Common \(f\)-endpoint for hybrid generalized multi-valued contraction mappings

Original Paper

Abstract

In this paper, we introduce common approximate \(f\)-endpoint property for multi-valued mapping \(T:X\rightarrow P_{cl,bd}(X)\) to obtain a necessary and sufficient condition for existence of a unique common \(f\)-endpoint for such multi-valued mappings. Our results extend and unify comparable results in existing literature (see for example, Kamran in Nonlinear Anal (TMA) 67:2289–2296, 2007; Sintunavarat and Kumam in Appl Math Lett 25:52–57, 2012 and references mentioned therein). We also provide an example to support our results.

Keywords

Multi-valued mapping Approximate \(f\)-endpoint property  Common endpoint Hybrid generalized contraction  Hausdorff metric 

Mathematics Subject Classification (2000)

47H10 54C60 

1 Introduction and preliminaries

Let \((X,d)\) be a metric space and \(P(X)\) denotes the class of all subsets of \(X\). Define
$$\begin{aligned} P_{p}(X)=\{A\subseteq X:A\ne \emptyset \,\,\mathrm{has\ a\ property}\ p\}. \end{aligned}$$
Thus \(P_{bd}(X),P_{cl}(X),P_{cp}(X)\) and \(P_{cl,bd}(X)\) denote the classes of bounded, closed, compact and closed bounded subsets of \(X\), respectively.
Let \(T:X\longrightarrow P_{f}(X)\) be a multi-valued mapping on \(X\) and \(f\) be a single valued self map on \(X\). A point \(x\in X\) is said to be:
  1. (i)

    a fixed point of \(T\) if \(x\in Tx\).

     
  2. (ii)

    an endpoint of \(T\) if \(T(x)=\{x\}\).

     
  3. (iii)

    a \(f\)-fixed point of \(T\) or coincidence point of hybrid pair \(\{f,T\}\) if \(fx\in Tx\).

     
  4. (iv)

    an \(f\)-endpoint of \(T\) or coincidence endpoint of hybrid pair \(\{f,T\}\) if \(Tx=\{fx\}\).

     
\(Fix(T),End(T),Fix_{f}(T)\) and \(End_{f}(T)\) denote set of all fixed points of \(T,\) set of all end points of \(T,\) set of all \(f\)-fixed points of \(T\) and set of all \(f\)-endpoints of \(T\) respectively. Obviously, \(End(T)\subseteq Fix(T)\) and \(End_{f}(T)\subseteq Fix_{f}(T)\).
For \(A,B\in P_{cl,bd}(X),H(A,B)\) denotes the distance between \(A\) and \(B\) in the Hausdorff metric induced by \(d\) on \(P_{cl,bd}(X),\) that is,
$$\begin{aligned} H(A,B):=\max \left\{ \sup _{x\in B}d(x,A),\sup _{x\in A}d(x,B)\right\} , \end{aligned}$$
(1)
where \(d(x,A)=\inf \{d(x,a):a\in A\}\) is the distance of the point \(x\) from the set \(A\).
A mapping \(T:X\rightarrow P_{cl,bd}(X)\) is said to be contraction if for some \(0\le \alpha <1\), we have
$$\begin{aligned} H(Tx,Ty)\le \alpha d(x,y), \end{aligned}$$
(2)
for all \(x,y\in X\).

Banach’s contraction principle [1] was extended to multi-valued contractions by Nadler [16] in the following theorem.

Theorem 1

Let \((X,d)\) be a complete metric space and \(T:X\rightarrow P_{cl,bd}(X)\) be a contraction mapping. Then there exists a point \(x\in X\) such that \(x\in Tx\).

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [12]. Later, an interesting and rich fixed point theory for such maps was developed (see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). The theory of multivalued maps has application in control theory, convex optimization, differential equations and economics. Kamran [9] further extended this notion for a hybrid pair of \(f:X\longrightarrow X\) and \(T:X\longrightarrow P_{cl,bd}(X)\) and obtained results regarding coincidence points of the hybrid pair \(\{f,T\}.\) Recently, Sintunavarat and Kumam [30] extended the notion of a generalized multi-valued contraction mappings given in [9] to a generalized multi-valued \((f,\theta ,\beta )\)-weak contraction mapping. They also obtained the common fixed point results for such mappings.

Theorem 2

([30], Theorem 4.5) Let \((X,d)\) be a metric space, \(f:X\rightarrow X\) be a single-valued mapping and \(T:X\rightarrow P_{cl,bd}(X)\) be a generalized multi-valued \((f,\alpha ,\beta )\)-weak contraction mapping. If \(fX\) is complete subspace of \(X\) and \(TX\subset fX\), then \(f\) and \(T\) have a coincidence point \(u\in X\). Moreover, if \(ffu=fu\), then \(f\) and \(T\) have a common fixed point.

Above Theorem 3 extended, improved, unified and generalized several fixed point results given in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Moreover, it also provided a partial answer to the problem posed by Reich [21].

Sintunavarat and Kumam [31] introduced the notion of a hybrid generalized multi-valued contraction mapping and established a common fixed point result for this class of mappings as follows.

Definition 1

Let \((X, d)\) be a metric space, \(f : X \rightarrow X\) be a single-valued mapping and \(T : X \rightarrow P_{cl,bd}(X)\) be a multi-valued mapping. \(T\) is said to be a hybrid generalized multi-valued contraction mapping if and only if there exist two functions \(\phi : [0,+\infty )\rightarrow [0, 1)\) satisfying \(\limsup _{r\rightarrow t^+}\phi (r)<1\) for every \(t\in [0,+\infty )\) and \(\psi : [0,+\infty ) \rightarrow [0,+\infty )\), such that
$$\begin{aligned} H(Tx,Ty)\le \phi (M(x,y))M(x,y)+\psi (N(x,y))N(x,y) \end{aligned}$$
(3)
for all \(x,y\in X\), where
$$\begin{aligned} M(x,y)=\max \{d(fx,fy),d(fy,Tx)\} \end{aligned}$$
(4)
and
$$\begin{aligned} N(x,y)=\min \{d(fx,fy),d(fy,Ty),d(fx,Tx),d(fy,Tx),d(fx,Ty)\}. \end{aligned}$$
(5)
 

 

Theorem 3

([31], Theorem 3.4) Let \((X,d)\) be a metric space, \(f:X\rightarrow X\) be a single-valued mapping and \(T:X\rightarrow P_{cl,bd}(X) \) be a a hybrid generalized multi-valued contraction mapping. If \(fX\) is complete subspace of \(X\) and \(TX\subset fX\), then \(f\) and \(T\) have a coincidence point \(z\in X\). Moreover, if \(ffz=fz\), then \(f\) and \(T\) have a common fixed point.

We say that \(T\) has the approximation endpoint property [15], if
$$\begin{aligned} \inf _{x\in X}\sup _{y\in Tx}d(x,y)=0. \end{aligned}$$
(6)
Equivalently, if there exists a sequence \(\{x_{n}\}\) such that \(H(\{x_{n}\},T\{x_{n}\})\rightarrow 0\). Clearly, a single-valued mapping \(T\) has the approximate endpoint property if and only if \(T\) has the approximate fixed point property, i.e.,
$$\begin{aligned} \inf _{x\in X}d(x,Tx)=0. \end{aligned}$$
(7)
Many authors have studied the existence and uniqueness of endpoints and fixed points for multi-valued mappings (see, for example [28, 29, 30] and the references therein). The purpose of this paper is to define the hybrid generalized approximation multi-valued contraction mapping which is more general than mappings existing in comparable literature. We give some properties of such mappings and also establish the common fixed point theorem.

2 Main result

In this section we discuss our main results. The following definition play a crucial role throughout this work.

Definition 2

Let \(f:X\rightarrow X\) be a single-valued and \(T,S:X\rightarrow P_{cl,bd}(X)\) be two multi-valued mapping. We say that \(T,S:X\rightarrow P_{cl,bd}(X)\) have the approximate common \(f\)-endpoint property if there exists a sequence \(\{x_n\}\subset X\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }H(\{fx_n\},Tx_n)=0 \quad \mathrm{and} \quad \lim _{n\rightarrow \infty }H(\{fx_n\},Sx_n)=0. \end{aligned}$$
(8)
Clearly, two single-valued mappings \(T,S\) have the approximate common \(f\)-endpoint property if and only if \(T,S\) have the approximate common \(f\)-fixed point property, i.e., there exists a sequence \(\{x_n\}\subset X\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }d(fx_n,Tx_n)=0 \quad \mathrm{and} \quad \lim _{n\rightarrow \infty }d(fx_n,Sx_n)=0. \end{aligned}$$
(9)

 

Theorem 4

Let \((X, d)\) be a metric space, \(f: X\rightarrow X\) be a single-valued mapping and \(T,S : X\rightarrow P_{cl,bd}(X)\) be two multi-valued mappings such that there exist two non-decreasing functions \(\phi ,\psi : [0,+\infty )\rightarrow [0, +\infty )\) and \(L \ge 1\) such that for every \(t,s\ge 0\),
$$\begin{aligned} 0\le \phi (t)+\psi (s)\le \frac{L-1}{2L+1}. \end{aligned}$$
(10)
Also,
$$\begin{aligned} H(Tx,Sy)\le \phi (M_{T,S}(x,y))M_{T,S}(x,y)+ \psi (N_{T,S}(x,y))N_{T,S}(x,y) \end{aligned}$$
(11)
for all \(x,y\in X\), where
$$\begin{aligned} M_{T,S}(x,y)=\max \{d(fx,fy),d(fy,Tx),d(fx,Sy)\} \end{aligned}$$
(12)
and
$$\begin{aligned} N_{T,S}(x,y)=\max \{d(fx,fy),d(fy,Ty),d(fx,Sx),d(fy,Tx),d(fx,Sy)\}. \end{aligned}$$
(13)
If \(fX\) is complete subspace of \(X\), then \(T,S\) have the common approximate \(f\)-endpoint property if and only if \(T\) and \(S\) have a common \(f\)-endpoint. Moreover, \(Fix_f(T)=End_f(T)=End_f(S)=Fix_f(S)\).

 

Proof

It is clear that if \(T\) and \(S\) have common \(f\)-endpoint, then \(T\) and \(S\) have the approximate common \(f\)-endpoint property. Conversely, suppose that \(T\) and \(S\) have the approximate common \(f\)-endpoint property; then there exists a sequence \(\{x_n\}\subset X\) such that \({\lim _n}H(\{fx_n\},Tx_n)=0\) and \({\lim _n}H(\{fx_n\},Sx_n)=0\). For all \(m,n\in N\) we have
$$\begin{aligned}&d(fx_n,fx_m)\le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+H(Tx_n,Sx_m)\nonumber \\&\quad \le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+\phi (M_{T,S}(x_n,x_m))M_{T,S}(x_n,x_m)\nonumber \\&\qquad +\psi (N_{T,S}(x_n,x_m))N_{T,S}(x_n,x_m)\nonumber \\&\quad \le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+\phi (M_{T,S}(x_n,x_m))\nonumber \\&\qquad \times \max \{d(fx_n,fx_m),d(fx_m,Tx_n),d(fx_n,Sx_m)\} +\psi (N_{T,S}(x_n,x_m))\nonumber \\&\qquad \times \max \{d(fx_n,fx_m),d(fx_m,Tx_m),d(fx_n,Sx_n),d(fx_m,Tx_n),d(fx_n,Sx_m)\}\nonumber \\&\quad \le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+\phi (M_{T,S}(x_n,x_m))\nonumber \\&\qquad \times \max \{d(fx_n,fx_m),H(\{fx_m\},Tx_n),H(\{fx_n\},Sx_m)\}+\psi (N_{T,S}(x_n,x_m))\nonumber \\&\qquad \times \max \{d(fx_n,fx_m),H(\{fx_m\},Tx_m),H(\{fx_n\},Sx_n), H(\{fx_m\},Tx_n), x_n\}, H(\{fx_m\}, Sx_m)\}\nonumber \\&\quad \le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+\phi (M_{T,S}(x_n,x_m))\nonumber \\&\qquad \times (2d(fx_n,fx_m)+H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m))+\psi (N_{T,S}(x_n,x_m))\nonumber \\&\qquad \times \max \{d(fx_n,fx_m),H(\{fx_m\},Tx_m), H(\{fx_n\},Sx_n),d(fx_m,fx_n)\nonumber \\&\quad \quad +H(\{fx_n\},Tx_n), d(fx_m,fx_n)+H(\{fx_m\},Sx_m)\} \end{aligned}$$
(14)
Therefore,
$$\begin{aligned} d(fx_n,fx_m)&\le H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)+\phi (M_{T,S}(x_m,x_n))\\&\times (2d(fx_n,fx_m)+H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m))\\&+\psi (N_{T,S}(x_n,x_m))(2d(fx_n,fx_m)+H(\{fx_n\},Tx_n)+H(\{fx_m\},Sx_m)\\&+H(\{fx_m\},Tx_m)+H(\{fx_n\},Sx_n)). \end{aligned}$$
Taking \(\alpha _n=H(\{fx_n\},Tx_n)\) and \(\beta _n=H(\{fx_n\},Sx_n)\). Thus,
$$\begin{aligned} d(fx_n,fx_m)&\le \alpha _n+\beta _m+\phi (M_{T,S}(x_m,x_n))\\&\times (2d(fx_n,fx_m)+\alpha _n+\beta _m)\\&+\psi (N_{T,S}(x_n,x_m))(2d(fx_n,fx_m)+\alpha _n+\beta _m+\alpha _m+\beta _n). \end{aligned}$$
So,
$$\begin{aligned}&d(fx_n,fx_m)-2\phi (M_{T,S}(x_m,x_n))d(fx_n,fx_m)\nonumber \\&\qquad -2\psi (N_{T,S}(x_n,x_m))d(fx_n,fx_m)\nonumber \\&\quad \le \alpha _n+\beta _m+\alpha _n(\phi (M_{T,S}(x_n,x_m))) +\beta _m(\phi (M_{T,S}(x_n,x_m))\nonumber \\&\qquad +\alpha _n\psi (N_{T,S}(x_m,x_n))+ \beta _m\psi (N_{T,S}(x_n,x_m)))+\alpha _m\psi (N_{T,S}(x_m,x_n))\nonumber \\&\qquad +\beta _n\psi (N_{T,S}(x_n,x_m)) \end{aligned}$$
(15)
It means that
$$\begin{aligned} d(fx_n,fx_m)&\le \left[\frac{1+\phi (M_{T,S}(x_n,x_m))+ \psi (N_{T,S}(x_n,x_m))}{1-2\phi (M_{T,S}(x_n,x_m)) -2\psi (N_{T,S}(x_n,x_m))}\right]\alpha _n\nonumber \\&+\left[\frac{1+\phi (M_{T,S}(x_n,x_m))+ \psi (N_{T,S}(x_n,x_m))}{1-2\phi (M_{T,S}(x_n,x_m)) -2\psi (N_{T,S}(x_n,x_m))}\right]\beta _n\nonumber \\&+[\alpha _m+\beta _n]\left[\frac{\psi (N_{T,S}(x_n,x_m))}{1-2\phi (M_{T,S}(x_n,x_m))-2\psi (N_{T,S}(x_n,x_m))}\right]. \end{aligned}$$
(16)
Since (10) holds thus for each \(t,s\ge 0\)
$$\begin{aligned} \frac{1+\phi (t)+\psi (s)}{1-2\phi (t)-2\psi (s)}\le L. \end{aligned}$$
(17)
Therefore
$$\begin{aligned} d(fx_n,fx_m)\le L\alpha _n+L\beta _n+L[\alpha _m+\beta _n]. \end{aligned}$$
(18)
Thus on taking limit on both side of (18) we deduce \(\{fx_n\}\) is a Cauchy sequence and \(fX\) is complete, so \(\{fx_n\}\) is convergent to \(fu\) for some \(u\in X\). Now we claim that \(Tu=Su=\{fu\}\). Suppose that \(H(\{fu\},Tu)\ne 0\). We have
$$\begin{aligned}&H(\{fu\},Tu)-d(fu,fx_n)-H(\{fx_n\},Sx_n) \nonumber \\&\quad \le H(Sx_n,Tu)\le \phi (M_{T,S}(u,x_n))M_{T,S}(u,x_n)+\psi (N_{T,S}(u,x_n))N_{T,S}(u,x_n).\qquad \end{aligned}$$
(19)
Since
$$\begin{aligned} N_{T,S}(u,x_n)\!=\!\max \{d(fu,fx_n),d(fx_n,Tx_n),d(fu,Su),d(fx_n,Tu),d(fu,Sx_n)\}\quad \quad \end{aligned}$$
(20)
and
$$\begin{aligned} M_{T,S}(u,x_n)=\max \{d(fu,fx_n),d(fx_n,Tu),d(fu,Sx_n)\} \end{aligned}$$
(21)
there exists \(N_1>0\) such that for each \(n\ge N_1\)
$$\begin{aligned} N_{T,S}(u,x_n)\le \max \{H(\{fu\},Tu),H(\{fu\},Su)\} \end{aligned}$$
(22)
there exists \(N_2>0\) such that for each \(n\ge N_2\)
$$\begin{aligned} M_{T,S}(u,x_n)\le H(\{fu\},Tu). \end{aligned}$$
(23)
Hence by taking \(N=\max \{N_1,N_2\}\), for each \(n\ge N\), (22) and (23) hold.
Since \(\phi \) and \(\psi \) are non-decreasing, by applying (22) and (23) and using (19) we have
$$\begin{aligned}&H(\{fu\},Tu)\le \phi (H(\{fu\},Tu))H(\{fu\},Tu)\nonumber \\&\qquad +\psi (\max \{H(\{fu\},Tu),H(\{fu\},Su)\})\max \{H(\{fu\},Tu),H(\{fu\},Su)\}. \end{aligned}$$
(24)
Now if \(H(\{fu\},Tu)>H(\{fu\},Su)\) then we have
$$\begin{aligned} H(\{fu\},Tu)&\le \phi (H(\{fu\},Tu))H(\{fu\},Tu)\nonumber \\&+\psi (H(\{fu\},Tu))H(\{fu\},Tu)\nonumber \\&< H(\{fu\},Tu) \end{aligned}$$
and this is a contradiction. Also, by similar argument we have
$$\begin{aligned}&H(\{fu\},Su)-d(fu,fx_n)-H(\{fx_n\},Tx_n)\\&\quad \le H(Tx_n,Su)\le \phi (M_{T,S}(x_n,u))M_{T,S}(x_n,u)\psi (N_{T,S}(x_n,u))N_{T,S}(x_n,u). \end{aligned}$$
By assuming \(H(\{fu\},Tu)\le H(\{fu\},Su)\) and Repeating the above proof for \(S\), we obtain another contradiction. Thus \(H(\{fu\},Tu)=0\) and \(H(\{fu\},Su)=0\). It means that, \(Tu=Su=\{fu\}\). Now suppose that \(fz\in Fix_f(T)\backslash End_f(T)\), then \(fz\in Tz\) and \(Tz\ne \{fz\}\). Thus
$$\begin{aligned} H(Tz,\{fu\})= H(Tz,Su)\le \phi (M_{T,S}(z,u))M_{T,S}(z,u)+\phi (N_{T,S}(z,u))N_{T,S}(z,u). \end{aligned}$$
Since \(M_{T,S}(z,u)=N_{T,S}(z,u)=d(fz,fu)\) hence \(d(fz,fu)\le H(Tz,\{fu\})<d(fz,fu)\) and this is a contradiction. Therefore, \(Fix_f(T)=End_f(T)=End_f(S)=Fix_f(S)\).

 

Example 1

\(X=\{0,\frac{1}{4},\frac{1}{3}\}\) endowed with Euclidean space and \( f:X\rightarrow X\) defined by
$$\begin{aligned} f(0)=\frac{1}{3},\quad f\left(\frac{1}{3}\right)=\frac{1}{4},\quad f\left(\frac{1}{4}\right)=0 \end{aligned}$$
(25)
\(T:X\rightarrow 2^{X}\) and \(S:X\rightarrow 2^{X}\) defined by
$$\begin{aligned} T(0)=\left\{ \frac{1}{3}\right\} ,\quad T\left(\frac{1}{3}\right)=0,\quad T\left(\frac{1}{4}\right)=\frac{1}{3}\end{aligned}$$
(26)
$$\begin{aligned} S(0)=\left\{ \frac{1}{3}\right\} ,\quad S\left(\frac{1}{3}\right)=\frac{1}{3},\quad S\left(\frac{1}{4}\right)=\frac{1}{4}. \end{aligned}$$
(27)
Since \(T(0)=\{f0\}\) and \(S(0)=\{f0\}\) thus \(0\) is an \(f\)-endpoint. If we define
$$\begin{aligned} \phi (t)&= {\left\{ \begin{array}{ll} \frac{4}{3}t&0\le t\le \frac{1}{4} \\ \frac{17}{48}&t>\frac{1}{4} \end{array}\right.}\end{aligned}$$
(28)
$$\begin{aligned} \psi (t)&= {\left\{ \begin{array}{ll} \frac{7}{4}t&0\le t\le \frac{1}{3} \\ \frac{5}{8}&t>\frac{1}{3} \end{array}\right.} \end{aligned}$$
(29)
It can easily seen that \(T,S\) satisfies in all conditions of Theorem 4 by taking \(L=72\).

 

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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  1. 1.The University of Birmingham School of MathematicsBirminghamUK
  2. 2.Department of MathematicsLahore University of Management SciencesLahorePakistan
  3. 3.Department of MathematicsIslamic Azad UniversityArakIran

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