Abstract
In this paper, the existence results of positive ωperiodic solutions are obtained for the secondorder functional differential equation
where
MSC: 34C25, 47H10.
Keywords:
functional differential equation; positive periodic solution; cone; fixed point index1 Introduction
In this paper, we discuss the existence of positive ωperiodic solutions of the secondorder functional differential equation with the delay terms of firstorder derivative in nonlinearity,
where
For the secondorder differential equation without delay and the firstorder derivative term in nonlinearity,
the existence problems of periodic solutions have attracted many authors’ attention and concern. Many theorems and methods of nonlinear functional analysis have been applied to research the periodic problems of Equation (2), such as the upper and lower solutions method and monotone iterative technique [14], the continuation method of topological degree [57], variational method and critical point theory [810], the theory of the fixed point index in cones [1116], etc.
In recent years, the existence of periodic solutions for the secondorder delayed differential equations have also been researched by many authors; see [1724] and the references therein. In some practice models, only positive periodic solutions are significant. In [20,21,23], the authors obtained the existence of positive periodic solutions for some delayed secondorder differential equations as a special form of the following equation:
by using Krasnoselskii’s fixed point theorem of cone mapping or the theory of the fixed point index in cones. In these works, the positivity of Green’s function of the corresponding linear secondorder periodic problems plays an important role. The positivity guarantees that the integral operators of the secondorder periodic problems are conepreserving in the cone
in the Banach space
The purpose of this paper is to discuss the existence of positive periodic solutions of Equation (1). We will use a different method to treat Equation (1). Our main results will be given in Section 3. Some preliminaries to discuss Equation (1) are presented in Section 2.
2 Preliminaries
Let
Generally,
Let
The ωperiodic solutions of Equation (5) are closely related to the linear secondorder boundary value problem
see [14]. It is easy to see that problem (6) has a unique solution which is explicitly given by
where
Lemma 2.1Let
Moreover,
Since
Define a set K in
It is easy to verify that K is a closed convex cone in
Lemma 2.2Let
Proof Let
and therefore,
Using (8), we obtain that
For every
we have
Hence,
Now we consider the nonlinear delay equation (1). Hereafter, we assume that the nonlinearity f satisfies the condition
(F0) There exists
Let
For every
Then
By the definition of the operator S, the positive ωperiodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. From assumption (F0), Lemma 2.1 and Lemma 2.2, we easily see that
Lemma 2.3
We will find the nonzero fixed point of A by using the fixed point index theory in cones. We recall some concepts and conclusions
on the fixed point index in [25,26]. Let E be a Banach space and
Lemma 2.4 ([26])
Let Ω be a bounded open subset ofEwith
Lemma 2.5 ([26])
Let Ω be a bounded open subset ofEand
In the next section, we will use Lemma 2.4 and Lemma 2.5 to discuss the existence of positive ωperiodic solutions of Equation (1).
3 Main results
We consider the existence of positive ωperiodic solutions of the functional differential equation (1). Let
Our main results are as follows.
Theorem 3.1Let
(F1)
then Equation (1) has at least one positiveωperiodic solution.
Theorem 3.2Let
(F2)
then Equation (1) has at least one positiveωperiodic solution.
In Theorem 3.1, the condition (F1) allows
satisfies (F0) with
In Theorem 3.2, the condition (F2) allows
satisfies (F0) with
Proof of Theorem 3.1 Choose the working space
We show that the operator A has a fixed point in
By
Let
Since
Hence, from (15) it follows that
By this, (16) and the definition of
Integrating both sides of this inequality from 0 to ω and using the periodicity of
Since
On the other hand, since
Choose
Since
By the latter inequality of (21), we have that
By (22) and the former inequality of (21), we have
From this, the latter inequality of (21) and (19), it follows that
By this inequality, (20) and the definition of
Integrating this inequality on I and using the periodicity of
Since
Now, by the additivity of fixed point index, (18) and (23), we have
Hence, A has a fixedpoint in
Proof of Theorem 3.2 Let
By
Let
Since
By this, (25) and the definition of
Integrating this inequality on
Since
Since
Choosing
Since
From this, the second inequality of (21) and (27), it follows that
By this and (28), we have
Integrating this inequality on
Since
Now, from (26) and (29), it follows that
Hence, A has a fixedpoint in
Example 1 Consider the following secondorder differential equation with delay:
where
satisfies the conditions (F0) and (F1) for
Example 2 Consider the functional differential equation
where
satisfies the conditions (F0) and (F2) for
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YL carried out the main part of this article. All authors read and approved the final manuscript.
Acknowledgements
The research was supported by the NNSFs of China (11261053, 11061031).
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