Abstract and Applied Analysis
Volume 2018 (2018), Article ID 5321314, 6 pages
https://doi.org/10.1155/2018/5321314

Research Article

On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

Nai-Sher Yeh

Department of Mathematics, Fu Jen Catholic University, Xinzhuang District, New Taipei City 24205, Taiwan

Correspondence should be addressed to Nai-Sher Yeh; 038300@mail.fju.edu.tw

Received 24 October 2017; Accepted 28 January 2018; Published 1 April 2018

Academic Editor: Chun-Lei Tang

Copyright © 2018 Nai-Sher Yeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 condition

for 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 and

Since ,   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 have

for 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.

By slightly modifying the proof of Theorem 3, we can apply Lemma 2 to obtain an existence theorem to (2) when condition is replaced by and either (8) with or (9) with is satisfied, which has been established in [20] for the case when (9) with is satisfied and in [9] for the case when (8) with is satisfied.

Theorem 4. Let and be a Caratheodory function satisfying . Then for each problem (2) has a solution , provided that either (8) with or (9) with holds.

Conflicts of Interest

There are no conflicts of interest involved.

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