JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 962058 10.1155/2013/962058 962058 Research Article Approximate Coincidence Point of Two Nonlinear Mappings 0000-0003-3978-4116 Dey Debashis 1 Kumar Laha Amit 2 Saha Mantu 2 Mukandavire Zindoga 1 Koshigram Union Institution Koshigram Burdwan West Bengal 713150 India 2 Department of Mathematics The University of Burdwan Burdwan West Bengal 713104 India buruniv.ac.in 2013 19 3 2013 2013 07 01 2013 13 02 2013 2013 Copyright © 2013 Debashis Dey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the approximate coincidence point of two nonlinear functions introduced by Geraghty in 1973 and Mizoguchi and Takahashi (𝒯-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. , Berinde  established some fundamental approximate fixed point theorems in metric space. In a recent paper, Dey and Saha  studied the existence of approximate fixed point for the Reich operator  which in turn generalizes the results of Berinde . Coincidence point theory has a vast literature, and many generalizations have been emerged so far (see ). 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  in 1973 and Mizoguchi and Takahashi  (𝒯-function) in 1989.

2. Approximate Coincidence Point Definition 1.

Let (X,d) be a metric space and f:XX, ϵ>0, x0X. 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)={xXd(f(x),x)<ϵ}.

Definition 2.

Let f:XX. 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:XX be two single valued maps. The maps f and g are said to have coincidence point x if fx=gx=wX 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 .

In 1973, Geraghty  (see also ) 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)1implies  tn0. 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:XX be two single-valued self-maps. The maps f and g are said to have approximate coincidence point property provided (4)infxXd(fx,gx)=0, or, equivalently, for any ϵ>0, there exists zX 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:XX be two self-mappings such that fXgX satisfying (6)d(fx,fy)β(d(gx,gy))d(gx,gy), for all x,yX and β𝒮. Then the following statements hold.

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

There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limnd(zn,zn+1)=infxXd(fx,gx)=0.

Proof.

Let x0X be arbitrary. Since fXgX, we can choose x1X 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, 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 q0. 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)limnβ(d(fxn,fxn+1))1. Now, using property of the function β, we conclude that limnd(fxn,fxn+1)=0. Assume zn=fxn for n, so limnd(zn,zn+1)=0. It implies that infxXd(fx,gx)d(fxn,gxn)d(zn-1,zn) for all n. Since d(zn,zn+1)0 as n, it follows that infxXd(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 0λ<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 with m>n, it follows from (11) that (12)d(zn,zm)i=nm-1d(zi,zi+1)λn-11-λd(z1,z2). Since 0λ<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,xX. 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,yX. 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  introduced 𝒯-function as follows.

A function φ:[0,)[0,1) is said to be an 𝒯-function, if (14)limst+supφ(s)<1,t[0,).

It is obvious that if φ:[0,)[0,1) is a monotone, then φ is an 𝒯-function.

Definition 7.

Let (X,d) be a metric space and φ:[0,)[0,1) is an 𝒯-function. Then T:XX is said to be an 𝒯-type mapping if (15)d(Tx,Ty)φ(d(x,y))d(x,y).

Using this function, Mizoguchi and Takahashi  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 ). But we only restrict ourselves in proving results for single valued mapping. In this aspect, we formulate our next results using 𝒯-function. For the properties and characterizations of 𝒯-function one can see [16, 17] for details.

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

Theorem 8.

Let (X,d) be a metric space and let f,g:XX be two single valued self-maps such that fXgX satisfying (16)d(fx,fy)φ(d(gx,gy))d(gx,gy), for all x,yX where φ:[0,)[0,1) is an 𝒯-function.

Then the following statements hold.

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

There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limnd(zn,zn+1)=infxXd(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 𝒯-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,xX. 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:XX be a single valued self-map satisfying (18)d(fx,fy)φ(d(x,y))d(x,y), for all x,yX where φ:[0,)[0,1) is an 𝒯-function.

Then the following statements hold.

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

There exists a sequence {zn} in (X,d)   such that {zn} is a Cauchy sequence and limnd(zn,zn+1)=infxXd(x,fx)=0.

Corollary 11.

Let (X,d) be a metric space and let f:XX be two single valued self-mapping satisfying (19)d(fx,fy)β(d(x,y))d(x,y), for all x,yX and β𝒮. Then the following statements hold.

( B 1 ) f has the approximate fixed point on X, that is, (infxXd(fx,gx)=0).

( B 2 ) There exists a sequence {zn} in (X,d) such that {zn} is a Cauchy sequence and limnd(zn,zn+1)=infxXd(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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