An Extension of Darbo’s Theorem and Its Application

Abstract

Here, some extensions of Darbo fixed point theorem associated with measures of noncompactness are proved. Then, as an application, our attention is focused on the existence of solutions of the integral equation ,  , ,  /,  ,   in the space of real functions defined and continuous on the interval .

1. Introduction

The concept of measure of noncompactness plays very important role in describing differential and integral equations. It was introduced by Kuratowski [1] as follows: for bounded subsets of a metric space . Darbo [2] used Kuratowski measure of noncompactness to generalize Shauder fixed point theorem to -set contractive operators, which satisfy the condition for some . Up to now, other measures of noncompactness have been defined. In recent years many papers have been devoted to the problem of existence of solutions of integral equations, using the technique of measures of noncompactness and Darbo fixed point theorem (cf. [3–6]). Recently, the technique of measure of noncompactness has been used to obtain some extensions of Darbo fixed point theorem and the obtained results have important applications [3, 4]. As some applications of the technique of measures of noncompactness and Darbo fixed point theorem, the following integral equations have been considered in [5, 6], respectively: Huang and Cao [7] have given a result to find the solution of the integral equation

In this paper, motivated and inspired by the integral equations (2), (3), and (4), we are going to prove a theorem on the existence of solutions of the integral equation in the Banach space . Note that (5) has a rather general form and extends the integral equation (2). Our aim will be realized with help of the technique of measure of noncompactness. In Section 2, we present some definitions and preliminary results about the concept of measure of noncompactness. In Section 3, using the existent contractive condition in [8, Theorem ] and the notion of shifting distance functions of [9], some generalizations of Darbo fixed point theorem are proved. In the last section, a result is proved concerning the existence of solutions for the integral equation (5).

2. Preliminaries

In this section, some definitions, notions, and results are presented which will be used in the next sections.

Assume that is a real Banach space with zero element . The closed ball centered at with radius and the ball are denoted by and , respectively. If is a nonempty subset of , then we denote by and the closure and closed convex hull of , respectively. Moreover, let indicate the family of all nonempty bounded subsets of and its subfamily consisting of all relatively compact subsets of .

In our considerations, we use the following definition of the concept of measure of noncompactness.

Definition 1 (see [10]). A mapping is called a measure of noncompactness if it satisfies the following conditions. (1)The family Ker is nonempty and Ker.(2). (3). (4).(5) for .(6)If is a sequence of closed sets from such that for and , then is nonempty.

The family Ker described in (1) is said to be the kernel of the measure of noncompactness and since , we infer that . So, .

Let denote the Banach space of all real functions defined and continuous on the interval equipped with the norm Fix a nonempty subset of . For and define Banas and Lecko [11] showed that the function is a measure of noncompactness in . Now, we state the following two important theorems which play a key role in the fixed point theory.

Theorem 2 (Schauder [12]). Let be a nonempty, bounded, closed, and convex subset of a Banach space E. Then, every continuous and compact map has at least one fixed point in .

Theorem 3 (Darbo [10]). Let be a nonempty, closed, bounded, and convex subset of a Banach space and let be a continuous mapping. Assume that there exists a constant such that for any nonempty subset of . Then, has a fixed point in .

The following definition of the concept of shifting distance functions will be used to generalize Darbo fixed point theorem.

Definition 4 (see [9]). Let be two functions. The pair of functions is said to be a pair of shifting distance functions, if the following conditions hold: (i)for if , then ;(ii)for with , if for all , then .

Example 5 (see [9]). The conditions (i) and (ii) of the above definition are fulfilled for the functions defined by and .

3. Some Generalizations of Darbo Fixed Point Theorem

In this section, we prove some generalizations of Darbo fixed point theorem.

Now, using the notion of shifting distance functions of [9], we obtain our generalization of Darbo fixed point theorem as follows.

Theorem 6. Let be a nonempty, bounded, closed, and convex subset of the Banach space . Moreover, assume that is a continuous function such that for any nonempty subset of , where is an arbitrary measure of noncompactness and are the pair of shifting distance functions. Then, has a fixed point in .

Proof. Define the sequence by and for all . If there exists an integer such that , then is relatively compact and since , thus Theorem 2 implies that has a fixed point. So suppose that for all . Using (8) we get Due to condition (i) of Definition 4 and (9) we infer that is a decreasing sequence of positive real numbers. Thus, there exists such that as . So, in view of (9) and condition (ii) of Definition 4, we get and hence . Now, since , , and as , condition (6) of Definition 1 implies that is nonempty, closed, convex, and invariant under the operator and belongs to . So, Theorem 2 completes the proof.

Taking in Theorem 6, we have the following result.

Corollary 7. Let be a nonempty, bounded, closed, and convex subset of the Banach space . Moreover, assume that is a continuous function such that for any nonempty subset of , where is an arbitrary measure of noncompactness and is a function such that (a)for with , if for all n, then ;(b)for if , then .Then, T has a fixed point in .

Using Proposition proved in [9] and Theorem 6, we have the following result.

Corollary 8. Let be a nonempty, bounded, closed, and convex subset of the Banach space . Moreover, assume that is a continuous function such that for any nonempty subset of , where is an arbitrary measure of noncompactness and are two nondecreasing and continuous functions satisfying if and only if . Then, T has a fixed point in .

Now, motivated and inspired by the contractive condition in [8, Theorem ], we present another generalization of Darbo fixed point theorem as follows.

Theorem 9. Let be a nonempty, bounded, closed, and convex subset of the Banach space . Moreover, assume that is a continuous function such that for any nonempty subset of , where is an arbitrary measure of noncompactness and are three functions such that and are bounded on any bounded interval in and is continuous. Moreover, assume that (1);(2)for any sequence in with ,  .Then, has a fixed point.

Proof. Similarly as in the proof of Theorem 6, we construct the sequence by and for all . If there exists an integer such that , then is relatively compact. Hence, from Theorem 2 we conclude that has a fixed point in . Assume that for all . By applying (12) we get Since , thus from (13) we get , which by condition (1) implies that . Hence, is a decreasing sequence of positive real numbers. So, there exists such that We will prove that as . Taking limit supremum on both sides of (13) and using the properties of the functions , and , we have Consequently, which implies that So, from condition (2) we conclude that . Thus, as and . Now, since and , thus from condition (6) of Definition 1 we conclude that is nonempty, closed, convex, and invariant under the operator and belongs to . Consequently, Theorem 2 implies that has a fixed point in .

Corollary 10 (Darbo [10]). Let be a nonempty, closed, bounded, and convex subset of a Banach space and let be a continuous mapping. Assume that there exists a constant such that for any nonempty subset of . Then, has a fixed point in .

Proof. Take and in Theorem 9.

Taking and in Theorem 9, we have the following corollary.

Corollary 11. Let be a nonempty, bounded, closed, and convex subset of the Banach space . Moreover, assume that is a continuous function such that for any nonempty subset of , where is an arbitrary measure of noncompactness and is bounded on any bounded interval in . Moreover, assume that and, for any sequence in with , Then, has a fixed point.

4. The Solutions of the Integral Equation of Mixed Type

In this section, we consider the integral equation (5) and prove an existence theorem of solutions of that equation. First, we recall the following two Lemmas of [13] which will be used to prove the existence theorem of the integral equation (5).

Lemma 12. Suppose and is a continuous function. Define for . Then, .

Lemma 13. Let and . Suppose that in . Then, uniformly on .

Now, we list the hypotheses which will be used to prove the existence theorem of the integral equation (5).) The functions are continuous.() is continuous and nondecreasing on .() is a continuous function and there exist positive real numbers such that  for and . Moreover, assume that .() is continuous with . Further, there exists continuous and nondecreasing function with so that(a), for and .() is a continuous operator such that  for and , where is the existent function in the assumption and are positive real numbers.() is a continuous function and there exists a function being continuous on and a function being continuous and nondecreasing on with such that  for and .() is a continuous function and there exists a continuous function and a continuous function such that and are integrable over . Moreover, there exists the function such that is continuous and nondecreasing with and the following conditions hold.(a), ,  , for all , , and .(b) for all , where and are the existent functions in the assumption and Corollary 7, respectively. Moreover, assume that .()There exists a positive solution such that  where and .

Now, we can present and prove the main result of this section.

Theorem 14. Under the assumptions , (5) has at least one solution belonging to the Banach space .

Proof. Define the operator on the Banach space by the formula Using the imposed assumptions we infer that is continuous on for each . Moreover, by using our assumptions, for any , we get Since , , and are nondecreasing, then from (25) we conclude that So, in view of assumption , the operator transforms into itself. Next, we show that is continuous on the ball . To do this, assume that is a sequence in such that and show that . Indeed, for each , we have On the other hand, we have Due to (27), (28), and our assumptions, we derive that Using (29), Lemma 13, and the imposed assumptions, we have as .
Now, let be a nonempty subset of the ball , , and fix arbitrarily . Choose such that and . Then, taking into account our assumptions, we have Moreover, we have Thus,
Using (30), (32), and the assumptions of Theorem 14, we earn where The estimate (33) implies that It follows from assumptions , and that the functions , and are uniformly continuous on the sets , , and , respectively. Consequently, we infer that Hence, using assumptions and estimate (35), we get Since the functions and are continuous and , then from (36), (37), and the assumption , we conclude that Thus, Corollary 7 completes the proof.

Now, we present the following example which shows that Theorem 14 can be applied to obtain solutions of the integral equation (39) but the existent results in [5, 6] are inapplicable.

Example 15. Now, we investigate the following integral equation: for . Equation (39) is a special case of the integral equation (5), where Then, it is easily seen that , and satisfy the assumptions and of the Theorem 14. Further, the function satisfies assumption with and . Since , then, for all and , we have So, the assumption is satisfied with and . In this example, we have and this operator satisfies assumption with ,  , and . On the other hand, for all and , we get So, and . Moreover, . Thus, assumption holds. Now, notice that the function is continuous and satisfies the conditions and of the assumption . Indeed, for all , , and , we have So, , , and . Moreover, , , and the condition of the assumption holds with . Thus, the existent inequalities in assumption have the forms It is easily seen that the last inequalities have a positive solution. For example, . We see that all assumptions of Theorem 14 are satisfied. Consequently, from Theorem 14 the integral equation (39) has a solution in the space .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.