#### Abstract

Some new existence theorems concerning approximate coincidence point property and approximate fixed point property for nonlinear maps in metric spaces without global completeness are established in this paper. By exploiting these results, we prove some new coincidence point and fixed point theorems which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Kikkawa-Suzuki's fixed point theorem, and some well known results in the literature. Moreover, some applications of our results to the existence of coupled coincidence point and coupled fixed point are also presented.

#### 1. Introduction

Let us begin with some basic definitions and notations that will be needed in this paper. The symbols and are used to denote the sets of positive integers and real numbers, respectively. Let be a metric space. Denote by the family of all nonempty subsets of , the class of all nonempty closed subsets of , and the family of all nonempty closed and bounded subsets of . For each and , let . A function defined by is said to be the Hausdorff metric on induced by the metric on .

Let be a self-map and be a multivalued map. A point in is said to be a coincidence point (see, for instance, [14]) of and if . The set of coincidence points of and is denoted by . If , then is called a fixed point of . The set of fixed points of is denoted by . The maps and are said to have an approximate coincidence point property [1, 4] on provided . The map is said to have the approximate fixed point property [15] on provided .

It is obvious that (resp., ) implies that has the approximate coincidence point property (resp., has the approximate fixed point property). Hussain et al. [1, Theorem 2.6] showed that a generalized multivalued almost contraction in a metric space have provided either is compact and the function is l.s.c. or is closed and compact. In [1, Lemma 2.2], the authors had also shown that every generalized multivalued almost contraction in a metric space has the approximate fixed point property.

The rapid growth of fixed point theory and its applications over the past decades has led to a number of scholarly essays that examine its nature and its importance in nonlinear analysis, applied mathematical analysis, economics, game theory, and so forth; see [147] and references therein. Many authors devoted their attention to investigate its generalizations in various different directions of the celebrated Banach contraction principle. In 2008, Suzuki [6] presented a new type of generalization of the celebrated Banach contraction principle and does characterize the metric completeness.

Theorem 1 (Suzuki [6]). Define a nonincreasing function from onto by Then for a metric space , the following are equivalent:(1)is complete.(2)Every mapping on satisfying the following has a fixed point:there exists such that implies for all . (3)There exists such that every mapping on satisfying the following has a fixed point: implies for all .

Remark 2 (see [6]). For every , is the best constant.

Later, Kikkawa and Suzuki [8] proved an interesting generalization of both Theorem 1 and the Nadler fixed point theorem [9] which is an extension of the Banach contraction principle to multivalued maps.

Theorem 3 (Kikkawa and Suzuki [8]). Define a strictly decreasing function from onto by Let be a complete metric space and let be a map from into . Assume that there exists such that for all . Then .

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

Definition 4 (see [3, 4, 1019]). A function is said to be an -function (or -function) if for all .

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.

In 1989, Mizoguchi and Takahashi [19] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer of Problem 9 in Reich [20]. It is worth to mention that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult. Recently, Suzuki [21] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem 5 (Mizoguchi and Takahashi [19]). Let be a complete metric space, be a -function and be a multivalued map. Assume that for all . Then .

In 2007, M. Berinde and V. Berinde [22] proved the following interesting fixed point theorem which generalized and improved Mizoguchi-Takahashi’s fixed point theorem.

Theorem 6 (M. Berinde and V. Berinde [22]). Let be a complete metric space, be a -function, be a multivalued map and . Assume that for all . Then .

Very recently, Du et al. [4] studied the existence of the approximate coincidence point property and the approximate fixed point property for some new nonlinear maps and applied them to metric fixed theory. Some new generalizations of Kikkawa-Suzuki’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, and some well-known results in the literature were established in [4]; for more detail, one can refer to [4].

The paper is organized as follows. In Section 3, we first present some new existence theorems concerning approximate coincidence point property, approximate fixed point property, coincidence point and fixed point for various types of nonlinear maps in metric spaces without global completeness. Section 4 is dedicated to the study of some new coincidence point, and fixed point theorems given by exploiting our results. We establish some generalizations of Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem and others. Some applications of our results to a generalizations of Kikkawa-Suzuki’s fixed point theorem and the existence of coupled coincidence point and coupled fixed point are also given in Section 5. Consequently, in this paper, some of our results are original in the literature and we obtain many results in the literature as special cases; see for example, [410, 13, 14, 1723, 30] and references therein.

#### 2. Preliminaries

Recall that a function is called a -distance [5, 7, 10, 14, 15, 2429, 3840], if the following are satisfied: for any ; for any , is l.s.c.; for any , there exists such that and imply .

A function is said to be a -function [5, 10, 14, 15, 2729], first introduced and studied by Lin and Du, if the following conditions hold: for all ; if and in with such that for some , then ; for any sequence in with , if there exists a sequence in such that , then ; for , and imply .

Note that not either of the implications necessarily holds and is nonsymmetric in general. It is well known that the metric is a -distance and any -distance is a -function, but the converse is not true; see [27, 29] for more detail.

Definition 7 (see [4]). Let be a metric space, be a -function, be a single-valued self-map and be a multivalued map.(1)The maps and are said to have the -approximate coincidence point property on provided (2)The map is said to have the -approximate fixed point property on provided The following results are crucial in this paper.

Lemma 8 (see [29, Lemma 2.1]). Let be a metric space and be a function. Assume that satisfies the condition . If a sequence in with , then is a Cauchy sequence in .

For each and , we denote .

Lemma 9 (see [10]). Let be a closed subset of a metric space and be any function. Suppose that satisfies and there exists such that . Then   if and only if .

The concepts of -functions and -metrics were introduced in [10] as follows.

Definition 10 (see [10]). Let be a metric space. A function is called a -function if it is a -function on with for all .

Remark 11. If is a -function, then, from , if and only if .

Example 12 (see [10]). Let with the metric and . Define the function by Then is nonsymmetric and hence is not a metric. It is easy to see that is a -function.

Definition 13 (see [10]). Let be a metric space and be a -function. For any , , define a function by where , then is said to be the -metric on induced by .

Clearly, any Hausdorff metric is a -metric, but the reverse is not true.

Lemma 14 (see [10]). Let be a metric space and be a -metric on induced by a -function . Then every -metric is a metric on .

The following characterizations of -functions is quite useful for proving our main results.

Lemma 15 (see [18]). 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 , one has .(g) is a function of contractive factor [12]; that is, for any strictly decreasing sequence in , one has .

#### 3. New Nonlinear Conditions for -Approximate Coincidence Point Property

In Section 3, we will establish some new existence theorems concerning approximate coincidence point property, approximate fixed point property, coincidence point and fixed point for various types of nonlinear maps in metric spaces without global completeness.

Theorem 16. Let be a metric space, be a -function, be a multivalued map, and be a self-map. Suppose that(S1) there exist a nondecreasing function and an -function such that for each , if with and , then it holds (S2).
Then the following statements hold.(a)There exists a sequence in such that (b); that is, and have the -approximate coincidence point property and approximate coincidence point property on .(c)If one further assumes the following conditions hold:(L1) is a complete subspace of ,(L2) for each sequence in with , and , one has as a closed subset of and ,then .

Proof. Let . By (S2), there exists such that . If , then and so which implies . Clearly, . Let for all . Then So, the conclusions (a) and (b) hold in this case. Otherwise, if , since is a -function, . Let be defined by . Clearly, for all . By [3, Lemma 2.1], we know that is also an -function. From (S1), we get Since , there exists such that Using (S2) again, there exists such that . Hence, from (17), we have If , then, following a similar argument as above, we can prove the conclusions (a) and (b). Otherwise, if , then there exists such that and By induction, we can obtain a sequences in satisfying Since for all , we deduces from the inequality (21) that the sequence ,  is strictly decreasing in . Hence Since is nondecreasing, is a nonincreasing sequence in . Since is an -function, by (f) of Lemma 15, we have Let . So . Put . Then . By (21), we get Since , and hence it follows from (24) that According to (22) and (25), we obtain
Next, we verify that is a Cauchy sequence in . Let for all . We claim that . Put
Then for all . For with , by (24), we have Since , and hence By Lemma 8, is a Cauchy sequence in . Hence Since for all and , one also obtain Hence (a) is proved. To see (b), since for each , we have for all . Since , combining (32), we get Moreover, if we further assume that conditions (L1) and (L2) hold, we want to show . Since is a Cauchy sequence in , by (L1), there exists such that as . So, by (L2), we have is a closed subset of and . Since , there exists with and such that . By , . Since as and , it implies as . By the closedness of , we have or . The proof is completed.

Theorem 17. In Theorem 16, if is the identity map on , then the following statements hold.(a)There exists a sequence in such that (b); that is, has the -approximate fixed point property and approximate fixed point property on .(c)If one further assumes the following conditions hold:(L1) is a complete subspace of ,(L3) for each sequence in with , and , one has as a closed subset of and ,then .

As an application of Theorems 16, we can establish the following new existence of approximate coincidence point property easily.

Theorem 18. Let be a metric space, be a -function, be a -metric on induced by , be a multivalued map and be a self-map. Suppose that (S2) as in Theorem 16 is satisfied and further satisfies one of the following conditions:(H1) there exist a nondecreasing function , an -function and a function such that (H2) there exist a nondecreasing function , an -function and a function such that
Then the following statements hold.(a)There exists a sequence in such that (b).If one further assumes the following conditions hold:(L1) is a complete subspace of ,(L2) for each sequence in with , and , one has as a closed subset of and ,then .

Proof. Suppose that (H1) holds. We first notice that for each , by (S2), the set . Let be given and let with be arbitrary. Since , we have Hence (H1) implies (S1). Therefore the conclusion follows from Theorem 16. Similarly, we can prove that (H2) implies (S1) and the desired result follows also from Theorem 16.

Theorem 19. In Theorem 18, if is the identity map on , then the conclusion following statements hold.(a)There exists a sequence in such that (b).(c)If one further assumes the following conditions hold: (L1) is a complete subspace of , (L3) for each sequence in with , and , one has as a closed subset of and ,then .

#### 4. New Generalizations of Berinde-Berinde’s Fixed Point Theorem and Mizoguchi-Takahashi’s Fixed Point Theorem

In this section, we will establish some new coincidence point theorems which generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem and some main results in [10, 14, 1719, 22].

Theorem 20. Let be a metric space, be a -function, be a -metric on induced by , be a multivalued map and be a self-map. Suppose that(i),(ii) is a complete subspace of ,(iii)there exists a nondecreasing function , an -function and a function such that then .

Moreover, if we further assume that for any , then and have a common fixed point in .

Proof. Let be given. Let with be arbitrary. Since , we have Hence (iii) implies (S1). So, following the same argument as the proof of Theorem 16, we can obtain two sequences and in satisfying the following:(i) for each ;(ii) for all ;(iii);(iv)There exists such that for each ;(v) for with , where for ;(vi);(vii) is a Cauchy sequence in .
By (ii), there exists such that as or . Since for with , from , we have So, for each , it follows from (40) and (42) that The last inequality implies that there exists such that Since , we get . By , we have . Since we obtain or as . Since for all and is closed, we have which means that . So .
Moreover, if we further assume that for all , then we have . For each , by (40) and (42) again, we have Therefore, there exists such that By (47) and , we have . By , we get . Since we have . Since is closed and for all , we get . Therefore, , which means that is a common fixed point of and in . The proof is completed.

Corollary 21. Let be a metric space, be a multivalued map and be a self-map. Suppose that(i),(ii) is a complete subspace of ,(iii)there exists a nondecreasing function , an -function and a function such that then .

The following results are immediate consequences of Theorem 20. They are generalizations of Berinde-Berinde’s fixed point and Mizoguchi-Takahashi’s fixed point theorem.

Theorem 22. Let be a complete metric space, be a -function, be a -metric on induced by , be a multivalued map, be a nondecreasing function, be an -function, and be a function. Suppose that then .

Corollary 23. Let be a complete metric space, be a multivalued map, be a nondecreasing function, be an -function, and be a function. Suppose that then .

Remark 24. (a) It is worth to mention that Theorem 22 is different from [10, Theorem 2.3]. Theorem 22 is comparable to [10, Theorem 2.3] in the following aspects.(1) In [10, Theorem 2.3], the map was assumed to satisfy where is an -function and is a given nonnegative real number. But in Theorem 22, we assume that where is a nondecreasing function, is an -function and is any function.(2) Notice that in [10, Theorem 2.3], the author assumed that further satisfies one of conditions (D1), (D2), (D3), (D4), and (D5), where(D1) is closed;(D2) the map defined by is l.s.c.;(D3) the map defined by is l.s.c.;(D4) for any sequence in with , and , we have ;(D5) for every . But Theorem 22 does not require the conditions (D1)–(D5).
(b) If we take , and for all in Theorem 22, then we obtain [17, Theorem 3.1].
(c) Theorems 20 and 22, and Corollary 21 all generalize Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle and some main results in [10, 14, 18, 19, 22] and references therein.
Here, we give a simple example illustrating Theorem 20.

Example 25. Let with the metric for . Then is a metric space. Let be defined by for all , . By Example 12, we know that is a -function. Let and be defined by and , respectively. Then is an -function and is a nondecreasing function. Let be defined by Then is a proper complete subspace of . Define by Clearly, . Let be defined by . We claim that for all . We consider the following six possible cases.

Case 1. Clearly, inequality holds for .

Case 2. If , then .

Case 3. If and , then and . So

Case 4. If and , then , and . Hence, we have

Case 5. If , then and . So, we have

Case 6. If , then , . Thus we obtain

By Cases 16, we verify that inequality holds for all . So all the hypotheses of Theorem 20 are fulfilled. It is therefore possible to apply Theorem 20 to get . In fact, .

Moreover, since for any , by Theorem 20 again, we know that and have a common fixed point in (precisely speaking, is the unique common fixed point of and ).

#### 5. Some Applications to New Coupled Coincidence Point Theorems and a Generalization of Kikkawa-Suzuki’s Fixed Point Theorem

Let be a metric space. We endow the product space with the metric defined by

Let be a self-map. Recall that an element is called a coupled coincidence point of the maps and if or In particular, if we take (the identity map) in (62) and (63), then is called a coupled fixed point of . The existence of coupled coincidence point and coupled fixed point has been investigated by several authors recently in [4, 11, 12, 23, 3037] and references therein.

As an interesting application of Corollary 21 (or Theorem 20), we establish the following new coupled coincidence point theorems. It is worth to mention that the technique in the proof of Theorem 26 is quite different in the well known literature.

Theorem 26. Let be a metric space, be a multivalued map and be a self-map satisfying and is a complete subspace of . Assume that there exists an -function such that for any ,