1. Introduction

In this paper, we are concerned with the existence and uniqueness of periodic solutions for the first-order functional differential equation (cf., e.g., [1–5]) where we will assume that and are continuous -periodic functions, that , that and -periodic with respect to the first variable, and that for .

Functional differential equations with periodic delays such as those stated above appear in a number of ecological, economical, control and physiological, and other models. One important question is whether these equations can support periodic solutions, and whether they are unique. The existence question has been studied extensively by many authors (see, e.g., [1–5]). The uniqueness problem seems to be more difficult, and less studies are known.

We will tackle the existence and uniqueness question by fixed point theorems for mixed monotone operators. We choose this approach because such fixed point methods, besides providing the usual existence and uniqueness results, sometimes may also provide additional numerical schemes for the computation of solutions.

We first recall some useful terminologies (see [6, 7]). Let be a real Banach space with zero element . A nonempty closed convex set is called a cone if it satisfies the following two conditions: (i) and (ii) and

Every cone induces an ordering in given by if and only if A cone is called normal if there is such that and . is said to be solid if the interior of is nonempty.

Assume that and The set is denoted by Assume that Let . Obviously if is a solid cone and then

Definition 1.1. Let be an ordered Banach space, and let . An operator is called mixed monotone on if and for any that satisfy and .Also, is called a fixed point of if

A function is said to be convex in if for any and any . We say that the function is a concave function if is a convex function.

Definition 1.2. Assume and Then, is said to be an -concave or -convex function if or, respectively, for and

Definition 1.3. Let , and let The operator is called (-concave)-(-convex) if there exist functions and such that() for and ,() for any and ,() for any and .
Assume that and Recall that a function is said to be left lower semicontinuous at if for any monotonically increasing sequence that converges to
The proof of the following theorem can be found in [7].

Theorem 1.4. Let be a normal cone of Let such that , and let be a mixed monotone operator. If is a (-concave)-(-convex) operator and satisfies the following three conditions: (A1) there exists such that ;(A2) and (A3) there exists such that for each , and is left lower semicontinuous at any ,then has a unique fixed point , that is, , and for any if we set and for then and

Remark 1.5. Condition (A3) in Theorem 1.4 can be replaced by (A3') is monotone in and left lower semicontinuous at any

2. Main Results

A real -periodic continuous function is said to be a -periodic solution of (1.1) if substitution of it into (1.1) yields an identity for all

It is well known (see, e.g., [1, 2]) that (1.1) has a -periodic solution if, and only if, is a -periodic solution of the equation where and (1.2) has a -periodic solution if, and only if, is a -periodic solution of the equation where Furthermore, the Cauchy function satisfies

Now let be the Banach space of all real -periodic continuous functions endowed with the usual linear structure as well as the norm Then is a normal cone of

Definition 2.1. The functions are said to form a pair of lower and upper quasisolutions of (1.1) if and as well as
We remark that the term quasi is used in the above definition to remind us that they are different from the traditional concept of lower and upper solutions (cf. (2.7) with ).
Let be defined by
We need two basic assumptions in the main results:() for any is an increasing function of , and is a decreasing function of () there exist such that and form a respective pair of lower and upper quasisolutions for (1.1).

Theorem 2.2. Suppose that conditions () and () hold, and (C1) for any is an -concave function, is a convex function;(C2) there exist such that Then (1.1) has a unique solution and for any if we set and then and

Proof. The mapping is a mixed monotone operator in view of (B1). Let Then Set . Then Next, we will prove that Suppose to the contrary that there exists such that Then which is a contradiction since Thus Similarly, we can prove Then we have From condition (C2), we know that Since we must have
We will prove that is a (-concave)-(-convex) operator, where In fact, for any , and we have thus so that Further we can prove for any and Indeed, since hence, we only need to prove From we know that for any therefore On the other hand, the function satisfies and From we have Then for Thus that is, Therefore, for any Finally, Therefore, is a (-concave)-(-convex) operator. From (2.20), is monotone in and is left lower semicontinuous at . By Theorem 1.4, we know that has a unique fixed point Hence (1.1) has a unique solution and for any if we set and then and The proof is complete.

Theorem 2.3. Suppose that conditions () and () hold, and (D1) there exist such that (D2) for any is an -concave function and for any and where satisfies the following conditions:() is monotone in and left lower semicontinuous in () for any Then (1.1) has a unique solution and for any if we set for then and

Proof. We assert that is a (-concave)-(-convex) mixed monotone operator, where In fact, for any and From (2.25), we know that Thus is a (-concave)-(-convex) mixed monotone operator. We may now complete our proof by Theorem 1.4.

Theorem 2.4. Suppose that conditions () and () hold, and (E1) for any is a concave function; for any and and satisfies the following conditions:() there exists such that () for any Then (1.1) has unique solution and for any if we set for then and

Proof. Set and for Then we know that From () we have Next we will prove that is a (-concave)-(-convex) operator, where In fact, for any and From (2.28), we know that Thus