Open Access

Recent Existence Results for Second-Order Singular Periodic Differential Equations

Boundary Value Problems20092009:540863

DOI: 10.1155/2009/540863

Received: 12 February 2009

Accepted: 29 April 2009

Published: 8 June 2009

Abstract

We present some recent existence results for second-order singular periodic differential equations. A nonlinear alternative principle of Leray-Schauder type, a well-known fixed point theorem in cones, and Schauder's fixed point theorem are used in the proof. The results shed some light on the differences between a strong singularity and a weak singularity.

1. Introduction

The main aim of this paper is to present some recent existence results for the positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq1_HTML.gif -periodic solutions of second order differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq2_HTML.gif are continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq3_HTML.gif -periodic functions. The nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq4_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq6_HTML.gif -periodic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq7_HTML.gif . We are mainly interested in the case that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq8_HTML.gif has a repulsive singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq9_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ2_HTML.gif
(1.2)
It is well known that second order singular differential equations describe many problems in the applied sciences, such as the Brillouin focusing system [1] and nonlinear elasticity [2]. Therefore, during the last two decades, singular equations have attracted many researchers, and many important results have been proved in the literature; see, for example, [310]. Recently, it has been found that a particular case of (1.1), the Ermakov-Pinney equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ3_HTML.gif
(1.3)

plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [1113].

In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach [1416], and the second one is topological methods. Because we mainly focus on the applications of topological methods to singular equations in this paper, here we try to give a brief sketch of this problem. As far as the authors know, this method was started with the pioneering paper of Lazer and Solimini [17]. They proved that a necessary and sufficient condition for the existence of a positive periodic solution for equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ4_HTML.gif
(1.4)

is that the mean value of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq10_HTML.gif is negative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq11_HTML.gif , here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq12_HTML.gif , which is a strong force condition in a terminology first introduced by Gordon [18]. Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq13_HTML.gif , which corresponds to a weak force condition, they found examples of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq14_HTML.gif with negative mean values and such that periodic solutions do not exist. Since then, the strong force condition became standard in the related works; see, for instance, [2, 810, 13, 1921], and the recent review [22]. With a strong singularity, the energy near the origin becomes infinity and this fact is helpful for obtaining the a priori bounds needed for a classical application of the degree theory. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity by topological methods is more recent but has also attracted many researchers [4, 6, 2328]. In [27], for the first time in this topic, Torres proved an existence result which is valid for a weak singularity whereas the validity of such results under a strong force assumption remains as an open problem. Among topological methods, the method of upper and lower solutions [6, 29, 30], degree theory [8, 20, 31], some fixed point theorems in cones for completely continuous operators [25, 3234], and Schauder's fixed point theorem [27, 35, 36] are the most relevant tools.

In this paper, we select several recent existence results for singular equation (1.1) via different topological tools. The remaining part of the paper is organized as follows. In Section 2, some preliminary results are given. In Section 3, we present the first existence result for (1.1) via a nonlinear alternative principle of Leray-Schauder. In Section 4, the second existence result is established by using a well-known fixed point theorem in cones. The condition imposed on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq15_HTML.gif in Sections 3 and 4 is that the Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq16_HTML.gif associated with the linear periodic equations is positive, and therefore the results cannot cover the critical case, for example, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq17_HTML.gif is a constant, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq20_HTML.gif is the first eigenvalue of the linear problem with Dirichlet conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq21_HTML.gif . Different from Sections 3 and 4, the results obtained in Section 5, which are established by Schauder's fixed point theorem, can cover the critical case because we only need that the Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq22_HTML.gif is nonnegative. All results in Sections 3–5 shed some lights on the differences between a strong singularity and a weak singularity.

To illustrate our results, in Sections 3–5, we have selected the following singular equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ5_HTML.gif
(1.5)

here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq25_HTML.gif is a given parameter. The corresponding results are also valid for the general case

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ6_HTML.gif
(1.6)

with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq26_HTML.gif . Some open problems for (1.5) or (1.6) are posed.

In this paper, we will use the following notation. Given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq27_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq28_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq29_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq30_HTML.gif and it is positive in a set of positive measure. For a given function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq31_HTML.gif essentially bounded, we denote the essential supremum and infimum of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq32_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq34_HTML.gif , respectively.

2. Preliminaries

Consider the linear equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ7_HTML.gif
(2.1)
with periodic boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ8_HTML.gif
(2.2)

In Sections 3 and 4, we assume that

(A)the Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq35_HTML.gif associated with (2.1)–(2.2), is positive for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq36_HTML.gif .

In Section 5, we assume that

(B)the Green function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq37_HTML.gif associated with (2.1)–(2.2), is nonnegative for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq38_HTML.gif

When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq39_HTML.gif condition (A) is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq40_HTML.gif and condition (B) is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq41_HTML.gif . In this case, we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ9_HTML.gif
(2.3)
For a nonconstant function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq42_HTML.gif , there is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq43_HTML.gif -criterion proved in [37], which is given in the following lemma for the sake of completeness. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq44_HTML.gif denote the best Sobolev constant in the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ10_HTML.gif
(2.4)

The explicit formula for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq45_HTML.gif is

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ11_HTML.gif
(2.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq46_HTML.gif is the Gamma function; see [21, 38]

Lemma 2.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq48_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq49_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ12_HTML.gif
(2.6)
then the condition (A) holds. Moreover, condition (B) holds if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ13_HTML.gif
(2.7)
When the hypothesis (A) is satisfied, we denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ14_HTML.gif
(2.8)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq51_HTML.gif .

Throughout this paper, we define the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq52_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ15_HTML.gif
(2.9)
which corresponds to the unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq53_HTML.gif -periodic solution of
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ16_HTML.gif
(2.10)

3. Existence Result (I)

In this section, we state and prove the first existence result for (1.1). The proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in [39]. This part can be regarded as the scalar version of the results in [4].

Lemma 3.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq54_HTML.gif is a relatively compact subset of a convex set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq55_HTML.gif in a normed space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq56_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq57_HTML.gif be a compact map with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq58_HTML.gif . Then one of the following two conclusions holds:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq59_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq60_HTML.gif

(b)thereexist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq62_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq63_HTML.gif

Theorem 3.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq64_HTML.gif satisfies (A) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq65_HTML.gif satisfies the following.

(H1)There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq67_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ17_HTML.gif
(3.1)

(H2)There exist continuous, nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq69_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ18_HTML.gif
(3.2)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq70_HTML.gif is nonincreasing and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq71_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq72_HTML.gif .

(H3)There exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq73_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq74_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ19_HTML.gif
(3.3)

Then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq75_HTML.gif , (1.1) has at least one positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq76_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq77_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq79_HTML.gif .

Proof.

The existence is proved using the Leray-Schauder alternative principle, together with a truncation technique. The idea is that we show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ20_HTML.gif
(3.4)
has a positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq80_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq81_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq83_HTML.gif If this is true, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq84_HTML.gif will be a positive periodic solution of (1.1) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq85_HTML.gif since
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ21_HTML.gif
(3.5)

Since ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq86_HTML.gif ) holds, we can choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq87_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq88_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ22_HTML.gif
(3.6)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq89_HTML.gif . Consider the family of equations
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ23_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq90_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ24_HTML.gif
(3.8)
Problem (3.7) is equivalent to the following fixed point problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ25_HTML.gif
(3.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq91_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ26_HTML.gif
(3.10)

We claim that any fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq92_HTML.gif of (3.9) for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq93_HTML.gif must satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq94_HTML.gif . Otherwise, assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq95_HTML.gif is a fixed point of (3.9) for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq96_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq97_HTML.gif . Note that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ27_HTML.gif
(3.11)
By the choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq99_HTML.gif . Hence, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq100_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ28_HTML.gif
(3.12)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ29_HTML.gif
(3.13)
Thus we have from condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq101_HTML.gif ), for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq102_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ30_HTML.gif
(3.14)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ31_HTML.gif
(3.15)

This is a contradiction to the choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq103_HTML.gif and the claim is proved.

From this claim, the Leray-Schauder alternative principle guarantees that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ32_HTML.gif
(3.16)
has a fixed point, denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq104_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq105_HTML.gif , that is, equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ33_HTML.gif
(3.17)

has a periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq106_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq107_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq108_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq110_HTML.gif is actually a positive periodic solution of (3.17).

In the next lemma, we will show that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq111_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ34_HTML.gif
(3.18)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq112_HTML.gif large enough.

In order to pass the solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq113_HTML.gif of the truncation equations (3.17) to that of the original equation (3.4), we need the following fact:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ35_HTML.gif
(3.19)
for some constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq114_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq115_HTML.gif . To this end, by the periodic boundary conditions, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq116_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq117_HTML.gif . Integrating (3.17) from 0 to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq118_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ36_HTML.gif
(3.20)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ37_HTML.gif
(3.21)

The fact https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq119_HTML.gif and (3.19) show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq120_HTML.gif is a bounded and equicontinuous family on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq121_HTML.gif . Now the Arzela-Ascoli Theorem guarantees that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq122_HTML.gif has a subsequence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq123_HTML.gif , converging uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq124_HTML.gif to a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq125_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq126_HTML.gif satisfies the integral equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ38_HTML.gif
(3.22)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq127_HTML.gif , we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ39_HTML.gif
(3.23)

where the uniform continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq128_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq129_HTML.gif is used. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq130_HTML.gif is a positive periodic solution of (3.4).

Lemma 3.3.

There exist a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq131_HTML.gif and an integer https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq132_HTML.gif such that any solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq133_HTML.gif of (3.17) satisfies (3.18) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq134_HTML.gif .

Proof.

The lower bound in (3.18) is established using the strong force condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq135_HTML.gif ) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq136_HTML.gif . By condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq137_HTML.gif ), there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq138_HTML.gif small enough such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ40_HTML.gif
(3.24)

Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq139_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq140_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq141_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq142_HTML.gif , let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ41_HTML.gif
(3.25)

We claim first that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq143_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq144_HTML.gif . Otherwise, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq145_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq146_HTML.gif . Then from (3.24), it is easy to verify

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ42_HTML.gif
(3.26)
Integrating (3.17) from 0 to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq147_HTML.gif , we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ43_HTML.gif
(3.27)

This is a contradiction. Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq148_HTML.gif .

Now we consider the minimum values https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq149_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq150_HTML.gif . Without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq151_HTML.gif , otherwise we have (3.18). In this case,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ44_HTML.gif
(3.28)
for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq152_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq153_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq154_HTML.gif (without loss of generality, we assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq155_HTML.gif ) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq157_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq158_HTML.gif By (3.24), it can be checked that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ45_HTML.gif
(3.29)

Thus for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq159_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq160_HTML.gif As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq161_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq162_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq163_HTML.gif and the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq164_HTML.gif is strictly increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq165_HTML.gif . We use https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq166_HTML.gif to denote the inverse function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq167_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq168_HTML.gif .

In order to prove (3.18) in this case, we first show that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq169_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ46_HTML.gif
(3.30)

Otherwise, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq170_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq171_HTML.gif . Then there would exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq172_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq173_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ47_HTML.gif
(3.31)
Multiplying (3.17) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq174_HTML.gif and integrating from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq175_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq176_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ48_HTML.gif
(3.32)
By the facts https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq178_HTML.gif one can easily obtain that the right side of the above equality is bounded. As a consequence, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq179_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ49_HTML.gif
(3.33)

On the other hand, by the strong force condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq180_HTML.gif ), we can choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq181_HTML.gif large enough such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ50_HTML.gif
(3.34)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq182_HTML.gif . So (3.30) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq183_HTML.gif

Finally, multiplying (3.17) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq184_HTML.gif and integrating from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq185_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq186_HTML.gif , we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ51_HTML.gif
(3.35)
(We notice that the estimate (3.30) is used in the second equality above). In the same way, one may readily prove that the right-hand side of the above equality is bounded. On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq187_HTML.gif by ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq188_HTML.gif ),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ52_HTML.gif
(3.36)

if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq189_HTML.gif Thus we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq190_HTML.gif for some constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq191_HTML.gif .

From the proof of Theorem 3.2 and Lemma 3.3, we see that the strong force condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq192_HTML.gif ) is only used when we prove (3.18). From the next theorem, we will show that, for the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq193_HTML.gif , we can remove the strong force condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq194_HTML.gif ), and replace it by one weak force condition.

Theorem 3.4.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq195_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq196_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq197_HTML.gif ) are satisfied. Suppose further that

(H4)for each constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq198_HTML.gif , there exists a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq199_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq200_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq201_HTML.gif .

Then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq202_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq203_HTML.gif (1.1) has at least one positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq204_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq205_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq207_HTML.gif .

Proof.

We only need to show that (3.18) is also satisfied under condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq208_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq209_HTML.gif The rest parts of the proof are in the same line of Theorem 3.2. Since ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq210_HTML.gif ) holds, there exists a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq211_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq212_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq213_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq214_HTML.gif be the unique periodic solution to the problems (2.1)–(2.2) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq215_HTML.gif . That is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ53_HTML.gif
(3.37)
Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ54_HTML.gif
(3.38)
here
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ55_HTML.gif
(3.39)

Corollary 3.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq216_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq217_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq218_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq219_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq220_HTML.gif . Then

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq221_HTML.gif then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq222_HTML.gif (1.5) has at least one positive periodic solution for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq223_HTML.gif ;

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq224_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq225_HTML.gif (1.5) has at least one positive periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq226_HTML.gif here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq227_HTML.gif is some positive constant.

(iii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq228_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq229_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq230_HTML.gif (1.5) has at least one positive periodic solution for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq231_HTML.gif ;

(iv)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq232_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq233_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq234_HTML.gif (1.5) has at least one positive periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq235_HTML.gif .

Proof.

We apply Theorems 3.2 and 3.4. Take
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ56_HTML.gif
(3.40)
then ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq236_HTML.gif ) is satisfied, and the existence condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq237_HTML.gif ) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ57_HTML.gif
(3.41)
for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq238_HTML.gif . Note that condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq239_HTML.gif ) is satisfied when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq240_HTML.gif , while ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq241_HTML.gif ) is satisfied when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq242_HTML.gif . So (1.5) has at least one positive periodic solution for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ58_HTML.gif
(3.42)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq243_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq245_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq246_HTML.gif . Thus we have (i)–(iv).

4. Existence Result (II)

In this section, we establish the second existence result for (1.1) using a well-known fixed point theorem in cones. We are mainly interested in the superlinear case. This part is essentially extracted from [24].

First we recall this fixed point theorem in cones, which can be found in [40]. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq247_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq248_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq249_HTML.gif is a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq250_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq252_HTML.gif

Theorem 4.1 (see [40]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq253_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq254_HTML.gif a cone in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq255_HTML.gif . Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq256_HTML.gif are open bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq257_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq258_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ59_HTML.gif
(4.1)

be a completely continuous operator such that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq259_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq260_HTML.gif

(b)There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq261_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq262_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq263_HTML.gif

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq264_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq265_HTML.gif

In applications below, we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq266_HTML.gif with the supremum norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq267_HTML.gif and define

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ60_HTML.gif
(4.2)

Theorem 4.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq268_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq269_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq270_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq271_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq272_HTML.gif ). Furthermore, assume that

(H5)there exist continuous nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq273_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq274_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ61_HTML.gif
(4.3)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq275_HTML.gif is nonincreasing and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq276_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq277_HTML.gif

(H6)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq278_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq279_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ62_HTML.gif
(4.4)

Then (1.1) has one positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq280_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq281_HTML.gif .

Proof.

As in the proof of Theorem 3.2, we only need to show that (3.4) has a positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq282_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq284_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq285_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq286_HTML.gif defined by (4.2). Define the open sets

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ63_HTML.gif
(4.5)
and the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq287_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ64_HTML.gif
(4.6)

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq288_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq289_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq290_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq291_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq292_HTML.gif is continuous, then the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq293_HTML.gif is well defined and is continuous and completely continuous. Next we claim that:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq294_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq295_HTML.gif and

(ii)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq296_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq297_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq298_HTML.gif

We start with (i). In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq299_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq301_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq302_HTML.gif Thus we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ65_HTML.gif
(4.7)

Next we consider (ii). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq303_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq304_HTML.gif Next, suppose that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq306_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq307_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq308_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq309_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq310_HTML.gif As a result, it follows from ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq311_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq312_HTML.gif ) that, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq313_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ66_HTML.gif
(4.8)

Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq314_HTML.gif this is a contradiction and we prove the claim.

Now Theorem 4.1 guarantees that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq315_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq316_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq317_HTML.gif Note https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq318_HTML.gif by (4.7).

Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.

Theorem 4.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq319_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq320_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq321_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq322_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq323_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq324_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq325_HTML.gif ). Then (1.1) has two different positive periodic solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq326_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq327_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq328_HTML.gif .

Theorem 4.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq329_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq330_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq331_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq332_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq333_HTML.gif ). Then (1.1) has two different positive periodic solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq334_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq335_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq336_HTML.gif .

Corollary 4.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq337_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq338_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq339_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq340_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq341_HTML.gif . Then

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq342_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq343_HTML.gif (1.5) has at least two positive periodic solutions for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq344_HTML.gif ;

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq345_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq346_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq347_HTML.gif (1.5) has at least two positive periodic solutions for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq348_HTML.gif .

Proof.

Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq349_HTML.gif Then ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq350_HTML.gif ) is satisfied and the existence condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq351_HTML.gif ) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ67_HTML.gif
(4.9)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq352_HTML.gif , it is easy to see that the right-hand side goes to 0 as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq353_HTML.gif . Thus, for any given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq354_HTML.gif , it is always possible to find such https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq355_HTML.gif that (4.9) is satisfied. Thus, (1.5) has an additional positive periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq356_HTML.gif .

5. Existence Result (III)

In this section, we prove the third existence result for (1.1) by Schauder's fixed point theorem. We can cover the critical case because we assume that the condition (B) is satisfied. This part comes essentially from [35], and the results for the vector version can be found in [4].

Theorem 5.1.

Assume that conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq357_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq358_HTML.gif ), ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq359_HTML.gif ) are satisfied. Furthermore, suppose that

(H7)there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq360_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq361_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq362_HTML.gif here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq363_HTML.gif

Then (1.1) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq364_HTML.gif -periodic solution.

Proof.

A https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq365_HTML.gif -periodic solution of (1.1) is just a fixed point of the map https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq366_HTML.gif defined by (4.6). Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq367_HTML.gif is a completely continuous map.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq368_HTML.gif be the positive constant satisfying ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq369_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq370_HTML.gif Then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq371_HTML.gif . Now we define the set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ68_HTML.gif
(5.1)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq372_HTML.gif is a closed convex set. Next we prove https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq373_HTML.gif

In fact, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq374_HTML.gif , using that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq375_HTML.gif and condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq376_HTML.gif ),

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ69_HTML.gif
(5.2)
On the other hand, by conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq377_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq378_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ70_HTML.gif
(5.3)

In conclusion, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq379_HTML.gif . By a direct application of Schauder's fixed point theorem, the proof is finished.

As an application of Theorem 5.1, we consider the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq380_HTML.gif . The following corollary is a direct result of Theorem 5.1.

Corollary 5.2.

Assume that conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq381_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq382_HTML.gif ), ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq383_HTML.gif ) are satisfied. Furthermore, assume that

(H8)there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq384_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq385_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ71_HTML.gif
(5.4)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq386_HTML.gif then (1.1) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq387_HTML.gif -periodic solution.

Corollary 5.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq388_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq389_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq390_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq391_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq392_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq393_HTML.gif one hasthe following:

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq394_HTML.gif then (1.5) has at least one positive periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq395_HTML.gif .

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq396_HTML.gif then (1.5) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq397_HTML.gif -periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq398_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq399_HTML.gif is some positive constant.

Proof.

We apply Corollary 3.5 and follow the same notation as in the proof of Corollary 3.5. Then ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq400_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq401_HTML.gif ) are satisfied, and the existence condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq402_HTML.gif ) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ72_HTML.gif
(5.5)
for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq403_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq404_HTML.gif . Note that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ73_HTML.gif
(5.6)
Therefore, (5.5) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ74_HTML.gif
(5.7)

for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq405_HTML.gif .

So (1.5) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq406_HTML.gif -periodic solution for

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ75_HTML.gif
(5.8)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq407_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq408_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq409_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq410_HTML.gif . We have the desired results (i) and (ii).

Remark 5.4.

The validity of (ii) in Corollary 5.3 under strong force conditions remains still open to us. Such an open problem has been partially solved by Corollary 3.5. However, we do not solve it completely because we need the positivity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq411_HTML.gif in Corollary 3.5, and therefore it is not applicable to the critical case. The validity for the critical case remains open to the authors.

The next results explore the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq412_HTML.gif .

Theorem 5.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq413_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq414_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq415_HTML.gif satisfies condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq416_HTML.gif ). Furthermore, assume that

(H9)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq417_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ76_HTML.gif
(5.9)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq418_HTML.gif then (1.1) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq419_HTML.gif -periodic solution.

Proof.

We follow the same strategy and notation as in the proof of Theorem 5.1. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq420_HTML.gif be the positive constant satisfying ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq421_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq422_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq423_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq424_HTML.gif . Next we prove https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq425_HTML.gif

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq426_HTML.gif , by the nonnegative sign of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq427_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq428_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ77_HTML.gif
(5.10)
On the other hand, by ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq429_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq430_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ78_HTML.gif
(5.11)

In conclusion, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq431_HTML.gif and the proof is finished by Schauder's fixed point theorem.

Corollary 5.6.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq432_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq433_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq434_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq435_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq436_HTML.gif , one has the following:

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq437_HTML.gif then (1.5) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq438_HTML.gif -periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq439_HTML.gif

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq440_HTML.gif , then (1.5) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq441_HTML.gif -periodic solution for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq442_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq443_HTML.gif is some positive constant.

Proof.

We apply Theorem 5.5 and follow the same notation as in the proof of Corollary 3.5. Then ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq444_HTML.gif ) is satisfied, and the existence condition ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq445_HTML.gif ) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ79_HTML.gif
(5.12)
for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq446_HTML.gif . So (1.5) has at least one positive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq447_HTML.gif -periodic solution for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ80_HTML.gif
(5.13)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq448_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq449_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq450_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq451_HTML.gif . We have the desired results (i) and (ii).

Declarations

Acknowledgments

The authors express their thanks to the referees for their valuable comments and suggestions. The research of J. Chu is supported by the National Natural Science Foundation of China (Grant no. 10801044) and Jiangsu Natural Science Foundation (Grant no. BK2008356). The research of J. J. Nieto is partially supported by Ministerio de Education y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, Hohai University
(2)
Department of Mathematics, Pusan National University
(3)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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© J. Chu and J. J. Nieto. 2009

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