Approximate Coincidence Point of Two Nonlinear Mappings

Abstract

We study the approximate coincidence point of two nonlinear functions introduced by Geraghty in 1973 and Mizoguchi and Takahashi (ℳ𝒯-function) in 1989.

1. Introduction

Fixed point theory has been an important tool for solving various problems in nonlinear functional analysis as well as a useful tool for proving the existence theorems for nonlinear differential and integral equations. However, in many practical situations, the conditions in the fixed point theorems are too strong and so the existence of a fixed point is not guaranteed. In that situation, one can consider nearly fixed points what we call as approximate fixed points. By an approximate fixed point 𝑥 of a function 𝑓 we mean in a sense that 𝑓(𝑥) is “near to” 𝑥. The study of approximate fixed point 𝑥 of a function 𝑓 we mean in a sense that 𝑓(𝑥) is “near to” 𝑥. The study of approximate fixed point theorems is equally interesting to that of fixed point theorems. Motivated by the article of Tijs et al. [1], Berinde [2] established some fundamental approximate fixed point theorems in metric space. In a recent paper, Dey and Saha [3] studied the existence of approximate fixed point for the Reich operator [4] which in turn generalizes the results of Berinde [2]. Coincidence point theory has a vast literature, and many generalizations have been emerged so far (see [5–11]). The aim of this paper is to define approximate coincidence point for a pair of single valued self-mappings to obtain some important results on approximate coincidence point using two nonlinear functions by Geraghty [12] in 1973 and Mizoguchi and Takahashi [13] (ℳ𝒯-function) in 1989.

2. Approximate Coincidence Point

Definition 1. Let (𝑋,𝑑) be a metric space and 𝑓∶𝑋→𝑋, 𝜖>0, 𝑥₀∈𝑋. Then 𝑥₀ is an 𝜖-fixed point (approximate fixed point) of 𝑓 if 𝑑(𝑓(𝑥₀),𝑥₀)<𝜖.

The set of all 𝜖-fixed points of 𝑓, for a given 𝜖, is denoted by 𝐹_𝜖𝑓=𝑥∈𝑋∣𝑑𝑓(𝑥),𝑥<𝜖. (1)

Definition 2. Let 𝑓∶𝑋→𝑋. Then 𝑓 has the approximate fixed point property if ∀𝜖>0,𝐹_𝜖𝑓≠𝜙. (2)

Definition 3. Let (𝑋,𝑑) be a metric space, and let  𝑓,𝑔∶𝑋→𝑋 be two single valued maps. The maps 𝑓 and 𝑔 are said to have coincidence point 𝑥 if 𝑓𝑥=𝑔𝑥=𝑤∈𝑋 say, 𝑤 is called a point of coincidence of 𝑓 and 𝑔. If 𝑤=𝑥, then 𝑥 is called a common fixed point of 𝑓 and 𝑔.

3. Approximate Coincidence Point Results for Two Nonlinear Maps

In this section, we establish existence of some results concerning approximate coincidence point for various types of nonlinear contractive maps in the setting of general metric spaces. For this purpose, we first define approximate coincidence point for two self-maps in metric space and prove results on approximate coincidence point using the idea of the Geraghty-type contractive condition [12].

In 1973, Geraghty [12] (see also [14]) introduced the following class of functions called the Geraghty function as follows.

Let 𝒮 denote the class of real functions 𝛽∶[0,∞)→[0,1) satisfying condition 𝛽𝑡_𝑛⟶1implies𝑡_𝑛⟶0. (3) An example of a function in 𝒮 may be given by 𝛽(𝑡)=𝑒^−2𝑡 for 𝑡>0 and 𝛽(0)∈[0,1). We now prove our result using this 𝛽-function.

Definition 4. Let (𝑋,𝑑) be a metric space, and let  𝑓,𝑔∶𝑋→𝑋 be two single-valued self-maps. The maps 𝑓 and 𝑔 are said to have approximate coincidence point property provided inf_𝑥∈𝑋𝑑𝑓𝑥,𝑔𝑥=0, (4) or, equivalently, for any 𝜖>0, there exists 𝑧∈𝑋 such that 𝑑𝑓𝑧,𝑔𝑧<𝜖. (5) The set of approximate coincidence point of 𝑓 and 𝑔 is denoted by 𝒞𝒪𝒫_𝜖(𝑓,𝑔).

Theorem 5. Let (𝑋,𝑑) be a metric space and let 𝑓,𝑔∶𝑋→𝑋 be two self-mappings such that 𝑓𝑋⊂𝑔𝑋 satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝛽𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑥,𝑔𝑦, (6) for all 𝑥,𝑦∈𝑋 and 𝛽∈𝒮. Then the following statements hold.(A₁)  𝑓 and 𝑔 have the approximate coincidence point property on 𝑋, that is, (inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0).(A₂) There exists a sequence {𝑧_𝑛} in (𝑋,𝑑) such that {𝑧_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0.

Proof. Let 𝑥₀∈𝑋 be arbitrary. Since 𝑓𝑋⊂𝑔𝑋, we can choose 𝑥₁∈𝑋 such that 𝑔𝑥₁=𝑓𝑥₀. Continuing this process, we obtain a sequence {𝑥_𝑛} in 𝑋 as follows: 𝑔𝑥_𝑛+1=𝑓𝑥_𝑛,𝑛=0,1,2,…. (7) We will suppose 𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)>0 for all 𝑛∈ℕ, since if 𝑓𝑥_𝑛=𝑓𝑥_𝑛+1 for some 𝑛, then for every 𝜖>0, 𝑑(𝑔𝑥_𝑛+1,𝑓𝑥_𝑛+1)=𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)=0<𝜖 implying that 𝑓,𝑔 have approximate coincidence point 𝑥_𝑛+1 and thus completes the proof. So we suppose that 𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)>0. Then by (6), 𝑑𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+2≤𝛽𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2𝑑𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2=𝛽𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1, (8) and so 𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)<𝑑(𝑓𝑥_𝑛−1,𝑓𝑥_𝑛). Hence, {𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)} is a strictly decreasing and bounded below, thus converging to some 𝑞≥0. Without loss of generality, let 𝑞>0. Then using (8), we get 𝑑𝑓𝑥_𝑛+1,𝑓𝑥_𝑛+2𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤𝛽𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1<1. (9) Now passing limit as 𝑛→∞ on (9), we get lim_𝑛→∞𝛽𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛+1⟶1. (10) Now, using property of the function 𝛽, we conclude that lim_𝑛→∞𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)=0. Assume 𝑧_𝑛=𝑓𝑥_𝑛 for 𝑛∈ℕ, so lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=0. It implies that inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)≤𝑑(𝑓𝑥_𝑛,𝑔𝑥_𝑛)≤𝑑(𝑧_𝑛−1,𝑧_𝑛) for all 𝑛∈ℕ. Since 𝑑(𝑧_𝑛,𝑧_𝑛+1)→0 as 𝑛→∞, it follows that inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0. So 𝑓 and 𝑔 have approximate coincidence point 𝑥 in 𝑋. So (A₁) follows.
Now it suffices to prove that {𝑧_𝑛} is a Cauchy sequence in (𝑋,𝑑). Let 𝜆=𝛽(𝑑(𝑧_𝑛,𝑧_𝑛+1)). Then using the property of 𝛽 and from (9), we have 0≤𝜆<1. Again using (9), we get 𝑑(𝑧_𝑛+1,𝑧_𝑛+2)≤𝜆𝑑(𝑧_𝑛,𝑧_𝑛+1). In this process, we obtain 𝑑𝑧_𝑛+1,𝑧_𝑛+2≤𝜆^𝑛𝑑𝑧₁,𝑧₂. (11) Thus, for 𝑚,𝑛∈ℕ with 𝑚>𝑛, it follows from (11) that 𝑑𝑧_𝑛,𝑧_𝑚≤^𝑚−1_𝑖=𝑛𝑑𝑧_𝑖,𝑧_𝑖+1≤𝜆^𝑛−1𝑑𝑧1−𝜆¹,𝑧₂. (12) Since 0≤𝜆<1, 𝑑(𝑧_𝑛,𝑧_𝑚)→0 as 𝑛→∞ and so {𝑧_𝑛} is a Cauchy sequence. Hence, (A₂) follows.

Example 6. Let 𝑋=[1,∞) with 𝑑 be usual metric and 𝑓,𝑔∶[1,∞)→[1,∞) be defined by 𝑓(𝑥)=1(𝑥)=14𝑥,𝑔2𝑥,∀𝑥∈𝑋. (13) Also take 𝛽(𝑡)=𝑒^−2𝑡 for 𝑡>0 and 𝛽(0)∈[0,1). Then one can check the inequality (6) satisfied with 𝑥>𝑦,  𝑥,𝑦∈𝑋. Also it is easy to see that all the conditions of Theorem 5 are satisfied having the approximate coincidence point property. In fact 𝑑(𝑓𝑥₀,𝑔𝑥₀)=|1/4𝑥₀−1/2𝑥₀|=1/4𝑥₀<𝜖, if we select 𝑥₀∈[1,∞) such that 𝑥₀>1/4𝜖 with 0<𝜖<1/4. On the contrary, it is clear that 𝑓 and 𝑔 have no coincidence point in [1,∞).

In 1989, Mizoguchi and Takahashi [13] introduced ℳ𝒯-function as follows.

A function 𝜑∶[0,∞)→[0,1) is said to be an ℳ𝒯-function, if lim_𝑠→𝑡+sup𝜑(𝑠)<1,∀𝑡∈[0,∞). (14)

It is obvious that if 𝜑∶[0,∞)→[0,1) is a monotone, then 𝜑 is an ℳ𝒯-function.

Definition 7. Let (𝑋,𝑑) be a metric space and 𝜑∶[0,∞)→[0,1) is an ℳ𝒯-function. Then 𝑇∶𝑋→𝑋 is said to be an ℳ𝒯-type mapping if 𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦. (15)

Using this function, Mizoguchi and Takahashi [13] proved a fixed point theorem for multivalued mapping, which is a generalization of Nadler’s fixed point theorem which extends the Banach contraction principle for multivalued maps, but its primitive proof is different (see [15]). But we only restrict ourselves in proving results for single valued mapping. In this aspect, we formulate our next results using ℳ𝒯-function. For the properties and characterizations of ℳ𝒯-function one can see [16, 17] for details.

Now, we establish the following approximate coincidence point property using the concept of Mizoguchi and Takahashi (ℳ𝒯)-type mappings.

Theorem 8. Let (𝑋,𝑑) be a metric space and let 𝑓,𝑔∶𝑋→𝑋 be two single valued self-maps such that 𝑓𝑋⊂𝑔𝑋 satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝜑𝑑𝑔𝑥,𝑔𝑦𝑑𝑔𝑥,𝑔𝑦, (16) for all 𝑥,𝑦∈𝑋 where 𝜑∶[0,∞)→[0,1) is an ℳ𝒯-function.
Then the following statements hold.(A^′₁)𝑓 and 𝑔 have the approximate coincidence point property on 𝑋, that is, (inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0).(A^′₂) There exists a sequence {𝑧_𝑛} in (𝑋,𝑑) such that {𝑧_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0.

Proof. Proof is similar to that of Theorem 5 and is left out. The only difference lies in the character of the 𝛽-function and ℳ𝒯-function which are supplementary to each other.

Example 9. Let 𝑋=[0,∞) with 𝑑 be usual metric and 𝑓,𝑔∶[0,∞)→[0,1) be defined by 𝑓(𝑥)=1(𝑥)=14𝑥,𝑔2𝑥,∀𝑥∈𝑋. (17) If we take 𝜑∶[0,∞)→[0,1) defined by 𝜑(𝑥)=2/3, then all the conditions of the Theorem 8 are satisfied.

It is easy to arrive at the following corollaries.

Corollary 10. Let (𝑋,𝑑) be a metric space and let 𝑓∶𝑋→𝑋 be a single valued self-map satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦, (18) for all 𝑥,𝑦∈𝑋 where 𝜑∶[0,∞)→[0,1) is an ℳ𝒯-function.
Then the following statements hold.(B₁)𝑓 has the approximate fixed point on 𝑋, that is, (inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥)=0).(B₂)There exists a sequence {𝑧_𝑛} in (𝑋,𝑑) such that {𝑧_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥)=0.

Corollary 11. Let (𝑋,𝑑) be a metric space and let 𝑓∶𝑋→𝑋 be two single valued self-mapping satisfying 𝑑𝑓𝑥,𝑓𝑦≤𝛽𝑑𝑥,𝑦𝑑𝑥,𝑦, (19) for all 𝑥,𝑦∈𝑋 and 𝛽∈𝒮. Then the following statements hold.
(B^′₁)  𝑓 has the approximate fixed point on 𝑋, that is, (inf_𝑥∈𝑋𝑑(𝑓𝑥,𝑔𝑥)=0).
(B^′₂) There exists a sequence {𝑧_𝑛} in (𝑋,𝑑) such that {𝑧_𝑛} is a Cauchy sequence and lim_𝑛→∞𝑑(𝑧_𝑛,𝑧_𝑛+1)=inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥)=0.

Remark 12. Corollaries 10 and 11 are the generalizations of the Banach contraction principle in approximate version.

Acknowledgment

The authors are thankful to the reviewers for their valuable suggestions.