This operator is completely continuous.
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-
ITERATIVE METHODS
F o r all n C N , a n + l
I n i t i a l s t e p 9w
=
>_ a n .
Ta-
oL 1 - o L 0 -
a > O. Notice that w solves the problem
w"Nlw' I - Mw = f(t,a(t),a'(t)) - a " ( t ) < O, w ( b ) - w ( a ) - 0, w ' ( b ) - w ' ( a ) = a ' ( a ) - a ' ( b ) >_ O.
The claim follows from the maximum principle argument used in Claim A. R e c u r r e n c e s t e p - A s s u m e a n + l - a n >_ 0 a n d p r o v e w = an+2 - a,~+l >_ 0.
The function w satisfies w" - Nlw'l - Mw ! ! -- f(t,O~n+l,Oen+l)- f ( t , a n , O l n ) - M ( o ~ n + l --
an) -
N]an+'
1 -
ten! I
<0 and w(a)
w(b), w'(a)
-
-
w'(b).
The claim follows from the same maximum principle argument. C 9 Claim - an
The proof repeats the argument of the previous
< ft.
claims. D 9 C l a i m - T h e s e q u e n c e (an),~ is b o u n d e d in X .
The functions an solve
problems u ( a ) - u ( b ) , u' (a) - u' (b),
u" + Nlu'[ - Mu = gn(t),
where I
I
I
+Nlo/n_ll q- f ( t ,
an +f(t,a~_l,0)-Ma~_l
n- l) _
1
~CI~ I
n-l)
-
-
f(t, an
_
1 O)
>_ f ( t , a ~ _ x , 0) - Ma~_~ > f(t, f l , 0 ) - Mfl. As the gn are uniformly lower bounded, the boundedness of the an follows now from Proposition I-4.5. E 9 C o n c l u s i o n - The convergence of the sequence (an)n to a solution Umin of (2.6) follows from Theorem 1.1. To prove Umin is a minimum solution, let u be any solution of (2.6) such that a < u
Similarly, we prove the convergence of the sequence (fin)n to a maximal solution using Theorem 1.2. [3 2.2
The Neumann
Problem
The Neumann problem u" - f(t, u), u' (a) = O, u' (b) = O,
(2.18)
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where f is a continuous function, can be investigated along the lines of the periodic problem. T h e o r e m 2.5. Let a and ~ C C2([a, b]), a _ 13 and let E be defined from (2.5). A s s u m e f " E ~ IR is a continuous function, there exists M > 0 such that for all (t, u 1), (t, u2) e E, Ul ~ U2 i m p l i e s f ( t , u 2 ) -
f(t, Ul) ~
M(u2-Ul)
and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , ~"(t) < f(t,13(t)),
a'(a) >_ O, a'(b) <_ O, ~'(a) < O, ~'(b) > O.
Then the sequences (an)n and (~n)n defined by ao ~ a~
a n''- M a n -- f ( t , a n - l ) - - M a n - 1, I I a n (a) -- O, a n (b) - O, and ~o = ~,
~-
MZn - f(t,~n-1) - M~n_l, 13~(a) = O, 13~n(b) - O,
converge monotonically in Cl([a,b]) to solutions Umin and Umax of (2.18) such that a <_ umi~ <_ u m ~
<_ 3.
Further, any solution u of (2.18) with graph in E verifies Umi n ~__ U ~ Urea x.
E x e r c i s e 2.1. Prove the above theorem. Hint 9 See the proof of Theorem 2.1.
To deal with derivative dependent equations u" = f (t, u, u'), u' (a) = 0, u' (b) - 0,
(2 19)
we can use the ideas of Theorem 2.2. T h e o r e m 2.6. Let a and ~ E C2([a, b]), a < 13 and let E be defined from (2.9). A s s u m e f " E --~ • is a continuous function, there exists M >_ 0 such that for all (t, Ul, v), (t, u2, v) c E, Ul <_ U2
implies
f ( t , u2,v) - f ( t , u l , v ) <_ M(u2 - ul);
254
V. M O N O T O N E
ITERATIVE METHODS
there exists N > 0 such that for all (t, u, vl), (t, u, v2) E E, If(t, u, v2) - f ( t , u, Vl)l ___Nlv2 - vii; and for all t E [a, b] a"(t) >_ f ( t , a ( t ) , a ' ( t ) ) , [3"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a'(a) > O, c~'(b) < O, [3'(a) < O, [Y(b) >_ O.
Then, if L is large enough, the sequences (an)n and (~n)n defined by Ol0 --" Ol~ I 1 ) - Lc~n-1, c~n'' - L c~n - f ( t, an-1, OLn! i a n (a) = O, a,~ (b) - 0
and flo -- fl, !
~
- Lfln = / ( t , ~n-1, f ~ n - 1 ) -~'~ (a) : O, ~ (b) = 0
L[Jn-1,
converge monotonically in cl([a, b]) to solutions Um~n and such that
Umax
of
(2.19)
<_ Umin < Um.~ <_ ~.
Further, any solution u of (2.19), such that ~ < u < ~, verifies Umin
~ U < Umax.
E x e r c i s e 2.2. Prove the above theorem. Hint " See the proofs of Theorem 2.2 and Corollary 2.3, or [65].
We can investigate problem (2.19) along the lines of Theorem 2.4. T h e o r e m 2.7. Let a and fl E C2([a, b]), a <_ ~ and let E be defined from (2.9). A s s u m e f " E --+ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) E E, Ul < u2
implies
f (t, u2, v) - f (t,
U l , 1)) ~__ M(u2
- Ul );
there exists N > 0 such that for all (t, u, Vl), (t, u, v2) C E, If(t, u, v2) - f ( t , u, vx)l ~ NIv2 - vii; and for all t E [a, b]
>_ f(t, ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
>_ O,
<_ O,
~' (a) <_ O, ~' (b) > O.
2. T H E W E L L - O R D E R E D
255
CASE
Let ao = a and/30 = ~. Then the problems /
/
i
an"- Nian -an-llM a n - f ( t , C~n_l,Ctn_l)- M a n - 1 I l an (a) = O, a n (b) - 0 and
r + NIr
- fl'~-i [ - M/3n - f(t,/3n-1, ~ ' _ 1 ) - M/3n_l, /3~ (a) = 0, /3" (b) = 0
define sequences (O~n)n and (~n)n that converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.19) such that
Further, any solution u of (2.19), such that a <_ u <_ ~, verifies U m i n ~_~ U ~ U m a x .
E x e r c i s e 2.3. Prove the above theorem. Hint 9 See the proof of Theorem 2.4 or [234].
2.3
The Dirichlet
Problem
Consider the Dirichlet problem (2.20)
u" = f ( t , u), u(a) = O, u(b) = O,
where f is a continuous function. T h e o r e m 2.8. Let a and ~ C C2([a, b]), a < ~ and let E be defined from (2.5). Assume f " E ~ R is a continuous function, there exists M > 0 such that for all (t, u 1), (t, u2) c E, Ul
~
U2 implies f ( t , u 2 ) - f ( t , ul) <_ M ( u 2 -
and for all t E [a, b]
a"(t) >__f(t, a(t)),
a(a) <_ O, a(b) <_ O, ~(a) >_ O, ~(b) >_ O.
~"(t) <_ f ( t , ~ ( t ) ) ,
Then the sequences (an)n and (~n)n defined by Ol0
--
Ol~
a~" - M a n : f ( t , O~n--1 ) -- Mozn-1, an (a) - O, an (b) : 0
Ul),
256
V. M O N O T O N E I T E R A T I V E M E T H O D S
and
Z0=Z, 13~ - Mt3~ = f ( t , ~ n - 1 ) - M 3 n - 1 , ~,~ (a) = O, ~n (b) = 0 converge monotonically in C l ([a, b]) to solutions Umin and umax of (2.20) such that Further, any solution u of (2.20) with graph in E verifies U m i n <___U <~__Urea x.
E x e r c i s e 2.4. Prove the above theorem. Hint 9 See the proof of Theorem 2.1.
In case of derivative dependent equations (2.21)
u" = f ( t , u, u'), u(a) = O, u(b) = O,
approximation schemes similar to (2.7) and (2.8) do not work. Indeed if we try to repeat the argument of Theorem 2.2, we have to prove a priori bounds on the derivatives of lower and upper solutions (see Parts D and E). As noted in Remark 2.1, it implies we use one-sided Nagumo condition which, for the Dirichlet problem, imposes the lower and upper solutions to satisfy the boundary conditions. We can think this condition is not very restrictive since the first iterates al and ~1 already satisfy such conditions. However L must also verify (2.13) and this might not be the case even for large values of L. We have then to consider alternative approximation schemes. T h e o r e m 2.9. Let a and ~ C C2([a, b]), a <_ ~ and let E be defined from (2.9). A s s u m e f 9 E ~ N is a continuous function, there exists M >_ 0 such that for all (t, ul, v), (t, u2, v) e E, Ul <_ u2
implies
f ( t , u2,v) - f ( t , U l , V) <__ M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, v2) c E, If(t, u, v2)-- f ( t , u, Vl)[ ~ NIv2 - vii; and for all t C [a, b]
> I(t, < f(t,
= O,
= O,
13(a) = O, 13(b) = O.
2. T H E W E L L - O R D E R E D
257
CASE
At last, let Ko e C([a,b]) be such that Ko(a) > 0 and for all t e [a,b], Ko(t) = - K o ( b + a - t). Then, for L large enough, the sequences (an)n and (~n)n defined by ao - c~, Zo - Z, an
"+
l
--
r
K0 (t)c~n+ 1 - - Lan + 1 = f ( t, an, OLn) ' - r
Ko(t)a~n - Lan,
GCn+x(a) -- O, ~ n + x ( b ) - 0 and ~-bx
--
r
- L~n+l - f ( t , ~ n , ~ n ) -- r ~n+l (a) = 0, 3~+1 (b) - 0
- L~n,
converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.21) such that a < u.~
< Um~ <
~.
Further, any solution u of (2.21) with graph in E verifies U m i n ~_ U ~_~ Urea x.
E x e r c i s e 2.5. Prove the above theorem. Hint 9 See the proofs of Theorem 2.2 and Corollary 2.3 or [63].
2.4
Bounded
Solutions
In this section we consider bounded solutions of the differential equation ~" + c ~ ' -
/ ( t , ~).
(2.22)
To this end, we define the set BC(R) = {u e C(R) [ u e L~(R)}. T h e o r e m 2.10. Let a and ~ c C2(IR) N L ~ ( N ) , a ~ ~ and let E := {(t,u) c N 2 [ a ( t ) _ u __ ~(t)}. Assume c E N, f : E ~ N is a continuous bounded function and there exists M > 0 such that for all (t, ul), (t, u2) c E, Ul ~__ U2 ::~ f ( t , u 2 ) - f ( t , Ul) ~ M ( u 2 - Ul). Assume further that for all t c IR
~"(t) + ~'(t) >_ f(t, ~(t)), 3"(t) + c3'(t) <_ f(t, 3(t)). Let ao = a and 13o = 13. Then the equations " O~n+l -[- eCgtn+l - Man+ 1 "-- f(t, a ~ ) -
3x+1 + c/3'~+1 - M 3 n + l = f ( t , 3 ~ ) -
Ma~, M~
V. M O N O T O N E
258
ITERATIVE METHODS
define sequences (an)n and (~n)n C BC(R) that converge monotonically and uniformly on bounded intervals of R to solutions Umin and Umax of (2.22) such that <_ u . . ~ < u m ~ <_ ~. Further, any solution u of (2.22) with graph in E verifies U m i n ~_ U ~ Urea x .
Proof. First recall that given p E BC(R), the problem y" + cy' - Ay = p(t) with A > 0 has a unique solution in BC(R) given by
y(t) = / + 5
G(t, s)p(s) ds,
where
G(t, s) = - ~ with
-
+ ~.
exp(~(s - t ) ) e x p ( - v l t - ~l),
This result implies the (an)n and (~n)n are uniquely
defined.
Claim 1 - If an is such that for all t C R I! a~(t) + c a ' ( t ) >_ f ( t , a ~ ( t ) ) , ~ ( t ) <__Z(t),
then the function
defined by
an+ 1 E Be(R) I!
a n + 1 + C a t n + l --
Man+l = f(t, an)
- Man,
satisfies for all t C R, II
I
a n + l ( t ) + can+l(t ) > f(t, an+l(t)), c~n(t) _< a n + l ( t ) ___~(t). It is enough to observe that an and fl satisfy for all t E ]1(
a~(t) + ca~(t) - M a n ( t ) >_ f(t, an(t)) - Man(t), fl"(t) + cfl'(t) - M ~ ( t ) <_ f ( t , ~ ( t ) ) - M~(t) < f(t, an(t)) - M a n ( t ) . Hence, by Theorem II-5.6, the solution an+ 1 of
u" + cu' - M u = f (t, an) - M a n
3. T H E R E V E R S E D is such that a n _( a n + l
259
ORDER CASE
_( ~ on R and then satisfies also for all t c IR
a ~ + l ( t ) + Ca~n+l(t) - M a n + l ( t )
= f(t, an(t)) - Man(t) >_ f ( t , a n + l ( t ) ) - M a n + l ( t ) .
Claim 2 - If/3~ is such that f o r all t E R
~'n'(t) + C~'n(t) <_ f(t,3n(t)), ~(t) < Z~(t) then the f u n c t i o n ~n+l defined by 13~+ 1 -[- C~In+ 1 - Ml3n+ 1 - f ( t , 3n) - Ml3n , satisfies f o r all t E R,
~"+ i ( t ) + ~ Z ' + l (t) <_ f(t , Zn+i(t)) ~(t) <_ Zn+i(t) <__Zn (t). The proof of this claim is similar to the proof of Claim 1. Conclusion - For every bounded interval I c R, we deduce from Proposition I-4.4, the existence of K such that for all n, Ilanllcl(/) _ K and II~nllc'(/) - g . Hence, we deduce from the Arzel~-Ascoli Theorem the convergence to solutions umin and umax. As usually if u is a solution such that a <_ u <_ ~, we can take u as a lower solution and prove that Umaz = limn~c~ ~n >_ U. Similarly we have Umin <_ u. ['7
3
Lower
and
Upper
Solutions
in Reversed
Order
The monotone approximation method can be used in case the lower and upper solutions are in the reversed order ~ < a. This method works for any boundary value problem such t h a t a uniform anti-maximum principle holds. This is the case for the periodic and the Neumann problems. We can also work bounded solutions on IR. However the method does not apply to the Dirichlet problem since, in this case, we only have a non-uniform anti-maximum principle. 3.1
The Periodic
Problem
Consider the periodic boundary value problem
u " - f(t,u), ~(a) = u(b), ~'(a) - u'(b),
(3.~)
260
V. M O N O T O N E
ITERATIVE
METHODS
where f is a continuous function. In Section 2, we have built an approximation scheme for solutions of (3.1) based on the maximum principle. Here, we consider a similar approach based on the anti-maximum principle. Given continuous functions a and ~, and M > 0, we consider the sequences (an)n and ( ~ n ) n defined by a 0 = a, a n, , - ~ a n
(3.2)
M a n = f(t, an-1) + Man-l, ! (a) - a n (b), a n (a) - a~ (b)
and fl~ + Ml3~ - f(t, f l ~ - l ) + M~n-1, fl~ (a) - fin (b), fl~ (a) = 13' (b).
(3.3)
If M is not an eigenvalue of the periodic problem, i.e. M ~: (2n~ g-s-d)2 with n E N, the functions an and 13n, solutions of (3.2) and (3.3) can be written explicitly
an(t) -- jfo b G ( t , s ) ( f ( S , an_l(S)) + M a n _ l ( s ) ) d s , n(t) =
+
where G(t, s) is the Green function of the problem
u" + M u = f (t), u(a) = u(b), u'(a) = u'(b).
a n
(3.4)
The next theorem indicates a framework to obtain convergence of the and fin to extremal solutions of (3.1).
Theorem
3.1. Let a and 13 C C2([a, b]), 13 _ a and E "= {(t,u) e [a,b] • I ~ l ~ ( t ) _< u < a(t)}.
(3.5)
Assume f . E --~ R is a continuous function, there exists M e ]0, (b_--~)2] such that for all (t, Ul), (t, u2) e E, Ul <__u2 implies f(t, u 2 ) - f(t, ul) > - M ( u 2 and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , ~"(t) <_ f ( t , ~ ( t ) ) ,
ul)
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) <_ ~'(b).
3. THE
REVERSED
ORDER
CASE
261
Then the sequences (an)n and (~n)n defined by (3.2) and (3.3) converge monotonically in C l([a, b]) to solutions Umax and Umin of (3.1) such that
Further, any solution u of (3.1) with graph in E verifies U m i n ~__ ~t ~_~ U m a x .
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u r Z lu(t) >_ 0 on [a,b]} be the ordered cone in Z and (3.6) The operator T" C --, X, defined by
Tu(t) =
jfa b
G(t, s)(f(s, u(s)) + Mu(s)) ds,
where G(t, s) is the Green function of (3.4), is continuous in X and monotone increasing (see Corollary A-6.3). Further, T($) is relatively compact in X, fl <_ T~ and a >_ Ta. The proof follows now from Theorem 1.3, where a and/3 have to be interchanged. FI Next, we consider the derivative dependent problem
u" = f(t, u, u'), u(a) = u(b), u' (a) = u'(b).
(3.7)
As above, given a,/3 E CS([a, b]) and L > 0, we consider the approximation schemes OZ0 m O~,
, a n + Lc~n - f (t, c~n- 1, c~n1) + L(~n- 1, ! c~n(a) - C~n (b), a~ (a) - c~n (b) ,,
(3.8)
and /~0 --" /~,
~ + L~n : f(t, ~n_l, ~ln_l) + L~n-1, /~n(a) -/3n(b), /3"(a) =/3"(b).
(3.9)
The following result paraphrases Theorem 2.2 in a case where the antimaximum principle applies. T h e o r e m 3.2. Let a and 13 E C2([a, b]),/~ <_ a and E "= {(t, u, v) e [a, b] • I~2 I/3(t) <_ u <__a(t)}.
(3.10)
262
V. M O N O T O N E
ITERATIVE
METHODS
Assume f " E ~ I~ is a continuous function, there exists M c ]0, (~-:-5 ] such that for all (t, ux, v), (t, u2, v) E E, Ul ~_~ u2
implies
f ( t , U2, V) -- f ( t , Ul, V) ~__ - M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, V2) E E , If(t, u, v2) - f(t, u, Vl)] _< N]v2 - vii; and for all t E [a, b] a"(t) > f(t, a(t), a'(t)),
a(a) = a(b), a'(a) > a'(b),
Z"(t) < f(t, Z(t), Z'(t)),
Z(a) = ~(b), Z'(a) < Z'(b).
At last, let L e [M, (b_-~) 2] be such that (L -- M) cos v ~ ( b---y-)-a N v ~ s i n v / L ( - ~ ) _> 0
(3.11)
and f ( t , a ( t ) , a ' ( t ) ) - f ( t , ~ ( t ) , ~ ' ( t ) ) + L(a(t) - ~(t)) >_ O. Then, the sequences (an)n and (~n)n defined by (3.8) and (3.9) converge monotonically in Cl([a, b]) to solutions u and v of (3.7) such that
f3<_v 0 on [a,b]} as the ordered cone in Z. Let s be defined from (3.6). The operator T" s ~ X, defined by
T~(t) -
a(t, s)(Y(~, ~(s), ~'(s)) + Lu(s)) as,
where G(t, s) is the Green function of (3.4) with M replaced by L, is completely continuous in X. With these notations, the approximation schemes (3.8) and (3.9) are equivalent to (1.3) and (1.2). A " Claim - Let L e [M, (b---~) 2] satisfy (3.11). Then the functions oLn defined recursively by (3.8) are such that for all n c N, (a) an is a lower solution, i.e. c~n(t) _> y(t, c~,, (t),a,~(t)), ' " (3 12) t an(a) = an(b), a~(a) > an(b ), (b) an+l <_an. The proof is by recurrence.
3. T H E R E V E R S E D
ORDER
263
CASE
Initial step 9 n = 0. The condition (3.12) for n - 0 is an assumption. Next, w = a0 - a l is a solution of w" + Lw = a~(t) - f(t, ao(t), a'o(t)) >__O, w(a) - w(b), w'(a) >_ w'(b). Hence, we deduce (b) from the a n t i - m a x i m u m principle (Corollary A-6.3).
Recurrence step - 1st part: assume (a) and (b) hold for some n and let us prove that l! a n + l ( t ) > f(t, an+l(t) a' ,
_
a n + 1(a)
a~+l(b)
- -
a n' + l ( a ) __ a n + l (b).
,
Let w = a~ - an+ 1 > O. We have II
I
--an+ 1 + f(t, an+l,an+l) !
= --f(t, an,C/)-[- f(t, an+l,an+X) -- L(an -C~n+l) < M(an a n + I ) + N[ a n ' +l-anl-n(an-an+l) =(M-L)w+NIw' I. -
-
On the other hand, w satisfies
w" + Lw - h(t),
w(a) - w(b), w'(a) - w'(b) - C,
(3.13)
I with h(t) := a nI I ( t ) - f(t, an(t),an(t)) >_ 0 and C >_ 0. Observe that
1
w(t) -- 2V/~ sin v / L ( L ~ )
[C cos v ~ ( ~
m t)
t
b
Hence, using (3.11) and denoting D - 2 v ~ s i n v / L ( L ~ ) , we compute
(M - L ) w ( t ) + NIw'(t)l 1
< - - [ C [(M - L) cos v~(~--~2b - t) + Nv/L[ sin v/-L(~2-~ - t)[] -D
+
h ( s ) [ ( M - L)cos v / L ( k ~ + s - t) + Nv/L[ sin v/-L(b-~ + s - t)]]ds b
+ ft h(s)[(M - L)cos v ~ ( ~
+ t-
s)
+ N vZ-LI sin v / L ( L ~ + t - s)[] ds] <0. m
Hence a n + 1 is a lower solution.
264
V. M O N O T O N E
ITERATIVE METHODS
Recurrence step - 2d p a r t " assume (a) and (b) hold f o r some n and let us prove that an+2 _< an+l. The function w - an+l -a,~+2 satisfies (3.13), where h(t)" =
" n+l(t) -
an+ ' 1(t))
f(t,
and
C - 0.
From the previous step h(t) >_ 0 and the claim follows from the antimaximum principle (Corollary A-6.3). B 9 Claim - Let L C [ M , ( ~_~) '~ 2 ] satisfy (3.11). Then the f u n c t i o n s ~n defined recursively by (3.9) are such that f o r all n E N, (a) 13n is an upper solution, i.e. ~ " ( t ) <_ f ( t , [ 3 n ( t ) , ~ ( t ) ) , 13n(a) = 13n(b), fl~(a) <_/3~(b),
(b) The proof of this claim parallels the proof of Claim A. C 9 Claim - an >_ ~,~. Define, for all i E N, wi = ai - 13i and hi(t) := f ( t , a i ( t ) , a ' , ( t ) ) - f ( t , ~ i ( t ) , ~ ( t ) )
+ L ( a i ( t ) - ~i(t)).
The proof of the claim is by recurrence. Initial step 9 a l >_ 131. The function wl is a solution of (3.13) with h = h0 >_ 0 and C - 0. Using the anti-maximum principle (Corollary A-6.3), we deduce that wx >__0, i.e. a l >_ ~1. Recurrence step 9 Let n >_ 2. I f h n - 2 >__0 and a n - 1 >_ 13,~-1, then h n - 1 >>_0 and an >_ ~n. First, let us prove that for all t E [a, b], the function hn-1 is nonnegative. Indeed, we have l
h n - 1 = f ( t , o l n - l , 0 Z n _ l ) -- f(t, ~n-1,/31n_1) 4- L(ctn-1 - / 3 n - 1 ) >_ - M ( a n - 1 - / 3 n - 1 ) -- NI a n' _ 1 -- ~ n' _ l l 4- n ( o l n - l -- ~ n - 1 ) ! = (n- M)wn-1glwn_l[.
Recall that Wn-1 is a solution of (3.13) with h(t) - h~_2(t) > 0 and C = 0. Hence, we can proceed as in the proof of Claim A to show that h n - 1 >_ O. It follows then from the anti-maximum principle (Corollary A-6.3) that Wn is nonnegative, i.e. an > ~n. D " Claim - There exists R > 0 such that any solution u of u" >_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with 13 <_ u <__ a satisfies Ilu'l]~ < R. We deduce from the assumptions that u" - f (t, u, u') + h(t),
3. T H E R E V E R S E D
ORDER
CASE
265
where h(t) > O, : ( t , u, u') + h(t) >__ - m a x F If(t, u, O)l -- Nlu'l and F =
{(t, u) l t c [a, b], ~(t) <_ u < ~(t)}. The proof follows now using Proposi-
tion 1-4.5. E 9 Claim - There exists R > 0 such that any solution u of u" <_ f ( t , u, u'),
u(a)
-
u(b), u'(a)
-
u'(b),
with ~ < u <_ a satisfies Ilu'll~ < R. The proof repeats the argument of Claim D. F 9 Conclusion - W e deduce from Theorems 1.2 and 1.1, where a and have to be interchanged, t h a t the sequences (Oln) n and (fl~)n converge monotonically in CX([a,b]) to functions u and v such that fl _ v _< a and < u < a. Further Claim C implies v _ u. [:3 E x a m p l e 3.1. Consider the problem u" - k arctan u' + cu 3 = sin t,
~(o) - ~ ( 2 ~ ) ,
~'(o) = ~'(2~),
where c > 0 and k > 0. This problem has a - c -1/3 and f l - - c -1/3 as lower and upper solutions. To satisfy the assumptions of Theorem 3.2, let M - 3c 1/3 and N - k, choose L E ]3c 1/3, 1/4[ and assume c < 12 -3
and
k <- - L-3cl/3 cotanv~lr. v~
As in Section 2, we can modify the analysis in case we agree to compute the approximations an and ~n from piecewise linear problems. The following result adapts Theorem 2.4 to lower and upper solutions in reversed order. T h e o r e m 3.3. Let (~ and ~ E C2([a, b]), ~ < a and let E be defined by (3.10). A s s u m e f 9E ~ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) C E, Ul ~ U2
implies
f ( t , u2, v) - f ( t , Ul, V) ~ - M ( u 2 - Ul),
there exists N >__0 such that for all (t, u, Vl), (t, u, v2) c E, I f ( t , u, v2) - f ( t , u, Vl)l ~ N I v 2 - vii
and for all t E [a, b] a"(t) > f ( t , a ( t ) , c ~ ' ( t ) ) ,
~"(t) <_ f(t, ~(t), ~'(t)),
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) < ~'(b).
266
V. M O N O T O N E
ITERATIVE
METHODS
Assume also b - a < 20(M, N / 2 ) , where Y/1~2'1 M a r c t a n h
O(M, 1V) =
s
./~-r},,2_ M
fil
"'
if O < M
if M = 1V2
1(.
N ~
v/M_.~2
< ~2
g -- arctan ~/M-~?~-
'
(3.14)
--
Let C~o = ~ and ~o = ~. Then the problems OZn" -
Man : f(t, an-l,an_l) + Man-l, N IOZn' --OLn_l[-Jr, ! ! an (a) : oLn (b), oLn (a) = a n (b)
and ~ + NI~
- fl~-ll + M ~ n = f ( t , ~ n - l , ~ - l ) ~n (a) = ~n (b), ~ (a) = ~ (b),
+ M~n-1,
define sequences (an)n and (~n)n that converge monotonically in Cl([a,b]) to solutions Umax and Umin of (3.7) such that
Further, any solution u of
(3.7),
such that ~ <_ u <_ a , verifies
Umin ~ U ~ Umax.
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u e Z l u ( t ) >_ 0 on [a,b]} be the ordered cone in Z and E be defined from (1.1). A " Definition of a completely continuous operator T. Claim 1 - For any u E X the problem v" - NIv' - u'(t)l + M v = f ( t , u(t), u'(t)) + M u ( t ) , v(a) = v(b), v'(a) = v'(b),
(3.15)
has a solution v. Consider the problem v"-
s l y ' - u'(t)[ + M y = a(t),
v(a) = v(b), v'(a) = v'(b),
(3.16)
with s e [0, N] as an homotopy parameter and a e C([a, b]). Assume that solutions of (3.16) are not a priori bounded in X. Hence, there exists sequences (Sn),~ C [0, N] and (Vn)n C X such t h a t vn solves (3.16) with s = sn and l i m n _ ~ IlVnliC, = +c~. We define then ~n = Vn/llvnllc 1 and going to subsequences we can assume C1
~n--*~,
i, L 2
~n--~v"
and
Sn---+s.
3. T H E R E V E R S E D
ORDER CASE
267
Going to the limit in (3.16), we obtain that ~ solves V " - slY' I + MY = O,
V(a) = V(b), V'(a) = V'(b).
If M <_ s 2 / 4 , Theorem A-6.2 implies ~ = 0. On the other hand, if M > s 2 / 4 , we have b - a < 2 0 ( M , N / 2 ) < 2 0 ( M , s / 2 ) < 2 x ( M , - s / 2 ) (see (A6.3) and (A-6.4)) and the conclusion ~ = 0 follows from the same theorem. Hence, in all cases, this contradicts limcl ~n - ~. The proof of the claim follows now from the a priori bounds and classical arguments in degree theory. C l a i m 2 - The solution of p r o b l e m (3.15) is unique Let Vl and v2 be two solutions of (3.15). Define w = v2 - vl and compute w " + N I w ' [ + M w - N ( l v ~ - v'21 + Iv'2 - u'l - [v~ - u'[) >_ 0.
Hence, we deduce from the anti-maximum principle (Theorem A-6.2) that w > 0. Similarly we prove w <_ 0, which implies w = 0, and the solution of (3.15) is unique. Now, we can define the operator T" g ~ X,u ~ Tu,
where $ is defined from (1.1) and T u is the solution of (3.15). This operator is completely continuous. B 9 Claim - For all n c N, a n >_ a n + l . Initial step 9w = a o -
al - a-
T a > O. Notice that w solves the problem
w " + N I w ' I + M w = a " ( t ) - f ( t , a ( t ) , a ' ( t ) ) >_ O, w ( a ) - w(b) = O, w ' ( a ) - w'(b) = a ' ( a ) - a'(b) >_ O.
The claim follows from the anti-maximum principle (Theorem A-6.2). R e c u r r e n c e step - A s s u m e a n -
a n + l >_ 0 and prove w - a n + l -
an+2 >_ O.
The function w satisfies w" + N[w' I + Mw ! -- - f(t, a,-,, an) - f ( t , o+,+1~ Ol/,,+1) + Yl a n'+ l - a n [ + M ( a n - a n + l ) >0
and the claim follows from the anti-maximum principle (Theorem A-6.2). C 9 Claim - a n >_ ~.
claims.
The proof repeats the argument of the previous
V. M O N O T O N E
268
ITERATIVE METHODS
D 9 Claim - The sequence (Oln) n is bounded in X . The functions c~n, n >_ 1, solve the problems u" + g l u ' ] + M u = gn(t),
u(a) = u(b), u'(a) = u'(b),
where
g~ = N(l~'nb + I~" - ~ ' - ~ 1 -
+Nl~'n-ll +
i~'n-~l) I
f ( t , Oln_l, C e n _ l ) - f ( t , OZn_l, O)
+ f ( t , a n - 1 , O) + M a n - 1
> f(t, a~_x, O) + Mc~_~ > f(t, ~, O) + M~. As the g~ are uniformly lower bounded, the boundedness of the an follows now from Proposition I-4.5.
E 9 Conclusion - The convergence of the sequence (Oln) n t o a solution Umax of (3.7) follows from Theorem 1.2, where a and/~ have to be interchanged. To prove Umax is the maximum solution, let u be any solution of (3.7) such that ~ < u < a. We apply then Theorem 1.2 with the a and f l o f t h i s theorem replaced respectively by u and a and obtain umax - lim an >_ u. u-,co
Similarly, we prove the convergence of the sequence (~n)n to a mimimal solution using Theorem 1.1. KI 3.2
The Neumann
Problem
The Neumann problem
u" = f ( t , u), u' (a) = 0, u' (b) = 0,
(3.17)
where f is a continuous function, can be investigated along the lines of the periodic problem. T h e o r e m 3.4. Let c~ and ~ C C2([a,b]), ~ <__ ~ and let E be defined from (3.5). A s s u m e f : E ~ R is a continuous function, there exists M C ]0, (2(b'_a)) 2] such that for all (t, Ul), (t, u 2 ) E E,
Ul ~_ U2 implies f ( t , u 2 ) - f(t, ul) > - M ( u 2 and for all t C [a, b] c~'(t) >_ f ( t , c~(t)), ~ ' ( t ) <_ f ( t , ~(t)),
a'(a) >_ O, a'(b) <_ O, ~'(a) <_ O, ~'(b) >_ O.
Then the s e q u e n c e s (OZn) n and ([3n),~ defined by Ol0 : Ol~ r ! ! Ozn ( a ) : O, Oln ( b ) -- O,
Ul)
3. THE
REVERSED
ORDER
269
CASE
and ~0 ~ ~
/3~ + M[3,~ = f(t,/3n-1) + M/~n-1, [3'~(a) = O, /~'~(b) = O, converge monotonically in Cl([a, b]) to solutions umax and Umin of (3.17) such that ~__ Umin ~ Umax ~__Or.
Further, any solution u of (3.17) with graph in E verifies Umi n ~ U ~ Urea x.
E x e r c i s e 3.1. Deduce the above theorem from Theorem 1.3. To deal with derivative dependent equations
u" = f (t, u, u'), u'(a) = O, u'(b) = O,
(3 18)
we can use the ideas of Theorem 3.2. T h e o r e m 3.5. Let a and /3 c C2([a,b]), ~ < a and let E be defined from (3.10). Assume f 9 E ~ R is a continuous ]unction, there exists M e]O,( 2(b-~) ~ )2] such that for all (t, Ul, v) (t, u2, v) E E,
ul <_ u2
implies
f(t, u2, v) - f(t, ul, v) > - M ( u 2
-
Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, U, V2) E E , If(t, u, v2) - f(t, u, Vl) [ ~ N[v2 - vii; and for all t e [a, b] a"(t) > f(t, a(t), a'(t)), /~"(t) <_ f(t,/3(t),/3'(t)),
a'(a) >_ O, a'(b) <__O, /~' (a) <_ O, ~' (b) >_ O.
At last, let L C [M, (2(b'-'~))2] be such that (i
-
M ) c o s v/L(b - a) - N x / ~ sin v ~ ( b - a) >__0
and f ( t , a ( t ) , a ' ( t ) ) - f(t,/3(t),/3'(t)) + i(c~(t) -/3(t)) >_ O.
270
V. M O N O T O N E
ITERATIVE
METHODS
Then, the sequences (an)n and (~n)n defined by aO = Ol~ II
O~n ~- Lan = f (t, Oln--1, a'n - l ) + L a n - 1 , a n' ( a ) = 0, a n' (b) = 0, and
Z0=Z, ~ + L~n = f ( t , fin-l, f l ~ - l ) + Lfln-1, fl~ (a) = 0, fl~ (b) = 0, converge monotonically in C l([a, b]) to solutions u and v of (3.18) such that
~
We can investigate problem (3.18) using an analog of Theorem 3.3. T h e o r e m 3.6. Let a and Z c C2([a, b]), Z <_ a and let E be defined from (3.10). Assume f " E ~ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) e E, Ul <_ u2
implies
f ( t , u2, v) - f (t, Ul, v) > - M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, vl), (t, u, v2) C E, If(t, u, v2) - f(t, u, vl)l <_ NIv2 - vii; and for all t C [a, b] a ' ( t ) >_ f ( t , a ( t ) , a'(t)), ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a'(a) >_ O, a'(b) <_ O, ~'(a) <_ O, ~'(b) >_ O.
Assume also b - a <_ O(M, N / 2 ) , where 0 is defined from (3.14). Let ao = a and/3o = ~. Then the problems t
t
a n " - N[an - a n -
at
M a n - f(t, a n - 1, n - l ) -[- M a n - l , a n (a) = O, a~ (b) = O,
11 + I
and
3~ + N[~tn - f~'n-l[ + M~n : f(t, ~n-1, ~tn_l) + M~n-1, /3~n(a) = 0, ~n (b) = 0, define sequences (an)n and (~n)n that converge monotonically in Cl([a, b]) to solutions Umax and umin of (3.18) such that
3. T H E R E V E R S E D O R D E R CASE
271
Further, any solution u of (3.18), such that 13 <_ u <_ a verifies Umin
~__ U ~ Urea x .
E x e r c i s e 3.3. Prove the above theorem along the lines of the proof of Theorem 3.3. Hint 9 See [51].
3.3
Bounded Solutions
Existence of bounded solutions of (2.22) can be deduced from lower and upper solutions in the reversed order. The following result parallels Theorem 2.10. Here we use/~C2(ll~) = {u c C2(PQ I u, u', u" E L~(R)}. T h e o r e m 3.7. Let a and Z E B e 2(R), a >_ ~ and let E "= {(t,u) E IR2 I ~(t) ~_ u < a(t)}. Assume c > O, f " E ~ R is a continuous bounded function such that for all (t, ul), (t, u2) e E, c2
ul <_ u2 ~ f ( t , u2) - f ( t , ul) >__-~-(u2- Ul). Assume further that for all t c R
~"(t) + r >__/(t, ~(t)), Z"(t) + cZ'(t) <__/(t, Z(t)). Let ao = a and ~o =/3. Then the equations II
c2
Oln+ 1 -~-CC~ln.i_l -~- -TO~n+l = , c ~-
~-+1 +
f ( t , an) +
Cfln+l + T ~ n + l -- f ( t , ~ n ) +
c2
"-~OLn,
~ , -~n
define sequences (an)n and (~n)n C 13C(I~) that converge monotonically and uniformly on bounded intervals of R to solutions Umaz and Umin of (2.22) such that Further, any solution u of (2.22), such that ~ < u < a verifies U m i n ~ U ~ Urea x .
Proof. First observe that given p c/~C(I~), the unique solution in BC(I~) of c2 y" + cy' + - ~ y = p ( t )
is given by y(t) -
G(t,s)p(s)ds, oo
272
V. M O N O T O N E
ITERATIVE METHODS
where
G(t, s) - ( t - s ) e x p ( - ~ ( t =0,
s)),
if s < t, ifs>t.
Claim - If an is such that for all t C I~ a~(t) + ca~(t) > f(t, an(t)), an(t) >__~(t) then the function an+l defined by a n"+ 1 -~- Catn+ 1
+
C2 "-~an+ 1
= f(t , an)+
C2 ~'an ,
satisfies for all t C ]~, a n!1+ l ( t ) + c O/n+l(t) >__f(t, an+l(t)), Oln(t) ~__ a n + l ( t ) ~ ~(t). It is enough to observe that w = c~n - a n + 1 solves c2 W l! -~- CW' Jr- " u
It
--- OLn -Jr- COln -- f ( t , a n ) .
As a nl/ + ca~ - f (t, an (t)) > 0 we deduce from the form of the solution that w >_ 0, i.e. an >_ an+l and also, for all t E R, a n1+' ' (t) -}- Catn+l
(t)
:
f(t,
c2
O~n(t))zF -~(an(t)-
O~n+l (t)) _> f(t, a n + l ( t ) ) .
The inequality ~ <_ c~n+l can be proved in the same way. The rest of the proof follows as in Theorem 2.10.
4
A Mixed
Approximation
Scheme
In the previous section, we considered sequences (an)n and (/~n)n which converge to solutions of a boundary value problem such as (2.1). The reader might have noticed that computing an and/~n can be a difficult problem. In this section, we give a method of approximations which is simple but provides only bounds on the solutions. We also give assumptions which imply these bounds to be equal, in which case they are solutions. Consider a Dirichlet problem
u" = f (t, u, u),
u(a) = O, u(b) = O.
(4.1)
To simplify the argument, we assume here f is independent of the derivative u ~, which is not essential and could be developed as in the previous sections. We shall use the following definition.
4. A M I X E D A P P R O X I M A T I O N
SCHEME
273
D e f i n i t i o n 4.1. Functions c~ and/3 c C([a, b]) are coupled lower and upper quasi-solutions of (4.1) if (a) for any t e [a, b], a(t) <_ ~(t); (b) for any to c]a,b[, either D _ a ( t o ) < D+a(to), or there exists an open interval Io c ]a,b[ such that to c Io, a c W 2,1 (io), and for a.e. t c Io
>__f(t,
Z(t));
(c) for any to e ]a, b[, either D - ~ ( t o ) > D+~(to), or there exists an open interval Io c ]a,b[ such that to c Io, ~ c W2'1(Io), and for a.e. t C Io
Z"(t) _< I(t, Z(t), (d) a(a) <_ 0 <_ ~(a),
a(b) <_ 0 <_ ~(b).
Consider the following auxiliary problem u" - f (t, u, v),
u(a) = O, u(b) - O,
v" - f (t, v, u),
v(a) = O, v(b) - O.
(4.2)
P r o p o s i t i o n 4.1. Let C~o, /~o C C([a, b]), E "= {(t, u, v) I t e [a, b], u, v e [C~o(t), ~o(t)]} and f " E increasing lower and Then,
(4.3)
~ R be an L1-Carathdodory function such that f(t, u, v) is nonin u and non-decreasing in v. Assume (~o and/~o are coupled upper quasi-solutions of (4.1). the sequences (an)n and (/3n)n, defined for n >_ 1 by t! O~n : f ( t, Otn-1, /3n-1 ), /~: : f ( t, /~n- l , Oln-1),
an (a) = 0, an (b) - 0, fin (a) - 0, fin (b) - 0,
(4.4)
converge monotonically in Cl([a, b]) to functions Umin and Urea x. The pair (umin, Umax) is a solution of (4.2) such that Olo ~- Umin ~- Umax ~---130.
Moreover, any solution (u, v) of (4.2) with so <_ u < ~o, (~o <_ v
Um~ <_ V <_ Um~.
274
V. M O N O T O N E
ITERATIVE METHODS
Proof. Let X = Cl([a,b]) xCl([a,b]), Z - C([a,b]) xC([a,b]), K - {(u,v) e Z i u >_ O,v <_ 0} and s {(u,v) E X l a o _< u _< ~o, ao <_ v _< ~o}. We define T " s ~ X , (u, v) H T(u, v), where T(u, v) is the solution (x, y) of =
x" - f ( t , u, v), y" - f ( t , v, u),
x(a) - O, x(b) = O, y(a) = O, y(b) - 0
and verify T is continuous, monotone increasing, T(s is relatively compact in X and (ao, 13o) <_ T(ao, ~o), (~o, ao) >_ T(~o, ao). Theorem 1.3 applies with a - (a0,~0) and/7 - (~0, a0), and the claims follow. [El Notice that if u is a solution of the given problem (4.1), then (u, u) is a solution of the auxiliary problem (4.2), thus Umin and Umaz are bounds on solutions of (4.1). If the bounds Umi n and umaz given in Proposition 4.1 are equal, Umin is a solution of the initial problem (4.1), and this theorem provides an approximation scheme to a solution of (4.1). In particular, this will be the case if solutions of the auxiliary problem (4.2) are unique. Unfortunately, these solutions are in general not unique as follows from the problem u n + se u - 0, v u d- se v O, -
-
u(0) - u(1) - 0, v(0) - v ( 1 ) - 0
and Theorem VIII-3.6, if we choose s > 0 small enough. The next proposition proves, under appropriate assumptions, the uniqueness of solutions of the auxiliary problem (4.2). Hence, it also proves convergence of the sequences defined in Proposition 4.1 to the unique solution of the given problem (4.1). T h e o r e m 4.2. Suppose the assumptions of Proposition ~.1 hold. A s s u m e moreover (i) there exists ~ > 0 such that C~o >_ c~o; (ii) for every s e [~, 1[, almost every t e [a, b] and every u, v e [c~0,~0] with sv<_u<_v,
> f (t,
v).
Then, the functions Umin and Um~x defined in Proposition ~. 1 are equal, i.e. are solutions of (4.1). Proof. From assumption (i), we deduce CUmax ~ Umin ~ ~max.
4. A M I X E D
APPROXIMATION
275
SCHEME
Let so = s u p ( s I ~ . ~ -< u~}. It is obvious t h a t so e [c, 1] and t h a t SoUma~ <_ Umin. From the definition of so, we deduce the existence of to E [a, b] such t h a t Umin(tO) -- SOUmax(to) -- O,
' n ( t o ) -- SOUmax(to ' Umi ) -- O.
If not we have
U m i n ( t ) - SOUm~(t) > 0, for t e In, b[, / / / Umin(a ) - sou'm~x(a ) > O, urn, n (b) - SOUma~ (b) < O. Hence there exists e > 0 so t h a t u m i n ( t ) - SOUm~(t) >_ eUm~(t) on [a, b]. This contradicts the definition of so. If to ~: b, we also have t l > to such t h a t Umi n ' (tl) -- 8 0 U mx 'a (t l) _> 0. Assume now t h a t so < 1. Hence, we can write 80Umax
"
--- s o l ( t ,
Umax, Umin)
> f(t, um~
~
sof(t, ~o Umin, SoUmax)
u m o ~ ) - u!'rain
which leads to the contradiction
0
<
--
I n (Umi
i
80Umax
) Ii t,to -- f~o 1 ( ~ ~" ( t ) -
~ o ~ oii ~
(t)) d t < O.
A similar a r g u m e n t holds if to = b. Hence So = 1 and Umax
= Umin.
[-]
Conditions of T h e o r e m 4.2 can be checked on any pair of lower and upper quasi-solutions. In some cases, it is useful to use some iterate (an, ~n) from the sequence defined in (4.4). T h e o r e m 4.3. Let no, 13o C C([a,b]) and E be defined from (4.3). Suppose f 9 E ~ R is an L1-Carathdodory function such that for some g > --(b_--~) 2, f ( t , u , v ) g v is non-increasing in u, non-decreasing in v and for almost every t and all u, v with no(t) < u < v < ~o(t), 7r2
[f(t, u, v) - f ( t , v, u)](v - u) < ((b-a)2 + 2 K ) ( v - u) 2. Assume ao and ~o are coupled lower and upper quasi-solutions of (4.1). Then, both sequences (an)n and (~n)n, defined for n >_ 1 by a n'' - K a n : f ( t , an-l, ~n-1) K~n-1, a n ( a ) : O, a n ( b ) - O, ~ : - K ~ n - f ( t , ~ n - l , a n - 1 ) - K a n _ , , ~n(a) - O, ~n(b) : O, -
-
converge to the same solution u of (4.1).
276
V. M O N O T O N E
ITERATIVE
METHODS
Proof. As in Proposition 4.1, we can prove t h a t the sequences (O~n)n and (/3n)n converge respectively to functions u = Umin and v = uma~ ___ u solutions of ~ ( a ) = 0, ~(b) - 0, v ( a ) = O, v(b) = O.
u ! ' - K u - f ( t , u, v) - K v , v !' - K v - f (t, v, u) - K u ,
If u r v, we c o m p u t e
~
u)](u - v ) d t
b[(v" -- u") -- K ( v =
[ f ( t , v,
u) -
7r2
< ((b-a)~ + K )
and also
f ( t , u, v) - K ( u
-
v)](u
-
v)dt
~ab
(u - v) 2 dt
b
fa
-
[(~/' -
u") -
- /
K(v
-
u)](v
-
u)dt
b
[ ( v ' - u') 2 +
g(v
-
u) 2] dt
Ja
~r2
--- ((t~-a)2 + K )
fab (v -
u) 2 dt,
which is a contradiction.
[2]
Another result in this direction uses a one-sided Lipschitz condition on the function f. T h e o r e m 4.4. Let ao, ~o E C([a,b]) and g be defined f r o m (4.3). Ass u m e g : E ~ IR is an L l - C a r a t h d o d o r y f u n c t i o n such that g ( t , u , v ) is non-increasing in u and non-decreasing in v and f o r s o m e L > (b-a)~' g(t, u, V) -- L v is non-increasing in v. A s s u m e ao and ~o are coupled lower and upper quasi-solutions of (4.1) with 1 f (t, u, v) = ~[g(t, u, u ) + g ( t , u , v )
- L ( u - v)].
Then, both sequences ( a n ) ~ and (/3n)n, defined f o r n > 1 by I/
OZn: f(t, an-l,~n-1), ~
= f(t,~n-l,an-1),
an (a) = O, a n (b) = O, /3,~(a) = O, ~,~ (b) - O,
converge to the s a m e solution u of u" = g(t, u, u),
u(a) = u(b) = O.
4. A M I X E D
APPROXIMATION
SCHEME
277
Proof. By Proposition 4.1, the limit functions u-
lim ~n
n---,c~
and
v =
l i m ~n >__ u
n---,c~
exist and are such t h a t
u" - 89
u, u) + g(t, u, v) - L ( u - v)],
l)// __ ~[g(t,v, 1 V) + g ( t , v ,
u)-L(v-u)],
u(a) = O, u(b) = O, v(a) = O, v(b) = O.
Hence, w - v - u is a nonnegative solution of
w" + L w - h(t),
w(a) - w(b) = O,
where h(t) = 89
v, v) - g(t, u, v) + g(t, v, u) - g(t, u, u)] _< 0. Such a ~2 nontrivial nonnegative solution does not exists if L > (b-a)~ as otherwise we have the contradiction
jfa b 7r 2
0 < (L - (b-a):)
-
71"
w(s) sin ~_a(S - a) ds
" ( s - a) ds w " ( s ) + n w ( s ) ) s i n b_--~ b
= ~
h ( s ) s i n b~_a(S -- a ) d s <_ O.
KI
Monotone Iterative Methods 1
An Abstract Formulation
Let Z be a B a n a c h space. An order cone K C Z is a closed set such t h a t for all u and v E K , u + v E K , for a l l t E R + a n d u E K , t u E K , if u E K and - u E K t h e n u - 0. Such a cone K induces an order on Z" u<_v
if and only if
v-uEK.
We write equivalently u < v or v >_ u. T h e cone is said to be normal if t h e r e exists c > 0 such t h a t 0 < u < v implies [[ul[ <_ cl{v[I. T h e following t h e o r e m gives conditions for an increasing sequence ( a n ) n to converge to a fixed point of T. T h e o r e m 1.1. Let X C Z be continuously included Banach spaces so that Z has a normal order cone. Let a and ~ E X , a < ~, E = {u E X l a < u < Z}
and let T 9 s ---, X be completely continuous in X . sequence (an)n defined by ao = a,
an = T a n - 1
(1.1)
Assume that the
(1.2)
is bounded in X and for all n E N a n ~__ C~n+l ~--- Z.
Then the sequence (~n)n converges monotonically in X to a fixed point u of T such that
a
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Proof. Claim - The sequence (an)n converges in X . The sequence (an)n is increasing and included in C. As the set A = {an ] n E N} is bounded in X , T ( A ) is relatively compact in Z . Hence, any sequence (ank)k C (an)n has a converging subsequence in X and therefore in Z. As the order cone is normal and the sequence is monotone, the sequence itself converges in Z, i.e. there exists u E Z so t h a t Z
a _<_u <_/~ and an --* u. It follows t h a t all such subsequence converging in X have the same limit u, which implies X
an----~U.
T.
Next, we deduce from the continuity of T that u is a fixed point of [::] A similar result holds to prove the convergence of the sequence (/~n)n.
T h e o r e m 1.2. Let X C Z be continuously included Banach spaces so that Z has a normal order cone. Let a and ~ E X , a < ~, C be defined by (1.1) and T " E ~ X be completely continuous in X . Assume the sequence (13n)n defined by ~0=~, ~n=T~n_l, (1.3)
is bounded in X and for all n E N ~>~.+1>~. Then the sequence (~n)n converges monotonically in X to a fixed point v of T such that ~
a < Ta
and
T ~ < ~.
Then, the sequence (an)n and (~n)n defined by (1.2) and (1.3) converge monotonically in X to fixed points Umin and Umax of T such that a <_ Umi~ < Um~ < ~.
Further, any fixed point u E E of T verifies Umin ~_ U ~ Umax.
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Proof. Claim 1 - The sequence (an)n converges in X to a fixed point Umin of T such that a < Umin <_ ~. As T is monotone increasing, we prove by recurrence that for any n C N, an ~_ an+l <_ ~. Hence, (an)n C s and since T(s is relatively compact in X the sequence (an)n is bounded in X. The claim follows now from Theorem 1.1. Claim 2 - The sequence (~n)n converges in X to a fixed point Umax of T such that Umin <__Uma~ <_ ~. As for Claim 1 we prove existence of a fixed point u , ~ such t h a t a _< U m ~ <_ ~. Next, as T is monotone increasing and a <__ ~ we deduce by recurrence t h a t an <__ ~n. At last, going to the limit we obtain Umin <_ U m ~ . Claim 3 - A n y fixed point u c g of T verifies Umin <_ u <_ Umax. Since a _< u < /3, we deduce by recurrence an = T a n - 1 <_ T u = u <_ T ~ n - I ~n. The claim follows now going to the limit. [::] -
-
R e m a r k 1.1. In Theorem 1.1, we cannot prove the fixed point u is minimal since we only have some control on T ( u ) for u C {an [ n C N}. Consider for example any continuous function T : [ - 1 / 2 , 1/2] ~ R such that 1 T ( _ 2~+1) = 1 T ( - ~ )1 = 2(n+1), 2n+1" The assumptions of the theorem are satisfied for a = - 1 / 2 and ~ - 1/2. 1 The sequence (an)n - (--~-~)n converges to the fixed point u - 0. However 1 1 E [ - ~1 ' ~] are fixed points u - 0 is not minimal since the points Un 2n+1 ofT.
2 2.1
Well-ordered The
Lower
Periodic
and
Upper
Solutions
Problem
Consider the periodic b o u n d a r y value problem u"= f(t,u), u(a) = u(b), u' (a) = u'(b),
(2.1)
where f is a continuous function. Our aim is to build an approximation scheme, easy to compute, that converges to solutions of (2.1). To this end, given continuous functions a and ~, and M > 0, we consider the sequences (an)n and (~n)n defined by OZ0 :
C~,
Ol nt, - - M o~n - - f ( t , Oln -- 1 ) - - M O~n - 1 , / / : : n(b)
(2.2)
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V. M O N O T O N E
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and
0=Z, ~ - M~n = f(t, ~ n - s ) - Mfln-1, ~n (a) = ~,~ (b), ~" (a) = ~" (b).
(2.3)
The approximations an and/~n are "easy to compute", in the sense that for every n, the problems (2.2) and (2.3) are linear and have unique solutions which read explicitely
an(t) = n(t)
L
G(t,s)(f(S, an_l(S)) - M a n _ l ( S ) ) d s ,
=
-
where G(t, s) is the Green function of the problem
u " - M u = f(t), u(a) = u(b), u'(a) = u'(b).
(2.4)
Clearly this method does not avoid numerical difficulties such as those related to stiff systems. The next theorem proves the convergence of the a~ and/3n. T h e o r e m 2.1. Let a and/3 C C2([a, b]), a < ~ and E := {(t, u) e [a, b] x R I ~(t) _ u _ fl(t)}.
(2.5)
Assume f : E ~ ]R is a continuous function, there exists M > 0 such that aZl
(t, ux),
e E,
Ul ~ U2 implies f(t, u 2 ) - f(t, UX) ~ M ( u 2 - Ul) and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , a(a) = a(b), a'(a) >_ a'(b), ~"(t) <_ f ( t , ~ ( t ) ) , ~(a) = ~(b), ~'(a) <_ ~'(b). Then the sequences (an),~ and (fln)n defined by (2.2) and (2.3) converge monotonically in Cl([a,b]) to solutions Umin and Umax of (2.1)such that a ~__ Umi n ~ Urea x ~ ~.
Further, any solution u of (2.1) with graph in E verifies Umi n ~_~U ~_ Uma x.
Proof. Let X = CS([a,b]), Z = C([a,b]), K = {u E Z l u ( t ) >_ 0 on [a,b]} be the ordered cone in Z and $ be defined from (1.1). Define the operator T'$---+ X by Tu(t) = Sa b G(t, s)(f(s, u(s)) - Mu(s)) ds,
2. T H E W E L L - O R D E R E D
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245
where G(t, s) is the Green function of (2.4). This operator is continuous in X and monotone increasing. Further, T(E) is relatively compact in X, a < T~ and ~ >_ T~. The proof follows now from Theorem 1.3. K] R e m a r k Notice that the assumption a and ~ E C2([a, b]) is not restrictive. If these functions are only C2-10wer and upper solutions, the first iterates al and ~1 satisfy the assumptions of the theorem and are such that a <_
al <~1 <~. R e m a r k Recall that existence of the minimal and maximal solutions and Umax follows from Theorem I-2.4.
Umi n
Next, we consider a derivative dependent problem (2.6)
u" - f (t, u, u'), u(a) - u(b), u'(a) = u'(b).
As above, given a, 13 C C~([a, b]) and L > O, we consider the approximation schemes Ol0
--
Ol~
, 1) - L a n _ 1, a n'' -- L a n -- f (t, an-- 1, an-Ol n ( a )
- - oL n ( b ) ,
! oL n ( a )
(2.r)
! - - oL n ( b )
and ~0 : ~
3 ~ - L3n = f(t, Z n - 1 , 3 ~ _ 1 ) - L3n-1, ~ (a) = Zn (b), /3" (a) = Z~ (b).
(2.8)
Such problems lead to a major difficulty. A straightforward application of the previous ideas would be to assume that for any ul, u2, Vl and v2, ul _ u2 implies f ( t , u2, v2) - f ( t , ul, vl) <_ L(u2 - ul). This would mean that f does not depend on derivatives. The following result turns out the difficulty. T h e o r e m 2.2. Let a and Z C
C2([a, b]),
a < 13 and
E "= {(t, u, v) e [a, b] x R2 I a(t) _< u <__Z(t)}.
(2.9)
A s s u m e f " E -o R is a continuous function, there exists M >_ 0 such that f o r all (t, ul, v), (t, u2, v) C E, ul <_ u2
implies
f ( t , u2, v) - f ( t , ul, v) ~ M ( u 2 - Ul);
(2.10)
there exists N >_ 0 such that f o r all (t, u, vl), (t, u, v2) c E,
If(t, u, v2) - f(t, u, Vl)l ~
Nlv2 -
Vl];
(2.11)
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and for all t E [a, b]
a"(t) >_f(t,a(t),a'(t)), Z"(t) <_ f(t, Z(t), Z'(t)),
a(a) = a(b), a'(a) >_ a'(b), ~(a) - ~(b), ~'(a) <_ ~'(b).
At last, let L > 0 be such that N 2 + -~ N v/N2 Jr 4M L >_ M + --~-
(2.12)
and for all t E [a, b] f ( t , a ( t ) , a ' ( t ) ) - f ( t , ~ ( t ) , ~ ' ( t ) ) + L(~(t) - a ( t ) ) >_ O.
(2.13)
Then, the sequences (an)n and (13n)n defined by (2.7) and (2.8) converge monotonically in Cl([a, b]) to solutions u and v of (2.6) such that
a
(a) The function w - ~ -
~ _ 0 satisfies
- w " + Niw' I + (M + 1)w = h(t) > O,
w(a) = w(b), w'(b) > w'(a). Thus, using the m a x i m u m principle (Theorem A-5.3 with p(t) - - N s i g n w ' and q(t) - - ( M + 1)) we can prove that if a ~=/3, our assumptions imply a < /3 on [a,b]. Also i f u i s a solution of (2.6) such t h a t a ~
Proof of Theorem 2.2 9 The proof uses Theorems 1.1 and 1.2 with X = Cl([a, b]), Z = C([a, b]) and g = {u E Z Iu(t) > 0 on [a, b]} as the ordered cone in Z. Let E be defined from (1.1). The operator T " E ~ X, defined by Tu(t) =
G(t, s)(f(s, u(s), u'(s)) - Lu(s)) ds,
where G(t, s) is the Green function of (2.4) with M - L, is completely continuous in X. With these notations, the approximation schemes (2.7) and (2.8) are equivalent to (1.2) and (1.3).
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CASE
A : Claim - Let L > 0 satisfy (2.12). Then the functions an defined recursively by (2.7) are such that for all n E N, ( a ) a n is a lower solution, i.e. It
I
an(t) >_ f ( t , a n ( t ) , a n ( t ) ) , ! ! an (a) = an (b), a n (a) >_ a n (b),
(2.14)
(b) a n + l ~ a n . The proof is by recurrence.
Initial step 9 n = O. The condition (2.14) for n - 0 is an assumption. Next, w = a l - a 0 is a solution of - w " + L w = a~(t) - f ( t , do(t), a'o(t)) >_ O, w(a) = w(b), w'(a) <_ w'(b). Hence, we deduce (b) from the m a x i m u m principle (Theorem A-5.3).
Recurrence step - 1st part : assume (a) and (b) hold for some n and let us prove that 1!
a n + l ( t ) >_ f ( t , a n + , ( t ) , a n!+ 1 ( t ) ) , a n + l ( a ) = an+l(b) , a n' + l ( a ) _> an+ l(b). Let w = a n + l - an. We have I!
!
-an+l + f(t, an+l,an+l) - f ( t , ~ + 1 , ~'n+l) - f ( t , an, an) - L ( a n + l - a n ) -<_ M ( a n + l - an) + NI an+ ' 1 - anl - L ( a n + l - an) =(M-L)w+g[w'[. On the other hand, w satisfies
- w " + L w = h(t),
w(a) = w(b), w'(b) - w'(a) = A,
(2.15)
with h ( t ) " = a nI ! ( t ) - f ( t , a n ( t ) , a nI ( t ) ) >_ 0 and A >_ 0. Its solution w reads
w(t) = k
h(s) cosh v/L(k~ -~ + s -
+
h(s) c o s h , / - s
b-a
t)ds + t - s)
+ A cosh v
(t -
where k = ( 2 v ~ sinh v ~ - a )
-1 .
Hence, to prove an+ 1 is a lower solution, we only have to verify
~t [(M - L)cosh v ~ ( - ~ + s - t ) + N v / L I sinh x / T ( L ~ + s - t)l ] h ( s ) d s < O,
]
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V. M O N O T O N E
248
ITERATIVE METHODS
b
ft
[(M - L)cosh v / L ( L ~ + t - s) + N v ~ I sinh v / - L ( ~ + t - s)l ] h(s) ds < 0
and (M - L) cosh V~(t - ~2-q~)+ NvrLI sinh v ~ ( t - ~ ) 1
-< 0.
Since h is nonpositive and (M-
L ) c o s h x + N v ~ I sinh x l _ ( M -
L + N v ~ ) I sinhxl
for all x C R, we obtain ( M - L ) w + N I w ' ! <_ 0 if M - L + N V ~ < 0, which follows from (2.12). Recurrence step - 2d part 9 assume (a) and (b) hold for some n and let us prove that an+2 _> an+l. The function w = a n + 2 - an+l satisfies (2.15),
where I!
h(t) ": a n + l ( t ) - f ( t , a n + l ( t ) , an+l(t)) '
and
A = 0.
From the previous step h(t) >_ 0 and the claim follows from the maximum principle (Theorem A-5.3). B : Claim - Let L > 0 satisfy (2.12). Then the functions 13n defined recursively by (2.8) are such that for all n c N, (a) ~n is an upper solution, i.e.
Z"(t) _< f(t, Zn (a) = (b),
Z'(t)), (a) _< Z" (b),
(b) Z +I The proof of this claim parallels the proof of Claim A. C : Claim - a n
< 3n. Define, for all i E N, wi = 3i - a i
and
hi(t) := f ( t , a i ( t ) , a:(t)) - f ( t , ~i(t), ~ ( t ) ) + L(~i(t) - a i ( t ) ) .
The proof of the claim is by recurrence. Initial step 9 a l _ 0 and A -- 0. Using the maximum principle (Theorem A-5.3), we deduce that Wl _> 0, i.e. a l __ 2. I f hn-2 >_ 0 and a n - 1 <_ ~n-1, then hn-1 >_~0 and an <_ ~n. First, let us prove that for all t C [a, b], the function hn_l is
nonnegative. Indeed, we have t
h n - i - f ( ' , a n - l , a n _ 1) - f(', ~n-1,/3tn_i)+ L ( ~ n - i - a n - i ) ! >_ - i ( ~ n - i -an-I)NI/3tn_i _ _ a n--l[ + L(/3n-1 - - a n - I ) -- ( L - M ) w n - i - N[w~n_ i [.
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249
Notice that wn-1 is a solution of (2.15) with h(t) - hn-2(t) _ 0 and A = 0. Hence, we can proceed as in the proof of Claim A to show that hn-1 _> 0. It follows then from the maximum principle (Theorem A-5.3) that Wn is nonnegative, i.e. an _ ~n. D " Claim - There exists R > 0 such that any solution u of u" >_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with ~ <_ u <_ 13 satisfies [lu'lloo < R. We deduce from the assumptions that u" - f (t, u, u') + h(t), where h (t) _> 0, f ( t , u, u') + h(t) >_ - max If(t, u, 0)1 - Nlu' I F
(2.16)
and F = {(t, u) l t C [a, b], a(t) <_ u <_/3(t)}. The proof follows now using Proposition I-4.5. E 9 Claim - There exists R > 0 such that any solution u of u" <_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with a <__u <_ 13 satisfies I[u'lloo < R. The proof repeats the argument of Claim D. F 9 Conclusion - We deduce from Theorems 1.1 and 1.2 that the sequences (an)n and (~n)n converge monotonically in Cl([a, b]) to functions u and v which are solutions of (2.6) such that a _< u __ fl and a _< v <_/3. Further, as an _< ~n for any n, we have u _ v. K] R e m a r k 2.1. In Claim D or E, the a priori bound on ilu'll~ has to be independent of h. We cannot deduce it from an usual Nagumo condition, the control we have on f ( t , u, u') + h(t) being basically one-sided (see (2.16)), which is why we used a one-sided Nagumo condition. Convergence to minimal and maximal solutions are easy to prove in case L is large enough. C o r o l l a r y 2.3. Let the assumptions of Theorem 2.2 be satisfied. Then, for L large enough, the sequences ((~n)n and (~n)n defined by (2.7) and (2.8)converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.6) such that < Umi~ <_ U m ~ <_ ~. Further, any solution u of (2.6), such that a < u < Z, verifies U m i n ~ U ~__ U m a x .
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Proof. Existence of extremal solutions umin and Umax follows from Theorem I-5.6. If a is not a solution, we deduce from the above Remark (a) t h a t umin > c~. Hence, choosing L > 0 large enough so that (2.12) is satisfied and f ( t , a ( t ) , a ' ( t ) ) - f(t, Umin(t),U~min(t)) + L(umin(t) - a ( t ) )
>_ 0
on [a, b], we can apply Theorem 2.2 with fl - Umin and obtain t h a t u = lim an is such t h a t a __ u <_ umin, whence u = Umin. n---~oo
Similarly, we prove lim
~ n --" U m a x .
[-1
n---,cx~
Remark
Observe t h a t the problem u" - u + (u') 2 - sin t, -
u'(0)
-
cannot be worked out from Theorem 2.2 as it does not satisfy (2.11). However, we can proceed as follows. Notice first that such a problem satisfies a Nagumo condition. Next, we know that lower and upper solutions, c~ and E [-1, 1], of problems t h a t satisfy this Nagumo condition have a pr/o r / b o u n d e d derivatives: Ila'll~ and I1~'11~r - R. We can modify then the equation for lull > R so t h a t the same a prior/bound on the derivatives can be obtained for the modified problem together with (2.11). It follows then t h a t the approximations defined from (2.7) and (2.8) are the corresponding approximations for the modified problem so that convergence follows from Theorem 2.2. The following theorem uses an approximation scheme which, from a computational point of view, is more involved since it uses piecewise linear equations. On the other hand, the analysis is simplified since the coefficients in the approximation scheme are the Lipschitz constants on the nonlinearity. Notice also t h a t in this theorem we use different approximation schemes for the c~ and for the ~n. 2.4. Let ~ and ~ E C2([a, b]), r < ~ and E be defined by (2.9). Assume f 9E ~ R is a continuous function, there exists M > 0 such that for all (t, ul, v), (t, u2, v) E E, Theorem
Ul ~ U2
implies
f(t, u2, v) -- f(t, ul, v) < M(u2 - ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, v2) E E, If(t, u, v2) - f(t, u, Vl)l <_ g[v2 - vii; and for all t E [a, b] a"(t) >_ f(t, a(t), a'(t)), ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) <_ ~'(b).
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Let c~o = ~ and ~o = 13. Then the problems I
gl~
c~,~" -
!
I
- c~n_l[ - Mc~n = f ( t , a ~ - l , ~ ! ! a n ( a ) = C~n(b), C~n(a) - an(b)
1)
--
MOln-1,
-
M~n-1,
and
~
+ NI~'~ - ~'~-~1- M~n
=
f(t,~n-l,~-l)
~n(a) = ~n(b), ~ ' ( a ) - ~ ' ( b ) , define sequences (C~n)n and (13n)n that converge monotonically in Cl([a, bl) to solutions Umin and Umax of (2.6) such that OL < U m i n < Uma x ~_< /3.
Furthermore, any solution u of (2.6), such that c~ < u
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u C Z l u ( t ) > 0 on [a,b]} be the ordered cone in Z and g be defined from (1.1). A " Definition of a completely continuous operator T. u C X the problem v"
-
NIv' -
u'(t)l-
Claim - For any
M y = f ( t , u(t), u'(t)) - M u ( t ) , v(a) = v(b), v'(a) = v'(b),
(2.17)
has a unique solution v. Notice t h a t if k > 0 is large enough, - k and k are well-ordered lower and upper solutions of (2.17). Existence of a solution of this problem follows then from Theorem I-5.3 with ~(s) = K ( s + 1) and K > 0 large enough. Let Vl and v2 be two solutions of (2.17). Define w = v 2 - Vl and assume t h a t for some to E [a, b] w(to) = m a x w(t) > O. tE[a,b]
We compute w'(to) = v ~ ( t o ) - v~(to) - 0 and obtain the contradiction w ' ( t o ) = M w ( t o ) + g ( I v ~ ( t o ) - u'(to)[ - I v y ( t 0 ) - u'(to)[) > 0. Hence w <_ 0. Similarly we prove w _ 0, which implies w = 0, and the solution of (2.17) is unique. Now, we can define the operator T'E~X,
uHTu,
where s is defined from (1.1) and T u is the solution of (2.17). This operator is completely continuous.
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-
ITERATIVE METHODS
F o r all n C N , a n + l
I n i t i a l s t e p 9w
=
>_ a n .
Ta-
oL 1 - o L 0 -
a > O. Notice that w solves the problem
w"Nlw' I - Mw = f(t,a(t),a'(t)) - a " ( t ) < O, w ( b ) - w ( a ) - 0, w ' ( b ) - w ' ( a ) = a ' ( a ) - a ' ( b ) >_ O.
The claim follows from the maximum principle argument used in Claim A. R e c u r r e n c e s t e p - A s s u m e a n + l - a n >_ 0 a n d p r o v e w = an+2 - a,~+l >_ 0.
The function w satisfies w" - Nlw'l - Mw ! ! -- f(t,O~n+l,Oen+l)- f ( t , a n , O l n ) - M ( o ~ n + l --
an) -
N]an+'
1 -
ten! I
<0 and w(a)
w(b), w'(a)
-
-
w'(b).
The claim follows from the same maximum principle argument. C 9 Claim - an
The proof repeats the argument of the previous
< ft.
claims. D 9 C l a i m - T h e s e q u e n c e (an),~ is b o u n d e d in X .
The functions an solve
problems u ( a ) - u ( b ) , u' (a) - u' (b),
u" + Nlu'[ - Mu = gn(t),
where I
I
I
+Nlo/n_ll q- f ( t ,
an +f(t,a~_l,0)-Ma~_l
n- l) _
1
~CI~ I
n-l)
-
-
f(t, an
_
1 O)
>_ f ( t , a ~ _ x , 0) - Ma~_~ > f(t, f l , 0 ) - Mfl. As the gn are uniformly lower bounded, the boundedness of the an follows now from Proposition I-4.5. E 9 C o n c l u s i o n - The convergence of the sequence (an)n to a solution Umin of (2.6) follows from Theorem 1.1. To prove Umin is a minimum solution, let u be any solution of (2.6) such that a < u
Similarly, we prove the convergence of the sequence (fin)n to a maximal solution using Theorem 1.2. [3 2.2
The Neumann
Problem
The Neumann problem u" - f(t, u), u' (a) = O, u' (b) = O,
(2.18)
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253
where f is a continuous function, can be investigated along the lines of the periodic problem. T h e o r e m 2.5. Let a and ~ C C2([a, b]), a _ 13 and let E be defined from (2.5). A s s u m e f " E ~ IR is a continuous function, there exists M > 0 such that for all (t, u 1), (t, u2) e E, Ul ~ U2 i m p l i e s f ( t , u 2 ) -
f(t, Ul) ~
M(u2-Ul)
and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , ~"(t) < f(t,13(t)),
a'(a) >_ O, a'(b) <_ O, ~'(a) < O, ~'(b) > O.
Then the sequences (an)n and (~n)n defined by ao ~ a~
a n''- M a n -- f ( t , a n - l ) - - M a n - 1, I I a n (a) -- O, a n (b) - O, and ~o = ~,
~-
MZn - f(t,~n-1) - M~n_l, 13~(a) = O, 13~n(b) - O,
converge monotonically in Cl([a,b]) to solutions Umin and Umax of (2.18) such that a <_ umi~ <_ u m ~
<_ 3.
Further, any solution u of (2.18) with graph in E verifies Umi n ~__ U ~ Urea x.
E x e r c i s e 2.1. Prove the above theorem. Hint 9 See the proof of Theorem 2.1.
To deal with derivative dependent equations u" = f (t, u, u'), u' (a) = 0, u' (b) - 0,
(2 19)
we can use the ideas of Theorem 2.2. T h e o r e m 2.6. Let a and ~ E C2([a, b]), a < 13 and let E be defined from (2.9). A s s u m e f " E --~ • is a continuous function, there exists M >_ 0 such that for all (t, Ul, v), (t, u2, v) c E, Ul <_ U2
implies
f ( t , u2,v) - f ( t , u l , v ) <_ M(u2 - ul);
254
V. M O N O T O N E
ITERATIVE METHODS
there exists N > 0 such that for all (t, u, vl), (t, u, v2) E E, If(t, u, v2) - f ( t , u, Vl)l ___Nlv2 - vii; and for all t E [a, b] a"(t) >_ f ( t , a ( t ) , a ' ( t ) ) , [3"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a'(a) > O, c~'(b) < O, [3'(a) < O, [Y(b) >_ O.
Then, if L is large enough, the sequences (an)n and (~n)n defined by Ol0 --" Ol~ I 1 ) - Lc~n-1, c~n'' - L c~n - f ( t, an-1, OLn! i a n (a) = O, a,~ (b) - 0
and flo -- fl, !
~
- Lfln = / ( t , ~n-1, f ~ n - 1 ) -~'~ (a) : O, ~ (b) = 0
L[Jn-1,
converge monotonically in cl([a, b]) to solutions Um~n and such that
Umax
of
(2.19)
<_ Umin < Um.~ <_ ~.
Further, any solution u of (2.19), such that ~ < u < ~, verifies Umin
~ U < Umax.
E x e r c i s e 2.2. Prove the above theorem. Hint " See the proofs of Theorem 2.2 and Corollary 2.3, or [65].
We can investigate problem (2.19) along the lines of Theorem 2.4. T h e o r e m 2.7. Let a and fl E C2([a, b]), a <_ ~ and let E be defined from (2.9). A s s u m e f " E --+ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) E E, Ul < u2
implies
f (t, u2, v) - f (t,
U l , 1)) ~__ M(u2
- Ul );
there exists N > 0 such that for all (t, u, Vl), (t, u, v2) C E, If(t, u, v2) - f ( t , u, vx)l ~ NIv2 - vii; and for all t E [a, b]
>_ f(t, ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
>_ O,
<_ O,
~' (a) <_ O, ~' (b) > O.
2. T H E W E L L - O R D E R E D
255
CASE
Let ao = a and/30 = ~. Then the problems /
/
i
an"- Nian -an-llM a n - f ( t , C~n_l,Ctn_l)- M a n - 1 I l an (a) = O, a n (b) - 0 and
r + NIr
- fl'~-i [ - M/3n - f(t,/3n-1, ~ ' _ 1 ) - M/3n_l, /3~ (a) = 0, /3" (b) = 0
define sequences (O~n)n and (~n)n that converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.19) such that
Further, any solution u of (2.19), such that a <_ u <_ ~, verifies U m i n ~_~ U ~ U m a x .
E x e r c i s e 2.3. Prove the above theorem. Hint 9 See the proof of Theorem 2.4 or [234].
2.3
The Dirichlet
Problem
Consider the Dirichlet problem (2.20)
u" = f ( t , u), u(a) = O, u(b) = O,
where f is a continuous function. T h e o r e m 2.8. Let a and ~ C C2([a, b]), a < ~ and let E be defined from (2.5). Assume f " E ~ R is a continuous function, there exists M > 0 such that for all (t, u 1), (t, u2) c E, Ul
~
U2 implies f ( t , u 2 ) - f ( t , ul) <_ M ( u 2 -
and for all t E [a, b]
a"(t) >__f(t, a(t)),
a(a) <_ O, a(b) <_ O, ~(a) >_ O, ~(b) >_ O.
~"(t) <_ f ( t , ~ ( t ) ) ,
Then the sequences (an)n and (~n)n defined by Ol0
--
Ol~
a~" - M a n : f ( t , O~n--1 ) -- Mozn-1, an (a) - O, an (b) : 0
Ul),
256
V. M O N O T O N E I T E R A T I V E M E T H O D S
and
Z0=Z, 13~ - Mt3~ = f ( t , ~ n - 1 ) - M 3 n - 1 , ~,~ (a) = O, ~n (b) = 0 converge monotonically in C l ([a, b]) to solutions Umin and umax of (2.20) such that Further, any solution u of (2.20) with graph in E verifies U m i n <___U <~__Urea x.
E x e r c i s e 2.4. Prove the above theorem. Hint 9 See the proof of Theorem 2.1.
In case of derivative dependent equations (2.21)
u" = f ( t , u, u'), u(a) = O, u(b) = O,
approximation schemes similar to (2.7) and (2.8) do not work. Indeed if we try to repeat the argument of Theorem 2.2, we have to prove a priori bounds on the derivatives of lower and upper solutions (see Parts D and E). As noted in Remark 2.1, it implies we use one-sided Nagumo condition which, for the Dirichlet problem, imposes the lower and upper solutions to satisfy the boundary conditions. We can think this condition is not very restrictive since the first iterates al and ~1 already satisfy such conditions. However L must also verify (2.13) and this might not be the case even for large values of L. We have then to consider alternative approximation schemes. T h e o r e m 2.9. Let a and ~ C C2([a, b]), a <_ ~ and let E be defined from (2.9). A s s u m e f 9 E ~ N is a continuous function, there exists M >_ 0 such that for all (t, ul, v), (t, u2, v) e E, Ul <_ u2
implies
f ( t , u2,v) - f ( t , U l , V) <__ M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, v2) c E, If(t, u, v2)-- f ( t , u, Vl)[ ~ NIv2 - vii; and for all t C [a, b]
> I(t, < f(t,
= O,
= O,
13(a) = O, 13(b) = O.
2. T H E W E L L - O R D E R E D
257
CASE
At last, let Ko e C([a,b]) be such that Ko(a) > 0 and for all t e [a,b], Ko(t) = - K o ( b + a - t). Then, for L large enough, the sequences (an)n and (~n)n defined by ao - c~, Zo - Z, an
"+
l
--
r
K0 (t)c~n+ 1 - - Lan + 1 = f ( t, an, OLn) ' - r
Ko(t)a~n - Lan,
GCn+x(a) -- O, ~ n + x ( b ) - 0 and ~-bx
--
r
- L~n+l - f ( t , ~ n , ~ n ) -- r ~n+l (a) = 0, 3~+1 (b) - 0
- L~n,
converge monotonically in Cl([a,b]) to solutions Umin and Umaz of (2.21) such that a < u.~
< Um~ <
~.
Further, any solution u of (2.21) with graph in E verifies U m i n ~_ U ~_~ Urea x.
E x e r c i s e 2.5. Prove the above theorem. Hint 9 See the proofs of Theorem 2.2 and Corollary 2.3 or [63].
2.4
Bounded
Solutions
In this section we consider bounded solutions of the differential equation ~" + c ~ ' -
/ ( t , ~).
(2.22)
To this end, we define the set BC(R) = {u e C(R) [ u e L~(R)}. T h e o r e m 2.10. Let a and ~ c C2(IR) N L ~ ( N ) , a ~ ~ and let E := {(t,u) c N 2 [ a ( t ) _ u __ ~(t)}. Assume c E N, f : E ~ N is a continuous bounded function and there exists M > 0 such that for all (t, ul), (t, u2) c E, Ul ~__ U2 ::~ f ( t , u 2 ) - f ( t , Ul) ~ M ( u 2 - Ul). Assume further that for all t c IR
~"(t) + ~'(t) >_ f(t, ~(t)), 3"(t) + c3'(t) <_ f(t, 3(t)). Let ao = a and 13o = 13. Then the equations " O~n+l -[- eCgtn+l - Man+ 1 "-- f(t, a ~ ) -
3x+1 + c/3'~+1 - M 3 n + l = f ( t , 3 ~ ) -
Ma~, M~
V. M O N O T O N E
258
ITERATIVE METHODS
define sequences (an)n and (~n)n C BC(R) that converge monotonically and uniformly on bounded intervals of R to solutions Umin and Umax of (2.22) such that <_ u . . ~ < u m ~ <_ ~. Further, any solution u of (2.22) with graph in E verifies U m i n ~_ U ~ Urea x .
Proof. First recall that given p E BC(R), the problem y" + cy' - Ay = p(t) with A > 0 has a unique solution in BC(R) given by
y(t) = / + 5
G(t, s)p(s) ds,
where
G(t, s) = - ~ with
-
+ ~.
exp(~(s - t ) ) e x p ( - v l t - ~l),
This result implies the (an)n and (~n)n are uniquely
defined.
Claim 1 - If an is such that for all t C R I! a~(t) + c a ' ( t ) >_ f ( t , a ~ ( t ) ) , ~ ( t ) <__Z(t),
then the function
defined by
an+ 1 E Be(R) I!
a n + 1 + C a t n + l --
Man+l = f(t, an)
- Man,
satisfies for all t C R, II
I
a n + l ( t ) + can+l(t ) > f(t, an+l(t)), c~n(t) _< a n + l ( t ) ___~(t). It is enough to observe that an and fl satisfy for all t E ]1(
a~(t) + ca~(t) - M a n ( t ) >_ f(t, an(t)) - Man(t), fl"(t) + cfl'(t) - M ~ ( t ) <_ f ( t , ~ ( t ) ) - M~(t) < f(t, an(t)) - M a n ( t ) . Hence, by Theorem II-5.6, the solution an+ 1 of
u" + cu' - M u = f (t, an) - M a n
3. T H E R E V E R S E D is such that a n _( a n + l
259
ORDER CASE
_( ~ on R and then satisfies also for all t c IR
a ~ + l ( t ) + Ca~n+l(t) - M a n + l ( t )
= f(t, an(t)) - Man(t) >_ f ( t , a n + l ( t ) ) - M a n + l ( t ) .
Claim 2 - If/3~ is such that f o r all t E R
~'n'(t) + C~'n(t) <_ f(t,3n(t)), ~(t) < Z~(t) then the f u n c t i o n ~n+l defined by 13~+ 1 -[- C~In+ 1 - Ml3n+ 1 - f ( t , 3n) - Ml3n , satisfies f o r all t E R,
~"+ i ( t ) + ~ Z ' + l (t) <_ f(t , Zn+i(t)) ~(t) <_ Zn+i(t) <__Zn (t). The proof of this claim is similar to the proof of Claim 1. Conclusion - For every bounded interval I c R, we deduce from Proposition I-4.4, the existence of K such that for all n, Ilanllcl(/) _ K and II~nllc'(/) - g . Hence, we deduce from the Arzel~-Ascoli Theorem the convergence to solutions umin and umax. As usually if u is a solution such that a <_ u <_ ~, we can take u as a lower solution and prove that Umaz = limn~c~ ~n >_ U. Similarly we have Umin <_ u. ['7
3
Lower
and
Upper
Solutions
in Reversed
Order
The monotone approximation method can be used in case the lower and upper solutions are in the reversed order ~ < a. This method works for any boundary value problem such t h a t a uniform anti-maximum principle holds. This is the case for the periodic and the Neumann problems. We can also work bounded solutions on IR. However the method does not apply to the Dirichlet problem since, in this case, we only have a non-uniform anti-maximum principle. 3.1
The Periodic
Problem
Consider the periodic boundary value problem
u " - f(t,u), ~(a) = u(b), ~'(a) - u'(b),
(3.~)
260
V. M O N O T O N E
ITERATIVE
METHODS
where f is a continuous function. In Section 2, we have built an approximation scheme for solutions of (3.1) based on the maximum principle. Here, we consider a similar approach based on the anti-maximum principle. Given continuous functions a and ~, and M > 0, we consider the sequences (an)n and ( ~ n ) n defined by a 0 = a, a n, , - ~ a n
(3.2)
M a n = f(t, an-1) + Man-l, ! (a) - a n (b), a n (a) - a~ (b)
and fl~ + Ml3~ - f(t, f l ~ - l ) + M~n-1, fl~ (a) - fin (b), fl~ (a) = 13' (b).
(3.3)
If M is not an eigenvalue of the periodic problem, i.e. M ~: (2n~ g-s-d)2 with n E N, the functions an and 13n, solutions of (3.2) and (3.3) can be written explicitly
an(t) -- jfo b G ( t , s ) ( f ( S , an_l(S)) + M a n _ l ( s ) ) d s , n(t) =
+
where G(t, s) is the Green function of the problem
u" + M u = f (t), u(a) = u(b), u'(a) = u'(b).
a n
(3.4)
The next theorem indicates a framework to obtain convergence of the and fin to extremal solutions of (3.1).
Theorem
3.1. Let a and 13 C C2([a, b]), 13 _ a and E "= {(t,u) e [a,b] • I ~ l ~ ( t ) _< u < a(t)}.
(3.5)
Assume f . E --~ R is a continuous function, there exists M e ]0, (b_--~)2] such that for all (t, Ul), (t, u2) e E, Ul <__u2 implies f(t, u 2 ) - f(t, ul) > - M ( u 2 and for all t C [a, b] a"(t) >_ f ( t , a ( t ) ) , ~"(t) <_ f ( t , ~ ( t ) ) ,
ul)
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) <_ ~'(b).
3. THE
REVERSED
ORDER
CASE
261
Then the sequences (an)n and (~n)n defined by (3.2) and (3.3) converge monotonically in C l([a, b]) to solutions Umax and Umin of (3.1) such that
Further, any solution u of (3.1) with graph in E verifies U m i n ~__ ~t ~_~ U m a x .
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u r Z lu(t) >_ 0 on [a,b]} be the ordered cone in Z and (3.6) The operator T" C --, X, defined by
Tu(t) =
jfa b
G(t, s)(f(s, u(s)) + Mu(s)) ds,
where G(t, s) is the Green function of (3.4), is continuous in X and monotone increasing (see Corollary A-6.3). Further, T($) is relatively compact in X, fl <_ T~ and a >_ Ta. The proof follows now from Theorem 1.3, where a and/3 have to be interchanged. FI Next, we consider the derivative dependent problem
u" = f(t, u, u'), u(a) = u(b), u' (a) = u'(b).
(3.7)
As above, given a,/3 E CS([a, b]) and L > 0, we consider the approximation schemes OZ0 m O~,
, a n + Lc~n - f (t, c~n- 1, c~n1) + L(~n- 1, ! c~n(a) - C~n (b), a~ (a) - c~n (b) ,,
(3.8)
and /~0 --" /~,
~ + L~n : f(t, ~n_l, ~ln_l) + L~n-1, /~n(a) -/3n(b), /3"(a) =/3"(b).
(3.9)
The following result paraphrases Theorem 2.2 in a case where the antimaximum principle applies. T h e o r e m 3.2. Let a and 13 E C2([a, b]),/~ <_ a and E "= {(t, u, v) e [a, b] • I~2 I/3(t) <_ u <__a(t)}.
(3.10)
262
V. M O N O T O N E
ITERATIVE
METHODS
Assume f " E ~ I~ is a continuous function, there exists M c ]0, (~-:-5 ] such that for all (t, ux, v), (t, u2, v) E E, Ul ~_~ u2
implies
f ( t , U2, V) -- f ( t , Ul, V) ~__ - M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, u, V2) E E , If(t, u, v2) - f(t, u, Vl)] _< N]v2 - vii; and for all t E [a, b] a"(t) > f(t, a(t), a'(t)),
a(a) = a(b), a'(a) > a'(b),
Z"(t) < f(t, Z(t), Z'(t)),
Z(a) = ~(b), Z'(a) < Z'(b).
At last, let L e [M, (b_-~) 2] be such that (L -- M) cos v ~ ( b---y-)-a N v ~ s i n v / L ( - ~ ) _> 0
(3.11)
and f ( t , a ( t ) , a ' ( t ) ) - f ( t , ~ ( t ) , ~ ' ( t ) ) + L(a(t) - ~(t)) >_ O. Then, the sequences (an)n and (~n)n defined by (3.8) and (3.9) converge monotonically in Cl([a, b]) to solutions u and v of (3.7) such that
f3<_v
T~(t) -
a(t, s)(Y(~, ~(s), ~'(s)) + Lu(s)) as,
where G(t, s) is the Green function of (3.4) with M replaced by L, is completely continuous in X. With these notations, the approximation schemes (3.8) and (3.9) are equivalent to (1.3) and (1.2). A " Claim - Let L e [M, (b---~) 2] satisfy (3.11). Then the functions oLn defined recursively by (3.8) are such that for all n c N, (a) an is a lower solution, i.e. c~n(t) _> y(t, c~,, (t),a,~(t)), ' " (3 12) t an(a) = an(b), a~(a) > an(b ), (b) an+l <_an. The proof is by recurrence.
3. T H E R E V E R S E D
ORDER
263
CASE
Initial step 9 n = 0. The condition (3.12) for n - 0 is an assumption. Next, w = a0 - a l is a solution of w" + Lw = a~(t) - f(t, ao(t), a'o(t)) >__O, w(a) - w(b), w'(a) >_ w'(b). Hence, we deduce (b) from the a n t i - m a x i m u m principle (Corollary A-6.3).
Recurrence step - 1st part: assume (a) and (b) hold for some n and let us prove that l! a n + l ( t ) > f(t, an+l(t) a' ,
_
a n + 1(a)
a~+l(b)
- -
a n' + l ( a ) __ a n + l (b).
,
Let w = a~ - an+ 1 > O. We have II
I
--an+ 1 + f(t, an+l,an+l) !
= --f(t, an,C/)-[- f(t, an+l,an+X) -- L(an -C~n+l) < M(an a n + I ) + N[ a n ' +l-anl-n(an-an+l) =(M-L)w+NIw' I. -
-
On the other hand, w satisfies
w" + Lw - h(t),
w(a) - w(b), w'(a) - w'(b) - C,
(3.13)
I with h(t) := a nI I ( t ) - f(t, an(t),an(t)) >_ 0 and C >_ 0. Observe that
1
w(t) -- 2V/~ sin v / L ( L ~ )
[C cos v ~ ( ~
m t)
t
b
Hence, using (3.11) and denoting D - 2 v ~ s i n v / L ( L ~ ) , we compute
(M - L ) w ( t ) + NIw'(t)l 1
< - - [ C [(M - L) cos v~(~--~2b - t) + Nv/L[ sin v/-L(~2-~ - t)[] -D
+
h ( s ) [ ( M - L)cos v / L ( k ~ + s - t) + Nv/L[ sin v/-L(b-~ + s - t)]]ds b
+ ft h(s)[(M - L)cos v ~ ( ~
+ t-
s)
+ N vZ-LI sin v / L ( L ~ + t - s)[] ds] <0. m
Hence a n + 1 is a lower solution.
264
V. M O N O T O N E
ITERATIVE METHODS
Recurrence step - 2d p a r t " assume (a) and (b) hold f o r some n and let us prove that an+2 _< an+l. The function w - an+l -a,~+2 satisfies (3.13), where h(t)" =
" n+l(t) -
an+ ' 1(t))
f(t,
and
C - 0.
From the previous step h(t) >_ 0 and the claim follows from the antimaximum principle (Corollary A-6.3). B 9 Claim - Let L C [ M , ( ~_~) '~ 2 ] satisfy (3.11). Then the f u n c t i o n s ~n defined recursively by (3.9) are such that f o r all n E N, (a) 13n is an upper solution, i.e. ~ " ( t ) <_ f ( t , [ 3 n ( t ) , ~ ( t ) ) , 13n(a) = 13n(b), fl~(a) <_/3~(b),
(b) The proof of this claim parallels the proof of Claim A. C 9 Claim - an >_ ~,~. Define, for all i E N, wi = ai - 13i and hi(t) := f ( t , a i ( t ) , a ' , ( t ) ) - f ( t , ~ i ( t ) , ~ ( t ) )
+ L ( a i ( t ) - ~i(t)).
The proof of the claim is by recurrence. Initial step 9 a l >_ 131. The function wl is a solution of (3.13) with h = h0 >_ 0 and C - 0. Using the anti-maximum principle (Corollary A-6.3), we deduce that wx >__0, i.e. a l >_ ~1. Recurrence step 9 Let n >_ 2. I f h n - 2 >__0 and a n - 1 >_ 13,~-1, then h n - 1 >>_0 and an >_ ~n. First, let us prove that for all t E [a, b], the function hn-1 is nonnegative. Indeed, we have l
h n - 1 = f ( t , o l n - l , 0 Z n _ l ) -- f(t, ~n-1,/31n_1) 4- L(ctn-1 - / 3 n - 1 ) >_ - M ( a n - 1 - / 3 n - 1 ) -- NI a n' _ 1 -- ~ n' _ l l 4- n ( o l n - l -- ~ n - 1 ) ! = (n- M)wn-1glwn_l[.
Recall that Wn-1 is a solution of (3.13) with h(t) - h~_2(t) > 0 and C = 0. Hence, we can proceed as in the proof of Claim A to show that h n - 1 >_ O. It follows then from the anti-maximum principle (Corollary A-6.3) that Wn is nonnegative, i.e. an > ~n. D " Claim - There exists R > 0 such that any solution u of u" >_ f ( t , u, u'),
u(a) = u(b), u'(a) = u'(b),
with 13 <_ u <__ a satisfies Ilu'l]~ < R. We deduce from the assumptions that u" - f (t, u, u') + h(t),
3. T H E R E V E R S E D
ORDER
CASE
265
where h(t) > O, : ( t , u, u') + h(t) >__ - m a x F If(t, u, O)l -- Nlu'l and F =
{(t, u) l t c [a, b], ~(t) <_ u < ~(t)}. The proof follows now using Proposi-
tion 1-4.5. E 9 Claim - There exists R > 0 such that any solution u of u" <_ f ( t , u, u'),
u(a)
-
u(b), u'(a)
-
u'(b),
with ~ < u <_ a satisfies Ilu'll~ < R. The proof repeats the argument of Claim D. F 9 Conclusion - W e deduce from Theorems 1.2 and 1.1, where a and have to be interchanged, t h a t the sequences (Oln) n and (fl~)n converge monotonically in CX([a,b]) to functions u and v such that fl _ v _< a and < u < a. Further Claim C implies v _ u. [:3 E x a m p l e 3.1. Consider the problem u" - k arctan u' + cu 3 = sin t,
~(o) - ~ ( 2 ~ ) ,
~'(o) = ~'(2~),
where c > 0 and k > 0. This problem has a - c -1/3 and f l - - c -1/3 as lower and upper solutions. To satisfy the assumptions of Theorem 3.2, let M - 3c 1/3 and N - k, choose L E ]3c 1/3, 1/4[ and assume c < 12 -3
and
k <- - L-3cl/3 cotanv~lr. v~
As in Section 2, we can modify the analysis in case we agree to compute the approximations an and ~n from piecewise linear problems. The following result adapts Theorem 2.4 to lower and upper solutions in reversed order. T h e o r e m 3.3. Let (~ and ~ E C2([a, b]), ~ < a and let E be defined by (3.10). A s s u m e f 9E ~ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) C E, Ul ~ U2
implies
f ( t , u2, v) - f ( t , Ul, V) ~ - M ( u 2 - Ul),
there exists N >__0 such that for all (t, u, Vl), (t, u, v2) c E, I f ( t , u, v2) - f ( t , u, Vl)l ~ N I v 2 - vii
and for all t E [a, b] a"(t) > f ( t , a ( t ) , c ~ ' ( t ) ) ,
~"(t) <_ f(t, ~(t), ~'(t)),
a(a) = a(b), a'(a) >_ a'(b), ~(a) = ~(b), ~'(a) < ~'(b).
266
V. M O N O T O N E
ITERATIVE
METHODS
Assume also b - a < 20(M, N / 2 ) , where Y/1~2'1 M a r c t a n h
O(M, 1V) =
s
./~-r},,2_ M
fil
"'
if O < M
if M = 1V2
1(.
N ~
v/M_.~2
< ~2
g -- arctan ~/M-~?~-
'
(3.14)
--
Let C~o = ~ and ~o = ~. Then the problems OZn" -
Man : f(t, an-l,an_l) + Man-l, N IOZn' --OLn_l[-Jr, ! ! an (a) : oLn (b), oLn (a) = a n (b)
and ~ + NI~
- fl~-ll + M ~ n = f ( t , ~ n - l , ~ - l ) ~n (a) = ~n (b), ~ (a) = ~ (b),
+ M~n-1,
define sequences (an)n and (~n)n that converge monotonically in Cl([a,b]) to solutions Umax and Umin of (3.7) such that
Further, any solution u of
(3.7),
such that ~ <_ u <_ a , verifies
Umin ~ U ~ Umax.
Proof. Let X = Cl([a,b]), Z = C([a,b]), K = {u e Z l u ( t ) >_ 0 on [a,b]} be the ordered cone in Z and E be defined from (1.1). A " Definition of a completely continuous operator T. Claim 1 - For any u E X the problem v" - NIv' - u'(t)l + M v = f ( t , u(t), u'(t)) + M u ( t ) , v(a) = v(b), v'(a) = v'(b),
(3.15)
has a solution v. Consider the problem v"-
s l y ' - u'(t)[ + M y = a(t),
v(a) = v(b), v'(a) = v'(b),
(3.16)
with s e [0, N] as an homotopy parameter and a e C([a, b]). Assume that solutions of (3.16) are not a priori bounded in X. Hence, there exists sequences (Sn),~ C [0, N] and (Vn)n C X such t h a t vn solves (3.16) with s = sn and l i m n _ ~ IlVnliC, = +c~. We define then ~n = Vn/llvnllc 1 and going to subsequences we can assume C1
~n--*~,
i, L 2
~n--~v"
and
Sn---+s.
3. T H E R E V E R S E D
ORDER CASE
267
Going to the limit in (3.16), we obtain that ~ solves V " - slY' I + MY = O,
V(a) = V(b), V'(a) = V'(b).
If M <_ s 2 / 4 , Theorem A-6.2 implies ~ = 0. On the other hand, if M > s 2 / 4 , we have b - a < 2 0 ( M , N / 2 ) < 2 0 ( M , s / 2 ) < 2 x ( M , - s / 2 ) (see (A6.3) and (A-6.4)) and the conclusion ~ = 0 follows from the same theorem. Hence, in all cases, this contradicts limcl ~n - ~. The proof of the claim follows now from the a priori bounds and classical arguments in degree theory. C l a i m 2 - The solution of p r o b l e m (3.15) is unique Let Vl and v2 be two solutions of (3.15). Define w = v2 - vl and compute w " + N I w ' [ + M w - N ( l v ~ - v'21 + Iv'2 - u'l - [v~ - u'[) >_ 0.
Hence, we deduce from the anti-maximum principle (Theorem A-6.2) that w > 0. Similarly we prove w <_ 0, which implies w = 0, and the solution of (3.15) is unique. Now, we can define the operator T" g ~ X,u ~ Tu,
where $ is defined from (1.1) and T u is the solution of (3.15). This operator is completely continuous. B 9 Claim - For all n c N, a n >_ a n + l . Initial step 9w = a o -
al - a-
T a > O. Notice that w solves the problem
w " + N I w ' I + M w = a " ( t ) - f ( t , a ( t ) , a ' ( t ) ) >_ O, w ( a ) - w(b) = O, w ' ( a ) - w'(b) = a ' ( a ) - a'(b) >_ O.
The claim follows from the anti-maximum principle (Theorem A-6.2). R e c u r r e n c e step - A s s u m e a n -
a n + l >_ 0 and prove w - a n + l -
an+2 >_ O.
The function w satisfies w" + N[w' I + Mw ! -- - f(t, a,-,, an) - f ( t , o+,+1~ Ol/,,+1) + Yl a n'+ l - a n [ + M ( a n - a n + l ) >0
and the claim follows from the anti-maximum principle (Theorem A-6.2). C 9 Claim - a n >_ ~.
claims.
The proof repeats the argument of the previous
V. M O N O T O N E
268
ITERATIVE METHODS
D 9 Claim - The sequence (Oln) n is bounded in X . The functions c~n, n >_ 1, solve the problems u" + g l u ' ] + M u = gn(t),
u(a) = u(b), u'(a) = u'(b),
where
g~ = N(l~'nb + I~" - ~ ' - ~ 1 -
+Nl~'n-ll +
i~'n-~l) I
f ( t , Oln_l, C e n _ l ) - f ( t , OZn_l, O)
+ f ( t , a n - 1 , O) + M a n - 1
> f(t, a~_x, O) + Mc~_~ > f(t, ~, O) + M~. As the g~ are uniformly lower bounded, the boundedness of the an follows now from Proposition I-4.5.
E 9 Conclusion - The convergence of the sequence (Oln) n t o a solution Umax of (3.7) follows from Theorem 1.2, where a and/~ have to be interchanged. To prove Umax is the maximum solution, let u be any solution of (3.7) such that ~ < u < a. We apply then Theorem 1.2 with the a and f l o f t h i s theorem replaced respectively by u and a and obtain umax - lim an >_ u. u-,co
Similarly, we prove the convergence of the sequence (~n)n to a mimimal solution using Theorem 1.1. KI 3.2
The Neumann
Problem
The Neumann problem
u" = f ( t , u), u' (a) = 0, u' (b) = 0,
(3.17)
where f is a continuous function, can be investigated along the lines of the periodic problem. T h e o r e m 3.4. Let c~ and ~ C C2([a,b]), ~ <__ ~ and let E be defined from (3.5). A s s u m e f : E ~ R is a continuous function, there exists M C ]0, (2(b'_a)) 2] such that for all (t, Ul), (t, u 2 ) E E,
Ul ~_ U2 implies f ( t , u 2 ) - f(t, ul) > - M ( u 2 and for all t C [a, b] c~'(t) >_ f ( t , c~(t)), ~ ' ( t ) <_ f ( t , ~(t)),
a'(a) >_ O, a'(b) <_ O, ~'(a) <_ O, ~'(b) >_ O.
Then the s e q u e n c e s (OZn) n and ([3n),~ defined by Ol0 : Ol~ r ! ! Ozn ( a ) : O, Oln ( b ) -- O,
Ul)
3. THE
REVERSED
ORDER
269
CASE
and ~0 ~ ~
/3~ + M[3,~ = f(t,/3n-1) + M/~n-1, [3'~(a) = O, /~'~(b) = O, converge monotonically in Cl([a, b]) to solutions umax and Umin of (3.17) such that ~__ Umin ~ Umax ~__Or.
Further, any solution u of (3.17) with graph in E verifies Umi n ~ U ~ Urea x.
E x e r c i s e 3.1. Deduce the above theorem from Theorem 1.3. To deal with derivative dependent equations
u" = f (t, u, u'), u'(a) = O, u'(b) = O,
(3 18)
we can use the ideas of Theorem 3.2. T h e o r e m 3.5. Let a and /3 c C2([a,b]), ~ < a and let E be defined from (3.10). Assume f 9 E ~ R is a continuous ]unction, there exists M e]O,( 2(b-~) ~ )2] such that for all (t, Ul, v) (t, u2, v) E E,
ul <_ u2
implies
f(t, u2, v) - f(t, ul, v) > - M ( u 2
-
Ul);
there exists N >_ 0 such that for all (t, u, Vl), (t, U, V2) E E , If(t, u, v2) - f(t, u, Vl) [ ~ N[v2 - vii; and for all t e [a, b] a"(t) > f(t, a(t), a'(t)), /~"(t) <_ f(t,/3(t),/3'(t)),
a'(a) >_ O, a'(b) <__O, /~' (a) <_ O, ~' (b) >_ O.
At last, let L C [M, (2(b'-'~))2] be such that (i
-
M ) c o s v/L(b - a) - N x / ~ sin v ~ ( b - a) >__0
and f ( t , a ( t ) , a ' ( t ) ) - f(t,/3(t),/3'(t)) + i(c~(t) -/3(t)) >_ O.
270
V. M O N O T O N E
ITERATIVE
METHODS
Then, the sequences (an)n and (~n)n defined by aO = Ol~ II
O~n ~- Lan = f (t, Oln--1, a'n - l ) + L a n - 1 , a n' ( a ) = 0, a n' (b) = 0, and
Z0=Z, ~ + L~n = f ( t , fin-l, f l ~ - l ) + Lfln-1, fl~ (a) = 0, fl~ (b) = 0, converge monotonically in C l([a, b]) to solutions u and v of (3.18) such that
~
We can investigate problem (3.18) using an analog of Theorem 3.3. T h e o r e m 3.6. Let a and Z c C2([a, b]), Z <_ a and let E be defined from (3.10). Assume f " E ~ R is a continuous function, there exists M > 0 such that for all (t, Ul, v), (t, u2, v) e E, Ul <_ u2
implies
f ( t , u2, v) - f (t, Ul, v) > - M ( u 2 - Ul);
there exists N >_ 0 such that for all (t, u, vl), (t, u, v2) C E, If(t, u, v2) - f(t, u, vl)l <_ NIv2 - vii; and for all t C [a, b] a ' ( t ) >_ f ( t , a ( t ) , a'(t)), ~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) ,
a'(a) >_ O, a'(b) <_ O, ~'(a) <_ O, ~'(b) >_ O.
Assume also b - a <_ O(M, N / 2 ) , where 0 is defined from (3.14). Let ao = a and/3o = ~. Then the problems t
t
a n " - N[an - a n -
at
M a n - f(t, a n - 1, n - l ) -[- M a n - l , a n (a) = O, a~ (b) = O,
11 + I
and
3~ + N[~tn - f~'n-l[ + M~n : f(t, ~n-1, ~tn_l) + M~n-1, /3~n(a) = 0, ~n (b) = 0, define sequences (an)n and (~n)n that converge monotonically in Cl([a, b]) to solutions Umax and umin of (3.18) such that
3. T H E R E V E R S E D O R D E R CASE
271
Further, any solution u of (3.18), such that 13 <_ u <_ a verifies Umin
~__ U ~ Urea x .
E x e r c i s e 3.3. Prove the above theorem along the lines of the proof of Theorem 3.3. Hint 9 See [51].
3.3
Bounded Solutions
Existence of bounded solutions of (2.22) can be deduced from lower and upper solutions in the reversed order. The following result parallels Theorem 2.10. Here we use/~C2(ll~) = {u c C2(PQ I u, u', u" E L~(R)}. T h e o r e m 3.7. Let a and Z E B e 2(R), a >_ ~ and let E "= {(t,u) E IR2 I ~(t) ~_ u < a(t)}. Assume c > O, f " E ~ R is a continuous bounded function such that for all (t, ul), (t, u2) e E, c2
ul <_ u2 ~ f ( t , u2) - f ( t , ul) >__-~-(u2- Ul). Assume further that for all t c R
~"(t) + r >__/(t, ~(t)), Z"(t) + cZ'(t) <__/(t, Z(t)). Let ao = a and ~o =/3. Then the equations II
c2
Oln+ 1 -~-CC~ln.i_l -~- -TO~n+l = , c ~-
~-+1 +
f ( t , an) +
Cfln+l + T ~ n + l -- f ( t , ~ n ) +
c2
"-~OLn,
~ , -~n
define sequences (an)n and (~n)n C 13C(I~) that converge monotonically and uniformly on bounded intervals of R to solutions Umaz and Umin of (2.22) such that Further, any solution u of (2.22), such that ~ < u < a verifies U m i n ~ U ~ Urea x .
Proof. First observe that given p c/~C(I~), the unique solution in BC(I~) of c2 y" + cy' + - ~ y = p ( t )
is given by y(t) -
G(t,s)p(s)ds, oo
272
V. M O N O T O N E
ITERATIVE METHODS
where
G(t, s) - ( t - s ) e x p ( - ~ ( t =0,
s)),
if s < t, ifs>t.
Claim - If an is such that for all t C I~ a~(t) + ca~(t) > f(t, an(t)), an(t) >__~(t) then the function an+l defined by a n"+ 1 -~- Catn+ 1
+
C2 "-~an+ 1
= f(t , an)+
C2 ~'an ,
satisfies for all t C ]~, a n!1+ l ( t ) + c O/n+l(t) >__f(t, an+l(t)), Oln(t) ~__ a n + l ( t ) ~ ~(t). It is enough to observe that w = c~n - a n + 1 solves c2 W l! -~- CW' Jr- " u
It
--- OLn -Jr- COln -- f ( t , a n ) .
As a nl/ + ca~ - f (t, an (t)) > 0 we deduce from the form of the solution that w >_ 0, i.e. an >_ an+l and also, for all t E R, a n1+' ' (t) -}- Catn+l
(t)
:
f(t,
c2
O~n(t))zF -~(an(t)-
O~n+l (t)) _> f(t, a n + l ( t ) ) .
The inequality ~ <_ c~n+l can be proved in the same way. The rest of the proof follows as in Theorem 2.10.
4
A Mixed
Approximation
Scheme
In the previous section, we considered sequences (an)n and (/~n)n which converge to solutions of a boundary value problem such as (2.1). The reader might have noticed that computing an and/~n can be a difficult problem. In this section, we give a method of approximations which is simple but provides only bounds on the solutions. We also give assumptions which imply these bounds to be equal, in which case they are solutions. Consider a Dirichlet problem
u" = f (t, u, u),
u(a) = O, u(b) = O.
(4.1)
To simplify the argument, we assume here f is independent of the derivative u ~, which is not essential and could be developed as in the previous sections. We shall use the following definition.
4. A M I X E D A P P R O X I M A T I O N
SCHEME
273
D e f i n i t i o n 4.1. Functions c~ and/3 c C([a, b]) are coupled lower and upper quasi-solutions of (4.1) if (a) for any t e [a, b], a(t) <_ ~(t); (b) for any to c]a,b[, either D _ a ( t o ) < D+a(to), or there exists an open interval Io c ]a,b[ such that to c Io, a c W 2,1 (io), and for a.e. t c Io
>__f(t,
Z(t));
(c) for any to e ]a, b[, either D - ~ ( t o ) > D+~(to), or there exists an open interval Io c ]a,b[ such that to c Io, ~ c W2'1(Io), and for a.e. t C Io
Z"(t) _< I(t, Z(t), (d) a(a) <_ 0 <_ ~(a),
a(b) <_ 0 <_ ~(b).
Consider the following auxiliary problem u" - f (t, u, v),
u(a) = O, u(b) - O,
v" - f (t, v, u),
v(a) = O, v(b) - O.
(4.2)
P r o p o s i t i o n 4.1. Let C~o, /~o C C([a, b]), E "= {(t, u, v) I t e [a, b], u, v e [C~o(t), ~o(t)]} and f " E increasing lower and Then,
(4.3)
~ R be an L1-Carathdodory function such that f(t, u, v) is nonin u and non-decreasing in v. Assume (~o and/~o are coupled upper quasi-solutions of (4.1). the sequences (an)n and (/3n)n, defined for n >_ 1 by t! O~n : f ( t, Otn-1, /3n-1 ), /~: : f ( t, /~n- l , Oln-1),
an (a) = 0, an (b) - 0, fin (a) - 0, fin (b) - 0,
(4.4)
converge monotonically in Cl([a, b]) to functions Umin and Urea x. The pair (umin, Umax) is a solution of (4.2) such that Olo ~- Umin ~- Umax ~---130.
Moreover, any solution (u, v) of (4.2) with so <_ u < ~o, (~o <_ v
Um~ <_ V <_ Um~.
274
V. M O N O T O N E
ITERATIVE METHODS
Proof. Let X = Cl([a,b]) xCl([a,b]), Z - C([a,b]) xC([a,b]), K - {(u,v) e Z i u >_ O,v <_ 0} and s {(u,v) E X l a o _< u _< ~o, ao <_ v _< ~o}. We define T " s ~ X , (u, v) H T(u, v), where T(u, v) is the solution (x, y) of =
x" - f ( t , u, v), y" - f ( t , v, u),
x(a) - O, x(b) = O, y(a) = O, y(b) - 0
and verify T is continuous, monotone increasing, T(s is relatively compact in X and (ao, 13o) <_ T(ao, ~o), (~o, ao) >_ T(~o, ao). Theorem 1.3 applies with a - (a0,~0) and/7 - (~0, a0), and the claims follow. [El Notice that if u is a solution of the given problem (4.1), then (u, u) is a solution of the auxiliary problem (4.2), thus Umin and Umaz are bounds on solutions of (4.1). If the bounds Umi n and umaz given in Proposition 4.1 are equal, Umin is a solution of the initial problem (4.1), and this theorem provides an approximation scheme to a solution of (4.1). In particular, this will be the case if solutions of the auxiliary problem (4.2) are unique. Unfortunately, these solutions are in general not unique as follows from the problem u n + se u - 0, v u d- se v O, -
-
u(0) - u(1) - 0, v(0) - v ( 1 ) - 0
and Theorem VIII-3.6, if we choose s > 0 small enough. The next proposition proves, under appropriate assumptions, the uniqueness of solutions of the auxiliary problem (4.2). Hence, it also proves convergence of the sequences defined in Proposition 4.1 to the unique solution of the given problem (4.1). T h e o r e m 4.2. Suppose the assumptions of Proposition ~.1 hold. A s s u m e moreover (i) there exists ~ > 0 such that C~o >_ c~o; (ii) for every s e [~, 1[, almost every t e [a, b] and every u, v e [c~0,~0] with sv<_u<_v,
> f (t,
v).
Then, the functions Umin and Um~x defined in Proposition ~. 1 are equal, i.e. are solutions of (4.1). Proof. From assumption (i), we deduce CUmax ~ Umin ~ ~max.
4. A M I X E D
APPROXIMATION
275
SCHEME
Let so = s u p ( s I ~ . ~ -< u~}. It is obvious t h a t so e [c, 1] and t h a t SoUma~ <_ Umin. From the definition of so, we deduce the existence of to E [a, b] such t h a t Umin(tO) -- SOUmax(to) -- O,
' n ( t o ) -- SOUmax(to ' Umi ) -- O.
If not we have
U m i n ( t ) - SOUm~(t) > 0, for t e In, b[, / / / Umin(a ) - sou'm~x(a ) > O, urn, n (b) - SOUma~ (b) < O. Hence there exists e > 0 so t h a t u m i n ( t ) - SOUm~(t) >_ eUm~(t) on [a, b]. This contradicts the definition of so. If to ~: b, we also have t l > to such t h a t Umi n ' (tl) -- 8 0 U mx 'a (t l) _> 0. Assume now t h a t so < 1. Hence, we can write 80Umax
"
--- s o l ( t ,
Umax, Umin)
> f(t, um~
~
sof(t, ~o Umin, SoUmax)
u m o ~ ) - u!'rain
which leads to the contradiction
0
<
--
I n (Umi
i
80Umax
) Ii t,to -- f~o 1 ( ~ ~" ( t ) -
~ o ~ oii ~
(t)) d t < O.
A similar a r g u m e n t holds if to = b. Hence So = 1 and Umax
= Umin.
[-]
Conditions of T h e o r e m 4.2 can be checked on any pair of lower and upper quasi-solutions. In some cases, it is useful to use some iterate (an, ~n) from the sequence defined in (4.4). T h e o r e m 4.3. Let no, 13o C C([a,b]) and E be defined from (4.3). Suppose f 9 E ~ R is an L1-Carathdodory function such that for some g > --(b_--~) 2, f ( t , u , v ) g v is non-increasing in u, non-decreasing in v and for almost every t and all u, v with no(t) < u < v < ~o(t), 7r2
[f(t, u, v) - f ( t , v, u)](v - u) < ((b-a)2 + 2 K ) ( v - u) 2. Assume ao and ~o are coupled lower and upper quasi-solutions of (4.1). Then, both sequences (an)n and (~n)n, defined for n >_ 1 by a n'' - K a n : f ( t , an-l, ~n-1) K~n-1, a n ( a ) : O, a n ( b ) - O, ~ : - K ~ n - f ( t , ~ n - l , a n - 1 ) - K a n _ , , ~n(a) - O, ~n(b) : O, -
-
converge to the same solution u of (4.1).
276
V. M O N O T O N E
ITERATIVE
METHODS
Proof. As in Proposition 4.1, we can prove t h a t the sequences (O~n)n and (/3n)n converge respectively to functions u = Umin and v = uma~ ___ u solutions of ~ ( a ) = 0, ~(b) - 0, v ( a ) = O, v(b) = O.
u ! ' - K u - f ( t , u, v) - K v , v !' - K v - f (t, v, u) - K u ,
If u r v, we c o m p u t e
~
u)](u - v ) d t
b[(v" -- u") -- K ( v =
[ f ( t , v,
u) -
7r2
< ((b-a)~ + K )
and also
f ( t , u, v) - K ( u
-
v)](u
-
v)dt
~ab
(u - v) 2 dt
b
fa
-
[(~/' -
u") -
- /
K(v
-
u)](v
-
u)dt
b
[ ( v ' - u') 2 +
g(v
-
u) 2] dt
Ja
~r2
--- ((t~-a)2 + K )
fab (v -
u) 2 dt,
which is a contradiction.
[2]
Another result in this direction uses a one-sided Lipschitz condition on the function f. T h e o r e m 4.4. Let ao, ~o E C([a,b]) and g be defined f r o m (4.3). Ass u m e g : E ~ IR is an L l - C a r a t h d o d o r y f u n c t i o n such that g ( t , u , v ) is non-increasing in u and non-decreasing in v and f o r s o m e L > (b-a)~' g(t, u, V) -- L v is non-increasing in v. A s s u m e ao and ~o are coupled lower and upper quasi-solutions of (4.1) with 1 f (t, u, v) = ~[g(t, u, u ) + g ( t , u , v )
- L ( u - v)].
Then, both sequences ( a n ) ~ and (/3n)n, defined f o r n > 1 by I/
OZn: f(t, an-l,~n-1), ~
= f(t,~n-l,an-1),
an (a) = O, a n (b) = O, /3,~(a) = O, ~,~ (b) - O,
converge to the s a m e solution u of u" = g(t, u, u),
u(a) = u(b) = O.
4. A M I X E D
APPROXIMATION
SCHEME
277
Proof. By Proposition 4.1, the limit functions u-
lim ~n
n---,c~
and
v =
l i m ~n >__ u
n---,c~
exist and are such t h a t
u" - 89
u, u) + g(t, u, v) - L ( u - v)],
l)// __ ~[g(t,v, 1 V) + g ( t , v ,
u)-L(v-u)],
u(a) = O, u(b) = O, v(a) = O, v(b) = O.
Hence, w - v - u is a nonnegative solution of
w" + L w - h(t),
w(a) - w(b) = O,
where h(t) = 89
v, v) - g(t, u, v) + g(t, v, u) - g(t, u, u)] _< 0. Such a ~2 nontrivial nonnegative solution does not exists if L > (b-a)~ as otherwise we have the contradiction
jfa b 7r 2
0 < (L - (b-a):)
-
71"
w(s) sin ~_a(S - a) ds
" ( s - a) ds w " ( s ) + n w ( s ) ) s i n b_--~ b
= ~
h ( s ) s i n b~_a(S -- a ) d s <_ O.
KI