Abstract and Applied Analysis

Volume 2012 (2012), Article ID 632628, 14 pages

doi:10.1155/2012/632628

Research Article

Fixed Points and Endpoints of Set-Valued Contractions in Cone Metric Spaces

Ing-Jer Lin,¹ Chi-Ming Chen,² Mirko Jovanović,³ and Tzi-Huei Wu²

¹Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

²Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 30014, Taiwan

³Faculty of Electrical Engineering, University of Belgrade, Kralja Aleksandra 73, 11120 Beograd, Serbia

Received 23 March 2012; Revised 25 May 2012; Accepted 9 June 2012

Academic Editor: Gabriel Turinici

Copyright © 2012 Ing-Jer Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 .

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