Recent Existence Results for Second-Order Singular Periodic Differential Equations

  • Jifeng Chu1, 2Email author and

    Affiliated with

    • JuanJ Nieto3

      Affiliated with

      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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq2_HTML.gif are continuous and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq3_HTML.gif -periodic functions. The nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq4_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq6_HTML.gif -periodic in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq8_HTML.gif has a repulsive singularity at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq9_HTML.gif :
      http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq10_HTML.gif is negative, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq11_HTML.gif , here http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq17_HTML.gif is a constant, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq19_HTML.gif , and http://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 http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ5_HTML.gif
      (1.5)

      here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq24_HTML.gif and http://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

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

      with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq27_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq28_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq29_HTML.gif for a.e. http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq32_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq33_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ7_HTML.gif
      (2.1)
      with periodic boundary conditions
      http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq38_HTML.gif

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ9_HTML.gif
      (2.3)
      For a nonconstant function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq42_HTML.gif , there is an http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ10_HTML.gif
      (2.4)

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

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

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq48_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq49_HTML.gif . If
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ14_HTML.gif
      (2.8)

      Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq50_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq52_HTML.gif by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq53_HTML.gif -periodic solution of
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq55_HTML.gif in a normed space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq56_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq57_HTML.gif be a compact map with http://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) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq60_HTML.gif

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

      Theorem 3.2.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq64_HTML.gif satisfies (A) and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq67_HTML.gif such that

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq69_HTML.gif such that

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

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

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

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

      Then for each http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq76_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq77_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq78_HTML.gif and http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq80_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq81_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq82_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq85_HTML.gif since
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ21_HTML.gif
      (3.5)

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ22_HTML.gif
      (3.6)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq89_HTML.gif . Consider the family of equations
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ23_HTML.gif
      (3.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq90_HTML.gif and
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ25_HTML.gif
      (3.9)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq91_HTML.gif is defined by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq92_HTML.gif of (3.9) for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq93_HTML.gif must satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq94_HTML.gif . Otherwise, assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq96_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq97_HTML.gif . Note that

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq104_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq105_HTML.gif , that is, equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ33_HTML.gif
      (3.17)

      has a periodic solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq106_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq107_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq108_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq109_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq111_HTML.gif such that

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

      for http://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 http://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:

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

      The fact http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq119_HTML.gif and (3.19) show that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq122_HTML.gif has a subsequence, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq123_HTML.gif , converging uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq124_HTML.gif to a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq125_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq126_HTML.gif satisfies the integral equation

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ38_HTML.gif
      (3.22)
      Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq127_HTML.gif , we arrive at
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq128_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq129_HTML.gif is used. Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq131_HTML.gif and an integer http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq132_HTML.gif such that any solution http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq135_HTML.gif ) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq136_HTML.gif . By condition ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq137_HTML.gif ), there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq138_HTML.gif small enough such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ40_HTML.gif
      (3.24)

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

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

      We claim first that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq143_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq144_HTML.gif . Otherwise, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq145_HTML.gif for some http://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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq147_HTML.gif , we deduce that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq148_HTML.gif .

      Now we consider the minimum values http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq149_HTML.gif . Let http://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 http://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,

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

      Thus for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq159_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq160_HTML.gif As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq162_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq163_HTML.gif and the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq164_HTML.gif is strictly increasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq165_HTML.gif . We use http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq166_HTML.gif to denote the inverse function of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq167_HTML.gif restricted to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq169_HTML.gif ,

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

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ47_HTML.gif
      (3.31)
      Multiplying (3.17) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq174_HTML.gif and integrating from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq175_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq176_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ48_HTML.gif
      (3.32)
      By the facts http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq177_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq179_HTML.gif such that
      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq180_HTML.gif ), we can choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq181_HTML.gif large enough such that

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq187_HTML.gif by ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq188_HTML.gif ),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ52_HTML.gif
      (3.36)

      if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq189_HTML.gif Thus we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq190_HTML.gif for some constant http://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 ( http://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 http://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 ( http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq195_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq196_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq198_HTML.gif , there exists a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq199_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq200_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq201_HTML.gif .

      Then for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq202_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq204_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq205_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq206_HTML.gif and http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq208_HTML.gif ) and http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq210_HTML.gif ) holds, there exists a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq211_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq212_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq213_HTML.gif . Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq215_HTML.gif . That is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ53_HTML.gif
      (3.37)
      Then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ54_HTML.gif
      (3.38)
      here
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq216_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq217_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq218_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq219_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq220_HTML.gif . Then

      (i)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq221_HTML.gif then for each http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq223_HTML.gif ;

      (ii)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq224_HTML.gif , then for each http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq226_HTML.gif here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq227_HTML.gif is some positive constant.

      (iii)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq228_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq229_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq231_HTML.gif ;

      (iv)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq232_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq233_HTML.gif with http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ56_HTML.gif
      (3.40)
      then ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq236_HTML.gif ) is satisfied, and the existence condition ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq237_HTML.gif ) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ57_HTML.gif
      (3.41)
      for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq238_HTML.gif . Note that condition ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq239_HTML.gif ) is satisfied when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq240_HTML.gif , while ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq241_HTML.gif ) is satisfied when http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ58_HTML.gif
      (3.42)

      Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq243_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq245_HTML.gif if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq247_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq248_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq249_HTML.gif is a subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq250_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq252_HTML.gif

      Theorem 4.1 (see [40]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq253_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq254_HTML.gif a cone in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq255_HTML.gif . Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq256_HTML.gif are open bounded subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq257_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq258_HTML.gif Let
      http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq259_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq260_HTML.gif

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq268_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq269_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq270_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq271_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq273_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq274_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ61_HTML.gif
      (4.3)

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

      (H6)there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq278_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq279_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq280_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq282_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq283_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq284_HTML.gif

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq285_HTML.gif be a cone in http://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

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

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq288_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq289_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq290_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq291_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq292_HTML.gif is continuous, then the operator http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq294_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq295_HTML.gif and

      (ii)there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq296_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq297_HTML.gif and all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq299_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq301_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq302_HTML.gif Thus we have

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

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

      Hence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq315_HTML.gif has at least one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq316_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq317_HTML.gif Note http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq319_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq320_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq321_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq322_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq323_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq324_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq327_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq328_HTML.gif .

      Theorem 4.4.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq329_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq330_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq331_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq332_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq334_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq335_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq336_HTML.gif .

      Corollary 4.5.

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

      (i)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq342_HTML.gif , then for each http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq344_HTML.gif ;

      (ii)if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq345_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq346_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq348_HTML.gif .

      Proof.

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

      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq353_HTML.gif . Thus, for any given http://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 http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq357_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq358_HTML.gif ), ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq360_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq361_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq362_HTML.gif here http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq364_HTML.gif -periodic solution.

      Proof.

      A http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq366_HTML.gif defined by (4.6). Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq367_HTML.gif is a completely continuous map.

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

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

      Obviously, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq373_HTML.gif

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

      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq377_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq378_HTML.gif ), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ70_HTML.gif
      (5.3)

      In conclusion, http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq381_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq382_HTML.gif ), ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq384_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq385_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ71_HTML.gif
      (5.4)

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

      (i)if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq395_HTML.gif .

      (ii)if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq397_HTML.gif -periodic solution for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq398_HTML.gif where http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq400_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq401_HTML.gif ) are satisfied, and the existence condition ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq402_HTML.gif ) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ72_HTML.gif
      (5.5)
      for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq403_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq404_HTML.gif . Note that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ73_HTML.gif
      (5.6)
      Therefore, (5.5) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ74_HTML.gif
      (5.7)

      for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq406_HTML.gif -periodic solution for

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

      Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq407_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq408_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq409_HTML.gif if http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq412_HTML.gif .

      Theorem 5.5.

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

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

      If http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq420_HTML.gif be the positive constant satisfying ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq421_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq422_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq423_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq424_HTML.gif . Next we prove http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq425_HTML.gif

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

      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq429_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq430_HTML.gif ), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ78_HTML.gif
      (5.11)

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

      (i)if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq438_HTML.gif -periodic solution for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq439_HTML.gif

      (ii)if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq441_HTML.gif -periodic solution for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq442_HTML.gif where http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq444_HTML.gif ) is satisfied, and the existence condition ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq445_HTML.gif ) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ79_HTML.gif
      (5.12)
      for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq447_HTML.gif -periodic solution for
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_Equ80_HTML.gif
      (5.13)

      Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq448_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq449_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F540863/MediaObjects/13661_2009_Article_856_IEq450_HTML.gif if http://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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