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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 302830, 17 pages
http://dx.doi.org/10.1155/2012/302830
Research Article

On Approximate Coincidence Point Properties and Their Applications to Fixed Point Theory

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

Received 29 October 2011; Revised 21 January 2012; Accepted 23 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Wei-Shih Du. 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

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the literature.

1. Introduction

Let be a metric space. For each and , let . Denote by the class of all nonempty subsets of , the collection of all nonempty closed subsets of , and the family of all nonempty closed and bounded subsets of . A function defined by is said to be the Hausdorff metric on induced by the metric on . Let be a multivalued map. A point in is a fixed point of if . The set of fixed points of is denoted by . Let be a single-valued self-map and be a multivalued map. A point in is said to be a coincidence point (see, for instance, [1, 2]) of and if . The set of coincidence point of and is denoted by . Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively.

Let be a real-valued function defined on . For , we recall that

Definition 1.1 (see [29]). A function is said to be an -function (or -function) if
It is obvious that if is a nondecreasing function or a nonincreasing function, then is an -function. So the set of -functions is a rich class.
Very recently, Du [7] first proved some characterizations of -functions.

Theorem D (see [7]). Let be a function. Then, the following statements are equivalent.(a) is an -function.(b) For each , there exist and such that for all .(c) For each , there exist and such that for all .(d) For each , there exist and such that for all .(e) For each , there exist and such that for all .(f) For any nonincreasing sequence in , we have (g) is a function of contractive factor [5]; that is, for any strictly decreasing sequence in , we have .

It is worth to mention that there exist functions which are not -functions. For example, let be defined by Since , is not an -function.

Let be a single-valued self-map. Recall that a multivalued map is called(1) a multivalued -contraction [1, 10, 11], if there exists a number such that (2) a multivalued -almost contraction [1, 12], if there exist two constants and such that(3) a generalized multivalued almost contraction [1, 12], if there exists an -function and such that(4) a multivalued almost -contraction [1, 12, 13], if there exist two constants and such that(5) a generalized multivalued almost -contraction [1], if there exists an -function and such that

In 1989, Mizoguchi and Takahashi [14] proved the following fixed point theorem which is a generalization of Nadler’s fixed point theorem [10] and the celebrated Banach contraction principle (see, e.g., [11]). It is worth to mention that Mizoguchi-Takahashi’s fixed point theorem gave a partial answer of Problem 9 in Reich [15] and it’s primitive proof is difficult. Recently, Suzuki [16] presented a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem MT (Mizoguchi and Takahashi). Let be a complete metric space, be a multivalued map, and be a -function. Assume that Then, .

Since then a number of generalizations in various different directions of Mizoguchi-Takahashi’s fixed point theorem have been investigated by several authors in the past. In 2007, M. Berinde and V. Berinde [12] proved the following interesting fixed point theorem to generalize Mizoguchi-Takahashi’s fixed point theorem.

Theorem BB (M. Berinde and V. Berinde). Let be a complete metric space, be a multivalued map, be a -function, and . Assume that that is, is a generalized multivalued almost contraction. Then, .

In [3], the author established some new fixed point theorems for nonlinear multivalued contractive maps by using -metrics (see [3, Def. 1.2]), -metrics (see [3, Def. 1.3]), and -functions. Applying those results, the author gave the generalizations of Berinde-Berinde‘s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach contraction principle, Kannan’s fixed point theorems, and Chatterjea’s fixed point theorems for nonlinear multivalued contractive maps in complete metric spaces; for more detail, one can refer to [3].

Very recently, Du [17] first introduced the concepts of TVS-cone metric and TVS-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [18].

Definition 1.2 (see [17]). Let be a nonempty set and a locally convex Hausdorff t.v.s. with its zero vector , a proper, closed, convex, and pointed cone in , and a partial ordering with respect to defined by A vector-valued function is said to be a -, if the following conditions hold:(C1) for all and if and only if ,(C2) for all ,(C3) for all . The pair is then called a TVS-cone metric space.

In this paper, we first establish some existence results concerning approximate coincidence point property and approximate fixed point property for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde’s fixed point theorem and Mizoguchi-Takahashi’s fixed point theorem. Our results generalize and improve some recent results in [16, 1019] and references therein.

2. Preliminaries

Let be a topological vector space (t.v.s. for short) with its zero vector . A nonempty subset of is called a convex cone if and for . A convex cone is said to be pointed if . For a given proper, pointed, and convex cone in , we can define a partial ordering with respect to by will stand for and , while will stand for , where denotes the interior of .

In the following, unless otherwise specified, we always assume that is a locally convex Hausdorff t.v.s. with its zero vector , is a proper, closed, convex, and pointed cone in with , is a partial ordering with respect to and is fixed.

Recall that the nonlinear scalarization function is defined by

Theorem 2.1 (see [6, 17, 20, 21]). For each and , the following statements are satisfied:(i),(ii),(iii),(iv),(v) is positively homogeneous and continuous on ,(vi) if (i.e., ), then ,(vii) for all .
Clearly, . Notice that the reverse statement of (vi) in Theorem 2.1 (i.e., ) does not hold in general. We illustrate the truth with the following simple example.

Example A. Let , , and . Then is a proper, closed, convex, and pointed cone in with and . For , it is easy to see that , and . By applying (iii) and (iv) of Theorem 2.1, we have while .

Definition 2.2 (see [17]). Let be a TVS-cone metric space, , and a sequence in .(i) is said to TVS-cone converge to if, for every with , there exists a natural number such that for all . We denote this by cone- or as and call the TVS-cone limit of .(ii) is said to be a TVS-cone Cauchy sequence if, for every with , there is a natural number such that for all .(iii) is said to be TVS-cone complete if every TVS-cone Cauchy sequence in is TVS-cone convergent in .
In [17], the author proved the following important results.

Theorem 2.3 (see [17]). Let be a TVS-cone metric spaces. Then, defined by is a metric.

Example B. Let , , , , and . Define by Then, is a TVS-cone complete metric space and So is a metric on and is a complete metric space.

Theorem 2.4 (see [17]). Let be a TVS-cone metric space, , and a sequence in . Then, the following statements hold.(a) If   TVS-cone converges to as , then as (i.e., as ).(b) If is a TVS-cone Cauchy sequence in , then is a Cauchy sequence (in usual sense) in .

Definition 2.5 (see [1, 22]). Let be a metric space. A multivalued map is said to have an approximate fixed point property provided or, equivalently, for any , there exists such that or, equivalently, for any , there exists such that
where denotes a closed ball of radius centered at .
It is known that every generalized multivalued almost contraction in a metric space has the approximate fixed point property (see [1, Lemma 2.2]).

Remark 2.6. It is obvious that implies that has the approximate fixed point property.

Definition 2.7 (see [1]). Let be a metric space, a single-valued self-map, and a multivalued map. The maps and are said to have an approximate coincidence point property provided or, equivalently, for any , there exists such that
It is known that every generalized multivalued almost -contraction in a metric space has the approximate coincidence point property provided each is -invariant (i.e., ) for each (see [1, Theorem 2.7]).

3. Main Results

For any locally convex Hausdorff t.v.s. with its zero vector , let denote the topology of and let be the base at consisting of all absolutely convex neighborhood of . Let Then, is a family of seminorms on . For each , put and denote Then, is a base at and the topology generated by is the weakest topology for such that all seminorms in are continuous and . Moreover, given any neighborhood of , there exists such that ; for more detail, we refer the reader to ([23, Theorem 12.4 in II.12, Page 113]).

The following lemmas are very crucial to our main results.

Lemma 3.1. Let be a neighborhood of , , and with . Suppose that satisfies , then there exists , such that for all .

Proof . Clearly, if , then for all , and we are done. Suppose . Since , for some , and . Let We need consider two possible cases.Case 1. If , since each is a seminorm, we have and for all and all .Case 2. If , since , there exists such that for all . So, for each and any , we obtain
Hence, by Cases 1 and 2, we get that, for any , for all . Therefore, for all .

Lemma 3.2 (see [24, Lemma 2.5]). Let be a t.v.s., a convex cone with in , and . Then, the following statements hold.(a).(b) If and , then .

In this section, we first establish an existence theorem related to approximate coincidence point property for maps in TVS-cone complete metric space which is one of the main results of this paper. It will have many applications to study metric fixed point theory.

It is worth observing that the following existence theorem does not require the TVS-cone completeness assumption on TVS-cone metric space .

Theorem 3.3. Let be a TVS-cone metric space, a multivalued map, a self-map, and . Suppose that(D1) there exists an -function such that, for each , if with then there exists such that(D2) is -invariant (i.e., ) for each . Then, the following statements hold.(a) and have the -approximate coincidence point property on (i.e., ).(b) There exists a sequence in such that is a TVS-cone Cauchy sequence and .

Proof. By Theorem 2.3, we know that is a metric on . Let and . If , then, by (D2), we have . Since we have . Let for all . Then, is a TVS-cone Cauchy sequence and . Hence, all conclusions are proved in the case of . If or , then, by (D1), there exists such that If , then the conclusions also hold by following a similar argument as above. If , then there exists such that By induction, we can obtain a sequence in satisfying the following: for each , Applying (v) and (vi) of Theorem 2.1, the inequality (3.12) implies Since for all , the sequence is strictly decreasing in . Since is an -function, by (g) of Theorem D, Let . So . From (3.12), we get It follows from (3.15) that Let , . Applying (v) and (vi) of Theorem 2.1, the inequality (3.16) implies Since , by (3.17), we have . By (3.12) and (D2), we obtain , . It implies that Since as , it follows from (3.18) that and the conclusion (a) is proved.
To see (b), it suffices to prove that is a TVS-cone Cauchy sequence in . For with , it follows from (3.16) that Given with (i.e, , there exists a neighborhood of such that . Therefore, there exists with such that Let and , . Since , we have and . Applying Lemma 3.1, there exists , such that for all . So, by (3.20), we obtain or for all . For with , since from (3.19) and , it follows from Lemma 3.2 that Hence, we prove that is a TVS-cone Cauchy sequence in . The proof is completed.

Theorem 3.4. Let be a TVS-cone metric space, a multivalued map, and . Suppose that(D3) there exists an -function such that, for each , if with , then there exists such that Then, there exists a nonempty proper subset of , such that (i.e., has the -approximate fixed point property on and .)

Proof. Let be the identity map on . Then, the conditions (D1) and (D2) as in Theorem 3.3 hold. Following the same argument as in the proof of Theorem 3.3, we obtain two sequences and in such that(i), and for each ,(ii).
Since , and are identical, we have . Put . Since we obtain . Hence, has the -approximate fixed point property on and .

New, we establish the following approximate coincidence point property for maps in general metric spaces by applying Theorem 3.3.

Theorem 3.5. Let be a metric space, a multivalued map, and a self-map. Suppose that(A1) there exists an -function such that, for each , if with , then there exists such that(A2) is -invariant (i.e., ) for each .Then, and have the approximate coincidence point property on .

Proof . In Theorem 3.3, let , , and . Therefore, the conclusion is immediate from Theorem 3.3.

The following approximate fixed point property for maps is immediate from Theorem 3.4.

Theorem 3.6. Let be a metric space and a multivalued map. Suppose that(A3) there exists an -function such that, for each , if with , then there exists such that Then, has the approximate fixed point property on .

Theorem 3.7. In Theorem 3.6, if the condition (A3) is replaced by (A3)H, where(A3)H there exists an -function such that, for each , then, has the approximate fixed point property on .

Proof. Define by . Then, by [3, Lemma 2.1], is also an -function. For each , let with . Then, . By (A3)H, we have It follows that there exists such that which shows that (A3) holds. Therefore, the conclusion follows from Theorem 3.6.

Theorem 3.8. Theorems 3.6 and 3.7 are equivalent.

Proof. We have shown that Theorem 3.6 implies Theorem 3.7. So it suffices to prove that Theorem 3.7 implies Theorem 3.6. If (A3) holds, then it is easy to verify that (A3)H also holds. Hence, Theorem 3.7 implies Theorem 3.6 and we get the desired result.

Remark 3.9 (see [1, Lemma 2.2]). Is a special cases of Theorems 3.6 and 3.7.
Let be a TVS-cone metric space. By Theorem 2.3, we know that is a metric on . So we can obtain the topology on induced by and hence define -open subsets, -closed subsets, and -compact subsets of .
Here, we denote by the collection of all nonempty -closed subsets of .

Theorem 3.10. Let be a TVS-cone complete metric space, a multivalued map, a self-map, and . Suppose that the conditions (D1) and (D2) as in Theorem 3.3 hold and further assume one of the following conditions hold:(L1) is -compact and the function defined by is l.s.c.,(L2) is a -closed subset of .Then, the following statements hold.(a) There exists a nonempty subset of , such that is a complete metric space.(b).

Proof. Following the same argument as in the proof of Theorem 3.3, one can obtain that there exist two sequences and in such that for any , , , is a TVS-cone Cauchy sequence in and . By the TVS-cone completeness of , there exists , such that   TVS-cone converges to . On the other hand, applying Theorem 2.4, we obtain that is a Cauchy sequence in and or as . Let . Then is a complete metric space and the conclusion (a) holds.
Now, we verify the conclusion (b). Suppose that (L1) holds. Then the infimum is attained. Since , there exists such that Since is a -closed subset of , it implies from (3.31) that . Hence, .
Suppose that (L2) holds. For any , since , we know . Since is -closed in and as , we have . Hence, there exists such that and , which say that . The proof is completed.

Theorem 3.11. Let be a TVS-cone complete metric space, a multivalued map, and . Suppose that the condition (D3) as in Theorem 3.4 holds and further assume one of the following conditions hold:(H1) is -closed; that is, , the graph of , is a -closed subset of ,(H2) the function defined by is l.s.c.,(H3) for every . Then, there exists a nonempty subset of , such that(a) is a complete metric space,(b).

Proof. Following a similar argument as in the proof of Theorem 3.3, we can obtain a sequence in such that, for any ,(i),(ii), where ,(iii), for with .(iv) is a TVS-cone Cauchy sequence in .
Applying Theorem 2.1, the inequality (iii) implies that, for with , we have Since , it follows from (3.32) that is a Cauchy sequence in . By the TVS-cone completeness of and (iv), there exists , such that   TVS-cone converges to . Applying Theorem 2.4, we have or as . Let . Then, is a complete metric space and the conclusion (a) holds.
Finally, in order to complete the proof, it suffices to show that . Suppose that (H1) holds. Since , we have for each . By (H1) and as , we get .
If (H2) holds, by the lower semicontinuity of and (i), we obtain which implies . Since is a -closed subset of , .
Let (H3) holds. Suppose . Since is a metric on and as , by (3.32), we get From (3.32) and (3.34), we have which leads a contradiction. Therefore, . The proof is completed.

The following result is immediate from Theorem 3.10.

Theorem 3.12. Let be a complete metric space, a multivalued map, and a self-map. Suppose that the conditions (A1) and (A2) as in Theorem 3.5 hold and further assume one of the following conditions hold:(i) is compact and the function defined by is l.s.c.;(ii) is a closed subset of .Then, .

Theorem 3.13. Let be a complete metric space and a multivalued map. Suppose that the condition (A3) or (A3)H holds and further assume one of the following conditions hold:(h1) is closed,(h2) the function defined by is l.s.c.,(h3) for every ,(h4) for each sequence in with , and , we have .Then, .

Proof. The conclusion is immediate from Theorem 3.11 if , , , and one of conditions (h1), (h2), and (h3) holds. Suppose that (h4) holds. Following a similar argument as in the proof of Theorem 3.3, we can construct a Cauchy sequence in such that , , and converge to some point . Since the function is continuous, and as , by (h4), we get which implies . The proof is completed.

Remark 3.14. (a)Let and be two topological vector spaces and a multivalued map. Recall that is u.s.c. if and only if for any open set in , is open in . It is known that if is u.s.c. with closed values, then is closed (see [25]). Hence, Theorem 3.13 is true if is a Banach space and is u.s.c.;(b)let be a nonempty subset of a metric space and u.s.c. Then, the function defined by is l.s.c.; for detail, see [26, Lemma 2] or [27, Lemma 3.1];(c)it is known that any single-valued map of Kannan’s type or Chatterjea’s type satisfies (h3); for more detail, one can see [28, Corollary 3] or [3, Remark 3.1].
Applying Theorem 3.13, we can prove the following generalization of Berinde-Berinde’s fixed point theorem [12].

Theorem 3.15. Let be a complete metric space, a multivalued map, and a function. Suppose that there exists an -function such that Then, .

Proof. Let . If , then . So (3.37) implies the inequality Hence, the condition (A3)H of Theorem 3.13 holds. Let in with , , and . By (3.37), we obtain which says that the condition (h4) of Theorem 3.13 also holds. Therefore, the conclusion follows from Theorem 3.13.

Remark 3.16. (a)Theorems 3.11, 3.13, and 3.15 all generalize Berinde-Berinde’s fixed point theorem [12].(b)In Theorem 3.15, if for all , then we can obtain Mizoguchi-Takahashi’s fixed point theorem [14].(c)In Theorem 3.15, if for all , and is defined by for some , then we can obtain Nadler’s fixed point theorem [10].(d)[1, Theorem 2.6] is a special case of Theorem 3.15.(e)Notice that, in [1, Theorem 2.6], the authors showed that a generalized multivalued almost contraction in a metric space has provided either is compact and the function is l.s.c. or is closed and compact. But reviewing Theorem 3.15, we know that the conditions in [1, Theorem 2.6] are redundant.

Acknowledgments

The author would like to express his sincere thanks to the anonymous referee for their valuable comments and useful suggestions in improving the paper. This research was supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.

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