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

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 (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 collection 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 byℋ(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. Let 𝑇∶𝑋→𝒩(𝑋) be a multivalued map. A point 𝑥 in 𝑋 is a fixed point of 𝑇 if 𝑥∈𝑇𝑥. The set of fixed points of 𝑇 is denoted by ℱ(𝑇). Let 𝑔∶𝑋→𝑋 be a single-valued 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, 2]) of 𝑔 and 𝑇 if 𝑔𝑥∈𝑇𝑥. The set of coincidence point of 𝑔 and 𝑇 is denoted by 𝒞𝒪𝒫(𝑔,𝑇). Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively.

Let 𝑓 be a real-valued function defined on ℝ. For 𝑐∈ℝ, we recall thatlimsup_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 [2–9]). 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.
Very recently, Du [7] first proved some characterizations of ℳ𝒯-functions.

Theorem D (see [7]). 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,∞), we have 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.(g)𝜑 is a function of contractive factor [5]; that is, for any strictly decreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), we have 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.

It is worth to mention that there exist functions which are not ℳ𝒯-functions. For example, let 𝜑∶[0,∞)→[0,1) be defined by𝜑(𝑡)∶=sin𝑡𝑡𝜋,if𝑡∈0,2,0,otherwise. (1.4) Since limsup_𝑠→0+𝜑(𝑠)=1, 𝜑 is not an ℳ𝒯-function.

Let 𝑔∶𝑋→𝑋 be a single-valued self-map. Recall that a multivalued map 𝑇∶𝑋→𝒩(𝑋) is called(1) a multivalued 𝑘-contraction [1, 10, 11], if there exists a number 0<𝑘<1 such that ℋ𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (1.5)(2) a multivalued (𝜃,𝐿)-almost contraction [1, 12], if there exist two constants 𝜃∈(0,1) and 𝐿≥0 such thatℋ𝑇𝑥,𝑇𝑦≤𝜃𝑑𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋, (1.6)(3) a generalized multivalued almost contraction [1, 12], if there exists an ℳ𝒯-function 𝜑 and 𝐿≥0 such thatℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋, (1.7)(4) a multivalued almost 𝑔-contraction [1, 12, 13], if there exist two constants 𝜃∈(0,1) and 𝐿≥0 such thatℋ𝑇𝑥,𝑇𝑦≤𝜃𝑑𝑔𝑥,𝑔𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋, (1.8)(5) a generalized multivalued almost 𝑔-contraction [1], if there exists an ℳ𝒯-function 𝜑 and 𝐿≥0 such thatℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑥,𝑔𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋, (1.9)

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 (𝑋,𝑑) be a complete metric space, 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, (1.10) Then, ℱ(𝑇)≠∅.

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 (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, (1.11) that is, 𝑇 is a generalized multivalued almost contraction. Then, ℱ(𝑇)≠∅.

In [3], the author established some new fixed point theorems for nonlinear multivalued contractive maps by using 𝜏⁰-metrics (see [3, Def. 1.2]), 𝜏⁰-metrics (see [3, Def. 1.3]), and ℳ𝒯-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 𝑋 be a nonempty set and 𝑌 a locally convex Hausdorff t.v.s. with its zero vector 𝜃, 𝐾 a proper, closed, convex, and pointed cone in 𝑌, and ⪷_𝐾 a partial ordering with respect to 𝐾 defined by 𝑥⪷_𝐾𝑦⟺𝑦−𝑥∈𝐾. (1.12) A vector-valued function 𝑝∶𝑋×𝑋→𝑌 is said to be a 𝑇𝑉𝑆-𝑐𝑜𝑛𝑒𝑚𝑒𝑡𝑟𝑖𝑐, if the following conditions hold:(C1)𝜃⪷_𝐾𝑝(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑝(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(C2)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(C3)𝑝(𝑥,𝑧)⪷_𝐾𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋. The pair (𝑋,𝑝) 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 [1–6, 10–19] and references therein.

2. Preliminaries

Let 𝐸 be a topological vector space (t.v.s. for short) with its zero vector 𝜃_𝐸. A nonempty subset 𝐾 of 𝐸 is called a convex cone if 𝐾+𝐾⊆𝐾 and 𝜆𝐾⊆𝐾 for 𝜆≥0. A convex cone 𝐾 is said to be pointed if 𝐾∩(−𝐾)={𝜃_𝐸}. For a given proper, pointed, and convex cone 𝐾 in 𝐸, we can define a partial ordering ⪷_𝐾 with respect to 𝐾 by𝑥⪷_𝐾𝑦⟺𝑦−𝑥∈𝐾. (2.1)𝑥≺_𝐾𝑦 will stand for 𝑥⪷_𝐾𝑦 and 𝑥≠𝑦, while 𝑥≪_𝐾𝑦 will stand for 𝑦−𝑥∈int𝐾, where int𝐾 denotes the interior of 𝐾.

In the following, unless otherwise specified, we always assume that 𝑌 is a locally convex Hausdorff t.v.s. with its zero vector 𝜃, 𝐾 is a proper, closed, convex, and pointed cone in 𝑌 with int𝐾≠∅, ⪷_𝐾 is a partial ordering with respect to 𝐾 and 𝑒∈int𝐾 is fixed.

Recall that the nonlinear scalarization function 𝜉_𝑒∶𝑌→ℝ is defined by𝜉_𝑒𝑦=inf𝑟∈ℝ∶𝑦∈𝑟𝑒−𝐾,∀𝑦∈𝑌. (2.2)

Theorem 2.1 (see [6, 17, 20, 21]). For each 𝑟∈ℝ and 𝑦∈𝑌, the following statements are satisfied:(i)𝜉_𝑒(𝑦)≤𝑟⇔𝑦∈𝑟𝑒−𝐾,(ii)𝜉_𝑒(𝑦)>𝑟⇔𝑦∉𝑟𝑒−𝐾,(iii)𝜉_𝑒(𝑦)≥𝑟⇔𝑦∉𝑟𝑒−int𝐾,(iv)𝜉_𝑒(𝑦)<𝑟⇔𝑦∈𝑟𝑒−int𝐾,(v)𝜉_𝑒(⋅) is positively homogeneous and continuous on 𝑌,(vi) if 𝑦₁∈𝑦₂+𝐾 (i.e., 𝑦₂≲_𝐾𝑦₁), then 𝜉_𝑒(𝑦₂)≤𝜉_𝑒(𝑦₁),(vii)𝜉_𝑒(𝑦₁+𝑦₂)≤𝜉_𝑒(𝑦₁)+𝜉_𝑒(𝑦₂) for all 𝑦₁,𝑦₂∈𝑌.
Clearly, 𝜉_𝑒(𝜃)=0. Notice that the reverse statement of (vi) in Theorem 2.1 (i.e., 𝜉_𝑒(𝑦₂)≤𝜉_𝑒(𝑦₁)⇒𝑦₁∈𝑦₂+𝐾) does not hold in general. We illustrate the truth with the following simple example.

Example A. Let 𝑌=ℝ², 𝐾=ℝ²₊={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦≥0}, and 𝑒=(1,1). Then 𝐾 is a proper, closed, convex, and pointed cone in 𝑌 with int𝐾={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦>0}≠∅ and 𝑒∈int𝐾. For 𝑟=1, it is easy to see that 𝑦₁=(10,−30)∉𝑟𝑒−int𝐾, and 𝑦₂=(0,0)∈𝑟𝑒−int𝐾. By applying (iii) and (iv) of Theorem 2.1, we have 𝜉_𝑒(𝑦₂)<1≤𝜉_𝑒(𝑦₁) while 𝑦₁∉𝑦₂+𝐾.

Definition 2.2 (see [17]). Let (𝑋,𝑝) be a TVS-cone metric space, 𝑥∈𝑋, and {𝑥_𝑛}_𝑛∈ℕ a sequence in 𝑋.(i){𝑥_𝑛} is said to TVS-cone converge to 𝑥 if, for every 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there exists a natural number ℕ₀ such that 𝑝(𝑥_𝑛,𝑥)≪_𝐾𝑐 for all 𝑛≥ℕ₀. We denote this by cone-lim_𝑛→∞𝑥_𝑛=𝑥 or 𝑥−−−→_𝑛^cone𝑥 as 𝑛→∞ and call 𝑥 the TVS-cone limit of {𝑥_𝑛}.(ii){𝑥_𝑛} is said to be a TVS-cone Cauchy sequence if, for every 𝑐∈𝑌 with 𝜃≪_𝐾𝑐, there is a natural number ℕ₀ such that 𝑝(𝑥_𝑛,𝑥_𝑚)≪_𝐾𝑐 for all 𝑛,𝑚≥ℕ₀.(iii)(𝑋,𝑝) is said to be TVS-cone complete if every TVS-cone Cauchy sequence in 𝑋 is TVS-cone convergent in 𝑋.
In [17], the author proved the following important results.

Theorem 2.3 (see [17]). Let (𝑋,𝑝) be a TVS-cone metric spaces. Then, 𝑑_𝑝∶𝑋×𝑋→[0,∞) defined by 𝑑_𝑝∶=𝜉_𝑒∘𝑝 is a metric.

Example B. Let 𝑋=[0,1], 𝑌=ℝ², 𝐾=ℝ²₊={(𝑥,𝑦)∈ℝ²∶𝑥,𝑦≥0}, 𝑒=(1,1), and 𝜃=(0,0). Define 𝑝∶𝑋×𝑋→𝑌 by 𝑝𝑥,𝑦=𝑥−𝑦,8𝑥−𝑦. (2.3) Then, (𝑋,𝑝) is a TVS-cone complete metric space and 𝑑_𝑝𝑥,𝑦=𝜉_𝑒𝑝𝑥,𝑦=inf𝑟∈ℝ∶𝑝𝑥,𝑦∈𝑟𝑒−𝐾=8𝑥−𝑦. (2.4) So 𝑑_𝑝 is a metric on 𝑋 and (𝑋,𝑑_𝑝) is a complete metric space.

Theorem 2.4 (see [17]). Let (𝑋,𝑝) be a TVS-cone metric space, 𝑥∈𝑋, and {𝑥_𝑛}_𝑛∈ℕ a sequence in 𝑋. Then, the following statements hold.(a) If {𝑥_𝑛}  TVS-cone converges to 𝑥(i.e.,𝑥−−−→_𝑛^cone𝑥 as 𝑛→∞), then 𝑑_𝑝(𝑥_𝑛,𝑥)→0 as 𝑛→∞(i.e., 𝑥−→_𝑛^𝑑𝑝𝑥 as 𝑛→∞).(b) If {𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝), then {𝑥_𝑛} is a Cauchy sequence (in usual sense) in (𝑋,𝑑_𝑝).

Definition 2.5 (see [1, 22]). Let (𝑋,𝑑) be a metric space. A multivalued map 𝑇∶𝑋→𝒩(𝑋) is said to have an approximate fixed point property provided inf_𝑥∈𝑋𝑑(𝑥,𝑇𝑥)=0, (2.5) or, equivalently, for any 𝜖>0, there exists 𝑧∈𝑋 such that 𝑑(𝑧,𝑇𝑧)≤𝜖, (2.6) or, equivalently, for any 𝜖>0, there exists 𝑥_𝜖∈𝑋 such that 𝑇𝑥_𝜖∩𝐵𝑥_𝜖,𝜖≠∅, (2.7)
where 𝐵(𝑥,𝑟) denotes a closed ball of radius 𝑟 centered at 𝑥.
It is known that every generalized multivalued almost contraction in a metric space (𝑋,𝑑) has the approximate fixed point property (see [1, Lemma 2.2]).

Remark 2.6. It is obvious that ℱ(𝑇)≠∅ implies that 𝑇 has the approximate fixed point property.

Definition 2.7 (see [1]). Let (𝑋,𝑑) be a metric space, 𝑔∶𝑋→𝑋 a single-valued self-map, and 𝑇∶𝑋→𝒩(𝑋) a multivalued map. The maps 𝑔 and 𝑇 are said to have an approximate coincidence point property provided inf_𝑥∈𝑋𝑑𝑔𝑥,𝑇𝑥=0, (2.8) or, equivalently, for any 𝜖>0, there exists 𝑧∈𝑋 such that 𝑑𝑔𝑧,𝑇𝑧≤𝜖. (2.9)
It is known that every generalized multivalued almost 𝑔-contraction in a metric space (𝑋,𝑑) has the approximate coincidence point property provided each 𝑇𝑥 is 𝑔-invariant (i.e., 𝑔(𝑇𝑥)⊆𝑇𝑥) for each 𝑥∈𝑋 (see [1, Theorem 2.7]).

3. Main Results

For any locally convex Hausdorff t.v.s. 𝑌 with its zero vector 𝜃, let 𝜏 denote the topology of 𝑌 and let 𝒰_𝜏 be the base at 𝜃 consisting of all absolutely convex neighborhood of 𝜃. Letℒ=ℓ∶ℓisaMinkowskifunctionalof𝑈for𝑈∈𝒰_𝜏. (3.1) Then, ℒ is a family of seminorms on 𝑌. For each ℓ∈ℒ, put𝑉(ℓ)=𝑦∈𝑌∶ℓ𝑦<1 (3.2) and denote𝒰_ℒ=𝑈∶𝑈=𝑟₁𝑉ℓ₁∩𝑟₂𝑉ℓ₂∩⋯∩𝑟_𝑛𝑉ℓ_𝑛,𝑟_𝑘>0,ℓ_𝑘∈ℒ,1≤𝑘≤𝑛,𝑛∈ℕ. (3.3) Then, 𝒰_ℒ is a base at 𝜃 and the topology Γ_ℒ generated by 𝒰_ℒ is the weakest topology for 𝑌 such that all seminorms in ℒ are continuous and 𝜏=Γ_ℒ. Moreover, given any neighborhood 𝒪_𝜃 of 𝜃, there exists 𝑈∈𝒰_ℒ such that 𝜃∈𝑈⊂𝒪_𝜃; 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 𝑂_𝜃 be a neighborhood of 𝜃, 𝜔∈𝐾, and {𝛼_𝑛}⊂[0,∞) with lim_𝑛→∞𝛼_𝑛=0. Suppose that 𝑈∈𝒰_ℒ satisfies 𝜃∈𝑈⊂𝑂_𝜃, then there exists 𝑛₀∈ℕ, such that ±𝛼_𝑛𝜔∈𝑈 for all 𝑛≥𝑛₀.

Proof . Clearly, if 𝜔=𝜃, then ±𝛼_𝑛𝜔=𝜃∈𝑈 for all 𝑛∈ℕ, and we are done. Suppose 𝜔∈𝐾⧵{𝜃}. Since 𝑈∈𝒰_ℒ, 𝑈=𝑟₁𝑉ℓ₁∩𝑟₂𝑉ℓ₂∩⋯∩𝑟_𝑠𝑉ℓ_𝑠, (3.4) for some 𝑟_𝑖>0, ℓ_𝑖∈ℒ and 1≤𝑖≤𝑠. Let 𝛿=min𝑟_𝑖ℓ∶1≤𝑖≤𝑠>0,𝜌=max_𝑖(𝜔)∶1≤𝑖≤𝑠. (3.5) We need consider two possible cases.Case 1. If 𝜌=0, since each ℓ_𝑖 is a seminorm, we have ℓ_𝑖(𝜔)=0 and ℓ_𝑖±𝛼_𝑛𝜔=𝛼_𝑛ℓ_𝑖(𝜔)=0<𝑟_𝑖 (3.6) for all 1≤𝑖≤𝑠 and all 𝑛∈ℕ.Case 2. If 𝜌>0, since lim_𝑛→∞𝛼_𝑛=0, there exists 𝑛₀∈ℕ such that 𝛼_𝑛<𝛿/𝜌 for all 𝑛≥𝑛₀. So, for each 𝑖∈{1,2,…,𝑠} and any 𝑛≥𝑛₀, we obtain ℓ_𝑖±𝛼_𝑛𝜔=𝛼_𝑛ℓ_𝑖(𝜔)<𝛿𝜌ℓ_𝑖(𝜔)≤𝛿≤𝑟_𝑖. (3.7)
Hence, by Cases 1 and 2, we get that, for any 𝑛≥𝑛₀, ±𝛼_𝑛𝜔∈𝑟_𝑖𝑉(ℓ_𝑖) for all 1≤𝑖≤𝑠. Therefore, ±𝛼_𝑛𝜔∈𝑈 for all 𝑛≥𝑛₀.

Lemma 3.2 (see [24, Lemma 2.5]). Let 𝐸 be a t.v.s., 𝐾 a convex cone with int𝐾≠∅ in 𝐸, and 𝑎,𝑏,𝑐∈𝐸. Then, the following statements hold.(a)int𝐾+𝐾⊆int𝐾.(b) If 𝑎⪷_𝐾𝑏 and 𝑏≪_𝐾𝑐, then 𝑎≪_𝐾𝑐.

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 (𝑋,𝑝).

Theorem 3.3. Let (𝑋,𝑝) be a TVS-cone metric space, 𝑇∶𝑋→𝒩(𝑋) a multivalued map, ℎ∶𝑋→𝑋 a self-map, and 𝑑_𝑝∶=𝜉_𝑒∘𝑝. Suppose that(D1) there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that, for each 𝑥∈𝑋, if 𝑦∈𝑇𝑥 with 𝑦≠𝑥 then there exists 𝑧∈𝑇𝑦 such that𝑝ℎ𝑦,ℎ𝑧⪷_𝐾𝜇𝑑_𝑝ℎ𝑥,ℎ𝑦𝑝ℎ𝑥,ℎ𝑦, (3.8)(D2)𝑇𝑥 is ℎ-invariant (i.e., ℎ(𝑇𝑥)⊆𝑇𝑥) for each 𝑥∈𝑋. Then, the following statements hold.(a)ℎ and 𝑇 have the 𝑑_𝑝-approximate coincidence point property on 𝑋 (i.e., inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0).(b) There exists a sequence {𝑧_𝑛} in (𝑋,𝑝) such that {𝑧_𝑛} is a TVS-cone Cauchy sequence and lim_𝑛→∞𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0.

Proof. By Theorem 2.3, we know that 𝑑_𝑝 is a metric on 𝑋. Let 𝑥₁∈𝑋 and 𝑥₂∈𝑇𝑥₁. If 𝑥₁=𝑥₂, then, by (D2), we have ℎ𝑥₁∈𝑇𝑥₁. Since inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)≤𝑑_𝑝ℎ𝑥₁,𝑇𝑥₁=0, (3.9) we have inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0. Let 𝑧_𝑛=ℎ𝑥₁ for all 𝑛∈ℕ. Then, {𝑧_𝑛}⊂𝑋 is a TVS-cone Cauchy sequence and lim_𝑛→∞𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)=0=inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥). Hence, all conclusions are proved in the case of 𝑥₁=𝑥₂. If 𝑥₂≠𝑥₁ or 𝑝(𝑥₁,𝑥₂)≠𝜃, then, by (D1), there exists 𝑥₃∈𝑇𝑥₂ such that 𝑝ℎ𝑥₂,ℎ𝑥₃⪷_𝐾𝜇𝑑_𝑝ℎ𝑥₁,ℎ𝑥₂𝑝ℎ𝑥₁,ℎ𝑥₂. (3.10) If 𝑥₂=𝑥₃, then the conclusions also hold by following a similar argument as above. If 𝑥₃≠𝑥₂, then there exists 𝑥₄∈𝑇𝑥₃ such that 𝑝ℎ𝑥₃,ℎ𝑥₄⪷_𝐾𝜇𝑑_𝑝ℎ𝑥₂,ℎ𝑥₃𝑝ℎ𝑥₂,ℎ𝑥₃. (3.11) By induction, we can obtain a sequence {𝑥_𝑛} in 𝑋 satisfying the following: for each 𝑛∈ℕ, 𝑥_𝑛+1∈𝑇𝑥_𝑛,𝑝ℎ𝑥_𝑛+1,ℎ𝑥_𝑛+2⪷_𝐾𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1. (3.12) Applying (v) and (vi) of Theorem 2.1, the inequality (3.12) implies 𝑑_𝑝ℎ𝑥_𝑛+1,ℎ𝑥_𝑛+2=𝜉_𝑒𝑝ℎ𝑥_𝑛+1,ℎ𝑥_𝑛+2≤𝜉_𝑒𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1≤𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1𝜉_𝑒𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1=𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1. (3.13) Since 𝜇(𝑡)<1 for all 𝑡∈[0,∞), the sequence {𝑑_𝑝(ℎ𝑥_𝑛,ℎ𝑥_𝑛+1)}_𝑛∈ℕ is strictly decreasing in [0,∞). Since 𝜇 is an ℳ𝒯-function, by (g) of Theorem D, 0≤sup_𝑛∈ℕ𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1<1. (3.14) Let 𝜆∶=sup_𝑛∈ℕ𝜇(𝑑_𝑝(ℎ𝑥_𝑛,ℎ𝑥_𝑛+1)). So 𝜆∈[0,1). From (3.12), we get 𝑝ℎ𝑥_𝑛+1,ℎ𝑥_𝑛+2⪷_𝐾𝜇𝑑_𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1⪷_𝐾𝜆𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1. (3.15) It follows from (3.15) that 𝑝ℎ𝑥_𝑛+1,ℎ𝑥_𝑛+2⪷_𝐾𝜆𝑝ℎ𝑥_𝑛,ℎ𝑥_𝑛+1⪷_𝐾⋯⪷_𝐾𝜆^𝑛𝑝ℎ𝑥₁,ℎ𝑥₂foreach𝑛∈ℕ. (3.16) Let 𝑧_𝑛∶=ℎ𝑥_𝑛, 𝑛∈ℕ. Applying (v) and (vi) of Theorem 2.1, the inequality (3.16) implies 𝑑_𝑝𝑧_𝑛+1,𝑧_𝑛+2≤𝜆^𝑛𝑑_𝑝𝑧₁,𝑧₂foreach𝑛∈ℕ. (3.17) Since 𝜆∈[0,1), by (3.17), we have lim_𝑛→∞𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)=0. By (3.12) and (D2), we obtain 𝑧_𝑛+1=ℎ𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ. It implies that inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)≤𝑑_𝑝ℎ𝑥_𝑛,𝑇𝑥_𝑛≤𝑑_𝑝𝑧_𝑛,𝑧_𝑛+1,∀𝑛∈ℕ. (3.18) Since 𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)→0 as 𝑛→∞, it follows from (3.18) that inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0 and the conclusion (a) is proved.
To see (b), it suffices to prove that {𝑧_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝). For 𝑚,𝑛∈ℕ with 𝑚>𝑛, it follows from (3.16) that 𝑝𝑧_𝑛,𝑧_𝑚⪷_𝐾^𝑚−1_𝑗=𝑛𝑝𝑧_𝑗,𝑧_𝑗+1⪷_𝐾𝜆^𝑛−1𝑝𝑧1−𝜆¹,𝑧₂. (3.19) Given 𝑐∈𝑌 with 𝜃≪_𝐾𝑐 (i.e, 𝑐∈int𝐾=int(int𝐾)), there exists a neighborhood 𝑁_𝜃 of 𝜃 such that 𝑐+𝑁_𝜃⊆int𝐾. Therefore, there exists 𝑈_𝑐∈𝒰_ℒ with 𝑈_𝑐⊆𝑁_𝜃 such that 𝑐+𝑈_𝑐⊆𝑐+𝑁_𝜃⊆int𝐾. (3.20) Let 𝜔=𝑝(𝑧₁,𝑧₂) and 𝛼_𝑛=𝜆^𝑛−1/(1−𝜆), 𝑛∈ℕ. Since 𝜆∈[0,1), we have {𝛼_𝑛}⊂[0,∞) and lim_𝑛→∞𝛼_𝑛=0. Applying Lemma 3.1, there exists 𝑛₀∈ℕ, such that −𝛼_𝑛𝜔∈𝑈_𝑐 for all 𝑛≥𝑛₀. So, by (3.20), we obtain 𝑐−𝛼_𝑛𝜔∈𝑐+𝑈_𝑐⊆int𝐾 (3.21) or 𝛼_𝑛𝜔≪_𝐾𝑐 (3.22) for all 𝑛≥𝑛₀. For 𝑚,𝑛∈ℕ with 𝑚>𝑛≥𝑛₀, since 𝑝(𝑧_𝑛,𝑧_𝑚)⪷_𝐾𝛼_𝑛𝜔 from (3.19) and 𝛼_𝑛𝜔≪_𝐾𝑐, it follows from Lemma 3.2 that 𝑝𝑧_𝑚,𝑧_𝑛≪_𝐾𝑐. (3.23) Hence, we prove that {𝑧_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝). The proof is completed.

Theorem 3.4. Let (𝑋,𝑝) be a TVS-cone metric space, 𝑇∶𝑋→𝒩(𝑋) a multivalued map, and 𝑑_𝑝∶=𝜉_𝑒∘𝑝. Suppose that(D3) there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that, for each 𝑥∈𝑋, if 𝑦∈𝑇𝑥 with 𝑦≠𝑥, then there exists 𝑧∈𝑇𝑦 such that𝑝𝑦,𝑧⪷_𝐾𝜇𝑑_𝑝𝑥,𝑦𝑝𝑥,𝑦. (3.24) Then, there exists a nonempty proper subset 𝑉 of 𝑋, such that inf_𝑥∈𝑋𝑑_𝑝(𝑥,𝑇𝑥)=inf_𝑥∈𝑉𝑑_𝑝(𝑥,𝑇𝑥)=0 (i.e., 𝑇 has the 𝑑_𝑝-approximate fixed point property on 𝑉 and 𝑋.)

Proof. Let ℎ≡𝑖𝑑 be the identity map on 𝑋. 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 {𝑥_𝑛} and {𝑧_𝑛} in 𝑋 such that(i)𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑧_𝑛=ℎ𝑥_𝑛 and 𝑧_𝑛+1∈𝑇𝑥_𝑛 for each 𝑛∈ℕ,(ii)lim_𝑛→∞𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)=0.
Since ℎ≡𝑖𝑑, {𝑥_𝑛} and {𝑧_𝑛} are identical, we have lim_𝑛→∞𝑑_𝑝(𝑥_𝑛,𝑥_𝑛+1)=0. Put 𝑉={𝑥_𝑛}_𝑛∈ℕ⊂𝑋. Since inf_𝑥∈𝑋𝑑_𝑝(𝑥,𝑇𝑥)≤inf_𝑥∈𝑉𝑑_𝑝(𝑥,𝑇𝑥)≤𝑑_𝑝𝑥_𝑛,𝑇𝑥_𝑛≤𝑑_𝑝𝑥_𝑛,𝑥_𝑛+1,∀𝑛∈ℕ, (3.25) we obtain inf_𝑥∈𝑋𝑑_𝑝(𝑥,𝑇𝑥)=inf_𝑥∈𝑉𝑑_𝑝(𝑥,𝑇𝑥)=0. Hence, 𝑇 has the 𝑑_𝑝-approximate fixed point property on 𝑉 and 𝑋.

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

Theorem 3.5. Let (𝑋,𝑑) be a metric space, 𝑇∶𝑋→𝒩(𝑋) a multivalued map, and ℎ∶𝑋→𝑋 a self-map. Suppose that(A1) there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that, for each 𝑥∈𝑋, if 𝑦∈𝑇𝑥 with 𝑦≠𝑥, then there exists 𝑧∈𝑇𝑦 such that𝑑ℎ𝑦,ℎ𝑧≤𝜇𝑑ℎ𝑥,ℎ𝑦𝑑ℎ𝑥,ℎ𝑦, (3.26)(A2)𝑇𝑥 is ℎ-invariant (i.e., ℎ(𝑇𝑥)⊆𝑇𝑥) for each 𝑥∈𝑋.Then, ℎ and 𝑇 have the approximate coincidence point property on 𝑋.

Proof . In Theorem 3.3, let 𝑌=ℝ, 𝐾=[0,∞)⊂ℝ, and 𝑒=1. 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 (𝑋,𝑑) be a metric space and 𝑇∶𝑋→𝒩(𝑋) a multivalued map. Suppose that(A3) there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that, for each 𝑥∈𝑋, if 𝑦∈𝑇𝑥 with 𝑦≠𝑥, then there exists 𝑧∈𝑇𝑦 such that𝑑𝑦,𝑧≤𝜇𝑑𝑥,𝑦𝑑𝑥,𝑦. (3.27) Then, 𝑇 has the approximate fixed point property on 𝑋.

Theorem 3.7. In Theorem 3.6, if the condition (A3) is replaced by (A3)_H, where(A3)_H there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that, for each 𝑥∈𝑋,𝑑𝑦,𝑇𝑦≤𝜇𝑑𝑥,𝑦𝑑𝑥,𝑦,∀𝑦∈𝑇𝑥, (3.28) then, 𝑇 has the approximate fixed point property on 𝑋.

Proof. Define 𝜏∶[0,∞)→[0,1) by 𝜏(𝑡)=(1+𝜇(𝑡))/2. Then, by [3, Lemma 2.1], 𝜏 is also an ℳ𝒯-function. For each 𝑥∈𝑋, let 𝑦∈𝑇𝑥 with 𝑦≠𝑥. Then, 𝑑(𝑥,𝑦)>0. By (A3)_H, we have 𝑑𝑦,𝑇𝑦<𝜏𝑑𝑥,𝑦𝑑𝑥,𝑦. (3.29) It follows that there exists 𝑧∈𝑇𝑦 such that 𝑑𝑦,𝑧<𝜏𝑑𝑥,𝑦𝑑𝑥,𝑦, (3.30) 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 (𝑋,𝑝) be a TVS-cone metric space. By Theorem 2.3, we know that 𝑑_𝑝∶=𝜉_𝑒∘𝑝 is a metric on 𝑋. So we can obtain the topology Γ_𝑑𝑝 on 𝑋 induced by 𝑑_𝑝 and hence define 𝑑_𝑝-open subsets, 𝑑_𝑝-closed subsets, and 𝑑_𝑝-compact subsets of 𝑋.
Here, we denote 𝒞_𝑑𝑝(𝑋) by the collection of all nonempty 𝑑_𝑝-closed subsets of 𝑋.

Theorem 3.10. Let (𝑋,𝑝) be a TVS-cone complete metric space, 𝑇∶𝑋→𝒞_𝑑𝑝(𝑋) a multivalued map, ℎ∶𝑋→𝑋 a self-map, and 𝑑_𝑝∶=𝜉_𝑒∘𝑝. Suppose that the conditions (D1) and (D2) as in Theorem 3.3 hold and further assume one of the following conditions hold:(L1)(𝑋,𝑑_𝑝) is 𝑑_𝑝-compact and the function 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑑_𝑝(ℎ𝑥,𝑇𝑥) is l.s.c.,(L2)𝒢={(ℎ𝑥,𝑦)∈𝑋×𝑋∶𝑥∈𝑋,𝑦∈𝑇𝑥} is a 𝑑_𝑝-closed subset of 𝑋×𝑋.Then, the following statements hold.(a) There exists a nonempty subset ℳ of 𝑋, such that (ℳ,𝑑_𝑝) is a complete metric space.(b)𝒞𝒪𝒫(ℎ,𝑇)≠∅.

Proof. Following the same argument as in the proof of Theorem 3.3, one can obtain that there exist two sequences {𝑥_𝑛} and {𝑧_𝑛} in 𝑋 such that for any 𝑛∈ℕ, 𝑧_𝑛=ℎ𝑥_𝑛, 𝑧_𝑛+1∈𝑇𝑥_𝑛, {𝑧_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝) and lim_𝑛→∞𝑑_𝑝(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0. By the TVS-cone completeness of (𝑋,𝑝), there exists 𝑣∈𝑋, such that {𝑧_𝑛}  TVS-cone converges to 𝑣. On the other hand, applying Theorem 2.4, we obtain that {𝑧_𝑛} is a Cauchy sequence in (𝑋,𝑑_𝑝) and 𝑑_𝑝(𝑧_𝑛,𝑣)→0 or 𝑧−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞. Let ℳ={𝑧_𝑛}_𝑛∈ℕ∪{𝑣}. Then (ℳ,𝑑_𝑝) is a complete metric space and the conclusion (a) holds.
Now, we verify the conclusion (b). Suppose that (L1) holds. Then the infimum inf_𝑥∈𝑋𝑓(𝑥) is attained. Since inf_𝑥∈𝑋𝑓(𝑥)=inf_𝑥∈𝑋𝑑_𝑝(ℎ𝑥,𝑇𝑥)=0, there exists 𝜁∈𝑋 such that 𝑑_𝑝(ℎ𝜁,𝑇𝜁)∶=𝑓(𝜁)=inf_𝑥∈𝑋𝑓(𝑥)=0. (3.31) Since 𝑇𝜁 is a 𝑑_𝑝-closed subset of 𝑋, it implies from (3.31) that ℎ𝜁∈𝑇𝜁. Hence, 𝜁∈𝒞𝒪𝒫(ℎ,𝑇).
Suppose that (L2) holds. For any 𝑛∈ℕ, since 𝑧_𝑛+1∈𝑇𝑥_𝑛, we know (𝑧_𝑛,𝑧_𝑛+1)∈𝒢. Since 𝒢 is 𝑑_𝑝-closed in 𝑋×𝑋 and 𝑧−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞, we have (𝑣,𝑣)∈𝒢. Hence, there exists 𝑎∈𝑋 such that 𝑣=ℎ𝑎 and 𝑣∈𝑇𝑎, which say that 𝑎∈𝒞𝒪𝒫(ℎ,𝑇). The proof is completed.

Theorem 3.11. Let (𝑋,𝑝) be a TVS-cone complete metric space, 𝑇∶𝑋→𝒞_𝑑𝑝(𝑋) a multivalued map, and 𝑑_𝑝∶=𝜉_𝑒∘𝑝. Suppose that the condition (D3) as in Theorem 3.4 holds and further assume one of the following conditions hold:(H1)𝑇 is 𝑑_𝑝-closed; that is, 𝐺𝑟𝑇∶={(𝑥,𝑦)∈𝑋×𝑋∶𝑦∈𝑇𝑥}, the graph of 𝑇, is a 𝑑_𝑝-closed subset of 𝑋×𝑋,(H2) the function 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑑_𝑝(𝑥,𝑇𝑥) is l.s.c.,(H3)inf{𝑑_𝑝(𝑥,𝑧)+𝑑_𝑝(𝑥,𝑇𝑥)∶𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇). Then, there exists a nonempty subset 𝑊 of 𝑋, such that(a)(𝑊,𝑑_𝑝) is a complete metric space,(b)ℱ(𝑇)∩𝑊≠∅.

Proof. Following a similar argument as in the proof of Theorem 3.3, we can obtain a sequence {𝑥_𝑛} in 𝑋 such that, for any 𝑛∈ℕ,(i)𝑥_𝑛+1∈𝑇𝑥_𝑛,(ii)𝑝(𝑥_𝑛+1,𝑥_𝑛+2)⪷_𝐾𝛾𝑝(𝑥_𝑛,𝑥_𝑛+1), where 𝛾∶=sup_𝑛∈ℕ𝜇(𝑑_𝑝(𝑥_𝑛,𝑥_𝑛+1))∈[0,1),(iii)𝑝(𝑥_𝑛,𝑥_𝑚)⪷_𝐾(𝛾^𝑛−1/(1−𝛾))𝑝(𝑥₁,𝑥₂), for 𝑚,𝑛∈ℕ with 𝑚>𝑛.(iv){𝑥_𝑛} is a TVS-cone Cauchy sequence in (𝑋,𝑝).
Applying Theorem 2.1, the inequality (iii) implies that, for 𝑚,𝑛∈ℕ with 𝑚>𝑛, we have 𝑑_𝑝𝑥_𝑛,𝑥_𝑚=𝜉_𝑒𝑝𝑥_𝑛,𝑥_𝑚≤𝜉_𝑒𝛾^𝑛−1𝑝𝑥1−𝛾¹,𝑥₂=𝛾^𝑛−1𝑑1−𝛾^𝑝𝑥₁,𝑥₂. (3.32) Since lim_𝑛→∞𝛾^𝑛−1/(1−𝛾)=0, it follows from (3.32) that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑_𝑝). By the TVS-cone completeness of (𝑋,𝑝) and (iv), there exists 𝑣∈𝑋, such that {𝑥_𝑛}  TVS-cone converges to 𝑣. Applying Theorem 2.4, we have 𝑑_𝑝(𝑥_𝑛,𝑣)→0 or 𝑥−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞. Let 𝑊={𝑥_𝑛}_𝑛∈ℕ∪{𝑣}. Then, (𝑊,𝑑_𝑝) is a complete metric space and the conclusion (a) holds.
Finally, in order to complete the proof, it suffices to show that 𝑣∈ℱ(𝑇). Suppose that (H1) holds. Since 𝑥_𝑛+1∈𝑇𝑥_𝑛, we have (𝑥_𝑛,𝑥_𝑛+1)∈𝐺𝑟𝑇 for each 𝑛∈ℕ. By (H1) and 𝑥−→_𝑛^𝑑𝑝𝑣 as 𝑛→∞, we get 𝑣∈ℱ(𝑇).
If (H2) holds, by the lower semicontinuity of 𝑓 and (i), we obtain 𝑑_𝑝(𝑣,𝑇𝑣)=𝑓(𝑣)≤liminf_𝑛→∞𝑓𝑥_𝑛≤liminf_𝑛→∞𝑑_𝑝𝑥_𝑛,𝑇𝑥_𝑛≤lim_𝑛→∞𝑑_𝑝𝑥_𝑛,𝑥_𝑛+1=0, (3.33) which implies 𝑑_𝑝(𝑣,𝑇𝑣)=0. Since 𝑇𝑣 is a 𝑑_𝑝-closed subset of 𝑋, 𝑣∈ℱ(𝑇).
Let (H3) holds. Suppose 𝑣∉ℱ(𝑇). Since 𝑑_𝑝 is a metric on 𝑋 and 𝑥−→_𝑚^𝑑𝑝𝑣 as 𝑚→∞, by (3.32), we get 𝑑_𝑝𝑥_𝑛≤𝛾,𝑣^𝑛−1𝑑1−𝛾^𝑝𝑥₁,𝑥₂forany𝑛∈ℕ. (3.34) From (3.32) and (3.34), we have 0<inf_𝑥∈𝑋𝑑_𝑝(𝑥,𝑣)+𝑑_𝑝(𝑥,𝑇𝑥)≤inf_𝑛∈ℕ𝑑_𝑝𝑥_𝑛,𝑣+𝑑_𝑝𝑥_𝑛,𝑇𝑥_𝑛≤inf_𝑛∈ℕ𝑑_𝑝𝑥_𝑛,𝑣+𝑑_𝑝𝑥_𝑛,𝑥_𝑛+1≤lim_𝑛→∞2𝛾^𝑛−1𝑑1−𝛾^𝑝𝑥₁,𝑥₂=0, (3.35) which leads a contradiction. Therefore, 𝑣∈ℱ(𝑇). The proof is completed.

The following result is immediate from Theorem 3.10.

Theorem 3.12. Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞(𝑋) a multivalued map, and ℎ∶𝑋→𝑋 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)(𝑋,𝑑) is compact and the function 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑑(ℎ𝑥,𝑇𝑥) is l.s.c.;(ii){(ℎ𝑥,𝑦)∈𝑋×𝑋∶𝑥∈𝑋,𝑦∈𝑇𝑥} is a closed subset of 𝑋×𝑋.Then, 𝒞𝒪𝒫(ℎ,𝑇)≠∅.

Theorem 3.13. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝒞(𝑋) a multivalued map. Suppose that the condition (A3) or (A3)_H holds and further assume one of the following conditions hold:(h1)𝑇 is closed,(h2) the function 𝑓∶𝑋→[0,∞) defined by 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥) is l.s.c.,(h3)inf{𝑑(𝑥,𝑧)+𝑑(𝑥,𝑇𝑥)∶𝑥∈𝑋}>0 for every 𝑧∉ℱ(𝑇),(h4) for each sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝑣, we have lim_𝑛→∞ℋ(𝑇𝑥_𝑛,𝑇𝑣)=0.Then, ℱ(𝑇)≠∅.

Proof. The conclusion is immediate from Theorem 3.11 if 𝑌=ℝ, 𝐾=[0,∞)⊂ℝ, 𝑒=1, 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 {𝑥_𝑛} in (𝑋,𝑑) such that 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ, and {𝑥_𝑛} converge to some point 𝑣∈𝑋. Since the function 𝑥↦𝑑(𝑥,𝑇𝑣) is continuous, 𝑥_𝑛+1∈𝑇𝑥_𝑛 and 𝑥−→_𝑛^𝑑𝑣 as 𝑛→∞, by (h4), we get 𝑑(𝑣,𝑇𝑣)=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑣≤lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑣=0, (3.36) which implies 𝑣∈ℱ(𝑇). The proof is completed.

Remark 3.14. (a)Let 𝐴 and 𝐵 be two topological vector spaces and 𝑇∶𝐴→2^𝐵 a multivalued map. Recall that 𝑇 is u.s.c. if and only if for any open set 𝑉 in 𝐵, 𝑇⁺(𝑉)∶={𝑥∈𝐴∶𝑇(𝑥)⊆𝑉} is open in 𝐴. It is known that if 𝑇 is u.s.c. with closed values, then 𝑇 is closed (see [25]). Hence, Theorem 3.13 is true if (𝑋,‖⋅‖) is a Banach space and 𝑇∶𝑋→𝐶(𝑋) is u.s.c.;(b)let 𝐺 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇∶𝑋→𝒞(𝑋) u.s.c. Then, the function 𝜙∶𝐺→[0,∞) defined by 𝜙(𝑥)=𝑑(𝑥,𝑇𝑥) 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 (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) a multivalued map, and 𝐺∶𝑋→[0,∞) a function. Suppose that there exists an ℳ𝒯-function 𝜇∶[0,∞)→[0,1) such that ℋ𝑇𝑥,𝑇𝑦≤𝜇𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐺𝑦𝑑𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋. (3.37) Then, ℱ(𝑇)≠∅.

Proof. Let 𝑥∈𝑋. If 𝑦∈𝑇𝑥, then 𝑑(𝑦,𝑇𝑥)=0. So (3.37) implies the inequality 𝑑𝑦,𝑇𝑦≤𝜇𝑑𝑥,𝑦𝑑𝑥,𝑦,∀𝑦∈𝑇𝑥. (3.38) Hence, the condition (A3)_H of Theorem 3.13 holds. Let {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ, and lim_𝑛→∞𝑥_𝑛=𝑣. By (3.37), we obtain lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑣≤lim_𝑛→∞𝜑𝑑𝑥_𝑛,𝑣𝑑𝑥_𝑛,𝑣+𝐺(𝑣)𝑑𝑣,𝑥_𝑛+1=0, (3.39) 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 𝐺(𝑥)=0 for all 𝑥∈𝑋, then we can obtain Mizoguchi-Takahashi’s fixed point theorem [14].(c)In Theorem 3.15, if 𝐺(𝑥)=0 for all 𝑥∈𝑋, and 𝜇∶[0,∞)→[0,1) is defined by 𝜇(𝑡)=𝛾 for some 𝛾∈[0,1), 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 𝑇 in a metric space (𝑋,𝑑) has 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. 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.