End-Point Results for Multivalued Mappings in Partially Ordered Metric Spaces

Abstract

The purpose of this paper is to prove end-point theorems for multivalued mappings satisfying comparatively a more general contractive condition in ordered complete metric spaces. Afterwards, we extend the results of previous sections and prove common end-point results for a pair of 𝒯-weakly isotone increasing multivalued mappings in the underlying spaces. Finally, we present common end point for a pair of 𝒯-weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

1. Introduction and Preliminaries

Fixed-point theory for multivalued mappings was originally initiated by Von Neumann in the study of game theory. Fixed-point theorem for multivalued mappings is quite useful in control theory and has been frequently used in solving the problem of economics and game theory.

The theory of multivalued nonexpansive mappings is comparatively complicated as compare to the corresponding theory of single-valued nonexpansive mappings. It is therefore natural to expect that the theory of noncontinuous nonself-multivalued mappings would be much more complicated.

The study of fixed-points for multivalued contraction mappings was equally an active topic as single-valued mappings. The development of geometric fixed-point theory for multivalued was initiated with the work of Nadler Jr. [1] in the year 1969. He used the concept of Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case, as following.

Theorem 1.1. Let (𝒳,𝑑) be a complete metric space and 𝒯 is a mapping from 𝒳 into 𝐶𝐵(𝒳) such that for all 𝑥,𝑦∈𝒳, ℋ𝒯𝑥,𝒯𝑦≤𝜆𝑑𝑥,𝑦, (1.1) where 0≤𝜆<1. Then 𝒯 has a fixed-point.

Since then, this discipline has been further developed, and many profound concepts and results have been established; for example, the work of Border [2], Ćirić [3], Corley [4], Itoh and Takahashi [5], Mizoguchi and Takahashi [6], Petruşel and Luca [7], Rhoades [8], Tarafdar and Yuan [9], and references cited therein.

Let (𝒳,𝑑) be a metric space. We denote the class of nonempty and bounded subsets of 𝒳 by 𝐵(𝒳). For 𝒜, ℬ∈𝐵(𝒳), functions 𝐷(𝒜,ℬ), and 𝛿(𝒜,ℬ) are defined as follows: 𝐷(){𝑑()},𝛿(){𝑑()}.𝒜,ℬ=inf𝑎,𝑏∶𝑎∈𝒜,𝑏∈ℬ𝒜,ℬ=sup𝑎,𝑏∶𝑎∈𝒜,𝑏∈ℬ (1.2) If 𝒜={𝑎}, then we write 𝐷(𝒜,ℬ)=𝐷(𝑎,ℬ) and 𝛿(𝒜,ℬ)=𝛿(𝑎,ℬ). Also in addition, if ℬ={𝑏}, then 𝐷(𝒜,ℬ)=𝑑(𝑎,𝑏) and 𝛿(𝒜,ℬ)=𝑑(𝑎,𝑏). Obviously, 𝐷(𝒜,ℬ)≤𝛿(𝒜,ℬ).

For all 𝒜,ℬ,𝒞∈𝐵(𝒳), the definition of 𝛿(𝒜,ℬ) yields the following: 𝛿()(),𝛿()()(),𝛿(){𝑎},𝛿()𝒜,ℬ=𝛿ℬ,𝒜𝒜,ℬ≤𝛿𝒜,𝒞+𝛿𝒞,ℬ𝒜,ℬ=0iff𝒜=ℬ=𝒜,𝒜=diam𝒜. (1.3)

A point 𝑥∈𝒳 is called a fixed-point of a multivalued mapping 𝒯∶𝒳→𝐵(𝒳) if 𝑥∈𝒯𝑥. If there exists a point 𝑥∈𝒳 such that 𝒯𝑥={𝑥}, then 𝑥 is called an end-point of 𝒯 [10].

Definition 1.2. Let 𝒳 be a nonempty set. Then (𝒳,𝑑,⪯) is called an ordered metric space if and only if:(i)(𝒳,𝑑) is a metric space,(ii)(𝒳,⪯) is a partially ordered set.

Definition 1.3. Let (𝒳,⪯) be a partial ordered set. Then 𝑥,𝑦∈𝒳 are called comparable if 𝑥⪯𝑦 or 𝑦⪯𝑥 holds.

Definition 1.4 (see [11]). Let 𝒜 and ℬ be two nonempty subsets of a partially ordered set (𝒳,⪯). The relation between 𝒜 and ℬ is denoted and defined as follows: 𝒜≺₁ℬ,ifforevery𝑎∈𝒜thereexists𝑏∈ℬsuchthat𝑎⪯𝑏. (1.4)

Definition 1.5 (see [12]). A function 𝜓∶[0,∞)→[0,∞) is called an altering distance function if the following properties are satisfied: (i)𝜓 is monotone increasing and continuous,(ii)𝜓(𝑡)=0 if and only if 𝑡=0.

On the other hand, fixed-point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [13, Theorem  2.1] who presented its applications to matrix equations. Subsequently, Nieto and Rodríguez-López [14] extended the result of Ran and Reurings for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed-point theorems in ordered metric spaces. For detail see [14–28] and references cited therein. Beg and Butt [11, 17, 29] worked on set-valued mappings and proved common fixed-point for mapping satisfying implicit relation in partially ordered metric space. Recently, Choudhury and Metiya [30] proved fixed-point theorems for multivalued mappings in the framework of a partially ordered metric space.

The results of this paper are divided in three sections. In the first section we establish the existence of end-points for a multivalued mapping under a more general contractive condition in partially ordered metric spaces. The consequences of the main theorem are also given. The second section is devoted for common end-point results for a pair of weakly isotone increasing multivalued mappings. In the third section, we present common end-point results for a pair of weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

2. End-Point Theorems for a Multivalued Mapping

In this section, we prove end-point theorems for a multivalued mapping in ordered complete metric space.

Theorem 2.1. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯∶𝒳→𝐵(𝒳) be such that the following conditions are satisfied: (i)there exists 𝑥₀∈𝒳 such that {𝑥₀}≺₁𝒯𝑥₀, (ii)for 𝑥,𝑦∈𝒳, 𝑥⪯𝑦 implies 𝒯𝑥≺₁𝒯𝑦,(iii)𝜓𝛿𝒯𝑥,𝒯𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝐷(𝑥,𝒯𝑥),𝐷𝑦,𝒯𝑦,𝐷𝑥,𝒯𝑦,𝐷𝑦,𝒯𝑥, (2.1)for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝐷𝑥,𝑦=max𝑥,𝑦,𝐷𝑥,𝒯𝑥,𝐷𝑦,𝒯𝑦𝑥,𝒯𝑦+𝐷𝑦,𝒯𝑥2. (2.2) If the property 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛≺𝑧∀𝑛 (2.3) holds, then 𝒯 has a end-point.

Proof. By the assumption (i), there exists 𝑥₁∈𝒯𝑥₀ such that 𝑥₀⪯𝑥₁. By the assumption (ii), 𝒯𝑥₀≺₁𝒯𝑥₁. Then there exists 𝑥₂∈𝒯𝑥₁ such that 𝑥₁⪯𝑥₂. Continuing this process we construct a monotone increasing sequence {𝑥_𝑛} in 𝒳 such that 𝑥_𝑛+1∈𝒯𝑥_𝑛, for all 𝑛≥0. Thus we have 𝑥₀⪯𝑥₁⪯𝑥₂⪯𝑥₃⪯⋯⪯𝑥_𝑛⪯𝑥_𝑛+1⪯⋯. (2.4) If 𝑥_𝑛0∈𝒯𝑥_𝑛0 for some 𝑛₀, then the proof is finished. So assume 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛≥0.
Using the monotone property of 𝜓 and the condition (iii), we have for all 𝑛≥0, 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓𝛿𝒯𝑥_𝑛,𝒯𝑥_𝑛+1≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷𝑥_𝑛+1,𝒯𝑥_𝑛+1,𝐷𝑥_𝑛,𝒯𝑥_𝑛+1+𝐷𝑥_𝑛+1,𝒯𝑥_𝑛2+𝐿min𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷𝑥_𝑛+1,𝒯𝑥_𝑛+1,𝐷𝑥_𝑛,𝒯𝑥_𝑛+1,𝐷𝑥_𝑛+1,𝒯𝑥_𝑛≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+2+𝑑𝑥_𝑛+1,𝑥_𝑛+12+𝐿min𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+2,𝑑𝑥_𝑛+1,𝑥_𝑛+1. (2.5) Since 𝑑(𝑥_𝑛,𝑥_𝑛+2)/2≤ max{𝑑(𝑥_𝑛,𝑥_𝑛+1),𝑑(𝑥_𝑛+1,𝑥_𝑛+2)}, it follows that 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2. (2.6) Suppose that 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝑑(𝑥_𝑛+1,𝑥_𝑛+2), for some positive integer 𝑛.
Then from (2.6), we have 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛼𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2, (2.7) it implies that 𝑑(𝑥_𝑛+1,𝑥_𝑛+2)=0, or that 𝑥_𝑛+1=𝑥_𝑛+2, contradicting our assumption that 𝑥_𝑛≠𝑥_𝑛+1, for each 𝑛.
Therefore, 𝑑(𝑥_𝑛+1,𝑥_𝑛+2)<𝑑(𝑥_𝑛,𝑥_𝑛+1), for all 𝑛≥0 and {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an 𝑟≥0 such that 𝑑𝑥_𝑛,𝑥_𝑛+1⟶𝑟as𝑛⟶∞. (2.8) Taking the limit as 𝑛→∞ in (2.6) and using the continuity of 𝜓, we have 𝜓(𝑟)≤𝛼𝜓(𝑟), which is a contradiction unless 𝑟=0. Hence lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (2.9) Next we show that {𝑥_𝑛} is a Cauchy sequence. If otherwise, there exists an 𝜖>0 for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that for all positive integers 𝑘, 𝑛(𝑘)>𝑚(𝑘)>𝑘 and 𝑑(𝑥_𝑚(𝑘),𝑥_𝑛(𝑘))≥𝜖.
Assuming that 𝑛(𝑘) is the smallest such positive integer, we get 𝑛(𝑘)>𝑚(𝑘)>𝑘, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜖,𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜖. (2.10) Now, 𝜖≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘), (2.11) that is, 𝜖≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)<𝜖+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘). (2.12) Taking the limit as 𝑘→∞ in the above inequality and using (2.9), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)=𝜖. (2.13) Again, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1+𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1. (2.14) Taking the limit as 𝑘→∞ in the above inequalities and using (2.9) and (2.13), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜖. (2.15) Again, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1. (2.16) Letting 𝑘→∞ in the above inequalities and using (2.9) and (2.13), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1=𝜖. (2.17) Similarly, we have that lim_𝑘→∞𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1=𝜖. (2.18) For each positive integer 𝑘, 𝑥_𝑚(𝑘) and 𝑥_𝑛(𝑘) are comparable. Then using the monotone property of 𝜓 and the condition (iii), we have 𝜓𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝜓𝛿𝒯𝑥_𝑚(𝑘),𝒯𝑥_𝑛(𝑘)≤𝛼𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑚(𝑘),𝐷𝑥_𝑛(𝑘),𝒯𝑥_𝑛(𝑘),𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑛(𝑘)+𝐷𝑥_𝑛(𝑘),𝒯𝑥_𝑚(𝑘)2+𝐿min𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑚(𝑘),𝐷𝑥_𝑛(𝑘),𝒯𝑥_𝑛(𝑘),𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑛(𝑘),𝐷𝑥_𝑛(𝑘),𝒯𝑥_𝑚(𝑘)≤𝛼𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘),𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+12+𝐿min𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1. (2.19) Letting 𝑘→∞ in above inequality, using (2.9), (2.13), (2.15), (2.17), and (2.18) and the continuity of 𝜓, we have 𝜓(𝜖)≤𝛼𝜓(𝜖), (2.20) which is a contradiction by virtue of a property of 𝜓.
Hence {𝑥_𝑛} is a Cauchy sequence. From the completeness of 𝒳, there exists a 𝑧∈𝒳 such that 𝑥_𝑛⟶𝑧as𝑛⟶∞. (2.21) By the assumption (2.3), 𝑥_𝑛⪯𝑧, for all 𝑛.
Then by the monotone property of 𝜓 and the condition (iii), we have 𝜓𝛿𝑥_𝑛+1,𝒯𝑧≤𝜓𝛿𝒯𝑥_𝑛𝑑𝑥,𝒯𝑧≤𝛼𝜓max_𝑛,𝑧,𝐷𝑥_𝑛,𝒯𝑥_𝑛(),𝐷𝑥,𝐷𝑧,𝒯𝑧^𝑛,𝒯𝑧+𝐷𝑧,𝒯𝑥_𝑛2+𝐿min𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷(𝑧,𝒯𝑧),𝐷𝑥_𝑛,𝒯𝑧,𝐷𝑧,𝒯𝑥_𝑛≤𝛼𝜓max𝑑𝑥_𝑛,𝑧,𝑑𝑥_𝑛,𝑥_𝑛+1(),𝐷𝑥,𝐷𝑧,𝑇𝑧^𝑛,𝑇𝑧+𝑑𝑧,𝑥_𝑛+12+𝐿min𝑑𝑥_𝑛,𝑥_𝑛+1,𝐷(𝑧,𝒯𝑧),𝐷𝑥_𝑛,𝒯𝑧,𝑑𝑧,𝑥_𝑛+1. (2.22) Taking the limit as 𝑛→∞ in the above inequality, using (2.9) and (2.21) and the continuity of 𝜓, we have 𝜓(𝛿(𝑧,𝒯𝑧))≤𝛼𝜓(𝐷(𝑧,𝒯𝑧))≤𝛼𝜓(𝛿(𝑧,𝒯𝑧)), (2.23) which implies that 𝛿(𝑧,𝒯𝑧)=0, or that {𝑧}=𝒯𝑧. Moreover, 𝑧 is a end-point of 𝒯.

Taking 𝜓 an identity function in Theorem 2.1, we have the following result.

Corollary 2.2. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯∶𝒳→𝐵(𝒳) be such that the following conditions are satisfied: (i)there exists 𝑥₀∈𝒳 such that {𝑥₀}≺₁𝒯𝑥₀, (ii)for 𝑥,𝑦∈𝒳, 𝑥⪯𝑦 implies 𝒯𝑥≺₁𝒯𝑦, (iii)𝛿𝒯𝑥,𝒯𝑦≤𝛼𝑀𝑥,𝑦+𝐿min𝐷(𝑥,𝒯𝑥),𝐷𝑦,𝒯𝑦,𝐷𝑥,𝒯𝑦,𝐷𝑦,𝒯𝑥, (2.24)for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝑀𝑑(),𝐷𝑥,𝑦=max𝑥,𝑦,𝐷𝑥,𝒯𝑥,𝐷𝑦,𝒯𝑦𝑥,𝒯𝑦+𝐷𝑦,𝒯𝑥2. (2.25) If the property 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛⪯𝑧∀𝑛 (2.26) holds, then 𝒯 has a end-point.

The following corollary is a special case of Theorem 2.1 when 𝒯 is a single-valued mapping.

Corollary 2.3. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯∶𝒳→𝒳 be such that the following conditions are satisfied: (i)there exists 𝑥₀∈𝒳 such that 𝑥₀⪯𝒯𝑥₀, (ii)for 𝑥,𝑦∈𝒳, 𝑥⪯𝑦 implies 𝒯𝑥⪯𝒯𝑦, (iii)𝜓𝑑𝒯𝑥,𝒯𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝑑(𝑥,𝒯𝑥),𝑑𝑦,𝒯𝑦,𝑑𝑥,𝒯𝑦,𝑑𝑦,𝒯𝑥, (2.27)for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝑑𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝒯𝑥,𝑑𝑦,𝒯𝑦𝑥,𝒯𝑦+𝑑𝑦,𝒯𝑥2. (2.28) If the property (2.3) holds, then 𝒯 has a fixed-point.

In the following theorem we replace condition (2.3) of the above corollary by requiring 𝒯 to be continuous.

Theorem 2.4. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯∶𝒳→𝒳 be a continuous mapping such that the following conditions are satisfied: (i)there exists 𝑥₀∈𝒳 such that {𝑥₀}≺₁𝒯𝑥₀, (ii)for 𝑥,𝑦∈𝒳, 𝑥⪯𝑦 implies 𝒯𝑥≺₁𝒯𝑦,(iii)𝜓𝑑𝒯𝑥,𝒯𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝑑(𝑥,𝒯𝑥),𝑑𝑦,𝒯𝑦,𝑑𝑥,𝒯𝑦,𝑑𝑦,𝒯𝑥, (2.29)for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝑑𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝒯𝑥,𝑑𝑦,𝒯𝑦𝑥,𝒯𝑦+𝑑𝑦,𝒯𝑥2. (2.30) Then 𝒯 has a end-point.

Proof. If we assume 𝒯 as a multivalued mapping in which 𝒯𝑥 is a singleton set for every 𝑥∈𝒳. Then we consider the same sequence {𝑥_𝑛} as in the proof of Theorem 2.1. Follows the line of proof of Theorem 2.1, we have that {𝑥_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑥_𝑛=𝑧. (2.31)
Then, the continuity of 𝒯 implies that 𝑧=lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞𝒯𝑥_𝑛=𝒯𝑧 (2.32) and this proves that 𝑧 is a end-point of 𝒯.

3. Common End-Point Theorems for a Pair of Multivalued Mappings

In this section, we prove common end-point theorems for a pair of 𝒯-weakly isotone increasing multivalued mappings.

To complete the result, we need notion of 𝒯-weakly isotone increasing for multivalued mappings given by Vetro [31, Definition  4.2].

Definition 3.1. Let (𝒳,⪯) be a partially ordered set and 𝒮,𝒯∶𝒳→𝐵(𝒳) be two maps. The mapping 𝒮 is said to be 𝒯-weakly isotone increasing if 𝒮𝑥⪯₂𝒯𝑦⪯₂𝒮𝑧 for all any 𝑥∈𝒳, 𝑦∈𝒮𝑥 and 𝑧∈𝒯𝑦.

Note that, in particular, for single-valued mappings 𝒯,𝒮∶𝒳→𝒳, mapping 𝒮 is said to be 𝒯-weakly isotone increasing if [31, Definition  2.2] if for each 𝑥∈𝒳 we have 𝒮𝑥⪯𝒯𝒮𝑥⪯𝒮𝒯𝒮𝑥.

Theorem 3.2. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯,𝒮∶𝒳→𝐵(𝒳) be such that 𝜓𝛿𝒯𝑥,𝒮𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝐷(𝑥,𝒯𝑥),𝐷𝑦,𝒮𝑦,𝐷𝑥,𝒮𝑦,𝐷𝑦,𝒯𝑥, (3.1) for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝐷𝑥,𝑦=max𝑥,𝑦,𝐷𝑥,𝒯𝑥,𝐷𝑦,𝒮𝑦𝑥,𝒮𝑦+𝐷𝑦,𝒯𝑥2. (3.2) Also suppose that 𝒮 is 𝒯-weakly isotone increasing and there exists an 𝑥₀∈𝒳 such that {𝑥₀}≺₂𝒮𝑥₀. If the property 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛⪯𝑧∀𝑛 (3.3) holds, then 𝒮 and 𝒯 have a common end-point.

Proof. Define a sequence {𝑥_𝑛}⊂𝒳 and prove that the limit point of that sequence is a unique common end-point for 𝒯 and 𝒮. For a given 𝑥₀∈𝒳 and nonnegative integer 𝑛 let 𝑥₀=𝑥,𝑥_2𝑛+1∈𝒮𝑥_2𝑛,𝑥_2𝑛+2∈𝒯𝑥_2𝑛+1for𝑛≥0. (3.4) If 𝑥_𝑛0∈𝒮𝑥_𝑛0 or 𝑥_𝑛0∈𝒯𝑥_𝑛0 for some 𝑛₀, then the proof is finished. So assume 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛.
Since {𝑥₀}⪯₂𝒮𝑥₀, 𝑥₁∈𝒮𝑥₀ can be chosen so that 𝑥₀⪯𝑥₁. Since 𝒮 is 𝒯-weakly isotone increasing, it is 𝒮𝑥₀⪯₂𝒯𝑥₁; in particular, 𝑥₂∈𝒯𝑥₁ can be chosen so that 𝑥₁⪯𝑥₂. Now, 𝒯𝑥₁⪯₂𝒮𝑥₂ (since 𝑥₂∈𝒯𝑥₁); in particular, 𝑥₃∈𝒮𝑥₂ can be chosen so that 𝑥₂⪯𝑥₃.
Continuing this process we construct a monotone increasing sequence {𝑥_𝑛} in 𝒳 such that 𝑥₀⪯𝑥₁⪯𝑥₂⪯𝑥₃⪯⋯⪯𝑥_𝑛⪯𝑥_𝑛+1⪯⋯. (3.5) If 𝑥_𝑛0∈𝒮𝑥_𝑛0 or 𝑥_𝑛0∈𝒯𝑥_𝑛0 for some 𝑛₀, then the proof is finished. So assume 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛.
Suppose that 𝑛 is an odd number. Substituting 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛+1 in (3.1) and using properties of function 𝜓, we have for all 𝑛≥0, 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓𝛿𝒯𝑥_𝑛,𝒮𝑥_𝑛+1≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷𝑥_𝑛+1,𝒮𝑥_𝑛+1,𝐷𝑥_𝑛,𝒮𝑥_𝑛+1+𝐷𝑥_𝑛+1,𝒯𝑥_𝑛2+𝐿min𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷𝑥_𝑛+1,𝒮𝑥_𝑛+1,𝐷𝑥_𝑛,𝒮𝑥_𝑛+1,𝐷𝑥_𝑛+1,𝒯𝑥_𝑛≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+2+𝑑𝑥_𝑛+1,𝑥_𝑛+12+𝐿min𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+2,𝑑𝑥_𝑛+1,𝑥_𝑛+1. (3.6) Since 𝑑(𝑥_𝑛,𝑥_𝑛+2)/2≤ max{𝑑(𝑥_𝑛,𝑥_𝑛+1),𝑑(𝑥_𝑛+1,𝑥_𝑛+2)}, it follows that 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛼𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2. (3.7) Suppose that 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝑑(𝑥_𝑛+1,𝑥_𝑛+2), for some positive integer 𝑛.
Then from (3.7), we have 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝛼𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2, (3.8) it implies that 𝑑(𝑥_𝑛+1,𝑥_𝑛+2)=0, or that 𝑥_𝑛+1=𝑥_𝑛+2, contradicting our assumption that 𝑥_𝑛≠𝑥_𝑛+1, for each 𝑛 and so we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2<𝑑𝑥_𝑛,𝑥_𝑛+1. (3.9) In the similar fashion, we can also show inequalities (3.9) when 𝑛 is an even number. Therefore, the sequence {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an 𝑟≥0 such that 𝑑𝑥_𝑛,𝑥_𝑛+1⟶𝑟as𝑛⟶∞. (3.10) Taking the limit as 𝑛→∞ in (3.7) and using the continuity of 𝜓, we have 𝜓(𝑟)≤𝛼𝜓(𝑟), which is a contradiction unless 𝑟=0. Hence lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (3.11) Next we show that {𝑥_𝑛} is a Cauchy sequence. If otherwise, there exists an 𝜖>0 for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that for all positive integers 𝑘, 𝑛(𝑘)>𝑚(𝑘)>𝑘 and 𝑑(𝑥_𝑚(𝑘),𝑥_𝑛(𝑘))≥𝜖.
Assuming that 𝑛(𝑘) is the smallest such positive integer, we get 𝑛(𝑘)>𝑚(𝑘)>𝑘, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≥𝜖,𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜖. (3.12) Now, 𝜖≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘), (3.13) that is, 𝜖≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)<𝜖+𝑑𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘). (3.14) Taking the limit as 𝑘→∞ in the above inequality and using (3.11), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)=𝜖. (3.15) Again, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1+𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑚(𝑘)+𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1. (3.16) Taking the limit as 𝑘→∞ in the above inequalities and using (3.11) and (3.15), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜖. (3.17) Again, 𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1+𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1≤𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+𝑑𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1. (3.18) Letting 𝑘→∞ in the above inequalities and using (2.9) and (3.15), we have lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1=𝜖. (3.19) Similarly, we have that lim_𝑘→∞𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1=𝜖. (3.20) For each positive integer 𝑘, 𝑥_𝑚(𝑘) and 𝑥_𝑛(𝑘) are comparable. Then using the monotone property of 𝜓 and the condition (3.1), we have 𝜓𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+2≤𝜓𝛿𝒯𝑥_𝑚(𝑘),𝒮𝑥_𝑛(𝑘)+1≤𝛼𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑚(𝑘),𝐷𝑥_𝑛(𝑘)+1,𝒮𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑚(𝑘),𝒮𝑥_𝑛(𝑘)+1+𝐷𝑥_𝑛(𝑘)+1,𝒯𝑥_𝑚(𝑘)2+𝐿min𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑚(𝑘),𝐷𝑥_𝑛(𝑘)+1,𝒮𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑚(𝑘),𝒮𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑛(𝑘)+1,𝒯𝑥_𝑚(𝑘)=𝛼𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+12+𝐿min𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+2,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+2,𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+1. (3.21) Letting 𝑘→∞ in above inequality, using (3.11), (3.15), (3.17), (3.19), and (3.20) and using the continuity of 𝜓, we have 𝜓(𝜖)≤𝛼𝜓(𝜖), (3.22) which is a contradiction by virtue of a property of 𝜓.
Hence {𝑥_𝑛} is a Cauchy sequence. From the completeness of 𝒳, there exists a 𝑧∈𝒳 such that 𝑥_𝑛⟶𝑧as𝑛⟶∞. (3.23) By the assumption (3.3), 𝑥_𝑛⪯𝑧, for all 𝑛.
Then by the monotone property of 𝜓 and the condition (3.1), we have 𝜓𝛿𝑥_𝑛+1,𝒮𝑧≤𝜓𝛿𝒯𝑥_𝑛𝑑𝑥,𝒮𝑧≤𝛼𝜓max_𝑛,𝑧,𝐷𝑥_𝑛,𝒯𝑥_𝑛(),𝐷𝑥,𝐷𝑧,𝒮𝑧^𝑛,𝒮𝑧+𝐷𝑧,𝒯𝑥_𝑛2+𝐿min𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷(𝑧,𝒮𝑧),𝐷𝑥_𝑛,𝒮𝑧,𝐷𝑧,𝒯𝑥_𝑛≤𝛼𝜓max𝑑𝑥_𝑛,𝑧,𝑑𝑥_𝑛,𝑥_𝑛+1(),𝐷𝑥,𝐷𝑧,𝒮𝑧^𝑛,𝒮𝑧+𝑑𝑧,𝑥_𝑛+12+𝐿min𝑑𝑥_𝑛,𝑥_𝑛+1,𝐷(𝑧,𝒮𝑧),𝐷𝑥_𝑛,𝒮𝑧,𝑑𝑧,𝑥_𝑛+1. (3.24) Taking the limit as 𝑛→∞ in the above inequality, using (3.11) and (3.23) and the continuity of 𝜓, we have 𝜓(𝛿(𝑧,𝒮𝑧))≤𝛼𝜓(𝐷(𝑧,𝒮𝑧))≤𝛼𝜓(𝛿(𝑧,𝒮𝑧)), (3.25) it implies that 𝛿(𝑧,𝒮𝑧)=0, or that {𝑧}=𝒮𝑧. Similarly {𝑧}=𝒯𝑧. Moreover, 𝑧 is a common end-point of 𝒯 and 𝒮.

Putting 𝒮=𝒯 in Theorem 3.2, we immediately obtain the following result.

Corollary 3.3. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯∶𝒳→𝐵(𝒳) be such that 𝜓𝛿𝒯𝑥,𝒯𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝐷(𝑥,𝒯𝑥),𝐷𝑦,𝒯𝑦,𝐷𝑥,𝒯𝑦,𝐷𝑦,𝒯𝑥, (3.26) for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝐷𝑥,𝑦=max𝑥,𝑦,𝐷𝑥,𝒯𝑥,𝐷𝑦,𝒯𝑦𝑥,𝒯𝑦+𝐷𝑦,𝒯𝑥2. (3.27) Also suppose that 𝒯𝑥⪯₁𝒯(𝒯𝑥) for all 𝑥∈𝒳 and there is 𝑥₀∈𝒳 such that {𝑥₀}≺₁𝒯𝑥₀. If the property 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛≺𝑧∀𝑛 (3.28) holds, then 𝒯 has a end-point.

In Theorem 3.2, if 𝒯,𝒮 are single valued mappings, then we have the following result.

Theorem 3.4. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯,𝒮∶𝒳→𝒳 be such that 𝜓𝑑𝒯𝑥,𝒮𝑦≤𝛼𝜓𝑀𝑥,𝑦+𝐿min𝑑(𝑥,𝒯𝑥),𝑑𝑦,𝒮𝑦,𝑑𝑥,𝒮𝑦,𝑑𝑦,𝒯𝑥, (3.29) for all comparable 𝑥,𝑦∈𝒳, where 𝐿≥0, 0<𝛼<1 and 𝜓 is an altering distance function and 𝑀𝑑(),𝑑𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝒯𝑥,𝑑𝑦,𝒮𝑦𝑥,𝒮𝑦+𝑑𝑦,𝒯𝑥2. (3.30) Also suppose that 𝒮 and 𝒯 are weakly isotone increasing. If 𝒮𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 (3.31) or 𝒯𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 (3.32) or 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛⪯𝑧∀𝑛 (3.33) holds, then 𝒮 and 𝒯 have a common end-point.

Proof. If we assume 𝒯 and 𝒮 as a multivalued mapping in which 𝒯𝑥 and 𝒮𝑥 are a singleton set for every 𝑥∈𝒳. Then we consider the same sequence {𝑥_𝑛} as in the proof of Theorem 3.4. Follows the line of proof of Theorem 3.4, we have that {𝑥_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑥_𝑛=𝑧. (3.34) Then, if 𝒯 is continuous, we have 𝑧=lim_𝑛→∞𝑥_𝑛+1=lim_𝑛→∞𝒯𝑥_𝑛=𝒯𝑧 (3.35) and this proves that 𝑧 is a end-point of 𝒯 and so 𝑧 is a end-point of 𝒮. Similarly, if 𝒮 is continuous, we have the result. Thus it is immediate to conclude that 𝒯 and 𝒮 have a common end-point.

4. Common End-Point Theorems for a Pair of Multivalued Mappings Satisfying Weakly Contractive Condition

In this section, we prove common end-point theorems for a pair of weakly isotone increasing multivalued mappings under weakly contractive condition.

To complete the result, we need notion of weakly contractive condition given by Rhoades [32].

Definition 4.1 (Weakly Contractive Mapping). Let 𝒳 be a metric space. A mapping 𝒯∶𝒳→𝒳 is called weakly contractive if and only if 𝑑𝒯𝑥,𝒯𝑦≤𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦,∀𝑥,𝑦∈𝒳, (4.1) where 𝜑 is an altering distance function.

Theorem 4.2. Let (𝒳,𝑑,⪯) be an ordered complete metric space. Let 𝒯,𝒮∶𝒳→𝐵(𝒳) be such that 𝜓𝛿𝑑(),𝐷𝒯𝑥,𝒮𝑦≤𝜓max𝑥,𝑦,𝐷𝑥,𝒯𝑥,𝐷𝑦,𝒮𝑦𝑥,𝒮𝑦+𝐷𝑦,𝒯𝑥2−𝜙max𝑑𝑥,𝑦,𝛿𝑦,𝒮𝑦, (4.2) for all comparable 𝑥,𝑦∈𝒳, where 𝜓,𝜙∶[0,+∞)→[0,+∞) are an altering distance functions.
Also suppose that 𝒮 is 𝒯-weakly isotone increasing and there exists an 𝑥₀∈𝒳 such that {𝑥₀}≺₂𝒮𝑥₀. If the property 𝑥_𝑛⊂𝒳𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑧𝑖𝑛𝒳,𝑡ℎ𝑒𝑛𝑥_𝑛⪯𝑧∀𝑛 (4.3) holds, then 𝒮 and 𝒯 have a common end-point.

Proof. Define a sequence {𝑥_𝑛}⊂𝒳 and prove that the limit point of that sequence is a unique common end-point for 𝒯 and 𝒮. For a given 𝑥₀∈𝒳 and nonnegative integer 𝑛 let 𝑥₀=𝑥,𝑥_2𝑛+1∈𝒮𝑥_2𝑛,𝑥_2𝑛+2∈𝒯𝑥_2𝑛+1for𝑛≥0. (4.4) Since {𝑥₀}⪯₂𝒮𝑥₀, 𝑥₁∈𝒮𝑥₀ can be chosen so that 𝑥₀⪯𝑥₁. Since 𝒮 is 𝒯-weakly isotone increasing, it is 𝒮𝑥₀⪯₂𝒯𝑥₁; in particular, 𝑥₂∈𝒯𝑥₁ can be chosen so that 𝑥₁⪯𝑥₂. Now, 𝒯𝑥₁⪯₂𝒮𝑥₂ (since 𝑥₂∈𝒯𝑥₁); in particular, 𝑥₃∈𝒮𝑥₂ can be chosen so that 𝑥₂⪯𝑥₃.
Continuing this process, we conclude that {𝑥_𝑛} can be an increasing sequence in 𝒳: 𝑥₁⪯𝑥₂⪯⋯⪯𝑥_𝑛⪯𝑥_𝑛+1⪯⋯. (4.5) If there exists a positive integer 𝑁 such that 𝑥_𝑁=𝑥_𝑁+1, then 𝑥_𝑁 is a common end-point of 𝒯 and 𝒮. Hence we will assume that 𝑥_𝑛≠𝑥_𝑛+1, for all 𝑛≥0.
Suppose that 𝑛 is an odd number. Substituting 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛+1 in (2.6) and using properties of functions 𝜓 and 𝜙, we have for all 𝑛≥0, 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓𝛿𝒯𝑥_𝑛,𝒮𝑥_𝑛+1≤𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝐷𝑥_𝑛,𝒯𝑥_𝑛,𝐷𝑥_𝑛+1,𝒮𝑥_𝑛+1,𝐷𝑥_𝑛,𝒮𝑥_𝑛+1+𝐷𝑥_𝑛+1,𝒯𝑥_𝑛2−𝜙max𝑑𝑥_𝑛,𝑥_𝑛+1,𝛿𝑥_𝑛+1,𝒮𝑥_𝑛+1≤𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+2+𝑑𝑥_𝑛+1,𝑥_𝑛+12−𝜙max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2. (4.6) Since 𝑑(𝑥_𝑛,𝑥_𝑛+2)/2≤max{𝑑(𝑥_𝑛,𝑥_𝑛+1),𝑑(𝑥_𝑛+1,𝑥_𝑛+2)}, it follows that 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2−𝜙max𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2. (4.7) Suppose that 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝑑(𝑥_𝑛+1,𝑥_𝑛+2), for some positive integer 𝑛. Then from (4.7), we have 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2−𝜙𝑑𝑥_𝑛+1,𝑥_𝑛+2, (4.8) that is, 𝜙(𝑑(𝑥_𝑛+1,𝑥_𝑛+2))≤0, which implies that 𝑑(𝑥_𝑛+1,𝑥_𝑛+2)=0, or that 𝑥_𝑛+1=𝑥_𝑛+2, contradicting our assumption that 𝑥_𝑛≠𝑥_𝑛+1. So we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2<𝑑𝑥_𝑛,𝑥_𝑛+1. (4.9) In the similar fashion, we can also show inequalities (4.9) when 𝑛 is an even number. Therefore, for all 𝑛≥0 and {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is a monotone decreasing sequence of nonnegative real numbers. Hence there exists an 𝑟≥0 such that lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=𝑟. (4.10) In view of the above facts, from (4.7) we have for all 𝑛≥0, 𝜓𝑑𝑥_𝑛+1,𝑥_𝑛+2≤𝜓𝑑𝑥_𝑛,𝑥_𝑛+1−𝜙𝑑𝑥_𝑛,𝑥_𝑛+1. (4.11) Taking the limit as 𝑛→∞ in the above inequality, using (4.10) and the continuities of 𝜙 and 𝜓, we have 𝜓(𝑟)≤𝜓(𝑟)−𝜙(𝑟), (4.12) which is a contradiction unless 𝑟=0. Hence lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (4.13) Next we show that {𝑥_𝑛} is a Cauchy sequence. If {𝑥_𝑛} is not a Cauchy sequence, then using an argument similar to that given in Theorem 3.2, we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} for which lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)=𝜖,lim_𝑘→∞𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜖,lim_𝑘→∞𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1=𝜖,lim_𝑘→∞𝑑𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+1=𝜖. (4.14) For each positive integer 𝑘, 𝑥_𝑚(𝑘) and 𝑥_𝑛(𝑘) are comparable. Then using the monotone property of 𝜓 and (4.2), we have 𝜓𝑑𝑥_𝑚(𝑘)+1,𝑥_𝑛(𝑘)+2≤𝜓𝛿𝒯𝑥_𝑚(𝑘),𝒮𝑥_𝑛(𝑘)+1≤𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑚(𝑘),𝒯𝑥_𝑚(𝑘),𝐷𝑥_𝑛(𝑘)+1,𝒮𝑥_𝑛(𝑘)+1,𝐷𝑥_𝑚(𝑘),𝒮𝑥_𝑛(𝑘)+1+𝐷𝑥_𝑛(𝑘)+1,𝒯𝑥_𝑚(𝑘)2−𝜙max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝛿𝑥_𝑛(𝑘)+1,𝒮𝑥_𝑛(𝑘)+1,=𝜓max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1,𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+2,𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+2+𝑑𝑥_𝑛(𝑘)+1,𝑇𝑥_𝑚(𝑘)+12−𝜙max𝑑𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)+1,𝑑𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+2. (4.15) Letting 𝑘→∞ in the above inequality, using (4.14) and the continuities of 𝜓 and 𝜙, we have 𝜓(𝜖)≤𝜓(𝜖)−𝜙(𝜖), (4.16) which is a contradiction by virtue of a property of 𝜙. Hence {𝑥_𝑛} is a Cauchy sequence. From the completeness of 𝒳, there exists a 𝑧∈𝒳 such that 𝑥_𝑛⟶𝑧as𝑛⟶∞. (4.17) By the condition (4.3), 𝑥_𝑛⪯𝑧, for all 𝑛. Then by the monotone property of 𝜓 and (4.2), we have 𝜓𝛿𝑥_𝑛+1,𝒮𝑧≤𝜓𝛿𝒯𝑥_𝑛𝑑𝑥,𝒮𝑧≤𝜓max_𝑛,𝑧,𝐷𝑥_𝑛,𝒯𝑥_𝑛(),𝐷𝑥,𝐷𝑧,𝒮𝑧^𝑛,𝒮𝑧+𝐷𝑧,𝒯𝑥_𝑛2−𝜙max𝑑𝑥_𝑛(𝑑𝑥,𝑧,𝛿𝑧,𝒮𝑧)≤𝜓max_𝑛,𝑧,𝑑𝑥_𝑛,𝑥_𝑛+1(),𝐷𝑥,𝐷𝑧,𝒮𝑧^𝑛,𝒮𝑧+𝑑𝑧,𝑥_𝑛+12−𝜙max𝑑𝑥_𝑛,𝑧,𝛿(𝑧,𝒮𝑧). (4.18) Taking the limit as 𝑛→∞ in the above inequality, using (4.13), (4.17) and the continuities of 𝜓 and 𝜙, we have 𝜓(𝛿(𝑧,𝒮𝑧))≤𝜓(𝐷(𝑧,𝒮𝑧))−𝜙(𝛿(𝑧,𝒮𝑧)), (4.19) which implies that 𝜓(𝛿(𝑧,𝒮𝑧))≤𝜓(𝛿(𝑧,𝒮𝑧))−𝜙(𝛿(𝑧,𝒮𝑧)), (4.20) which is a contradiction unless 𝛿(𝑧,𝒮𝑧)=0, or that {𝑧}=𝒮𝑧; that is, 𝑧 is a end-point of 𝒮. Similarly {𝑧}=𝒯𝑧. Moreover, 𝑧 is a common end-point of 𝒯 and 𝒮.

Similar corollaries can be derived from Theorem 4.2.

Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees.