# Set-valued Prešić–Reich type mappings in metric spaces

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DOI: 10.1007/s13398-012-0114-2

- Cite this article as:
- Shukla, S. & Sen, R. RACSAM (2014) 108: 431. doi:10.1007/s13398-012-0114-2

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## Abstract

The purpose of this paper is to establish some coincidence and common fixed point theorems for a set-valued and a single-valued mapping satisfying Prešić–Reich type contraction conditions in metric spaces. Our results generalize and extend some known results in metric spaces. An example is included which illustrate the results.

### Keywords

Set-valued mappingCoincidence pointCommon fixed pointPrešić type mapping### Mathematics Subject Classification (2000)

47H1054H25## 1 Introduction

*x*is called the fixed point of mapping

*T*.

In 1965, Prešić [20, 21] generalized the Banach contraction mapping theorem in product spaces and proved following theorem.

**Theorem 1**

*k*a positive integer and \(T:X^{k}\rightarrow X\) a mapping satisfying the following contractive type condition:

*X*and for \(n\in \mathbb N \), \(x_{n+k}=T(x_{n},x_{n+1},\ldots ,x_{n+k-1})\), then the sequence \(\{x_{n}\}\) is convergent and \(\lim x_{n}=T(\lim x_{n},\lim x_{n},\ldots ,\lim x_{n})\).

Note that condition (4) in the case \(k = 1\) reduces to the well-known Banach contraction mapping principle. So, Theorem 1 is a generalization of the Banach fixed point theorem. Some generalizations and applications of Theorem 1 can be seen in [6, 7, 12, 15, 18, 19, 26].

On the other hand Nadler [17] generalized the Banach contraction mapping principle to set-valued functions and proved the following fixed point theorem.

**Theorem 2**

*T*be a mapping from

*X*into \(CB(X)\)\((\)here \(CB(X)\) denotes the set of all nonempty closed bounded subset of

*X*\()\) such that for all \(x,y\in X\),

After the work of Nadler, several authors proved fixed point results for set-valued mappings (see, e.g. [1–5, 8–11, 16, 22, 24, 25, 29]).

Recently in [26] authors introduced the notion of weak compatibility of set-valued Prešić type mappings with a single-valued mapping and proved some coincidence and common fixed point theorems for such mappings in product spaces. Following theorem was the main result of [26].

**Theorem 3**

*k*a positive integer. Let \(f:X^{k}\rightarrow CB(X)\) and \(g:X\rightarrow X\) be two mappings such that \(g(X)\) is a closed subspace of \(X\) and \(f(x_{1},x_{2},\ldots ,x_{k})\subset g(X)\) for all \(x_{1},x_{2},\ldots ,x_{k}\in X\). Suppose following condition holds:

*f*and

*g*have a point of coincidence \(v\in X\).

Above theorem generalizes the results of Prešić and Nadler in product spaces. In the present paper we prove coincidence point and common fixed point theorems for two maps satisfying Prešić–Reich type contraction conditions in metric spaces. Our results generalize the results of Prešić [21], Nadler [17], Reich [23], Pǎcurar [19], recent results of [26] and several known results in metric spaces.

## 2 Preliminaries

Following definitions and assumptions will be needed in sequel.

*A*be any nonempty subset of a metric space \((X,d)\). For \(x\in X\), define

*X*. For \(A,B\in CB(X)\), define

*H*is a metric on \(CB(X)\) and called Hausdorff metric.

*Remark 1*

- 1.
for all \(\epsilon >0\) and \(a\in A\) there exists a point \(b\in B\) such that \(d(a,b)\le H(A,B)+\epsilon \);

- 2.
for each \(a\in X\), \(a\in A\) if and only if \(d(a,A)=0\).

*k*be a positive integer and \(f:X^{k}\rightarrow CB(X)\) be a mapping. Then

*f*is said to be Lipschitzian if there exist nonnegative constants \(\alpha _{i}\) such that

If \(\sum \nolimits _{i=1}^{k}\alpha _{i}<1\), then the mapping \(f\) is said to be a set-valued Prešić type contraction.

**Definition 1**

- (a)
If \(x\in f(x,\ldots ,x)\), then \(x\in X\) is called a fixed point of \(f\).

- (b)
An element \(x\in X\) said to be a coincidence point of \(f\) and \(g\) if \(gx\in f(x,\ldots ,x)\).

- (c)
If \(w=gx\in f(x,\ldots ,x)\), then \(w\) is called a point of coincidence of

*f*and*g*. - (d)
If \(x=gx\in f(x,\ldots ,x)\), then \(x\) is called a common fixed point of

*f*and*g*. - (e)
Mappings

*f*and*g*are said to be commuting if \(g(f(x,\ldots ,x))=f(gx,\ldots ,gx)\) for all \(x\in X\). - (f)
Mappings

*f*and*g*are said to be weakly compatible if \(gx\in f(x,\ldots ,x)\) implies \(g(f(x,\ldots ,x))\subseteq f(gx,\ldots ,gx)\).

Now we can state our main results.

## 3 Main results

The following theorem is one of the main results in this article.

**Theorem 4**

*k*a positive integer. Let \(f:X^{k}\rightarrow CB(X)\) and \(g:X\rightarrow X\) be two mappings such that \(g(X)\) is a closed subspace of

*X*and \(f(x_{0},x_{1},\ldots ,x_{k-1})\subset g(X)\) for all \(x_{0},x_{1},\ldots ,x_{k-1}\in X\). Suppose following condition holds:

*f*and

*g*have a point of coincidence \(v\in X\).

*Proof*

*H*is a metric on \(CB(X)\), for any \(n\in \mathbb N \) it follows from (11) that

*u*is coincidence point of

*f*and

*g*.

*Remark 2*

Note that above theorem is a generalization of Theorems 1, 2 and 3 in product spaces. Also for \(k=1\) above theorem reduces into a coincidence point result for set-valued *g*-weak contraction (see [27]) in metric spaces.

Taking \(g=I_{X}\) in Theorem 4, we obtain the following fixed point result for set-valued Prešić–Reich type contraction.

**Corollary 1**

Let \((X,d)\) be any complete metric space, *k* a positive integer. Let \(f:X^{k}\rightarrow CB(X)\) be a set-valued Prešić–Reich type contraction (i.e. \(f\) satisfies (8)). Then *f* has a fixed point \(v\in X\).

Following theorem provides a sufficient condition for the uniqueness of fixed point.

**Theorem 5**

Let \((X,d)\) be any complete metric space, \(k\) a positive integer. Let \(f:X^{k}\rightarrow CB(X)\) and \(g:X\rightarrow X\) be two mappings such that, all the conditions of Theorem 4 are satisfied. Suppose in addition that *f* and *g* are weakly compatible in such a way that, for any coincidence point \(u\) of \(f\) and \(g\) we have \(f(u,\ldots ,u)=\{gu\}\), then *f* and *g* have a unique common fixed point.

*Proof*

The existence of coincidence point *u* and point of coincidence \(v=gu\) follows from Theorem 4. Suppose \(f\) and \(g\) are weakly compatible in such a way that, for any coincidence point \(u\) of \(f\) and \(g\) we have \(f(u,\ldots ,u)=\{gu\}=\{v\}\). We shall show that the point of coincidence \(v\) is unique.

As \(v=gu\in f(u,\ldots ,u)\) and \(v^{\prime }=gu^{\prime }\in f(u^{\prime },\ldots ,u^{\prime })\), it follows from above inequality that \(d(v,v^{\prime })=0\) i.e. \(v=v^{\prime }\). So point of coincidence of \(f\) and \(g\) is unique.

*gv*is another point of coincidence of

*f*and

*g*and by uniqueness we have \(v=gv\in f(v,\ldots ,v)\). Thus \(v\) is unique common fixed point of \(f\) and \(g\).

Following is a simple example which illustrate above theorems, also that the fixed point of set-valued Prešić–Reich type contraction may not be unique.

*Example 1*

On the other hand, it is easy to see that \(f\) is a set-valued Prešić–Reich type contraction on \(X\) with \(\alpha _{1}=\alpha _{2}=\frac{1}{10}\) and \(\beta _{0}=\beta _{1}=\beta _{2}=\frac{1}{12}\). All the conditions of Corollary 1 are satisfied and the set of fixed points of \(f\) is \(\mathcal F (f)=\left\{ 0,\frac{1}{13}\right\} \). Note that conditions of Theorem 5 (with \(gx=x\) for all \(x\in X\)) are not satisfied and fixed point of \(f\) is not unique.

Taking \(\alpha _{i}=0\) for \(i=1,2,\ldots ,k\) in Theorem 4, we obtain following generalization of result of Pǎcurar [19] for set-valued mappings.

**Corollary 2**

*Remark*

Note that fixed point result for set-valued Prešić-Kannan type contraction is obtain by taking \(g=I_X\) (i.e. identity mapping of \(X\)) and \(\beta _{i}=a\) for \(i=0,1,\ldots ,k\) in Corollary 2.

**Theorem 6**

*Proof*

**Theorem 7**

Let \((X,d)\) be any metric space, \(k\) a positive integer. Let \(f:X^{k}\rightarrow CB(X)\) and \(g:X\rightarrow X\) be two mappings such that, all the conditions of Theorem 6 are satisfied. Suppose in addition that \(f\) and \(g\) are weakly compatible in such a way that, for any coincidence point \(u\) of \(f\) and \(g\) we have \(f(u,\ldots ,u)=\{gu\}\), then \(f\) and \(g\) have a unique common fixed point.

*Proof*

The proof of this theorem is followed by a similar process as used in Theorem 5.