#### 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 [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan [2] and Chatterjea [3] introduced two conditions that can replace (1.1) in Banach's theorem as follows:

Kannan [2]: There exists such that, for all ,

Chatterjea [3]: 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 [4] showed the following fixed-point theorem.

Theorem 1.1 (see [4]). *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 [5], using a result of Chu and Diaz [6], 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 [5]). *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., [7–10]). 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 [11] 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 [11]). *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 [12] proved the following fixed-point theorem.

Theorem 1.6 (see [12]). *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 [13] 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 [13].

Theorem 1.8 (see [13]). *Let be a complete metric space and a multi valued operator. Suppose that *(i)* for all *(ii)*there exist a comparison function (see [14]) and a Picard sequence , such that
where . ** Then as and is a unique strict fixed-point of . *

Recently, Wardowski [15] 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 [15]). *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 [15]). * 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 [16] 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 [17] 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 [18] have generalized the results of Huang and Zhang [17] 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 [19] 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, 19–39]).

*Definition 1.12 (see [17]). *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 [40]). *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 [41]). *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 [17]). *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 [17]). * 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 [17]). *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 [17]). *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 [17]). * 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 [42–44] and Wardowski [15].

#### 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 [15], 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 .*