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# Fixed point theorems for *α*-*ψ*-quasi contractive mappings in metric spaces

- Seong-Hoon Cho
^{1}Email author and - Jong-Sook Bae
^{2}

**2013**:268

https://doi.org/10.1186/1687-1812-2013-268

© Cho and Bae; licensee Springer. 2013

**Received: **4 March 2013

**Accepted: **28 August 2013

**Published: **7 November 2013

## Abstract

A notion of *α*-*ψ*-quasi contractive mappings is introduced. Some new fixed point theorems for *α*-*ψ*-quasi contractive mappings are established. An application to integral equations is given.

**MSC:**47H10, 54H25.

## Keywords

*α*-

*ψ*-quasi contractive mappingmetric spaceordered metric space

## 1 Introduction and preliminaries

Banach’s contraction principle [1] is one of the pivotal results in nonlinear analysis. Banach’s contraction principle and its generalizations have many applications in solving nonlinear functional equations. In a metric space setting, it can be stated as follows.

**Theorem 1.1**

*Let*$(X,d)$

*be a complete metric space*.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*where* $0\le k<1$.

*Then* *T* *has a unique fixed point in* *X*.

Ćirić [2] introduced quasi contraction, which is one of the most general contraction conditions.

**Theorem 1.2**

*Let*$(X,d)$

*be a complete metric space*.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$, *where* $0\le k<1$.

*Then* *T* *has a unique fixed point in* *X*.

In [3], Berinde generalized Ćirić’s result.

**Theorem 1.3**

*Let*$(X,d)$

*be a complete metric space*.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$, *where* $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is nondecreasing and continuous such that* ${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(t)=0$ *for all* $t>0$.

*If* *T* *has bounded orbits*, *then* *T* *has a unique fixed point in* *X*.

Recently, Samet *et al.* [4] introduced the notion of *α*-*ψ* contractive mapping and gave some fixed point theorems for such mappings. Then, Asl *et al.* [5] gave generalizations of some of the results in [4], and Mohammadi *et al.* [6] generalized the results in [5].

The purpose of the paper is to introduce a concept of *α*-*ψ*-quasi contractive mappings and to give some new fixed point theorems for such mappings.

## 2 Fixed point theorems

for all $t>0$.

**Lemma 2.1**

*If*$\psi \in \mathrm{\Psi}$,

*then the following are satisfied*.

- (a)
$\psi (t)<t$

*for all*$t>0$; - (b)
$\psi (0)=0$;

- (c)
*ψ**is right continuous at*$t=0$.

**Remark 2.1**

- (a)
If $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is nondecreasing such that ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for each $t>0$, then $\psi \in \mathrm{\Psi}$.

- (b)
If $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is upper semicontinuous such that $\psi (t)<t$ for all $t>0$, then ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(t)=0$ for all $t>0$.

Let $(X,d)$ be a metric space, and let $\alpha :X\times X\to [0,\mathrm{\infty})$ be a function.

*α-ψ-quasi contractive*if there exists $\psi \in \mathrm{\Psi}$ such that, for all $x,y\in X$,

where $M(x,y)=max\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}$.

For $A\subset X$, we denote by $\delta (A)$ the diameter of *A*.

Let Λ be the family of all functions $\alpha :X\times X\to [0,\mathrm{\infty})$.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$

*be such that*

*ψ*

*is upper semicontinuous*.

*Suppose that*$T:X\to X$

*is*

*α*-

*ψ*-

*quasi contractive*.

*Assume that there exists*${x}_{0}\in X$

*such that*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either*

*T*

*is continuous or*

*for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

*Proof* Let ${x}_{0}\in X$ be such that $O({x}_{0},T;\mathrm{\infty})$ is bounded and $\alpha ({T}^{i}{x}_{0},{T}^{j}{x}_{0})\ge 1$ for all $i,j\in \mathbb{N}\cup \{0\}$ with $i<j$.

Define a sequence $\{{x}_{n}\}\subset X$ by ${x}_{n+1}=T{x}_{n}$ for $n\in \mathbb{N}\cup \{0\}$.

If ${x}_{n}={x}_{n+1}$ for some $n\in \mathbb{N}\cup \{0\}$, then ${x}_{n}$ is a fixed point.

Assume that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}\cup \{0\}$.

We now show that $\{{x}_{n}\}$ is a Cauchy sequence.

Let ${B}_{n}=\{{x}_{i}:i\ge n\}$, for $n=0,1,2,\dots $ , and let $\delta ({B}_{0})=B$.

If $n=0$, then obviously, (2.3) holds.

Suppose that (2.3) holds when $n=k$, *i.e.*, $\delta ({B}_{k})\le {\psi}^{k}(B)$.

Therefore, (2.3) is true for $n=0,1,2,\dots $ .

*X*. It follows from the completeness of

*X*that there exists

If *T* is continuous, then ${lim}_{n\to \mathrm{\infty}}{x}_{n}=T{x}_{\ast}$, and so ${x}_{\ast}=T{x}_{\ast}$.

Assume that (2.2) is satisfied.

Assume that $d({x}_{\ast},T{x}_{\ast})>0$.

We obtain ${lim}_{n\to \mathrm{\infty}}M({x}_{n},{x}_{\ast})=d({x}_{\ast},T{x}_{\ast})$.

*ψ*, we have

which is a contradiction. Hence, $d({x}_{\ast},T{x}_{\ast})=0$, and hence, ${x}_{\ast}=T{x}_{\ast}$. □

**Example 2.1**Let $X=[0,\mathrm{\infty})$ and $d(x,y)=|x-y|$ for all $x,y\in X$, and let

Then $\psi \in \mathrm{\Psi}$.

Note that *ψ* is not continuous at $t=1$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(\frac{1}{2})={\sum}_{n=1}^{\mathrm{\infty}}\frac{1}{2+n}=\mathrm{\infty}$.

Clearly, *T* is an *α*-*ψ* quasi contractive mapping. Condition (2.1) holds with ${x}_{0}=\frac{3}{2}$, and $O({x}_{0},T;\mathrm{\infty})$ is bounded. Obviously, (2.2) is satisfied.

Applying Theorem 2.1, *T* has a fixed point. Note that $\frac{3}{2}$ and 3 are two fixed points of *T*.

The following example shows that if we do not have the condition of which $O({x}_{0},T;\mathrm{\infty})$ is bounded for some ${x}_{0}\in X$, then Theorem 2.1 does not hold. Thus, we have to have the condition above.

**Example 2.2**Let $X=\mathbb{N}$, and let

for all $x,y\in X$.

Then $(X,d)$ is a complete metric space.

Let $T:X\to X$ be a mapping defined by $T(n)=n+1$, and let $\alpha (x,y)=1$ for all $x,y\in X$.

Let ${t}_{k}={\sum}_{i=1}^{k}\frac{1}{{i}^{2}}$ for $k=1,2,3,\dots $ .

Then, it easy to see that $\psi \in \mathrm{\Psi}$.

We show that *T* is *α*-*ψ*-quasi contractive.

Hence, *T* is *α*-*ψ*-quasi contractive. But the orbits are not bounded, and *T* has no fixed points.

**Corollary 2.2**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$

*be such that*

*ψ*

*is upper semicontinuous*.

*Suppose that*$T:X\to X$

*is*

*α*-

*ψ*-

*quasi contractive*.

*Assume that there exists*${x}_{0}\in X$

*such that*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous or* ${lim}_{n\to \mathrm{\infty}}inf\alpha ({T}^{n}{x}_{0},x)\ge 1$ *for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

*Proof* Define a sequence $\{{x}_{n}\}\subset X$ by ${x}_{n}={T}^{n}{x}_{0}$ for $n=1,2,\dots $ .

where ${v}_{1}=max\{d({x}_{i-1},{x}_{j-1}),d({x}_{i-1},{x}_{i}),d({x}_{j-1},{x}_{j}),d({x}_{i-1},{x}_{j}),d({x}_{j-1},{x}_{i})\}\le \delta (O({x}_{i-1},T;n-i+1))$.

From (2.5) and (2.7), we have $\delta (O({x}_{0},T;n))\le M$ for all $n\in \mathbb{N}$. Hence, $O({x}_{0},T;n)$ is bounded. By Theorem 2.1, *T* has a fixed point in *X*. □

**Corollary 2.3**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$

*be such that*

*ψ*

*is upper semicontinuous*.

*Suppose that*$T:X\to X$

*is*

*α*-

*ψ*-

*quasi contractive*.

*Assume that*${lim}_{t\to \mathrm{\infty}}(t-\psi (t))=\mathrm{\infty}$,

*and there exists*${x}_{0}\in X$

*such that*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous or* ${lim}_{n\to \mathrm{\infty}}inf\alpha ({T}^{n}{x}_{0},x)\ge 1$ *for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

**Corollary 2.4** *Let* $(X,\u2aaf,d)$ *be a complete ordered metric space*, *and let* $\psi \in \mathrm{\Psi}$ *be such that* *ψ* *is upper semicontinuous*.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all comparable elements* $x,y\in X$. *Assume that there exists* ${x}_{0}\in X$ *such that* $O({x}_{0},T;n)$ *is bounded*, *and* ${T}^{i}{x}_{0}$ *and* ${T}^{j}{x}_{0}$ *are comparable for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous*, *or* ${T}^{n}{x}_{0}$ *and* *x* *are comparable for all* $n\in \mathbb{N}\cup \{0\}$ *and for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

*Proof*Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Using Theorem 2.1, *T* has a fixed point in *X*. □

**Remark 2.2** Let $(X,d)$ be a metric space, and let $\alpha \in \mathrm{\Lambda}$.

- (1)
for each $x,y,z\in X$, $\alpha (x,y)\ge 1$ and $\alpha (y,z)\ge 1$ implies $\alpha (x,z)\ge 1$;

- (2)
for each $x,y\in X$, $\alpha (x,y)\ge 1$ implies $\alpha (Tx,Ty)\ge 1$;

- (3)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge 1$;

- (4)
if $\{{x}_{n}\}$ is a sequence with $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x\in X$, then $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$;

- (5)
there exists ${x}_{0}\in X$ such that $\alpha ({T}^{i}{x}_{0},{T}^{j}{x}_{0})\ge 1$ for all $i,j\in \mathbb{N}\cup \{0\}$ with $i<j$;

- (6)
$liminf\alpha ({T}^{n}{x}_{0},x)\ge 1$ for all cluster point

*x*of $\{{T}^{n}{x}_{0}\}$.

Then conditions (1), (2) and (3) imply (5), and condition (4) implies (6).

**Remark 2.3** If we replace condition (2.1) of Theorem 2.1 with the conditions (1), (2) and (3) above and replace condition (2.2) of Theorem 2.1 with the condition (4) above, then *T* has a fixed point.

**Corollary 2.5**

*Let*$(X,\u2aaf,d)$

*be a complete ordered metric space*,

*and let*$\psi \in \mathrm{\Psi}$

*be such that*

*ψ*

*is upper semicontinuous*.

*Suppose that a nondecreasing mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$ *with* $x\u2aafy$.

*Assume that there exists* ${x}_{0}\in X$ *such that* $O({x}_{0},T;n)$ *is bounded*, *and* ${x}_{0}\u2aafT{x}_{0}$. *Suppose that either* *T* *is continuous or if* $\{{x}_{n}\}$ *is a sequence in* *X* *such that* ${x}_{n}\u2aaf{x}_{n+1}$ *for all* $n\in \mathbb{N}$ *and* ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$.

*Then* *T* *has a fixed point in* *X*.

*Proof*Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Using Remark 2.3, *T* has a fixed point in *X*. □

In Theorem 2.1, if $\alpha (x,y)=1$ for all $x,y\in X$, we have the following corollary.

**Corollary 2.6**

*Let*$(X,d)$

*be a complete metric space*,

*and let*$\psi \in \mathrm{\Psi}$

*be such that*

*ψ*

*is upper semicontinuous*.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$. *If there exists* ${x}_{0}\in X$ *such that* $O({x}_{0},T;n)$ *is bounded*, *then* *T* *has a fixed point in* *X*.

**Remark 2.4** Corollary 2.6 is a generalization of Theorem 2 in [3].

**Theorem 2.7**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$.

*Assume that there exists*${x}_{0}\in X$

*such that*$O({x}_{0},T;\mathrm{\infty})$

*is bounded and*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either*

*T*

*is continuous*,

*or*

*for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

*Proof*Define a sequence $\{{x}_{n}\}\subset X$ by ${x}_{n+1}=T{x}_{n}$ for all $n\in \mathbb{N}\cup \{0\}$. As in the proof of Theorem 2.1, $\{{x}_{n}\}$ is a Cauchy sequence in

*X*. Since

*X*is complete, there exists ${x}_{\ast}\in X$ such that

If *T* is continuous, then ${x}_{\ast}$ is a fixed point of *T*.

Assume that (2.8) is satisfied.

Then, $L:=liminf\alpha ({x}_{n},{x}_{\ast})>0$.

Let $\u03f5>0$ be given.

Then there exists $N\in \mathbb{N}$ such that $d({x}_{n},{x}_{\ast})<L\u03f5$ and $\alpha ({x}_{n},{x}_{\ast})>0$ for all $n>N$.

Since *ψ* is nondecreasing, $\psi (d({x}_{n},{x}_{\ast}))\le \psi (L\u03f5)$ for all $n>N$.

for all $n>N$.

and so ${x}_{\ast}=T{x}_{\ast}$. □

**Remark 2.5** Theorem 2.7 is a generalization of Theorem 2.1 and Theorem 2.2 in [4].

**Corollary 2.8**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$.

*Assume that there exists*${x}_{0}\in X$

*such that*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous*, *or* ${lim}_{n\to \mathrm{\infty}}inf\alpha ({T}^{n}{x}_{0},x)>0$ *for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

*Proof* Define a sequence $\{{x}_{n}\}\subset X$ by ${x}_{n}={T}^{n}{x}_{0}$ for $n=1,2,\dots $ .

As in the proof of Corollary 2.2, $O({x}_{0},T;n)$ is bounded. By Theorem 2.7, *T* has a fixed point in *X*. □

**Corollary 2.9**

*Let*$(X,d)$

*be a complete metric space*, $\alpha \in \mathrm{\Lambda}$,

*and let*$\psi \in \mathrm{\Psi}$.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all* $x,y\in X$.

*Assume that*${lim}_{t\to \mathrm{\infty}}(t-\psi (t))=\mathrm{\infty}$,

*and there exists*${x}_{0}\in X$

*such that*

*for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous or* ${lim}_{n\to \mathrm{\infty}}inf\alpha ({T}^{n}{x}_{0},x)>0$ *for any cluster point* *x* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

**Corollary 2.10**

*Let*$(X,\u2aaf,d)$

*be a complete ordered metric space*,

*and let*$\psi \in \mathrm{\Psi}$.

*Suppose that a mapping*$T:X\to X$

*satisfies*

*for all comparable elements* $x,y\in X$.

*Assume that there exists* ${x}_{0}\in X$ *such that* $O({x}_{0},T;\mathrm{\infty})$ *is bounded*, *and* ${T}^{i}{x}_{0}$ *and* ${T}^{j}{x}_{0}$ *are comparable for all* $i,j\in \mathbb{N}\cup \{0\}$ *with* $i<j$.

*Suppose that either* *T* *is continuous or* ${T}^{n}{x}_{0}$ *and* *x* *are comparable for all* $n\in \mathbb{N}\cup \{0\}$ *and for any cluster point* *X* *of* $\{{T}^{n}{x}_{0}\}$.

*Then* *T* *has a fixed point in* *X*.

**Corollary 2.11** *Let* $(X,\u2aaf,d)$ *be a complete ordered metric space*, *and let* $\psi \in \mathrm{\Psi}$. *Suppose that a mapping* $T:X\to X$ *is nondecreasing such that* $d(Tx,Ty)\le \psi (d(x,y))$ *for all* $x,y\in X$ *with* $x\u2aafy$.

*Assume that there exists* ${x}_{0}\in X$ *such that* $O({x}_{0},T;\mathrm{\infty})$ *is bounded*, *and* ${x}_{0}\u2aafT{x}_{0}$. *Suppose that either* *T* *is continuous*, *or if* $\{{x}_{n}\}$ *is a sequence in* *X* *such that* ${x}_{n}\u2aaf{x}_{n+1}$ *for all* $n\in \mathbb{N}$ *and* ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$, *then* ${x}_{n}\u2aafx$ *for all* $n\in \mathbb{N}$.

*Then* *T* *has a fixed point in* *X*.

**Remark 2.6** Corollary 2.10 and Corollary 2.11 are generalizations of the results of [7].

## 3 An application to integral equations

where $K:I\times I\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ and $g:I\to {\mathbb{R}}^{n}$ are continuous.

*X*,

where $\tau >0$, is arbitrarily chosen.

Let ${d}_{B}(x,y)={\parallel x-y\parallel}_{B}={max}_{t\in [a,b]}|x(t)-y(t)|{e}^{-\tau (t-a)}$ for all $x,y\in X$.

Then it is well known that $(X,{d}_{B})$ is a complete metric space.

**Theorem 3.1**

*Let*$\xi :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to \mathbb{R}$

*be a function*.

*Suppose that the following conditions*(1)-(5)

*are satisfied*:

- (1)
*for all*$u,v,w\in {\mathbb{R}}^{n}$, $\xi (u,v)\ge 0$*and*$\xi (v,w)\ge 0$*implies*$\xi (u,w)\ge 0$; - (2)
*for all*$s,t\in I$*and for all*$u,v\in {\mathbb{R}}^{n}$*with*$\xi (u,v)\ge 0$$|K(t,s,u)-K(t,s,v)|\le \psi (|u-v|),$*where*$\psi \in \mathrm{\Psi}$*such that*${lim}_{t\to \mathrm{\infty}}(t-\psi (t))=\mathrm{\infty}$*and*$\psi (\lambda t)\le \lambda \psi (t)$*for all*$t\ge 0$*and for all*$\lambda \ge 1$; - (3)
*there exists*${x}_{0}\in X$*such that for all*$t\in I$,$\xi ({x}_{0}(t),{\int}_{a}^{t}K(t,s,{x}_{0}(s))\phantom{\rule{0.2em}{0ex}}ds+g(t)\phantom{\rule{0.2em}{0ex}}ds)\ge 0;$ - (4)
*for all*$x,y\in X$*and for all*$t\in I$,$\xi (x(t),y(t))\ge 0\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}\xi ({\int}_{a}^{t}K(t,s,x(s))\phantom{\rule{0.2em}{0ex}}ds+g(t),{\int}_{a}^{t}K(t,s,y(s))\phantom{\rule{0.2em}{0ex}}ds+g(t))\ge 0;$ - (5)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${lim}_{n\to \mathrm{\infty}}{x}_{n}=x\in X$*and*$\xi ({x}_{n},{x}_{n+1})\ge 0$*for all*$n\in \mathbb{N}$,*then*$\xi ({x}_{n},x)\ge 0$*for all*$n\in \mathbb{N}$.

*Then the integral equation* (3.1) *has at least one solution* ${x}^{\ast}\in X$.

*Proof*

Let $x,y\in X$ such that $\xi (x(t),y(t))\ge 0$ for all $t\in I$.

Hence, for $\tau \ge 1$, we obtain ${d}_{B}(Tx,Ty)\le \psi ({d}_{B}(x,y))$ for all $x,y\in X$ with $\xi (x,y)\ge 0$.

It is easy to see that conditions (1)-(4) of Remark 2.2 are satisfied. By Corollary 2.9, *T* has a fixed point in *X*. □

## Declarations

## Authors’ Affiliations

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