# On fixed points of α-ψ-contractive multifunctions

## Abstract

Recently Samet, Vetro and Vetro introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems in complete metric spaces. In this paper, we introduce the notion of ${\alpha }_{\ast }\text{-}\psi$-contractive multifunctions and give a fixed point result for these multifunctions. We also obtain a fixed point result for self-maps in complete metric spaces satisfying a contractive condition.

## 1 Introduction

Fixed point theory has many applications in different branches of science. During the last few decades, there has been a lot of activity in this area and several well-known fixed point theorems have been extended by a number of authors in different directions (see, for example, ). Recently Samet, Vetro and Vetro introduced the notion of α-ψ-contractive type mappings . Denote with Ψ the family of nondecreasing functions $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ such that ${\sum }_{n=1}^{\mathrm{\infty }}{\psi }^{n}\left(t\right)<\mathrm{\infty }$ for all $t>0$, where ${\psi }^{n}$ is the n th iterate of ψ. It is known that $\psi \left(t\right) for all $t>0$ and $\psi \in \mathrm{\Psi }$. Let $\left(X,d\right)$ be a metric space, T be a self-map on X, $\psi \in \mathrm{\Psi }$ and $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ be a function. Then T is called an α-ψ-contraction mapping whenever $\alpha \left(x,y\right)d\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right)\right)$ for all $x,y\in X$. Also, we say that T is α-admissible whenever $\alpha \left(x,y\right)\ge 1$ implies $\alpha \left(Tx,Ty\right)\ge 1$. Also, we say that α has the property (B) if $\left\{{x}_{n}\right\}$ is a sequence in X such that $\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$ for all $n\ge 1$ and ${x}_{n}\to x$, then $\alpha \left({x}_{n},x\right)\ge 1$ for all $n\ge 1$. Let $\left(X,d\right)$ be a complete metric space and T be an α-admissible α-ψ-contractive mapping on X. Suppose that there exists ${x}_{0}\in X$ such that $\alpha \left({x}_{0},T{x}_{0}\right)\ge 1$. If T is continuous or T has the property (B), then T has a fixed point (see ; Theorems 2.1 and 2.2). Finally, we say that X has the property (H) whenever for each $x,y\in X$ there exists $z\in X$ such that $\alpha \left(x,z\right)\ge 1$ and $\alpha \left(y,z\right)\ge 1$. If X has the property (H) in Theorems 2.1 and 2.2, then X has a unique fixed point (; Theorem 2.3). It is considerable that the results of Samet et al. generalize similar ordered results in the literature (see the results of the third section in ). The aim of this paper is to introduce the notion of ${\alpha }_{\ast }\text{-}\psi$-contractive multifunctions and give a fixed point result about the multifunctions. Let $\left(X,d\right)$ be a metric space, $T:X\to {2}^{X}$ be a closed-valued multifunction, $\psi \in \mathrm{\Psi }$ and $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ be a function. In this case, we say that T is an ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction whenever ${\alpha }_{\ast }\left(Tx,Ty\right)H\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right)\right)$ for $x,y\in X$, where H is the Hausdorff generalized metric, ${\alpha }_{\ast }\left(A,B\right)=inf\left\{\alpha \left(a,b\right):a\in A,b\in B\right\}$ and ${2}^{X}$ denotes the family of all nonempty subsets of X. Also, we say that T is ${\alpha }_{\ast }$-admissible whenever $\alpha \left(x,y\right)\ge 1$ implies ${\alpha }_{\ast }\left(Tx,Ty\right)\ge 1$.

Example 1.1 Let $X=\left[0,\mathrm{\infty }\right)$, $d\left(x,y\right)=|x-y|$ and $\delta \in \left(0,1\right)$ be a fixed number. Define $T:X\to {2}^{X}$ by $Tx=\left[0,\delta x\right]$ for all $x\in X$ and $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ by $\alpha \left(x,y\right)=1$ whenever $x,y\in \left[0,1\right]$ and $\alpha \left(x,y\right)=0$ whenever $x\notin \left[0,1\right]$ or $y\notin \left[0,1\right]$. Now, we show that T is ${\alpha }_{\ast }$-admissible. If $\alpha \left(x,y\right)\ge 1$, then $x,y\in \left[0,1\right]$ and so Tx and Ty are subsets of $\left[0,1\right]$. Thus, $a,b\in \left[0,1\right]$ for all $a\in Tx$ and $b\in Ty$. Hence, $\alpha \left(a,b\right)=1$ for all $a\in Tx$ and $b\in Ty$. This implies that

${\alpha }_{\ast }\left(Tx,Ty\right)=inf\left\{\alpha \left(a,b\right):a\in Tx,b\in Ty\right\}=1.$

Therefore, T is ${\alpha }_{\ast }$-admissible. Now, we show that T is an ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction, where $\psi \left(t\right)=\delta t$ for all $t\ge 0$. If $x\notin \left[0,\frac{1}{\delta }\right]$ or $y\notin \left[0,\frac{1}{\delta }\right]$, then an easy calculation shows us that ${\alpha }_{\ast }\left(Tx,Ty\right)=0$. If $0\le x,y\le \frac{1}{\delta }$, then ${\alpha }_{\ast }\left(Tx,Ty\right)=1$. By using the definition of the Hausdorff metric, it is easy to see that $H\left(Tx,Ty\right)\le \delta d\left(x,y\right)$ for $x,y\in \left[0,\frac{1}{\delta }\right]$. Thus, ${\alpha }_{\ast }\left(Tx,Ty\right)H\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right)\right)$ for $x,y\in X$. Therefore, T is an ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction.

Let $\left(X,⪯,d\right)$ be an ordered metric space and $A,B\subseteq X$. We say that $A⪯B$ whenever for each $a\in A$ there exists $b\in B$ such that $a⪯b$. Also, we say that $A{⪯}_{r}B$ whenever for each $a\in A$ and $b\in B$ we have $a⪯b$.

## 2 Main results

Now, we are ready to state and prove our main results. In the following result, we use the argument similar to that in the proof of Theorem 3.1 in .

Theorem 2.1 Let$\left(X,d\right)$be a complete metric space, $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$be a function, $\psi \in \mathrm{\Psi }$be a strictly increasing map and T be a closed-valued, ${\alpha }_{\ast }$-admissible and${\alpha }_{\ast }\text{-}\psi$-contractive multifunction on X. Suppose that there exist${x}_{0}\in X$and${x}_{1}\in T{x}_{0}$such that$\alpha \left({x}_{0},{x}_{1}\right)\ge 1$. Assume that if$\left\{{x}_{n}\right\}$is a sequence in X such that$\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$for all n and${x}_{n}\to x$, then$\alpha \left({x}_{n},x\right)\ge 1$for all n. Then T has a fixed point.

Proof If ${x}_{1}={x}_{0}$, then we have nothing to prove. Let ${x}_{1}\ne {x}_{0}$. If ${x}_{1}\in T{x}_{1}$, then ${x}_{1}$ is a fixed point of T. Let ${x}_{1}\notin T{x}_{1}$ and $q>1$ be given. Then

$0

Hence, there exists ${x}_{2}\in T{x}_{1}$ such that

$0

It is clear that ${x}_{2}\ne {x}_{1}$ and $\alpha \left({x}_{1},{x}_{2}\right)\ge 1$. Thus, ${\alpha }_{\ast }\left(T{x}_{1},T{x}_{2}\right)\ge 1$. Now, put ${t}_{0}=d\left({x}_{0},{x}_{1}\right)$. Then, ${t}_{0}>0$ and $d\left({x}_{1},{x}_{2}\right). Since ψ is strictly increasing, $\psi \left(d\left({x}_{1},{x}_{2}\right)\right)<\psi \left(q\psi \left({t}_{0}\right)\right)$. Put ${q}_{1}=\frac{\psi \left(q\psi \left({t}_{0}\right)\right)}{\psi \left(d\left({x}_{1},{x}_{2}\right)\right)}$. Then ${q}_{1}>1$. If ${x}_{2}\in T{x}_{2}$, then ${x}_{2}$ is a fixed point of T. Assume that ${x}_{2}\notin T{x}_{2}$. Then

$0

Hence, there exists ${x}_{3}\in T{x}_{2}$ such that

$0

It is clear that ${x}_{3}\ne {x}_{2}$, $\alpha \left({x}_{2},{x}_{3}\right)\ge 1$ and $\psi \left(d\left({x}_{2},{x}_{3}\right)\right)<{\psi }^{2}\left(q\psi \left({t}_{0}\right)\right)$. Now, put ${q}_{2}=\frac{{\psi }^{2}\left(q\psi \left({t}_{0}\right)\right)}{\psi \left(d\left({x}_{2},{x}_{3}\right)\right)}$. Then ${q}_{2}>1$. If ${x}_{3}\in T{x}_{3}$, then ${x}_{3}$ is a fixed point of T. Assume that ${x}_{3}\notin T{x}_{3}$. Then

$0

Thus, there exists ${x}_{4}\in T{x}_{3}$ such that

$0

By continuing this process, we obtain a sequence $\left\{{x}_{n}\right\}$ in X such that ${x}_{n}\in T{x}_{n-1}$, ${x}_{n}\ne {x}_{n-1}$, $\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$ and $d\left({x}_{n},{x}_{n+1}\right)\le {\psi }^{n-1}\left(q\psi \left({t}_{0}\right)\right)$ for all n. Now, for each $m>n$, we have

$d\left({x}_{n},{x}_{m}\right)\le \sum _{i=n}^{m-1}d\left({x}_{i},{x}_{i+1}\right)\le \sum _{i=n}^{m-1}{\psi }^{i-1}\left(q\psi \left({t}_{0}\right)\right).$

Hence, $\left\{{x}_{n}\right\}$ is a Cauchy sequence in X. Choose ${x}^{\ast }\in X$ such that ${x}_{n}\to {x}^{\ast }$. Since $\alpha \left({x}_{n},{x}^{\ast }\right)\ge 1$ for all n and T is ${\alpha }_{\ast }$-admissible, ${\alpha }_{\ast }\left(T{x}_{n},T{x}^{\ast }\right)\ge 1$ for all n, thus

$\begin{array}{rcl}d\left({x}^{\ast },T{x}^{\ast }\right)& \le & H\left(T{x}^{\ast },T{x}_{n}\right)+d\left({x}_{n+1},{x}^{\ast }\right)\le {\alpha }_{\ast }\left(T{x}_{n},T{x}^{\ast }\right)H\left(T{x}_{n},T{x}^{\ast }\right)+d\left({x}_{n+1},{x}^{\ast }\right)\\ \le & \psi \left(d\left({x}_{n},{x}^{\ast }\right)\right)+d\left({x}_{n+1},{x}^{\ast }\right)\end{array}$

for all n. Therefore, $d\left({x}^{\ast },T{x}^{\ast }\right)=0$ and so ${x}^{\ast }\in T{x}^{\ast }$. □

Example 2.1 Let $X=\left[0,\mathrm{\infty }\right)$ and $d\left(x,y\right)=|x-y|$. Define $T:X\to {2}^{X}$ by $Tx=\left[2x-\frac{3}{2},\mathrm{\infty }\right)$ for all $x>1$, $Tx=\left[0,\frac{x}{2}\right]$ for all $0\le x\le 1$ and $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ by $\alpha \left(x,y\right)=1$ whenever $x,y\in \left[0,1\right]$ and $\alpha \left(x,y\right)=0$ whenever $x\notin \left[0,1\right]$ or $y\notin \left[0,1\right]$. Then it is easy to check that T is an ${\alpha }_{\ast }$-admissible and ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction, where $\psi \left(t\right)=\frac{t}{2}$ for all $t\ge 0$. Put ${x}_{0}=1$ and ${x}_{1}=\frac{1}{2}$. Then $\alpha \left({x}_{0},{x}_{1}\right)\ge 1$. Also, if $\left\{{x}_{n}\right\}$ is a sequence in X such that $\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$ for all n and ${x}_{n}\to x$, then $\alpha \left({x}_{n},x\right)\ge 1$ for all n. Note that T has infinitely many fixed points.

Corollary 2.2 Let$\left(X,⪯,d\right)$be a complete ordered metric space, $\psi \in \mathrm{\Psi }$be a strictly increasing map and T be a closed-valued multifunction on X such that

$H\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right)\right)$

for all$x,y\in X$with$x⪯y$. Suppose that there exists${x}_{0}\in X$and${x}_{1}\in T{x}_{0}$such that${x}_{0}⪯{x}_{1}$. Assume that if$\left\{{x}_{n}\right\}$is a sequence in X such that${x}_{n}⪯{x}_{n+1}$for all n and${x}_{n}\to x$, then${x}_{n}⪯x$for all n. If$x⪯y$implies$Tx{⪯}_{r}Ty$, then T has a fixed point.

Proof Define $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ by $\alpha \left(x,y\right)=1$ whenever $x⪯y$ and $\alpha \left(x,y\right)=0$ whenever $x⋠y$. Since $x⪯y$ implies $Tx{⪯}_{r}Ty$, $\alpha \left(x,y\right)=1$ implies ${\alpha }_{\ast }\left(Tx,Ty\right)=1$. Thus, it is easy to check that T is an ${\alpha }_{\ast }$-admissible and ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction on X. Now, by using Theorem 2.1, T has a fixed point. □

Now, we prove the following result for self-maps.

Theorem 2.3 Let$\left(X,d\right)$be a complete metric space, $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$be a function, $\psi \in \mathrm{\Psi }$and T be a self-map on X such that$\alpha \left(x,y\right)d\left(Tx,Ty\right)\le \psi \left(m\left(x,y\right)\right)$for all$x,y\in X$, where$m\left(x,y\right)=max\left\{d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),\frac{1}{2}\left[d\left(x,Ty\right)+d\left(y,Tx\right)\right]\right\}$. Suppose that T is α-admissible and there exists${x}_{0}\in X$such that$\alpha \left({x}_{0},T{x}_{0}\right)\ge 1$. Assume that if$\left\{{x}_{n}\right\}$is a sequence in X such that$\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$for all n and${x}_{n}\to x$, then$\alpha \left({x}_{n},x\right)\ge 1$for all n. Then T has a fixed point.

Proof Take ${x}_{0}\in X$ such that $\alpha \left({x}_{0},T{x}_{0}\right)\ge 1$ and define the sequence $\left\{{x}_{n}\right\}$ in X by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$. If ${x}_{n}={x}_{n+1}$ for some n, then ${x}^{\ast }={x}_{n}$ is a fixed point of T. Assume that ${x}_{n}\ne {x}_{n+1}$ for all n. Since T is α-admissible, it is easy to check that $\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$ for all natural numbers n. Thus, for each natural number n, we have

$\begin{array}{rcl}d\left({x}_{n},{x}_{n+1}\right)& =& d\left(T{x}_{n-1},T{x}_{n}\right)\le \alpha \left({x}_{n-1},{x}_{n}\right)d\left(T{x}_{n-1},T{x}_{n}\right)\\ \le & \psi \left(max\left\{d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}_{n-1},{x}_{n}\right),\frac{1}{2}\left[d\left({x}_{n},{x}_{n}\right)+d\left({x}_{n-1},{x}_{n+1}\right)\right]\right\}\right)\\ \le & \psi \left(max\left\{d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right),\frac{1}{2}\left[d\left({x}_{n},{x}_{n-1}\right)+d\left({x}_{n},{x}_{n+1}\right)\right]\right\}\right)\\ =& \psi \left(max\left\{d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right)\right\}\right).\end{array}$

If $max\left\{d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n+1}\right)\right\}=d\left({x}_{n},{x}_{n+1}\right)$, then

$d\left({x}_{n+1},{x}_{n}\right)\le \psi \left(d\left({x}_{n},{x}_{n+1}\right)\right)

which is contradiction. Thus, $max\left\{d\left({x}_{n},{x}_{n-1}\right),d\left({x}_{n},{x}_{n-1}\right)\right\}=d\left({x}_{n},{x}_{n+1}\right)$ for all n. Hence, $d\left({x}_{n+1},{x}_{n}\right)\le \psi \left(d\left({x}_{n},{x}_{n-1}\right)\right)$ and so $d\left({x}_{n+1},{x}_{n}\right)\le {\psi }^{n}\left(d\left({x}_{1},{x}_{0}\right)\right)$ for all n. It is easy to check that $\left\{{x}_{n}\right\}$ is a Cauchy sequence. Thus, there exists ${x}^{\ast }\in X$ such that ${x}_{n}\to {x}^{\ast }$. By using the assumption, we have $\alpha \left({x}_{n},{x}^{\ast }\right)\ge 1$ for all n. Thus,

$\begin{array}{rcl}d\left(T{x}^{\ast },{x}^{\ast }\right)& \le & d\left(T{x}^{\ast },T{x}_{n}\right)+d\left({x}_{n+1},{x}^{\ast }\right)\le \alpha \left({x}_{n},{x}^{\ast }\right)d\left(T{x}^{\ast },T{x}_{n}\right)+d\left({x}_{n+1},{x}^{\ast }\right)\\ \le & \psi \left(max\left\{d\left({x}_{n},{x}^{\ast }\right),d\left({x}_{n},{x}_{n+1}\right),d\left({x}^{\ast },T{x}^{\ast }\right),\\ \frac{1}{2}\left[d\left({x}_{n},T{x}^{\ast }\right)+d\left({x}^{\ast },{x}_{n+1}\right)\right]\right\}\right)+d\left({x}_{n+1},{x}^{\ast }\right)\\ \le & \psi \left(d\left({x}^{\ast },T{x}^{\ast }\right)\right)+d\left({x}_{n+1},{x}^{\ast }\right)\end{array}$

for sufficiently large n. Hence, $d\left(T{x}^{\ast },{x}^{\ast }\right)=0$ and so $T{x}^{\ast }={x}^{\ast }$. □

Example 2.2 Let $X=\left[0,\mathrm{\infty }\right)$ and $d\left(x,y\right)=|x-y|$. Define the self-map T on X by $Tx=2x-\frac{5}{3}$ for $x>1$, $Tx=\frac{x}{3}$ for $0\le x\le 1$ and $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$ by $\alpha \left(x,y\right)=1$ whenever $x,y\in \left[0,1\right]$ and $\alpha \left(x,y\right)=0$ whenever $x\notin \left[0,1\right]$ or $y\notin \left[0,1\right]$. Then it is easy to check that T is α-admissible and $\alpha \left(x,y\right)d\left(Tx,Ty\right)\le \psi \left(m\left(x,y\right)\right)$ for all $x,y\in X$, where $\psi \left(t\right)=\frac{t}{3}$ for all $t\ge 0$. Also, $\alpha \left(1,T1\right)=1$ and if $\left\{{x}_{n}\right\}$ is a sequence in X such that $\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$ for all n and ${x}_{n}\to x$, then $\alpha \left({x}_{n},x\right)\ge 1$ for all n. Note that, T has two fixed points.

Corollary 2.4 Let$\left(X,⪯,d\right)$be a complete ordered metric space, $\psi \in \mathrm{\Psi }$and T be a self-map on X such that$d\left(Tx,Ty\right)\le \psi \left(m\left(x,y\right)\right)$for all$x,y\in X$with$x⪯y$. Suppose that there exists${x}_{0}\in X$such that${x}_{0}⪯T{x}_{0}$. If$\left\{{x}_{n}\right\}$is a sequence in X such that${x}_{n}⪯{x}_{n+1}$for all n and${x}_{n}\to x$, then${x}_{n}⪯x$for all n. If$x⪯y$implies$Tx⪯Ty$, then T has a fixed point.

If we substitute a partial metric ρ for the metric d in Theorem 2.3, it is easy to check that a similar result holds for the partial metric case as follows.

Theorem 2.5 Let$\left(X,\rho \right)$be a complete partial metric space, $\alpha :X×X\to \left[0,\mathrm{\infty }\right)$be a function, $\psi \in \mathrm{\Psi }$and T be a self-map on X such that$\alpha \left(x,y\right)\rho \left(Tx,Ty\right)\le \psi \left(m\left(x,y\right)\right)$for all$x,y\in X$, where$m\left(x,y\right)=max\left\{\rho \left(x,y\right),\rho \left(x,Tx\right),\rho \left(y,Ty\right),\frac{1}{2}\left[\rho \left(x,Ty\right)+\rho \left(y,Tx\right)\right]\right\}$. Suppose that T is α-admissible and there exists${x}_{0}\in X$such that$\alpha \left({x}_{0},T{x}_{0}\right)\ge 1$. Assume that if$\left\{{x}_{n}\right\}$is a sequence in X such that$\alpha \left({x}_{n},{x}_{n+1}\right)\ge 1$for all n and${x}_{n}\to x$, then$\alpha \left({x}_{n},x\right)\ge 1$for all n. Then T has a fixed point.

## References

1. Abbas M, Nazir T, Radenovic S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038

2. Aleomraninejad SMA, Rezapour S, Shahzad N: On generalizations of the Suzuki’s method. Appl. Math. Lett. 2011, 24: 1037–1040. 10.1016/j.aml.2010.12.025

3. Aleomraninejad SMA, Rezapour S, Shahzad N: Fixed points of hemi-convex multifunctions. Topol. Methods Nonlinear Anal. 2011, 37(2):383–389.

4. Aleomraninejad SMA, Rezapour S, Shahzad N: Some fixed point results on a metric space with a graph. Topol. Appl. 2012, 159: 659–663. 10.1016/j.topol.2011.10.013

5. Alghamdi MA, Alnafei SH, Radenovic S, Shahzad N: Fixed point theorems for convex contraction mappings on cone metric spaces. Math. Comput. Model. 2011, 54: 2020–2026. 10.1016/j.mcm.2011.05.010

6. Altun I, Damjanovic B, Djoric D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett. 2010, 23: 310–316. 10.1016/j.aml.2009.09.016

7. Aydi H, Damjanovic B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059

8. Aydi H, Nashine HK, Samet B, Yazidi H: Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 2011, 74: 6814–6825. 10.1016/j.na.2011.07.006

9. Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 7347–7355. 10.1016/j.na.2011.07.053

10. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4889–4897. 10.1016/j.na.2011.03.032

11. Berinde V, Vetro F: Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory Appl. 2012., 2012: Article ID 105

12. Borcut M, Berinde V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012. doi:10.1016/j.amc.2011.11.049

13. Derafshpour M, Rezapour S, Shahzad N: Best proximity points of cyclic φ -contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 2011, 37: 193–202.

14. Di Bari C, Vetro P: Fixed points for weak φ -contractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci. 2011, 1: 4–9.

15. Di Bari C, Vetro P: φ -pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 279–285. doi:10.1007/s12215–008–0020–9 10.1007/s12215-008-0020-9

16. Di Bari C, Vetro P: Weakly φ -pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 125–132. doi:10.1007/s12215–009–0012–4 10.1007/s12215-009-0012-4

17. Ding HS, Lu L: Coupled fixed point theorems in partially ordered cone metric spaces. Filomat 2011, 25(2):137–149. 10.2298/FIL1102137D

18. Du W-S: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasi-ordered metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 876372

19. Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052

20. Hu X-Q, Ma X-Y: Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces. Nonlinear Anal. 2011, 74: 6451–6458. 10.1016/j.na.2011.06.028

21. Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 2011, 74: 768–774. 10.1016/j.na.2010.09.025

22. Lazar VL: Fixed point theory for multivalued ϕ -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 50

23. Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055

24. Nashine HK, Samet B, Vetro C: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math. Comput. Model. 2011, 54: 712–720. 10.1016/j.mcm.2011.03.014

25. Nashine HK, Samet B: Fixed point results for mappings satisfying $\left(\psi ,\phi \right)$-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024

26. Nashine HK, Shatanawi W: Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. 2011, 62: 1984–1993. 10.1016/j.camwa.2011.06.042

27. Rezapour S, Amiri P: Some fixed point results for multivalued operators in generalized metric spaces. Comput. Math. Appl. 2011, 61: 2661–2666. 10.1016/j.camwa.2011.03.014

28. Rezapour S, Haghi RH: Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’. J. Math. Anal. Appl. 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049

29. Rezapour S, Haghi RH, Rhoades BE: Some results about T -stability and almost T -stability. Fixed Point Theory 2011, 12(1):179–186.

30. Rezapour S, Haghi RH, Shahzad N: Some notes on fixed points of quasi-contraction maps. Appl. Math. Lett. 2010, 23: 498–502. 10.1016/j.aml.2010.01.003

31. Rus MD: Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric. Nonlinear Anal. 2011, 74: 1804–1813. 10.1016/j.na.2010.10.053

32. Samet B, Vetro C: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4260–4268. 10.1016/j.na.2011.04.007

33. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014

34. Shatanawi W, Samet B: On $\left(\psi ,\phi \right)$-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 2011, 62: 3204–3214. 10.1016/j.camwa.2011.08.033

35. Vetro F: On approximating curves associated with nonexpansive mappings. Carpath. J. Math. 2011, 27: 142–147.

36. Vetro P: Common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2007, 56: 464–468. doi:10.1007/BF03032097 10.1007/BF03032097

37. Zhang X: Fixed point theorems of multivalued monotone mappings in ordered metric spaces. Appl. Math. Lett. 2010, 23: 235–240. 10.1016/j.aml.2009.06.011

38. Zhilong L: Fixed point theorems in partially ordered complete metric spaces. Math. Comput. Model. 2011, 54: 69–72. 10.1016/j.mcm.2011.01.035

## Acknowledgements

The authors are grateful to the reviewers for their useful comments.

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### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Asl, J.H., Rezapour, S. & Shahzad, N. On fixed points of α-ψ-contractive multifunctions. Fixed Point Theory Appl 2012, 212 (2012). https://doi.org/10.1186/1687-1812-2012-212

• ${\alpha }_{\ast }\text{-}\psi$-contractive multifunction 