New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness

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 ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let (𝑋,𝑑) be a metric space. Denote by 𝒩(𝑋) the family of all nonempty subsets of 𝑋, 𝐶(𝑋) the class of all nonempty closed subsets of 𝑋, and 𝐶𝐵(𝑋) the family of all nonempty closed and bounded subsets of 𝑋. For each 𝑥∈𝑋 and 𝐴⊆𝑋, let 𝑑(𝑥,𝐴)=inf_𝑦∈𝐴𝑑(𝑥,𝑦). A function ℋ∶𝐶𝐵(𝑋)×𝐶𝐵(𝑋)→[0,∞) defined by ℋ(𝐴,𝐵)=maxsup_𝑥∈𝐵𝑑(𝑥,𝐴),sup_𝑥∈𝐴𝑑(𝑥,𝐵) (1) is said to be the Hausdorff metric on 𝐶𝐵(𝑋) induced by the metric 𝑑 on 𝑋.

Let 𝑔∶𝑋→𝑋 be a self-map and 𝑇∶𝑋→𝒩(𝑋) be a multivalued map. A point 𝑣 in 𝑋 is said to be a coincidence point (see, for instance, [1–4]) of 𝑔 and 𝑇 if 𝑔𝑣∈𝑇𝑣. The set of coincidence points of 𝑔 and 𝑇 is denoted by 𝐶𝑂𝑃(𝑔,𝑇). If 𝑣∈𝑇𝑣, then 𝑣 is called a fixed point of 𝑇. The set of fixed points of 𝑇 is denoted by ℱ(𝑇). The maps 𝑔 and 𝑇 are said to have an approximate coincidence point property [1, 4] on 𝑋 provided inf_𝑥∈𝑋𝑑(𝑔𝑥,𝑇𝑥)=0. The map 𝑇 is said to have the approximate fixed point property [1–5] on 𝑋 provided inf_𝑥∈𝑋𝑑(𝑥,𝑇𝑥)=0.

It is obvious that 𝐶𝑂𝑃(𝑔,𝑇)≠∅ (resp., ℱ(𝑇)≠∅) implies that 𝑇 has the approximate coincidence point property (resp., 𝑇 has the approximate fixed point property). Hussain et al. [1, Theorem 2.6] showed that a generalized multivalued almost contraction 𝑇 in a metric space (𝑋,𝑑) have ℱ(𝑇)≠∅ provided either (𝑋,𝑑) is compact and the function 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c. or 𝑇 is closed and compact. In [1, Lemma 2.2], the authors had also shown that every generalized multivalued almost contraction in a metric space (𝑋,𝑑) 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 [1–47] 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 𝜃 from [0,1) onto (1/2,1] by 𝜃(𝑟)=11,𝑖𝑓0≤𝑟≤2√5−1,1−𝑟𝑟²1,𝑖𝑓2√5−1≤𝑟≤1√2,111+𝑟,𝑖𝑓√2≤𝑟<1. (2) Then for a metric space (𝑋,𝑑), the following are equivalent:(1)𝑋is complete.(2)Every mapping 𝑇 on 𝑋 satisfying the following has a fixed point: there exists 𝑟∈[0,1) such that 𝜃(𝑟)𝑑(𝑥,𝑇𝑥)≤𝑑(𝑥,𝑦) implies 𝑑(𝑇𝑥,𝑇𝑦)≤𝑟𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. (3)There exists 𝑟∈[0,1) such that every mapping 𝑇 on 𝑋 satisfying the following has a fixed point: (1/10000)𝑑(𝑥,𝑇𝑥)≤𝑑(𝑥,𝑦) implies 𝑑(𝑇𝑥,𝑇𝑦)≤𝑟𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋.

Remark 2 (see [6]). For every 𝑟∈[0,1), 𝜃(𝑟) 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 𝜂 from [0,1) onto (1/2,1] by 𝜂(𝑟)=1.1+𝑟 (3) Let (𝑋,𝑑) be a complete metric space and let 𝑇 be a map from 𝑋 into 𝐶𝐵(𝑋). Assume that there exists 𝑟∈[0,1) such that 𝜂(𝑟)𝑑(𝑥,𝑇𝑥)≤𝑑𝑥,𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠ℋ𝑇𝑥,𝑇𝑦≤𝑟𝑑𝑥,𝑦 (4) for all 𝑥,𝑦∈𝑋. Then ℱ(𝑇)≠∅.

Let 𝑓 be a real-valued function defined on ℝ. For 𝑐∈ℝ, we recall that limsup_𝑥→𝑐+𝑓(𝑥)=inf_𝜀>0sup_{𝑐<𝑥<𝜀+𝑐}𝑓(𝑥). (5)

Definition 4 (see [3, 4, 10–19]). A function 𝜑∶ [0,∞)→ [0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsup_𝑠→𝑡+𝜑(𝑠)<1 for all 𝑡∈[0,∞).

It is obvious that if 𝜑∶ [0,∞)→ [0,1) is a nondecreasing function or a nonincreasing function, then 𝜑 is an ℳ𝒯-function. So the set of ℳ𝒯-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 (𝑋,𝑑) be a complete metric space, 𝛼∶ [0,∞)→ [0,1) be a ℳ𝒯-function and 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦, (6) for all 𝑥,𝑦∈𝑋. Then ℱ(𝑇)≠∅.

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 (𝑋,𝑑) be a complete metric space, 𝛼∶ [0,∞)→ [0,1) be a ℳ𝒯-function, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map and 𝐿≥0. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥, (7) for all 𝑥,𝑦∈𝑋. Then ℱ(𝑇)≠∅.

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, [4–10, 13, 14, 17–23, 30] and references therein.

2. Preliminaries

Recall that a function 𝑝∶𝑋×𝑋→[0,∞) is called a 𝑤-distance [5, 7, 10, 14, 15, 24–29, 38–40], if the following are satisfied:(𝑤1)𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for any 𝑥,𝑦,𝑧∈𝑋;(𝑤2) for any 𝑥∈𝑋, 𝑝(𝑥,⋅)∶𝑋→[0,∞) is l.s.c.;(𝑤3) for any 𝜀>0, there exists 𝛿>0 such that 𝑝(𝑧,𝑥)≤𝛿 and 𝑝(𝑧,𝑦)≤𝛿 imply 𝑑(𝑥,𝑦)≤𝜀.

A function 𝑝∶𝑋×𝑋→[0,∞) is said to be a 𝜏-function [5, 10, 14, 15, 27–29], first introduced and studied by Lin and Du, if the following conditions hold:(𝜏1)𝑝(𝑥,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋;(𝜏2) if 𝑥∈𝑋 and {𝑦_𝑛} in 𝑋 with lim_𝑛→∞𝑦_𝑛=𝑦 such that 𝑝(𝑥,𝑦_𝑛)≤𝑀 for some 𝑀=𝑀(𝑥)>0, then 𝑝(𝑥,𝑦)≤𝑀;(𝜏3) for any sequence {𝑥_𝑛} in 𝑋 with lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0, if there exists a sequence {𝑦_𝑛} in 𝑋 such that lim_𝑛→∞𝑝(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑦_𝑛)=0;(𝜏4) for 𝑥,𝑦,𝑧∈𝑋, 𝑝(𝑥,𝑦)=0 and 𝑝(𝑥,𝑧)=0 imply 𝑦=𝑧.

Note that not either of the implications 𝑝(𝑥,𝑦)=0 ⇔𝑥=𝑦 necessarily holds and 𝑝 is nonsymmetric in general. It is well known that the metric 𝑑 is a 𝑤-distance and any 𝑤-distance is a 𝜏-function, but the converse is not true; see [27, 29] for more detail.

Definition 7 (see [4]). Let (𝑋,𝑑) be a metric space, 𝑝 be a 𝜏-function, 𝑔∶𝑋→𝑋 be a single-valued self-map and 𝑇∶𝑋→𝒩(𝑋) be a multivalued map.(1)The maps 𝑔 and 𝑇 are said to have the 𝑝-approximate coincidence point property on 𝑋 provided inf_𝑥∈𝑋𝑝𝑔𝑥,𝑇𝑥=0. (8)(2)The map 𝑇 is said to have the 𝑝-approximate fixed point property on 𝑋 provided inf_𝑥∈𝑋𝑝(𝑥,𝑇𝑥)=0. (9)The following results are crucial in this paper.

Lemma 8 (see [29, Lemma 2.1]). Let (𝑋,𝑑) be a metric space and 𝑝∶𝑋×𝑋→[0,∞) be a function. Assume that 𝑝 satisfies the condition (𝜏3). If a sequence {𝑥_𝑛} in 𝑋 with lim_𝑛→∞sup{𝑝(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0, then {𝑥_𝑛} is a Cauchy sequence in 𝑋.

For each 𝑥∈𝑋 and 𝐴⊆𝑋, we denote 𝑝(𝑥,𝐴)=inf_𝑦∈𝐴𝑝(𝑥,𝑦).

Lemma 9 (see [10]). Let 𝐴 be a closed subset of a metric space (𝑋,𝑑) and 𝑝∶𝑋×𝑋→[0,∞) be any function. Suppose that 𝑝 satisfies (𝜏3) and there exists 𝑢∈𝑋 such that 𝑝(𝑢,𝑢)=0. Then 𝑝(𝑢,𝐴)=0  if and only if 𝑢∈𝐴.

The concepts of 𝜏⁰-functions and 𝜏⁰-metrics were introduced in [10] as follows.

Definition 10 (see [10]). Let (𝑋,𝑑) be a metric space. A function 𝑝∶𝑋×𝑋→[0,∞) is called a 𝜏⁰-function if it is a 𝜏-function on 𝑋 with 𝑝(𝑥,𝑥)=0 for all 𝑥∈𝑋.

Remark 11. If 𝑝 is a 𝜏⁰-function, then, from (𝜏4), 𝑝(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Example 12 (see [10]). Let 𝑋=ℝ with the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| and 0<𝑎<𝑏. Define the function 𝑝∶𝑋×𝑋→[0,∞) by 𝑝𝑥,𝑦=max𝑎𝑦−𝑥,𝑏𝑥−𝑦. (10) Then 𝑝 is nonsymmetric and hence 𝑝 is not a metric. It is easy to see that 𝑝 is a 𝜏⁰-function.

Definition 13 (see [10]). Let (𝑋,𝑑) be a metric space and 𝑝 be a 𝜏⁰-function. For any 𝐴, 𝐵∈𝐶𝐵(𝑋), define a function 𝒟_𝑝∶𝐶𝐵(𝑋)×𝐶𝐵(𝑋)→[0,∞) by 𝒟_𝑝(𝐴,𝐵)=max𝛿_𝑝(𝐴,𝐵),𝛿_𝑝(𝐵,𝐴), (11) where 𝛿_𝑝(𝐴,𝐵)=sup_𝑥∈𝐴𝑝(𝑥,𝐵), then 𝒟_𝑝 is said to be the 𝜏⁰-metric on 𝐶𝐵(𝑋) induced by 𝑝.

Clearly, any Hausdorff metric is a 𝜏⁰-metric, but the reverse is not true.

Lemma 14 (see [10]). Let (𝑋,𝑑) be a metric space and 𝒟_𝑝 be a 𝜏⁰-metric on 𝐶𝐵(𝑋) induced by a 𝜏⁰-function 𝑝. Then every 𝜏⁰-metric 𝒟_𝑝 is a metric on 𝐶𝐵(𝑋).

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

Lemma 15 (see [18]). Let 𝜑∶ [0,∞)→ [0,1) be a function. Then the following statements are equivalent. (a)𝜑 is an ℳ𝒯-function.(b)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽¹⁾∈[0,1) and 𝜀_𝑡⁽¹⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽¹⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽¹⁾).(c)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽²⁾∈[0,1) and 𝜀_𝑡⁽²⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽²⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽²⁾].(d)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽³⁾∈[0,1) and 𝜀_𝑡⁽³⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽³⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽³⁾].(e)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽⁴⁾∈[0,1) and 𝜀_𝑡⁽⁴⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽⁴⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽⁴⁾).(f)For any nonincreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), one has 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.(g)𝜑 is a function of contractive factor [12]; that is, for any strictly decreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), one has 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.

3. New Nonlinear Conditions for 𝑝-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 (𝑋,𝑑) be a metric space, 𝑝 be a 𝜏⁰-function, 𝑇∶𝑋→𝒩(𝑋) be a multivalued map, and 𝑓∶𝑋→𝑋 be a self-map. Suppose that(S1) there exist a nondecreasing function 𝜏∶[0,∞)→ [0,∞) and an ℳ𝒯-function 𝜑∶ [0,∞)→ [0,1) such that for each 𝑥∈𝑋, if 𝑦∈𝑋 with 𝑓𝑦≠𝑓𝑥 and 𝑓𝑦∈𝑇𝑥, then it holds 𝑝𝑓𝑦,𝑇𝑦≤𝜑𝜏𝑝𝑓𝑥,𝑓𝑦𝑝𝑓𝑥,𝑓𝑦. (12)(S2)𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋).
Then the following statements hold.(a)There exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 such that inf_𝑛∈ℕ𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=inf_𝑛∈ℕ𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=0. (13)(b) inf_𝑥∈𝑋𝑝(𝑓𝑥,𝑇𝑥)=inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑇𝑥)=0; that is, 𝑓 and 𝑇 have the 𝑝-approximate coincidence point property and approximate coincidence point property on 𝑋.(c)If one further assumes the following conditions hold:(L1)𝑓(𝑋) is a complete subspace of 𝑋,(L2) for each sequence {𝑥_𝑛} in 𝑋 with 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤, one has 𝑇𝑤 as a closed subset of 𝑋 and lim_𝑛→∞𝑝(𝑓𝑥_𝑛,𝑇𝑤)=0, then 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

Proof. Let 𝑥₁∈𝑋. By (S2), there exists 𝑥₂∈𝑋 such that 𝑓𝑥₂∈𝑇𝑥₁. If 𝑓𝑥₁=𝑓𝑥₂, then 𝑓𝑥₁∈𝑇𝑥₁ and so inf_𝑥∈𝑋𝑝𝑓𝑥,𝑇𝑥≤𝑝𝑓𝑥₁,𝑇𝑥₁≤𝑝𝑓𝑥₁,𝑓𝑥₁=0, (14) which implies inf_𝑥∈𝑋𝑝(𝑓𝑥,𝑇𝑥)=0. Clearly, inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑇𝑥)=0. Let 𝑤_𝑛=𝑥₁ for all 𝑛∈ℕ. Then lim_𝑛→∞𝑝𝑓𝑤_𝑛,𝑓𝑤_𝑛+1=inf_𝑛∈ℕ𝑝𝑓𝑤_𝑛,𝑓𝑤_𝑛+1=𝑝𝑓𝑥₁,𝑓𝑥₁=0,lim_𝑛→∞𝑑𝑓𝑤_𝑛,𝑓𝑤_𝑛+1=inf_𝑛∈ℕ𝑑𝑓𝑤_𝑛,𝑓𝑤_𝑛+1=𝑑𝑓𝑥₁,𝑓𝑥₁=0. (15) So, the conclusions (a) and (b) hold in this case. Otherwise, if 𝑓𝑥₂≠𝑓𝑥₁, since 𝑝 is a 𝜏⁰-function, 𝑝(𝑓𝑥₁,𝑓𝑥₂)>0. Let 𝜇∶[0,∞)→[0,1) be defined by 𝜇(𝑡)=(1+𝜑(𝑡))/2. Clearly, 0≤𝜑(𝑡)<𝜇(𝑡)<1 for all 𝑡∈[0,∞). By [3, Lemma 2.1], we know that 𝜇 is also an ℳ𝒯-function. From (S1), we get 𝑝𝑓𝑥₂,𝑇𝑥₂≤𝜑𝜏𝑝𝑓𝑥₁,𝑓𝑥₂𝑝𝑓𝑥₁,𝑓𝑥₂<𝜇𝜏𝑝𝑓𝑥₁,𝑓𝑥₂𝑝𝑓𝑥₁,𝑓𝑥₂. (16) Since 𝜇(𝜏(𝑝(𝑓𝑥₁,𝑓𝑥₂)))𝑝(𝑓𝑥₁,𝑓𝑥₂)>0, there exists 𝜉∈𝑇𝑥₂ such that 𝑝𝑓𝑥₂,𝜉<𝜇𝜏𝑝𝑓𝑥₁,𝑓𝑥₂𝑝𝑓𝑥₁,𝑓𝑥₂. (17) Using (S2) again, there exists 𝑥₃∈𝑋 such that 𝑓𝑥₃=𝜉∈𝑇𝑥₂. Hence, from (17), we have 𝑝𝑓𝑥₂,𝑓𝑥₃<𝜇𝜏𝑝𝑓𝑥₁,𝑓𝑥₂𝑝𝑓𝑥₁,𝑓𝑥₂. (18) If 𝑓𝑥₂=𝑓𝑥₃∈𝑇𝑥₂, then, following a similar argument as above, we can prove the conclusions (a) and (b). Otherwise, if 𝑓𝑥₃≠𝑓𝑥₂, then there exists 𝑥₄∈𝑋 such that 𝑓𝑥₄∈𝑇𝑥₃ and 𝑝𝑓𝑥₃,𝑓𝑥₄<𝜇𝜏𝑝𝑓𝑥₂,𝑓𝑥₃𝑝𝑓𝑥₂,𝑓𝑥₃. (19) By induction, we can obtain a sequences {𝑥_𝑛} in 𝑋 satisfying 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛,𝑝𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+2<𝜇𝜏𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1,foreach𝑛∈ℕ. (20)(21) Since 𝜇(𝑡)<1 for all 𝑡∈[0,∞), we deduces from the inequality (21) that the sequence {𝑝(𝑓𝑥_𝑛, 𝑓𝑥_𝑛+1)}_𝑛∈ℕ is strictly decreasing in [0,∞). Hence lim_𝑛→∞𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=inf_𝑛∈ℕ𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≥0exists. (22) Since 𝜏 is nondecreasing, {𝜏(𝑝(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1))}_𝑛∈ℕ is a nonincreasing sequence in [0,∞). Since 𝜇 is an ℳ𝒯-function, by (f) of Lemma 15, we have 0≤sup_𝑛∈ℕ𝜇𝜏𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<1. (23) Let 𝛾∶=sup_𝑛∈ℕ𝜇(𝜏(𝑝(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1))). So 𝛾∈[0,1). Put 𝜆∶=(1+𝛾)/2. Then 0≤𝛾<𝜆<1. By (21), we get 𝑝𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+2<𝜇𝜏𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤𝛾𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<𝜆𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<⋯<𝜆^𝑛𝑝𝑓𝑥₁,𝑓𝑥₂foreach𝑛∈ℕ. (24) Since 𝜆∈(0,1), lim_𝑛→∞𝜆^𝑛=0 and hence it follows from (24) that lim_𝑛→∞𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=0. (25) According to (22) and (25), we obtain inf_𝑛∈ℕ𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=0. (26)
Next, we verify that {𝑓𝑥_𝑛} is a Cauchy sequence in 𝑓(𝑋). Let 𝑣_𝑛∶=𝑓𝑥_𝑛 for all 𝑛∈ℕ. We claim that lim_𝑛→∞sup{𝑝(𝑣_𝑛,𝑣_𝑚)∶𝑚>𝑛}=0. Put𝛼_𝑛=𝜆^𝑛−1𝑝𝑣1−𝜆¹,𝑣₂,𝑛∈ℕ. (27)
Then 𝛼_𝑛>0 for all 𝑛∈ℕ. For 𝑚,𝑛∈ℕ with 𝑚>𝑛, by (24), we have 𝑝𝑣_𝑛,𝑣_𝑚≤^𝑚−1_𝑗=𝑛𝑝𝑣_𝑗,𝑣_𝑗+1<𝛼_𝑛. (28) Since 𝜆∈(0,1), lim_𝑛→∞𝛼_𝑛=0 and hence lim_𝑛→∞sup𝑝𝑣_𝑛,𝑣_𝑚∶𝑚>𝑛=0. (29) By Lemma 8, {𝑣_𝑛} is a Cauchy sequence in 𝑓(𝑋). Hence lim_𝑛→∞𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑑𝑣_𝑛,𝑣_𝑛+1=0. (30) Since inf_𝑛∈ℕ𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)≤𝑑(𝑓𝑥_𝑚,𝑓𝑥_𝑚+1) for all 𝑚∈ℕ and lim_𝑚→∞𝑑(𝑓𝑥_𝑚,𝑓𝑥_𝑚+1)=0, one also obtain lim_𝑛→∞𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=inf_𝑛∈ℕ𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=0. (31) Hence (a) is proved. To see (b), since 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛 for each 𝑛∈ℕ, we have inf_𝑥∈𝑋𝑝𝑓𝑥,𝑇𝑥≤𝑝𝑓𝑥_𝑛,𝑇𝑥_𝑛≤𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1,inf_𝑥∈𝑋𝑑𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1, (32) for all 𝑛∈ℕ. Since lim_𝑛→∞𝑝(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)=lim_𝑛→∞𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)=0, combining (32), we get inf_𝑥∈𝑋𝑝𝑓𝑥,𝑇𝑥=inf_𝑥∈𝑋𝑑𝑓𝑥,𝑇𝑥=0. (33) Moreover, if we further assume that conditions (L1) and (L2) hold, we want to show 𝐶𝑂𝑃(𝑓,𝑇)≠∅. Since {𝑣_𝑛} is a Cauchy sequence in 𝑓(𝑋), by (L1), there exists 𝑣̂∈𝑋 such that 𝑣_𝑛≡𝑓𝑥_𝑛→ 𝑓𝑣̂ as 𝑛→∞. So, by (L2), we have 𝑇𝑣̂ is a closed subset of 𝑋 and lim_𝑛→∞𝑝(𝑣_𝑛,𝑇𝑣)̂=0. Since lim_𝑛→∞sup{𝑝(𝑣_𝑛,𝑣_𝑚)∶𝑚>𝑛}=0, there exists {𝑎_𝑛}⊂{𝑣_𝑛} with lim_𝑛→∞sup{𝑝(𝑎_𝑛,𝑎_𝑚)∶𝑚>𝑛}=0 and {𝑏_𝑛}⊂𝑇𝑣̂ such that lim_𝑛→∞𝑝(𝑎_𝑛,𝑏_𝑛)=0. By (𝜏3), lim_𝑛→∞𝑑(𝑎_𝑛,𝑏_𝑛)=0. Since 𝑎_𝑛→𝑓𝑣̂ as 𝑛→∞ and 𝑑(𝑏_𝑛,𝑓𝑣)̂≤𝑑(𝑏_𝑛,𝑎_𝑛)+𝑑(𝑎_𝑛,𝑓𝑣)̂, it implies 𝑏_𝑛→𝑓𝑣̂ as 𝑛→∞. By the closedness of 𝑇𝑣̂, we have 𝑓𝑣̂∈𝑇𝑣̂ or 𝑣̂∈𝐶𝑂𝑃(𝑓,𝑇). The proof is completed.

Theorem 17. In Theorem 16, if 𝑓≡𝑖𝑑 is the identity map on 𝑋, then the following statements hold.(a)There exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 such that inf_𝑛∈ℕ𝑝𝑥_𝑛,𝑥_𝑛+1=lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=inf_𝑛∈ℕ𝑑𝑥_𝑛,𝑥_𝑛+1=0. (34)(b)inf_𝑥∈𝑋𝑝(𝑥,𝑇𝑥)=inf_𝑥∈𝑋𝑑(𝑥,𝑇𝑥)=0; that is, 𝑇 has the 𝑝-approximate fixed point property and approximate fixed point property on 𝑋.(c)If one further assumes the following conditions hold:(L1)𝑓(𝑋) is a complete subspace of 𝑋,(L3) for each sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑤, one has 𝑇𝑤 as a closed subset of 𝑋 and lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑤)=0, then ℱ(𝑇)≠∅.

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

Theorem 18. Let (𝑋,𝑑) be a metric space, 𝑝 be a 𝜏⁰-function, 𝒟_𝑝 be a 𝜏⁰-metric on 𝐶𝐵(𝑋) induced by 𝑝, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map and 𝑓∶𝑋→𝑋 be a self-map. Suppose that (S2) as in Theorem 16 is satisfied and 𝑇 further satisfies one of the following conditions:(H1) there exist a nondecreasing function 𝜏∶[0,∞)→ [0,∞), an ℳ𝒯-function 𝜑∶ [0,∞)→ [0,1) and a function 𝐿∶ 𝑋×𝑋→ [0,∞) such that 𝒟_𝑝𝜏𝑝𝑝𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+𝐿𝑓𝑥,𝑓𝑦𝑓𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (35)(H2) there exist a nondecreasing function 𝜏∶[0,∞)→ [0,∞), an ℳ𝒯-function 𝜑∶ [0,∞)→ [0,1) and a function 𝐿∶ 𝑋×𝑋→ [0,∞) such that 𝑝𝜏𝑝𝑝𝑝𝑓𝑦,𝑇𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+𝐿𝑓𝑥,𝑓𝑦𝑓𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (36)
Then the following statements hold.(a)There exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 such that inf_𝑛∈ℕ𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑝𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=lim_𝑛→∞𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=inf_𝑛∈ℕ𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1=0. (37)(b)inf_𝑥∈𝑋𝑝(𝑓𝑥,𝑇𝑥)=inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑇𝑥)=0. If one further assumes the following conditions hold:(L1)𝑓(𝑋) is a complete subspace of 𝑋,(L2) for each sequence {𝑥_𝑛} in 𝑋 with 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤, one has 𝑇𝑤 as a closed subset of 𝑋 and lim_𝑛→∞𝑝(𝑓𝑥_𝑛,𝑇𝑤)=0, then 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

Proof. Suppose that (H1) holds. We first notice that for each 𝑥∈𝑋, by (S2), the set {𝑦∈𝑋∶𝑓𝑦∈𝑇𝑥}≠∅. Let 𝑥∈𝑋 be given and let 𝑦∈𝑋 with 𝑓𝑦∈𝑇𝑥 be arbitrary. Since 𝑝(𝑓𝑦,𝑇𝑥)=0, we have 𝑝𝑓𝑦,𝑇𝑦≤𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝜏𝑝𝑓𝑥,𝑓𝑦𝑝𝑓𝑥,𝑓𝑦. (38) 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 𝑓≡𝑖𝑑 is the identity map on 𝑋, then the conclusion following statements hold.(a)There exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 such that inf_𝑛∈ℕ𝑝𝑥_𝑛,𝑥_𝑛+1=lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=inf_𝑛∈ℕ𝑑𝑥_𝑛,𝑥_𝑛+1=0. (39)(b)inf_𝑥∈𝑋𝑝(𝑥,𝑇𝑥)=inf_𝑥∈𝑋𝑑(𝑥,𝑇𝑥)=0.(c)If one further assumes the following conditions hold: (L1)𝑓(𝑋) is a complete subspace of 𝑋, (L3) for each sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑤, one has 𝑇𝑤 as a closed subset of 𝑋 and lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑤)=0, then ℱ(𝑇)≠∅.

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, 17–19, 22].

Theorem 20. Let (𝑋,𝑑) be a metric space, 𝑝 be a 𝜏⁰-function, 𝒟_𝑝 be a 𝜏⁰-metric on 𝐶𝐵(𝑋) induced by 𝑝, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map and 𝑓∶𝑋→𝑋 be a self-map. Suppose that(i)𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋),(ii)𝑓(𝑋) is a complete subspace of 𝑋,(iii)there exists a nondecreasing function 𝜏∶[0,∞)→ [0,∞), an ℳ𝒯-function 𝜑∶ [0,∞)→ [0,1) and a function ℎ∶ 𝑋→ [0,∞) such that 𝒟_𝑝𝜏𝑝𝑝𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (40)then 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

Moreover, if we further assume that 𝑓𝑤=𝑓𝑓𝑤 for any 𝑤∈𝐶𝑂𝑃(𝑓,𝑇), then 𝑇 and 𝑓 have a common fixed point in 𝑋.

Proof. Let 𝑥∈𝑋 be given. Let 𝑦∈𝑋 with 𝑓𝑦∈𝑇𝑥 be arbitrary. Since 𝑑(𝑓𝑦,𝑇𝑥)=0, we have 𝑝𝑓𝑦,𝑇𝑦≤𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝜏𝑝𝑓𝑥,𝑓𝑦𝑝𝑓𝑥,𝑓𝑦. (41) Hence (iii) implies (S1). So, following the same argument as the proof of Theorem 16, we can obtain two sequences {𝑥_𝑛} and {𝑣_𝑛} in 𝑋 satisfying the following:(i)𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛 for each 𝑛∈ℕ;(ii)𝑣_𝑛∶=𝑓𝑥_𝑛 for all 𝑛∈ℕ;(iii)𝑝(𝑣₁,𝑣₂)>0;(iv)There exists 𝜆∈(0,1) such that 𝑝(𝑣_𝑛+1,𝑣_𝑛+2)≤𝜆^𝑛𝑝(𝑣₁,𝑣₂) for each 𝑛∈ℕ;(v)𝑝(𝑣_𝑛,𝑣_𝑚)<𝛼_𝑛 for 𝑚,𝑛∈ℕ with 𝑚>𝑛, where 𝛼_𝑛=(𝜆^𝑛−1/(1−𝜆))𝑝(𝑣₁,𝑣₂)>0 for 𝑛∈ℕ;(vi)lim_𝑛→∞sup{𝑝(𝑣_𝑛,𝑣_𝑚)∶𝑚>𝑛}=0;(vii){𝑣_𝑛} is a Cauchy sequence in 𝑓(𝑋).
By (ii), there exists 𝑣∈𝑋 such that 𝑓𝑥_𝑛≡𝑣_𝑛→ 𝑓𝑣 as 𝑛→∞ or lim_𝑛→∞𝑑(𝑣_𝑛,𝑓𝑣)=0. Since 𝑝(𝑣_𝑛,𝑣_𝑚)<𝛼_𝑛 for 𝑚,𝑛∈ℕ with 𝑚>𝑛, from (𝜏2), we have 𝑝𝑓𝑥_𝑛,𝑓𝑣=𝑝𝑣_𝑛,𝑓𝑣≤𝛼_𝑛∀𝑛∈ℕ. (42) So, for each 𝑛∈ℕ, it follows from (40) and (42) that 𝑝𝑣_𝑛+1,𝑇𝑣=𝑝𝑓𝑥_𝑛+1,𝑇𝑣≤sup_{𝑦∈𝑇𝑥𝑛}𝑝𝑦,𝑇𝑣≤𝒟_𝑝𝑇𝑥_𝑛,𝑇𝑣≤𝜑𝜏𝑝𝑓𝑥_𝑛,𝑓𝑣𝑝𝑓𝑥_𝑛,𝑓𝑣+ℎ(𝑣)𝑑𝑓𝑣,𝑇𝑥_𝑛<𝑝𝑓𝑥_𝑛,𝑓𝑣+ℎ(𝑣)𝑑𝑓𝑣,𝑓𝑥_𝑛+1≤𝛼_𝑛+ℎ(𝑣)𝑑𝑓𝑣,𝑓𝑥_𝑛+1. (43) The last inequality implies that there exists 𝑦_𝑛+1∈𝑇𝑣 such that 𝑝𝑣_𝑛+1,𝑦_𝑛+1<𝛼_𝑛+ℎ(𝑣)𝑑𝑓𝑣,𝑓𝑥_𝑛+1foreach𝑛∈ℕ. (44) Since lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝑑(𝑓𝑣,𝑓𝑥_𝑛+1)=0, we get lim_𝑛→∞𝑝(𝑣_𝑛+1,𝑦_𝑛+1)=0. By (𝜏3), we have lim_𝑛→∞𝑑(𝑣_𝑛+1,𝑦_𝑛+1)=0. Since 0≤𝑑𝑦_𝑛+1,𝑓𝑣≤𝑑𝑦_𝑛+1,𝑣_𝑛+1+𝑑𝑣_𝑛+1,𝑓𝑣∀𝑛∈ℕ, (45) we obtain lim_𝑛→∞𝑑(𝑦_𝑛+1,𝑓𝑣)=0 or 𝑦_𝑛+1→ 𝑓𝑣 as 𝑛→∞. Since 𝑦_𝑛+1∈𝑇𝑣 for all 𝑛∈ℕ and 𝑇𝑣 is closed, we have 𝑓𝑣∈𝑇𝑣 which means that 𝑣∈𝐶𝑂𝑃(𝑓,𝑇). So 𝐶𝑂𝑃(𝑓,𝑇)≠∅.
Moreover, if we further assume that 𝑓𝑤=𝑓𝑓𝑤 for all 𝑤∈𝐶𝑂𝑃(𝑓,𝑇), then we have 𝑓𝑣=𝑓𝑓𝑣. For each 𝑛∈ℕ, by (40) and (42) again, we have 𝑝𝑣_𝑛+1,𝑇𝑓𝑣=𝑝𝑓𝑥_𝑛+1,𝑇𝑓𝑣≤𝒟_𝑝𝑇𝑥_𝑛,𝑇𝑓𝑣≤𝜑𝜏𝑝𝑓𝑥_𝑛,𝑓𝑓𝑣𝑝𝑓𝑥_𝑛,𝑓𝑓𝑣+ℎ𝑓𝑣𝑑𝑓𝑓𝑣,𝑇𝑥_𝑛<𝑝𝑓𝑥_𝑛,𝑓𝑣+ℎ𝑓𝑣𝑑𝑓𝑣,𝑓𝑥_𝑛+1≤𝛼_𝑛+ℎ𝑓𝑣𝑑𝑓𝑣,𝑓𝑥_𝑛+1. (46) Therefore, there exists 𝑧_𝑛+1∈𝑇𝑓𝑣 such that 𝑝𝑣_𝑛+1,𝑧_𝑛+1<𝛼_𝑛+ℎ𝑓𝑣𝑑𝑓𝑣,𝑓𝑥_𝑛+1∀𝑛∈ℕ. (47) By (47) and lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝑑(𝑓𝑣,𝑓𝑥_𝑛+1)=0, we have lim_𝑛→∞𝑝(𝑣_𝑛+1,𝑧_𝑛+1)=0. By (𝜏3), we get lim_𝑛→∞𝑑(𝑣_𝑛+1,𝑧_𝑛+1)=0. Since 𝑑𝑓𝑣,𝑧_𝑛+1≤𝑑𝑓𝑣,𝑣_𝑛+1+𝑑𝑣_𝑛+1,𝑧_𝑛+1, (48) we have lim_𝑛→∞𝑑(𝑓𝑣,𝑧_𝑛+1)=0. Since 𝑇𝑓𝑣 is closed and 𝑧_𝑛∈𝑇𝑓𝑣 for all 𝑛∈ℕ, we get 𝑓𝑣∈𝑇𝑓𝑣. Therefore, 𝑓𝑣=𝑓𝑓𝑣∈𝑇𝑓𝑣, which means that 𝑓𝑣 is a common fixed point of 𝑓 and 𝑇 in 𝑋. The proof is completed.

Corollary 21. Let (𝑋,𝑑) be a metric space, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map and 𝑓∶𝑋→𝑋 be a self-map. Suppose that(i)𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋),(ii)𝑓(𝑋) is a complete subspace of 𝑋,(iii)there exists a nondecreasing function 𝜏∶[0,∞)→ [0,∞), an ℳ𝒯-function 𝜑∶ [0,∞)→ [0,1) and a function ℎ∶ 𝑋→ [0,∞) such that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝜏𝑑𝑓𝑥,𝑓𝑦𝑑𝑓𝑥,𝑓𝑦+ℎ𝑦𝑑𝑓𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (49)then 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

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 (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, 𝒟_𝑝 be a 𝜏⁰-metric on 𝐶𝐵(𝑋) induced by 𝑝, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map, 𝜏∶[0,∞)→ [0,∞) be a nondecreasing function, 𝜑∶ [0,∞)→ [0,1) be an ℳ𝒯-function, and ℎ∶ 𝑋→ [0,∞) be a function. Suppose that 𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝜏𝑝𝑥,𝑦𝑝𝑥,𝑦+ℎ𝑦𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (50) then ℱ(𝑇)≠∅.

Corollary 23. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map, 𝜏∶[0,∞)→ [0,∞) be a nondecreasing function, 𝜑∶ [0,∞)→ [0,1) be an ℳ𝒯-function, and ℎ∶ 𝑋→ [0,∞) be a function. Suppose that ℋ𝜏𝑑𝑑𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑥,𝑦+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (51) then ℱ(𝑇)≠∅.

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 𝑇 was assumed to satisfy 𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝𝑥,𝑦+𝐿𝑝𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋 (52) where 𝜑 is an ℳ𝒯-function and 𝐿 is a given nonnegative real number. But in Theorem 22, we assume that 𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝜏𝑝𝑥,𝑦𝑝𝑥,𝑦+ℎ𝑦𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (53) where 𝜏 is a nondecreasing function, 𝜑 is an ℳ𝒯-function and ℎ∶ 𝑋→ [0,∞) is any function.(2) Notice that in [10, Theorem 2.3], the author assumed that 𝑇 further satisfies one of conditions (D1), (D2), (D3), (D4), and (D5), where (D1) 𝑇 is closed; (D2) the map 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑝(𝑥,𝑇𝑥) is l.s.c.; (D3) the map 𝑔∶𝑋→[0,∞) defined by 𝑔(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.; (D4) for any sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑣, we have lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑣)=0; (D5) inf{𝑝(𝑥,𝑧)+𝑝(𝑥,𝑇𝑥)∶ 𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇).  But Theorem 22 does not require the conditions (D1)–(D5).
(b) If we take 𝜏(𝑡)=𝑡, 𝑡∈[0,∞) and ℎ(𝑡)=0 for all 𝑥∈𝑋 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 𝑋=(−9876,−10)∪[0,∞) with the metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| for 𝑥,𝑦∈𝑋. Then (𝑋,𝑑) is a metric space. Let 𝑝∶𝑋×𝑋→[0,∞) be defined by 𝑝𝑥,𝑦=max2𝑥−𝑦,3𝑦−𝑥, (54) for all 𝑥, 𝑦∈𝑋. By Example 12, we know that 𝑝 is a 𝜏⁰-function. Let 𝜑∶ [0,∞)→ [0,1) and 𝜏∶[0,∞)→ [0,∞) be defined by 𝜑(𝑡)=1/2 and 𝜏(𝑡)=5, respectively. Then 𝜑 is an ℳ𝒯-function and 𝜏 is a nondecreasing function. Let 𝑓∶𝑋→𝑋 be defined by 𝑓(𝑥)(),[).∶=0,if𝑥∈−9876,−102𝑥,if𝑥∈0,∞ (55) Then 𝑓(𝑋)=[0,∞) is a proper complete subspace of 𝑋. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑇(𝑥){0}(),[][).∶=,if𝑥∈−9876,−100,𝑥,if𝑥∈0,∞ (56) Clearly, 𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋). Let ℎ∶ 𝑋→ [0,∞) be defined by ℎ(𝑥)=𝑒^𝑥. We claim that 𝒟_𝑝𝜏𝑝𝑝𝑦𝑑,𝑇𝑥,𝑇𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥 (∗) for all 𝑥,𝑦∈𝑋. We consider the following six possible cases.

Case 1. Clearly, inequality (∗) holds for 𝑥=𝑦∈𝑋.

Case 2. If 𝑥,𝑦∈(−9876,−10), then 𝒟_𝑝(𝑇𝑥,𝑇𝑦)=0=𝜑(𝜏(𝑝(𝑓𝑥,𝑓𝑦)))𝑝(𝑓𝑥,𝑓𝑦)+ℎ(𝑦)𝑑(𝑓𝑦,𝑇𝑥).

Case 3. If 𝑥∈(−9876,−10) and 𝑦∈[0,∞), then 𝑇𝑥={0} and 𝑇𝑦=[0,𝑦]. So 𝑝2𝒟𝑓𝑥,𝑓𝑦=max𝑓𝑥−𝑓𝑦,3𝑓𝑦−𝑓𝑥=max−4𝑦,6𝑦=6𝑦,_𝑝𝑇𝑥,𝑇𝑦=maxsup_{𝑧∈𝑇𝑥}𝑝𝑧,𝑇𝑦,sup_{𝑧∈𝑇𝑦}𝑝(1𝑧,𝑇𝑥)=2𝑦=3𝑝𝜏𝑝𝑝𝑦𝑑.𝑓𝑥,𝑓𝑦<𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥 (57)

Case 4. If 𝑥∈[0,∞) and 𝑦∈(−9876,−10), then 𝑇𝑥=[0,𝑥], 𝑇𝑦={0} and 𝑝(𝑓𝑦,𝑇𝑥)=0. Hence, we have 𝑝2{}𝒟𝑓𝑥,𝑓𝑦=max𝑓𝑥−𝑓𝑦,3𝑓𝑦−𝑓𝑥=max4𝑥,−6𝑥=4𝑥,_𝑝𝑇𝑥,𝑇𝑦=maxsup_{𝑧∈𝑇𝑥}𝑝𝑧,𝑇𝑦,sup_{𝑧∈𝑇𝑦}𝑝(1𝑧,𝑇𝑥)=2𝑥=2𝑝𝜏𝑝𝑝𝑦𝑑.𝑓𝑥,𝑓𝑦=𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥 (58)

Case 5. If 0≤𝑥<𝑦, then 𝑇𝑥=[0,𝑥] and 𝑇𝑦=[0,𝑦]. So, we have 𝑝2,𝒟𝑓𝑥,𝑓𝑦=max𝑓𝑥−𝑓𝑦,3𝑓𝑦−𝑓𝑥=6𝑦−𝑥_𝑝𝑇𝑥,𝑇𝑦=maxsup_{𝑧∈𝑇𝑥}𝑝𝑧,𝑇𝑦,sup_{𝑧∈𝑇𝑦}𝑝(<1𝑧,𝑇𝑥)=2𝑦−𝑥2𝑝𝜏𝑝𝑝𝑦𝑑.𝑓𝑥,𝑓𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥 (59)

Case 6. If 0≤𝑦<𝑥, then 𝑇𝑥=[0,𝑥], 𝑇𝑦=[0,𝑦]. Thus we obtain 𝑝2,𝒟𝑓𝑥,𝑓𝑦=max𝑓𝑥−𝑓𝑦,3𝑓𝑦−𝑓𝑥=4𝑥−𝑦_𝑝𝑇𝑥,𝑇𝑦=maxsup_{𝑧∈𝑇𝑥}𝑝𝑧,𝑇𝑦,sup_{𝑧∈𝑇𝑦}𝑝(=1𝑧,𝑇𝑥)=2𝑥−𝑦2𝑝𝜏𝑝𝑝𝑦𝑑.𝑓𝑥,𝑓𝑦≤𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦+ℎ𝑓𝑦,𝑇𝑥 (60)

By Cases 1–6, we verify that inequality (∗) holds for all 𝑥,𝑦∈𝑋. So all the hypotheses of Theorem 20 are fulfilled. It is therefore possible to apply Theorem 20 to get 𝐶𝑂𝑃(𝑓,𝑇)≠∅. In fact, 𝐶𝑂𝑃(𝑓,𝑇)=(−9876,−10)∪{0}.

Moreover, since 𝑓𝑤=𝑓𝑓𝑤=0 for any 𝑤∈𝐶𝑂𝑃(𝑓,𝑇), by Theorem 20 again, we know that 𝑓 and 𝑇 have a common fixed point in 𝑋 (precisely speaking, 0 is the unique common fixed point of 𝑓 and 𝑇).

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

Let (𝑋,𝑑) be a metric space. We endow the product space 𝑋×𝑋 with the metric 𝜌 defined by 𝜌,()(),()𝑥,𝑦𝑢,𝑣=𝑑𝑥,𝑢+𝑑𝑦,𝑣forany𝑥,𝑦𝑢,𝑣∈𝑋×𝑋. (61)

Let 𝑔∶𝑋→𝑋 be a self-map. Recall that an element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled coincidence point of the maps 𝐹 and 𝑔 if 𝐹𝑥,𝑦=𝑔𝑥,𝐹𝑦,𝑥=𝑔𝑦when𝐹∶𝑋×𝑋→𝑋isasingle-valuedmap (62) or 𝑔𝑥∈𝐹𝑥,𝑦,𝑔𝑦∈𝐹𝑦,𝑥when𝐹∶𝑋×𝑋→𝑋isamultivaluedmap. (63) In particular, if we take 𝑔≡𝑖𝑑 (the identity map) in (62) and (63), then (𝑥,𝑦) is called a coupled fixed point of 𝐹. The existence of coupled coincidence point and coupled fixed point has been investigated by several authors recently in [4, 11, 12, 23, 30–37] 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 (𝑋,𝑑) be a metric space, 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) be a multivalued map and 𝑓∶𝑋→𝑋 be a self-map satisfying ⋃_𝑥∈𝑋𝐹(𝑥,𝑥)⊆𝑓(𝑋) and 𝑓(𝑋) is a complete subspace of 𝑋. Assume that there exists an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) such that for any (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, ℋ𝐹()≤1𝑥,𝑦,𝐹𝑢,𝑣2𝜑𝜌𝑓𝑥,𝑓𝑦,𝑓𝑢,𝑓𝑣𝜌𝑓𝑥,𝑓𝑦,𝑓𝑢,𝑓𝑣, (64) where 𝜌((𝑥,𝑦),(𝑢,𝑣))=𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)for (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋. Then there exists 𝑣∈𝑋 such that 𝑓𝑣∈𝐹(𝑣,𝑣); that is, (𝑣,𝑣)∈𝑋×𝑋 is a coupled coincidence point of 𝐹 and 𝑓.

Proof. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑇(𝑥)=𝐹(𝑥,𝑥). (65) Then 𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋). Let 𝜏(𝑡)=2𝑡, ∀𝑡∈[0,∞) and ℎ(𝑥)=0, ∀𝑥∈𝑋. So the inequality (64) implies ℋ(𝑇(𝑥)(𝑢𝑑𝜏𝑑𝑑(𝑢)𝑑,𝑇))≤𝜑2𝑑𝑓𝑥,𝑓𝑢𝑓𝑥,𝑓𝑢=𝜑𝑓𝑥,𝑓𝑢𝑓𝑥,𝑓𝑢+ℎ𝑓𝑢,𝑇𝑥∀𝑥,𝑢∈𝑋. (66) Therefore, all the assumptions of Corollary 21 are fulfilled. Hence we can apply Corollary 21 to show that there exists 𝑣∈𝑋 such that 𝑓𝑣∈𝑇𝑣=𝐹(𝑣,𝑣).

The following conclusion is immediate from Theorem 26 with 𝑓≡𝑖𝑑 (the identity map).

Corollary 27 (see [4]). Let (𝑋,𝑑) be a complete metric space and 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) be a multivalued map. Assume that there exists an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) such that for any (𝑥,𝑦), (𝑢,𝑣)∈𝑋×𝑋, ℋ𝐹()≤1𝑥,𝑦,𝐹𝑢,𝑣2𝜑𝜌𝑥,𝑦,(𝑢,𝑣)𝜌𝑥,𝑦,(𝑢,𝑣), (67) where 𝜌((𝑥,𝑦),(𝑢,𝑣))=𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)for (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋. Then there exists 𝑣∈𝑋 such that 𝑣∈𝐹(𝑣,𝑣); that is, (𝑣,𝑣)∈𝑋×𝑋 is a coupled fixed point of 𝐹.

Applying Theorem 26, we obtain the following coupled coincidence point theorem.

Theorem 28. Let (𝑋,𝑑) be a metric space, 𝐹∶𝑋×𝑋→𝑋 be a map and 𝑓∶𝑋→𝑋 be a self-map satisfying ⋃_𝑥∈𝑋𝐹(𝑥,𝑥)⊆𝑓(𝑋) and 𝑓(𝑋) is a complete subspace of 𝑋. Assume that there exists an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) such that for any (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, 𝑑𝐹()≤1𝑥,𝑦,𝐹𝑢,𝑣2𝜑𝜌𝑓𝑥,𝑓𝑦,𝑓𝑢,𝑓𝑣𝜌𝑓𝑥,𝑓𝑦,𝑓𝑢,𝑓𝑣. (68) Then there exists 𝑣∈𝑋 such that (𝑣,𝑣) is the unique coupled coincidence point of 𝐹 and 𝑓.

Proof. Applying Theorem 26, there exists 𝑣̂∈𝑋 such that 𝑓𝑣̂=𝐹(𝑣,̂𝑣)̂. To complete the proof, it suffices to show the uniqueness of the coupled coincidence point of 𝐹 and 𝑓. On the contrary, suppose that there exists (𝑥,𝑦)∈𝑋×𝑋, such that 𝑓𝑥=𝐹(𝑥,𝑦) and 𝑓𝑦=𝐹(𝑦,𝑥). By (68), we have 𝑑𝑓̂𝑥𝐹(̂𝑣)̂𝑦<1𝑣,𝑓=𝑑𝑣,,𝐹𝑥,2𝑑𝑓̂𝑥𝑓̂𝑦,𝑑𝑓̂𝑦𝐹(̂𝑣)̂𝑥<1𝑣,𝑓+𝑑𝑣,𝑓𝑣,𝑓=𝑑𝑣,,𝐹𝑦,2𝑑𝑓𝑣,̂𝑓𝑦+𝑑𝑓𝑣,̂𝑓𝑥. (69) So, combining (69), we get 𝑑𝑓𝑣,̂𝑓𝑥+𝑑𝑓𝑣,̂𝑓𝑦<𝑑𝑓𝑣,̂𝑓𝑥+𝑑𝑓𝑣,̂𝑓𝑦, (70) which leads a contradiction. The proof is completed.

Corollary 29 (see [4]). Let (𝑋,𝑑) be a complete metric space and 𝐹∶𝑋×𝑋→𝑋 be a map. Assume that there exists an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) such that for any (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, 𝑑𝐹()≤1𝑥,𝑦,𝐹𝑢,𝑣2𝜑𝜌𝑥,𝑦,(𝑢,𝑣)𝜌𝑥,𝑦,(𝑢,𝑣). (71) Then there exists 𝑣∈𝑋 such that (𝑣,𝑣) is the unique coupled fixed point of 𝐹.

Applying Theorem 18, we can prove the following coincidence point theorem which is a generalization of Kikkawa-Suzuki’s fixed point theorem.

Theorem 30. Define a strictly decreasing function 𝜂 from [0,1) onto (1/2,1] by 𝜂(𝑟)=1.1+𝑟 (72) Let (𝑋,𝑑) be a metric space, 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map, and 𝑓∶𝑋→𝑋 be an injective self-map such that 𝑇(𝑋)=⋃_𝑥∈𝑋𝑇(𝑥)⊆𝑓(𝑋). Assume that(KS) there exists 𝛾∈[0,1) such that 𝜂𝛾𝑑,𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥,𝑓𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠ℋ𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑓𝑥,𝑓𝑦 (73)for all 𝑥,𝑦∈𝑋.

Then 𝑓 and 𝑇 have the approximate coincidence property on 𝑋.

Moreover, if we further assume 𝑓(𝑋) is a complete subspace of 𝑋, then 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

Proof. Let 𝑥∈𝑋 be arbitrary. If 𝑓𝑥∈𝑇𝑥, then 𝑥∈𝐶𝑂𝑃(𝑓,𝑇) and hence 𝑓 and 𝑇 have the approximate coincidence property. So the theorem is finished in this case. Suppose 𝑓𝑥∉𝑇𝑥. Let 𝜏∶[0,∞)→ [0,∞) be defined by 𝜏(𝑡)=𝑡. Then 𝜏 is a nondecreasing function. By (KS), we can define an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) by 𝜑(𝑡)=𝛾, ∀𝑡∈[0,∞). Let 𝑦∈𝑋 with 𝑓𝑦≠𝑓𝑥 and 𝑓𝑦∈𝑇𝑥. Since 𝜂𝛾𝑑𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥,𝑓𝑦. (74) by (KS) again, we have 𝑑𝜏𝑑𝑑,𝑓𝑦,𝑇𝑦≤ℋ𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑓𝑥,𝑓𝑦=𝜑𝑓𝑥,𝑓𝑦𝑓𝑥,𝑓𝑦 (75) which shows that (H1) as in Theorem 18 holds. Therefore 𝑇 have the approximate coincidence property by applying Theorem 18.
Now, suppose that 𝑓(𝑋) is a complete subspace of 𝑋. We shall prove 𝐶𝑂𝑃(𝑓,𝑇)≠∅. It suffices to prove that the condition (L2) as in Theorem 18 holds. Let {𝑥_𝑛} in 𝑋 with 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤. Clearly, 𝑇𝑤 is a closed subset of 𝑋. We will proceed with the following claims to prove lim_𝑛→∞𝑑(𝑓𝑥_𝑛,𝑇𝑤)=0.
Claim 1. 𝑑(𝑓𝑤,𝑇𝑥)≤𝛾𝑑(𝑓𝑤,𝑓𝑥) for all 𝑥∈𝑋 with 𝑓𝑥≠𝑓𝑤.
For 𝑥∈𝑋 with 𝑓𝑥≠𝑓𝑤, since lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤, there exists 𝑛₀∈ℕ, such that 𝑑𝑓𝑤,𝑓𝑥_𝑛≤13𝑑𝑓𝑥,𝑓𝑤∀𝑛∈ℕwith𝑛≥𝑛₀. (76) For 𝑛∈ℕ with 𝑛≥𝑛₀, it follows from (76) that 𝜂𝛾𝑑𝑓𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑓𝑥_𝑛,𝑇𝑥_𝑛≤𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤𝑑𝑓𝑥_𝑛,𝑓𝑤+𝑑𝑓𝑤,𝑓𝑥_𝑛+1≤23𝑑−1𝑓𝑥,𝑓𝑤=𝑑𝑓𝑥,𝑓𝑤3𝑑𝑓𝑥,𝑓𝑤≤𝑑𝑓𝑥,𝑓𝑤−𝑑𝑓𝑥_𝑛,𝑓𝑤≤𝑑𝑓𝑥_𝑛,𝑓𝑥. (77) By our hypothesis (KS), we have 𝑑𝑓𝑥_𝑛+1,𝑇𝑥≤ℋ𝑇𝑥_𝑛,𝑇𝑥≤𝛾𝑑𝑓𝑥_𝑛,𝑓𝑥∀𝑛∈ℕwith𝑛≥𝑛₀. (78) Since lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤, the last inequality implies 𝑑𝑓𝑤,𝑇𝑥≤𝛾𝑑𝑓𝑤,𝑓𝑥∀𝑥∈𝑋with𝑓𝑥≠𝑓𝑤. (79)
Claim 2. ℋ(𝑇𝑥,𝑇𝑤)≤𝛾𝑑(𝑓𝑥,𝑓𝑤) for all 𝑥∈𝑋.
Let 𝑥∈𝑋. It is quite obvious that the desired inequality holds if 𝑥=𝑤. Suppose 𝑥≠𝑤. Since 𝑓 is one-to-one, we have 𝑓𝑥≠𝑓𝑤. For every 𝑛∈ℕ, there exists 𝑦_𝑛∈𝑇𝑥 such that 𝑑𝑓𝑤,𝑦_𝑛+1<𝑑𝑓𝑤,𝑇𝑥𝑛𝑑(𝑥,𝑤). (80) Using (80) and Claim 1, we obtain 𝑑𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥,𝑦_𝑛≤𝑑𝑓𝑥,𝑓𝑤+𝑑𝑓𝑤,𝑦_𝑛+1<𝑑𝑓𝑥,𝑓𝑤+𝑑𝑓𝑤,𝑇𝑥𝑛𝑑()+1𝑥,𝑤≤𝑑𝑓𝑥,𝑓𝑤+𝛾𝑑𝑓𝑤,𝑓𝑥𝑛𝑑()=𝑑+1𝑥,𝑤1+𝛾𝑓𝑥,𝑓𝑤𝑛𝑑(𝑥,𝑤)∀𝑛∈ℕ. (81) The last inequality implies 𝜂𝛾𝑑=1𝑓𝑥,𝑇𝑥𝑑.1+𝛾𝑓𝑥,𝑇𝑥≤𝑑𝑓𝑥,𝑓𝑤 (82) By (KS), we get ℋ(𝑇𝑥,𝑇𝑤)≤𝛾𝑑𝑓𝑥,𝑓𝑤∀𝑥∈𝑋. (83)
Claim 3. lim_𝑛→∞𝑑(𝑓𝑥_𝑛,𝑇𝑤)=0.
Since 𝑓𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑤, applying Claim 2, we have lim_𝑛→∞𝑑𝑓𝑥_𝑛+1,𝑇𝑤≤lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑤≤lim_𝑛→∞𝛾𝑑𝑓𝑥_𝑛,𝑓𝑤=0, (84) which implies lim_𝑛→∞𝑑(𝑓𝑥_𝑛,𝑇𝑤)=0.
By our Claims, we prove that (L2) holds. Therefore, by applying Theorem 18, we prove 𝐶𝑂𝑃(𝑓,𝑇)≠∅.

Take 𝑓≡𝑖𝑑 (the identity map) in Theorem 30, we obtain the following existence theorem which is also a generalized Kikkawa-Suzuki’s fixed point theorem.

Corollary 31. Define a strictly decreasing function 𝜂 from [0,1) onto (1/2,1] by 𝜂(𝑟)=1.1+𝑟 (85) Let (𝑋,𝑑) be a metric space and let 𝑇 be a map from 𝑋 into 𝐶𝐵(𝑋). Assume that there exists 𝛾∈[0,1) such that 𝜂𝛾𝑑(𝑥,𝑇𝑥)≤𝑑𝑥,𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠ℋ𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑥,𝑦, (86) for all 𝑥,𝑦∈𝑋. Then 𝑇 has the approximate fixed property.

Moreover, if we further assume 𝑋 is complete, then ℱ(𝑇)≠∅.

Acknowledgments

The author wishes to express his hearty thanks to the anonymous referees for their valuable suggestions and comments. This research was supported partially by Grant no. NSC 101-2115-M-017-001 of the National Science Council of China.