Abstract
In this paper, the existence results of positive ωperiodic solutions are obtained for the secondorder functional differential equation
where is a continuous function which is ωperiodic in t, is a ωperiodic function, . Our discussion is based on the fixed point index theory in cones.
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 is a continuous function which is ωperiodic in t and is a ωperiodic delay function, .
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 , where is a constant. Hence, the fixed point theorems of cone mapping can be applied to periodic problems of the secondorder delay equation (3) as well as Equation (2) (for Equation (2), see [1116]). However, few people consider the existence of positive periodic solutions of Equation (1). Since the nonlinearity of Equation (1) explicitly contains the delayed firstorder derivative term, the corresponding integral operator has no definition on the cone P. Thus, the argument methods used in [20,21,23] are not applicable to Equation (1).
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 denote the Banach space of all continuous ωperiodic function with the norm . Let be the Banach space of all continuous differentiable ωperiodic function with the norm
Generally, denotes the nthorder continuous differentiable ωperiodic function space for . Let be the cone of all nonnegative functions in .
Let be a constant. For , we consider the linear secondorder differential equation
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 . By [[14], Lemma 1], we have
Lemma 2.1Let. Then, for every, the linear equation (5) has a uniqueωperiodic solutionwhich is given by
Moreover, is a completely continuous linear operator.
Since , for every , by (8), if and , then the ωperiodic solution of Equation (5) for every , and we term it the positive ωperiodic solution. Let
It is easy to verify that K is a closed convex cone in .
Lemma 2.2Let. Then, for every, the positiveωperiodic solution of Equation (5) . Namely, .
Proof Let , . For every , from (8) it follows that
and therefore,
Using (8), we obtain that
we have
Now we consider the nonlinear delay equation (1). Hereafter, we assume that the nonlinearity f satisfies the condition
Let , then for , , , and Equation (1) is rewritten to
Then is continuous. We define an integral operator by
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 2.3andis completely continuous.
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 be a closed convex cone in E. Assume Ω is a bounded open subset of E with the boundary ∂Ω, and . Let be a completely continuous mapping. If for any , then the fixed point index has a definition. One important fact is that if , then A has a fixed point in . The following two lemmas are needed in our argument.
Lemma 2.4 ([26])
Let Ω be a bounded open subset ofEwithandbe a completely continuous mapping. Iffor everyand, then.
Lemma 2.5 ([26])
Let Ω be a bounded open subset ofEandbe a completely continuous mapping. If there exists ansuch thatfor everyand, then.
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 satisfy assumption (F0) and be ωperiodic in t. Let be the constant defined by (9) and . For convenience, we introduce the notations
Our main results are as follows.
Theorem 3.1Letandbeωperiodic int, . Iffsatisfies assumption (F0) and the condition
then Equation (1) has at least one positiveωperiodic solution.
Theorem 3.2Letandbeωperiodic int, . Iffsatisfies assumption (F0) and the conditions
then Equation (1) has at least one positiveωperiodic solution.
In Theorem 3.1, the condition (F1) allows to be superlinear growth on x and . For example,
satisfies (F0) with and (F1) with and .
In Theorem 3.2, the condition (F2) allows to be sublinear growth on x and . For example,
satisfies (F0) with and (F2) with and .
Proof of Theorem 3.1 Choose the working space . Let be the closed convex cone in defined by (10) and be the operator defined by (13). Then the positive ωperiodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. Let and set
We show that the operator A has a fixed point in when r is small enough and R is large enough.
By and the definition of , there exist and such that
Let . We now prove that A satisfies the condition of Lemma 2.4 in , namely for every and . In fact, if there exist and such that , then by the definition of A and Lemma 2.1, satisfies the delay differential equation
Since , by the definitions of K and , we have
Hence, from (15) it follows that
By this, (16) and the definition of , we have
Integrating both sides of this inequality from 0 to ω and using the periodicity of , we obtain that
Since , it follows that , which is a contradiction. Hence, A satisfies the condition of Lemma 2.4 in . By Lemma 2.4, we have
On the other hand, since , by the definition of , there exist and such that
Choose and let . Clearly, . We show that A satisfies the condition of Lemma 2.5 in , namely for every and . In fact, if there exist and such that , since , by the definition of A and Lemma 2.1, satisfies the differential equation
Since , by the definition of K, we have
By the latter inequality of (21), we have that . This implies that . Consequently,
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 , we have
Integrating this inequality on I and using the periodicity of , we get that
Since , from this inequality it follows that , which is a contradiction. This means that A satisfies the condition of Lemma 2.5 in . By Lemma 2.5,
Now, by the additivity of fixed point index, (18) and (23), we have
Hence, A has a fixedpoint in , which is a positive ωperiodic solution of Equation (1). □
Proof of Theorem 3.2 Let be defined by (14). We prove that the operator A defined by (13) has a fixed point in if r is small enough and R is large enough.
By and the definition of , there exist and such that
Let and . We prove that A satisfies the condition of Lemma 2.5 in , namely for every and . In fact, if there exist and such that , since , by the definition of A and Lemma 2.1, satisfies the delay differential equation
Since , by the definitions of K and , satisfies (17). From (17) and (24) it follows that
By this, (25) and the definition of , we have
Integrating this inequality on and using the periodicity of , we obtain that
Since , from this inequality it follows that , which is a contradiction. Hence, A satisfies the condition of Lemma 2.5 in . By Lemma 2.5, we have
Since , by the definition of , there exist and such that
Choosing , we show that A satisfies the condition of Lemma 2.4 in , namely for every and . In fact, if there exist and such that , then by the definition of A and Lemma 2.1, satisfies the differential equation
Since , by the definition of K, satisfies (21). From the second inequality of (21), it follows that (22) holds. By (22) and the first inequality of (21), we have
From this, the second inequality of (21) and (27), it follows that
By this and (28), we have
Integrating this inequality on and using the periodicity of , we obtain that
Since , from this inequality it follows that , which is a contradiction. This means that A satisfies the condition of Lemma 2.4 in . By Lemma 2.4,
Now, from (26) and (29), it follows that
Hence, A has a fixedpoint in , which is a positive ωperiodic solution of Equation (1). □
Example 1 Consider the following secondorder differential equation with delay:
where , . If and for , we can verify that
satisfies the conditions (F0) and (F1) for . By Theorem 3.1, the delay equation (30) has at least one positive ωperiodic solution.
Example 2 Consider the functional differential equation
where , , and . If and for . We easily see that
satisfies the conditions (F0) and (F2) for . By Theorem 3.2, the functional differential equation (31) has a positive ωperiodic solution.
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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