Journal of Applied Mathematics and Stochastic Analysis, 13:1 (2000), 41-50. PERIODIC SOLUTIONS OF SYSTEMS WITIt ASYMPTOTICALLY EVEN NONLINEARITIES

New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate "next order" terms are obtained for the 2-periodic problem for the scalar equation x" + n2x g(x I)+ f(t, x)+ b(t) with bounded g(u) and f(t,x)O as Ix I-0. The result is also applied to the solvability of a two-point boundary-value problem and to resonant problems for equations arising in control theory.


Introduction
Consider the problem of the existence of a 2r-periodic solution of a forced second order quasilinear equation x"= w(t,x), such as the nonlinear forced pendulum equation x"+ ax f(x)+ b(t), and suppose that its right-hand side is asymptotically linear, i.e., the following limit exists: lim w,(t, x), k.
1p.E.Kloeden was partially supported by the Australian Research Council Grant A 8913 2609 and A.M. Krasnosel'skii by Grants 97-01-00692 and 96-15-96048 of the Russian Foundation of Fundamental Research.42 PETER E. KLOEDEN and ALEXANDER M. KRASNOSEL'SKII If k is not the square of an integer, then the problem is not resonant and it can be easily investigated by, for example, the Schauder principle.Suppose instead that k-n 2 and w(t,x)--n2x + v(t,x) with bounded v(t,x).The usual way to study this case is to assume the Landesman-Lazer conditions, i.e., v(t,x)-V + (t)as V + (t), the problem can be handled by, for example, topological methods.If these non-degeneracy conditions fail, then it is necessary to use properties of terms that vanish at in- finity (see [5]).
In this paper we consider a situation without the Landesman-Lazer property: the 2r-periodic problem for the equation x" + n2x g( x I) + f(t,x) + b(t) where the bounded function g(u) does not have a limit at infinity and f(t,x) tends to zero as The 2r-periodic problem for linearized equation x" + n2x 0 is degenerate for an integer n and the behavior of perturbations of this equation depends on some delicate properties of the bounded nonlinearities g(u) and f(t,x).If g(u)= 0 and f(t,x)=_ 0 then the Fredholm alternative gives a complete answer on the solvability of the wr- periodic problem of x'-n2x b(t) in terms of the value of the complex number 27r bn-/ eintb(t)dt.
If b n 0, then there exists a 2-dimensional linear set of 2r-periodic solutions, while if b n 0, then such solutions do not exist.
Nonlinear terms change the situation.Suppose that we know only the asympto- tics of the nonlinearities at infinity, i.e. the behavior of the nonlinearities for suffi- ciently large Ix I.For example, consider the equation x"+ n2x f(t, x) + b(t) with f(t,x)-.O as Ix -oc.If bn 7 0, then the topological properties of the equations can- not guarantee the solvability of the 2r-periodic problem; the index at infinity of the corresponding vector fields is equal to 0. If b n 0, then, under some appropriate hypotheses, the index differs from 0 and topological methods are applicable.Related problems (for g(u)--0) were studied in [1, 2], in series of papers by J. Mawhin and his co-authors (see [1, 3, 7, 8] and the references therein).In addition, the equation x + n:x g( x I)+ v(t, x) with a non-degenerate Landesman-Lazer type term v(t, x) was considered in [4].
In the next section we present a new theorem on the solvability of our equation.This theorem follows from a general theorem, Theorem 2, on the calculation of the index at infinity that is formulated in Section 3. Proofs are given in Sections 4 and 5. Theorem 2 can also be applied to various problems on solvability and bifurcation at infinity, etc., for some other boundary value problems.Examples of possible appli- cations to two-point boundary value problems and problems of forced oscillations in control systems are presented in Section 6.

Main Result for the Second Order Equation
Consider the equation x" + n2x g( x I) + f(t,x) + b(t) with non-zero integer n and functions f(t,x) and b(t) that are 2r-periodic in t.
(1) These functions together with the function g(u) are supposed to be bounded and continuous with respect to all their variables.(This assumption can be weakened: It is possible to prove the assertion of Theorem 1 for Carathfiodorian functions f(t,x) and the integrable functions b(t).)The following hypotheses are supposed to be valid.
is differentiable, then this hypothesis is equivalent to If the function g(u) Theorem 1: Let b n -O, suppose that both hypotheses (A) and (B) hold, and let oo. (a) Then (1) has at least one 2r-periodic solution.
From the proof of Theorem 1 in Section 5 one can see that the function in conditions ( 2) and (3) can be replaced by the function o(t, Ix I), where e no, 0( t, It) 0, t t f0 and mes fo > 0.

An Abstract Generalization
In this section we give an abstract generalization of Theorem 1.Let f be some set of positive finite measure and let L 2 denote the Hilbert space of scalar square integrable functions x(t):a--,N with the usual scalar product (.,.) and norm I1" II.
We want to calculate the index at infinity (see, e.g.[6]) of the Hammerstein-type vector field Tx(.) x(.A(x(.+ f(. ,x(. ))+ g( x(.+ b(. )),in the space L2, where A is a completely continuous linear operator in L 2 and f,g and b are as above.If 1 is not an eigenvalue of the operator A, this index is defined and is equal (-1)a, where a is the sum of the multiplicities of all real eigenvalues of A which are greater than 1.We will consider the case for which 1 is an eigenvalue of a normal linear operator A, that is with AA* A*A.The normality of A guarantees that there are no genera- lized eigenvectors corresponding to the eigenvalue 1.Let E 0 Ker(I-A) denote the corresponding linear finite dimensional subspace of the eigenvectors of the eigenvalue 1.
The main restriction of the operator A that allows vector fields with an even term g(Ix I) to be considered is the identity: e(t),g( e(t) )dt O, e(t) e E o, (6) which was first mentioned in [4] and is valid for various important applications.

If
[0,2r] and E 0 is the 2-dimensional subspace containing the functions sin nt and cos nt, then (6) is valid; this example occurs in the investigation of the 2r- periodic problem to be considered in Section 6.The identity ( 6) is also valid for [0, r] and a l-dimensional subspace E 0 containing the function sin nt for even n; this case arises in the study of degenerate two-point boundary value problems.Suppose that (A') One of the following one-sided estimates f(t,x).signx>_(t, I 1), (B') or f(t,x).signx<_ -(t, xl), xl >_u0, ten (s) for some u 0 > 0, holds where (t,u)' {u >_ u0}--R + is a nonnegative Carathodorian nonincreasing function which is strictly positive for t E 0 with mes 0 > 0.
The asymptotical Lipschitz condition holds, where d(r)'R + + is some positive nonincreasing function satisfying The distribution function li_Lmd(r O. (10) X(5) X(5; e) mes {t E : (t) _ 5} (11) of a non-zero function e(t) E o plays an important role in the formulation and proof of the following theorem.Let P denote the orthogonal projector onto E 0 and in formula (13) below, let d(u)-d(uo) for 0 _< u < u 0. Theorem 2: Let Pb(t)-O.Suppose that both hypotheses (A') and (B') hold and that X(0) mes {t G 'e(t) 0} 0. (12) for any non-zero function e(t) G E o. Suppose also that the operator A maps square integrable functions into essentially bounded ones and that A is continuous as an operator from the space L 2 into the space L.Moreover, suppose for any R > 0 and u, >_ u o that lim sup f[e(t) ld(le(tl)dt ( ---cx e EO, II e II 1 f e(t) (t, u, + R e(t) )dt indT (-1) a with r o=r+ dimE o for the case (7), and r O--r for the case (8), where r is the sum of the multiplicities of all real eigenvalues of the operator A that are greater than 1.
Condition (12) has been used by many authors, while the functions (11) were con- sidered in a related context in Chapter 25 of [1] and were later used systematically in [3].The combination of ( 6) with the condition that f(t,x)-O has not been consider- ed before.The assumption about the operator A:L2---L is technical and can be omitted, but this then makes proof much more cumbersome.It is usually valid in many applications.
In specific examples, conditions (13) and (14) are often not as awkward (see [3]) as may at first seem, e.g. in the case of Theorem 1, the relation (5) guarantees both (13) and ( 14).If the function d(u) has the form d(u) cufor some c E (0, 1) and (14) holds, then (13)is equivalent to lim sup c = 0.
---,c e EO, II e II 1 f a e(t) (t, u, + 4. Proof of Theorem 2 Consider the homotopy where A has a fixed sign.Let us prove for I1 _< "0 with positive A0 small enough that the vector field (I)(x, ) is non-zero for sufficiently large values of I I x I I under the assumptions of Theorem 2. We consider only non-positive values of for the case where (7) holds and non-negative , for the opposite case.
This is a prior estimate will prove Theorem 2: for small :/: 0, the linear part I-(1-)A of the field (x,,) is non-degenerate and its index at infinity has exactly the value given in Theorem 2; the number # (1-,)-1 is an eigenvalue of the oper- ator (1 A)A with # > 1 iff A < 0. The general properties of the index then complete the proof.The proof of a common estimate will be given only for the case (7), , E [-Suppose that (I)(x,A)-0 for some xEL 2 and AE[0, A0].Denote by E 1CL 2 the orthogonal to E 0 subspace and let Q-I-P.The projectors P and Q commute with the operator A and the projector P can be easily represented as Px-(ej, x)ej, where {ej} is an orthonormal basis in E 0 and AP-P.
For | small enough, the linear operators B(A)-Q(I-(1-A)A) are contin- uously invertible in E 1 for any A.Moreover, these inverse operators have uniformly bounded norms [I B(,)-l ll EI_.,E 1 _ Ca, _ 0" (16) Here and below c j denote constants for which only the existence and not their exact value is of importance.In particular, these constants do not depend on A and x.
The equation (I)(x,A)-0 can be rewritten as the pair of equations: Q(x,)-0 and P(x, A) O.
The first equation, rewritten as B(1)x AQ(f(t, ) + g( x I) + b(t)), together with the continuity of the operator A: L2--,L implies that The second equation can be rewritten as APx,-P(f(t,x) + g( x )) (18) (recall that Pb 0 is an assumption of Theorem 2).This last formula will now be considered in some detail Let Px=e(t)where [[e[[ =1, _>0 and h=Qx.In view of (17)we have to obtain an a priori estimate for the scalar positive .
The scheme of the proof below is as follows.