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

## 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.

## 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, [2â€“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 Î±-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

Let â„± be the set of functions $F:{\left[0,+\mathrm{âˆž}\right)}^{6}â†’\mathbb{R}$ satisfying the following conditions:

1. (i)

F is continuous;

2. (ii)

F is non-decreasing in the first variable;

3. (iii)

F is non-increasing in the fifth variable;

4. (iv)

$\mathrm{âˆƒ}hâˆˆ\left(0,1\right)âˆ£F\left(u,v,v,u,u+v,0\right)â‰¤0âŸ¹uâ‰¤hv$.

Example Let $F:{\left[0,+\mathrm{âˆž}\right)}^{6}â†’\mathbb{R}$ be the function defined by

$F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right):={t}_{1}âˆ’qmax\left\{{t}_{2},{t}_{3},{t}_{4},\frac{{t}_{5}+{t}_{6}}{2}\right\},$

where $qâˆˆ\left(0,1\right)$. We can check easily that $Fâˆˆ\mathcal{F}$.

Let X be a nonempty set endowed with two metrics d and ${d}^{â€²}$. If ${x}_{0}âˆˆX$ and $r>0$, let

$B\left({x}_{0},r\right):=\left\{xâˆˆX:d\left({x}_{0},x\right)

We denote by ${\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$ the ${d}^{â€²}$-closure of $B\left({x}_{0},r\right)$.

Let $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}â†’X$ and $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$. We say that T is Î±-admissible (see [11]) if the following condition holds: for all $x,yâˆˆB\left({x}_{0},r\right)$, we have

$\mathrm{Î±}\left(x,y\right)â‰¥1\phantom{\rule{1em}{0ex}}âŸ¹\phantom{\rule{1em}{0ex}}\mathrm{Î±}\left(Tx,Ty\right)â‰¥1.$

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

If ${lim}_{nâ†’\mathrm{âˆž}}d\left({x}_{n},x\right)=0$ for some $xâˆˆX$ and $\mathrm{Î±}\left({x}_{n},{x}_{n+1}\right)â‰¥1$ for all n, then there exist a positive integer Îº and a subsequence $\left\{{x}_{n\left(k\right)}\right\}$ of $\left\{{x}_{n}\right\}$ such that $\mathrm{Î±}\left({x}_{n\left(k\right)},x\right)â‰¥1$ for all $kâ‰¥\mathrm{Îº}$.

Our first result is the following.

Theorem 2.1 Let $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, ${x}_{0}âˆˆX$, $r>0$, $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}â†’X$, and $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$, we have

$F\left(\mathrm{Î±}\left(x,y\right)d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$
(1)

In addition, assume the following properties hold:

1. (I)

$d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$ and $\mathrm{Î±}\left({x}_{0},T{x}_{0}\right)â‰¥1$;

2. (II)

3. (III)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(B\left({x}_{0},r\right),d\right)$ into $\left(X,{d}^{â€²}\right)$;

4. (IV)

if $d={d}^{â€²}$, assume X satisfies the property (H) with respect to the metric d;

5. (V)

if , assume T is continuous from $\left({\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}},{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

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

$d\left({x}_{0},{x}_{1}\right)=d\left({x}_{0},T{x}_{0}\right)â‰¤\left(1âˆ’h\right)r

which implies that ${x}_{1}âˆˆB\left({x}_{0},r\right)$. Let ${x}_{2}=T{x}_{1}$. From (1), we have

$F\left(\mathrm{Î±}\left({x}_{0},{x}_{1}\right)d\left(T{x}_{0},T{x}_{1}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{0},{x}_{2}\right),0\right)â‰¤0.$

From (I), we have

$d\left(T{x}_{0},T{x}_{1}\right)â‰¤\mathrm{Î±}\left({x}_{0},{x}_{1}\right)d\left(T{x}_{0},T{x}_{1}\right).$

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

$F\left(d\left({x}_{1},{x}_{2}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{0},{x}_{2}\right),0\right)â‰¤0.$

Since $d\left({x}_{0},{x}_{2}\right)â‰¤d\left({x}_{0},{x}_{1}\right)+d\left({x}_{1},{x}_{2}\right)$, using (iii), we obtain

$F\left(d\left({x}_{1},{x}_{2}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{0},{x}_{1}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{0},{x}_{1}\right)+d\left({x}_{1},{x}_{2}\right),0\right)â‰¤0,$

which implies from (iv) that

$d\left({x}_{1},{x}_{2}\right)â‰¤hd\left({x}_{0},{x}_{1}\right)â‰¤h\left(1âˆ’h\right)r

Now, we have

$d\left({x}_{0},{x}_{2}\right)â‰¤d\left({x}_{0},{x}_{1}\right)+hd\left({x}_{0},{x}_{1}\right)=\left(1+h\right)d\left({x}_{0},{x}_{1}\right)â‰¤\left(1+h\right)\left(1âˆ’h\right)r

This implies that ${x}_{2}âˆˆB\left({x}_{0},r\right)$. Again, let ${x}_{3}=T{x}_{2}$. Since T is Î±-admissible and $\mathrm{Î±}\left({x}_{0},{x}_{1}\right)â‰¥1$, we have

$d\left({x}_{2},{x}_{3}\right)â‰¤\mathrm{Î±}\left({x}_{1},{x}_{2}\right)d\left(T{x}_{1},T{x}_{2}\right).$

Then, from (1), we obtain

$F\left(d\left({x}_{2},{x}_{3}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{2},{x}_{3}\right),d\left({x}_{1},{x}_{3}\right),0\right)â‰¤0.$

Using (iii), we obtain

$F\left(d\left({x}_{2},{x}_{3}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{1},{x}_{2}\right),d\left({x}_{2},{x}_{3}\right),d\left({x}_{1},{x}_{2}\right)+d\left({x}_{2},{x}_{3}\right),0\right)â‰¤0,$

which implies from (iv) that

$d\left({x}_{2},{x}_{3}\right)â‰¤hd\left({x}_{1},{x}_{2}\right)â‰¤{h}^{2}\left(1âˆ’h\right)r

Now, we have

$d\left({x}_{0},{x}_{3}\right)â‰¤d\left({x}_{0},{x}_{2}\right)+d\left({x}_{2},{x}_{3}\right)â‰¤\left(1+h\right)\left(1âˆ’h\right)r+{h}^{2}\left(1âˆ’h\right)r=\left(1âˆ’{h}^{3}\right)r

This implies that ${x}_{3}âˆˆB\left({x}_{0},r\right)$. Continuing this process, by induction, we can define the sequence $\left\{{x}_{n}\right\}$ by

${x}_{n+1}=T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N}.$

Such sequence satisfies the following property:

${x}_{n}âˆˆB\left({x}_{0},r\right),\phantom{\rule{1em}{0ex}}\mathrm{Î±}\left({x}_{n},{x}_{n+1}\right)â‰¥1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}d\left({x}_{n},{x}_{n+1}\right)â‰¤{h}^{n}\left(1âˆ’h\right)r,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N}.$
(2)

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

Now, we shall prove that $\left\{{x}_{n}\right\}$ is also a Cauchy sequence with respect to ${d}^{â€²}$. If ${d}^{â€²}â‰¤d$, the result follows immediately from (2). If $dâ‰±{d}^{â€²}$, from (III), given $\mathrm{Îµ}>0$, there exists Î´> such that

$x,yâˆˆB\left({x}_{0},r\right),\phantom{\rule{1em}{0ex}}d\left(x,y\right)<\mathrm{Î´}\phantom{\rule{1em}{0ex}}âŸ¹\phantom{\rule{1em}{0ex}}{d}^{â€²}\left(Tx,Ty\right)<\mathrm{Îµ}.$
(3)

On the other hand, since $\left\{{x}_{n}\right\}$ is Cauchy with respect to d, there exists a positive integer N such that

$d\left({x}_{n},{x}_{m}\right)<\mathrm{Î´},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}n,mâ‰¥N.$

Using (3), we have

${d}^{â€²}\left({x}_{n+1},{x}_{m+1}\right)<\mathrm{Îµ},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}n,mâ‰¥N.$

Thus we proved that $\left\{{x}_{n}\right\}$ is Cauchy with respect to ${d}^{â€²}$.

Since $\left(X,{d}^{â€²}\right)$ is complete, there exists $zâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$ such that

$\underset{nâ†’\mathrm{âˆž}}{lim}{d}^{â€²}\left({x}_{n},z\right)=0.$
(4)

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

Case 1. If $d={d}^{â€²}$.

From (IV), there exist a positive integer Îº and a subsequence $\left\{{x}_{n\left(k\right)}\right\}$ of $\left\{{x}_{n}\right\}$ such that

$\mathrm{Î±}\left({x}_{n\left(k\right)},z\right)â‰¥1,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}kâ‰¥\mathrm{Îº}.$
(5)

Using (1), for all $kâ‰¥\mathrm{Îº}$, we obtain

$\begin{array}{c}F\left(\mathrm{Î±}\left({x}_{n\left(k\right)},z\right)d\left(T{x}_{n\left(k\right)},Tz\right),d\left({x}_{n\left(k\right)},z\right),d\left({x}_{n\left(k\right)},{x}_{n\left(k\right)+1}\right),d\left(z,Tz\right),d\left({x}_{n\left(k\right)},Tz\right),d\left(z,{x}_{n\left(k\right)+1}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤0.\hfill \end{array}$

Using (5) and condition (ii), for all $kâ‰¥\mathrm{Îº}$, we obtain

$F\left(d\left({x}_{n\left(k\right)+1},Tz\right),d\left({x}_{n\left(k\right)},z\right),d\left({x}_{n\left(k\right)},{x}_{n\left(k\right)+1}\right),d\left(z,Tz\right),d\left({x}_{n\left(k\right)},Tz\right),d\left(z,{x}_{n\left(k\right)+1}\right)\right)â‰¤0.$

Letting $kâ†’\mathrm{âˆž}$, using (4) and the continuity of F, we obtain

$F\left(d\left(z,Tz\right),0,0,d\left(z,Tz\right),d\left(z,Tz\right),0\right)â‰¤0,$

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

Case 2. If .

In this case, using (V) and (4), we obtain

$\underset{nâ†’\mathrm{âˆž}}{lim}{d}^{â€²}\left(T{x}_{n},Tz\right)=\underset{nâ†’\mathrm{âˆž}}{lim}{d}^{â€²}\left({x}_{n+1},Tz\right)=0.$

The uniqueness of the limit gives $z=Tz$.â€ƒâ–¡

Taking $d={d}^{â€²}$ in Theorem 2.1, we obtain the following result.

Theorem 2.2 Let $\left(X,d\right)$ be a complete metric space, ${x}_{0}âˆˆX$, $r>0$, $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}â†’X$, and $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}$, we have

$F\left(\mathrm{Î±}\left(x,y\right)d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

$d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$ and $\mathrm{Î±}\left({x}_{0},T{x}_{0}\right)â‰¥1$;

2. (II)

3. (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 $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, $T:Xâ†’X$, and $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆX$, we have

$F\left(\mathrm{Î±}\left(x,y\right)d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

there exists ${x}_{0}âˆˆX$ such that $\mathrm{Î±}\left({x}_{0},T{x}_{0}\right)â‰¥1$;

2. (II)

T is Î±-admissible ($x,yâˆˆX$, $\mathrm{Î±}\left(x,y\right)â‰¥1âŸ¹\mathrm{Î±}\left(Tx,Ty\right)â‰¥1$);

3. (III)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(X,d\right)$ into $\left(X,{d}^{â€²}\right)$;

4. (IV)

if $d={d}^{â€²}$, assume X satisfies the property (H) with respect to the metric d;

5. (V)

if , assume T is continuous from $\left(X,{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

Proof We take $r>0$ such that $d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$. From Theorem 2.1, T has a fixed point in ${\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$.â€ƒâ–¡

Taking $d={d}^{â€²}$ in Theorem 2.3, we obtain the following result.

Theorem 2.4 Let $\left(X,d\right)$ be a complete metric space, $T:Xâ†’X$, and $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆX$, we have

$F\left(\mathrm{Î±}\left(x,y\right)d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

there exists ${x}_{0}âˆˆX$ such that $\mathrm{Î±}\left({x}_{0},T{x}_{0}\right)â‰¥1$;

2. (II)

T is Î±-admissible ($x,yâˆˆX$, $\mathrm{Î±}\left(x,y\right)â‰¥1âŸ¹\mathrm{Î±}\left(Tx,Ty\right)â‰¥1$);

3. (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 $\mathrm{Î±}\left(x,y\right)=1$

Taking $\mathrm{Î±}\left(x,y\right):=1$ for all $x,yâˆˆX$, 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 $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, ${x}_{0}âˆˆX$, $r>0$, and $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}â†’X$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

$d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$;

2. (II)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(B\left({x}_{0},r\right),d\right)$ into $\left(X,{d}^{â€²}\right)$;

3. (III)

if , assume T is continuous from $\left({\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}},{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

Corollary 3.2 Let $\left(X,d\right)$ be a complete metric space, ${x}_{0}âˆˆX$, $r>0$, and $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}â†’X$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume that $d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$. Then T has a fixed point.

Corollary 3.3 Let $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, and $T:Xâ†’X$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆX$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(X,d\right)$ into $\left(X,{d}^{â€²}\right)$;

2. (II)

if , assume T is continuous from $\left(X,{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

Corollary 3.4 Let $\left(X,d\right)$ be a complete metric space and $T:Xâ†’X$. Suppose there exists $Fâˆˆ\mathcal{F}$ such that for $x,yâˆˆX$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

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 Rhoadesâ€™ classification [6].

### 3.2 The case of a partial ordered set

Let âª¯ be a partial order on X. Let âŠ² be the binary relation on X defined by

$\left(x,y\right)âˆˆXÃ—X,\phantom{\rule{1em}{0ex}}xâŠ²y\phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}xâª¯y\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}yâª¯x.$

We say that $\left(X,âŠ²\right)$ satisfies the property (H) with respect to the metric d if the following condition holds:

If ${lim}_{nâ†’\mathrm{âˆž}}d\left({x}_{n},x\right)=0$ for some $xâˆˆX$ and ${x}_{n}âŠ²{x}_{n+1}$ for all n, then there exist a positive integer Îº and a subsequence $\left\{{x}_{n\left(k\right)}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{n\left(k\right)}âŠ²x$ for all $kâ‰¥\mathrm{Îº}$.

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].

At first, we denote by $\stackrel{Ëœ}{\mathcal{F}}$ the set of functions $F:{\left[0,+\mathrm{âˆž}\right)}^{6}â†’\mathbb{R}$ satisfying the following conditions:

1. (j)

$Fâˆˆ\mathcal{F}$;

(jj) $F\left(0,{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)â‰¤0$ for all ${t}_{i}â‰¥0$, $i=2,â€¦,6$.

Corollary 3.5 Let $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, ${x}_{0}âˆˆX$, $r>0$, and $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}â†’X$. Suppose there exists $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$ with $xâŠ²y$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

$d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$ and ${x}_{0}âŠ²T{x}_{0}$;

2. (II)

$x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$, $xâŠ²yâŸ¹TxâŠ²Ty$;

3. (III)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(B\left({x}_{0},r\right),d\right)$ into $\left(X,{d}^{â€²}\right)$;

4. (IV)

if $d={d}^{â€²}$, assume $\left(X,âŠ²\right)$ satisfies the property (H) with respect to the metric d;

5. (V)

if , assume T is continuous from $\left({\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}},{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

Proof It follows from Theorem 2.1 by taking

â€ƒâ–¡

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

Corollary 3.6 Let $\left(X,d\right)$ be a complete metric space, ${x}_{0}âˆˆX$, $r>0$, and $T:{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}â†’X$. Suppose there exists $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$ such that for $x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{d}$ with $xâŠ²y$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

$d\left({x}_{0},T{x}_{0}\right)<\left(1âˆ’h\right)r$ and ${x}_{0}âŠ²T{x}_{0}$;

2. (II)

$x,yâˆˆ{\stackrel{Â¯}{B\left({x}_{0},r\right)}}^{{d}^{â€²}}$, $xâŠ²yâŸ¹TxâŠ²Ty$;

3. (III)

$\left(X,âŠ²\right)$ 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 $\left(X,{d}^{â€²}\right)$ be a complete metric space, d another metric on X, and $T:Xâ†’X$. Suppose there exists $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$ such that for $x,yâˆˆX$ with $xâŠ²y$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

there exists ${x}_{0}âˆˆX$ such that ${x}_{0}âŠ²T{x}_{0}$;

2. (II)

$x,yâˆˆX$, $xâŠ²yâŸ¹TxâŠ²Ty$;

3. (III)

if $dâ‰±{d}^{â€²}$, assume T is uniformly continuous from $\left(X,d\right)$ into $\left(X,{d}^{â€²}\right)$;

4. (IV)

if $d={d}^{â€²}$, assume $\left(X,âŠ²\right)$ satisfies the property (H) with respect to the metric d;

5. (V)

if , assume T is continuous from $\left(X,{d}^{â€²}\right)$ into $\left(X,{d}^{â€²}\right)$.

Then T has a fixed point.

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

Corollary 3.8 Let $\left(X,d\right)$ be a complete metric space and $T:Xâ†’X$. Suppose there exists $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$ such that for $x,yâˆˆX$ with $xâŠ²y$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume the following properties hold:

1. (I)

there exists ${x}_{0}âˆˆX$ such that ${x}_{0}âŠ²T{x}_{0}$;

2. (II)

$x,yâˆˆX$, $xâŠ²yâŸ¹TxâŠ²Ty$;

3. (III)

$\left(X,âŠ²\right)$ 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 $\left(Y,d\right)$ be a complete metric space, $\left\{A,B\right\}$ a pair of nonempty closed subsets of Y, and $T:AâˆªBâ†’AâˆªB$. Suppose there exists $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$ such that for $xâˆˆA$, $yâˆˆB$, we have

$F\left(d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

In addition, assume that $T\left(A\right)âŠ†B$ and $T\left(B\right)âŠ†A$.

Then T has a fixed point in $Aâˆ©B$.

Proof Let $X:=AâˆªB$. Clearly (since A and B are closed), $\left(X,d\right)$ is a complete metric space. Define $\mathrm{Î±}:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right)$ by

Clearly (since $Fâˆˆ\stackrel{Ëœ}{\mathcal{F}}$), for all $x,yâˆˆX$, we have

$F\left(\mathrm{Î±}\left(x,y\right)d\left(Tx,Ty\right),d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right)â‰¤0.$

Taking any point ${x}_{0}âˆˆA$, since $T\left(A\right)âŠ†B$, we have $T{x}_{0}âˆˆB$, which implies that $\mathrm{Î±}\left({x}_{0},T{x}_{0}\right)â‰¥1$.

Now, let $\left(x,y\right)âˆˆXÃ—X$ such that $\mathrm{Î±}\left(x,y\right)â‰¥1$. We have two cases.

Case 1. $\left(x,y\right)âˆˆAÃ—B$.

Since $T\left(A\right)âŠ†B$ and $T\left(B\right)âŠ†A$, we have $\left(Tx,Ty\right)âˆˆBÃ—A$, which implies that $\mathrm{Î±}\left(Tx,Ty\right)â‰¥1$.

Case 2. $\left(x,y\right)âˆˆBÃ—A$.

In this case, we have $\left(Tx,Ty\right)âˆˆAÃ—B$, which implies that $\mathrm{Î±}\left(Tx,Ty\right)â‰¥1$.

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

Let $\left\{{x}_{n}\right\}$ be a sequence in X such that ${lim}_{nâ†’\mathrm{âˆž}}d\left({x}_{n},x\right)=0$ for some $xâˆˆX$, and $\mathrm{Î±}\left({x}_{n},{x}_{n+1}\right)â‰¥1$ for all n. From the definition of Î±, this implies that $\left({x}_{n},{x}_{n+1}\right)âˆˆ\left(AÃ—B\right)âˆª\left(BÃ—A\right)$ for all n. Since A and B are closed, we get $xâˆˆAâˆ©B$. Then we have $\mathrm{Î±}\left({x}_{n},x\right)=1$ 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 $zâˆˆAâˆªB$ such that $Tz=z$. Since $T\left(A\right)âŠ†B$ and $T\left(B\right)âŠ†A$, obviously, we have $zâˆˆAâˆ©B$.â€ƒâ–¡

## Authorâ€™s contributions

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## References

1. Banach S: Sur les opÃ©rations dans les ensembles abstraits et leur application aux Ã©quations intÃ©grales. Fundam. Math. 1922, 3: 133-181.

2. Hardy GE, Rogers TG: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 1973, 16: 201-206. 10.4153/CMB-1973-036-0

3. Kannan R: Some remarks on fixed points. Bull. Calcutta Math. Soc. 1960, 60: 71-76.

4. Karapinar E: Some fixed point theorems on the class of comparable partial metric spaces. Appl. Gen. Topol. 2011,12(2):187-192.

5. Reich S: Kannanâ€™s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1-11.

6. Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1997, 226: 257-290.

7. Turinici M: Fixed points of implicit contraction mappings. An. ÅžtiinÅ£. Univ. â€˜Al.I. Cuzaâ€™ IaÅŸi, Mat. 1976, 22: 177-180.

8. Popa V: Fixed point theorems for implicit contractive mappings. Stud. Cercet. ÅžtiinÅ£. - Univ. BacÄƒu, Ser. Mat. 1997, 7: 127-133.

9. Popa V: Some fixed point theorems for compatible mappings satisfying an implicit relation. Demonstr. Math. 1999, 32: 157-163.

10. Berinde V: Stability of Picard iteration for contractive mappings satisfying an implicit relation. Carpath. J. Math. 2011,27(1):13-23.

11. Samet B, Vetro C, Vetro P: Fixed point theorems for Î± - Ïˆ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154-2165. 10.1016/j.na.2011.10.014

12. Agarwal RP, Oâ€™Regan D: Fixed point theory for generalized contractions on spaces with two metrics. J. Math. Anal. Appl. 2000, 248: 402-414. 10.1006/jmaa.2000.6914

13. Maia MG: Unâ€™osservazione sulle contrazioni metrich. Rend. Semin. Mat. Univ. Padova 1968, 40: 139-143.

14. Nieto JJ, RodrÃ­guez-LÃ³pez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205-2212. 10.1007/s10114-005-0769-0

15. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4

16. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003,4(1):79-89.

## Acknowledgements

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

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Samet, B. Fixed point results for implicit contractions on spaces with two metrics. J Inequal Appl 2014, 84 (2014). https://doi.org/10.1186/1029-242X-2014-84