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Perturbations with several independent parameters
JOURNALOFMATHEMATICALANALYSISAND
Perturbations
APPLICATIONS
50, 3&i-414(1975)
with Several Independent
Parameters
STEPHEN BANCROFT* University
of Rhode Island, Kingston, Rhode Island Submitted by J. P. LaSalle
In this paper we discuss the problem 3c*(-, A) of the differential equation
of determining
a T-periodic
solution
3E= A(+ + f (t, x, A) + b(t), where the perturbation parameter A is a vector in a parameter-space Rk. The customary approach assumes that h = A(e), E E R. One then establishes the existence of an co > 0 such that the differential equation has a T-periodic solution x*(-, A(e)) for all E satisfying 0 < l < cg . More specifically it is usually assumed that A(c) has the form A(e) = e&where X0 is a fixed vector in Rk. This means that attention is confined in the perturbation procedure to examining the dependence of x*(., A) on A as h varies along a line segment terminating at the origin in the parameter-space Rk. The results established here generalize this previous work by allowing one to study the dependence of x*(3, A) on h as h varies through a “conical-horn” whose vertex rests at the origin in Rk. In the process an implicit-function formula is developed which is of some interest in its own right.
1.
INTRODUCTION
AND
NOTATION
In this paper we discuss the problem of determining a T-periodic solution x*(., A) of the differential equation 9 = A(t) x +f(t,
x, A) + b(t), h E R”
(1.1)
It is assumed that the parameter dependent term f(t, x, A) satisfies f(t, x, 0) = 0 for all (t, x) E R x R”, so that equation (1.1) may be regarded as a perturbation of the nonhomogeneous linear problem 2 = A(t) x + b(t). Moreover, the vector-field h: R * The author is currently 01984.
affiliated
x
Rn x Rk + Rn defined by h(t, x, A) =
with Gordon
384 Copyright AI1 rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
U-2)
College, Wenham,
Massachusetts
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385
A(t) x + f(t, X, h) + b(t) and the functions aflax, aj/iah, a2f18xaXare assumed continuous, with h(t + T, x, X) = h(t, x, X), for all (t, x, X) E R x R" x Rk. We will let II denote the parameter space Rk, equipped with the usual euclidean norm, and 8, , denote the Banach space of continuous T-periodic functions defined on the real-line with values in Rn. 8, is equipped with the usual uniform norm:
A T-periodic solution x*(., h) for (1.1) may be regarded as a map x*: 9 -+ glr , where 9 C (1 is any appropriate domain of parameters for which (1.1) has a T-periodic solution. We shall see that if one requires x*(., X) to be uniformly bounded in Pr for X E 9, then 9 will in general be a “horn”shaped region of LI, which terminates in a cone whose vertex resides at 0 E (1:
FIGURE
1
Actually we shall examine only the tip of such a domain, providing a conical linear approximation for 9 in a neighborhood of 0 E A. Since (1.1) may depend on X in a nonlinear fashion, a global analysis would in general be expected to reveal a more detailed picture of 9, such as that depicted in Fig. 1. To be more specific, we will show that one may expect to find a conical shaped region C(E, u) of the form
where E is a compact set on the unit-sphere in .4, such that (1 .l) has a uniformly bounded T-periodic solution x*(., h) for all h E C(E, u):
386
STEPHEN
BANCROFT
FIGURE 2
Our results are a generalization of the more or less standard results available for the differential equation k = A(t) x + eg(t, x, E) + b(t),
EE R.
(1.5)
Equation (1.5) is obviously a special case of (l.l), the conical domains described above reducing to intervals on the real-line of the form [0, c,,] with E,,> 0. The problem of determining a T-periodic solution x*(*, l ) of (1.5) has received much attention, owing to the importance of nonlinear oscillation theory in certain areas of physics and engineering, [l-3]. It will be convenient at this time for us to review the type of results which one may expect in connection with problem (1.5). In order to present these, we recall that if the unperturbed problem (1.2) has a particular T-periodic solution x9(.), then it has a family of T-periodic solutions forming a pdimensional linear manifold L C 9’* . Indeed, if CD(.)is an 12x p matrixvalued function whose columns {Gi(.) 1i = 1,2,..., p> form a basis for the vectorspace of T-periodic solutions of the homogeneous unperturbed problem, then L = {+(-) E 6- I 4(e) = G(*> + @(.)a0 , a, E R”>.
W)
Proceding on the idea that a T-periodic solution x*(*, C) for problem (1.5) should reduce to a T-periodic solution of the unperturbed problem (1.2) as E+ Of, we see that it is natural to try to exhibit an co > 0 and an a0 E Rp such that (1.5) has a T-periodic solution x*(., C) for 0 < E <
Ey+x*(0,l ) = x0(-) + @(a)a, .
U-7)
To see how one might determine the appropriate vector a, E RP, recall
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387
that (1.2) will have a T-periodic solution xv(.) if and only if the function b( *) E YT satisfies 1 r Y*(t) b(t) dt = 0, -Tos
U.8)
where Y*(t) is Y(t)-transpose, !P(.) being the rz x p matrix-valued function whose columns {&(.) 1i = 1, 2,..., p} f orm a basis for the T-periodic solutions of the homogeneous adjoint problem 2 =
--A”(t)z.
(1.9)
Hence we conclude that if (1.3) h as a T-periodic solution x*(., c) for 0 < E < Q, , then b(-) satisfies (1.8) and x*(., e) satisfies 1 T Y*(t)g(t, To s
(1.10)
x*(t, E), C) dt = 0.
Moreover if x*(., e) satisfies (1.7), then a, E RP must satisfy &a)
(1.11)
= 0,
where h: RP --f R* is defined by &a) = f j-’ y*(t) g(t, Xp(t> + Q(t)
aa
>0) dt.
(1.12)
0
Condition (1.11) is necessary and “almost” sufficient. In fact we have the following theorem which summarizes the results available for problem (1 S). THEOREM 1.I. Let h: Rp --f Rp be the function de$ned by (1.12), and suppose that b(.) E 8, satisjies (1.8). Then 3 co > 0 such that the perturbation
problem 3i = A(t) x +
(1.5)
has a T-periodic solution x*(*, 6) depending continuously on Efor 0 < E < co , provided an a, E R@ can be found satisfying H,: h(a,) = 0,
H,: det / &
(uo) / # 0.
If conditions HI , H2 are met, x*(., E) satis$es $
x*(-Y c) = %I(-> + @(*> a0 7
where x,( .) and @( .) are described in connection with (1.6) above.
(1.7)
388
STEPHEN
BANCROFT
We are now in position to reveal the nature of the difficulties encountered in trying to determine a T-periodic solution x*(*, A) of the perturbation problem (1.1). Indeed, let us observe that the perturbation term f(t, x, A) in (1.1) satisfies fk
af
x2 4 = x
(4 x, 0) A + R(t, x, A),
where R(t, X, A) satisfies lim R(t’ ” ‘) = 0 A+0 IhI ’ uniformly for x in compact subsets of R*, t E R. In order to investigate the behavior of a T-periodic solution x*(., A) as h -+ 0, we will let h -+ 0 along a differentiable arc terminating at 0 E A. If we let E measure arc-length from 0 along such a curve, we may suppose that the curve is described by a function A: [0, a] -+ Rk, 01> 0, satisfying h(r) = YE + o(e)
(1.15)
where v = (A/de)(O) is a unit-vector giving the direction of approach that the curve A(E)takes toward 0 E A: (1.16) and O(E)is a function satisfying (1.17) Let us examine the perturbation term f (t, x, A) along such a curve. In fact consulting equations (1.13)-(1.17) we see that
f (4 % W) = &, where gv(t, x, E) is defined by
x, 4,
(1.18)
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Hence if X -+ 0 in A along differentiable arc approaching we can study problem (1.1) by examining the problem &+= A(t) x + q,(t,
0 in the direction V,
x, 6) + b(t),
as E --f O+. We now make preparations for applying Theorem We let Z C A be the surface of the unit-sphere in A,
389
(I.i9)” 1.1 to problem
(1.19).
(1.20)
C={XFAIIhI=l}, and H: RP x L: + Rp be the map defined by
H(a, , u) = f J’ y*(t) g (t, up, + Q(t) a, ) 0) u dt.
(1.21)
0
We now con&de from Theorem (1.19)” has a T-periodic solution x,*(.,
I.1 that 1 E,, = Q(V) > 0 such that c) for 0 < E < c,, satisfying
iiF* x,*(‘, 4 = qa(*) + a’(->4J(~>,
(1.22)
if uo(v) satisfies H,: H(a, , v) = 0,
HE: det 1 g
(a,, V) ) # 0.
These observations convince us that should problem (1 .l) have a Tperiodic solution x*(*, A) we can not expect x*(., A) to depend continuously on A at A = 0. Indeed, what we can expect is that x*(., A) satisfy a relation of the sort (1.23) in case limA,, X/l h 1 exists. Here a, : Z+ Rp is the function defined implicitly by HI and H, . We mention that in general one must restrict u,, : 2 4 Rp to compact subsets E of Z to guarantee that a,(v) depend continuously on u with a uniform bound. (This remark should shed light on the discussion surrounding Fig. 1 and 2). We see that as h -+ 0, x*(*, A) will tend to various different solutions of the unperturbed equation (1.2), depending on the direction of approach adopted by h as it moves toward 0 E R”. Equation (1.23) suggests the for X # 0 we may expect a T-periodic solution x*(-, A) of (1.1) to be representable in the form
STEPHEN BANCROFT
390 where w(*, A) E Pr satisfies
(1.25)
ljl$ W(‘, A) = 0. On the basis of these remarks and observations, of this paper.
we state the main result
THEOREM 1.2. Let EC Z be a compact set on the surface of the unitsphere in A. DeJ;ne C(E, u) to be the cone of rays from 0 E .A to E, intersect the sphere of radius o in A. Let H: Rp x Z + Rp be deJined by (1.21), and supposethat b(e) E 8, satisfies (1.8). Then 3 a u = o(E) > 0 such that the perturbation problem ff = A(t) x + f (t, x, 4 + b(t),
(1-l)
has a uniformly bounded, T-periodic solution x*(., A) for all X E C(E, u), depending continuously on X E C(E, u) - {0), p rovided only that 3 a continuous map a,, : E--f Rp satisfying H,: H(a,(h), A) = 0,
H2: det / E
(a,(% 4 / f 0,
for all h E E. If conditions HI and H2 are met, then x*(., A) sats@es
~*(a, A) = x,(-j + @(-I a0 ($-j)
+ wt., 4,
for all X E C(E, u) - {0}, where w(*, A) E 8, satisfies l&l W(‘, A) = 0.
Remark. We point out that because the function H(a, , A) depends linearly on A, the equation H(a, , A) = 0 will always determine implicitly a function so(A) satisfying a,(h) = a,(h/j h I). EXAMPLE. We provide a trivial but instructive example to illustrate Theorem 1.2. We will apply this theorem to the problem of determining a 2+-periodic solution of the forced damped harmonic oscilaltor as the periodic forcing term tends toward resonance:
(1.26)
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391
It is well-known from elementary theory that in case 01~+ y” # 0 (1.26) has a 2n/w-periodic solution u*(T; (Y,j3, y) given by (1.27)
u*(T; % p, y) = p COS(WT + 4) with p = (a2(1 + YB,+ Y2)‘i2 4 = tan-i
(
a(1 + Yy Y
1.
Moreover, if 01~+ y2 = 0 but /3 # 0, then resonance prevents (1.26) from having any T-periodic solution. We now show how similar conclusions may be arrived at through an application of Theorem 1.2. We make the change of variable t = WT,fixing the period of the differential equation at T = 27~.Hence the problem of finding a 2rr/w-periodic solution of (1.23) is equivalent to determining a 2n-periodic solution for the problem
Y zi+ 1+BYcos ii+u=lfyu- (1+ay)l,z t.
(1.29)
Letting u = x1 , ti = x2 we convert (1.29) to a first order system:
R, = -x1
Y + -x1 1 +r
-
(1 +Oiyy)‘i2 x2 +
-
P
cos t.
1tr
(1.30)
Equation (1.30) has the form k = Ax +f(t,
x, A),
A = (01, B, Y),
A
(1.31)
with x=
Xl ( x2 )*
=
(-‘:
;I’
0 f(4 x, A) =
Y
i I+yxl
-
(1 +ay)‘y)‘/2x2 +&cost)
We observe that (1.31) is a special case of (1 .l) with b(.) = 0. Hence condition (1.8) is satisfied. We must now find the function
H(a, ) A) = &J’” 0
Y*(t) g (t, x,(t) + @5(t) a, >0) Adt.
392
STEPHEN
BANCROFT
We notice that x,(e) = 0 is a particular solution of the unperturbed problem. Moreover, we may take the matrix-valued functions @(.) and Y(a) associated with the homogeneous unperturbed problem and the homogeneous adjoint problem to be
Writing a, E R2 in polar coordinates, a, =
I cos * ( r sin * 1 ’
we have
G(t) a, =
y co+ + $) ( -r sin(t + 4) 1 ’
Y*(t) = (
sin
t
-cost
cos
t
sint 1 ’
Finally, we observe that
g (t, x, 0) h = ;z f(t, x, hh) - f(t, x, 0) h
=
(‘yx, -
0 “x2 + p cos t ) *
Thus, we have
w,
$6 4 =
1 2n cos t(yr cos(t + #) + 011sin(t + #) + j? cos t} dt -1 :” ‘“2, sin t{yr cos(t + #) + a~ sin(t + #) + /3 cos t} dt 2no I
Let h(r, #, A) = Hl(r, 4, A) + iH&r, 4, A) where Hi(y, #, A), i = 1,2, is the ith row of H(Y, 4, A). Evidently the conditions H(a, , A) = 0, det ~(~H/&z,)(a, , A)( # 0 of Theorem 1.2 are equivalent to requiring (9
(ii)
h(r, ~4 4 = 0
det
F&f?!!&-I-E aY r a+
Imah ar
ImIE r a*
(t.,gL, A) # 0.
Moreover, we have h(r, $, A) = &
h” eit{yr cos(t + (CI)+ 01~sin(t + #) + /3 cos t} dt.
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Converting the trigonometric fact that
SEVERAL
INDEPENDENT
PARAMETERS
393
functions to exponential form and using the
1 2n ei(m-n)t dt = a,, , --.I 297 0
m, n integers,
we can evaluate the above integral by inspection. We find that h(r, I), A) = &(ye@& + ictreeib + /3}. Also (r, $, A) = e-i@(az+ y2).
Conditions (i) and (ii) above are obviously equivalent to (i)
v + iorr + /3ei+ = 0,
(ii)
a2 + y2 # 0.
Equating the real and imaginary parts to zero in (i) we find (i) is equivalent to y + p cm # = 0, ar + #I sin $ = 0. Hence if 01~+ y2 # 0 (condition ii), r = /3/(a2 + y2), I/J = tan-l(+). We apply the conclusion of Theorem 1.2 to get the result that if E is a compact set on the surface of the unit-sphere in OL- /I - y space, with (0, &l, 0) 4 E, then 3 u = a(E) > 0 such that (1.30) has a 27-periodic solution x*(., A) uniformly bounded for h E C(E, U) and continuous for h E C(E, u) - (O}, satisfying
x*(-9A>= @‘(.Ia, (A)
+ w(., A),
x E C(E, u) - (01,
where lim,,, w(., X) = 0. Since r cos(t @(*)ao
C&J
= t-r
+ #)
sin(t + 9) ) ’
we see that our Theorem enables us to conclude that Eq. (1.26) has a 27r/o-
394
STEPHEN BANCROFT
periodic solution u*(., A) uniformly bounded for h E C(E, C) and continuous for A E C(E, o) - (0) satisfying u(T; A) = r COS(WT + I/%)+ W&UT, A). We can actually determine Z+(WT,A) and verify that lim,, wr(w7, A) = 0, since the 2r/w-periodic solution of (1.26) given by (1.27)-(1.28) is unique. Thus Wl(T, A) = p cos(UJT+ ‘j> - ?’cos(WT+ #>, with p = (G(1 + y”, + Y2Y2 ’ y=
(a2
B
+ y2)-1/2
'
#J= tan-l a(1 + yy2 Y ’ *=tan-lz.
Y
We see that
lpl=l, -f Hence lim,,,
wl(T,
limk = 1. A-0 i/J
h) = 0, as claimed.
2. ABSTRACT PERTURBATION PROCEDURE Let X be a Banach space and B: X + X be a densely defined closed linear operator. We shall say that B is admissible for perturbation if the dimension of the nullity of B is equal to the codimension of the range of B. We denote the nullity and range of B by N(B) and R(B). Let A denote a parameter space which we also take to be a Banach space. Let N: X x A -+ X be a map which is continuous on X x A together with its derivatives aNlax, aN/aA, a2N/axah. We say that N: X x A -+ X is weakly nonlinear over the sphere S,(X) if 3 a continuous function ~(p, a) with ~(p, 0) = 0 such that II N(*> 4ll.s,cx,
G rl(p, 4,
for all h E S,(A), where 11N(., A)jjSP(r)is defined by
with S,(X) = {x E X / j x 1 < p}.
(2.1)
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395
We say that N: X x A -+ X is a weak nonlinearity if N is weakly nonlinear over every sphere S,(X) in X. A perturbation problem is defined to be an expression of the sort Bx = N(x, A),
(2.3)
where B: X -+ X is admissible for perturbation and N: X x A -+ X is a weak nonlinearity. The unperturbedproblem associated with (2.3) is the homogeneous linear problem Bx = 0. (2.4) Remark. Our discussion can easily be extended to include perturbations of the nonhomogeneous linear problem Bx = d,
(2.5)
as we shall show following an investigation of (2.3). PROPOSITION 2.1. Let B: X -+ X be admissible for perturbation. Then there are boundedprojection operators P: X + X, Q: X -+ X satisfying
(i)
R(P) = N(B)
(2.6)
(ii)
N(Q) = R(B).
(2.7)
Moreover there is a bounded linear operator K: X + X which is a rightinverse for B : (iii)
BKx = x,
x E R(B).
W-4
Proof. Let X = Xi @ X, , where Xi and X, are closed subspaces of X. Then the closed graph theorem assures us that the projection operators P:X+Xand(I -P):X ---f X associated with this direct sum decomposition are bounded. We now observe that if X, is a finite dimensional subspace of X, then X is closed in X together with any complementary subspace X, satisfying X = Xi @ X, . We conclude that if Xr is finite dimensional, then there is a bounded projection operator P: X-t X with R(P) = X, . If B is admissible for perturbation, then there are subspaces N,(B), R,(B) complementary to N(B), R(B) such that (9 (ii)
X = NW 0 N,(B), X = R(B) @ R,(B),
with dim N(B) = dim R,(B) < co. Hence there exist bounded projections P: X + X, Q: X-t X with R(P) = N(B), R(Q) = R,(B). Evidently N(Q) = R(B). This finishes the proof of (2.6) and (2.7) in the proposition.
396
STEPHEN BANCROFT
To prove the existence of a bounded linear operator satisfying (2.8), let X = N(B) @ N,(B). Let 8: N,(B) -+ X be the restriction of B: X -+ X to N,(B). Then 8: N,(B) -+ X is nonsingular. As such fi has an inverse map 8-r defined on I@). But R(B) = R(B) is closed. Hence the closed graph theorem asserts that 8-l is bounded. Evidently K = k’( 1 - Q) is a bounded linear extension of 8-l to all of X which satisfies (2.8). PROPOSITION 2.2. Let P, Q and K be the bounded linear operators of Proposition 2.1. Let X0 = N(B). Then the linear problem
Bx =d, Px = x, ,
(2.9
XoE-&,
has a (unique) solution ;f and only ;f (2.10)
Qd = 0. In case(2.10) is satisJied, this solution is provided by
(2.11)
x=(1-P)Kd+x,,
Proof. The solution set of (2.9) is nonempty iff d E R(B). But d E R(B) iff Qd = 0. A solution to (2.9)is unique because the operators B, P are linear and N(B) n N(P) = (0). Finally, x = (1 - P) Kd + x0 solves (2.9) if d E R(B) by direct substitution. PROPOSITION 2.3. Let Q: X-+
X be any projection operator satisfring
(2.7). Then Bx = N(x, A) o I&;(; Proof. (i) =>: Multiply that QB = 0. (ii)
zl N(x’ ‘)
(2.12)
Bx = N(x, h) by (1 - Q) and Q, using the fact
+: Add Bx = (1 - Q) B(x, h) and 0 = QN(x, A).
Remark. We shall refer to the problem Bx = (1 - Q) N(x, A),
(2.13)
as the modiJied perturbation problem associated with Bx = N(x, A). We shall refer to the problem 0 = QjW, 4,
(2.14)
PERTURBATIONS WITH SEVERAL INDEPENDENT PARAMETERS
397
as the bifurcation condition associated with the modified perturbation problem (2.13). PROPOSITION 2.4. Let X,, = N(B) and P, Q, K be the operators referred to in (2.6)-(2.8). Then
;I
z y - ‘) N(xt “‘1 e x = (1 - P) K(1 - Q) N(x, X) + x,, .
(2.15)
0
Proof. The proof follows immediately from proposition 2.2 together with the observation that Q(1 - Q) N(x, A) = 0. PROPOSITION 2.5. Let T: X x X0 x A -+ X be the map defined by
T(x, xo , h)=(l-P)K(l-Q)N(x,h)+x,.
(2.16)
Then x*(h) is a solution of the perturbation problem Bx = N(x, X) if and only if 3 x0 E X0 such that x*(h) = %(x0, h), where 2(x0 , X) is a fixed point of T(., xo >A) satisfying 0 = QN(n(x, , A), A). Proof.
(2.17)
The result follows directly from Propositions 2.3-2.4.
PROPOSITION 2.6. Let U, = {x 1x E S,(X), Px E S&X,)). Let T: X x X0 x A --f X be the map dejned by (2.16). Then 3 u. = o,(p) > 0 such that T( ‘, x0 , h) is unaformly contracting on U, with respect to the parameter set S,,,(X,) x SJA). The unique fixed point 5(x,, A) of this contraction is continuous on S,,,(X,) x S,,(A) together with its derivatives %/&G,, &Z/ah, a%/ax,aX. Moreover
P(x, , 0) = x0 .
(2.18)
Proof. To show that T: X x X0 x A + X is uniformly contracting on U,, with respect to the parameter set S,,,(X,) x SJA), one must verify that (i)
T(., x0, X): U,, -+ U,
(ii)
( T(x, x0 , h) - T(x, x0 , h)( < 0 / x - 5 (
for all (x0, A) E S&X,)
x SJA),
where 0 < 1 is independent of (x0, A).
Using the estimate (2.1) on N together with appropriate constants providing bounds for the linear operators P, Q, K, it is easily verified that there is a o. = uo(r) > 0 for which (i) and (ii) are satisfied. The smoothness of 2(x0, A) is a consequence of the smoothness of T(x, x0 , A) together with the uniform contraction mapping principle. Finally the estimate (2.1) on N shows that
398
STEPHEN BANCROFT
N(x, 0) = 0, from which it follows immediately that x0 is a fixed point of q-9 x0 9 0). Since this fixed point belongs to U, , we conclude that 2(x, ) 0) = x0 . PROPOSITION 2.7. Let Y, = R,(B), the complementary subspacefor R(B) projected by Q. Define G: S&X,) x S&4) - Y. , H: S,,,(X,) x &,,(A) -+ Y. bY
(i) G(xo,A) = QW+, ,A), 4, (ii)
H(xo ,A) = Q g
(2.19) (2.20)
(x0 , 0) A,
where 2(x0, A) is the Jixed point of T(*, x0, A) referred to in Proposition 2.6. Then 3 a continuous function k(p, u) with k(p, 0) = 0 such that the map Z: SoI, x S,&l) -+ Y, defined by (iii)
(2.21)
G(x, , 4 = H(x, ,A) + Z(x, ,A),
satisjes II Z(., 4ll~~,,(~,) ,< k(p>u> I X I >
(2.22)
for all h E S&l).
Heye 1)Z(., h)llSp,a(xo)is dejined as in (2.2). Proof. From the smoothness of N(x, A) and 2(x,, A) we see that G(x, , A) x SUa(A) together with its derivatives aG/ax, , is continuous on S&X,) aG/ah, a2Gjaxoah. We conclude in particular that 3 a continuous map, c(xo , u) with l (xo , 0) = 0, such that the function 2(x0 , A) defined by
z@o 9A)= G(xo,A) -
G(xo , 0) - $
(x0 , 0) A,
satisfies 12(x0, X)1 ,< ‘(x0, u) 1h j for all (x0, A) E S,,,(X,) X S,o(A). Since X0 is finite dimensional by hypothesis, S,,,(X,) is compact. Hence the function kI(, , u) defined by kl(p, u) =
sup +i, , a), w%,,(-Q
is continuous with k,(, , 0) = 0. Evidently 2(x0, A) satisfies E@~,&J
I -Do 79 G kh, 4 1X I
for X E S&l).
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
399
Next notice that because PG/ax,ah is continuous we may write a2G/ax,i3X= aaG/aXax, . It is thus clear that (aZ/ax,)(xO , A) satisfies
By an argument precisely the same as above, we conclude the existence of a continuous function Ka(p, cr) with k&p, 0) = 0 such that
u) we Letting k(p,4 = kdp,4 + k2(p,
see that k(p, a) is a continuous
function with k(p, 0) = 0 such that
for h E &&I). We have only to show that this function 2(x,, A) is the function set forth in (2.21). To see this, notice that (2.1) implies that G(x, , 0) = 0. Finally we have
But (2.1) again implies that (&V/8x)(5(x0, 0), 0) = 0. Using the fact that 2(x,, 0) = x0, we see that (aG/ah)(x, , 0) = Q(aN/ah)(x, , 0). Hence
z(% 3A>= ‘3x,, ,A) - Q g
(x0 , o) A,
verifying that Z(x, , A) is given by (2.21). Remark. In light of proposition 2.5, we see that we will have a solution x*(h) of the perturbation problem Bx = N(x, A) if we can determine a function x,(X) satisfying the bifurcation condition Ghl 3 A) = 0.
(2.23)
Since G(xO, A) = H(xO , A) + Z(x, , A), we must find x,,(h) satisfying H(xo , A) = -Z(x,
, A).
(2.24)
But Z(x,, , A) gets “small” much more rapidly with h then does H(x,, , A). Hence for X small, the problem w% PA) = 0,
(2.25)
STEPHEN BANCROFT
400
represents an “approximate” bifurcation condition. If we can find a function u,(h) satisfying (2.25) and if the function H(a,(h) + w, A) maps a neighborhood of OE X0 onto a neighborhood of 0 E Y,, , for each h in a neighborhood of 0 E A, then we should be able to find a “correction” term w(h) such that q,(A) + w(X) solves (2.24), or q,(A) = a,(A) + w(A) solves (2.23). Evidently the condition that H(u,,(h) + w, A) maps a neighborhood of 0 E X,, into a neighborhood of 0 E Y,, is met if and only if
det1g (a,(h),A)1# 0.
(2.26)
0
We make one last observation. If u,(h) satisfies (2.25) we may assume that %I(4 = ao(M h I), owing to the fact that H(x, , A) depends linearly on A. With these ideas in mind, we procede to our most important proposition. PROPOSITION 2.8. Let E be a compact set on the surface of the unit sphere in A. Defke C(E, u) to be the cone of rays from (0) E A to E intersect the sphere S&l) of radius u in A.
Then 3 o(E) > 0 and a map x0: C(E, U) +X0 which is uniformly bounded on C(E, u) and continuous on C(E, a) - (0) satisfying G(xo(4,4
= 0, X E C(-G 4,
(2.27)
provided only that a continuous map a,: E--f X0 can be found which satisfies HI: H(a,(A), A) = 0,
AEE,
(2.28)
H,: det1g (a,@), 4 1f 0,
h E E.
(2.29)
In case the map a,: E -+ X0 can be found, the map x0: C(E, u) + X0 satisfies A E C(E, u) - (0} where lim,,, w(A) = 0. Proof. Suppose that a continuous map a, : E -+ X0 has been found satisfying (2.28)-(2.29). Because E is compact a0 : E + X0 is bounded. Choose p large enough so that u,(h) belongs to a compact set 1M contained in the interior of S&X,). Let M + &(X0) be the set defined by M + &(X0) = {x,, I x0 = a, + w, a, E M, w E fW3>.
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
401
It is clear that 3 q, > 0 such that M + SJX,,) C S,,,(X,). Define a map I? Se0 x (C(J% 4 - V% + Yo by
F(w,X) = G(a, (A)
+ w, A).
We will have proved Proposition (2.8) if we can find a 0 = o(E) > 0 and a continuous map w: C(E, u) - (0) -+ Se0such that (9
(ii)
F(WN~
4 = 01 (j E C(E,
u) _ (O),
lim,+y(h) = 0
In fact the map x0 : C(E, u) --t X,, defined by
xEw, 0)- {O),
x0(4=
a, E S,,2(Xo), arbitrary,
h = 0,
will then satisfy (2.27). Let us define a map R: Scox (C(E,u) -{O))-+
Y. by
F(w,X) = F(O,X) + g (0,X)w + R(w,X). Assuming for a moment that (aF/aw)(O,h) is invertible for A E C(E, u)-(O}, we see that
F(w,X) = Oow = - ?$O,h)-'{R(w,h) Defining the map J: S<,(X,)
x
J(w A> = - g
(C(E,
U)
+F(O,h)).
- (0)) --f X0 by
(0, WV+
4 + F(O, 41,
we see that w(h) satisfies F(w, A) = 0 if and only if w(X) is a fixed point of J(., h). Our proof will be finished if we can find a fixed point w(A) depending continuously on X E C(E, u) - (0) satisfying lim,, w(h) = 0. To determine such a function, we will show that for o = U(E) > 0 sufficiently small, 3 a continuous function e(u) with lim,,,+ C(U) = 0 such that I(*, h) is uniformly contracting on S6(,JXo) for h E C(E, u) - (0): (9 (4
.J(., 4: SddVo) + Sdd(Xo) , I I@4 4 - J&4 41 < 0 I w - W 0
where 8 < 1 is independent of X E C(E,
U)
-
(0).
402
STEPHEN
BANCROFT
To verify that J satisfies these properties we must investigate the functions F(O, 4, (W~w)(O, A), and R(w, A) which enter into the definition of J. Returning to the definition of F(w, A), we see that F(0, A) = G(u,(h/( h j, A) and (8F/8w)(O,A) = (BG[~x,)(a,,(hfjh I), A). Recalling that G(x, , A) = fwo 9 4 + qxo > A) (Proposition 2.7), and using the fact that H(a,(h/j h I), A) = 0 for h E C(E, u) - {0}, we see that
(9
F@4 = 2 (00(6) , A),
(ii) Since R(w, A) = F(w, A) - F(0, A) - (aF/aw)(O,h)w, it is evident that
(iii) R(w, A)= H(og(&) + w,x>- H(a0(&) ,A) - g (a0(i&J 9 ‘miq~o(+J+w>q -~~o(~),h)-~(.o(~)‘h;).w (3 $ (w,4 = g (a0(+-J+ w,x>- g (ob($-j),A)
Using hypothesis H, above (Eq. exists and is uniformly bounded for the estimates of Proposition 2.7 and assures that 3 continuous functions constant p > 0 such that
I F(O, 41 < +) g
(0, q-1
u sufficiently small, with
(2.29)), we see that (i?H-l/8xo) (a,(A), A) all h E E compact. This fact together with an application of the mean value theorem R(u), k(e) with K(0) = K(0) = 0 and a
IXI >
h E x&q,
exists for X E C(E, U) - (01,
(*) (**I
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
403
and I qw, 4 - w,
41 < &)
I A I I w - 23I,
c***>
for all w, G E S,(X,). Combining these estimates, it is readily seen that I(., A) satisfies the two conditions cited above with regard to the uniform contraction mapping principle. THEOREM 2.1. Let E be a compact set on the surface of the unit sphere in A. De@ C(E, U) C A by
C(E, u) = [A / 1h 1
E E ifh # 01 .
(2.30)
X be a projection operator whose range is a to R(B). Define the map H: X0 x A --f Y, by
Wx, , A) = Q $! (x,, , 0) A.
(2.31)
Then 3a = o(E) > 0 such that the perturbation problem Bx = N(x, A), has a solution x*: C(E, u) -+ X which is unaformly bounded on C(E, u) and continuous on C(E, u) - (0}, provided that a continuous map a,,: E + X,, can befound such that H,: H(a,(X), A) 10,
HGdet12 (a,(%A)1# 0,
(2.32) (2.33)
all X E E. In casesuch a map can befound, the solution x*: C(E, a) --f X of the perturbation problem satis$es
for
x*(x) = a, (+ 1 + w*(q,
(2.34)
where limo-, w*(X) = 0. Proof. If a, : E---f X0 can be found satisfying HI and H, then , 3 w: C(E, u) - (0) -+ X0 such that x,(h) = a,@/\ X 1) + w(X) satisfies 0 = QN(3;: (x,(Q A), 4, X E C(E, u> - {O), where lim,,, w(h) = 0.
404
STEPHEN BANCROFT
We see from Proposition 2.5 that the map x*: C(E, o) - (0} -+ X defined bY x*(A) = f(x&), A>, satisfies Bx = N(x, A) for h E C(E, U) - (0). Since Z(x,, , A) is continuous, 2(x, , 0) = x,, , and x,(h) is uniformly bounded on C(E, a) - {0}, we see that G(A) = Ji;(x,(h), A) - x,(h) satisfies lim A+0e?(h) = 0, h E C(E, u) - {O>. But x,(h) = a&i/l h I) + w(h) where lim,,, w(h) = 0, h E C(E, u) - {O}. Hence w*(h) = %(x,(h), A) - a&l h I) is given by w*(h) = f.qh) + w(h) and satisfies lim,,, w*(X) = 0, h E C(E, U) - {O}. We provide a few propositions which are useful in applications of Theorem 2.1. PROPOSITION 2.9. Let B: X + X be a densely defined closed linear operator, and let B*: X* + X* be th e adjoint operator for B deJned by
x) =
(B*x*,
where X* is the dual-space of continuous linear functions defined on X. Let {xi* ( i = I,2 ,..., p} be a basisfor N(B*). Then 3 {xi 1i = 1,2 ,..., p} such that (Xi*, Xj) = S$j ,
1 < i, j < p.
(2.35)
Moreover the operator Q: X -+ X defined by
QX = CXj 3
(2.36)
is a projection of X satzjfy*ng N(Q)
= R(B),
(2.37)
where R(B) denotesR(B)-closure. COROLLARY B: X + X is admissiblefor perturbation if and only if
(9
R(B) = R(B),
(ii)
dim N(B) = dim N(B*).
PROPOSITION 2.10.
Suppose that the unperturbed nonhomogeneousproblem Bx = d,
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
405
has a particular solution x9 . Then the transformation x = x, + y reduces the perturbation problem Bx = N(x, A) + d, to the form By = R(Y, 4, where R(y, A) = N(x, + y, A) is a weak nonlinearity. Proof.
Obvious.
2.2. Let E be a compact set in the surface of the unit spherein A. DeJine C(E, u) C A by THEOREM
C(E,o) = IAEA/
1x1
Let X0 = N(B) and define @: Rp -+ X,, by @a, = E fjjaj j=l
wbe @ = (A, A ,...,4sp)is a “matrix” whose columns form a basis for X,, and a,, = col(ai) is a column-vector in Rp.
Dejne !Pz X -+ R” by VP x> = col(
Let H: Rp
x
whose rows form a basis for Y, = N(B*).
A --+ Rp be dejined by
Waoy4
=
(
Y g
(x, + @a, , 0) A) ,
where xD is any particular solution of the unperturbed problem Bx = d. Then 3 u = o(E) > 0 such that the perturbation problem Bx = N(x, A) + d has a solution x*: C(E, u) --f X which is uniformly bounded on C(E, u) and
406
STEPHEN BANCROFT
continuous on C(E, u) - {0}, provided a continuous map a0 : E -+ Rp can be found such that H,: H(a,(h), A) = 0,
f&cdet)g (a,@), 4 1# 0, for all h E E. In casesuch a map can befound, the solution x*: C(E, u) + X of
the perturbation problem satis$es
x*(4 = xp+ @a,(j+) + w*(h), where Em,,, w*(A) = 0. Proof. The proof follows immediately from Theorem 2.1 and Propositions 2.9 and 2.10. We observe that the function H: RP x A -+ R’ defined here is simply a Cartesian coordinate representation of the map H: X,, x A --+ Y0 found in Theorem 2.1.
3. APPLICATION OF THE PERTURBATION PROCEDURE TO NONLINEAR OSCILLATION THEORY
We apply the results of Section 2 to the problem of finding a T-periodic solution of the differential equation A?= A(t) x + f(t,
x, 4 + d(t),
X E R”,
(3.1)
where the vector-field for (3.1) is described in Section 1. Let gr be the Banach space of T-periodic functions also described in Section 1. Define themapsB:B,-+grandN:Yr x A-tgrby W)(t) WA
= &I 4(t)
- 4)
C(t),
= f (4 dt>, 4.
(3.2) (3.3)
Evidently the problem of finding a T-periodic x(., A) for (3.1) can be formulated as a functional equation in 8, : Bx = N(x, A) + d. PROPOSITION 3.1.
perturbation.
(3.4)
The map B: .9r + 8, defined in (3.2) is admissible for
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
407
Proof. It is clear that B: YJT-+ 9, is a densely defined closed linear operator. We need only show that dim N(B*) = dim N(B) and R(B) = R(B). In this connection, let R” be the collection of all n-tuple row-vectors, and 8,+ be the Banach space of continuous T-periodic functions defined on the real line with values in RN. Brf may be embedded in Pr*, the dual space for 8, , by the identification
where (+, .) is the continuous linear functional defined on 9’r by
Furthermore, this embedding is dense in the weak-star topology for PIT*, i.e., given any +* E Pr*, 3 {#, 14, E 9r+, n = 1,2 ,... } such that
for every 4 E Pr . Next observe that the map B+: 8,+ + 8,+ defined by (B+vW) = h> + W
4%
satisfies @+A +) = (A Bfb), for all $ E 9’r , $ E 9’r+. We next show that R(B) = W(B+).
(3.5)
dim N(B+) = dim N(B).
(3.6)
We also will verify that
These remarks enable us to regard B: 9, --t 9, as an operator which is admissible for purturbation, and B+: Br+ -+ 8,+ may be used with confidence in place of B*: 9=* --f PIT*, where B* is the “true” adjoint for B, extending B+ to a maximal domain in Pr+. Let X(t) be a fundamental matrix solution of 3i = A(t)x. Then X-i(t) is a fundamental matrix solution of j = -yA(t). In particular we have
(9 N(B) = M.> I+(t) = X@h , [X-V) - Oq, = 01,
(3.7)
(ii)
(3.8)
N(B+) = {#(.) / 4(t) = y&?(t),
y,,[X-l(T)
- I] = O}.
We see from (3.7), (3.8) that dim N(B+) = dim N(B).
408
STEPHEN BANCROFT
It remains to be shown that R(B) = W(B+). In fact it suffices to show that the initial value problem k = A(t) x +f(t), x(O) = x0 has a T-periodic solution if and only if f( .) E IN(B+). Letting x(t, x0) denote the solution of this initial value problem, x(t, x0) is given by
Evidently x( *, x,,) is T-periodic if and only if x( T, x,,) = x,, , i.e., if and only if [X-l(T)
- I] x0 = loT F(s)
(3.9) is soluble if and only if b = sr X-l(s)f(s)
f(s) ds.
(3.9)
ds satisfies
(Yo 9 8 = 0,
(3.10)
for all y0 E Rn solving y,[X-l(T)
- I] = 0.
Condition (3.10) is just the condition that f(v) E lN(B+). PROPOSITION 3.2. Let f: R x Rn x Rk -+ R” satisfy f (t, x, 0) z 0, (t, x) E R x Rn. Then the map N: 8, x A+ ~9’~dejned in (3.3) is a weak nonlinearity.
Proof. Since f(t, x, 0) = 0, it is clear that (af/iYx)(t, x, 0) = 0. Thus a continuity argument using the compactness of the sphere SO(Rn) in R” shows the existence of a continuous function q(p, u) with ~(p, 0) = 0 such that
for all A E S&l). Now
= sup ] sup [I f(t, 9V)Y w + 12 (4 d(t), 3 I] 1 * lldII
Since II 4 (JBr < p implies I $(t)l < p, t E [0, T], we see that II Nl.9 4llS&)
G rl(P, 49
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
409
for all h E S,(A). Thus N: YT x A -+ 8, is weakly nonlinear over every sphere S,(B,) in 9)T, or N: L?~ x A -+ 8, is a weak nonlinearity. Consulting the proof of Proposition 3.1 and the statement of Theorem 2.2, we obtain the main theorem stated in Section 1. We restate this result here as Theorem 3. I. THEOREM 3.1. Let E be a compact set in the surface of the unit sphere in A. De$ne C(E, u) by
Let a(.)
be the T-periodic n x p matrix-valued function whose columns P)f orm a basis for the T-periodic solutions of
+A(*> Ii = 1,2...,
ji = A(t)x. Let
function ‘u(e) be the T-periodic p x n matrix-valued P}f orm a basis for the T-periodic solution of
whose rows
NJi(.) I i = 1, L,
j = -yA(t). DeJine H: Rp
x
A -+ Rp by
H(aO J ‘1 = f where x9( *) is a particular
IoT y(t> g (t, X,(t) + @(t) a, , 0) A dt, solution of the unperturbed problem 3i = A(t) x + d(t).
Then 3 a u = a(E) > 0 such that the perturbed problem * = 4)
x + f (t, x, A) + d(t),
has a T-periodic solution x*: C(E, U) -+ 8, uniformly bounded on C(E, o) and continuous on C(E, U) - {0}, p rovided that a continuous map ao: E --f Rr can be found satisfying HI: H(ao(A), A) = 0,
Hz: det1g (a,@),4 [ f 0,
410
STEPHEN BANCROFT
In casesuch a map can befound, x*( *, A) satisfies
x*(-Y4 = x21(-> + @(*Ia0(A)
+ w*(‘, A),
where limA,, w*(*, A) = 0. EXAMPLE. We discuss the problem of determining the 27+.+periodic solutions of the undamped Duffing equation:
(3.14)
(d2u/ch2) + u + yu3 = F cos WT,
where w2 = 1 + #!, y, F, /3 < 1. Making the change of time-variable t = wr, (3.14) becomes
B
Yu3+I+8
ii+u=G-Pu-
F 1 +
(3.15) B cos t-
Equation (3.15) is converted to system form by writing u = x, , zi = x2 : fi 3i2= ---xl + I+B%
3; = x2 )
(3.16)
-
(3.16) has the form (3.17)
ji =Ax+f(t,x,h) where x=
Xl
( X2 1 ,
h =
(Y, A F)>
We observe that
We may take
w>= (-sincos tt
t
sin cost )
and
Y(t)=
(-Fz:
Ez :) .
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
411
Letting a, =
r cos * ( r sin 1+4 1’
we have
H(ao 34 =
1 -1 :”
257 cos t{fiy cos(t - 4) - yr cos3(t - #) + F cos t} dt “2, sin t@r cos(t - #) - YY cos3(t - #) + F cost} dt
1 i -Y&F0
i
Writing h(r, +, X) = ~r(r, #, h) + &.(r, (i)
4, 4, we find that
h(r, I,!J,X) = $(/3rei@- $@eiti + F).
Evidently h(r, 16,h) = 0 if and only if r@ -
$yy2)
+ Fe-i* = 0.
Equating real and imaginary parts, we see that h(r, a,b,A) = 0 if and only if r(/?-
&w2) +Fcos+
=O,
Fsin$
=O.
This pair of equations is equivalent to the single equation
43 - by”> + F = 0, where
Y
is allowed to be positive or negative. Alternatively r(/? -
zyr2)
we must study
f F = 0,
depending on whether we take ~+4 = 0 or 4 = CTin the equation F sin 4 = 0. Evidently condition Hr and H2 of Theorem 3.1 are equivalent to the equations (i) (ii)
r(/3 - $r2y) + F = 0, (/3 - $yr2)(fl - $yr”) # 0.
412
STEPHEN
BANCROFT
In caseF # 0, we see that (p - QPy) # 0. Hence we need only study in this case the equations
(9 43 - 4r2r>+ F = 0, (ii) p - $r2 # 0.
(3.18)
A special investigation shows that these conditions are actually sufficient conditions for a solution of the bifurcation equations even when F = 0, but the treatment is slightly different from our above procedures owing to the fact that our differential equation is in this case autonomous. Suffice it to say that (3.14) has a 2rr/w-periodic solution depending continuously on h in case equations (3.18) are satisfied. We can display these results geometrically, using the unitsphere in (y, /3,F)space. Indeed, the equations (i) (ii)
r(P - b”r) + F = 0, /I - $yr2 = 0,
(iii)
6” + yr + F2 - 1 = 0,
(3.19)
define a Jordan curve r on the surface Z of the unit-sphere. If E is any compact set contained in Z with E n r = O, then (3.14) has a 27r/wperiodic solution depending continuously on h for A E C(E, u) - {0}, u sufficiently small. In order to get a preliminary notion of I’ from (3.19) it is convenient to eliminate F from (i) and (iii), leaving
(3.20)
(i) r(P - $r2y) * (1 - (8” + y2W2 = 0, (ii)
/3 - i r2y = 0.
If r = 0, then (7, & F) = (1, 0,O). If r # 0, then we have
(i) 18- 2~2~ f (’ - (fi,” + Y’))l” = 0, (ii)
(3.21)
p - $r2y = 0.
There is no bounded solution of this pair of equations in case y = 0. Therefore we may assume that y # 0, in which case we can eliminate I in (3.21). In fact if we use the second form equations to write r2 = (4/9)(/?/y) and substitute in the first equation we obtain
p - + f 3 (1 - (p” + y2))1/2(g2
= 0.
(3.22)
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
413
Simplifying, we have $B3+3zy
- ;
(y2 - 1) y = 0.
(3.23)
Notice that if (/3, y) solves (3.23) th en so does (-8, -7). If y = +l, then 6 = 0. Moreover if p(y) is a solution of (3.23) then /3(y) satisfies lim,,, &)=O. Since /3/y = (9/4)r2 for y # 0, weseethat&~>OwhenO
FIGURE
3
A similar analysis shows that the F - /? profile of r is invariant under a reflection about the p-axis in Fig. 4.
FIGURE 4 40915012”I3
414
STEPHEN BANCROFT
These projected views of the sphere enable us to construct a total picture of r, Fig. 5.
FIGURE 5
We summarize this example by pointing out that the conclusions of Theorem 3.1 apply to the Duffing equation (3.14) in case the set E used in defining C(E, u) in that theorem does not intersect the curve r displayed above in the (y, /3,F)-parameter space.
REFERENCES 1. S. BANCROFT, J. K. HALE, AND D. SWFZET,Alternative problems for nonlinear functional equations, J. Dzyerential Equations 4 (1968), 40-56. 2. J. K. HALE, “Ordinary Differential Equations,” Interscience, New York, 1969. 3. N. MINORSKY, “Nonlinear Oscillations,” van Nostrand, Princeton, NJ, 1962.
Perturbations
APPLICATIONS
50, 3&i-414(1975)
with Several Independent
Parameters
STEPHEN BANCROFT* University
of Rhode Island, Kingston, Rhode Island Submitted by J. P. LaSalle
In this paper we discuss the problem 3c*(-, A) of the differential equation
of determining
a T-periodic
solution
3E= A(+ + f (t, x, A) + b(t), where the perturbation parameter A is a vector in a parameter-space Rk. The customary approach assumes that h = A(e), E E R. One then establishes the existence of an co > 0 such that the differential equation has a T-periodic solution x*(-, A(e)) for all E satisfying 0 < l < cg . More specifically it is usually assumed that A(c) has the form A(e) = e&where X0 is a fixed vector in Rk. This means that attention is confined in the perturbation procedure to examining the dependence of x*(., A) on A as h varies along a line segment terminating at the origin in the parameter-space Rk. The results established here generalize this previous work by allowing one to study the dependence of x*(3, A) on h as h varies through a “conical-horn” whose vertex rests at the origin in Rk. In the process an implicit-function formula is developed which is of some interest in its own right.
1.
INTRODUCTION
AND
NOTATION
In this paper we discuss the problem of determining a T-periodic solution x*(., A) of the differential equation 9 = A(t) x +f(t,
x, A) + b(t), h E R”
(1.1)
It is assumed that the parameter dependent term f(t, x, A) satisfies f(t, x, 0) = 0 for all (t, x) E R x R”, so that equation (1.1) may be regarded as a perturbation of the nonhomogeneous linear problem 2 = A(t) x + b(t). Moreover, the vector-field h: R * The author is currently 01984.
affiliated
x
Rn x Rk + Rn defined by h(t, x, A) =
with Gordon
384 Copyright AI1 rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
U-2)
College, Wenham,
Massachusetts
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
385
A(t) x + f(t, X, h) + b(t) and the functions aflax, aj/iah, a2f18xaXare assumed continuous, with h(t + T, x, X) = h(t, x, X), for all (t, x, X) E R x R" x Rk. We will let II denote the parameter space Rk, equipped with the usual euclidean norm, and 8, , denote the Banach space of continuous T-periodic functions defined on the real-line with values in Rn. 8, is equipped with the usual uniform norm:
A T-periodic solution x*(., h) for (1.1) may be regarded as a map x*: 9 -+ glr , where 9 C (1 is any appropriate domain of parameters for which (1.1) has a T-periodic solution. We shall see that if one requires x*(., X) to be uniformly bounded in Pr for X E 9, then 9 will in general be a “horn”shaped region of LI, which terminates in a cone whose vertex resides at 0 E (1:
FIGURE
1
Actually we shall examine only the tip of such a domain, providing a conical linear approximation for 9 in a neighborhood of 0 E A. Since (1.1) may depend on X in a nonlinear fashion, a global analysis would in general be expected to reveal a more detailed picture of 9, such as that depicted in Fig. 1. To be more specific, we will show that one may expect to find a conical shaped region C(E, u) of the form
where E is a compact set on the unit-sphere in .4, such that (1 .l) has a uniformly bounded T-periodic solution x*(., h) for all h E C(E, u):
386
STEPHEN
BANCROFT
FIGURE 2
Our results are a generalization of the more or less standard results available for the differential equation k = A(t) x + eg(t, x, E) + b(t),
EE R.
(1.5)
Equation (1.5) is obviously a special case of (l.l), the conical domains described above reducing to intervals on the real-line of the form [0, c,,] with E,,> 0. The problem of determining a T-periodic solution x*(*, l ) of (1.5) has received much attention, owing to the importance of nonlinear oscillation theory in certain areas of physics and engineering, [l-3]. It will be convenient at this time for us to review the type of results which one may expect in connection with problem (1.5). In order to present these, we recall that if the unperturbed problem (1.2) has a particular T-periodic solution x9(.), then it has a family of T-periodic solutions forming a pdimensional linear manifold L C 9’* . Indeed, if CD(.)is an 12x p matrixvalued function whose columns {Gi(.) 1i = 1,2,..., p> form a basis for the vectorspace of T-periodic solutions of the homogeneous unperturbed problem, then L = {+(-) E 6- I 4(e) = G(*> + @(.)a0 , a, E R”>.
W)
Proceding on the idea that a T-periodic solution x*(*, C) for problem (1.5) should reduce to a T-periodic solution of the unperturbed problem (1.2) as E+ Of, we see that it is natural to try to exhibit an co > 0 and an a0 E Rp such that (1.5) has a T-periodic solution x*(., C) for 0 < E <
Ey+x*(0,l ) = x0(-) + @(a)a, .
U-7)
To see how one might determine the appropriate vector a, E RP, recall
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
387
that (1.2) will have a T-periodic solution xv(.) if and only if the function b( *) E YT satisfies 1 r Y*(t) b(t) dt = 0, -Tos
U.8)
where Y*(t) is Y(t)-transpose, !P(.) being the rz x p matrix-valued function whose columns {&(.) 1i = 1, 2,..., p} f orm a basis for the T-periodic solutions of the homogeneous adjoint problem 2 =
--A”(t)z.
(1.9)
Hence we conclude that if (1.3) h as a T-periodic solution x*(., c) for 0 < E < Q, , then b(-) satisfies (1.8) and x*(., e) satisfies 1 T Y*(t)g(t, To s
(1.10)
x*(t, E), C) dt = 0.
Moreover if x*(., e) satisfies (1.7), then a, E RP must satisfy &a)
(1.11)
= 0,
where h: RP --f R* is defined by &a) = f j-’ y*(t) g(t, Xp(t> + Q(t)
aa
>0) dt.
(1.12)
0
Condition (1.11) is necessary and “almost” sufficient. In fact we have the following theorem which summarizes the results available for problem (1 S). THEOREM 1.I. Let h: Rp --f Rp be the function de$ned by (1.12), and suppose that b(.) E 8, satisjies (1.8). Then 3 co > 0 such that the perturbation
problem 3i = A(t) x +
(1.5)
has a T-periodic solution x*(*, 6) depending continuously on Efor 0 < E < co , provided an a, E R@ can be found satisfying H,: h(a,) = 0,
H,: det / &
(uo) / # 0.
If conditions HI , H2 are met, x*(., E) satis$es $
x*(-Y c) = %I(-> + @(*> a0 7
where x,( .) and @( .) are described in connection with (1.6) above.
(1.7)
388
STEPHEN
BANCROFT
We are now in position to reveal the nature of the difficulties encountered in trying to determine a T-periodic solution x*(*, A) of the perturbation problem (1.1). Indeed, let us observe that the perturbation term f(t, x, A) in (1.1) satisfies fk
af
x2 4 = x
(4 x, 0) A + R(t, x, A),
where R(t, X, A) satisfies lim R(t’ ” ‘) = 0 A+0 IhI ’ uniformly for x in compact subsets of R*, t E R. In order to investigate the behavior of a T-periodic solution x*(., A) as h -+ 0, we will let h -+ 0 along a differentiable arc terminating at 0 E A. If we let E measure arc-length from 0 along such a curve, we may suppose that the curve is described by a function A: [0, a] -+ Rk, 01> 0, satisfying h(r) = YE + o(e)
(1.15)
where v = (A/de)(O) is a unit-vector giving the direction of approach that the curve A(E)takes toward 0 E A: (1.16) and O(E)is a function satisfying (1.17) Let us examine the perturbation term f (t, x, A) along such a curve. In fact consulting equations (1.13)-(1.17) we see that
f (4 % W) = &, where gv(t, x, E) is defined by
x, 4,
(1.18)
PERTURBATIONS
WITH
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PARAMETERS
Hence if X -+ 0 in A along differentiable arc approaching we can study problem (1.1) by examining the problem &+= A(t) x + q,(t,
0 in the direction V,
x, 6) + b(t),
as E --f O+. We now make preparations for applying Theorem We let Z C A be the surface of the unit-sphere in A,
389
(I.i9)” 1.1 to problem
(1.19).
(1.20)
C={XFAIIhI=l}, and H: RP x L: + Rp be the map defined by
H(a, , u) = f J’ y*(t) g (t, up, + Q(t) a, ) 0) u dt.
(1.21)
0
We now con&de from Theorem (1.19)” has a T-periodic solution x,*(.,
I.1 that 1 E,, = Q(V) > 0 such that c) for 0 < E < c,, satisfying
iiF* x,*(‘, 4 = qa(*) + a’(->4J(~>,
(1.22)
if uo(v) satisfies H,: H(a, , v) = 0,
HE: det 1 g
(a,, V) ) # 0.
These observations convince us that should problem (1 .l) have a Tperiodic solution x*(*, A) we can not expect x*(., A) to depend continuously on A at A = 0. Indeed, what we can expect is that x*(., A) satisfy a relation of the sort (1.23) in case limA,, X/l h 1 exists. Here a, : Z+ Rp is the function defined implicitly by HI and H, . We mention that in general one must restrict u,, : 2 4 Rp to compact subsets E of Z to guarantee that a,(v) depend continuously on u with a uniform bound. (This remark should shed light on the discussion surrounding Fig. 1 and 2). We see that as h -+ 0, x*(*, A) will tend to various different solutions of the unperturbed equation (1.2), depending on the direction of approach adopted by h as it moves toward 0 E R”. Equation (1.23) suggests the for X # 0 we may expect a T-periodic solution x*(-, A) of (1.1) to be representable in the form
STEPHEN BANCROFT
390 where w(*, A) E Pr satisfies
(1.25)
ljl$ W(‘, A) = 0. On the basis of these remarks and observations, of this paper.
we state the main result
THEOREM 1.2. Let EC Z be a compact set on the surface of the unitsphere in A. DeJ;ne C(E, u) to be the cone of rays from 0 E .A to E, intersect the sphere of radius o in A. Let H: Rp x Z + Rp be deJined by (1.21), and supposethat b(e) E 8, satisfies (1.8). Then 3 a u = o(E) > 0 such that the perturbation problem ff = A(t) x + f (t, x, 4 + b(t),
(1-l)
has a uniformly bounded, T-periodic solution x*(., A) for all X E C(E, u), depending continuously on X E C(E, u) - {0), p rovided only that 3 a continuous map a,, : E--f Rp satisfying H,: H(a,(h), A) = 0,
H2: det / E
(a,(% 4 / f 0,
for all h E E. If conditions HI and H2 are met, then x*(., A) sats@es
~*(a, A) = x,(-j + @(-I a0 ($-j)
+ wt., 4,
for all X E C(E, u) - {0}, where w(*, A) E 8, satisfies l&l W(‘, A) = 0.
Remark. We point out that because the function H(a, , A) depends linearly on A, the equation H(a, , A) = 0 will always determine implicitly a function so(A) satisfying a,(h) = a,(h/j h I). EXAMPLE. We provide a trivial but instructive example to illustrate Theorem 1.2. We will apply this theorem to the problem of determining a 2+-periodic solution of the forced damped harmonic oscilaltor as the periodic forcing term tends toward resonance:
(1.26)
PERTURBATIONS
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391
It is well-known from elementary theory that in case 01~+ y” # 0 (1.26) has a 2n/w-periodic solution u*(T; (Y,j3, y) given by (1.27)
u*(T; % p, y) = p COS(WT + 4) with p = (a2(1 + YB,+ Y2)‘i2 4 = tan-i
(
a(1 + Yy Y
1.
Moreover, if 01~+ y2 = 0 but /3 # 0, then resonance prevents (1.26) from having any T-periodic solution. We now show how similar conclusions may be arrived at through an application of Theorem 1.2. We make the change of variable t = WT,fixing the period of the differential equation at T = 27~.Hence the problem of finding a 2rr/w-periodic solution of (1.23) is equivalent to determining a 2n-periodic solution for the problem
Y zi+ 1+BYcos ii+u=lfyu- (1+ay)l,z t.
(1.29)
Letting u = x1 , ti = x2 we convert (1.29) to a first order system:
R, = -x1
Y + -x1 1 +r
-
(1 +Oiyy)‘i2 x2 +
-
P
cos t.
1tr
(1.30)
Equation (1.30) has the form k = Ax +f(t,
x, A),
A = (01, B, Y),
A
(1.31)
with x=
Xl ( x2 )*
=
(-‘:
;I’
0 f(4 x, A) =
Y
i I+yxl
-
(1 +ay)‘y)‘/2x2 +&cost)
We observe that (1.31) is a special case of (1 .l) with b(.) = 0. Hence condition (1.8) is satisfied. We must now find the function
H(a, ) A) = &J’” 0
Y*(t) g (t, x,(t) + @5(t) a, >0) Adt.
392
STEPHEN
BANCROFT
We notice that x,(e) = 0 is a particular solution of the unperturbed problem. Moreover, we may take the matrix-valued functions @(.) and Y(a) associated with the homogeneous unperturbed problem and the homogeneous adjoint problem to be
Writing a, E R2 in polar coordinates, a, =
I cos * ( r sin * 1 ’
we have
G(t) a, =
y co+ + $) ( -r sin(t + 4) 1 ’
Y*(t) = (
sin
t
-cost
cos
t
sint 1 ’
Finally, we observe that
g (t, x, 0) h = ;z f(t, x, hh) - f(t, x, 0) h
=
(‘yx, -
0 “x2 + p cos t ) *
Thus, we have
w,
$6 4 =
1 2n cos t(yr cos(t + #) + 011sin(t + #) + j? cos t} dt -1 :” ‘“2, sin t{yr cos(t + #) + a~ sin(t + #) + /3 cos t} dt 2no I
Let h(r, #, A) = Hl(r, 4, A) + iH&r, 4, A) where Hi(y, #, A), i = 1,2, is the ith row of H(Y, 4, A). Evidently the conditions H(a, , A) = 0, det ~(~H/&z,)(a, , A)( # 0 of Theorem 1.2 are equivalent to requiring (9
(ii)
h(r, ~4 4 = 0
det
F&f?!!&-I-E aY r a+
Imah ar
ImIE r a*
(t.,gL, A) # 0.
Moreover, we have h(r, $, A) = &
h” eit{yr cos(t + (CI)+ 01~sin(t + #) + /3 cos t} dt.
PERTURBATIONS
WITH
Converting the trigonometric fact that
SEVERAL
INDEPENDENT
PARAMETERS
393
functions to exponential form and using the
1 2n ei(m-n)t dt = a,, , --.I 297 0
m, n integers,
we can evaluate the above integral by inspection. We find that h(r, I), A) = &(ye@& + ictreeib + /3}. Also (r, $, A) = e-i@(az+ y2).
Conditions (i) and (ii) above are obviously equivalent to (i)
v + iorr + /3ei+ = 0,
(ii)
a2 + y2 # 0.
Equating the real and imaginary parts to zero in (i) we find (i) is equivalent to y + p cm # = 0, ar + #I sin $ = 0. Hence if 01~+ y2 # 0 (condition ii), r = /3/(a2 + y2), I/J = tan-l(+). We apply the conclusion of Theorem 1.2 to get the result that if E is a compact set on the surface of the unit-sphere in OL- /I - y space, with (0, &l, 0) 4 E, then 3 u = a(E) > 0 such that (1.30) has a 27-periodic solution x*(., A) uniformly bounded for h E C(E, U) and continuous for h E C(E, u) - (O}, satisfying
x*(-9A>= @‘(.Ia, (A)
+ w(., A),
x E C(E, u) - (01,
where lim,,, w(., X) = 0. Since r cos(t @(*)ao
C&J
= t-r
+ #)
sin(t + 9) ) ’
we see that our Theorem enables us to conclude that Eq. (1.26) has a 27r/o-
394
STEPHEN BANCROFT
periodic solution u*(., A) uniformly bounded for h E C(E, C) and continuous for A E C(E, o) - (0) satisfying u(T; A) = r COS(WT + I/%)+ W&UT, A). We can actually determine Z+(WT,A) and verify that lim,, wr(w7, A) = 0, since the 2r/w-periodic solution of (1.26) given by (1.27)-(1.28) is unique. Thus Wl(T, A) = p cos(UJT+ ‘j> - ?’cos(WT+ #>, with p = (G(1 + y”, + Y2Y2 ’ y=
(a2
B
+ y2)-1/2
'
#J= tan-l a(1 + yy2 Y ’ *=tan-lz.
Y
We see that
lpl=l, -f Hence lim,,,
wl(T,
limk = 1. A-0 i/J
h) = 0, as claimed.
2. ABSTRACT PERTURBATION PROCEDURE Let X be a Banach space and B: X + X be a densely defined closed linear operator. We shall say that B is admissible for perturbation if the dimension of the nullity of B is equal to the codimension of the range of B. We denote the nullity and range of B by N(B) and R(B). Let A denote a parameter space which we also take to be a Banach space. Let N: X x A -+ X be a map which is continuous on X x A together with its derivatives aNlax, aN/aA, a2N/axah. We say that N: X x A -+ X is weakly nonlinear over the sphere S,(X) if 3 a continuous function ~(p, a) with ~(p, 0) = 0 such that II N(*> 4ll.s,cx,
G rl(p, 4,
for all h E S,(A), where 11N(., A)jjSP(r)is defined by
with S,(X) = {x E X / j x 1 < p}.
(2.1)
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
395
We say that N: X x A -+ X is a weak nonlinearity if N is weakly nonlinear over every sphere S,(X) in X. A perturbation problem is defined to be an expression of the sort Bx = N(x, A),
(2.3)
where B: X -+ X is admissible for perturbation and N: X x A -+ X is a weak nonlinearity. The unperturbedproblem associated with (2.3) is the homogeneous linear problem Bx = 0. (2.4) Remark. Our discussion can easily be extended to include perturbations of the nonhomogeneous linear problem Bx = d,
(2.5)
as we shall show following an investigation of (2.3). PROPOSITION 2.1. Let B: X -+ X be admissible for perturbation. Then there are boundedprojection operators P: X + X, Q: X -+ X satisfying
(i)
R(P) = N(B)
(2.6)
(ii)
N(Q) = R(B).
(2.7)
Moreover there is a bounded linear operator K: X + X which is a rightinverse for B : (iii)
BKx = x,
x E R(B).
W-4
Proof. Let X = Xi @ X, , where Xi and X, are closed subspaces of X. Then the closed graph theorem assures us that the projection operators P:X+Xand(I -P):X ---f X associated with this direct sum decomposition are bounded. We now observe that if X, is a finite dimensional subspace of X, then X is closed in X together with any complementary subspace X, satisfying X = Xi @ X, . We conclude that if Xr is finite dimensional, then there is a bounded projection operator P: X-t X with R(P) = X, . If B is admissible for perturbation, then there are subspaces N,(B), R,(B) complementary to N(B), R(B) such that (9 (ii)
X = NW 0 N,(B), X = R(B) @ R,(B),
with dim N(B) = dim R,(B) < co. Hence there exist bounded projections P: X + X, Q: X-t X with R(P) = N(B), R(Q) = R,(B). Evidently N(Q) = R(B). This finishes the proof of (2.6) and (2.7) in the proposition.
396
STEPHEN BANCROFT
To prove the existence of a bounded linear operator satisfying (2.8), let X = N(B) @ N,(B). Let 8: N,(B) -+ X be the restriction of B: X -+ X to N,(B). Then 8: N,(B) -+ X is nonsingular. As such fi has an inverse map 8-r defined on I@). But R(B) = R(B) is closed. Hence the closed graph theorem asserts that 8-l is bounded. Evidently K = k’( 1 - Q) is a bounded linear extension of 8-l to all of X which satisfies (2.8). PROPOSITION 2.2. Let P, Q and K be the bounded linear operators of Proposition 2.1. Let X0 = N(B). Then the linear problem
Bx =d, Px = x, ,
(2.9
XoE-&,
has a (unique) solution ;f and only ;f (2.10)
Qd = 0. In case(2.10) is satisJied, this solution is provided by
(2.11)
x=(1-P)Kd+x,,
Proof. The solution set of (2.9) is nonempty iff d E R(B). But d E R(B) iff Qd = 0. A solution to (2.9)is unique because the operators B, P are linear and N(B) n N(P) = (0). Finally, x = (1 - P) Kd + x0 solves (2.9) if d E R(B) by direct substitution. PROPOSITION 2.3. Let Q: X-+
X be any projection operator satisfring
(2.7). Then Bx = N(x, A) o I&;(; Proof. (i) =>: Multiply that QB = 0. (ii)
zl N(x’ ‘)
(2.12)
Bx = N(x, h) by (1 - Q) and Q, using the fact
+: Add Bx = (1 - Q) B(x, h) and 0 = QN(x, A).
Remark. We shall refer to the problem Bx = (1 - Q) N(x, A),
(2.13)
as the modiJied perturbation problem associated with Bx = N(x, A). We shall refer to the problem 0 = QjW, 4,
(2.14)
PERTURBATIONS WITH SEVERAL INDEPENDENT PARAMETERS
397
as the bifurcation condition associated with the modified perturbation problem (2.13). PROPOSITION 2.4. Let X,, = N(B) and P, Q, K be the operators referred to in (2.6)-(2.8). Then
;I
z y - ‘) N(xt “‘1 e x = (1 - P) K(1 - Q) N(x, X) + x,, .
(2.15)
0
Proof. The proof follows immediately from proposition 2.2 together with the observation that Q(1 - Q) N(x, A) = 0. PROPOSITION 2.5. Let T: X x X0 x A -+ X be the map defined by
T(x, xo , h)=(l-P)K(l-Q)N(x,h)+x,.
(2.16)
Then x*(h) is a solution of the perturbation problem Bx = N(x, X) if and only if 3 x0 E X0 such that x*(h) = %(x0, h), where 2(x0 , X) is a fixed point of T(., xo >A) satisfying 0 = QN(n(x, , A), A). Proof.
(2.17)
The result follows directly from Propositions 2.3-2.4.
PROPOSITION 2.6. Let U, = {x 1x E S,(X), Px E S&X,)). Let T: X x X0 x A --f X be the map dejned by (2.16). Then 3 u. = o,(p) > 0 such that T( ‘, x0 , h) is unaformly contracting on U, with respect to the parameter set S,,,(X,) x SJA). The unique fixed point 5(x,, A) of this contraction is continuous on S,,,(X,) x S,,(A) together with its derivatives %/&G,, &Z/ah, a%/ax,aX. Moreover
P(x, , 0) = x0 .
(2.18)
Proof. To show that T: X x X0 x A + X is uniformly contracting on U,, with respect to the parameter set S,,,(X,) x SJA), one must verify that (i)
T(., x0, X): U,, -+ U,
(ii)
( T(x, x0 , h) - T(x, x0 , h)( < 0 / x - 5 (
for all (x0, A) E S&X,)
x SJA),
where 0 < 1 is independent of (x0, A).
Using the estimate (2.1) on N together with appropriate constants providing bounds for the linear operators P, Q, K, it is easily verified that there is a o. = uo(r) > 0 for which (i) and (ii) are satisfied. The smoothness of 2(x0, A) is a consequence of the smoothness of T(x, x0 , A) together with the uniform contraction mapping principle. Finally the estimate (2.1) on N shows that
398
STEPHEN BANCROFT
N(x, 0) = 0, from which it follows immediately that x0 is a fixed point of q-9 x0 9 0). Since this fixed point belongs to U, , we conclude that 2(x, ) 0) = x0 . PROPOSITION 2.7. Let Y, = R,(B), the complementary subspacefor R(B) projected by Q. Define G: S&X,) x S&4) - Y. , H: S,,,(X,) x &,,(A) -+ Y. bY
(i) G(xo,A) = QW+, ,A), 4, (ii)
H(xo ,A) = Q g
(2.19) (2.20)
(x0 , 0) A,
where 2(x0, A) is the Jixed point of T(*, x0, A) referred to in Proposition 2.6. Then 3 a continuous function k(p, u) with k(p, 0) = 0 such that the map Z: SoI, x S,&l) -+ Y, defined by (iii)
(2.21)
G(x, , 4 = H(x, ,A) + Z(x, ,A),
satisjes II Z(., 4ll~~,,(~,) ,< k(p>u> I X I >
(2.22)
for all h E S&l).
Heye 1)Z(., h)llSp,a(xo)is dejined as in (2.2). Proof. From the smoothness of N(x, A) and 2(x,, A) we see that G(x, , A) x SUa(A) together with its derivatives aG/ax, , is continuous on S&X,) aG/ah, a2Gjaxoah. We conclude in particular that 3 a continuous map, c(xo , u) with l (xo , 0) = 0, such that the function 2(x0 , A) defined by
z@o 9A)= G(xo,A) -
G(xo , 0) - $
(x0 , 0) A,
satisfies 12(x0, X)1 ,< ‘(x0, u) 1h j for all (x0, A) E S,,,(X,) X S,o(A). Since X0 is finite dimensional by hypothesis, S,,,(X,) is compact. Hence the function kI(, , u) defined by kl(p, u) =
sup +i, , a), w%,,(-Q
is continuous with k,(, , 0) = 0. Evidently 2(x0, A) satisfies E@~,&J
I -Do 79 G kh, 4 1X I
for X E S&l).
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
399
Next notice that because PG/ax,ah is continuous we may write a2G/ax,i3X= aaG/aXax, . It is thus clear that (aZ/ax,)(xO , A) satisfies
By an argument precisely the same as above, we conclude the existence of a continuous function Ka(p, cr) with k&p, 0) = 0 such that
u) we Letting k(p,4 = kdp,4 + k2(p,
see that k(p, a) is a continuous
function with k(p, 0) = 0 such that
for h E &&I). We have only to show that this function 2(x,, A) is the function set forth in (2.21). To see this, notice that (2.1) implies that G(x, , 0) = 0. Finally we have
But (2.1) again implies that (&V/8x)(5(x0, 0), 0) = 0. Using the fact that 2(x,, 0) = x0, we see that (aG/ah)(x, , 0) = Q(aN/ah)(x, , 0). Hence
z(% 3A>= ‘3x,, ,A) - Q g
(x0 , o) A,
verifying that Z(x, , A) is given by (2.21). Remark. In light of proposition 2.5, we see that we will have a solution x*(h) of the perturbation problem Bx = N(x, A) if we can determine a function x,(X) satisfying the bifurcation condition Ghl 3 A) = 0.
(2.23)
Since G(xO, A) = H(xO , A) + Z(x, , A), we must find x,,(h) satisfying H(xo , A) = -Z(x,
, A).
(2.24)
But Z(x,, , A) gets “small” much more rapidly with h then does H(x,, , A). Hence for X small, the problem w% PA) = 0,
(2.25)
STEPHEN BANCROFT
400
represents an “approximate” bifurcation condition. If we can find a function u,(h) satisfying (2.25) and if the function H(a,(h) + w, A) maps a neighborhood of OE X0 onto a neighborhood of 0 E Y,, , for each h in a neighborhood of 0 E A, then we should be able to find a “correction” term w(h) such that q,(A) + w(X) solves (2.24), or q,(A) = a,(A) + w(A) solves (2.23). Evidently the condition that H(u,,(h) + w, A) maps a neighborhood of 0 E X,, into a neighborhood of 0 E Y,, is met if and only if
det1g (a,(h),A)1# 0.
(2.26)
0
We make one last observation. If u,(h) satisfies (2.25) we may assume that %I(4 = ao(M h I), owing to the fact that H(x, , A) depends linearly on A. With these ideas in mind, we procede to our most important proposition. PROPOSITION 2.8. Let E be a compact set on the surface of the unit sphere in A. Defke C(E, u) to be the cone of rays from (0) E A to E intersect the sphere S&l) of radius u in A.
Then 3 o(E) > 0 and a map x0: C(E, U) +X0 which is uniformly bounded on C(E, u) and continuous on C(E, a) - (0) satisfying G(xo(4,4
= 0, X E C(-G 4,
(2.27)
provided only that a continuous map a,: E--f X0 can be found which satisfies HI: H(a,(A), A) = 0,
AEE,
(2.28)
H,: det1g (a,@), 4 1f 0,
h E E.
(2.29)
In case the map a,: E -+ X0 can be found, the map x0: C(E, u) + X0 satisfies A E C(E, u) - (0} where lim,,, w(A) = 0. Proof. Suppose that a continuous map a, : E -+ X0 has been found satisfying (2.28)-(2.29). Because E is compact a0 : E + X0 is bounded. Choose p large enough so that u,(h) belongs to a compact set 1M contained in the interior of S&X,). Let M + &(X0) be the set defined by M + &(X0) = {x,, I x0 = a, + w, a, E M, w E fW3>.
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
401
It is clear that 3 q, > 0 such that M + SJX,,) C S,,,(X,). Define a map I? Se0 x (C(J% 4 - V% + Yo by
F(w,X) = G(a, (A)
+ w, A).
We will have proved Proposition (2.8) if we can find a 0 = o(E) > 0 and a continuous map w: C(E, u) - (0) -+ Se0such that (9
(ii)
F(WN~
4 = 01 (j E C(E,
u) _ (O),
lim,+y(h) = 0
In fact the map x0 : C(E, u) --t X,, defined by
xEw, 0)- {O),
x0(4=
a, E S,,2(Xo), arbitrary,
h = 0,
will then satisfy (2.27). Let us define a map R: Scox (C(E,u) -{O))-+
Y. by
F(w,X) = F(O,X) + g (0,X)w + R(w,X). Assuming for a moment that (aF/aw)(O,h) is invertible for A E C(E, u)-(O}, we see that
F(w,X) = Oow = - ?$O,h)-'{R(w,h) Defining the map J: S<,(X,)
x
J(w A> = - g
(C(E,
U)
+F(O,h)).
- (0)) --f X0 by
(0, WV+
4 + F(O, 41,
we see that w(h) satisfies F(w, A) = 0 if and only if w(X) is a fixed point of J(., h). Our proof will be finished if we can find a fixed point w(A) depending continuously on X E C(E, u) - (0) satisfying lim,, w(h) = 0. To determine such a function, we will show that for o = U(E) > 0 sufficiently small, 3 a continuous function e(u) with lim,,,+ C(U) = 0 such that I(*, h) is uniformly contracting on S6(,JXo) for h E C(E, u) - (0): (9 (4
.J(., 4: SddVo) + Sdd(Xo) , I I@4 4 - J&4 41 < 0 I w - W 0
where 8 < 1 is independent of X E C(E,
U)
-
(0).
402
STEPHEN
BANCROFT
To verify that J satisfies these properties we must investigate the functions F(O, 4, (W~w)(O, A), and R(w, A) which enter into the definition of J. Returning to the definition of F(w, A), we see that F(0, A) = G(u,(h/( h j, A) and (8F/8w)(O,A) = (BG[~x,)(a,,(hfjh I), A). Recalling that G(x, , A) = fwo 9 4 + qxo > A) (Proposition 2.7), and using the fact that H(a,(h/j h I), A) = 0 for h E C(E, u) - {0}, we see that
(9
F@4 = 2 (00(6) , A),
(ii) Since R(w, A) = F(w, A) - F(0, A) - (aF/aw)(O,h)w, it is evident that
(iii) R(w, A)= H(og(&) + w,x>- H(a0(&) ,A) - g (a0(i&J 9 ‘miq~o(+J+w>q -~~o(~),h)-~(.o(~)‘h;).w (3 $ (w,4 = g (a0(+-J+ w,x>- g (ob($-j),A)
Using hypothesis H, above (Eq. exists and is uniformly bounded for the estimates of Proposition 2.7 and assures that 3 continuous functions constant p > 0 such that
I F(O, 41 < +) g
(0, q-1
u sufficiently small, with
(2.29)), we see that (i?H-l/8xo) (a,(A), A) all h E E compact. This fact together with an application of the mean value theorem R(u), k(e) with K(0) = K(0) = 0 and a
IXI >
h E x&q,
exists for X E C(E, U) - (01,
(*) (**I
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
403
and I qw, 4 - w,
41 < &)
I A I I w - 23I,
c***>
for all w, G E S,(X,). Combining these estimates, it is readily seen that I(., A) satisfies the two conditions cited above with regard to the uniform contraction mapping principle. THEOREM 2.1. Let E be a compact set on the surface of the unit sphere in A. De@ C(E, U) C A by
C(E, u) = [A / 1h 1
E E ifh # 01 .
(2.30)
X be a projection operator whose range is a to R(B). Define the map H: X0 x A --f Y, by
Wx, , A) = Q $! (x,, , 0) A.
(2.31)
Then 3a = o(E) > 0 such that the perturbation problem Bx = N(x, A), has a solution x*: C(E, u) -+ X which is unaformly bounded on C(E, u) and continuous on C(E, u) - (0}, provided that a continuous map a,,: E + X,, can befound such that H,: H(a,(X), A) 10,
HGdet12 (a,(%A)1# 0,
(2.32) (2.33)
all X E E. In casesuch a map can befound, the solution x*: C(E, a) --f X of the perturbation problem satis$es
for
x*(x) = a, (+ 1 + w*(q,
(2.34)
where limo-, w*(X) = 0. Proof. If a, : E---f X0 can be found satisfying HI and H, then , 3 w: C(E, u) - (0) -+ X0 such that x,(h) = a,@/\ X 1) + w(X) satisfies 0 = QN(3;: (x,(Q A), 4, X E C(E, u> - {O), where lim,,, w(h) = 0.
404
STEPHEN BANCROFT
We see from Proposition 2.5 that the map x*: C(E, o) - (0} -+ X defined bY x*(A) = f(x&), A>, satisfies Bx = N(x, A) for h E C(E, U) - (0). Since Z(x,, , A) is continuous, 2(x, , 0) = x,, , and x,(h) is uniformly bounded on C(E, a) - {0}, we see that G(A) = Ji;(x,(h), A) - x,(h) satisfies lim A+0e?(h) = 0, h E C(E, u) - {O>. But x,(h) = a&i/l h I) + w(h) where lim,,, w(h) = 0, h E C(E, u) - {O}. Hence w*(h) = %(x,(h), A) - a&l h I) is given by w*(h) = f.qh) + w(h) and satisfies lim,,, w*(X) = 0, h E C(E, U) - {O}. We provide a few propositions which are useful in applications of Theorem 2.1. PROPOSITION 2.9. Let B: X + X be a densely defined closed linear operator, and let B*: X* + X* be th e adjoint operator for B deJned by
x) =
(B*x*,
where X* is the dual-space of continuous linear functions defined on X. Let {xi* ( i = I,2 ,..., p} be a basisfor N(B*). Then 3 {xi 1i = 1,2 ,..., p} such that (Xi*, Xj) = S$j ,
1 < i, j < p.
(2.35)
Moreover the operator Q: X -+ X defined by
QX = C
(2.36)
is a projection of X satzjfy*ng N(Q)
= R(B),
(2.37)
where R(B) denotesR(B)-closure. COROLLARY B: X + X is admissiblefor perturbation if and only if
(9
R(B) = R(B),
(ii)
dim N(B) = dim N(B*).
PROPOSITION 2.10.
Suppose that the unperturbed nonhomogeneousproblem Bx = d,
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
405
has a particular solution x9 . Then the transformation x = x, + y reduces the perturbation problem Bx = N(x, A) + d, to the form By = R(Y, 4, where R(y, A) = N(x, + y, A) is a weak nonlinearity. Proof.
Obvious.
2.2. Let E be a compact set in the surface of the unit spherein A. DeJine C(E, u) C A by THEOREM
C(E,o) = IAEA/
1x1
Let X0 = N(B) and define @: Rp -+ X,, by @a, = E fjjaj j=l
wbe @ = (A, A ,...,4sp)is a “matrix” whose columns form a basis for X,, and a,, = col(ai) is a column-vector in Rp.
Dejne !Pz X -+ R” by VP x> = col(
Let H: Rp
x
whose rows form a basis for Y, = N(B*).
A --+ Rp be dejined by
Waoy4
=
(
Y g
(x, + @a, , 0) A) ,
where xD is any particular solution of the unperturbed problem Bx = d. Then 3 u = o(E) > 0 such that the perturbation problem Bx = N(x, A) + d has a solution x*: C(E, u) --f X which is uniformly bounded on C(E, u) and
406
STEPHEN BANCROFT
continuous on C(E, u) - {0}, provided a continuous map a0 : E -+ Rp can be found such that H,: H(a,(h), A) = 0,
f&cdet)g (a,@), 4 1# 0, for all h E E. In casesuch a map can befound, the solution x*: C(E, u) + X of
the perturbation problem satis$es
x*(4 = xp+ @a,(j+) + w*(h), where Em,,, w*(A) = 0. Proof. The proof follows immediately from Theorem 2.1 and Propositions 2.9 and 2.10. We observe that the function H: RP x A -+ R’ defined here is simply a Cartesian coordinate representation of the map H: X,, x A --+ Y0 found in Theorem 2.1.
3. APPLICATION OF THE PERTURBATION PROCEDURE TO NONLINEAR OSCILLATION THEORY
We apply the results of Section 2 to the problem of finding a T-periodic solution of the differential equation A?= A(t) x + f(t,
x, 4 + d(t),
X E R”,
(3.1)
where the vector-field for (3.1) is described in Section 1. Let gr be the Banach space of T-periodic functions also described in Section 1. Define themapsB:B,-+grandN:Yr x A-tgrby W)(t) WA
= &I 4(t)
- 4)
C(t),
= f (4 dt>, 4.
(3.2) (3.3)
Evidently the problem of finding a T-periodic x(., A) for (3.1) can be formulated as a functional equation in 8, : Bx = N(x, A) + d. PROPOSITION 3.1.
perturbation.
(3.4)
The map B: .9r + 8, defined in (3.2) is admissible for
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
407
Proof. It is clear that B: YJT-+ 9, is a densely defined closed linear operator. We need only show that dim N(B*) = dim N(B) and R(B) = R(B). In this connection, let R” be the collection of all n-tuple row-vectors, and 8,+ be the Banach space of continuous T-periodic functions defined on the real line with values in RN. Brf may be embedded in Pr*, the dual space for 8, , by the identification
where (+, .) is the continuous linear functional defined on 9’r by
Furthermore, this embedding is dense in the weak-star topology for PIT*, i.e., given any +* E Pr*, 3 {#, 14, E 9r+, n = 1,2 ,... } such that
for every 4 E Pr . Next observe that the map B+: 8,+ + 8,+ defined by (B+vW) = h> + W
4%
satisfies @+A +) = (A Bfb), for all $ E 9’r , $ E 9’r+. We next show that R(B) = W(B+).
(3.5)
dim N(B+) = dim N(B).
(3.6)
We also will verify that
These remarks enable us to regard B: 9, --t 9, as an operator which is admissible for purturbation, and B+: Br+ -+ 8,+ may be used with confidence in place of B*: 9=* --f PIT*, where B* is the “true” adjoint for B, extending B+ to a maximal domain in Pr+. Let X(t) be a fundamental matrix solution of 3i = A(t)x. Then X-i(t) is a fundamental matrix solution of j = -yA(t). In particular we have
(9 N(B) = M.> I+(t) = X@h , [X-V) - Oq, = 01,
(3.7)
(ii)
(3.8)
N(B+) = {#(.) / 4(t) = y&?(t),
y,,[X-l(T)
- I] = O}.
We see from (3.7), (3.8) that dim N(B+) = dim N(B).
408
STEPHEN BANCROFT
It remains to be shown that R(B) = W(B+). In fact it suffices to show that the initial value problem k = A(t) x +f(t), x(O) = x0 has a T-periodic solution if and only if f( .) E IN(B+). Letting x(t, x0) denote the solution of this initial value problem, x(t, x0) is given by
Evidently x( *, x,,) is T-periodic if and only if x( T, x,,) = x,, , i.e., if and only if [X-l(T)
- I] x0 = loT F(s)
(3.9) is soluble if and only if b = sr X-l(s)f(s)
f(s) ds.
(3.9)
ds satisfies
(Yo 9 8 = 0,
(3.10)
for all y0 E Rn solving y,[X-l(T)
- I] = 0.
Condition (3.10) is just the condition that f(v) E lN(B+). PROPOSITION 3.2. Let f: R x Rn x Rk -+ R” satisfy f (t, x, 0) z 0, (t, x) E R x Rn. Then the map N: 8, x A+ ~9’~dejned in (3.3) is a weak nonlinearity.
Proof. Since f(t, x, 0) = 0, it is clear that (af/iYx)(t, x, 0) = 0. Thus a continuity argument using the compactness of the sphere SO(Rn) in R” shows the existence of a continuous function q(p, u) with ~(p, 0) = 0 such that
for all A E S&l). Now
= sup ] sup [I f(t, 9V)Y w + 12 (4 d(t), 3 I] 1 * lldII
Since II 4 (JBr < p implies I $(t)l < p, t E [0, T], we see that II Nl.9 4llS&)
G rl(P, 49
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
409
for all h E S,(A). Thus N: YT x A -+ 8, is weakly nonlinear over every sphere S,(B,) in 9)T, or N: L?~ x A -+ 8, is a weak nonlinearity. Consulting the proof of Proposition 3.1 and the statement of Theorem 2.2, we obtain the main theorem stated in Section 1. We restate this result here as Theorem 3. I. THEOREM 3.1. Let E be a compact set in the surface of the unit sphere in A. De$ne C(E, u) by
Let a(.)
be the T-periodic n x p matrix-valued function whose columns P)f orm a basis for the T-periodic solutions of
+A(*> Ii = 1,2...,
ji = A(t)x. Let
function ‘u(e) be the T-periodic p x n matrix-valued P}f orm a basis for the T-periodic solution of
whose rows
NJi(.) I i = 1, L,
j = -yA(t). DeJine H: Rp
x
A -+ Rp by
H(aO J ‘1 = f where x9( *) is a particular
IoT y(t> g (t, X,(t) + @(t) a, , 0) A dt, solution of the unperturbed problem 3i = A(t) x + d(t).
Then 3 a u = a(E) > 0 such that the perturbed problem * = 4)
x + f (t, x, A) + d(t),
has a T-periodic solution x*: C(E, U) -+ 8, uniformly bounded on C(E, o) and continuous on C(E, U) - {0}, p rovided that a continuous map ao: E --f Rr can be found satisfying HI: H(ao(A), A) = 0,
Hz: det1g (a,@),4 [ f 0,
410
STEPHEN BANCROFT
In casesuch a map can befound, x*( *, A) satisfies
x*(-Y4 = x21(-> + @(*Ia0(A)
+ w*(‘, A),
where limA,, w*(*, A) = 0. EXAMPLE. We discuss the problem of determining the 27+.+periodic solutions of the undamped Duffing equation:
(3.14)
(d2u/ch2) + u + yu3 = F cos WT,
where w2 = 1 + #!, y, F, /3 < 1. Making the change of time-variable t = wr, (3.14) becomes
B
Yu3+I+8
ii+u=G-Pu-
F 1 +
(3.15) B cos t-
Equation (3.15) is converted to system form by writing u = x, , zi = x2 : fi 3i2= ---xl + I+B%
3; = x2 )
(3.16)
-
(3.16) has the form (3.17)
ji =Ax+f(t,x,h) where x=
Xl
( X2 1 ,
h =
(Y, A F)>
We observe that
We may take
w>= (-sincos tt
t
sin cost )
and
Y(t)=
(-Fz:
Ez :) .
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
411
Letting a, =
r cos * ( r sin 1+4 1’
we have
H(ao 34 =
1 -1 :”
257 cos t{fiy cos(t - 4) - yr cos3(t - #) + F cos t} dt “2, sin t@r cos(t - #) - YY cos3(t - #) + F cost} dt
1 i -Y&F0
i
Writing h(r, +, X) = ~r(r, #, h) + &.(r, (i)
4, 4, we find that
h(r, I,!J,X) = $(/3rei@- $@eiti + F).
Evidently h(r, 16,h) = 0 if and only if r@ -
$yy2)
+ Fe-i* = 0.
Equating real and imaginary parts, we see that h(r, a,b,A) = 0 if and only if r(/?-
&w2) +Fcos+
=O,
Fsin$
=O.
This pair of equations is equivalent to the single equation
43 - by”> + F = 0, where
Y
is allowed to be positive or negative. Alternatively r(/? -
zyr2)
we must study
f F = 0,
depending on whether we take ~+4 = 0 or 4 = CTin the equation F sin 4 = 0. Evidently condition Hr and H2 of Theorem 3.1 are equivalent to the equations (i) (ii)
r(/3 - $r2y) + F = 0, (/3 - $yr2)(fl - $yr”) # 0.
412
STEPHEN
BANCROFT
In caseF # 0, we see that (p - QPy) # 0. Hence we need only study in this case the equations
(9 43 - 4r2r>+ F = 0, (ii) p - $r2 # 0.
(3.18)
A special investigation shows that these conditions are actually sufficient conditions for a solution of the bifurcation equations even when F = 0, but the treatment is slightly different from our above procedures owing to the fact that our differential equation is in this case autonomous. Suffice it to say that (3.14) has a 2rr/w-periodic solution depending continuously on h in case equations (3.18) are satisfied. We can display these results geometrically, using the unitsphere in (y, /3,F)space. Indeed, the equations (i) (ii)
r(P - b”r) + F = 0, /I - $yr2 = 0,
(iii)
6” + yr + F2 - 1 = 0,
(3.19)
define a Jordan curve r on the surface Z of the unit-sphere. If E is any compact set contained in Z with E n r = O, then (3.14) has a 27r/wperiodic solution depending continuously on h for A E C(E, u) - {0}, u sufficiently small. In order to get a preliminary notion of I’ from (3.19) it is convenient to eliminate F from (i) and (iii), leaving
(3.20)
(i) r(P - $r2y) * (1 - (8” + y2W2 = 0, (ii)
/3 - i r2y = 0.
If r = 0, then (7, & F) = (1, 0,O). If r # 0, then we have
(i) 18- 2~2~ f (’ - (fi,” + Y’))l” = 0, (ii)
(3.21)
p - $r2y = 0.
There is no bounded solution of this pair of equations in case y = 0. Therefore we may assume that y # 0, in which case we can eliminate I in (3.21). In fact if we use the second form equations to write r2 = (4/9)(/?/y) and substitute in the first equation we obtain
p - + f 3 (1 - (p” + y2))1/2(g2
= 0.
(3.22)
PERTURBATIONS
WITH
SEVERAL
INDEPENDENT
PARAMETERS
413
Simplifying, we have $B3+3zy
- ;
(y2 - 1) y = 0.
(3.23)
Notice that if (/3, y) solves (3.23) th en so does (-8, -7). If y = +l, then 6 = 0. Moreover if p(y) is a solution of (3.23) then /3(y) satisfies lim,,, &)=O. Since /3/y = (9/4)r2 for y # 0, weseethat&~>OwhenO
FIGURE
3
A similar analysis shows that the F - /? profile of r is invariant under a reflection about the p-axis in Fig. 4.
FIGURE 4 40915012”I3
414
STEPHEN BANCROFT
These projected views of the sphere enable us to construct a total picture of r, Fig. 5.
FIGURE 5
We summarize this example by pointing out that the conclusions of Theorem 3.1 apply to the Duffing equation (3.14) in case the set E used in defining C(E, u) in that theorem does not intersect the curve r displayed above in the (y, /3,F)-parameter space.
REFERENCES 1. S. BANCROFT, J. K. HALE, AND D. SWFZET,Alternative problems for nonlinear functional equations, J. Dzyerential Equations 4 (1968), 40-56. 2. J. K. HALE, “Ordinary Differential Equations,” Interscience, New York, 1969. 3. N. MINORSKY, “Nonlinear Oscillations,” van Nostrand, Princeton, NJ, 1962.