New Nonlinear Conditions and Inequalities for the Existence of Coincidence Points and Fixed Points

Abstract

Some new fixed point theorems for nonlinear maps are established. By using these results, we can obtain some new coincidence point theorems. Our results are quite different in the literature and references therein.

1. Introduction and Preliminaries

Let us begin with some basic definitions and notation that will be needed in this paper. Throughout this paper, we denote by ℕ and ℝ, the sets of positive integers and real numbers, respectively. Let (𝑋,𝑑) be a metric space. For each 𝑥∈𝑋, and 𝐴⊆𝑋, let 𝑑(𝑥,𝐴)=inf_𝑦∈𝐴𝑑(𝑥,𝑦). Denote by 𝒩(𝑋) the class of all nonempty subsets of 𝑋, 𝒞(𝑋) the family of all nonempty closed subsets of 𝑋, and 𝒞ℬ(𝑋) the family of all nonempty closed and bounded subsets of 𝑋. A function ℋ∶𝐶𝐵(𝑋)×𝐶𝐵(𝑋)→[0,∞), defined by ℋ(𝐴,𝐵)=maxsup_𝑥∈𝐵𝑑(𝑥,𝐴),sup_𝑥∈𝐴𝑑(𝑥,𝐵), (1.1) is said to be the Hausdorff metric on 𝒞ℬ(𝑋) induced by the metric d on 𝑋.

A point 𝑣 in 𝑋 is a fixed point of a map 𝑇 if 𝑣=𝑇𝑣 (when 𝑇∶𝑋→𝑋 is a single-valued map) or 𝑣∈𝑇𝑣 (when 𝑇∶𝑋→𝒩(𝑋) is a multivalued map). The set of fixed points of 𝑇 is denoted by ℱ(𝑇).

Let 𝑔∶𝑋→𝑋 be a self-map and 𝑇∶𝑋→𝒩(𝑋) be a multivalued map. A point 𝑥 in 𝑋 is said to be a coincidence point (see, for instance, [1–4]) of 𝑔 and 𝑇 if 𝑔𝑥∈𝑇𝑥. The set of coincidence points of 𝑔 and 𝑇 is denoted by 𝒞𝒪𝒫(𝑔,𝑇).

The celebrated Banach contraction principle (see, e.g., [5]) plays an important role in various fields of applied mathematical analysis. Since then a number of generalizations in various different directions of the Banach contraction principle have been investigated by several authors in the past; see [6–20] and references therein. In 1969, Nadler [6] first proved a famous generalization of the Banach contraction principle for multivalued maps.

Let 𝑓 be a real-valued function defined on ℝ. For 𝑐∈ℝ, we recall that limsup_𝑥→𝑐𝑓(x)=inf_𝜀>0sup_{||0<𝑥−𝑐<𝜖}𝑓(𝑥),limsup_𝑥→𝑐+𝑓(𝑥)=inf_𝜀>0sup_{0<𝑥−𝑐<𝜀}𝑓(𝑥). (1.2)

Definition 1.1 (see [3, 4, 7–10, 21, 22]). A function 𝜑∶[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsup_𝑠→𝑡+𝜑(𝑠)<1∀𝑡∈[0,∞). (1.3)

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. But it is worth to mention that there exist functions which are not ℳ𝒯-functions.

Example A [see [4]]
Let 𝜑∶[0,∞)→[0,1) be defined by 𝜑(𝑡)∶=sin𝑡𝑡𝜋,if𝑡∈0,2,0,otherwise. (1.4) Since limsup_𝑠→0+𝜑(𝑠)=1,   𝜑 is not an ℳ𝒯-function.
Very recently, Du [4] first proved the following characterizations of ℳ𝒯-functions.

Theorem 1.2 (see [4]). 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 [10]; that is, for any strictly decreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), one has 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.

In 1989, Mizoguchi and Takahashi [11] proved the following fixed point theorem which is a generalization of Nadler’s fixed point theorem and gave a partial answer of Problem 9 in Reich [12]. It is worth to mention that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult. Recently, Suzuki [13] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.

Theorem MT (Mizoguchi and Takahashi). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map and 𝜑∶[0,∞)→[0,1) be a ℳ𝒯-function. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋, (1.5) then ℱ(𝑇)≠∅.

In 2007, M. Berinde and V. Berinde [14] proved the following interesting fixed point theorem which generalized Mizoguchi-Takahashi’s fixed point theorem.

Theorem BB (M. Berinde and V. Berinde). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map, 𝜑∶[0,∞)→[0,1) be a ℳ𝒯-function and 𝐿≥0. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (1.6) then ℱ(𝑇)≠∅.

Let (𝑋,𝑑) be a metric space. Recall that a function 𝑝∶𝑋×𝑋→[0,∞) is called a 𝑤-distance [3, 5, 7, 15–23], 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 [3, 7, 16, 18–22], 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.

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 [7, 16].

The following results are crucial in this paper.

Lemma 1.3 (see [7, 20]). 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.
Let 𝑝∶𝑋×𝑋→[0,∞) be any function. For each 𝑥∈𝑋 and 𝐴⊆𝑋, let 𝑝(𝑥,𝐴)=inf_𝑦∈𝐴𝑝(𝑥,𝑦).

Lemma 1.4 (see [7, 19, 20]). 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.
Recently, Du [7, 19] first introduced the concepts of 𝜏⁰-functions and 𝜏⁰-metrics as follows.

Definition 1.5 (see [7, 19]). 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 1.6. If 𝑝 is a 𝜏⁰-function, then, from (𝜏4), 𝑝(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Definition 1.7 (see [7, 19]). Let (𝑋,𝑑) be a metric space and 𝑝 be a 𝜏⁰-function (resp., 𝑤⁰-distance). 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), (1.7) where 𝛿_𝑝(𝐴,𝐵)=sup_𝑥∈𝐴𝑝(𝑥,𝐵), then 𝒟_𝑝 is said to be the 𝜏⁰-𝑚𝑒𝑡𝑟𝑖𝑐 (resp., 𝑤⁰-𝑚𝑒𝑡𝑟𝑖𝑐) on 𝒞ℬ(𝑋) induced by 𝑝.
Clearly, any Hausdorff metric is a 𝜏⁰-metric, but the reverse is not true.

Lemma 1.8 (see [7, 19]). Let (𝑋,𝑑) be a metric space and 𝒟_𝑝 be a 𝜏⁰-metri on 𝒞ℬ(𝑋) induced by a 𝜏⁰-function 𝑝. Then every 𝜏⁰-metric 𝒟_𝑝 is a metric on 𝒞ℬ(𝑋).
Recently, Du [7] established the following new fixed point theorems for 𝜏⁰-metric and ℳ𝒯-functions to extend Berinde-Berinde’s fixed point theorem.

Theorem D (Du [7, Theorem 2.1]). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞(𝑋) be a multivalued map and 𝜑∶[0,∞)→[0,1) a ℳ𝒯-function. Suppose that for each 𝑥∈𝑋𝑝𝑦,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝𝑥,𝑦∀𝑦∈𝑇𝑥, (1.8) and 𝑇 further satisfies one of the following conditions:(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 each sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_n=𝑣, one has lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑣)=0;(D5)inf{𝑝(𝑥,𝑧)+𝑝(𝑥,𝑇𝑥)∶𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇). Then ℱ(𝑇)≠∅.

In [7], Du also gave the generalizations of Kannan’s fixed point theorem, Chatterjea’s fixed point theorem and other new fixed point theorems for nonlinear multivalued contractive maps; see [7] for more detail.

In this paper, we first establish some new types of fixed point theorem. Some applications to the existence for coincidence point and others are also given. Our results are quite different in the literature and references therein.

2. New Inequalities and Nonlinear Conditions for Fixed Point Theorems

In this section, we first establish some new existence theorems for fixed point.

Theorem 2.1. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that(𝒜1) there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and for each 𝑥∈𝑋, it holds 3𝑝𝑦,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝜏𝑝𝑥,𝑦𝑝𝑥,𝑇𝑦∀𝑦∈𝑇𝑥. (2.1)(𝒜2)𝑇 further satisfies one of the following conditions:(H1)𝑇 is closed, that is, 𝐺𝑟𝑇={(𝑥,𝑦)∈𝑋×𝑋∶𝑦∈𝑇𝑥}, the graph of 𝑇 is closed in 𝑋×𝑋;(H2)the map 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑝(𝑥,𝑇𝑥) is l.s.c.;(H3)the map 𝑔∶𝑋→[0,∞) defined by 𝑔(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.;(H4)for any sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑣, one has lim_𝑛→∞𝑝(𝑥_𝑛,𝑇𝑣)=0;(H5)inf{𝑝(𝑥,𝑧)+𝑝(𝑥,𝑇𝑥)∶𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇).
Then ℱ(𝑇)≠∅.

Proof . Let 𝑢₁∈𝑋. If 𝑢₁∈𝑇𝑢₁, then we are done. If 𝑢₁∉𝑇𝑢₁, then 𝑝(𝑢₁,𝑇𝑢₁)>0 by Lemma 1.4. Choose 𝑢₂∈𝑇𝑢₁. If 𝑢₂∈𝑇𝑢₂, then 𝑢₂ is a fixed point of 𝑇. Otherwise, if 𝑢₂∉𝑇𝑢₂, then, by (𝒜1), we have 3𝑝𝑢₂,𝑇𝑢₂≤𝜑𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑇𝑢₁+𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑇𝑢₂≤𝜑𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₂+𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑇𝑢₂. (2.2) Since 𝜑(𝑝(𝑢₁,𝑢₂))+𝜏(𝑝(𝑢₁,𝑢₂))<2, there exists 𝑢₃∈𝑇𝑢₂ such that 3𝑝𝑢₂,𝑢₃<2−𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₂+𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₃≤2−𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₂+𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₂+𝑝𝑢₂,𝑢₃, (2.3) which implies 𝑝𝑢₂,𝑢₃<23−𝜏𝑝𝑢₁,𝑢₂𝑝𝑢₁,𝑢₂. (2.4) Since 𝑢₃≠𝑢₂ and 𝑝 is a 𝜏⁰-function, 𝑝(𝑢₂,𝑢₃)>0. Since 0≤𝜏(𝑝(𝑢₁,𝑢₂))<1, we have 2/(3−𝜏(𝑝(𝑢₁,𝑢₂)))∈[2/3,1) and hence 0<𝑝𝑢₂,𝑢₃<𝑝𝑢₁,𝑢₂. (2.5) Continuing in this way, we can construct inductively a sequence {𝑢_𝑛}_𝑛∈ℕ in 𝑋 satisfying 𝑢_𝑛+1∈𝑇𝑢_𝑛, {𝑝(𝑢_𝑛,𝑢_𝑛+1)} is strictly decreasing in (0,∞) and 𝑝𝑢_𝑛+1,𝑢_𝑛+2<23−𝜏𝑝𝑢_𝑛,𝑢_𝑛+1𝑝𝑢_𝑛,𝑢_𝑛+1 (2.6) for each 𝑛∈ℕ. Since 𝜏 is an ℳ𝒯-function, applying Theorem 1.2, we get 0≤sup_𝑛∈ℕ𝜏𝑝𝑢_𝑛,𝑢_𝑛+1<1. (2.7) Put 𝜆∶=sup_𝑛∈ℕ𝜏(𝑝(𝑢_𝑛,𝑢_𝑛+1)) and 𝜉∶=2/(3−𝜆). Then 𝜆∈[0,1), 𝜉∈[2/3,1) and 23−𝜏𝑝𝑢_𝑛,𝑢_𝑛+1≤𝜉∀𝑛∈ℕ. (2.8) By (2.6) and (2.8), we have 𝑝𝑢_𝑛+1,𝑢_𝑛+2<𝜉𝑝𝑢_𝑛,𝑢_𝑛+1<⋯<𝜉^𝑛𝑝𝑢₁,𝑢₂foreach𝑛∈ℕ. (2.9) We claim that lim_𝑛→∞sup{𝑝(𝑢_𝑛,𝑢_𝑚)∶𝑚>𝑛}=0. Let 𝛼_𝑛=(𝜉^𝑛−1/(1−𝜉))𝑝(𝑢₁,𝑢₂), 𝑛∈ℕ. For 𝑚, 𝑛∈ℕ with 𝑚>𝑛, we have 𝑝𝑢_𝑛,𝑢_𝑚≤^𝑚−1_𝑗=𝑛𝑝𝑢_𝑗,𝑢_𝑗+1<𝛼_𝑛. (2.10) Since 𝜉∈[2/3,1), lim_𝑛→∞𝛼_𝑛=0 and, from (2.10), we get lim_𝑛→∞sup𝑝𝑢_𝑛,𝑢_𝑚∶𝑚>𝑛=0. (2.11) Applying Lemma 1.3, {𝑢_𝑛} is a Cauchy sequence in 𝑋. By the completeness of 𝑋, there exists 𝑣∈𝑋 such that 𝑢_𝑛→𝑣 as 𝑛→∞. From (𝜏2) and (2.10), we have 𝑝𝑢_𝑛,𝑣≤𝛼_𝑛∀𝑛∈ℕ. (2.12) Now, we verify that 𝑣∈ℱ(𝑇). If (H1) holds, since 𝑇 is closed, 𝑢_𝑛∈𝑇𝑢_𝑛−1 and 𝑢_𝑛→𝑣 as 𝑛→∞, we have 𝑣∈𝑇𝑣.
If (H2) holds, by the lower semicontinuity of 𝑓, 𝑢_𝑛→𝑣 as 𝑛→∞ and (2.11), we obtain 𝑝(𝑣,𝑇𝑣)=𝑓(𝑣)≤liminf_𝑚→∞𝑓𝑢_𝑛=liminf_𝑚→∞𝑝𝑢_𝑛,𝑇𝑢_𝑛≤lim_𝑛→∞𝑝𝑢_𝑛,𝑢_𝑛+1=0, (2.13) which implies 𝑝(𝑣,𝑇𝑣)=0. By Lemma 1.4, we get 𝑣∈ℱ(𝑇).
Suppose that (H3) holds. Since {𝑥_𝑛} is convergent in 𝑋, lim_𝑛→∞𝑑(𝑢_𝑛,𝑢_𝑛+1)=0. Since 𝑑(𝑣,𝑇𝑣)=𝑔(𝑣)≤liminf_𝑚→∞𝑑𝑢_𝑛,𝑇𝑢_𝑛≤lim_𝑛→∞𝑑𝑢_𝑛,𝑢_𝑛+1=0, (2.14) we have 𝑑(𝑣,𝑇𝑣)=0 and hence 𝑣∈ℱ(𝑇).
If (H4) holds, by (2.11), there exists {𝑎_𝑛}⊂{𝑢_𝑛} with lim_𝑛→∞sup{𝑝(𝑎_𝑛,𝑎_𝑚)∶𝑚>𝑛}=0 and {𝑏_𝑛}⊂𝑇𝑣 such that lim_𝑛→∞𝑝(𝑎_𝑛,𝑏_𝑛)=0. By (𝜏3), lim_𝑛→∞𝑑(𝑎_𝑛,𝑏_𝑛)=0. Since 𝑑(𝑏_𝑛,𝑣)≤𝑑(𝑏_𝑛,𝑎_𝑛)+𝑑(𝑎_𝑛,𝑣), it follows that 𝑏_𝑛→𝑣 as 𝑛→∞. By the closedness of 𝑇𝑣, we get 𝑣∈𝑇𝑣 or 𝑣∈ℱ(𝑇).
Finally, assume that (H5) holds. On the contrary, suppose that 𝑣∉𝑇𝑣. Then, by (2.10) and (2.12), we obtain 0<inf_𝑥∈𝑋𝑝(𝑥,𝑣)+𝑝(𝑥,𝑇𝑥)≤inf_𝑛∈ℕ𝑝𝑢_𝑛,𝑣+𝑝𝑢_𝑛,𝑇𝑢_𝑛≤inf_𝑛∈ℕ𝑝𝑢_𝑛,𝑣+𝑝𝑢_𝑛,𝑢_𝑛+1≤lim_𝑛→∞2𝛼_𝑛=0, (2.15) a contradiction. Therefore 𝑣∈ℱ(𝑇). The proof is completed.

If we put 𝑝≡𝑑 in Theorem 2.1, then we have the following result.

Corollary 2.2. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that(ℬ1) there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and for each 𝑥∈𝑋, it holds 3𝑑𝑦,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑(𝑥,𝑇x)+𝜏𝑑𝑥,𝑦𝑑𝑥,𝑇𝑦∀𝑦∈𝑇𝑥. (2.16)(ℬ2)𝑇 further satisfies one of the following conditions: (h1)𝑇 is closed; (h2) the map 𝑔∶𝑋→[0,∞) defined by 𝑔(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.; (h3) for any sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑣, one has lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑣)=0; (h4)inf{𝑑(𝑥,𝑧)+𝑑(𝑥,𝑇𝑥)∶𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇).Then ℱ(𝑇)≠∅.

The following result is immediate from Theorem 2.1.

Theorem 2.3. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (𝒜2) holds and further assume that(𝒜3) there exists an ℳ𝒯-function 𝛼∶[0,∞)→[0,1) such that for each 𝑥∈𝑋, 3𝑝𝑦,𝑇𝑦≤𝛼𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝑝𝑥,𝑇𝑦∀𝑦∈𝑇𝑥, (2.17)then ℱ(𝑇)≠∅.

Corollary 2.4. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (ℬ2) holds and further assume that(ℬ3) there exists an ℳ𝒯-function 𝛼∶[0,∞)→[0,1) such that for each 𝑥∈𝑋, 3𝑑𝑦,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑(𝑥,𝑇𝑥)+𝑑𝑥,𝑇𝑦∀𝑦∈𝑇𝑥, (2.18)then ℱ(𝑇)≠∅.

Theorem 2.5. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (𝒜2) holds and further assume that(𝒜4) there exist 𝛼,𝛽∈[0,1) such that for each 𝑥∈𝑋, 3𝑝(𝑦,𝑇𝑦)≤𝛼𝑝(𝑥,𝑇𝑥)+𝛽𝑝(𝑥,𝑇𝑦) for all 𝑦∈𝑇𝑥, then ℱ(𝑇)≠∅.

Proof . Let 𝜑,𝜏∶[0,∞)→[0,1) be defined by 𝜑(𝑡)=𝛼 and 𝜏(𝑡)=𝛽 for all 𝑡∈[0,∞). Then (𝒜4) implies (𝒜1) and the conclusion follows from Theorem 2.1.

Corollary 2.6. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (ℬ2) holds and further assume that(ℬ4) there exist 𝛼,𝛽∈[0,1) such that for each 𝑥∈𝑋, 3𝑑(𝑦,𝑇𝑦)≤𝛼𝑑(𝑥,𝑇𝑥)+𝛽𝑑(𝑥,𝑇𝑦) for all 𝑦∈𝑇𝑥, then ℱ(𝑇)≠∅.

Theorem 2.7. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (𝒜2) holds and further assume that(𝒜5) there exists 𝛾∈[0,1) such that for each 𝑥∈𝑋, 3𝑝(𝑦,𝑇𝑦)≤𝛾(𝑝(𝑥,𝑇𝑥)+𝑝(𝑥,𝑇𝑦)) for all 𝑦∈𝑇𝑥, then ℱ(𝑇)≠∅.

Proof . Let 𝛼=𝛽=𝛾. Then (𝒜5) implies (𝒜4) and the conclusion follows from Theorem 2.5.

Remark 2.8. (𝒜4) and (𝒜5) are equivalent. Indeed, in the proof of Theorem 2.7, we have shown that (𝒜5) implies (𝒜4). If (𝒜4) holds, then put 𝛾=max{𝛼,𝛽}. So 𝛾∈[0,1) and (𝒜5) holds. Hence (𝒜4) and (𝒜5) are equivalent. Therefore Theorem 2.5 can also be proved by using Theorem 2.7 and we know that Theorems 2.5 and 2.7 are indeed equivalent.

Corollary 2.9. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞(𝑋) be a multivalued map. Suppose that the condition (ℬ2) holds and further assume that(ℬ5) there exists 𝛾∈[0,1) such that for each 𝑥∈𝑋, 3𝑑(𝑦,𝑇𝑦)≤𝛾(𝑑(𝑥,𝑇𝑥)+𝑑(𝑥,𝑇𝑦)) for all 𝑦∈𝑇𝑥,
then ℱ(𝑇)≠∅.

Remark 2.10. Corollaries 2.6 and 2.9 are equivalent.

3. Applications of Theorem 2.1 to the Existence of Coincidence Points

By applying Theorem 2.1, we can prove easily the following new coincidence point theorem.

Theorem 3.1. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, 𝑔∶𝑋→𝑋 be a self-map, 𝑇∶𝑋→𝒞(𝑋) be a multivalued map, and 𝐿≥0. Suppose that the condition (𝒜2) holds and further assume that(𝒜6)𝑇𝑥 is 𝑔-invariant (i.e., 𝑔(𝑇𝑥)⊆𝑇𝑥) for each 𝑥∈𝑋;(𝒜7) there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 3𝑝𝑦,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝜏𝑝𝑥,𝑦𝑝𝑥,𝑇𝑦+𝐿𝑝𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.1)then 𝒞𝒪𝒫(𝑔,𝑇)∩ℱ(𝑇)≠∅.

Proof . For each 𝑥∈𝑋, if 𝑦∈𝑇𝑥, from (𝒜6), we have 𝑔𝑦∈𝑇𝑥. So 𝑝(𝑔𝑦,𝑇𝑥)=0. Hence (𝒜7) implies (𝒜1). Applying Theorem 2.1, ℱ(𝑇)≠∅. So there exists 𝑣∈𝑋 such that 𝑣∈𝑇𝑣. By (𝒜6), 𝑔𝑣∈𝑇𝑣. Therefore, 𝑣∈𝒞𝒪𝒫(𝑔,𝑇)∩ℱ(𝑇) and the proof is complete.

The following result is immediate from Theorem 3.1.

Corollary 3.2. Let (𝑋,𝑑) be a complete metric space, 𝑔∶𝑋→𝑋 be a self-map, 𝑇∶𝑋→𝒞(𝑋) be a multivalued map, and 𝐿≥0. Suppose that the condition (ℬ2) holds and further assume(ℬ6)𝑇𝑥 is 𝑔-invariant (i.e., 𝑔(𝑇𝑥)⊆𝑇𝑥) for each 𝑥∈𝑋;(ℬ7) there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 3𝑑𝑦,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑(𝑥,𝑇𝑥)+𝜏𝑑𝑥,𝑦𝑑𝑥,𝑇𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.2)then 𝒞𝒪𝒫(𝑔,𝑇)∩ℱ(𝑇)≠∅.

As an application of Theorem 3.1, one has the following fixed point theorem.

Theorem 3.3. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, 𝑇∶𝑋→𝒞(𝑋) be a multivalued map, and 𝐿≥0. Suppose that the condition (𝒜2) holds and further assume that(𝒜8) there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 3𝑝𝑦,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝜏𝑝𝑥,𝑦𝑝𝑥,𝑇𝑦+𝐿𝑝𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.3) then ℱ(𝑇)≠∅.

Corollary 3.4. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞(𝑋) be a multivalued map and 𝐿≥0. Suppose that the condition (ℬ2) holds and further assume that(𝒜8)there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 3𝑑𝑦,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑(𝑥,𝑇𝑥)+𝜏𝑑𝑥,𝑦𝑑𝑥,𝑇𝑦+𝐿𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.4)then ℱ(𝑇)≠∅.

Theorem 3.5. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, 𝒟_𝑝 be a 𝜏⁰-metric on 𝒞ℬ(𝑋), 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map, 𝑔∶𝑋→𝑋 be a self-map, and 𝐿≥0. Suppose that the conditions (𝒜2) and (𝒜6) hold and further assume that(𝒜9)there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝜏𝑝𝑥,𝑦𝑝𝑥,𝑇𝑦+𝐿𝑝𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.5) then 𝒞𝒪𝒫(𝑔,𝑇)∩ℱ(𝑇)≠∅.

Corollary 3.6. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map, 𝑔∶𝑋→𝑋 be a self-map, and 𝐿≥0. Suppose that the conditions (ℬ2) and (ℬ6) hold and further assume that(ℬ9)there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑(𝑥,𝑇𝑥)+𝜏𝑑𝑥,𝑦𝑑𝑥,𝑇𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.6)then 𝒞𝒪𝒫(𝑔,𝑇)∩ℱ(𝑇)≠∅.

Theorem 3.7. Let (𝑋,𝑑) be a complete metric space, 𝑝 be a 𝜏⁰-function, 𝒟_𝑝 be a 𝜏⁰-metric on 𝒞ℬ(𝑋), 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map, and 𝐿≥0. Suppose that the condition (𝒜2) holds and further assume that(𝒜10)there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds 𝒟_𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑝𝑥,𝑦𝑝(𝑥,𝑇𝑥)+𝜏𝑝𝑥,𝑦𝑝𝑥,𝑇𝑦+𝐿𝑝𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.7)then ℱ(𝑇)≠∅.

Corollary 3.8. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) be a multivalued map and 𝐿≥0. Suppose that the condition (ℬ2) holds and further assume that(ℬ10)there exist two functions 𝜑,𝜏∶[0,∞)→[0,1) such that 𝜏 is an ℳ𝒯-function and it holds ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑(𝑥,𝑇𝑥)+𝜏𝑑𝑥,𝑦𝑑𝑥,𝑇𝑦+𝐿𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋, (3.8)then ℱ(𝑇)≠∅.

Acknowledgments

This research was supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.