Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

, Volume 108, Issue 2, pp 431–440

Set-valued Prešić–Reich type mappings in metric spaces

Authors

• Department of Applied MathematicsShri Vaishnav Institute of Technology and Science
• Ravindra Sen
• Department of Applied MathematicsShri Vaishnav Institute of Technology and Science
Original Paper

DOI: 10.1007/s13398-012-0114-2

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

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

47H1054H25

1 Introduction

The well known Banach contraction mapping principle states that if $$(X, d)$$ is a complete metric space and $$T:X\rightarrow X$$ is a self mapping such that
\begin{aligned} d(T x, T y)\le \lambda d(x, y), \end{aligned}
(1)
for all $$x,y\in X$$, where $$0\le \lambda <1$$, then there exists a unique $$x\in X$$ such that $$Tx =x$$. This point x is called the fixed point of mapping T.
On the other hand, for mappings $$T:X\rightarrow X$$ Kannan [13] introduced the contractive condition:
\begin{aligned} d(Tx,Ty)\le \lambda [d(x,Tx)+d(y,Ty)] \end{aligned}
(2)
for all $$x,y \in X$$, where $$\lambda \in [0,1/2)$$ is a constant, and proved a fixed point theorem using (2) instead of (1). The conditions (1) and (2) are independent, as it was shown by two examples in [14].
Reich [23], for mappings $$T:X\rightarrow X$$ generalized Banach and Kannan fixed point theorems, using contractive condition: for all $$x,y\in X$$,
\begin{aligned} d(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty) \end{aligned}
(3)
where $$\alpha ,\beta ,\gamma$$ are nonnegative reals with $$\alpha +\beta +\gamma <1$$. An example in [23] shows that the condition (3) is a proper generalization of (1) and (2).

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

Theorem 1

Let $$(X,d)$$ be a complete metric space, k a positive integer and $$T:X^{k}\rightarrow X$$ a mapping satisfying the following contractive type condition:
\begin{aligned} d(T(x_{1},x_{2},\ldots ,x_{k}),T(x_{2},x_{3},\ldots ,x_{k+1}))\le \sum _{i=1}^{k}q_{i}d(x_{i},x_{i+1}), \end{aligned}
(4)
for every $$x_{1},x_{2},\ldots ,x_{k+1}\in X$$, where $$q_{1},q_{2},\ldots ,q_{k}$$ are nonnegative constants such that $$q_{1}+q_{2}+\cdots +q_{k}<1$$. Then there exists a unique point $$x\in X$$ such that $$T(x,x,\ldots ,x)=x$$. Moreover if $$x_{1},x_{2},\ldots ,x_{k}$$ are arbitrary points in 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

Let $$(X,d)$$ be a complete metric space and let 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 $$x,y\in X$$,
\begin{aligned} H(Tx,Ty)\le \lambda d(x, y) \end{aligned}
where, $$0 \le \lambda < 1$$. Then T has a fixed point.

After the work of Nadler, several authors proved fixed point results for set-valued mappings (see, e.g. [15, 811, 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

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 $$g(X)$$ 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:
\begin{aligned} H(f(x_{1},x_{2},\ldots ,x_{k}),f(x_{2},x_{3},\ldots ,x_{k+1}))\le \sum _{i=1}^{k}\alpha _{i}d(gx_{i},gx_{i+1}), \end{aligned}
(5)
for all $$x_{1},x_{2},\ldots ,x_{k+1}\in X$$, where $$\alpha _{i}$$ are nonnegative constants such that $$\sum \nolimits _{i=1}^{k}\alpha _{i}<1$$. Then 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.

Let A be any nonempty subset of a metric space $$(X,d)$$. For $$x\in X$$, define
\begin{aligned} d(x,A)=\inf \{d(x,y):y\in A\}. \end{aligned}
Let $$CB(X)$$ denotes the set of all nonempty closed bounded subset of X. For $$A,B\in CB(X)$$, define
\begin{aligned} \delta (A,B)&= \sup \{d(x,B):x\in A\},\\ H(A,B)&= \max \{\delta (A,B),\delta (B,A)\}. \end{aligned}
Then H is a metric on $$CB(X)$$ and called Hausdorff metric.

Remark 1

Let $$(X,d)$$ be a metric space and $$A,B\in CB(X)$$. Then:
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. 2.

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

Let 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
\begin{aligned} H(f(x_{0},x_{1},\ldots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))\le \sum \limits _{i=1}^{k}\alpha _{i}d(x_{i-1},x_{i}), \end{aligned}
(6)
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$.

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

$$f$$ is said to be a set-valued Prešić–Kannan type contraction if there exists nonnegative constant $$a$$ such that, $$ak(k+1)<1$$ and
\begin{aligned} H(f(x_{0},x_{1},\ldots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))\le a\sum \limits _{i=0}^{k}d(x_{i},f(x_{i},x_{i},\ldots ,x_{i})), \end{aligned}
(7)
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$.
$$f$$ is said to be a set-valued Prešić–Reich type contraction if,
\begin{aligned}&H(f(x_{0},x_{1},\dots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))\nonumber \\&\quad \le \sum _{i=1}^{k}\alpha _{i}d(x_{i-1},x_{i})+\sum _{i=0}^{k}\beta _{i}d(x_{i},f(x_{i},x_{i},\ldots ,x_{i})) \end{aligned}
(8)
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$, where $$\alpha _{i},\beta _{i}$$ are nonnegative constants such that
\begin{aligned} \sum \limits _{i=1}^{k}\alpha _{i}+k\sum \limits _{i=0}^{k}\beta _{i}<1. \end{aligned}
Note that the set-valued Prešić–Reich type contraction in case $$\beta _{i}=0$$ for $$i=0,1,\ldots ,k$$ reduces to set-valued Prešić type contraction and in case $$\alpha _{i}=0$$ for $$i=1,2,\ldots ,k$$ and $$\beta _{i}=a$$ for $$i=0,1,\ldots ,k$$ it reduces to set-valued Prešić–Kannan type contraction. Also, set-valued Prešić–Kannan type contraction in case $$k=1$$ reduces to set-valued Kannan type contraction and set-valued Prešić–Reich type contraction reduces to set-valued Reich type contraction.

Definition 1

Let $$X$$ be a nonempty set, $$k$$ a positive integer, $$f:X^{k}\rightarrow CB(X)$$ and $$g:X\rightarrow X$$ be mappings.
1. (a)

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

2. (b)

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

3. (c)

If $$w=gx\in f(x,\ldots ,x)$$, then $$w$$ is called a point of coincidence of f and g.

4. (d)

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

5. (e)

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

6. (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

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 $$g(X)$$ 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:
\begin{aligned}&H(f(x_{0},x_{1},\dots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))\nonumber \\&\quad \le \sum _{i=1}^{k}\alpha _{i}d(gx_{i-1},gx_{i})+\sum _{i=0}^{k}\beta _{i}d(gx_{i},f(x_{i},x_{i},\ldots ,x_{i})) \end{aligned}
(9)
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$, where $$\alpha _{i},\beta _{i}$$ are nonnegative constants such that
\begin{aligned} \sum _{i=1}^{k}\alpha _{i}+k\sum _{i=0}^{k}\beta _{i}<1. \end{aligned}
(10)
Then f and g have a point of coincidence $$v\in X$$.

Proof

Let $$x_{0}\in X$$ be arbitrary. As $$f(x_{0},\ldots ,x_{0})\in CB(X)$$ and $$f(x_{0},\ldots ,x_{0})\subset g(X)$$, let $$y_{1}=gx_{1}\in f(x_{0},\ldots ,x_{0})$$ for some $$x_{1}\in X$$, so by Remark 1 there exists $$y_{2}=gx_{2}\in f(x_{1},\ldots ,x_{1})$$, $$x_{2}\in X$$ such that
\begin{aligned} d(gx_{1},gx_{2})\le H(f(x_{0},\ldots ,x_{0}),f(x_{1},\ldots ,x_{1}))+\theta , \end{aligned}
where $$\theta >0$$ is arbitrary. Similarly, there exists $$y_{3}=gx_{3}\in f(x_{2},\ldots ,x_{2})$$ such that
\begin{aligned} d(gx_{2},gx_{3})\le H(f(x_{1},\ldots ,x_{1}),f(x_{2},\ldots ,x_{2}))+\theta ^{2}. \end{aligned}
Continuing this procedure we obtain $$y_{n+1}=gx_{n+1}\in f(x_{n},\ldots ,x_{n})$$ and
\begin{aligned} d(gx_{n},gx_{n+1})\le H(f(x_{n-1},\ldots ,x_{n-1}),f(x_{n},\ldots ,x_{n}))+\theta ^{n} \end{aligned}
(11)
for all $$n\in \mathbb N$$.
As H is a metric on $$CB(X)$$, for any $$n\in \mathbb N$$ it follows from (11) that
\begin{aligned} d(y_{n},y_{n+1})&= d(gx_{n},gx_{n+1})\\&\le H(f(x_{n-1},\ldots ,x_{n-1}),f(x_{n},\ldots ,x_{n}))+\theta ^{n} \\&\le H(f(x_{n-1},\ldots ,x_{n-1}),f(x_{n-1},\ldots ,x_{n-1},x_{n}))\\&\quad +H(f(x_{n-1},\ldots ,x_{n-1},x_{n}),f(x_{n-1},\ldots ,x_{n-1},x_{n},x_{n}))\\&\quad +\cdots +H(f(x_{n-1},x_{n},\ldots ,x_{n}),f(x_{n},\ldots ,x_{n}))+\theta ^{n}. \end{aligned}
Using (9) in above inequality we obtain
\begin{aligned} d(y_{n},y_{n+1})&\le \alpha _{k}d(gx_{n-1},gx_{n})+\beta _{0}d(gx_{n-1},f(x_{n-1},\ldots ,x_{n-1}))+\cdots \\&\quad +\beta _{k-1}d(gx_{n-1},f(x_{n-1},\ldots ,x_{n-1}))+\beta _{k}d(gx_{n},f(x_{n},\ldots ,x_{n}))\\&\quad +\alpha _{k-1}d(gx_{n-1},gx_{n})+\beta _{0}d(gx_{n-1},f(x_{n-1},\ldots ,x_{n-1}))+\cdots \\&\quad +\beta _{k-1}d(gx_{n},f(x_{n},\ldots ,x_{n}))+\beta _{k}d(gx_{n},f(x_{n},\ldots ,x_{n}))\\&\quad +\cdots +\alpha _{1}d(gx_{n-1},gx_{n})+\beta _{0}d(gx_{n-1},f(x_{n-1},\ldots ,x_{n-1}))\\&\quad +\beta _{1}d(gx_{n},f(x_{n},\ldots ,x_{n}))+\cdots +\beta _{k}d(gx_{n},f(x_{n},\ldots ,x_{n}))+\theta ^{n}, \end{aligned}
as $$y_{n}=gx_{n}\in f(x_{n-1},\ldots ,x_{n-1})$$ for all $$n\in \mathbb N$$, it follows that
\begin{aligned} d(y_{n},y_{n+1})&\le \left[\sum _{i=1}^{k}\alpha _{i}\right]d(y_{n-1},y_{n})+\beta _{0}d(y_{n-1},y_{n}) +\cdots +\beta _{k-1}d(y_{n-1},y_{n})\\&+\beta _{k}d(y_{n},y_{n+1})+\beta _{0}d(y_{n-1},y_{n})+\cdots +\beta _{k-1}d(y_{n},y_{n+1})\\&+\beta _{k}d(y_{n},y_{n+1})+\cdots +\beta _{0}d(y_{n-1},y_{n})+\beta _{1}d(y_{n},y_{n+1})\\&+\cdots +\beta _{k}d(y_{n},y_{n+1})+\theta ^{n}\\&= \left[\sum _{i=1}^{k}\alpha _{i}\right]d(y_{n-1},y_{n})+[k\beta _{0}+(k-1)\beta _{1}+\cdots +\beta _{k-1}]d(y_{n-1},y_{n})\\&+[\beta _{1}+2\beta _{2}+\cdots +k\beta _{k}]d(y_{n},y_{n+1})+\theta ^{n} \end{aligned}
i.e.
\begin{aligned} d(y_{n},y_{n+1})&\le \left[\sum _{i=1}^{k}\alpha _{i}+\sum _{i=0}^{k-1}(k-i)\beta _{i}\right]d(y_{n-1},y_{n})+\left[\sum _{i=1}^{k}i\beta _{i}\right]d(y_{n},y_{n+1})+\theta ^{n}\nonumber \\ d(y_{n},y_{n+1})&\le \frac{\sum _{i=1}^{k}\alpha _{i}+\sum _{i=0}^{k}(k-i)\beta _{i}}{1-\sum _{i=0}^{k}i\beta _{i}} \ d(y_{n-1},y_{n})+\frac{1}{1-\sum _{i=0}^{k}i\beta _{i}} \ \theta ^{n}. \end{aligned}
(12)
For simplicity, set $$A=\sum _{i=1}^{k}\alpha _{i},B=k\sum _{i=0}^{k}\beta _{i},C=\sum _{i=0}^{k}i\beta _{i}$$ and $$\lambda =\frac{A+B-C}{1-C}$$, then in view of (10) we have,
\begin{aligned} A+B=\sum _{i=1}^{k}\alpha _{i}+k\sum _{i=0}^{k}\beta _{i}<1, \quad C<1, \quad \text{ also} C\le B, \end{aligned}
therefore $$0\le \lambda <1$$. As $$\theta$$ was arbitrary, choose $$\theta =\lambda$$, so from (12) it follows that
\begin{aligned} d(y_{n},y_{n+1})\le \lambda d(y_{n-1},y_{n})+\frac{\lambda ^{n}}{1-C} \text{ for} \text{ all} n\in \mathbb N . \end{aligned}
(13)
It follows from successive application of (13) that
\begin{aligned} d(y_{n},y_{n+1})&\le \lambda \left[\lambda d(y_{n-2},y_{n-1})+\frac{\lambda ^{n-1}}{1-C}\right]+\frac{\lambda ^{n}}{1-C}\\&= \lambda ^{2} d(y_{n-2},y_{n-1})+\frac{2\lambda ^{n}}{1-C}\\&\le \lambda ^{2}\left[\lambda d(y_{n-3},y_{n-2})+\frac{\lambda ^{n-2}}{1-C}\right]+\frac{2\lambda ^{n}}{1-C}\\&= \lambda ^{3}d(y_{n-3},y_{n-2})+\frac{3\lambda ^{n}}{1-C}\\&\vdots&\\&\le \lambda ^{n}d(y_{0},y_{1})+\frac{n\lambda ^{n}}{1-C}. \end{aligned}
As $$0\le \lambda <1$$, so $$\sum \nolimits _{n=0}^{\infty }\lambda ^{n}<\infty$$ and $$\sum \nolimits _{n=0}^{\infty }n\lambda ^{n}<\infty$$, we have
\begin{aligned} \sum _{n=0}^{\infty }d(y_{n},y_{n+1})\le d(y_{0},y_{1})\sum _{n=0}^{\infty }\lambda ^{n}+\frac{1}{1-C}\sum _{n=0}^{\infty }n\lambda ^{n}<\infty . \end{aligned}
Therefore $$\{y_{n}\}=\{gx_{n}\}$$ is a Cauchy sequence in $$g(X)$$. As $$g(X)$$ is closed and $$X$$ is complete, there exists $$u,v\in X$$ such that $$v=gu$$ and
\begin{aligned} \lim \limits _{n\rightarrow \infty }d(y_{n},v)=\lim \limits _{n\rightarrow \infty }d(gx_{n},gu)=0. \end{aligned}
(14)
We shall show that u is coincidence point of f and g.
As $$gx_{n+1}=y_{n+1}\in f(x_{n},\ldots ,x_{n})$$ we have
\begin{aligned} d(v,f(u,\ldots ,u))&\le d(v,y_{n+1})+d(y_{n+1},f(u,\ldots ,u))\\&\le d(v,y_{n+1})+H(f(x_{n},\ldots ,x_{n}),f(u,\ldots ,u))\\&\le d(v,y_{n+1})+ H(f(x_{n},\ldots ,x_{n}),f(x_{n},\ldots ,x_{n},u))\\&+H(f(x_{n},\ldots ,x_{n},u),f(x_{n},\ldots ,x_{n},u,u))\\&+\cdots +H(f(x_{n},u,\ldots ,u),f(u,\ldots ,u)), \end{aligned}
using (9) in above inequality we obtain
\begin{aligned} d(v,f(u,\ldots ,u))&\le \left[\sum _{i=1}^{k}\alpha _{i}\right]d(gx_{n},gu)+\left[\sum _{i=0}^{k}(k-i)\beta _{i}\right]d(gx_{n},f(x_{n},\ldots ,x_{n}))\\&+\left[\sum _{i=0}^{k}i\beta _{i}\right]d(gu,f(u,\ldots ,u))+d(v,y_{n+1})\\&= Ad(gx_{n},v)+(B-C)d(gx_{n},f(x_{n},\ldots ,x_{n}))\\&+Cd(v,f(u,\ldots ,u))+d(v,y_{n+1})\\&\le Ad(y_{n},v)+(B-C)d(y_{n},y_{n+1})+Cd(v,f(u,\ldots ,u))\\&+d(v,y_{n+1}) \;(\text{ as} y_{n+1}=gx_{n+1}\in f(x_n,\ldots ,x_n)) \end{aligned}
i.e.
\begin{aligned} d(v,f(u,\ldots ,u))\le \frac{A+B-C}{1-C}d(y_{n},v)+\frac{1+B-C}{1-C}d(v,y_{n+1}). \end{aligned}
In view of (14), from above inequality we obtain $$d(v,f(u,\ldots ,u))=0$$. As $$f(u,\ldots ,u)\in CB(X)$$, we obtain $$v=gu\in f(u,\ldots ,u)$$. Thus $$u$$ is coincidence point and $$v$$ is point of coincidence 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.

If $$v^{\prime }$$ is another point of coincidence with coincidence point $$u^{\prime }$$ of $$f$$ and $$g$$, then $$f(u^{\prime },\ldots ,u^{\prime })=\{gu^{\prime }\}=\{v^{\prime }\}$$. As $$H$$ is metric, we obtain
\begin{aligned} d(v,v^{\prime })&= H(\{v\},\{v^{\prime }\})\\&= H(f(u,\ldots ,u),f(u^{\prime },\ldots ,u^{\prime }))\\&\le H(f(u,\ldots ,u),f(u,\ldots ,u,u^{\prime }))+H(f(u,\ldots ,u,u^{\prime }),f(u,\ldots ,u,u^{\prime },u^{\prime }))\\&+\cdots +H(f(u,u^{\prime },\ldots ,u^{\prime }),f(u^{\prime },\ldots ,u^{\prime })), \end{aligned}
using (9) and the process as used several times before, we obtain
\begin{aligned} d(v,v^{\prime })&\le Ad(gu,gu^{\prime })+(B-C)d(gu,f(u,\ldots ,u))+Cd(gu^{\prime },f(u^{\prime },\ldots ,u^{\prime })) \\ d(v,v^{\prime })&\le Ad(v,v^{\prime })+(B-C)d(v,f(u,\ldots ,u))+Cd(v^{\prime },f(u^{\prime },\ldots ,u^{\prime })),\\ d(v,v^{\prime })&\le \frac{B-C}{1-A}d(v,f(u,\ldots ,u))+\frac{C}{1-A}d(v^{\prime },f(u^{\prime },\ldots ,u^{\prime })), \end{aligned}
where $$A=\sum \nolimits _{i=1}^{k}\alpha _{i},$$$$B=k\sum \nolimits _{i=0}^{k}\beta _{i}$$ and $$C=\sum \nolimits _{i=0}^{k}i\beta _{i}$$.

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.

Suppose $$f$$ and $$g$$ are weakly compatible, then we have
\begin{aligned} g(f(u,\ldots ,u))\subseteq f(gu,\ldots ,gu)=f(v,\ldots ,v) \text{ i.e.} \{gv\}\subseteq f(v,\ldots ,v). \end{aligned}
Therefore $$gv\in f(v,\ldots ,v)$$, which shows that 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

Let $$X=[0,1]$$ and $$d$$ is usual metric on $$X$$, then $$(X,d)$$ is complete metric spaces. For $$k=2$$ ,define $$T:X^{2}\rightarrow X$$ by
\begin{aligned} T(x,y)=\left\{ \begin{array}{l@{\quad }l} 0,&\text{ if} bad hbox\\ \frac{x+y+1}{15},&\text{ otherwise.} \end{array} \right. \end{aligned}
Define $$f:X^{2}\rightarrow CB(X)$$ by
\begin{aligned} f(x,y)=\{T(x,y)\}\cup \{0\}\quad \text{ for} \text{ all} x,y\in [0,1]. \end{aligned}
Note that $$f$$ is not a set-valued Prešić type contraction. For example, let $$x=\frac{9}{10},y=z=1$$ then
\begin{aligned} H(f(x,y),f(y,z))=H\left(f\left(\frac{9}{10},1\right),f(1,1)\right)=\frac{29}{150} \end{aligned}
and
\begin{aligned} \alpha _{1}d(x,y)+\alpha _{2}d(y,z)=\alpha _{1}d\left(\frac{9}{10},1\right)+\alpha _{2}d(1,1)=\frac{1}{10}\alpha _{1}. \end{aligned}
Therefore, we cannot find nonnegative constants $$\alpha _{1},\alpha _{2}$$ such that $$\alpha _{1}+\alpha _{2}<1$$ and
\begin{aligned} H(f(x,y),f(y,z))\le \alpha _{1}d(x,y)+\alpha _{2}d(y,z). \end{aligned}
So $$f$$ is not a set-valued Prešić type contraction.

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

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 $$g(X)$$ 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:
\begin{aligned} H(f(x_{0},x_{1},\dots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))\le \sum _{i=0}^{k}\beta _{i}d(gx_{i},f(x_{i},x_{i},\ldots ,x_{i})) \end{aligned}
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$, where $$\beta _{i}$$ are nonnegative constants such that $$k\sum \nolimits _{i=0}^{k}\beta _{i}<1$$. Then $$f$$ and $$g$$ have a point of coincidence $$v\in X$$.

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

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 $$f(x_{1},x_{2},\ldots ,x_{k})\subset g(X)$$ for all $$x_{1},x_{2},\ldots ,x_{k}\in X$$. Suppose following condition holds:
\begin{aligned} H(f(x_{0},x_{1},\dots ,x_{k-1}),f(x_{1},x_{2},\ldots ,x_{k}))&\le \sum _{i=1}^{k}\alpha _{i}d(gx_{i-1},gx_{i})\nonumber \\&\quad +\sum _{i=0}^{k}\beta _{i}d(gx_{i},f(x_{i},x_{i},\ldots ,x_{i})) \end{aligned}
(15)
for all $$x_{0},x_{1},\ldots ,x_{k}\in X$$, where $$\alpha _{i},\beta _{i}$$ are nonnegative constants such that
\begin{aligned} \sum _{i=1}^{k}\alpha _{i}+k\sum _{i=0}^{k}\beta _{i}<1. \end{aligned}
Suppose there exists $$u\in X$$ such that
\begin{aligned} d(gu,f(u,\ldots ,u))\le d(gx,f(x,\ldots ,x)) \quad \text{ for} \text{ all} x\in X. \end{aligned}
Then $$f$$ and $$g$$ have a point of coincidence $$v\in X$$.

Proof

Let $$G(x)=d(gx,f(x,\ldots ,x))$$ for all $$x\in X$$. Then by assumption we have
\begin{aligned} G(u)\le G(x)\quad \text{ for} \text{ all} x\in X. \end{aligned}
If $$v=gu\in f(u,\ldots ,u)$$ then $$u$$ is a coincidence point and $$v$$ is point of coincidence of $$f$$ and $$g$$. If not then $$G(u)=d(gu,f(u,\ldots ,u))>0$$. As $$f(u,\ldots ,u)\in CB(X)$$ and $$f(u,\ldots ,u)\subset g(X)$$, let $$gz\in f(u,\ldots ,u)$$ be arbitrary. We have from (5) that
\begin{aligned} G(z)&= d(gz,f(z,\ldots ,z))\nonumber \\&\le H(f(u,\ldots ,u),f(z,\ldots ,z))\nonumber \\&\le H(f(u,\ldots ,u),f(u,\ldots ,u,z))+H(f(u,\ldots ,u,z),f(u,\ldots ,u,z,z))\nonumber \\&+\cdots + H(f(u,z,\ldots ,z),f(z,\ldots ,z))\nonumber \\&\le Ad(gu,gz)+(B-C)d(gu,f(u,\ldots ,u))+Cd(gz,f(z,\ldots ,z))\nonumber \\&= Ad(gu,gz)+(B-C)G(u)+CG(z), \end{aligned}
(16)
where $$A=\sum \nolimits _{i=1}^{k}\alpha _{i}$$, $$B=k\sum \nolimits _{i=0}^{k}\beta _{i}$$ and $$C=\sum \nolimits _{i=0}^{k}i\beta _{i}$$.
As inequality (16) is true for all $$gz\in f(u,\ldots ,u)$$, therefore we obtain
\begin{aligned} G(z)\le \frac{A+B-C}{1-C} \ G(u). \end{aligned}
As $$A+B<1$$, we obtain $$G(z)<G(u), z\in X$$, a contradiction. Therefore we must have $$G(u)=0$$ i.e. $$d(gu,f(u,\ldots ,u))=0$$, so $$v=gu\in f(u,\ldots ,u)$$. Thus $$u$$ is a coincidence point and $$v$$ is point of coincidence of $$f$$ and $$g$$.

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.