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:
where is a positive parameter, and the nonlinearity . In particular, the nonlinearity may change sign and have a repulsive singularity at , which means that
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
where , is the Green function of (2.1), associated with (1.3), and we will prove its positiveness. Throughout this paper, we assume that the following condition is satisfied:
(H) , are continuous functions and for all .
Lemma 2.1Let be a continuous function. Then, for any nonnegative continuous function defined on , the integral equation
has a unique solution , which is continuous on and satisfies the following inequality:
Proof We solve equation (2.2) by the method of successive approximations. Let
One can easily verify that
which implies that the series converges uniformly with respect to . Obviously, is a continuous solution of (2.2). Moreover, inequality (2.3) holds because and are nonnegative functions.
Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is unique on with . Then the uniqueness of the solution on is direct using the continuation property.
On the contrary, suppose that (2.2) has two solutions and on . Then, for , we have
Hence it follows that for all . □
Let us denote by and the solutions of (1.2) satisfying the initial conditions
Lemma 2.2 and satisfy the following integral equations:
Proof Since is a solution of (1.2), we have
Integrating (2.6) from 0 to t and noticing , we obtain
To obtain (2.4), we only need to integrate the above equality from 0 to t and notice . In a similar way, we can prove (2.5). □
Lemma 2.3For the solution of boundary value problem (2.1)-(1.3), the formula
is the Green function, the numberDis defined by .
Proof It is easy to see that the general solution of equation (2.1) has the form
where and are arbitrary constants. Substituting this expression for in boundary condition (1.3), we can obtain
After not very complicated calculations, we can get (2.7) and (2.8). □
Remark 2.4 As a direct application of Lemma 2.3, if , , then the Green function of boundary value problem (2.1)-(1.3) has the form
Lemma 2.5Assume that (H) holds. Then the Green function associated with (2.1)-(1.3) is positive for all .
Proof Since , it is enough to prove that for . Recall that and satisfy integral equations (2.4) and (2.5). By condition (H) and Lemma 2.1, it follows that
Now from (2.9) we get . Setting
for , we have
Evidently, , for holds. Let us now show that
To prove (2.10), we note that for fixed ,
Hence it follows that for all , we have
Using Lemma 2.1, we get from (2.12) that for all .
Next, we prove (2.11), note that for fixed ,
Hence it follows that, for all , we have
Again using Lemma 2.1, we get from (2.13) that if , and the proof is completed. □
Under hypothesis (H), we always define
Thus and . When , , and a direct calculation shows that
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
(i) for all and all ,
(ii) implies .
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 .
LetXbe a Banach space andK (⊂X) be a cone. Assume that , are open subsets ofXwith , , and let
be a completely continuous operator such that either
(i) , and , ; or
(ii) , and , .
Thenhas a fixed point in .
Let and define
whereσis as in (2.14).
One may readily verify thatKis a cone inX. Now, suppose that is a continuous function. Define an operator
for and .
Lemma 3.3 is well defined.
Proof Let , then we have
This implies that and the proof is completed. □
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
(H1) is continuous and there exists a constant such that
(H2) and uniformly .
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
has a positive solution x satisfying (1.3) and for . If it is right, then is a solution of (1.1) since
where is used.
Problem (3.1)-(1.3) is equivalent to the following fixed point of the operator equation:
where is a completely continuous operator defined by
Since , there exists such that
For , let and note that .
First we show
Let and . For any and , we can verify that
Then we have
This implies .
In view of the assumption
then there is such that
Hence, we have
Next, we show that
To see this, let , then
It follows from Lemma 3.2 that has a fixed point such that , which is a positive periodic solution of (3.1) for satisfying
So, equation (1.1) has a positive solution .
On the other hand, since
hence, there exists a positive number such that
problem (1.1)-(1.3) is equivalent to the following fixed point of the operator equation:
where is a continuous and completely continuous operator defined by
And for any , define
Furthermore, for any , we have
Thus, from the above inequality, there exists such that
Since , then there is a positive number such that
where γ satisfies .
If , then
It follows from Lemma 3.2 that has a fixed point such that , which is a positive periodic solution of (1.1) for satisfying
we can conclude that and are the desired distinct positive periodic solutions of (1.1) for . □
Example Let the nonlinearity in (1.1) be
where , , , . It is clear that satisfies conditions (H1), (H2). Then (1.1) has at least two positive T-periodic solutions for sufficiently small λ.
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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