We study the existence and multiplicity of positive periodic solutions for second-order nonlinear damped differential equations by combing the analysis of positiveness of the Green function for a linear damped equation. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones.
MSC: 34B16, 34C25.
Keywords:positive solutions; singular; Guo-Krasnosel’skii fixed point theorem
In this paper, we study the existence and multiplicity of positive T-periodic solutions for the following second-order singular differential equation:
Equation (1.1) is a particular case of a more general class of Sturm equations of the type
where p is a strictly positive absolutely continuous function. Such equations, even in the case , where they are referred to as being of Schrödinger or Klein-Gordon type, appear in many scientific areas including quantum field theory, gas dynamics, fluid mechanics and chemistry.
Electrostatic or gravitational forces are the most important examples of singular interactions. During the last few decades, the study of the existence of positive solutions for singular differential equations has deserved the attention of many researchers [1-7]. Some strong force conditions introduced by Gordon  are standard in the related earlier works [6,7,9]. Compared with the case of strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent, but has also attracted many researchers [2,3,10,11]. In particular, the degree theory [6,7], the method of upper and lower solutions [11,12], Schauder’s fixed point theorem [2,10], some fixed point theorems in cones for completely continuous operators [13-16] and a nonlinear Leray-Schauder alternative principle [17-19] are the most relevant tools.
However, singular differential equation (1.1), in which the nonlinearity is dependent on the derivative and does not require f to be nonnegative, has not attracted much attention in the literature. There are not so many existence results for (1.1) even when the nonlinearity is independent of the derivative. In this paper, we try to fill this gap and establish the existence of positive T-periodic solutions of (1.1) using the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones, which has been used to study positive solutions for systems of ordinary, functional differential equations [14-16]. We remark that it is sufficient to prove that is continuous and completely continuous in Lemma 3.2 (Section 3). This point is essential and advantageous.
As mentioned above, this paper is mainly motivated by the recent paper [14,16]. The aim of this paper is to study the multiplicity of positive solutions to (1.1). It is proved that such a problem has at least two positive solutions under reasonable conditions (see Theorem 3.5). And the remaining part of this paper is organized as follows. In Section 2, we find the Green function of the linear damped equation
subject to periodic boundary conditions
and prove its positiveness. The fact is very crucial to our arguments. Moreover, the one-signed property of the Green function implies that a maximum principle and an anti-maximum principle hold for the corresponding linear differential equations subject to various boundary conditions, which is an important topic in differential equations (see [20,21]). In Section 3, by employing the Guo-Krasnosel’skii fixed point theorem, we prove the existence of twin positive solutions for (1.1) under the positiveness of the Green function associated with (1.2)-(1.3). To illustrate the new results, some applications are also given.
2 The Green function and its positiveness
In this section, we consider the nonhomogeneous equation
We say that (1.2)-(1.3) is nonresonant if its unique T-periodic solution is the trivial one. When (1.2)-(1.3) is nonresonant, as a consequence of Fredholm’s alternative, equation (2.1) admits a unique T-periodic solution which can be written as
Proof We solve equation (2.2) by the method of successive approximations. Let
One can easily verify that
Proof It is easy to see that the general solution of equation (2.1) has the form
After not very complicated calculations, we can get (2.7) and (2.8). □
Under hypothesis (H), we always define
3 Main results
In this section, we state and prove the new existence results for (1.1). The proof is based on the following well-known fixed point theorem on compression and expansion of cones, which we state here for the convenience of the reader, after introducing the definition of a cone.
Definition 3.1 Let X be a Banach space and let K be a closed, nonempty subset of X. K is a cone if
We also recall that a compact operator means an operator which transforms every bounded set into a relatively compact set. Let us define the function and use to denote the usual -norm over , by we denote the supremum norm of .
be a completely continuous operator such that either
whereσis as in (2.14).
It is easy to prove.
Lemma 3.4is continuous and completely continuous.
Now we present our main result.
Theorem 3.5Suppose that (1.1) satisfies (H). Furthermore, assume that
Then (1.1) has at least two positiveT-periodic solutions for sufficiently smallλ.
Proof To show that (1.1) has a positive solution, we should only show that
Problem (3.1)-(1.3) is equivalent to the following fixed point of the operator equation:
where is a completely continuous operator defined by
First we show
Then we have
In view of the assumption
Hence, we have
Next, we show that
On the other hand, since
problem (1.1)-(1.3) is equivalent to the following fixed point of the operator equation:
Example Let the nonlinearity in (1.1) be
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11161017, No. 11301001, No. 11301139), Hainan Natural Science Foundation (Grant No.113001), Excellent Youth Scholars Foundation and the Natural Science Foundation of Anhui Province of PR China (No. 2013SQRL030ZD).
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