Fixed Points and Endpoints of Set-Valued Contractions in Cone Metric Spaces

Abstract

The purpose of this paper is to present the fixed points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović, 2011; Wardowski, 2009.

1. Introduction and Preliminaries

Throughout this paper, by ℝ⁺, ℝ we denote the sets of all nonnegative real numbers and all real numbers, respectively, while ℕ is the set of all natural numbers. Let (𝑋,𝑑) be a metric space, 𝐷 a subset of 𝑋, and 𝑓∶𝐷→𝑋 a map. We say 𝑓 is contractive if there exists 𝛼∈[0,1) such that, for all 𝑥,𝑦∈𝐷, 𝑑𝑓𝑥,𝑓𝑦≤𝛼⋅𝑑𝑥,𝑦. (1.1) The well-known Banach fixed-point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed-point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan [2] and Chatterjea [3] introduced two conditions that can replace (1.1) in Banach's theorem as follows:

Kannan [2]: There exists 𝛼∈[0,1) such that, for all 𝑥,𝑦∈𝑋, 𝑑≤𝛼𝑓𝑥,𝑓𝑦2𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦. (1.2)

Chatterjea [3]: There exists 𝛼∈[0,1) such that for all 𝑥,𝑦∈𝑋, 𝑑≤𝛼𝑓𝑥,𝑓𝑦2𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥. (1.3)

After these three conditions, many papers have been written generalizing some of the conditions (1.1), (1.2), and (1.3). In 1969, Boyd and Wong [4] showed the following fixed-point theorem.

Theorem 1.1 (see [4]). Let (𝑋,𝑑) be a complete metric space and 𝑓∶𝑋→𝑋 a map. Suppose there exists a function 𝜙∶[0,∞)→[0,∞) satisfying 𝜙(0)=0, 𝜙(𝑡)<𝑡 for all 𝑡>0 and 𝜙 is right upper semicontinuous such that 𝑑𝑓𝑥,𝑓𝑦≤𝜙𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (1.4) Then 𝑓 has a unique fixed-point in 𝑋.

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wong's result to mappings satisfying the following more general condition: ∀𝜂>0∃𝛿>0suchthat𝜂≤𝑑𝑥,𝑦<𝜂+𝛿⟹𝑑𝑓𝑥,𝑓𝑦<𝜂, (1.5) and Meir-Keeler proved the following very interesting fixed-point theorem which is a generalization of the Banach contraction principle.

Theorem 1.2 (Meir-Keeler [5]). Let (𝑋,𝑑) be a complete metric space and let 𝑓 be a Meir-Keeler contraction; that is, for every 𝜂>0, there exists 𝛿>0 such that 𝑑(𝑥,𝑦)<𝜂+𝛿 implies 𝑑(𝑓𝑥,𝑓𝑦)<𝜂 for all 𝑥,𝑦∈𝑋. Then 𝑓 has a unique fixed-point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7–10]). In this paper, we introduce a new notion of the stronger Meir-Keeler-type mapping 𝜓∶[0,∞)→[0,𝑏), 𝑏∈(0,1], as follows.

Definition 1.3. A mapping 𝜓∶[0,∞)→[0,𝑏), 𝑏∈(0,1] is called a stronger Meir-Keeler-type mapping in a metric space 𝑋, if for each 𝜂>0, there exists 𝛿>0 such that for 𝑥,𝑦∈𝑋 with 𝜂≤𝑑(𝑥,𝑦)<𝜂+𝛿, there exists 𝛾_𝜂∈[0,𝑏) such that 𝜓(𝑑(𝑥,𝑦))<𝛾_𝜂.

Example 1.4. Let (𝑋,𝑑) be a metric space, and define 𝜓∶[0,∞)→[0,𝑏), 𝑏∈(0,1] by 𝜓𝑑𝑥,𝑦=𝑏𝑑𝑥,𝑦𝑑𝑥,𝑦+1,for𝑥,𝑦∈𝑋. (1.6) Then 𝜓 is a stronger Meir-Keeler-type mapping.

The existence of fixed-points for various multi valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [11] extended the famous Banach contraction principle from single-valued mapping to multi valued mapping and proved the following fixed-point theorem for multi valued contraction. Let 𝑁(𝑋) denote the collection of all nonempty subsets of 𝑋, 𝐵(𝑋) the collection of all nonempty bounded subsets of 𝑋, 𝐶(𝑋) the collection of all nonempty closed subsets of 𝑋, 𝐶𝐵(𝑋) the collection of all nonempty closed and bounded subsets of 𝑋, and 𝐾(𝑋) the collection of all nonempty sequentially compact subsets of 𝑋.

Theorem 1.5 (see [11]). Let (𝑋,𝑑) be a complete metric space, and, 𝑇 be a mapping from 𝑋 into 𝐶𝐵(𝑋). Assume that there exists 𝑐∈[0,1) such that ℋ𝑇𝑥,𝑇𝑦≤𝑐𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (1.7) Then 𝑇 has a fixed-point in 𝑋.

In 1989, Mizoguchi-Takahashi [12] proved the following fixed-point theorem.

Theorem 1.6 (see [12]). Let (𝑋,𝑑) be a complete metric space and 𝑇 a map from 𝑋 into 𝐶𝐵(𝑋). Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜉𝑑𝑥,𝑦⋅𝑑𝑥,𝑦 (1.8) for all 𝑥,𝑦∈𝑋, where 𝜉∶[0,∞)→[0,1) satisfies limsup_𝑠→𝑡+𝜉(𝑠)<1 for all 𝑡∈[0,∞). Then 𝑇 has a fixed-point in 𝑋.

Remark 1.7. It is clear that if the function 𝜉∶[0,∞)→[0,𝑏), 𝑏∈(0,1] satisfies limsup_𝑠→𝑡+𝜉(𝑠)<𝑏,∀𝑡∈[0,∞), (1.9) then 𝜉 is also a stronger Meir-Keeler-type function.

In 2003, Rus [13] introduced the notion of endpoints (strict fixed-points) and proved some results on strict fixed-point theory on multi valued operator. An element 𝑥∈𝑋 is said to be a fixed-point of 𝑇∶𝑋→𝑁(𝑋), if 𝑥∈𝑇𝑥. If {𝑥}=𝑇𝑥, then 𝑥 is called an endpoint of 𝑇. We will denote the sets of all fixed-points and endpoints of 𝑇 by Fix(𝑇) and End(𝑇), respectively. The following theorems were the main results of Rus [13].

Theorem 1.8 (see [13]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝐵(𝑋) a multi valued operator. Suppose that (i)𝑥∈𝑇𝑥 for all 𝑥∈𝑋(ii)there exist a comparison function 𝜑∶ℝ⁺→ℝ⁺ (see [14]) and a Picard sequence 𝑥_𝑛+1∈𝑇𝑥_𝑛, 𝑛∈ℕ such that 𝛿𝑇𝑥_𝑛+1≤𝜑𝛿𝑇𝑥_𝑛,𝑛∈ℕ, (1.10) where 𝛿(𝐴)=sup{𝑑(𝑥,𝑦)∶𝑥,𝑦∈𝐴}.
Then 𝑥_𝑛→𝑥^∗ as 𝑛→∞ and 𝑥^∗ is a unique strict fixed-point of 𝑇.

Recently, Wardowski [15] proved the following theorems concerning fixed-points and endpoints of set-valued contractions in complete cone metric spaces.

Definition 1.9. A function 𝐼∶𝑋→ℝ is called lower semicontinuous, if, for any sequence {𝑥_𝑛}⊂𝑋 and 𝑥∈𝑋, 𝑥_𝑛⟶𝑥implies𝑓(𝑥)≤liminf_𝑛→∞𝑓𝑥_𝑛. (1.11)
If (𝑋,𝑑) is a cone metric space and 𝑇∶𝑋→𝐶(𝑋), then, for 𝑥∈𝑋, we denote 𝒟()={𝑑()},𝒮()=()∶𝑦𝑧‖(.𝑥,𝑇𝑥𝑥,𝑧∶𝑧∈𝑇𝑥𝑥,𝑇𝑥𝑦∈𝒟𝑥,𝑇𝑥=inf{‖∶𝑧∈𝒟𝑥,𝑇𝑥)} (1.12)

Theorem 1.10 (see [15]). Let (𝑋,𝑑) be a complete cone metric space, let 𝑃 be a normal cone with normal constant 𝐾, and let 𝑇∶𝑋→𝐶(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. If there exist 𝜆∈[0,1), 𝑏∈(𝜆,1] such that (),∧,∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒟𝑦,𝑇𝑦∀𝑢∈𝒟𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜆𝑑𝑥,𝑦 (1.13) then Fix(𝑇)≠𝜙.

Theorem 1.11 (see [15]). Let (𝑋,𝑑) be a complete cone metric space, let 𝑃 be a normal cone with normal constant 𝐾, and let 𝑇∶𝑋→𝐾(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. The following hold(a) If there exist 𝜆∈[0,1), 𝑏∈(𝜆,1] such that (),∧,∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜆𝑑𝑥,𝑦 (1.14) then Fix(𝑇)≠𝜙.(b) If there exist 𝜆∈[0,1), 𝑏∈(𝜆,1] such that (),∧,∀𝑥∈𝑋∀𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜆𝑑𝑥,𝑦 (1.15) then Fix(𝑇)=End(𝑇)≠𝜙.

In 1997, Zabrejko [16] introduced the 𝐾-metric and 𝐾-normed linear spaces and showed the existence and uniqueness of fixed-points for operators which act in 𝐾-metric or 𝐾-normed linear spaces. Huang and Zhang [17] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed-point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors like Abbas and Jungck [18] have generalized the results of Huang and Zhang [17] and studied the existence of common fixed-points of a pair of self-mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Rezapour and Hamlbarani [19] studied the existence of common fixed-points of a pair of self- and non-self-mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed-point theory are proved (see, e.g., [12, 19–39]).

Definition 1.12 (see [17]). Let 𝐸 be a real Banach space and 𝑃 a nonempty subset of 𝐸. 𝑃≠{𝜃}, where 𝜃 denotes the zero element of 𝐸, is called a cone if and only if: (i)𝑃 is closed;(ii)𝑎,𝑏∈ℝ⁺, 𝑎,𝑏>0, 𝑥,𝑦∈𝑃⇒𝑎𝑥+𝑏𝑦∈𝑃;(iii)𝑥∈𝑃 and −𝑥∈𝑃⇒𝑥=𝜃.

For given a cone 𝑃⊂𝐸, we define a partial ordering with respect to 𝑃 by 𝑥≼𝑦 or 𝑥≽𝑦 if and only if 𝑦−𝑥∈𝑃 for all 𝑥,𝑦∈𝐸. The real Banach space 𝐸 equipped with the partial order induced by 𝑃 is denoted by (𝐸,≼). We shall write 𝑥≺𝑦 to indicate that 𝑥≼𝑦 but 𝑥≠𝑦, while 𝑥≪𝑦 or 𝑦≫𝑥 will stand for 𝑦−𝑥∈int(𝑃), where int(𝑃) denotes the interior of 𝑃.

Proposition 1.13 (see [40]). Suppose 𝑃 is a cone in a real Banach space 𝐸. Then one has the following. (i)If 𝑒≼𝑓 and 𝑓≪𝑔, then 𝑒≪𝑔.(ii)If 𝑒≪𝑓 and 𝑓≼𝑔, then 𝑒≪𝑔.(iii)If 𝑒≪𝑓 and 𝑓≪𝑔, then 𝑒≪𝑔. (iv)If 𝑎∈𝑃 and 𝑎≪𝑒 for each 𝑒∈int(𝑃), then 𝑎=𝜃.

Proposition 1.14 (see [41]). Suppose 𝑒∈int(𝑃), 𝜃≼𝑎_𝑛 and 𝑎_𝑛→𝜃. Then there exists 𝑛₀∈ℕ such that 𝑎_𝑛≪𝑒 for all 𝑛≥𝑛₀.

The cone 𝑃 is called normal if there exists a real number 𝐾>0 such that, for all 𝑥,𝑦∈𝐸, 𝜃≼𝑥≼𝑦implies‖𝑥‖≤𝐾‖𝑦‖. (1.16)

The least positive number 𝐾, satisfying the above is called the normal constant of 𝑃.

The cone 𝑃 is called regular if every increasing sequence which is bounded from above is convergent, that is, if {𝑥_𝑛} is a sequence such that 𝑥₁≼𝑥₂≼⋯≼𝑥_𝑛≼⋯≼𝑦, (1.17) for some 𝑦∈𝐸, then there is 𝑥∈𝐸 such that ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. Equivalently, the cone 𝑃 is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

Definition 1.15 (see [17]). Let 𝑋 be a nonempty set, 𝐸 a real Banach space, and 𝑃 a cone in 𝐸. Suppose that the mapping 𝑑∶𝑋×𝑋→(𝐸,≼) satisfies(i)𝜃≼𝑑(𝑥,𝑦), for all 𝑥,𝑦∈𝑋; (ii)𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦; (iii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), for all 𝑥,𝑦∈𝑋; (iv)𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)≽𝑑(𝑥,𝑧), for all 𝑥,𝑦,𝑧∈𝑋. Then 𝑑 is called a cone metric on 𝑋, and (𝑋,𝑑) is called a cone metric space.

Definition 1.16 (see [17]). Let (𝑋,𝑑) be a cone metric space, and let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋. If for every 𝑐∈𝐸 with 𝜃≪𝑐, there is 𝑛₀∈ℕ such that 𝑑𝑥_𝑛,𝑥≪𝑐,∀𝑛>𝑛₀, (1.18) then {𝑥_𝑛} is said to be convergent and {𝑥_𝑛} converges to 𝑥.

Definition 1.17 (see [17]). Let (𝑋,𝑑) be a cone metric space, and let {𝑥_𝑛} be a sequence in 𝑋. We say that {𝑥_𝑛} is a Cauchy sequence if, for any 𝑐∈𝐸 with 𝜃≪𝑐, there is 𝑛₀∈ℕ such that 𝑑𝑥_𝑛,𝑥_𝑚≪𝑐,∀𝑛,𝑚>𝑛₀, (1.19)

Definition 1.18 (see [17]). Let (𝑋,𝑑) be a cone metric space. If every Cauchy sequence is convergent in 𝑋, then 𝑋 is called a complete cone metric space.

Remark 1.19 (see [17]). If 𝑃 is a normal cone, then {𝑥_𝑛} converges to 𝑥 if and only if 𝑑(𝑥_𝑛,𝑥)→𝜃 as 𝑛→∞. Further, in the case {𝑥_𝑛} is a Cauchy sequence if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→𝜃 as 𝑚,𝑛→∞.

The purpose of this paper is to present the fixed-points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović [42–44] and Wardowski [15].

2. Main Results

In this section, we first introduce the following notion of stronger Meir-Keeler cone-type mapping.

Definition 2.1. Let (𝑋,𝑑) be a cone metric space with cone 𝑃, and let 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),where𝑏∈(0,1]. (2.1) Then the function 𝜓 is called a stronger Meir-Keeler cone-type mapping, if, for each 𝜂∈int(𝑃), there exists 𝛿≫𝜃 such that, for 𝑥,𝑦∈𝑋 with 𝜂≼𝑑(𝑥,𝑦)≪𝛿+𝜂, there exists 𝛾_𝜂∈[0,𝑏) such that 𝜓(𝑑(𝑥,𝑦))<𝛾_𝜂.

We now are in a position to present the following fixed-point theorem.

Theorem 2.2. Let (𝑋,𝑑) be a complete cone metric space, and let 𝑃 be a normal cone in 𝐸 and 𝑇∶𝑋→𝐶(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. The following holds.(*)  There exists a stronger Meir-Keeler cone-type mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1] (2.2) such that (),∧𝑑𝑑.∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒟𝑦,𝑇𝑦∀𝑢∈𝒟𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.3)Then Fix(𝑇)≠𝜙.

Proof. Given 𝑥₁∈𝑋, we define the sequence {𝑥_𝑛} recursively as follows. Take any 𝑢₁∈𝒟(𝑥₁,𝑇𝑥₁). By (∗), there exist 𝑥₂∈𝑇𝑥₁ and 𝑢₂∈𝒟(𝑥₂,𝑇𝑥₂) such that 𝑏𝑑𝑥₁,𝑥₂≼𝑢₁,𝑢₂≼𝜓𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂,𝜓𝑑𝑥₁,𝑥₂<𝑏. (2.4) Thus, 𝑢₁−𝑢₂≽𝑏−𝜓𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂,𝑢₂≼𝜓𝑑𝑥₁,𝑥₂𝑏𝑢₁. (2.5) If 𝑑(𝑥₁,𝑥₂)=𝜃, then 𝑥₁=𝑥₂∈𝑇𝑥₁, and we are done. Assume that 𝑑(𝑥₁,𝑥₂)≫𝜃. Put 𝑑(𝑥₁,𝑥₂)=𝜂₁. By the definition of the stronger Meir-Keeler cone-type mapping 𝜓, corresponding to 𝜂₁ use, there exists 𝛿₁≫𝜃 and 𝛾_𝜂1∈[0,𝑏) with 𝜂₁≼𝑑(𝑥₁,𝑥₂)≪𝛿₁+𝜂₁ such that 𝜓(𝑑(𝑥₁,𝑥₂))<𝛾_𝜂1. Taking into account (2.5), we have that 𝑢₁−𝑢₂≽𝑏−𝛾_𝜂1𝑑𝑥₁,𝑥₂,𝑢₂≼𝛾_𝜂1𝑏𝑢₁. (2.6)
By the same process, for 𝑥₂∈𝑋, there exist 𝑥₃∈𝑇𝑥₂ and 𝑢₃∈𝒟(𝑥₃,𝑇𝑥₃) such that 𝑏𝑑𝑥₂,𝑥₃≼𝑢₂,𝑢₃≼𝜓𝑑𝑥₂,𝑥₃𝑑𝑥₂,𝑥₃,𝜓𝑑𝑥₂,𝑥₃<𝑏. (2.7) Thus, 𝑢₂−𝑢₃≽𝑏−𝜓𝑑𝑥₂,𝑥₃𝑑𝑥₂,𝑥₃,𝑢₃≼𝜓𝑑𝑥₂,𝑥₃𝑏𝑢₂≼𝜓𝑑𝑥₁,𝑥₂⋅𝜓𝑑𝑥₂,𝑥₃𝑏²𝑢₁. (2.8)(2.9) Now, put 𝑑(𝑥₂,𝑥₃)=𝜂₂. By the definition of the stronger Meir-Keeler cone-type mapping 𝜓, corresponding to 𝜂₂ use, there exists 𝛿₂≫𝜃 and 𝛾_𝜂2∈[0,𝑏) with 𝜂₂≼𝑑(𝑥₂,𝑥₃)≪𝛿₂+𝜂₂ such that 𝜓(𝑑(𝑥₂,𝑥₃))<𝛾_𝜂2. And we put 𝛼₁=𝛾_𝜂1 and 𝛼₂=max{𝛾_𝜂1,𝛾_𝜂2}. Taking into account (2.9), we have that 𝑢₂−𝑢₃≽𝑏−𝛾_𝜂2𝑑𝑥₂,𝑥₃≽𝑏−𝛼₂𝑑𝑥₂,𝑥₃,𝑢₃≼𝛾_𝜂2𝑏𝑢₂≼𝛾_𝜂1⋅𝛾_𝜂2𝑏²𝑢₁≼𝛼²₂𝑏²𝑢₁. (2.10)(2.11)
We continue in this manner. Inductively, for 𝑥_𝑛∈𝑋, there exist 𝑥_𝑛+1∈𝑇𝑥_𝑛 and 𝑢_𝑛+1∈𝒟(𝑥_𝑛+1,𝑇𝑥_𝑛+1) such that 𝑏𝑑𝑥_𝑛,𝑥_𝑛+1≼𝑢_𝑛,𝑢_𝑛+1≼𝜓𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1,𝜓𝑑𝑥_𝑛,𝑥_𝑛+1<𝑏. (2.12) Thus, 𝑢_𝑛−𝑢_𝑛+1≽𝑏−𝜓𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1,𝑢_𝑛+1≼𝜓𝑑𝑥_𝑛,𝑥_𝑛+1𝑏𝑢_𝑛𝜓𝑑𝑥≼⋯≼¹,𝑥₂⋅𝜓𝑑𝑥₂,𝑥₃⋯𝜓𝑑𝑥_𝑛,𝑥_𝑛+1𝑏^𝑛𝑢₁. (2.13)(2.14) Put 𝑑(𝑥_𝑛,𝑥_𝑛+1)=𝜂_𝑛. By the definition of the stronger Meir-Keeler cone-type mapping 𝜓, corresponding to 𝜂_𝑛 use, there exist 𝛿_𝑛≫𝜃 and 𝛾_𝜂𝑛∈[0,𝑏) with 𝜂_𝑛≼𝑑(𝑥_𝑛,𝑥_𝑛+1)≪𝛿_𝑛+𝜂_𝑛 such that 𝜓(𝑑(𝑥_𝑛,𝑥_𝑛+1))<𝛾_𝜂𝑛. And we put 𝛼_𝑛=𝛾_𝜂𝑛 and 𝛼_𝑛=max{𝛾_𝜂1,𝛾_𝜂2,…,𝛾_𝜂𝑛}. Taking into account (2.14), we have that 𝑢_𝑛−𝑢_𝑛+1≽𝑏−𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛+1,𝑢_𝑛+1≼𝛼^𝑛_𝑛𝑏^𝑛𝑢₁. (2.15)(2.16)
Let 𝑚,𝑛∈ℕ such that 𝑚>𝑛. Taking into account (2.15) and (2.16), we obtain that 𝑑𝑥_𝑛,𝑥_𝑚≼^𝑚−1_𝑖=𝑛𝑑𝑥_𝑖,𝑥_𝑖+1≼^𝑚−1_𝑖=𝑛1𝑏−𝛼_𝑖𝑢_𝑖−𝑢_𝑖+1≼^𝑚−1_𝑖=𝑛1𝑏−𝛼_𝑚𝑢_𝑖−𝑢_𝑖+1≼1𝑏−𝛼_𝑚𝑢_𝑛−𝑢_𝑚≼1𝑏−𝛼_𝑚𝛼^𝑛_𝑛𝑏^𝑛𝑢₁. (2.17) Let 𝑒≫𝜃 be given. Since 𝛼_𝑛<𝑏 and 𝛼_𝑚<𝑏, we get lim_𝑛→∞1𝑏−𝛼_𝑚𝛼^𝑛_𝑛𝑏^𝑛𝑢₁=𝜃. (2.18) Thus, there exists 𝑛₀∈ℕ such that, for all 𝑚,𝑛≥𝑛₀, 1𝑏−𝛼_𝑚𝛼^𝑛_𝑛𝑏^𝑛𝑢₁≪𝑒, (2.19) and we also conclude that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑒 for all 𝑚,𝑛≥𝑛₀. This implies that {𝑥_𝑛} is a Cauchy sequence. Since 𝑋 is complete, there exists 𝜈∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝜈. (2.20)
By the definition of 𝒟(𝑥,𝑇𝑥), since 𝑢_𝑛∈𝒟(𝑥_𝑛,𝑇𝑥_𝑛), there exists a sequence {𝑧_𝑛} such that 𝑧_𝑛∈𝑇𝑥_𝑛, 𝑢_𝑛=𝑑(𝑥_𝑛,𝑧_𝑛) for all 𝑛∈ℕ. From the convergence of the sequence {𝑢_𝑛} and from the lower semicontinuity of the function 𝐼, we obtain that inf_{𝑦∈𝑇𝜈}𝑑𝜈,𝑦≤liminf_𝑛→∞𝑑𝑥_𝑛,𝑦≤liminf_𝑛→∞𝑑𝑥_𝑛,𝑧_𝑛=0. (2.21) Thus, inf_{𝑦∈𝑇𝑥}𝑑𝜈,𝑦=0. (2.22) We claim that 𝜈∈𝑇𝜈. To prove this, on the contrary, assume that 𝜈∉𝑇𝜈. Then by (2.22), there exists a sequence {𝑦_𝑛}⊂𝑇𝜈 such that lim_𝑛→∞‖𝑑(𝜈,𝑦_𝑛)‖=0, and hence lim_𝑛→∞𝑑(𝑦_𝑛,𝜈)=𝜃. Thus, for any 𝑚,𝑛≥0, 𝑑𝑦_𝑚,𝑦_𝑛≼𝑑𝑦_𝑚,𝜈+𝑑𝜈,𝑦_𝑛. (2.23) Let 𝜏≫𝜃 be given. Then there exists 𝑁∈ℕ such that for all 𝑚,𝑛>𝑁, 𝑑(𝑦_𝑚,𝑝)≪1/2𝜏 and 𝑑(𝑦_𝑛,𝑝)≪1/2𝜏, and so 𝑑𝑦_𝑚,𝑦_𝑛≼𝑑𝑦_𝑚,𝜈+𝑑𝜈,𝑦_𝑛≪121𝜏+2𝜏=𝜏. (2.24) This implies that {𝑦_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝜇∈𝑋 such that lim_𝑛→∞𝑦_𝑛=𝜇. (2.25) By the closedness of the 𝑇𝜈, we deduce that 𝜇∈𝑇𝜈. Then, for any 𝑛∈ℕ, 𝑑𝜇,𝜈≼𝑑𝜈,𝑦_𝑛+𝑑𝑦_𝑛,𝜇. (2.26) Let 𝜔≫𝜃 be given. Since lim_𝑛→∞𝑑(𝑦_𝑛,𝜈)=𝜃 and lim_𝑛→∞𝑦_𝑛=𝜇, we can deduce that there exists 𝑀∈ℕ such that, for all 𝑛>𝑀, 𝑑(𝑦_𝑛,𝜈)≪1/2𝜔 and 𝑑(𝑦_𝑛,𝜇)≪1/2𝜔, and hence 𝑑≪1𝜇,𝜈21𝜔+2𝜔=𝜔. (2.27) By Proposition 1.13, we obtain that 𝑑(𝜇,𝜈)=𝜃 that is, 𝜇=𝜈, which is a contradiction. Therefore, 𝜈∈𝑇𝜈.

Applying Remark 1.7, it is easy to establish the following corollary.

Corollary 2.3. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a normal cone in 𝐸, and 𝑇∶𝑋→𝐶(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by :𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. The following holds.(**) There exists a mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1] (2.28) such that limsup_𝑠→𝑡𝜓(𝑠)(𝑃)∪{𝜃},(),∧𝑑𝑑.<𝑏,∀𝑡∈int∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒟𝑦,𝑇𝑦∀𝑢∈𝒟𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.29)(2.30)Then Fix(𝑇)≠𝜙.

Corollary 2.4. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a normal cone in 𝐸, and 𝑇∶𝑋→𝑋. Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. The following holds.(***)There exists a stronger Meir-Keeler cone-type mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1], (2.31) such that 𝑑𝑇𝑥,𝑇²𝑥≼𝜓(𝑑(𝑥,𝑇𝑥))𝑑(𝑥,𝑇𝑥),∀𝑥∈𝑋. (2.32) Then Fix(𝑇)≠𝜙.

Example 2.5. Let 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸,𝑥,𝑦≥0}, 𝑋={𝑎,𝑏,𝑐}⊂𝐸 where 𝑎=(0,0), 𝑏=(0,1), 𝑐=(1,0), 𝑑∶𝑋×𝑋→𝐸 defined by 𝑑(𝑎,𝑏)=(1,2/3), 𝑑(𝑐,𝑎)=(4/3,1), 𝑑(𝑏,𝑐)=(7/3,5/3), with 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦. Let 𝑇∶𝑋→𝑋 be defined by 𝑇𝑎=𝑏, 𝑇𝑏=𝑏, and 𝑇𝑐=𝑎. Then 𝑇²𝑎=𝑏, 𝑇²𝑏=𝑏, 𝑇²𝑐=𝑏. Take 𝜓(𝑡)=5/6 for all 𝑡∈int(𝑃)∪{𝜃}. Calculation shows that, for 𝑥,𝑦∈𝑋, 𝑑𝑇𝑥,𝑇²𝑥≼56𝑑(𝑥,𝑇𝑥)=𝜓(𝑑(𝑥,𝑇𝑥))𝑑(𝑥,𝑇𝑥) (2.33) that is, according to Corollary 2.4 we have that Fix(𝑇)≠𝜙.

Theorem 2.6. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a normal cone in 𝐸, and 𝑇∶𝑋→𝐾(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋, is lower semicontinuous. The following hold:(A) There exists a stronger Meir-Keeler cone-type mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1] (2.34) such that (),∧𝑑𝑑.∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.35)
Then Fix(𝑇)≠𝜙.(B) There exists a stronger Meir-Keeler cone-type mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1] (2.36) such that (),∧𝑑𝑑.∀𝑥∈𝑋∀𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.37)
Then Fix(𝑇)=End(𝑇)≠𝜙.

Proof. (A) By the same proof of the part (i) of Theorem 3.2 in [15], we have that 𝒮(𝑥,𝑇𝑥)≠𝜙 for all 𝑥∈𝑋. The remainder proof is similar to Theorem 2.2, we can deduce that Fix(𝑇)≠𝜙.
(B) By (A), we obtain that the existence of 𝜈∈𝑋 such that, 𝜈∈𝑇𝜈. Taking any 𝑦∈𝑇𝜈, we have that, for all 𝑢∈𝒮(𝜈,𝑇𝜈), 𝑏𝑑(𝜈,𝑦)≼𝑢. Since 𝜈∈𝑇𝜈, we get 𝜃∈𝒮(𝜈,𝑇𝜈), and hence 𝑏𝑑(𝜈,𝑦)≼𝜃, which gives 𝜈=𝑦. Thus 𝑇𝜈={𝜈}.

Example 2.7. Let 𝑋=[0,1], 𝐸=ℝ², 𝑃={(𝑥,𝑦)∈𝐸,𝑥,𝑦≥0}. Let 𝑑(𝑥,𝑦)=(|𝑥−𝑦|,0) for all 𝑥,𝑦∈𝑋, and let 𝑇∶𝑋→𝐾(𝑋), 𝑇𝑥={(1/2)𝑥}. Then 𝒮(𝑥,𝑇𝑥)={((1/2)𝑥,0)∶𝑥∈𝑋}, 𝐼(𝑥)=(1/2)𝑥. Let 𝜓(𝑡)=2/3 for all 𝑡∈int(𝑃)∪{𝜃}. It is easy to check that all conditions of Theorem 2.6 are satisfied and that Fix(𝑇)=End(𝑇)={0}≠𝜙.

Corollary 2.8. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a normal cone in 𝐸, and 𝑇∶𝑋→𝐾(𝑋). Assume that a function 𝐼∶𝑋→ℝ defined by 𝐼(𝑥)=inf_{𝑦∈𝑇𝑥}‖𝑑(𝑥,𝑦)‖, 𝑥∈𝑋 is lower semicontinuous. The following hold:(C) There exists a mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1], (2.38) such that limsup_𝑠→𝑡𝜓(𝑠)(𝑃)∪{𝜃},(),∧𝑑𝑑.<𝑏,∀𝑡∈int∀𝑥∈𝑋∃𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.39)Then Fix(𝑇)≠𝜙.(D) There exists a mapping 𝜓∶int(𝑃)∪{𝜃}⟶[0,𝑏),𝑤ℎ𝑒𝑟𝑒𝑏∈(0,1], (2.40) such that limsup_𝑠→𝑡𝜓(𝑠)(𝑃)∪{𝜃},(),∧𝑑𝑑.<𝑏,∀𝑡∈int∀𝑥∈𝑋∀𝑦∈𝑇𝑥∃𝑣∈𝒮𝑦,𝑇𝑦∀𝑢∈𝒮𝑥,𝑇𝑥𝑏𝑑𝑥,𝑦≼𝑢𝑣≼𝜓𝑥,𝑦𝑥,𝑦 (2.41)Then Fix(𝑇)=End(𝑇)≠𝜙.