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# Fixed point results for implicit contractions on spaces with two metrics

- Bessem Samet
^{1}Email author

**2014**:84

https://doi.org/10.1186/1029-242X-2014-84

© Samet; licensee Springer. 2014

**Received:**6 September 2013**Accepted:**23 January 2014**Published:**20 February 2014

## Abstract

We establish new fixed-point results involving implicit contractions on a metric space endowed with two metrics. The main results in this paper extend and generalize several existing fixed-point theorems in the literature.

## Keywords

- fixed point
- implicit contraction
- two metrics

## 1 Introduction

Fixed-point theory is a major branch of nonlinear analysis because of its wide applicability. The existence problem of fixed points of mappings satisfying a given metrical contractive condition has attracted many researchers in past few decades. The Banach contraction principle [1] is one of the most important theorems in this direction. Many generalizations of this famous principle exist in the literature, see, for examples, [2–6] and references therein. On the other hand, several classical fixed-point theorems have been unified by considering general contractive conditions expressed by an implicit condition, see for examples, Turinici [7], Popa [8, 9], Berinde [10], and references therein.

This paper presents fixed-point theorems for implicit contractions on a metric space endowed with two metrics. This paper will be divided into two main sections. Section 2 presents local and global fixed-point results for implicit contractions involving *α*-admissible mappings, a recent concept introduced in [11]. Section 3 presents some interesting consequences that can be obtained from the results established in the previous section.

## 2 Main results

- (i)
*F*is continuous; - (ii)
*F*is non-decreasing in the first variable; - (iii)
*F*is non-increasing in the fifth variable; - (iv)
$\mathrm{\exists}h\in (0,1)\mid F(u,v,v,u,u+v,0)\le 0\u27f9u\le hv$.

**Example**Let $F:{[0,+\mathrm{\infty})}^{6}\to \mathbb{R}$ be the function defined by

where $q\in (0,1)$. We can check easily that $F\in \mathcal{F}$.

*X*be a nonempty set endowed with two metrics

*d*and ${d}^{\prime}$. If ${x}_{0}\in X$ and $r>0$, let

We denote by ${\overline{B({x}_{0},r)}}^{{d}^{\prime}}$ the ${d}^{\prime}$-closure of $B({x}_{0},r)$.

*T*is

*α*-admissible (see [11]) if the following condition holds: for all $x,y\in B({x}_{0},r)$, we have

We say that *X* satisfies the property (H) with respect to the metric *d* if the following condition holds:

If ${lim}_{n\to \mathrm{\infty}}d({x}_{n},x)=0$ for some $x\in X$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all *n*, then there exist a positive integer *κ* and a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha ({x}_{n(k)},x)\ge 1$ for all $k\ge \kappa $.

Our first result is the following.

**Theorem 2.1**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*, ${x}_{0}\in X$, $r>0$, $T:{\overline{B({x}_{0},r)}}^{{d}^{\prime}}\to X$,

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{{d}^{\prime}}$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
$d({x}_{0},T{x}_{0})<(1-h)r$

*and*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (II)
*T**is**α*-*admissible*; - (III)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(B({x}_{0},r),d)$*into*$(X,{d}^{\prime})$; - (IV)
*if*$d={d}^{\prime}$,*assume**X**satisfies the property*(H)*with respect to the metric**d*; - (V)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$({\overline{B({x}_{0},r)}}^{{d}^{\prime}},{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

*Proof*Let ${x}_{1}=T{x}_{0}$. From (I), we have

*F*is non-decreasing in the first variable (property (i)), we obtain

*T*is

*α*-admissible and $\alpha ({x}_{0},{x}_{1})\ge 1$, we have

Since $h\in (0,1)$, it follows from (2) that $\{{x}_{n}\}$ is a Cauchy sequence with respect to the metric *d*.

*δ*> such that

*d*, there exists a positive integer

*N*such that

Thus we proved that $\{{x}_{n}\}$ is Cauchy with respect to ${d}^{\prime}$.

We shall prove that *z* is a fixed point of *T*. We consider two cases.

Case 1. If $d={d}^{\prime}$.

*κ*and a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that

*F*, we obtain

which implies from (iv) that $d(z,Tz)=0$.

Case 2. If $d\ne {d}^{\prime}$.

The uniqueness of the limit gives $z=Tz$. □

Taking $d={d}^{\prime}$ in Theorem 2.1, we obtain the following result.

**Theorem 2.2**

*Let*$(X,d)$

*be a complete metric space*, ${x}_{0}\in X$, $r>0$, $T:{\overline{B({x}_{0},r)}}^{d}\to X$,

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{d}$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
$d({x}_{0},T{x}_{0})<(1-h)r$

*and*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (II)
*T**is**α*-*admissible*; - (III)
*X**satisfies the property*(H)*with respect to the metric**d*.

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

From Theorem 2.1, we can deduce the following global result.

**Theorem 2.3**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*, $T:X\to X$,

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in X$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (II)
*T**is**α*-*admissible*($x,y\in X$, $\alpha (x,y)\ge 1\u27f9\alpha (Tx,Ty)\ge 1$); - (III)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(X,d)$*into*$(X,{d}^{\prime})$; - (IV)
*if*$d={d}^{\prime}$,*assume**X**satisfies the property*(H)*with respect to the metric**d*; - (V)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$(X,{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

*Proof* We take $r>0$ such that $d({x}_{0},T{x}_{0})<(1-h)r$. From Theorem 2.1, *T* has a fixed point in ${\overline{B({x}_{0},r)}}^{{d}^{\prime}}$. □

Taking $d={d}^{\prime}$ in Theorem 2.3, we obtain the following result.

**Theorem 2.4**

*Let*$(X,d)$

*be a complete metric space*, $T:X\to X$,

*and*$\alpha :X\times X\to [0,\mathrm{\infty})$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in X$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (II)
*T**is**α*-*admissible*($x,y\in X$, $\alpha (x,y)\ge 1\u27f9\alpha (Tx,Ty)\ge 1$); - (III)
*X**satisfies the property*(H)*with respect to the metric**d*.

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

## 3 Consequences

We present here some interesting consequences that can be obtained from our main results.

### 3.1 The case $\alpha (x,y)=1$

Taking $\alpha (x,y):=1$ for all $x,y\in X$, from Theorems 2.1, 2.2, 2.3, and 2.4, we obtain the following results that are generalizations of the fixed-point results in [2, 3, 5, 8, 10, 12, 13].

**Corollary 3.1**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*, ${x}_{0}\in X$, $r>0$,

*and*$T:{\overline{B({x}_{0},r)}}^{{d}^{\prime}}\to X$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{{d}^{\prime}}$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
$d({x}_{0},T{x}_{0})<(1-h)r$;

- (II)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(B({x}_{0},r),d)$*into*$(X,{d}^{\prime})$; - (III)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$({\overline{B({x}_{0},r)}}^{{d}^{\prime}},{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

**Corollary 3.2**

*Let*$(X,d)$

*be a complete metric space*, ${x}_{0}\in X$, $r>0$,

*and*$T:{\overline{B({x}_{0},r)}}^{d}\to X$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{d}$,

*we have*

*In addition*, *assume that* $d({x}_{0},T{x}_{0})<(1-h)r$. *Then* *T* *has a fixed point*.

**Corollary 3.3**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*,

*and*$T:X\to X$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in X$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(X,d)$*into*$(X,{d}^{\prime})$; - (II)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$(X,{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

**Corollary 3.4**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$.

*Suppose there exists*$F\in \mathcal{F}$

*such that for*$x,y\in X$,

*we have*

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

Corollary 3.4 is an enriched version of Popa [8] that unifies the most important metrical fixed-point theorems for contractive mappings in Rhoades’ classification [6].

### 3.2 The case of a partial ordered set

*X*. Let ⊲ be the binary relation on

*X*defined by

We say that $(X,\u22b2)$ satisfies the property (H) with respect to the metric *d* if the following condition holds:

If ${lim}_{n\to \mathrm{\infty}}d({x}_{n},x)=0$ for some $x\in X$ and ${x}_{n}\u22b2{x}_{n+1}$ for all *n*, then there exist a positive integer *κ* and a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that ${x}_{n(k)}\u22b2x$ for all $k\ge \kappa $.

From Theorems 2.1, 2.2, 2.3, and 2.4, we obtain the following results that are extensions and generalizations of the fixed-point results in [14, 15].

- (j)
$F\in \mathcal{F}$;

(jj) $F(0,{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})\le 0$ for all ${t}_{i}\ge 0$, $i=2,\dots ,6$.

We start with the following fixed-point result.

**Corollary 3.5**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*, ${x}_{0}\in X$, $r>0$,

*and*$T:{\overline{B({x}_{0},r)}}^{{d}^{\prime}}\to X$.

*Suppose there exists*$F\in \tilde{\mathcal{F}}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{{d}^{\prime}}$

*with*$x\u22b2y$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
$d({x}_{0},T{x}_{0})<(1-h)r$

*and*${x}_{0}\u22b2T{x}_{0}$; - (II)
$x,y\in {\overline{B({x}_{0},r)}}^{{d}^{\prime}}$, $x\u22b2y\u27f9Tx\u22b2Ty$;

- (III)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(B({x}_{0},r),d)$*into*$(X,{d}^{\prime})$; - (IV)
*if*$d={d}^{\prime}$,*assume*$(X,\u22b2)$*satisfies the property*(H)*with respect to the metric**d*; - (V)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$({\overline{B({x}_{0},r)}}^{{d}^{\prime}},{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

*Proof*It follows from Theorem 2.1 by taking

□

Similarly, from Theorem 2.2, we obtain the following result.

**Corollary 3.6**

*Let*$(X,d)$

*be a complete metric space*, ${x}_{0}\in X$, $r>0$,

*and*$T:{\overline{B({x}_{0},r)}}^{d}\to X$.

*Suppose there exists*$F\in \tilde{\mathcal{F}}$

*such that for*$x,y\in {\overline{B({x}_{0},r)}}^{d}$

*with*$x\u22b2y$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
$d({x}_{0},T{x}_{0})<(1-h)r$

*and*${x}_{0}\u22b2T{x}_{0}$; - (II)
$x,y\in {\overline{B({x}_{0},r)}}^{{d}^{\prime}}$, $x\u22b2y\u27f9Tx\u22b2Ty$;

- (III)
$(X,\u22b2)$

*satisfies the property*(H)*with respect to the metric**d*;

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

From Theorem 2.3, we obtain the following global result.

**Corollary 3.7**

*Let*$(X,{d}^{\prime})$

*be a complete metric space*,

*d*

*another metric on*

*X*,

*and*$T:X\to X$.

*Suppose there exists*$F\in \tilde{\mathcal{F}}$

*such that for*$x,y\in X$

*with*$x\u22b2y$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u22b2T{x}_{0}$; - (II)
$x,y\in X$, $x\u22b2y\u27f9Tx\u22b2Ty$;

- (III)
*if*$d\ngeqq {d}^{\prime}$,*assume**T**is uniformly continuous from*$(X,d)$*into*$(X,{d}^{\prime})$; - (IV)
*if*$d={d}^{\prime}$,*assume*$(X,\u22b2)$*satisfies the property*(H)*with respect to the metric**d*; - (V)
*if*$d\ne {d}^{\prime}$,*assume**T**is continuous from*$(X,{d}^{\prime})$*into*$(X,{d}^{\prime})$.

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

Finally, from Theorem 2.4, we obtain the following fixed-point result.

**Corollary 3.8**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$.

*Suppose there exists*$F\in \tilde{\mathcal{F}}$

*such that for*$x,y\in X$

*with*$x\u22b2y$,

*we have*

*In addition*,

*assume the following properties hold*:

- (I)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u22b2T{x}_{0}$; - (II)
$x,y\in X$, $x\u22b2y\u27f9Tx\u22b2Ty$;

- (III)
$(X,\u22b2)$

*satisfies the property*(H)*with respect to the metric**d*.

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

### 3.3 The case of cyclic mappings

From Theorem 2.4, we obtain the following fixed-point result that is a generalization of Theorem 1.1 in [16].

**Corollary 3.9**

*Let*$(Y,d)$

*be a complete metric space*, $\{A,B\}$

*a pair of nonempty closed subsets of*

*Y*,

*and*$T:A\cup B\to A\cup B$.

*Suppose there exists*$F\in \tilde{\mathcal{F}}$

*such that for*$x\in A$, $y\in B$,

*we have*

*In addition*, *assume that* $T(A)\subseteq B$ *and* $T(B)\subseteq A$.

*Then* *T* *has a fixed point in* $A\cap B$.

*Proof*Let $X:=A\cup B$. Clearly (since

*A*and

*B*are closed), $(X,d)$ is a complete metric space. Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Taking any point ${x}_{0}\in A$, since $T(A)\subseteq B$, we have $T{x}_{0}\in B$, which implies that $\alpha ({x}_{0},T{x}_{0})\ge 1$.

Now, let $(x,y)\in X\times X$ such that $\alpha (x,y)\ge 1$. We have two cases.

Case 1. $(x,y)\in A\times B$.

Since $T(A)\subseteq B$ and $T(B)\subseteq A$, we have $(Tx,Ty)\in B\times A$, which implies that $\alpha (Tx,Ty)\ge 1$.

Case 2. $(x,y)\in B\times A$.

In this case, we have $(Tx,Ty)\in A\times B$, which implies that $\alpha (Tx,Ty)\ge 1$.

Then *T* is *α*-admissible.

Finally, we shall prove that *X* satisfies the property (H) with respect to the metric *d*.

Let $\{{x}_{n}\}$ be a sequence in *X* such that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},x)=0$ for some $x\in X$, and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all *n*. From the definition of *α*, this implies that $({x}_{n},{x}_{n+1})\in (A\times B)\cup (B\times A)$ for all *n*. Since *A* and *B* are closed, we get $x\in A\cap B$. Then we have $\alpha ({x}_{n},x)=1$ for all *n*. Thus, we proved that *X* satisfies the property (H) with respect to the metric *d*.

Now, from Theorem 2.4, *T* has a fixed point in *X*, that is, there exists $z\in A\cup B$ such that $Tz=z$. Since $T(A)\subseteq B$ and $T(B)\subseteq A$, obviously, we have $z\in A\cap B$. □

## Author’s contributions

The author read and approved the final manuscript.

## Declarations

### Acknowledgements

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

## Authors’ Affiliations

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