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Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems
Remarks on the Lower and Upper Solutions Method for Second- and Third-Order Periodic Boundary Value Problems Pierpaolo
Omari
Dipartimento
and Maurizio
Trombetta
di Scienze Matematiche
Universitd di Trieste Piazzale Europa 1, 34127
Trieste, ltalia
ABSTRACT We study the solvability and the approximation of the solutions by monotone iteration of the second- and third-order periodic boundary value problems (BVPs)
-u”+cu’=f(t,u), u(o) = 4274,
u’(0) = u’(27+
u(o) = u(27r), u’(0) = u’(27r), u”(0) = u”(27r),
in the presence of lower and upper solutions, which may not satisfy the usual ordering condition. To this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems.
1.
INTRODUCTION
This paper is concerned the solutions by monotone
APPLIEDMATHEMATZCSAND
with the solvability and the approximation of iteration of the second and the third order
COMPUTATION 50:1-Zl(1992)
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
1 0096-3003/92/$5.00
2 periodic
P. OMARI AND M. TROMBETTA boundary
value problems
- u”+ cd =
f( t, 24)’
u(o) = u(27r),
(1.1)
u’(0) = u’(27r),
(I .2)
and
Urn+ Qd + bu’ = f( t, u) , u(0) = u(27r),
(1.3) u’(0) = u’(27r),
in presence of lower and upper solutions. suppose that a, b, c are fixed real constants
u”(0) = u”(2a),
(1.4)
In the preceding equations, we and f satisfies the Caratheodory
conditions. As far as one is concerned with problems (l.l)-(I .2), there exists a vast literature (see, e.g., the bibliography in [ll] dealing with the case where f is continuous and the lower solution a! and the upper solution 0 satisfy the ordering
condition o(t)
f@(t),
for every
t.
(I .5)
Unlike this type of result, here we mainly concentrate our attention on the case where the inequality in (1.5) is reversed. To investigate this situation, we prove in Section 2 an elementary lemma from which the validity of an antimaximum principle for certain second-order linear differential operators associated with the equation in (1.1) can be immediately deduced. Necessary and sufficient conditions on the coefficients are also given so that such an antimaximum principle holds. The existence and approximaare then proved exploiting this result in tion theorems for (I.l)-(1.2) connection with a classical and very general principle given in [lo] (compare also [4]), which permits as well a simple and unified treatment of the problem in the Caratheodory case. The aforementioned lemma is further exploited to obtain some maximum and antimaximum principles for certain third-order linear differential operators associated with the equation in (1.3), and again necessary and sufficient conditions on the coefficients are given in order that they hold. These results constitute the basic tool to get mation theorems for problem (1.3)-(1.4) in solutions. We stress that, although there exist investigation of third-order boundary value
some existence and approxipresence of lower and upper several works devoted to the problems (compare [B], [3]),
Periodic Boundary Value Problems
3
almost no result in this direction seems available for the periodic problem; indeed, the only contributions we know are given in [Is], where several questions discussed in this paper have been also raised. Finally, we note that the results of Section 2 could be carried out as well to obtain similar statements for higher-order periodic BVPs. Moreover, extensions of our theorems to more general equations, which include a dependence of the right-hand side on the derivatives, can be performed and will appear elsewhere.
2.
MAXIMUM AND ANTIMAXIMUM PRINCIPLES OF LJNEAR DIFFERENTIAL OPERATORS
FOR
A CLASS
In this section we formulate and prove some criteria of positivity or negativity of the inverses of certain linear differential operators acting in spaces of periodic functions, which in turn imply the validity of suitable maximum and antimaximum principles. Let us denote by W&‘(O, 2 a), for k = 1,2,. . . , the Sobolev space of all absolutely continuous, functions u: [0,2 n] --) R of class Ck-‘, with &‘) satisfying the periodic boundary conditions U(~)(O)= u(‘)(2 ?r), for i = O,l,..., k-l. Let us consider the nth order linear differential operator L: W&‘(O, 2 ?r) + L’(O,2 ?r), defined by
L=
D”+a,D”-‘+
.a.
+a,_,D+a,l,
where D = d /dt, 1 is the identity operator and ai, for i = 1,. . . , n, are fixed real constants. Following [18], we say that L is inverse-positive (respectively, inverse-negative) if Lu > 0 implies u > 0 (respectively, 24,< 0), where the ordering between functions is the usual pointwise ordering. Of course, this is equivalent to saying that L is invertible and its inverse is positive (respectively, negative). To study the inverse-definiteness (that is the inverse-positivity or inverse-negativity) of L, we perform a simple reduction, by factorizing L as a product of first- and second-order operators. Precisely, let us denote by (Yl,. . . , UT> PI + i-y,, . . . , P, f i-i,, with (Y~,fij, ~~43 and rj > 0, the roots of the characteristic polynomial of L, having respective multiplicities Accordingly, the splitting /Jl,...,/+.. ul>.*.,LJs.
L=(D-cY,z)“‘...(
o_,,I)“[(D-~,I)“+y~Z]“‘...[(D-~,~)”+y,2Z]”’~
4
P. OMARI AND M. TROMBETTA
holds and hence to obtain information on the inverse definiteness of L, it is sufficient to study the inverse-definiteness of each factor at the right hand side. The following elementary result provides the basic tools.
LEMMA 2.1. The operator D - cxl : Wi;‘(O, 2 r) + L’(O,2 7r) is inversepositive (respectively, inverse-negative) if and only if cx< 0 (respectively, CY>O). The operator (D-flZ)2+y21:Wz;1(0,2?r)+L1(0,2?r) is inversedefinite, and precisely inverse-positive, i$ and only if (0 < )y < i.
PROOF. The proof is carried out studying the sign of the Green’s functions associated with the involved differential operators. Let us consider the operator D - 0~1: W,‘;‘(O,2 7r)+ L’(O,27r). It is invertible if and only if a # 0 and its inverse can be represented in the form
(D-+(t)=/
G(s)u(t+s)ds, [0,2*]
for tE[O,21r]. Here, u~L’(O,27r) and
is extended by 27r-periodicity
G(S) = ( eFzra - 1)
a.e. to W
-leeas,
for s~[O,2?r], is the solution of the boundary value problem G’+ CYG= 0,
G(24-G(O)=]. Accordingly, D - CY1 is inverse-positive (respectively, and only if CY< 0 (respectively, QL> 0). Now, let us consider the operator (D - 01)’ + L’(O,2 a). It is easily verified that it is invertible if and or p = 0 and (0 < )r+!RI; moreover, its inverse can be form [(D-fil)e+y21]-1v(t)=j-
inverse-negative)
if
y”I : Wz”;‘(O,2 7~)-+ only if either /3# 0 represented in the
G(s)u(t+s)ds, [0,2?rl
for t E[O, 2 ?r]. Here, UE L’(O,2 ?r) is again extended by 2 ?r-periodicity a.e.
Periodic Boundary Value Problems
to R and G(s) = [y(e4”B-2e2~Pcos(2*y)+l)]-1 Xe”(Z’+s)[sin(y(2n for SE[O, 2 ~1, is the solution
- s)) + eW2”‘sin(ys)]
of the boundary
value problem
G”+2/3G’+(P2+y2)G=0, G(O)=G(2+G’(O)-G’(2?r)=l. The preceding
conditions
on /3 and y imply that
y(e4”P-2e therefore
s”%os(2
74
+ 1) > 0;
we only need to study the sign of
for SE[O, 2 ~1. Moreover,
taking into account - S) = sin(y(2a
.Fg(27r
for sE[O, 2 a], it is not restrictive If (0 < )y Q i, we have
the identity
- s)) + e2”Osin(ys),
to assume
fi < 0.
g(s)asin(y(2?r-s))+sin(ys)=2sin(ya)cos(y(s-?r)) and then G(s)>O, for sE]O, 2 7r[. On the contrary,
we have, if $ < y < 1,
g(O)*g(?r)
= 2(sin(y*))2(1+
e-2”B)cos(ra)
< 0,
P. OMARI AND M. TROMBETTA
6 and, if
721, with y#Rl when /3=0,
*[sin(2y7r
Hence,
in both
cases,
G
changes
sign
- T)
on
- eCz”B]
[0,2a].
conclude that (D - /3I)2 + y21 is inverse-definite, positive, if and only if (0 < )y G i.
We
2.1,
when
they
are definite,
therefore inverse-
n
REMARK. It is clear that the inverses Lemma
can
and precisely
of the operators are actually strictly
considered in definite (i.e.,
strictly positive or strictly negative). This is owing to the fact that corresponding Green’s functions vanish only on sets of zero measure. Now we apply Lemma definiteness
properties
PROPOSITION2.2.
2.1to obtain a complete
of second-
The
operator
L’(O,2 ?r) is inverse-negative and only if 0 < b C (1+
and third-order
picture
of the inverse-
operators.
L = D2 + aD + b1 : W;;‘(O,
if and only if b
the
2n)
c 0 and is inverse-positive
+ if
a2)/4.
PROOF. We distinguish
between
two cases:
a2 - 4 b > 0 and a2 - 4 b <
0. In the first case, the splitting L = (D - CY~I)( D - a2 I) holds, with CY,,CY~ER and (Ye* (;y2 = b. Accordingly, when a2 - 4b > 0, L is inversenegative if and only if b < 0 and inverse-positive if and only if 0 < b G a2 14. In the second case, we have L = (Dp1)2 +y”l, with fi = - a/2 and if (0 <)y = (b - a2/4) ‘12. Hence, when a2 - 4 b < 0, L is inverse-positive q and only if a’/4 < b G (1 + a2)/4. Proposition 2.2 is related to a result in [6, sect. 51, where an antimaximum principle is proved for an operator similar to L, but subjected to certain Sturm-Liouville boundary conditions, not including the periodic ones. Moreover, in [6] only sufficient conditions on the coefficients that yield the mentioned antimaximum principle are produced.
Periodic Boundary Value Problems
7
PROPOSITION 2.3. The operator L = D3 + aD2 + bD + cl: Wz;‘(O, 2 ?r) + L’(O,2 ?r) is inuerse-positive (respectively, inverse-negative) if and only if c > 0
( respectioely
, c < 0))
b-&f and
By Lemma 2.1, it is clear that L is inverse-definite if and only PROOF. if c z 0 and its characteristic polynomial f(X) = A3+ up + bh + c has only real roots or, when a pair of complex roots /3f iy, with y > 0, exists, it is y < 4. The change
of variable
p = h+ t
transforms
f(h)
into the polyno-
mial g( cl) = 4 + pp + y, where
and q=c+
leaving unchanged the modulus y of the imaginary part of the possible complex roots. Then the classical Cardano’s formulas say that, setting
the polynomial g(p), and therefore f(X), has only real roots if and only if A < 0 and, when A > 0, the modulus y of the imaginary part of the complex
roots is given by
_,-A1~2)1’3~~ T=(a) “‘I( _;+A1,2)1’3-(
8
P. OMARI AND M. TROMBETTA
The condition
A < 0 is equivalent
p
On the contrary,
and
to
191 <2
<(l-p)
the condition
A > 0 is satisfied
(1;g)1’2
if and only if
P>O or
The additional equivalent to
condition
[
y
which
has to be imposed
when
A > 0, is
-- 9 + 2
-
-i
To solve this inequality
9 2
with respect
to 9, we set
- 912 = s and
+,-[s+(~~+(~)3)“2]1’3-[+~+(~)3)“21’i
=[s+(sa+(;)3)1’2~+(;)[~+(?+(~)J1’2]-’;’.
Since cp is an even function for sa0.
on its domain,
we restrict
ourselves
to study it
Periodic Boundary Value Problems Let us suppose a(s) 2 0. Setting
p > 0.
9
In this
case,
cp is defined
and taking into account that t is a strictly study of cp reduces to the study of
On this interval,
increasing
\k is strictly
for all
s > 0 and
function
of s, the
increasing condition
Under this necessary
which is equivalent
Hence,
condition,
we solve the inequality
to
we easily get
p (0 1 112 <
3
”
t<
l+(l-4p)1’2 2J3
and then, by simple calculations,
.
(o<)s<
4 . P-4pY2P6,f3
.
and attains
its
y < J-, at the
P. OMAN AND M. TROMBETTA
10
Finally, as C,O is even, we conclude that, when p > 0, it is A > 0 and y < i if and only if
Let us suppose now p
and
)9)>2
First we examine the case p = 0 and ) q 1 > 0. We immediately see that p is defined for all s and
I’p(s)J=(214y3. Accordingly, by the condition y < l/2, we find 2)sl<-
1 343
.
and then
(O-=)l91
G-.
1
343
Then, we turn to the case p
and
p Now, cp is defined for s > (- 3)
for s>(--$
’
312
,
we
191 >2 312
( > 0). Setting again
turn to the study of
Periodic Boundary Value Problems for t+
1’2. Since
vanishes
for
at (-
is strictly
increasing
for t 2 (-
:)‘I2
and
it follows that
‘I2 . Accordingly,
t>(-:)
equivalent
!)r12,
t + Z3t
11
imposing
the
condition
y
which
is
to
we find
--
i
p 1
3
112
l+(l-4p)1’2 2J3
and then
(I-
P)(l-4Pye
6J3 Finally,
the substitution
Hence,
we conclude
s = - f
’
and the even character
that the polynomial
of cp yield
g(p), and therefore
f(X),
has only
real roots or, in the opposite case, the modulus y of the imaginary part of the complex roots of g( CL),and therefore of f(X), satisfies y < i, if and only if
p
and
IqI
<
(l-
p)(l-4p)“’ 3J3
’
REMARK. Using Lemma 2.1, maximum and antimaximum principles for nth order differential operators can be obtained in a similar way.
12 3.
P. OMARI AND M. TROMBETTA LOWER ORDER
AND BVPS
Let us consider
UPPER
SOLUTIONS
the second-order
FOR
periodic
SECOND
boundary
value problem
-u”+cu’=f(t,u), u(0) where interval
c is a fixed
real
= u(27r),
constant
having a nonempty
(3.1) u’(O) = u/(27+
and
interior,
(3.2)
f: [0,2 ~1 x I-+ F%‘,with
satisfies
the Caratheodory
1 C R an conditions.
This means that f( * , s ) is measurable for every SE I, f(t, *) is continuous for a.e. te[0,2 ~1 and, for each compact subset K of I, there exists ~0,~L’(O,27r)
such
that
1f(t,.s)
( < pK(t),
for every
seK
and
a.e.
te
[0,2?r]. Solutions of (3.1)-(3.2) are also intended in the Caratheodory sense; precisely, a solution is a function ueW 2.1(0,2?r) such that
U(t)EL which satisfies
Equation
(3.1),
for every
te[0,2a],
(3 3
a.e. in [0,2 a], and the boundary
(3.2). Moreover, a function (3.1)-(3.2) if it satisfies (3.2),
(YEW 2*1(0,27r) (3.3) and
- cY”“+ cd
is
a
lower
conditions solution
of
(34
for a.e. te[0,2 ~1. An upper solution fl is defined similarly, but reversing the inequality in (3.4). In this context the following result is almost classical: we refer in particular to [9] and the surveys [4] and [ll], recent works [l] and [14], for f Caratheodory.
THEOREM 3.1. Suppose upper solution /3 such that
for f continuous,
that there exists a lower solution
cx(t)
w < 0 such that
and the
Q! and an
(3 4
Periodic BoundaryValueProblems f or a.e. possesses
t~[O,27r] and cx(t)
interval [cx, /3]. Moreouer, of the following
u,
and v, can be computed
-
wwk+l
=
wk+l(“)
If w0 = CY, the sequence
a natural
reversed,
f(t,t”k)
is whether
reversing
the inequality
example
reduces
the
in (3.5).
(3.8)
= W;+1(2’+
in W 2,1(0, 2~)
and converges inequality
at all. This problem
to
in W 2,1(O, 2 ?r)
in (3.5)
was raised explicitly
can
be
in [17,
and a linear counterexample one cannot expect solvability
In our framework,
to the following
(3.7)
and converges
{ wk} is decreasing
question
or even dropped
w;+~(“)
p. 9991, in the context of elliptic equations, was exhibited in [4], showing that generally tioned
by means
- wwk,
= wk+1(2+
{ wk) is increasing
lfwa = /3, the sequence
to v,. Now
iteratively
scheme:
- w;+l + ct”;+l
u,.
13
the previously
men-
equation:
- nn = u+sin(t),
(3.9)
which admits - 1 as an upper solution and 1 as a lower solution but which has no 2n-periodic solution. Nevertheless, some results in this direction have been obtained in recent years, by imposing further conditions on the right-hand side f. In [5] the function f was supposed bounded and the solvability was proved just assuming the existence of a pair of lower and upper solutions, which can be nonordered at all. However, unlike Theorem 3.1,the proof of the result in [5], as well as that of its recent extensions in [16]and [7], is not constructive. On the contrary, the monotone method was used in [I31 and [I2] to get extremal solutions, upper solution fi satisfy a(t)
> (3(t),
assuming
that the lower solution
for every tE[O,Z?r],
CYand the
(3.10)
and f is decreasing with respect to the second variable. Yet, this last condition seems rather unnatural when it is coupled with (3.10). Indeed, in the case where CYand /3 are constant functions, such an assumption implies that (Y = 0 and cr is a solution. In Theorem 3.2, by exploiting the antimaximum principle stated in Proposition 2.2, we remove such monotonicity hypotheses on f, replacing it by a more natural one-sided
P. OMARI AND M. TROMBETTA
14
Lipschitz condition from above. We stress that example an assumption of this type cannot be dropped at all.
THEOREM 3.2.
Suppose
that there (3.10).
upper solution B satisfying 1+ c2 such that <4
exists a lower
Assume
(3.9)
solution
that
CY and
that there is a constant
an
0 < o
(3.11)
f(t>sq)-f(t>s,)G+2-s,). f or a.e. possesses
suggests
tE[O,2n] and B(t)< s, < s2
interval [ /3, CY]. Moreover, u, and v, can be computed iteratively by means of the scheme defined by (3.7)-(3.8). lf wg = cy, the sequence {wk} is decreasing and converges in W 2,1(0, 2 T) to v,. lf wCJ= /3, the sequence (wk} is increasing and converges in W *,‘(O, 2~) to u,. We produce a unified proof of both Theorems 3.1 and 3.2, showing that they can be easily derived as consequences of the following very general principle
that we recall here
as a lemma.
LEMMA 3.3 ([lo], [4]). Let E be an ordered real Banach space with a normal positive cone. Let D be an order convex subset of E and suppose that T: D-t E is an increasing and completely continuous operator. Assume that there exists u(,, VIE D, with u0 < v(,, such that u0 < Tu,, and v, > TV,,. Then T has a minimal fixed point u, and a maximal fixed point v,, in the order
interval ug, vol. Moreover, u and the [ sequPnces IT u > a;d k
respective1
0
=
;Tlmkk-+co) vo
TkuO and vCo= lin++caTkvO, are increasing and decreasing,
y.
Now we are in a position
to prove Theorems
3.1 and 3.2.
For every o # 0, if c # 0, and w # n*, for each neM, if c = 0, PROOF. let K, denote the linear operator sending any function eE L’(O,2 n) onto the unique solution w E W *, ’ (0,2 a) of the problem
- w”+
cw’-
cow = e(t), w(0) = w(27r), w’(0) = w’(27r).
15
Periodic Boundary Value Problems The operator K, is well-defined C”([0,2?r]). Moreover, Proposition only if w < 0 and is negative
and compact from L’(O,2 n) to, say, 2.2 states that K, is positive if and 1+c2 if and only if 0 < w < -. Let D, be the
order convex subset of C’([O, 2 7r]) defined by D, = (n2CC”([0, 2 a]) : u( t)~ I, for t~[O,2 7r]}. Let F, : D, C C”([O, 2 T]) + L’(O,2 n) denote the Nemytskii operator which maps any function u E D, onto .f( *, u) - wu. The Carathkodory conditions on f imply that F, is continuous and maps bounded sets to bounded sets. Moreover, F, is increasing, by (3.6) if 1+ c2 w
T,= problem
(3.1)-(3.2)
K,F,:D,CC0([0,2?r])+Co([0,2~]), is equivalent
to the fixed point equation u = T,u,
where the operator
T, is increasing
and completely
continuous.
Moreover,
we have
if (3.5) holds and o < 0, or
cr>T,a
and
P,
1+ c2 Finally, recalling that the positive cone holds and 0 < w < 4 . in C”([O,2a]) is normal, we see that all the assumptions of Lemma 3.3 are fulfilled and therefore the conclusions of Theorems 3.1 and 3.2 follow. In particular, the convergence of the iterates in W231(0,2?r) is an immediate if (3.10)
consequence of the continuity K,: L’(o,27r)+w”z’(o,27r).
of F, : D, c C”([O, 2 rr]) -+ L’(O,2 n) and of
n
1+ c2 We notice that the bound w < seems optimal, in order that the 4 monotone method works. Indeed, as stated in Proposition 2.2, the antimaximum principle, which is required to get the monotonicity of the 1+ c2 associated operator T,, fails for w > -. Nevertheless, as far as one is only concerned
with the existence
of solutions
of problem
(3.1)-(3.2)
i.e.,
16
P. OMARI AND M. TROMBETTA
without the approximation have been proved recently and 2 given there,
scheme provided by (3.7)-(3.8), in [16]; for instance, combining
the following
theorem
sharper results Propositions 1
can be easily derived.
PROPOSITION3.4. Suppose thatf(t,s)= g(s)+ h(t), with g:R-+W and R continuous. Moreover assume that either c + 0, or, if c = 0,
h : [0,2 r]+
Finally, suppose that there exists a lower solution Q!and an upper solution fl ( possibly, nonordered at all). Then problem (3.1)-(3.2) admits at least one solution. We conclude this section by remarking that it is not known if the result of Proposition 3.4 carries over to the case where f does not admit the splitting
4.
f( t, s) = g(s) + h(t).
LOWER
AND
PERIODIC
UPPER
SOLUTIONS
FOR
THIRD-ORDER
BVPS
Let us consider
the third order periodic
BVP
u”‘+au”+bu’=f(t,u),
(4.1)
u(O) = u(27r),
u’(0) = u/(2+
where a, b are fixed real constants interval having a nonempty interior, By a solution
of (4.1)-(4.2)
(4.2)
and f: [0,2 7r] x I+ W, with I C W an satisfies the Caratheodory conditions.
we mean a function for every
+)a
u”(O) = u”(27r),
u~W~‘~(0,2n),
tE[O,27r],
such that
(4.3)
which satisfies Equation (4.1), a.e. in [0,2 ~1, and the boundary conditions (4.2). Moreover, we say that a function ar~W 3’1(0,2a) is a lower solution of (4.1),
(4.2),
if it satisfies
(4.2),
a”‘+ for a.e. te[O, 2 7~1. An upper the inequality in (4.4).
(4.3)
a&+ solution
and
bcz’
(4.4) similarly,
but reversing
Periodic Boundary Value Problems
17
Here, we are concerned with the solvability of (4.1)-(4.2) in the presence of a pair of lower and upper solutions. This study, which bears on the maximum and antimaximum principles proved in Section 2, is also motivated by the recent paper [15] where some partial results in this direction are produced. According to Proposition 2.3, we define the set
and the functions wl, w2 : A + W, by
w,(a, b) =
-2a3 27
and
Then the following results hold.
Suppose that (a, b)E A and there exists a lower solution THEOREM 4.1. (Y and an upper solution /3 such that a(t)
Assume that there is a constant that
forever-y te[0,27r].
(4.5)
w < 0, with wl(a, b) < - w < w2(a, b), such
f(t,s,)-f(t,sl)~W(S2-SI),
P-6)
for a.e. te[O,27r] and a(t)< s, < s2
18
I’. OMARI AND M. TROMBETTA
of the following
scheme:
w~+l+aw;+,+bw;+,-Wwk+l=f(t,wk)-w~t,
(4.7)
wk+l(O) = wk+,(2+ w;+l(O)
= w;+1(2+
w;+Jo)
= Wk”+J27r).
(4.8)
Zf wg = (Y, the sequence { wk} is increasing and converges in W3,‘(0, 2 a) to u,. lf w0 = 0, the sequence { wk) is decreasing and converges in W 3, ‘(0,2 7~)
to u,.
THEOREM 4.2. cx and an upper
Suppose that (a, b)E A and there exists a lower solution solution /3 such that cx(t)>fi(t),
Assume that there is a constant
foreverytE[0,27r].
(4.9)
w > 0, with wl(a, b) < - w < ~~(a, b), such
that (4.10)
f(t>sp)-f(t>s++,-s,), for
a.e.
tE[O,2?r]
and
P(t)<
sl<
s2 < cx(t).
Then
problem
(4.1)-(4.2)
possesses a minimal solution u, and a maximal solution v, in the order interval [ 0, CY]. Moreover, u, and v, can be computed iteratively by means of the scheme defined by (4.7)-(4.8). lf w0 = /3, the sequence {wk} is increasing and converges in W3s1(0, 2 n) to u,. lf w0 = CY, the sequence { wk) is decreasing
and converges
in W3x1(0,2n)
PROOF. The proof is similar to that of we sketch it for the sake of completeness. if b = n2 for some neM, let K, denote function eE L’(O,2 7r) onto the unique
to u,.
Theorems 3.1 and 3.2; however, For every w # 0, with w # - an’, the linear operator sending any solution WE W3,1(0, 2 T) of the
problem
w”‘+
au;“+ bw’-
ww = e(t), w(0)
= w(2a),
w’(0)
= w’(27r),
w”(0)
= w”(27r).
19
Periodic Boundary Value Problems
The operator K, is well-defined (compare [2]), is continuous from L’(O,2 ?r) to W3*‘(0, 2 ?T) and compact from L’(O,2 7r) to, say, C’([O, 2 ~1). Moreover, Proposition 2.3 states that K, is positive if and only if w c 0 and o,( a, h) G - fJ G wp( a, b), whereas it is negative if and only if u > 0 and ~,(a, b) G - w G wa(a, b). As before, we define in C”([0,27r]) the order convex set D, = { UEC~([O, 2 n]) : u( t)~ I, for te[O, 2 7r]} and the Neymtskii operator F, : D, C C’([O, 2 ~1) + L’(O,2 ?r) mapping any function UE D, onto f(*, u)- wu. By the Caratheodory conditions on f,F, is continuous and maps bounded sets to bounded sets. Moreover, F, is increasing, by (4.6), when w < 0 and wi(a, b) ,< - w G wr(a, b), or is decreasing, by (4.10), when w > 0 and ~~(a, b) G - w Q wa(u, b). Hence, setting T,=
problem
(4.1)-(4.2)
K,F,:D,CC0([0,2?r])+Co([0,27r]),
is equivalent
to the fixed point equation u = TUu,
where the operator we have
T, is increasing
and completely
continuous.
Moreover,
if (4.5) holds, and cxaT,cx
and
fi
if (4.9) holds. Accordingly, all the assumptions of Lemma 3.3 are fulfilled and therefore the conclusions of Theorems 4.1 and 4.2 easily follow. W We point out that Theorem 4.1 contains as a particular case the main result of [I5], where it is supposed a = b c 0 and w = u2. Indeed, it is easy, although cumbersome, to verify that wi(a, a) < - u2 < w2(u, a). Moreover, Theorem 4.2 gives a positive answer to a question raised in [15], about the existence of a solution to (4.1)-(4.2) when the lower and the upper solutions are in the reversed order (i.e. satisfy (4.9)). Finally, letting a = b = 0, we get the following immediate consequence of Theorems 4.1 and 4.2, which solves a further problem posed in [15], concerning the solvability of (4.11)
um =f(t,u), u(0) = u(2a), in the presence
u’(0) = u/(2+
of lower and upper
solutions.
u”(0) = u”(27r),
(4.12)
P. OMARI
20 4.3. Suppose that there solution 0. Assume that either
exists a lower- solution
COROLLARY
upper
a(t)
AND M. TROMBETTA
foreuery
CY and an
te[0,27r],
and
f(t>S+f(t,s,)>
for a.e.
tE[O,2*]
and
-+
a(t) < s1 < sg < /3(t), or
a(t)
for-every
2 0(t),
tE[O,27r],
and
for a.e.
tE[O,2?r]
and
p(t)<
has a minimal and a maximal attained by monotone iteration.
s1 < s2 < cx(t). Then problem (4.11)-(4.12) solution between CY and 0, which can be
We conclude by remarking that no result to now is known for third-order equations.
similar
to proposition
3.4 up
REFERENCES A. Adje, avec
Sur et sous solutions
conditions
Neuve,
aux
limites
dans les equations non-lineaires,
differentielles
Ph.D.
Thesis,
discontinues
Univ.
Louvain-la-
1987.
A. U. Afuwape, tial operators boundary
P. Omari, with
and F. Zanolin,
nontrivial
value problems,
Math.
Nonlinear
perturbations
and applications An&
Appl.
tions, World
Scientific,
H.
Fixed
143:35-56
periodic
(1989).
Singapore,
points
1986.
equations
and
nonlinear
eigenvalue
ordered Banach spaces, SZAM Rev. 18:620-709 (1976). H. Amann, A. Ambrosetti, and G. Mancini, Elliptic equations ible Fredholm
of differen-
to third-order
Boundary Value Problems for Higher Order Dijferential Equa-
R. P. Agarwal, Amann,
J.
kernel
linear
part and bounded
nonlinearities,
problems
in
with noninvert-
Math. 2. 158:179-194
(1978). P. Clement, elliptic J.-P.
Gossez,
semilinear M.
and L. A. Peletier,
operators,
Gregus,
1987.
and
elliptic
J.
An anti-maximum
principle
Differential Equations 34:218-229 P.
Omari,
equations,
Non-ordered
lower
for second-order
(1979). and
upper
solutions
to appear.
Third Order Linear
Differential
Equations,
Reidel,
Dordrecht,
in
Periodic Boundary Value Problems 9
10 11 12
13 14 15 16
21
H. W. Knobloch, Eine neue Methode zur Approximation periodischer LGsungen nicht-linearer Differentialgleichungen zweiter Ordnung, Math. Z. 82:177-197 (1963). M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Dqferential Equations, Pitman, Boston, 1985. V. Lakshmikantham, J. J. Nieto and Y. Sun, An existence result about periodic boundary value problems of second order differential equations, Appl. Anal. 40:1-10 (1990). J. J. Nieto, Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl. 130:22-29 (1988). J. J. Nieto, Nonlinear second order periodic boundary value problems with Garatheodory functions, Appl. Anal. 34:111-128 (1989). J. J. Nieto, Periodic Solutions for Third Order Ordinary Differential Equations, to appear. P. Omari, Non-ordered lower and upper solutions and solvability of the periodic problem for the Lienard and the Rayleigh equations, Rend. Zst. Mat. Univ. Trieste 20:54-64 (1988) (special issue dedicated to the memory of U. Barbuti) .
17 D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic ary value problems, Indiana Univ. Math. /. 21:979-1000 (1972). 18 J. Scbroder, Operator Inequalities, Academic Press, New York, 1980.
bound-
Omari
Dipartimento
and Maurizio
Trombetta
di Scienze Matematiche
Universitd di Trieste Piazzale Europa 1, 34127
Trieste, ltalia
ABSTRACT We study the solvability and the approximation of the solutions by monotone iteration of the second- and third-order periodic boundary value problems (BVPs)
-u”+cu’=f(t,u), u(o) = 4274,
u’(0) = u’(27+
u(o) = u(27r), u’(0) = u’(27r), u”(0) = u”(27r),
in the presence of lower and upper solutions, which may not satisfy the usual ordering condition. To this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems.
1.
INTRODUCTION
This paper is concerned the solutions by monotone
APPLIEDMATHEMATZCSAND
with the solvability and the approximation of iteration of the second and the third order
COMPUTATION 50:1-Zl(1992)
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
1 0096-3003/92/$5.00
2 periodic
P. OMARI AND M. TROMBETTA boundary
value problems
- u”+ cd =
f( t, 24)’
u(o) = u(27r),
(1.1)
u’(0) = u’(27r),
(I .2)
and
Urn+ Qd + bu’ = f( t, u) , u(0) = u(27r),
(1.3) u’(0) = u’(27r),
in presence of lower and upper solutions. suppose that a, b, c are fixed real constants
u”(0) = u”(2a),
(1.4)
In the preceding equations, we and f satisfies the Caratheodory
conditions. As far as one is concerned with problems (l.l)-(I .2), there exists a vast literature (see, e.g., the bibliography in [ll] dealing with the case where f is continuous and the lower solution a! and the upper solution 0 satisfy the ordering
condition o(t)
f@(t),
for every
t.
(I .5)
Unlike this type of result, here we mainly concentrate our attention on the case where the inequality in (1.5) is reversed. To investigate this situation, we prove in Section 2 an elementary lemma from which the validity of an antimaximum principle for certain second-order linear differential operators associated with the equation in (1.1) can be immediately deduced. Necessary and sufficient conditions on the coefficients are also given so that such an antimaximum principle holds. The existence and approximaare then proved exploiting this result in tion theorems for (I.l)-(1.2) connection with a classical and very general principle given in [lo] (compare also [4]), which permits as well a simple and unified treatment of the problem in the Caratheodory case. The aforementioned lemma is further exploited to obtain some maximum and antimaximum principles for certain third-order linear differential operators associated with the equation in (1.3), and again necessary and sufficient conditions on the coefficients are given in order that they hold. These results constitute the basic tool to get mation theorems for problem (1.3)-(1.4) in solutions. We stress that, although there exist investigation of third-order boundary value
some existence and approxipresence of lower and upper several works devoted to the problems (compare [B], [3]),
Periodic Boundary Value Problems
3
almost no result in this direction seems available for the periodic problem; indeed, the only contributions we know are given in [Is], where several questions discussed in this paper have been also raised. Finally, we note that the results of Section 2 could be carried out as well to obtain similar statements for higher-order periodic BVPs. Moreover, extensions of our theorems to more general equations, which include a dependence of the right-hand side on the derivatives, can be performed and will appear elsewhere.
2.
MAXIMUM AND ANTIMAXIMUM PRINCIPLES OF LJNEAR DIFFERENTIAL OPERATORS
FOR
A CLASS
In this section we formulate and prove some criteria of positivity or negativity of the inverses of certain linear differential operators acting in spaces of periodic functions, which in turn imply the validity of suitable maximum and antimaximum principles. Let us denote by W&‘(O, 2 a), for k = 1,2,. . . , the Sobolev space of all absolutely continuous, functions u: [0,2 n] --) R of class Ck-‘, with &‘) satisfying the periodic boundary conditions U(~)(O)= u(‘)(2 ?r), for i = O,l,..., k-l. Let us consider the nth order linear differential operator L: W&‘(O, 2 ?r) + L’(O,2 ?r), defined by
L=
D”+a,D”-‘+
.a.
+a,_,D+a,l,
where D = d /dt, 1 is the identity operator and ai, for i = 1,. . . , n, are fixed real constants. Following [18], we say that L is inverse-positive (respectively, inverse-negative) if Lu > 0 implies u > 0 (respectively, 24,< 0), where the ordering between functions is the usual pointwise ordering. Of course, this is equivalent to saying that L is invertible and its inverse is positive (respectively, negative). To study the inverse-definiteness (that is the inverse-positivity or inverse-negativity) of L, we perform a simple reduction, by factorizing L as a product of first- and second-order operators. Precisely, let us denote by (Yl,. . . , UT> PI + i-y,, . . . , P, f i-i,, with (Y~,fij, ~~43 and rj > 0, the roots of the characteristic polynomial of L, having respective multiplicities Accordingly, the splitting /Jl,...,/+.. ul>.*.,LJs.
L=(D-cY,z)“‘...(
o_,,I)“[(D-~,I)“+y~Z]“‘...[(D-~,~)”+y,2Z]”’~
4
P. OMARI AND M. TROMBETTA
holds and hence to obtain information on the inverse definiteness of L, it is sufficient to study the inverse-definiteness of each factor at the right hand side. The following elementary result provides the basic tools.
LEMMA 2.1. The operator D - cxl : Wi;‘(O, 2 r) + L’(O,2 7r) is inversepositive (respectively, inverse-negative) if and only if cx< 0 (respectively, CY>O). The operator (D-flZ)2+y21:Wz;1(0,2?r)+L1(0,2?r) is inversedefinite, and precisely inverse-positive, i$ and only if (0 < )y < i.
PROOF. The proof is carried out studying the sign of the Green’s functions associated with the involved differential operators. Let us consider the operator D - 0~1: W,‘;‘(O,2 7r)+ L’(O,27r). It is invertible if and only if a # 0 and its inverse can be represented in the form
(D-+(t)=/
G(s)u(t+s)ds, [0,2*]
for tE[O,21r]. Here, u~L’(O,27r) and
is extended by 27r-periodicity
G(S) = ( eFzra - 1)
a.e. to W
-leeas,
for s~[O,2?r], is the solution of the boundary value problem G’+ CYG= 0,
G(24-G(O)=]. Accordingly, D - CY1 is inverse-positive (respectively, and only if CY< 0 (respectively, QL> 0). Now, let us consider the operator (D - 01)’ + L’(O,2 a). It is easily verified that it is invertible if and or p = 0 and (0 < )r+!RI; moreover, its inverse can be form [(D-fil)e+y21]-1v(t)=j-
inverse-negative)
if
y”I : Wz”;‘(O,2 7~)-+ only if either /3# 0 represented in the
G(s)u(t+s)ds, [0,2?rl
for t E[O, 2 ?r]. Here, UE L’(O,2 ?r) is again extended by 2 ?r-periodicity a.e.
Periodic Boundary Value Problems
to R and G(s) = [y(e4”B-2e2~Pcos(2*y)+l)]-1 Xe”(Z’+s)[sin(y(2n for SE[O, 2 ~1, is the solution
- s)) + eW2”‘sin(ys)]
of the boundary
value problem
G”+2/3G’+(P2+y2)G=0, G(O)=G(2+G’(O)-G’(2?r)=l. The preceding
conditions
on /3 and y imply that
y(e4”P-2e therefore
s”%os(2
74
+ 1) > 0;
we only need to study the sign of
for SE[O, 2 ~1. Moreover,
taking into account - S) = sin(y(2a
.Fg(27r
for sE[O, 2 a], it is not restrictive If (0 < )y Q i, we have
the identity
- s)) + e2”Osin(ys),
to assume
fi < 0.
g(s)asin(y(2?r-s))+sin(ys)=2sin(ya)cos(y(s-?r)) and then G(s)>O, for sE]O, 2 7r[. On the contrary,
we have, if $ < y < 1,
g(O)*g(?r)
= 2(sin(y*))2(1+
e-2”B)cos(ra)
< 0,
P. OMARI AND M. TROMBETTA
6 and, if
721, with y#Rl when /3=0,
*[sin(2y7r
Hence,
in both
cases,
G
changes
sign
- T)
on
- eCz”B]
[0,2a].
conclude that (D - /3I)2 + y21 is inverse-definite, positive, if and only if (0 < )y G i.
We
2.1,
when
they
are definite,
therefore inverse-
n
REMARK. It is clear that the inverses Lemma
can
and precisely
of the operators are actually strictly
considered in definite (i.e.,
strictly positive or strictly negative). This is owing to the fact that corresponding Green’s functions vanish only on sets of zero measure. Now we apply Lemma definiteness
properties
PROPOSITION2.2.
2.1to obtain a complete
of second-
The
operator
L’(O,2 ?r) is inverse-negative and only if 0 < b C (1+
and third-order
picture
of the inverse-
operators.
L = D2 + aD + b1 : W;;‘(O,
if and only if b
the
2n)
c 0 and is inverse-positive
+ if
a2)/4.
PROOF. We distinguish
between
two cases:
a2 - 4 b > 0 and a2 - 4 b <
0. In the first case, the splitting L = (D - CY~I)( D - a2 I) holds, with CY,,CY~ER and (Ye* (;y2 = b. Accordingly, when a2 - 4b > 0, L is inversenegative if and only if b < 0 and inverse-positive if and only if 0 < b G a2 14. In the second case, we have L = (Dp1)2 +y”l, with fi = - a/2 and if (0 <)y = (b - a2/4) ‘12. Hence, when a2 - 4 b < 0, L is inverse-positive q and only if a’/4 < b G (1 + a2)/4. Proposition 2.2 is related to a result in [6, sect. 51, where an antimaximum principle is proved for an operator similar to L, but subjected to certain Sturm-Liouville boundary conditions, not including the periodic ones. Moreover, in [6] only sufficient conditions on the coefficients that yield the mentioned antimaximum principle are produced.
Periodic Boundary Value Problems
7
PROPOSITION 2.3. The operator L = D3 + aD2 + bD + cl: Wz;‘(O, 2 ?r) + L’(O,2 ?r) is inuerse-positive (respectively, inverse-negative) if and only if c > 0
( respectioely
, c < 0))
b-&f and
By Lemma 2.1, it is clear that L is inverse-definite if and only PROOF. if c z 0 and its characteristic polynomial f(X) = A3+ up + bh + c has only real roots or, when a pair of complex roots /3f iy, with y > 0, exists, it is y < 4. The change
of variable
p = h+ t
transforms
f(h)
into the polyno-
mial g( cl) = 4 + pp + y, where
and q=c+
leaving unchanged the modulus y of the imaginary part of the possible complex roots. Then the classical Cardano’s formulas say that, setting
the polynomial g(p), and therefore f(X), has only real roots if and only if A < 0 and, when A > 0, the modulus y of the imaginary part of the complex
roots is given by
_,-A1~2)1’3~~ T=(a) “‘I( _;+A1,2)1’3-(
8
P. OMARI AND M. TROMBETTA
The condition
A < 0 is equivalent
p
On the contrary,
and
to
191 <2
<(l-p)
the condition
A > 0 is satisfied
(1;g)1’2
if and only if
P>O or
The additional equivalent to
condition
[
y
which
has to be imposed
when
A > 0, is
-- 9 + 2
-
-i
To solve this inequality
9 2
with respect
to 9, we set
- 912 = s and
+,-[s+(~~+(~)3)“2]1’3-[+~+(~)3)“21’i
=[s+(sa+(;)3)1’2~+(;)[~+(?+(~)J1’2]-’;’.
Since cp is an even function for sa0.
on its domain,
we restrict
ourselves
to study it
Periodic Boundary Value Problems Let us suppose a(s) 2 0. Setting
p > 0.
9
In this
case,
cp is defined
and taking into account that t is a strictly study of cp reduces to the study of
On this interval,
increasing
\k is strictly
for all
s > 0 and
function
of s, the
increasing condition
Under this necessary
which is equivalent
Hence,
condition,
we solve the inequality
to
we easily get
p (0 1 112 <
3
”
t<
l+(l-4p)1’2 2J3
and then, by simple calculations,
.
(o<)s<
4 . P-4pY2P6,f3
.
and attains
its
y < J-, at the
P. OMAN AND M. TROMBETTA
10
Finally, as C,O is even, we conclude that, when p > 0, it is A > 0 and y < i if and only if
Let us suppose now p
and
)9)>2
First we examine the case p = 0 and ) q 1 > 0. We immediately see that p is defined for all s and
I’p(s)J=(214y3. Accordingly, by the condition y < l/2, we find 2)sl<-
1 343
.
and then
(O-=)l91
G-.
1
343
Then, we turn to the case p
and
p Now, cp is defined for s > (- 3)
for s>(--$
’
312
,
we
191 >2 312
( > 0). Setting again
turn to the study of
Periodic Boundary Value Problems for t+
1’2. Since
vanishes
for
at (-
is strictly
increasing
for t 2 (-
:)‘I2
and
it follows that
‘I2 . Accordingly,
t>(-:)
equivalent
!)r12,
t + Z3t
11
imposing
the
condition
y
which
is
to
we find
--
i
p 1
3
112
l+(l-4p)1’2 2J3
and then
(I-
P)(l-4Pye
6J3 Finally,
the substitution
Hence,
we conclude
s = - f
’
and the even character
that the polynomial
of cp yield
g(p), and therefore
f(X),
has only
real roots or, in the opposite case, the modulus y of the imaginary part of the complex roots of g( CL),and therefore of f(X), satisfies y < i, if and only if
p
and
IqI
<
(l-
p)(l-4p)“’ 3J3
’
REMARK. Using Lemma 2.1, maximum and antimaximum principles for nth order differential operators can be obtained in a similar way.
12 3.
P. OMARI AND M. TROMBETTA LOWER ORDER
AND BVPS
Let us consider
UPPER
SOLUTIONS
the second-order
FOR
periodic
SECOND
boundary
value problem
-u”+cu’=f(t,u), u(0) where interval
c is a fixed
real
= u(27r),
constant
having a nonempty
(3.1) u’(O) = u/(27+
and
interior,
(3.2)
f: [0,2 ~1 x I-+ F%‘,with
satisfies
the Caratheodory
1 C R an conditions.
This means that f( * , s ) is measurable for every SE I, f(t, *) is continuous for a.e. te[0,2 ~1 and, for each compact subset K of I, there exists ~0,~L’(O,27r)
such
that
1f(t,.s)
( < pK(t),
for every
seK
and
a.e.
te
[0,2?r]. Solutions of (3.1)-(3.2) are also intended in the Caratheodory sense; precisely, a solution is a function ueW 2.1(0,2?r) such that
U(t)EL which satisfies
Equation
(3.1),
for every
te[0,2a],
(3 3
a.e. in [0,2 a], and the boundary
(3.2). Moreover, a function (3.1)-(3.2) if it satisfies (3.2),
(YEW 2*1(0,27r) (3.3) and
- cY”“+ cd
is
a
lower
conditions solution
of
(34
for a.e. te[0,2 ~1. An upper solution fl is defined similarly, but reversing the inequality in (3.4). In this context the following result is almost classical: we refer in particular to [9] and the surveys [4] and [ll], recent works [l] and [14], for f Caratheodory.
THEOREM 3.1. Suppose upper solution /3 such that
for f continuous,
that there exists a lower solution
cx(t)
w < 0 such that
and the
Q! and an
(3 4
Periodic BoundaryValueProblems f or a.e. possesses
t~[O,27r] and cx(t)
interval [cx, /3]. Moreouer, of the following
u,
and v, can be computed
-
wwk+l
=
wk+l(“)
If w0 = CY, the sequence
a natural
reversed,
f(t,t”k)
is whether
reversing
the inequality
example
reduces
the
in (3.5).
(3.8)
= W;+1(2’+
in W 2,1(0, 2~)
and converges inequality
at all. This problem
to
in W 2,1(O, 2 ?r)
in (3.5)
was raised explicitly
can
be
in [17,
and a linear counterexample one cannot expect solvability
In our framework,
to the following
(3.7)
and converges
{ wk} is decreasing
question
or even dropped
w;+~(“)
p. 9991, in the context of elliptic equations, was exhibited in [4], showing that generally tioned
by means
- wwk,
= wk+1(2+
{ wk) is increasing
lfwa = /3, the sequence
to v,. Now
iteratively
scheme:
- w;+l + ct”;+l
u,.
13
the previously
men-
equation:
- nn = u+sin(t),
(3.9)
which admits - 1 as an upper solution and 1 as a lower solution but which has no 2n-periodic solution. Nevertheless, some results in this direction have been obtained in recent years, by imposing further conditions on the right-hand side f. In [5] the function f was supposed bounded and the solvability was proved just assuming the existence of a pair of lower and upper solutions, which can be nonordered at all. However, unlike Theorem 3.1,the proof of the result in [5], as well as that of its recent extensions in [16]and [7], is not constructive. On the contrary, the monotone method was used in [I31 and [I2] to get extremal solutions, upper solution fi satisfy a(t)
> (3(t),
assuming
that the lower solution
for every tE[O,Z?r],
CYand the
(3.10)
and f is decreasing with respect to the second variable. Yet, this last condition seems rather unnatural when it is coupled with (3.10). Indeed, in the case where CYand /3 are constant functions, such an assumption implies that (Y = 0 and cr is a solution. In Theorem 3.2, by exploiting the antimaximum principle stated in Proposition 2.2, we remove such monotonicity hypotheses on f, replacing it by a more natural one-sided
P. OMARI AND M. TROMBETTA
14
Lipschitz condition from above. We stress that example an assumption of this type cannot be dropped at all.
THEOREM 3.2.
Suppose
that there (3.10).
upper solution B satisfying 1+ c2 such that <4
exists a lower
Assume
(3.9)
solution
that
CY and
that there is a constant
an
0 < o
(3.11)
f(t>sq)-f(t>s,)G+2-s,). f or a.e. possesses
suggests
tE[O,2n] and B(t)< s, < s2
interval [ /3, CY]. Moreover, u, and v, can be computed iteratively by means of the scheme defined by (3.7)-(3.8). lf wg = cy, the sequence {wk} is decreasing and converges in W 2,1(0, 2 T) to v,. lf wCJ= /3, the sequence (wk} is increasing and converges in W *,‘(O, 2~) to u,. We produce a unified proof of both Theorems 3.1 and 3.2, showing that they can be easily derived as consequences of the following very general principle
that we recall here
as a lemma.
LEMMA 3.3 ([lo], [4]). Let E be an ordered real Banach space with a normal positive cone. Let D be an order convex subset of E and suppose that T: D-t E is an increasing and completely continuous operator. Assume that there exists u(,, VIE D, with u0 < v(,, such that u0 < Tu,, and v, > TV,,. Then T has a minimal fixed point u, and a maximal fixed point v,, in the order
interval ug, vol. Moreover, u and the [ sequPnces IT u > a;d k
respective1
0
=
;Tlmkk-+co) vo
TkuO and vCo= lin++caTkvO, are increasing and decreasing,
y.
Now we are in a position
to prove Theorems
3.1 and 3.2.
For every o # 0, if c # 0, and w # n*, for each neM, if c = 0, PROOF. let K, denote the linear operator sending any function eE L’(O,2 n) onto the unique solution w E W *, ’ (0,2 a) of the problem
- w”+
cw’-
cow = e(t), w(0) = w(27r), w’(0) = w’(27r).
15
Periodic Boundary Value Problems The operator K, is well-defined C”([0,2?r]). Moreover, Proposition only if w < 0 and is negative
and compact from L’(O,2 n) to, say, 2.2 states that K, is positive if and 1+c2 if and only if 0 < w < -. Let D, be the
order convex subset of C’([O, 2 7r]) defined by D, = (n2CC”([0, 2 a]) : u( t)~ I, for t~[O,2 7r]}. Let F, : D, C C”([O, 2 T]) + L’(O,2 n) denote the Nemytskii operator which maps any function u E D, onto .f( *, u) - wu. The Carathkodory conditions on f imply that F, is continuous and maps bounded sets to bounded sets. Moreover, F, is increasing, by (3.6) if 1+ c2 w
T,= problem
(3.1)-(3.2)
K,F,:D,CC0([0,2?r])+Co([0,2~]), is equivalent
to the fixed point equation u = T,u,
where the operator
T, is increasing
and completely
continuous.
Moreover,
we have
if (3.5) holds and o < 0, or
cr>T,a
and
P,
1+ c2 Finally, recalling that the positive cone holds and 0 < w < 4 . in C”([O,2a]) is normal, we see that all the assumptions of Lemma 3.3 are fulfilled and therefore the conclusions of Theorems 3.1 and 3.2 follow. In particular, the convergence of the iterates in W231(0,2?r) is an immediate if (3.10)
consequence of the continuity K,: L’(o,27r)+w”z’(o,27r).
of F, : D, c C”([O, 2 rr]) -+ L’(O,2 n) and of
n
1+ c2 We notice that the bound w < seems optimal, in order that the 4 monotone method works. Indeed, as stated in Proposition 2.2, the antimaximum principle, which is required to get the monotonicity of the 1+ c2 associated operator T,, fails for w > -. Nevertheless, as far as one is only concerned
with the existence
of solutions
of problem
(3.1)-(3.2)
i.e.,
16
P. OMARI AND M. TROMBETTA
without the approximation have been proved recently and 2 given there,
scheme provided by (3.7)-(3.8), in [16]; for instance, combining
the following
theorem
sharper results Propositions 1
can be easily derived.
PROPOSITION3.4. Suppose thatf(t,s)= g(s)+ h(t), with g:R-+W and R continuous. Moreover assume that either c + 0, or, if c = 0,
h : [0,2 r]+
Finally, suppose that there exists a lower solution Q!and an upper solution fl ( possibly, nonordered at all). Then problem (3.1)-(3.2) admits at least one solution. We conclude this section by remarking that it is not known if the result of Proposition 3.4 carries over to the case where f does not admit the splitting
4.
f( t, s) = g(s) + h(t).
LOWER
AND
PERIODIC
UPPER
SOLUTIONS
FOR
THIRD-ORDER
BVPS
Let us consider
the third order periodic
BVP
u”‘+au”+bu’=f(t,u),
(4.1)
u(O) = u(27r),
u’(0) = u/(2+
where a, b are fixed real constants interval having a nonempty interior, By a solution
of (4.1)-(4.2)
(4.2)
and f: [0,2 7r] x I+ W, with I C W an satisfies the Caratheodory conditions.
we mean a function for every
+)a
u”(O) = u”(27r),
u~W~‘~(0,2n),
tE[O,27r],
such that
(4.3)
which satisfies Equation (4.1), a.e. in [0,2 ~1, and the boundary conditions (4.2). Moreover, we say that a function ar~W 3’1(0,2a) is a lower solution of (4.1),
(4.2),
if it satisfies
(4.2),
a”‘+ for a.e. te[O, 2 7~1. An upper the inequality in (4.4).
(4.3)
a&+ solution
and
bcz’
(4.4) similarly,
but reversing
Periodic Boundary Value Problems
17
Here, we are concerned with the solvability of (4.1)-(4.2) in the presence of a pair of lower and upper solutions. This study, which bears on the maximum and antimaximum principles proved in Section 2, is also motivated by the recent paper [15] where some partial results in this direction are produced. According to Proposition 2.3, we define the set
and the functions wl, w2 : A + W, by
w,(a, b) =
-2a3 27
and
Then the following results hold.
Suppose that (a, b)E A and there exists a lower solution THEOREM 4.1. (Y and an upper solution /3 such that a(t)
Assume that there is a constant that
forever-y te[0,27r].
(4.5)
w < 0, with wl(a, b) < - w < w2(a, b), such
f(t,s,)-f(t,sl)~W(S2-SI),
P-6)
for a.e. te[O,27r] and a(t)< s, < s2
18
I’. OMARI AND M. TROMBETTA
of the following
scheme:
w~+l+aw;+,+bw;+,-Wwk+l=f(t,wk)-w~t,
(4.7)
wk+l(O) = wk+,(2+ w;+l(O)
= w;+1(2+
w;+Jo)
= Wk”+J27r).
(4.8)
Zf wg = (Y, the sequence { wk} is increasing and converges in W3,‘(0, 2 a) to u,. lf w0 = 0, the sequence { wk) is decreasing and converges in W 3, ‘(0,2 7~)
to u,.
THEOREM 4.2. cx and an upper
Suppose that (a, b)E A and there exists a lower solution solution /3 such that cx(t)>fi(t),
Assume that there is a constant
foreverytE[0,27r].
(4.9)
w > 0, with wl(a, b) < - w < ~~(a, b), such
that (4.10)
f(t>sp)-f(t>s++,-s,), for
a.e.
tE[O,2?r]
and
P(t)<
sl<
s2 < cx(t).
Then
problem
(4.1)-(4.2)
possesses a minimal solution u, and a maximal solution v, in the order interval [ 0, CY]. Moreover, u, and v, can be computed iteratively by means of the scheme defined by (4.7)-(4.8). lf w0 = /3, the sequence {wk} is increasing and converges in W3s1(0, 2 n) to u,. lf w0 = CY, the sequence { wk) is decreasing
and converges
in W3x1(0,2n)
PROOF. The proof is similar to that of we sketch it for the sake of completeness. if b = n2 for some neM, let K, denote function eE L’(O,2 7r) onto the unique
to u,.
Theorems 3.1 and 3.2; however, For every w # 0, with w # - an’, the linear operator sending any solution WE W3,1(0, 2 T) of the
problem
w”‘+
au;“+ bw’-
ww = e(t), w(0)
= w(2a),
w’(0)
= w’(27r),
w”(0)
= w”(27r).
19
Periodic Boundary Value Problems
The operator K, is well-defined (compare [2]), is continuous from L’(O,2 ?r) to W3*‘(0, 2 ?T) and compact from L’(O,2 7r) to, say, C’([O, 2 ~1). Moreover, Proposition 2.3 states that K, is positive if and only if w c 0 and o,( a, h) G - fJ G wp( a, b), whereas it is negative if and only if u > 0 and ~,(a, b) G - w G wa(a, b). As before, we define in C”([0,27r]) the order convex set D, = { UEC~([O, 2 n]) : u( t)~ I, for te[O, 2 7r]} and the Neymtskii operator F, : D, C C’([O, 2 ~1) + L’(O,2 ?r) mapping any function UE D, onto f(*, u)- wu. By the Caratheodory conditions on f,F, is continuous and maps bounded sets to bounded sets. Moreover, F, is increasing, by (4.6), when w < 0 and wi(a, b) ,< - w G wr(a, b), or is decreasing, by (4.10), when w > 0 and ~~(a, b) G - w Q wa(u, b). Hence, setting T,=
problem
(4.1)-(4.2)
K,F,:D,CC0([0,2?r])+Co([0,27r]),
is equivalent
to the fixed point equation u = TUu,
where the operator we have
T, is increasing
and completely
continuous.
Moreover,
if (4.5) holds, and cxaT,cx
and
fi
if (4.9) holds. Accordingly, all the assumptions of Lemma 3.3 are fulfilled and therefore the conclusions of Theorems 4.1 and 4.2 easily follow. W We point out that Theorem 4.1 contains as a particular case the main result of [I5], where it is supposed a = b c 0 and w = u2. Indeed, it is easy, although cumbersome, to verify that wi(a, a) < - u2 < w2(u, a). Moreover, Theorem 4.2 gives a positive answer to a question raised in [15], about the existence of a solution to (4.1)-(4.2) when the lower and the upper solutions are in the reversed order (i.e. satisfy (4.9)). Finally, letting a = b = 0, we get the following immediate consequence of Theorems 4.1 and 4.2, which solves a further problem posed in [15], concerning the solvability of (4.11)
um =f(t,u), u(0) = u(2a), in the presence
u’(0) = u/(2+
of lower and upper
solutions.
u”(0) = u”(27r),
(4.12)
P. OMARI
20 4.3. Suppose that there solution 0. Assume that either
exists a lower- solution
COROLLARY
upper
a(t)
AND M. TROMBETTA
foreuery
CY and an
te[0,27r],
and
f(t>S+f(t,s,)>
for a.e.
tE[O,2*]
and
-+
a(t) < s1 < sg < /3(t), or
a(t)
for-every
2 0(t),
tE[O,27r],
and
for a.e.
tE[O,2?r]
and
p(t)<
has a minimal and a maximal attained by monotone iteration.
s1 < s2 < cx(t). Then problem (4.11)-(4.12) solution between CY and 0, which can be
We conclude by remarking that no result to now is known for third-order equations.
similar
to proposition
3.4 up
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