Abstract
We study the existence and multiplicity of positive periodic solutions for secondorder 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 GuoKrasnosel’skii fixed point theorem on compression and expansion of cones.
MSC: 34B16, 34C25.
Keywords:
positive solutions; singular; GuoKrasnosel’skii fixed point theorem1 Introduction
In this paper, we study the existence and multiplicity of positive Tperiodic solutions for the following secondorder singular differential equation:
where
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
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 [17]. Some strong force conditions introduced by Gordon [8] 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 [1316] and a nonlinear LeraySchauder alternative principle [1719] 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 Tperiodic solutions of (1.1) using the GuoKrasnosel’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 [1416]. We remark that it is sufficient to prove that
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 onesigned property of the Green function implies that a maximum principle and an antimaximum 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 GuoKrasnosel’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 Tperiodic 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 Tperiodic solution which can be written as
where
(H)
Lemma 2.1Let
has a unique solution
Proof We solve equation (2.2) by the method of successive approximations. Let
Take
One can easily verify that
which implies that the series
Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is
unique on
On the contrary, suppose that (2.2) has two solutions
Hence it follows that
Let us denote by
Lemma 2.2
Proof Since
Integrating (2.6) from 0 to t and noticing
To obtain (2.4), we only need to integrate the above equality from 0 to t and notice
Lemma 2.3For the solution
holds, where
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
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
Lemma 2.5Assume that (H) holds. Then the Green function
Proof Since
Now from (2.9) we get
for
Evidently,
To prove (2.10), we note that for fixed
and
Hence it follows that for all
Using Lemma 2.1, we get from (2.12) that
Next, we prove (2.11), note that for fixed
and
Hence it follows that, for all
Again using Lemma 2.1, we get from (2.13) that
Under hypothesis (H), we always define
Thus
3 Main results
In this section, we state and prove the new existence results for (1.1). The proof is based on the following wellknown 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)
(ii)
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
Lemma 3.2[22]
LetXbe a Banach space andK (⊂X) be a cone. Assume that
be a completely continuous operator such that either
(i)
(ii)
Thenhas a fixed point in
Let
whereσis as in (2.14).
One may readily verify thatKis a cone inX. Now, suppose that
for
Lemma 3.3
Proof Let
This implies that
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
(H_{1})
(H_{2})
Then (1.1) has at least two positiveTperiodic 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
where
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
For
First we show
Let
Then we have
This implies
In view of the assumption
then there is
Hence, we have
Next, we show that
To see this, let
It follows from Lemma 3.2 that has a fixed point
So, equation (1.1) has a positive solution
On the other hand, since
hence, there exists a positive number
problem (1.1)(1.3) is equivalent to the following fixed point of the operator equation:
where
And for any
Furthermore, for any
Thus, from the above inequality, there exists
Since
where γ satisfies
If
It follows from Lemma 3.2 that
Noting that
we can conclude that
Example Let the nonlinearity in (1.1) be
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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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