# Fixed point theorems of generalized cyclic orbital Meir-Keeler contractions

## Abstract

In this paper, we introduce two class of generalized cyclic orbital Meir-Keeler contractions and we study the existence and uniqueness of fixed points for these mappings. Our results in this paper extend and generalize several existing fixed-point theorems in the literature.

MSC:47H10, 54C60, 54H25, 55M20.

## 1 Introduction and preliminaries

Throughout this paper, by ${\mathbb{R}}^{+}$, we denote the set of all non-negative numbers, while â„• is the set of all natural numbers. It is well known and easy to prove that if $\left(X,d\right)$ is a complete metric space, and if $f:Xâ†’X$ is continuous and f satisfies

then f has a fixed point in X. Using the above conclusion, Kirk, Srinivasan and Veeramani [1] proved the following fixed-point theorem.

Theorem 1 [1]

Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and suppose $f:AâˆªBâ†’AâˆªB$ satisfies

1. (i)

$f\left(A\right)âŠ‚B$ and $f\left(B\right)âŠ‚A$,

2. (ii)

$d\left(fx,fy\right)â‰¤kâ‹\dots d\left(x,y\right)$ for all $xâˆˆA$, $yâˆˆB$ and $kâˆˆ\left(0,1\right)$.

Then $Aâˆ©B$ is nonempty and f has a unique fixed point in $Aâˆ©B$.

The following definitions and results will be needed in the sequel. Let A and B be two nonempty subsets of a metric space $\left(X,d\right)$. A mapping $f:AâˆªBâ†’AâˆªB$ is called a cyclic map if $f\left(A\right)âŠ†B$ and $f\left(B\right)âŠ†A$. In 2010, Karpagam and Agrawal [2] introduced the notion of cyclic orbital contraction, and obtained a unique fixed point theorem for such a map.

Definition 1 [2]

Let A and B be nonempty subsets of a metric space $\left(X,d\right)$, $f:AâˆªBâ†’AâˆªB$ be a cyclic map such that for some $xâˆˆA$, there exists a ${\mathrm{Îº}}_{x}âˆˆ\left(0,1\right)$ such that

$d\left({f}^{2n}x,fy\right)â‰¤{\mathrm{Îº}}_{x}â‹\dots d\left({f}^{2nâˆ’1}x,y\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},yâˆˆA.$

Then f is called a cyclic orbital contraction.

Theorem 2 [2]

Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and let $f:AâˆªBâ†’AâˆªB$ be a cyclic orbital contraction. Then f has a fixed point in $Aâˆ©B$.

Further, many results dealing with cyclic contractions have appeared in the literature (see, e.g., [3â€“16]).

In 2012, Chen [17] introduced the below notion of cyclic orbital stronger Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

Definition 2 [17]

Let $\left(X,d\right)$ be a metric space. We call $\mathrm{Ïˆ}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ a stronger Meir-Keeler type mapping in X if the mapping Ïˆ satisfies the following condition:

$\mathrm{âˆ€}\mathrm{Î·}>0,\mathrm{âˆƒ}\mathrm{Î´}>0,\mathrm{âˆƒ}{\mathrm{Î³}}_{\mathrm{Î·}}âˆˆ\left[0,1\right),\mathrm{âˆ€}x,yâˆˆX\phantom{\rule{1em}{0ex}}\left(\mathrm{Î·}â‰¤d\left(x,y\right)<\mathrm{Î´}+\mathrm{Î·}â‡’\mathrm{Ïˆ}\left(d\left(x,y\right)\right)<{\mathrm{Î³}}_{\mathrm{Î·}}\right).$

Definition 3 [17]

Let A and B be nonempty subsets of a metric space $\left(X,d\right)$. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic map such that for some $xâˆˆA$, there exists a stronger Meir-Keeler type mapping ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ in X such that

$d\left({f}^{2n}x,fy\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,y\right)\right)â‹\dots d\left({f}^{2nâˆ’1}x,y\right),$

for all $nâˆˆ\mathbb{N}$ and $yâˆˆA$. Then f is called a cyclic orbital stronger Meir-Keeler ${\mathrm{Ïˆ}}_{x}$-contraction.

Clearly, if $f:AâˆªBâ†’AâˆªB$ is a cyclic orbital contraction, then f is a cyclic orbital stronger Meir-Keeler ${\mathrm{Ïˆ}}_{x}$-contraction, where ${\mathrm{Ïˆ}}_{x}\left(t\right)={k}_{x}$ for all $tâˆˆ{\mathbb{R}}^{+}$.

Theorem 3 [17]

Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and let ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ be a stronger Meir-Keeler type mapping in X. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic orbital stronger Meir-Keeler ${\mathrm{Ïˆ}}_{x}$-contraction. Then $Aâˆ©B$ is nonempty and f has a unique fixed point in $Aâˆ©B$.

Chen [17] also introduced the below notion of cyclic orbital weaker Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

Definition 4 [17]

Let $\left(X,d\right)$ be a metric space, and $\mathrm{Ïˆ}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$. Then Ïˆ is called a weaker Meir-Keeler type mapping in X, if the mapping Ïˆ satisfies the following condition:

$\mathrm{âˆ€}\mathrm{Î·}>0,\mathrm{âˆƒ}\mathrm{Î´}>0,\mathrm{âˆ€}x,yâˆˆX\phantom{\rule{1em}{0ex}}\left(\mathrm{Î·}â‰¤d\left(x,y\right)<\mathrm{Î´}+\mathrm{Î·}â‡’\mathrm{âˆƒ}{n}_{0}âˆˆ\mathbb{N}\phantom{\rule{0.25em}{0ex}}{\mathrm{Ïˆ}}^{{n}_{0}}\left(d\left(x,y\right)\right)<\mathrm{Î·}\right).$

Definition 5 [17]

Let $\left(X,d\right)$ be a metric space. We call $f:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ a Ïˆ-mapping in X if the function f satisfies the following conditions:

(${\mathrm{Ïˆ}}_{1}$) f is a weaker Meir-Keeler type mapping in X with $f\left(0\right)=0$;

(${\mathrm{Ïˆ}}_{2}$)

1. (a)

if ${lim}_{nâ†’\mathrm{âˆž}}{t}_{n}=\mathrm{Î³}>0$, then ${lim}_{nâ†’\mathrm{âˆž}}f\left({t}_{n}\right)â‰¤\mathrm{Î³}$, and

2. (b)

if ${lim}_{nâ†’\mathrm{âˆž}}{t}_{n}=0$, then ${lim}_{nâ†’\mathrm{âˆž}}f\left({t}_{n}\right)=0$;

(${\mathrm{Ïˆ}}_{3}$) ${\left\{{f}^{n}\left(t\right)\right\}}_{nâˆˆ\mathbb{N}}$ is decreasing, for each $tâˆˆ{\mathbb{R}}^{+}\mathrm{âˆ–}\left\{0\right\}$.

Definition 6 [17]

Let A and B be nonempty subsets of a metric space $\left(X,d\right)$. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic map such that for some $xâˆˆA$, there exists a Ïˆ-mapping ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ in X such that

$d\left({f}^{2n}x,fy\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,y\right)\right),$

for all $nâˆˆ\mathbb{N}$ and $yâˆˆA$. Then f is called a cyclic orbital weaker Meir-Keeler ${\mathrm{Ïˆ}}_{x}$-contraction.

Theorem 4 [17]

Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and let ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ be a Ïˆ-mapping in X. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic orbital weaker Meir-Keeler ${\mathrm{Ïˆ}}_{x}$-contraction. Then $Aâˆ©B$ is nonempty and f has a unique fixed point in $Aâˆ©B$.

## 2 Fixed-point theorems (I)

In this section, we will introduce the class of generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction and we study the existence and uniqueness of fixed points for such mappings. Our results in this section extend and generalize several existing fixed-point theorems in the literature, including Theorem 2 and Theorem 3.

In the sequel, we denote by Î˜ the class of functions $\mathrm{Ï†}:{{\mathbb{R}}^{+}}^{5}â†’{\mathbb{R}}^{+}$ satisfying the following conditions:

(${\mathrm{Ï†}}_{1}$) Ï† is a strictly increasing, continuous function in each coordinate;

(${\mathrm{Ï†}}_{2}$) for all $t>0$, $\mathrm{Ï†}\left(t,t,t,0,2t\right), $\mathrm{Ï†}\left(t,t,t,2t,0\right), $\mathrm{Ï†}\left(t,0,0,t,t\right), $\mathrm{Ï†}\left(0,0,t,t,0\right), and $\mathrm{Ï†}\left(0,0,0,0,0\right)=0$.

Example 1 Let $\mathrm{Ï†}:{{\mathbb{R}}^{+}}^{5}â†’{\mathbb{R}}^{+}$ denote

$\mathrm{Ï†}\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5}\right)=\frac{2}{3}â‹\dots max\left\{{t}_{1},{t}_{2},{t}_{3},\frac{1}{2}{t}_{4},\frac{1}{2}{t}_{5}\right\}.$

Then Ï† satisfies the above conditions $\left({\mathrm{Ï†}}_{1}\right)$ and $\left({\mathrm{Ï†}}_{2}\right)$.

We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction.

Definition 7 Let A and B be nonempty subsets of a metric space $\left(X,d\right)$. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic map such that for some $xâˆˆA$, there exist a stronger Meir-Keeler type mapping ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ in X and $\mathrm{Ï†}âˆˆ\mathrm{Î˜}$ such that

$d\left({f}^{2n}x,fy\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,y\right)\right)â‹\dots \mathrm{Î¸},$

where

$\mathrm{Î¸}=\mathrm{Ï†}\left(d\left({f}^{2nâˆ’1}x,y\right),d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right),d\left(fy,y\right),d\left({f}^{2nâˆ’1}x,fy\right),d\left({f}^{2n}x,y\right)\right)$

for all $nâˆˆ\mathbb{N}$ and $yâˆˆA$. Then f is called a generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction.

Our main result is the following.

Theorem 5 Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and let ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ be a stronger Meir-Keeler type mapping in X and $\mathrm{Ï†}âˆˆ\mathrm{Î˜}$. Suppose $f:AâˆªBâ†’AâˆªB$ is a generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction. Then $Aâˆ©B$ is nonempty and f has a unique fixed point in $Aâˆ©B$.

Proof Since $f:AâˆªBâ†’AâˆªB$ is a generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction and for $xâˆˆA$, we have ${f}^{2n}xâˆˆA$. Put $y={f}^{2n}x$, for $nâˆˆ\mathbb{N}$. Then we have that for each $nâˆˆ\mathbb{N}$

and by the conditions of the function Ï†, we get

$\mathrm{Î¸}

and

$\begin{array}{rl}d\left({f}^{2n}x,{f}^{2n+1}x\right)& <{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\right)â‹\dots d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\\ â‰¤d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right).\end{array}$
(2.1)

Similarly, we put $y={f}^{2n}x$ and for each $nâˆˆ\mathbb{N}$

and by the conditions of the function Ï†, we get

$\mathrm{Î¸}

and

$\begin{array}{rl}d\left({f}^{2n+1}x,{f}^{2n+2}x\right)& <{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right)â‹\dots d\left({f}^{2n}x,{f}^{2n+1}x\right)\\ â‰¤d\left({f}^{2n}x,{f}^{2n+1}x\right).\end{array}$
(2.2)

Using inequalities (2.1) and (2.2), we deduce that $\left\{d\left({f}^{n}x,{f}^{n+1}x\right)\right\}$ is a decreasing sequence and hence it is convergent. Let ${lim}_{nâ†’\mathrm{âˆž}}d\left({f}^{n}x,{f}^{n+1}x\right)=\mathrm{Î·}$. Then there exists ${\mathrm{Îº}}_{0}âˆˆ\mathbb{N}$ and $\mathrm{Î´}>0$ such that for all $nâ‰¥{\mathrm{Îº}}_{0}$,

$\mathrm{Î·}â‰¤d\left({f}^{n}x,{f}^{n+1}x\right)<\mathrm{Î·}+\mathrm{Î´}.$

Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ${\mathrm{Ïˆ}}_{x}$ in X, corresponding to Î· use, there exists ${\mathrm{Î³}}_{\mathrm{Î·}}âˆˆ\left[0,1\right)$ such that

(2.3)

Put ${n}_{0}=\left[\frac{{\mathrm{Îº}}_{0}+3}{2}\right]$, where $\left[\frac{{\mathrm{Îº}}_{0}+3}{2}\right]$ is the integer part of $\frac{{\mathrm{Îº}}_{0}+3}{2}$. It follows from (2.1), (2.2) and (2.3) that we deduce that for all $nâ‰¥{n}_{0}$,

$\begin{array}{rcl}d\left({f}^{2n}x,{f}^{2n+1}x\right)& <& {\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\right)â‹\dots d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\\ <& {\mathrm{Î³}}_{\mathrm{Î·}}â‹\dots d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right),\end{array}$
(2.4)

and

$\begin{array}{rcl}d\left({f}^{2n+1}x,{f}^{2n+2}x\right)& <& {\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right)â‹\dots d\left({f}^{2n}x,{f}^{2n+1}x\right)\\ <& {\mathrm{Î³}}_{\mathrm{Î·}}â‹\dots d\left({f}^{2n}x,{f}^{2n+1}x\right).\end{array}$
(2.5)

It follows from (2.4) and (2.5) that for each $nâˆˆ\mathbb{N}âˆª\left\{0\right\}$

$d\left({f}^{2{n}_{0}+n}x,{f}^{2{n}_{0}+n+1}x\right)<{\mathrm{Î³}}_{\mathrm{Î·}}^{n}â‹\dots d\left({f}^{2{n}_{0}âˆ’1}x,{f}^{2{n}_{0}}x\right).$
(2.6)

Since ${\mathrm{Î³}}_{\mathrm{Î·}}<1$, we get

$\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2{n}_{0}+n}x,{f}^{2{n}_{0}+n+1}x\right)=0.$

For $m,nâˆˆ\mathbb{N}$ with $m>n$, we have

$d\left({f}^{2{n}_{0}+n}x,{f}^{2{n}_{0}+m}x\right)â‰¤\underset{i=n}{\overset{mâˆ’1}{âˆ‘}}d\left({f}^{2{n}_{0}+i}x,{f}^{2{n}_{0}+i+1}x\right)<\frac{{\mathrm{Î³}}_{\mathrm{Î·}}^{mâˆ’1}}{1âˆ’{\mathrm{Î³}}_{\mathrm{Î·}}}d\left({f}^{2{n}_{0}}x,{f}^{2{n}_{0}+1}x\right),$

and hence $d\left({f}^{n}x,{f}^{m}x\right)â†’0$, since $0<{\mathrm{Î³}}_{\mathrm{Î·}}<1$. So, $\left\{{f}^{n}x\right\}$ is a Cauchy sequence. Since $\left(X,d\right)$ is a complete metric space, A and B are closed, $\left\{{f}^{n}x\right\}âŠ‚AâˆªB$, there exists $\mathrm{Î½}âˆˆAâˆªB$ such that ${lim}_{nâ†’\mathrm{âˆž}}{f}^{n}x=\mathrm{Î½}$. Now $\left\{{f}^{2n}x\right\}$ is a sequence in A and $\left\{{f}^{2n+1}x\right\}$ is a sequence in B, and also both converge to Î½. Since A and B are closed, $\mathrm{Î½}âˆˆAâˆ©B$, and so $Aâˆ©B$ is nonempty. Next, we want to show that Î½ is a fixed point of f. Suppose that Î½ is not a fixed point of f. Then $d\left(\mathrm{Î½},f\mathrm{Î½}\right)>0$. Since ${lim}_{nâ†’\mathrm{âˆž}}d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right)=0$ and

$d\left({f}^{2n}x,f\mathrm{Î½}\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right)\right)â‹\dots \mathrm{Î¸},$

where

$\mathrm{Î¸}=\mathrm{Ï†}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right),d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right),d\left(f\mathrm{Î½},\mathrm{Î½}\right),d\left({f}^{2nâˆ’1}x,f\mathrm{Î½}\right),d\left({f}^{2n}x,\mathrm{Î½}\right)\right),$

we obtain that

$\begin{array}{rl}d\left(\mathrm{Î½},f\mathrm{Î½}\right)& =\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2n}x,f\mathrm{Î½}\right)\\ â‰¤{\mathrm{Î³}}_{\mathrm{Î·}}â‹\dots \mathrm{Ï†}\left(d\left(\mathrm{Î½},\mathrm{Î½}\right),d\left(\mathrm{Î½},\mathrm{Î½}\right),d\left(f\mathrm{Î½},\mathrm{Î½}\right),d\left(\mathrm{Î½},f\mathrm{Î½}\right),d\left(\mathrm{Î½},\mathrm{Î½}\right)\right)\\ â‰¤\mathrm{Ï†}\left(0,0,d\left(\mathrm{Î½},f\mathrm{Î½}\right),d\left(\mathrm{Î½},f\mathrm{Î½}\right),0\right)\\

This leads to a contradiction. So, $d\left(\mathrm{Î½},f\mathrm{Î½}\right)=0$, that is, Î½ is a fixed point of f.

Finally, we want to show the uniqueness of the fixed point. Let Î¼ be another fixed point of f. By the cyclic character of f, we have $\mathrm{Î½},\mathrm{Î¼}âˆˆAâˆ©B$. Since f is a generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction, we have

$d\left(\mathrm{Î½},\mathrm{Î¼}\right)=d\left(\mathrm{Î½},f\mathrm{Î¼}\right)=\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2n}x,f\mathrm{Î¼}\right),$
(2.7)

and

$d\left({f}^{2n}x,f\mathrm{Î¼}\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î¼}\right)\right)â‹\dots \mathrm{Î¸}<{\mathrm{Î³}}_{\mathrm{Î·}}â‹\dots \mathrm{Î¸},$
(2.8)

where

$\mathrm{Î¸}=\mathrm{Ï†}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î¼}\right),d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right),d\left(f\mathrm{Î¼},\mathrm{Î¼}\right),d\left({f}^{2nâˆ’1}x,f\mathrm{Î¼}\right),d\left({f}^{2n}x,\mathrm{Î¼}\right)\right).$

It follows from (2.7), (2.8) and the condition $\left({\mathrm{Ï†}}_{2}\right)$ of the mapping Ï† that

$\begin{array}{rl}d\left(\mathrm{Î½},\mathrm{Î¼}\right)& <{\mathrm{Î³}}_{\mathrm{Î·}}â‹\dots \mathrm{Ï†}\left(d\left(\mathrm{Î½},\mathrm{Î¼}\right),d\left(\mathrm{Î½},\mathrm{Î½}\right),d\left(f\mathrm{Î¼},\mathrm{Î¼}\right),d\left(\mathrm{Î½},f\mathrm{Î¼}\right),d\left(\mathrm{Î½},\mathrm{Î¼}\right)\right)\\ â‰¤\mathrm{Ï†}\left(d\left(\mathrm{Î½},\mathrm{Î¼}\right),0,0,d\left(\mathrm{Î½},\mathrm{Î¼}\right),d\left(\mathrm{Î½},\mathrm{Î¼}\right)\right)\\

This leads to a contradiction. Therefore, $\mathrm{Î½}=\mathrm{Î¼}$, and so Î½ is the unique fixed point of f.â€ƒâ–¡

We give the following example to illustrate Theorem 5.

Example 2 Let $A=B=X={\mathbb{R}}^{+}$ and we define $d:XÃ—Xâ†’{\mathbb{R}}^{+}$ by

and let $f:Xâ†’X$ denote

We next define ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’\left[0,1\right)$ by

and let $\mathrm{Ï†}:{{\mathbb{R}}^{+}}^{5}â†’{\mathbb{R}}^{+}$ denote

$\mathrm{Ï†}\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5}\right)=\frac{1}{2}â‹\dots max\left\{{t}_{1},{t}_{2},{t}_{3},\frac{1}{2}{t}_{4},\frac{1}{2}{t}_{5}\right\}.$

Then f is a generalized cyclic orbital stronger Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï†}\right)$-contraction and 0 is the unique fixed point.

## 3 Fixed-point theorems (II)

In this section, we will introduce the class of generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction and we study the existence and uniqueness of fixed points for such mappings.

In the sequel, we denote by Î¦ the class of functions $\mathrm{Ï•}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ satisfying the following conditions:

(${\mathrm{Ï•}}_{1}$) Ï• is lower semi-continuous, and

(${\mathrm{Ï•}}_{2}$) $\mathrm{Ï•}\left(0\right)=0$ if and only if $t=0$.

Definition 8 Let A and B be nonempty subsets of a metric space $\left(X,d\right)$. Suppose $f:AâˆªBâ†’AâˆªB$ is a cyclic map such that for some $xâˆˆA$, there exist a Ïˆ-mapping ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ in X and $\mathrm{Ï•}âˆˆ\mathrm{Î¦}$ such that

$d\left({f}^{2n}x,fy\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,y\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2nâˆ’1}x,y\right)\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},yâˆˆA.$
(3.1)

Then f is called a generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction.

Our second main result is the following.

Theorem 6 Let A and B be two nonempty closed subsets of a complete metric space $\left(X,d\right)$, and let ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ be a Ïˆ-mapping in X and $\mathrm{Ï•}âˆˆ\mathrm{Î¦}$. Suppose $f:AâˆªBâ†’AâˆªB$ is a generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction. Then $Aâˆ©B$ is nonempty and f has a unique fixed point in $Aâˆ©B$.

Proof Since $f:AâˆªBâ†’AâˆªB$ is a generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction and for $xâˆˆX$, there exist a Ïˆ-mapping ${\mathrm{Ïˆ}}_{x}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ in X and $\mathrm{Ï•}âˆˆ\mathrm{Î¦}$ such that (3.1) is satisfied. Put $y={f}^{2n}x$ for all $nâˆˆ\mathbb{N}$. Then we have that for each $nâˆˆ\mathbb{N}$

$\begin{array}{rl}d\left({f}^{2n}x,{f}^{2n+1}x\right)& â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\right)\\ â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,{f}^{2n}x\right)\right),\end{array}$

and

$\begin{array}{rl}d\left({f}^{2n+1}x,{f}^{2n+2}x\right)& =d\left({f}^{2n+2}x,{f}^{2n+1}x\right)\\ â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right)\\ â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2n+1}x,{f}^{2n}x\right)\right).\end{array}$

Generally, we have that for each $nâˆˆ\mathbb{N}$

$d\left({f}^{n}x,{f}^{n+1}x\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{nâˆ’1}x,{f}^{n}x\right)\right),$

and so we conclude that for each $nâˆˆ\mathbb{N}$

$\begin{array}{rl}d\left({f}^{n}x,{f}^{n+1}x\right)& â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{nâˆ’1}x,{f}^{n}x\right)\right)\\ â‰¤{\mathrm{Ïˆ}}_{x}^{2}\left(d\left({f}^{nâˆ’2}x,{f}^{nâˆ’1}x\right)\right)\\ â‰¤â‹¯\\ â‰¤{\mathrm{Ïˆ}}_{x}^{n}\left(d\left(x,fx\right)\right).\end{array}$

Since ${\left\{{\mathrm{Ïˆ}}_{x}^{n}\left(d\left(x,fx\right)\right)\right\}}_{nâˆˆ\mathbb{N}}$ is decreasing, it must converge to some $\mathrm{Î·}â‰¥0$. We claim that $\mathrm{Î·}=0$. On the contrary, assume that $\mathrm{Î·}>0$. Then by the definition of weaker Meir-Keeler type mapping ${\mathrm{Ïˆ}}_{x}$ in X, there exists $\mathrm{Î´}>0$ such that for $x,yâˆˆX$ with $\mathrm{Î·}â‰¤d\left(x,y\right)<\mathrm{Î´}+\mathrm{Î·}$, there exists ${n}_{0}âˆˆ\mathbb{N}$ such that ${\mathrm{Ïˆ}}_{x}^{{n}_{0}}\left(d\left(x,y\right)\right)<\mathrm{Î·}$. Since ${lim}_{nâ†’\mathrm{âˆž}}{\mathrm{Ïˆ}}_{x}^{n}\left(d\left(x,fx\right)\right)=\mathrm{Î·}$, there exists ${m}_{0}âˆˆ\mathbb{N}$ such that $\mathrm{Î·}â‰¤{\mathrm{Ïˆ}}_{x}^{m}\left(d\left(x,fx\right)\right)<\mathrm{Î´}+\mathrm{Î·}$, for all $mâ‰¥{m}_{0}$. Thus, we conclude that ${\mathrm{Ïˆ}}_{x}^{{m}_{0}+{n}_{0}}\left(d\left({x}_{0},{x}_{1}\right)\right)<\mathrm{Î·}$, and we get a contradiction. So, ${lim}_{nâ†’\mathrm{âˆž}}{\mathrm{Ïˆ}}_{x}^{n}\left(d\left(x,fx\right)\right)=0$, that is,

$\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{n}x,{f}^{n+1}x\right)=0.$
(3.2)

We now claim that $\left\{{f}^{n}x\right\}$ is a Cauchy sequence. It is sufficient to show that $\left\{{f}^{2n}x\right\}$ is a Cauchy sequence. Suppose $\left\{{f}^{2n}x\right\}$ is not Cauchy. Then there exists $\mathrm{Îµ}>0$ such that for all $kâˆˆ\mathbb{N}$, there are ${m}_{k},{n}_{k}âˆˆ\mathbb{N}$ with ${m}_{k}>{n}_{k}â‰¥k$ satisfying:

1. (i)

$d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}}\right)â‰¥\mathrm{Îµ}$, and

2. (ii)

${m}_{k}$ is the smallest number greater than ${n}_{k}$ such that the condition (i) holds.

Using (3.2), we have

$\begin{array}{rl}\mathrm{Îµ}& â‰¤d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}}\right)â‰¤d\left({f}^{2{m}_{k}}x,{f}^{2{m}_{k}âˆ’1}\right)+d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{m}_{k}âˆ’2}\right)+d\left({f}^{2{m}_{k}âˆ’2}x,{f}^{2{n}_{k}}\right)\\ â‰¤d\left({f}^{2{m}_{k}}x,{f}^{2{m}_{k}âˆ’1}\right)+d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{m}_{k}âˆ’2}\right)+\mathrm{Îµ}.\end{array}$

Let $kâ†’\mathrm{âˆž}$, we get

$\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}}\right)=\mathrm{Îµ}.$
(3.3)

On the other hand, applying (3.1) with $y={f}^{2{n}_{k}}x$ for all $kâˆˆ\mathbb{N}$, we get

$d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}+1}\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)\right).$
(3.4)

Since for each $kâˆˆ\mathbb{N}$

$d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}+1}\right)â‰¤d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}}\right)+d\left({f}^{2{n}_{k}}x,{f}^{2{n}_{k}+1}\right),$
(3.5)

and

$d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)â‰¤d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{m}_{k}}\right)+d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}}\right),$
(3.6)

taking $kâ†’\mathrm{âˆž}$ and using the inequalities (3.3), (3.5) and (3.6), we get

$\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}+1}\right)=\mathrm{Îµ},$
(3.7)

and

$\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)=\mathrm{Îµ}.$
(3.8)

Taking into account the inequalities (3.4), (3.7) and (3.8), and by the definitions of the functions Ï• and ${\mathrm{Ïˆ}}_{x}$ , we get

$\begin{array}{rl}\mathrm{Îµ}& =\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2{m}_{k}}x,{f}^{2{n}_{k}+1}\right)\\ â‰¤\underset{nâ†’\mathrm{âˆž}}{lim}{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)\right)âˆ’\underset{nâ†’\mathrm{âˆž}}{lim}\mathrm{Ï•}\left(d\left({f}^{2{m}_{k}âˆ’1}x,{f}^{2{n}_{k}}\right)\right)\\ â‰¤\mathrm{Îµ}âˆ’\mathrm{Ï•}\left(\mathrm{Îµ}\right),\end{array}$

which implies that $\mathrm{Îµ}=0$. Thus, $\left\{{f}^{n}x\right\}$ is a Cauchy sequence.

Since $\left(X,d\right)$ is a complete metric space, A and B are closed, $\left\{{f}^{n}x\right\}âŠ‚AâˆªB$, there exists $\mathrm{Î½}âˆˆAâˆªB$ such that ${lim}_{nâ†’\mathrm{âˆž}}{f}^{n}x=\mathrm{Î½}$. Now $\left\{{f}^{2n}x\right\}$ is a sequence in A and $\left\{{f}^{2n+1}x\right\}$ is a sequence in B, and also both converge to Î½. Since A and B are closed, $\mathrm{Î½}âˆˆAâˆ©B$, and so $Aâˆ©B$ is nonempty. On the other hand, since ${lim}_{nâ†’\mathrm{âˆž}}d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right)=0$ and

$d\left({f}^{2n}x,f\mathrm{Î½}\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î½}\right)\right),$

taking $nâ†’\mathrm{âˆž}$,we obtain that

$d\left(\mathrm{Î½},f\mathrm{Î½}\right)â‰¤0âˆ’\mathrm{Ï•}\left(d\left(\mathrm{Î½},\mathrm{Î½}\right)\right)=0,$

and hence $d\left(\mathrm{Î½},f\mathrm{Î½}\right)=0$, that is, Î½ is a fixed point of f.

Finally, we want to show the uniqueness of the fixed point. Let Î¼ be another fixed point of f. By the cyclic character of f, we have $\mathrm{Î½},\mathrm{Î¼}âˆˆAâˆ©B$. Since f is a generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction, we have

$d\left({f}^{2n}x,f\mathrm{Î¼}\right)â‰¤{\mathrm{Ïˆ}}_{x}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î¼}\right)\right)âˆ’\mathrm{Ï•}\left(d\left({f}^{2nâˆ’1}x,\mathrm{Î¼}\right)\right).$

Letting $nâ†’\mathrm{âˆž}$, and by the definitions of the functions Ï• and ${\mathrm{Ïˆ}}_{x}$, we obtain that

$d\left(\mathrm{Î½},\mathrm{Î¼}\right)=d\left(\mathrm{Î½},f\mathrm{Î¼}\right)=\underset{nâ†’\mathrm{âˆž}}{lim}d\left({f}^{2n}x,f\mathrm{Î¼}\right)â‰¤d\left(\mathrm{Î½},\mathrm{Î¼}\right)âˆ’\mathrm{Ï•}\left(d\left(\mathrm{Î½},\mathrm{Î¼}\right)\right),$

which implies that $d\left(\mathrm{Î½},\mathrm{Î¼}\right)=0$. Therefore, $\mathrm{Î½}=\mathrm{Î¼}$, and so Î½ is the unique fixed point of f.â€ƒâ–¡

We give the following example to illustrate Theorem 6.

Example 3 Let $A=B=X={\mathbb{R}}^{+}$ and we define $d:XÃ—Xâ†’{\mathbb{R}}^{+}$ by

Define $f:Xâ†’X$ by

and define $\mathrm{Ïˆ},\mathrm{Ï•}:{\mathbb{R}}^{+}â†’{\mathbb{R}}^{+}$ by

Then f is a generalized cyclic orbital weaker Meir-Keeler $\left({\mathrm{Ïˆ}}_{x},\mathrm{Ï•}\right)$-contraction and 0 is the unique fixed point.

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## Acknowledgements

This research was supported by the National Science Council of the Republic of China. The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.

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Correspondence to Chi-Ming Chen.

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The author declares that they have no competing interests.

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Chen, CM. Fixed point theorems of generalized cyclic orbital Meir-Keeler contractions. Fixed Point Theory Appl 2013, 91 (2013). https://doi.org/10.1186/1687-1812-2013-91