Abstract

For each and , we obtain some existence theorems of periodic solutions to the two-point boundary value problem in with when is a Caratheodory function which grows linearly in as , and may satisfy a generalized Landesman-Lazer condition for all . Here denotes the subspace of spanned by and , , , and .

1. Introduction

Let and be fixed. We consider the following two-point boundary value problems:where is given and is a Caratheodory function; that is, is continuous in , for a.e. , is measurable in for all , and satisfies, for each , the fact that there exists an such that for a.e. and all Concerning the growth condition of the nonlinear term to (1) and (2), we assume that (H)there exist constants , , and and for a.e. with strict inequality on a positive measurable subset of , such that for a.e. and all  and for a.e. and all (G)there exist constants , , and and for a.e. with strict inequality on a positive measurable subset of , such that for a.e. and all  and for a.e. and all

respectively, and a generalized Landesman-Lazer condition for all , may be satisfied. Here denotes the subspace of spanned by and , , , and . Under assumptions and either with or without the Landesman-Lazer conditionfor all , the solvability of the problem (1) has been extensively studied if the nonlinearity has at most linear growth in as (see [1–13] for the case and [14–16] for the general case) or grows superlinearly in in one of directions and and may be bounded in the other (see [8, 17] for the case and [14] for the general case when ). Based on the well-known Leray-Schauder continuation method (see [18, 19]), we obtain solvability theorems to (1) (resp., (2)) when satisfies (resp., ) and either (8) with or (9) with is satisfied, which extends the results of [15] for the nonresonance case, and has been established in [9] for the case and grows sublinearly in as with . Unfortunately, it is still unknown when ,   grows linearly in as and the assumption of (8) is replaced by for all with . In the following we will make use of real Banach spaces and Sobolev spaces and . The norms of and are denoted by and , respectively. By a solution of (1), we mean a periodic function of period which belongs to and satisfies the differential equation in (1) a.e. .

2. Existence Theorems

For each with and , we write ,  , and . Here denotes the projection of on the eigenspace of spanned by and for . Just as an application of [11, Lemma 2] or [1, Lemma 2.2], we can modify slightly the proof of [15, Lemma 1] to obtain the next lemma.

Lemma 1. Let and be a nonnegative -function such that for a.e. with strict inequality on a positive measurable subset of . Then there exists a constant such that whenever with for a.e. and is a periodic function of period with .

Proof. Just as in [20, Lemma 1], we can modify slightly the proof of [11, Lemma 2] or [1, Lemma 2.2] to obtain the fact that there exists a constant such that whenever with for a.e. and with . Let us extend and periodically in to all of R and then use the same notations for the periodic extensions as for the original functions. In this case, we have andSince ,   for a.e. , and for all with and , we have and Combining (12) with (13), we have

Lemma 2. Let and be a nonnegative -function such that for a.e. with strict inequality on a positive measurable subset of . Then there exists a constant such that whenever with for a.e. and is a periodic function of period with . Here and for each with .

Theorem 3. Let and be a Caratheodory function satisfying . Then for each problem (1) has a solution , provided that either (8) with or (9) with holds.

Proof. Let be fixed and We consider the boundary value problems for , which becomes the original problem when . Since , we observe from Lemma 1 that (17) has only a trivial solution when . To apply the Leray-Schauder continuation method, it suffices to show that solutions to (17) for have an a priori bound in . To this end, let be a continuous function such that ,   for , and for . We define , and Then are Caratheodory functions, such that for a.e. and ,   If is a possible solution to (17) for some , then using (19), (20), and Lemma 1, we have which implies that for some constants independent of . It remains to show that solutions to (17) for have an a priori bound in . We argue by contradiction and suppose that there exists a sequence of periodic functions with period and a corresponding sequence in such that is a solution to (17) with and for all . Let ; then for all , and by (22) we have as . Since and for all , we have a bounded sequence in . For simplicity, we may assume that converges to in for some with . In particular, in . Clearly, and . It follows that for each with , and for each with . Since and , we have Multiplying each side of (17) by , and then integrating them over when and , we get By (19) and the assumption of , we havefor a.e. . Combining (22) with (25), we get that is bounded from below by an -function independent of . By (20) and the assumption of , we have for a.e. , In particular, is bounded from below by an -function independent of , which implies that is also so, , and for all with . Here if , if , and if . Applying Fatou’s lemma to the integral , we have which is a contradiction when either (8) with or (9) with is satisfied. Hence, the proof of this theorem is complete.