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 PointDefinition 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
(1)Fϵ(f)={x∈X∣d(f(x),x)<ϵ}.

Definition 2.

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

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
(3)β(tn)⟶1impliestn⟶0.
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
(4)infx∈Xd(fx,gx)=0,
or, equivalently, for any ϵ>0, there exists z∈X such that
(5)d(fz,gz)<ϵ.
The set of approximate coincidence point of f and g is denoted by 𝒞𝒪𝒫ϵ(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
(6)d(fx,fy)≤β(d(gx,gy))d(gx,gy),
for all x,y∈X and β∈𝒮. Then the following statements hold.

f and g have the approximate coincidence point property on X, that is, (infx∈Xd(fx,gx)=0).

There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limn→∞d(zn,zn+1)=infx∈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 {xn} in X as follows:
(7)gxn+1=fxn,n=0,1,2,….
We will suppose d(fxn,fxn+1)>0 for all n∈ℕ, since if fxn=fxn+1 for some n, then for every ϵ>0, d(gxn+1,fxn+1)=d(fxn,fxn+1)=0<ϵ implying that f,g have approximate coincidence point xn+1 and thus completes the proof. So we suppose that d(fxn,fxn+1)>0. Then by (6),
(8)d(fxn+1,fxn+2)≤β(d(gxn+1,gxn+2))d(gxn+1,gxn+2)=β(d(fxn,fxn+1))d(fxn,fxn+1)<d(fxn,fxn+1),
and so d(fxn,fxn+1)<d(fxn-1,fxn). Hence, {d(fxn,fxn+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
(9)d(fxn+1,fxn+2)d(fxn,fxn+1)≤β(d(fxn,fxn+1))<1.
Now passing limit as n→∞ on (9), we get
(10)limn→∞β(d(fxn,fxn+1))⟶1.
Now, using property of the function β, we conclude that limn→∞d(fxn,fxn+1)=0. Assume zn=fxn for n∈ℕ, so limn→∞d(zn,zn+1)=0. It implies that infx∈Xd(fx,gx)≤d(fxn,gxn)≤d(zn-1,zn) for all n∈ℕ. Since d(zn,zn+1)→0 as n→∞, it follows that infx∈Xd(fx,gx)=0. So f and g have approximate coincidence point x in X. So (A1) follows.

Now it suffices to prove that {zn} is a Cauchy sequence in (X,d). Let λ=β(d(zn,zn+1)). Then using the property of β and from (9), we have 0≤λ<1. Again using (9), we get d(zn+1,zn+2)≤λd(zn,zn+1). In this process, we obtain
(11)d(zn+1,zn+2)≤λnd(z1,z2).
Thus, for m,n∈ℕ with m>n, it follows from (11) that
(12)d(zn,zm)≤∑i=nm-1d(zi,zi+1)≤λn-11-λd(z1,z2).
Since 0≤λ<1, d(zn,zm)→0 as n→∞ and so {zn} 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
(13)f(x)=14x,g(x)=12x,∀x∈X.
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
(14)lims→t+supφ(s)<1,∀t∈[0,∞).

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
(15)d(Tx,Ty)≤φ(d(x,y))d(x,y).

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
(16)d(fx,fy)≤φ(d(gx,gy))d(gx,gy),
for all x,y∈X where φ:[0,∞)→[0,1) is an ℳ𝒯-function.

Then the following statements hold.

f and g have the approximate coincidence point property on X, that is, (infx∈Xd(fx,gx)=0).

There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limn→∞d(zn,zn+1)=infx∈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
(17)f(x)=14x,g(x)=12x,∀x∈X.
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
(18)d(fx,fy)≤φ(d(x,y))d(x,y),
for all x,y∈X where φ:[0,∞)→[0,1) is an ℳ𝒯-function.

Then the following statements hold.

f has the approximate fixed point on X, that is, (infx∈Xd(x,fx)=0).

There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limn→∞d(zn,zn+1)=infx∈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
(19)d(fx,fy)≤β(d(x,y))d(x,y),
for all x,y∈X and β∈𝒮. Then the following statements hold.

(B1′)f has the approximate fixed point on X, that is, (infx∈Xd(fx,gx)=0).

(B2′)
There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limn→∞d(zn,zn+1)=infx∈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.

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