#### Abstract

The purpose of this paper is to present the fixed points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović, 2011; Wardowski, 2009.

#### 1. Introduction and Preliminaries

Throughout this paper, by , we denote the sets of all nonnegative real numbers and all real numbers, respectively, while is the set of all natural numbers. Let be a metric space, a subset of , and a map. We say is contractive if there exists such that, for all , The well-known Banach fixed-point theorem asserts that if , is contractive and is complete, then has a unique fixed-point in . It is well known that the Banach contraction principle  is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan  and Chatterjea  introduced two conditions that can replace (1.1) in Banach's theorem as follows:

Kannan : There exists such that, for all ,

Chatterjea : There exists such that for all ,

After these three conditions, many papers have been written generalizing some of the conditions (1.1), (1.2), and (1.3). In 1969, Boyd and Wong  showed the following fixed-point theorem.

Theorem 1.1 (see ). Let be a complete metric space and a map. Suppose there exists a function satisfying , for all and is right upper semicontinuous such that Then has a unique fixed-point in .

Later, Meir-Keeler , using a result of Chu and Diaz , extended Boyd-Wong's result to mappings satisfying the following more general condition: and Meir-Keeler proved the following very interesting fixed-point theorem which is a generalization of the Banach contraction principle.

Theorem 1.2 (Meir-Keeler ). Let be a complete metric space and let be a Meir-Keeler contraction; that is, for every , there exists such that implies for all . Then has a unique fixed-point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., ). In this paper, we introduce a new notion of the stronger Meir-Keeler-type mapping , , as follows.

Definition 1.3. A mapping , is called a stronger Meir-Keeler-type mapping in a metric space , if for each , there exists such that for with , there exists such that .

Example 1.4. Let be a metric space, and define , by Then is a stronger Meir-Keeler-type mapping.

The existence of fixed-points for various multi valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler  extended the famous Banach contraction principle from single-valued mapping to multi valued mapping and proved the following fixed-point theorem for multi valued contraction. Let denote the collection of all nonempty subsets of , the collection of all nonempty bounded subsets of , the collection of all nonempty closed subsets of , the collection of all nonempty closed and bounded subsets of , and the collection of all nonempty sequentially compact subsets of .

Theorem 1.5 (see ). Let be a complete metric space, and, be a mapping from into . Assume that there exists such that Then has a fixed-point in .

In 1989, Mizoguchi-Takahashi  proved the following fixed-point theorem.

Theorem 1.6 (see ). Let be a complete metric space and a map from into . Assume that for all , where satisfies for all . Then has a fixed-point in .

Remark 1.7. It is clear that if the function , satisfies then is also a stronger Meir-Keeler-type function.

In 2003, Rus  introduced the notion of endpoints (strict fixed-points) and proved some results on strict fixed-point theory on multi valued operator. An element is said to be a fixed-point of , if . If , then is called an endpoint of . We will denote the sets of all fixed-points and endpoints of by and , respectively. The following theorems were the main results of Rus .

Theorem 1.8 (see ). Let be a complete metric space and a multi valued operator. Suppose that (i) for all (ii)there exist a comparison function (see ) and a Picard sequence , such that where .
Then as and is a unique strict fixed-point of .

Recently, Wardowski  proved the following theorems concerning fixed-points and endpoints of set-valued contractions in complete cone metric spaces.

Definition 1.9. A function is called lower semicontinuous, if, for any sequence and ,
If is a cone metric space and , then, for , we denote

Theorem 1.10 (see ). Let be a complete cone metric space, let be a normal cone with normal constant , and let . Assume that a function defined by , is lower semicontinuous. If there exist , such that then .

Theorem 1.11 (see ). Let be a complete cone metric space, let be a normal cone with normal constant , and let . Assume that a function defined by , is lower semicontinuous. The following hold(a) If there exist , such that then .(b) If there exist , such that then .

In 1997, Zabrejko  introduced the -metric and -normed linear spaces and showed the existence and uniqueness of fixed-points for operators which act in -metric or -normed linear spaces. Huang and Zhang  introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed-point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors like Abbas and Jungck  have generalized the results of Huang and Zhang  and studied the existence of common fixed-points of a pair of self-mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Rezapour and Hamlbarani  studied the existence of common fixed-points of a pair of self- and non-self-mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed-point theory are proved (see, e.g., [12, 1939]).

Definition 1.12 (see ). Let be a real Banach space and a nonempty subset of . , where denotes the zero element of , is called a cone if and only if: (i) is closed;(ii), , ;(iii) and .

For given a cone , we define a partial ordering with respect to by or if and only if for all . The real Banach space equipped with the partial order induced by is denoted by . We shall write to indicate that but , while or will stand for , where denotes the interior of .

Proposition 1.13 (see ). Suppose is a cone in a real Banach space . Then one has the following. (i)If and , then .(ii)If and , then .(iii)If and , then . (iv)If and for each , then .

Proposition 1.14 (see ). Suppose , and . Then there exists such that for all .

The cone is called normal if there exists a real number such that, for all ,

The least positive number , satisfying the above is called the normal constant of .

The cone is called regular if every increasing sequence which is bounded from above is convergent, that is, if is a sequence such that for some , then there is such that as . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

Definition 1.15 (see ). Let be a nonempty set, a real Banach space, and a cone in . Suppose that the mapping satisfies(i), for all ; (ii) if and only if ; (iii), for all ; (iv), for all . Then is called a cone metric on , and is called a cone metric space.

Definition 1.16 (see ). Let be a cone metric space, and let be a sequence in and . If for every with , there is such that then is said to be convergent and converges to .

Definition 1.17 (see ). Let be a cone metric space, and let be a sequence in . We say that is a Cauchy sequence if, for any with , there is such that

Definition 1.18 (see ). Let be a cone metric space. If every Cauchy sequence is convergent in , then is called a complete cone metric space.

Remark 1.19 (see ). If is a normal cone, then converges to if and only if as . Further, in the case is a Cauchy sequence if and only if as .

The purpose of this paper is to present the fixed-points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović  and Wardowski .

#### 2. Main Results

In this section, we first introduce the following notion of stronger Meir-Keeler cone-type mapping.

Definition 2.1. Let be a cone metric space with cone , and let Then the function is called a stronger Meir-Keeler cone-type mapping, if, for each , there exists such that, for with , there exists such that .

We now are in a position to present the following fixed-point theorem.

Theorem 2.2. Let be a complete cone metric space, and let be a normal cone in and . Assume that a function defined by , is lower semicontinuous. The following holds.(*)  There exists a stronger Meir-Keeler cone-type mapping such that Then .

Proof. Given , we define the sequence recursively as follows. Take any . By , there exist and such that Thus, If , then , and we are done. Assume that . Put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exists and with such that . Taking into account (2.5), we have that
By the same process, for , there exist and such that Thus, Now, put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exists and with such that . And we put and . Taking into account (2.9), we have that
We continue in this manner. Inductively, for , there exist and such that Thus, Put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exist and with such that . And we put and . Taking into account (2.14), we have that
Let such that . Taking into account (2.15) and (2.16), we obtain that Let be given. Since and , we get Thus, there exists such that, for all , and we also conclude that for all . This implies that is a Cauchy sequence. Since is complete, there exists such that
By the definition of , since , there exists a sequence such that , for all . From the convergence of the sequence and from the lower semicontinuity of the function , we obtain that Thus, We claim that . To prove this, on the contrary, assume that . Then by (2.22), there exists a sequence such that , and hence . Thus, for any , Let be given. Then there exists such that for all , and , and so This implies that is a Cauchy sequence in . Since is complete, there exists such that By the closedness of the , we deduce that . Then, for any , Let be given. Since and , we can deduce that there exists such that, for all , and , and hence By Proposition 1.13, we obtain that that is, , which is a contradiction. Therefore, .

Applying Remark 1.7, it is easy to establish the following corollary.

Corollary 2.3. Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by :, is lower semicontinuous. The following holds.(**) There exists a mapping such that Then .

Corollary 2.4. Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , is lower semicontinuous. The following holds.(***)There exists a stronger Meir-Keeler cone-type mapping such that Then .

Example 2.5. Let , , where , , , defined by , , , with for all and if and only if . Let be defined by , , and . Then , , . Take for all . Calculation shows that, for , that is, according to Corollary 2.4 we have that .

Theorem 2.6. Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , , is lower semicontinuous. The following hold:(A) There exists a stronger Meir-Keeler cone-type mapping such that
Then .(B) There exists a stronger Meir-Keeler cone-type mapping such that
Then .

Proof. (A) By the same proof of the part (i) of Theorem 3.2 in , we have that for all . The remainder proof is similar to Theorem 2.2, we can deduce that .
(B) By (A), we obtain that the existence of such that, . Taking any , we have that, for all , . Since , we get , and hence , which gives . Thus .

Example 2.7. Let , , . Let for all , and let , . Then , . Let for all . It is easy to check that all conditions of Theorem 2.6 are satisfied and that .

Corollary 2.8. Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , is lower semicontinuous. The following hold:(C) There exists a mapping such that Then .(D) There exists a mapping such that Then .