Abstract

The existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays , where is a ω-periodic function, is a continuous function, which is ω-periodic in , and are positive constants. Our discussion is based on the fixed point index theory in cones.

1. Introduction and Main Results

In this paper, we discuss the existence of positive -periodic solutions of the second-order differential equation with delays where is a -periodic function, is a continuous function, which is -periodic in , and are positive constants.

In recent years, the existence of periodic solutions for second-order delay differential equations has been researched by many authors, see [1–8] and references therein. In some practice models, only positive periodic solutions are significant. In [4, 5, 7], the authors obtained the existence of positive periodic solutions for some delay second-order differential equations by using Krasnoselskii's fixed-point theorem of cone mapping. For the second-order differential equations without delay, the existence of positive periodic solutions has been discussed by more authors, see [9–14].

Motivated by the papers mentioned above, we research the existence of positive periodic solutions of (1.1) with multiple delays. We aim to obtain the essential conditions on the existence of positive periodic solutions of (1.1) via the theory of the fixed-point index in cones. The conditions concern with the relation of the coefficient function and nonlinearity . Let Obviously, . Our main results are as follows.

Theorem 1.1. Let be a -periodic function, , and -periodic in . If satisfies the following conditions:(F1) there exist positive constants satisfying and such that for and ;(F2) there exist positive constants satisfying and such that for and ,
then (1.1) has at least one positive -periodic solution.

Theorem 1.2. Let be a -periodic function, , and -periodic in . If satisfies the following conditions:(F3) there exist positive constants satisfying and such that for and ;(F4) there exist positive constants satisfying and such that for and ,
then (1.1) has at least one positive -periodic solution.

In Theorem 1.1, the conditions (F1) and (F2) allow to be superlinear growth on . For example, satisfies (F1) and (F2), where are positive and continuous -periodic functions.

In Theorem 1.2, the conditions (F3) and (F4) allow to be sublinear growth on . For example, satisfies (F3) and (F4), where are positive and continuous -periodic functions.

Our results are different from those in the references mentioned above. The conditions (F1) and (F2) in Theorem 1.1 and the conditions (F3) and (F4) in Theorem 1.2 are optimal for the existence of positive periodic solutions of (1.1). This fact can been shown from the differential equation with linear delays where are positive constants and is a positive -periodic function. If satisfy Equation (1.9) has no positive -periodic solutions. In fact, if (1.9) has a positive -periodic solution, integrating the equation on and using the periodicity of , we can obtain that , which contradicts to the positivity of . Hence, (1.9) has no positive -periodic solution. For and , if Condition (1.10) holds, the conditions (F1) and (F2) in Theorem 1.1 and the conditions (F3) and (F4) in Theorem 1.2 have just not been satisfied. From this, we see that the conditions in Theorems 1.1–1.2 are optimal.

The proofs of Theorems 1.1–1.2 are based on the fixed point index theory in cones, which will be given in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

Let denote the Banach space of all continuous -periodic function with norm . Let be the cone of all nonnegative functions in . Generally, denotes the th-order continuous differentiable -periodic function space for .

Let be the positive constant defined by (1.2). For , we consider the linear second-order differential equation The -periodic solutions of (2.1) are can been expressed by the solution of the linear second-order boundary value problem see [11]. Problem (2.2) has a unique solution, which is explicitly given by where .

By a direct calculation, we easily prove the following lemma.

Lemma 2.1. Let . For every , the linear equation (2.1) has a unique -periodic solution , which is given by Moreover, is a completely continuous linear operator.

Since for every , if and , by (2.4) the -periodic solution of (2.1) is positive. Moreover, we can show that the -periodic solution has the following strong positivity: where , in which In fact, for and , from (2.4) it follows that and therefore, Using (2.4) and this inequality, we have that Hence, (2.5) holds.

Now we consider the periodic solution problem of the linear differential equation with variable coefficient

Lemma 2.2. Let be a positive -periodic function. For every , the linear equation (2.10) has a unique -periodic solution . Moreover, is a completely continuous linear operator and with strong positivity

Proof. Let and be the positive constants defined by (1.2). Then , . Let be the -periodic solution operator of (2.1) given by (2.4). We rewrite (2.10) to the form of Then it is easy to see that the -periodic solution problem of (2.10) is equivalent to the operator equation in Banach space where is the identity operator in and is the product operator defined by which is a positive linear bounded operator. We prove that the norm of in satisfies .
For every and , by the definition (2.4) of and the positivity of , we have Therefore, . By the arbitrariness of , we have .
Thus, has a bounded inverse operator given by the series with the norm estimate Consequently, (2.13), equivalently (2.10), has a unique -periodic solution where By the complete continuity of , is a completely continuous linear operator.
For every , by the expression (2.19) of , we have If , by the series expression of and the positivity of and , we have Hence, form (2.5) and (2.20), it follows that Namely, (2.11) holds.

Let . For every , set Then is continuous. Define a mapping by By the definition of operator , the -periodic solution of (1.1) is equivalent to the fixed point of . Choose a subcone of by From the strong positivity of in Lemma 2.2 and the definition of , we easily obtain the following lemma.

Lemma 2.3. , and is completely continuous.

Hence, the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . We will find the nonzero fixed point of by using the fixed point index theory in cones.

We recall some concepts and conclusions on the fixed point index in [15, 16]. Let be a Banach space and be a closed convex cone in . Assume is a bounded open subset of with boundary , and . Let be a completely continuous mapping. If for any , then the fixed point index has definition. One important fact is that if , then has a fixed point in . The following two lemmas are needed in our argument.

Lemma 2.4 (see [16]). Let be a bounded open subset of with and a completely continuous mapping. If for every and , then .

Lemma 2.5 (see [16]). Let be a bounded open subset of and a completely continuous mapping. If there exists an such that for every and , then .

In next section, we will use Lemmas 2.4 and 2.5 to prove Theorems 1.1 and 1.2.

3. Proofs of Main Results

Proof of Theorem 1.1. Choose the working space . Let be the closed convex cone in defined by (2.25) and the operator defined by (2.24). Then the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . Let and set We show that the operator has a fixed point in when is small enough and large enough.
Let , where is the positive constant in Condition (F1). We prove that 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 and Lemma 2.2, satisfies the delay differential equation Since , by the definitions of and , we have Hence from condition (F1), it follows that By this and (3.2), we get that Integrating both sides of this inequality from to and using the periodicity of , we have Hence, we obtain that By the definition of cone , . From (3.7), it follows that , which contradicts to the assumption in Condition (F1). Hence satisfies the condition of Lemma 2.4 in . By Lemma 2.4, we have On the other hand, choose , where is the positive constant in condition (F2), and let . Clearly, . We show that satisfies the condition of Lemma 2.5 in , namely, for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.2, satisfies the differential equation Since , by the definition of , we have From this and Condition (F2), it follows that By this inequality and (3.9), we have Integrating this inequality on and using the periodicity of , we obtain that Consequently, we have that Since , form this inequality it follows that , which contradicts to the assumption in Condition (F2). This means that satisfies the condition of Lemma 2.5 in . By Lemma 2.5, Now by the additivity of fixed point index, (3.8), and (3.15) we have Hence has a fixed point in , which is a positive -periodic solution of (1.1).