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 ℕ is the set of all natural numbers. It is well known and easy to prove that if $(X,d)$ is a complete metric space, and if $f:X\to X$ is continuous and f satisfies

$$d(fx,f^{2}x)\le k\cdot d(x,fx),\text{for all }x\in X\text{ and }k\in (0,1),$$

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 $(X,d)$, and suppose $f:A\cup B\to A\cup B$ satisfies

• (i) $f(A)\subset B$ and $f(B)\subset A$,

• (ii) $d(fx,fy)\le k\cdot d(x,y)$ for all $x\in A$, $y\in B$ and $k\in (0,1)$.

Then $A\cap B$ is nonempty and f has a unique fixed point in $A\cap B$.

The following definitions and results will be needed in the sequel. Let A and B be two nonempty subsets of a metric space $(X,d)$. A mapping $f:A\cup B\to A\cup B$ is called a cyclic map if $f(A)\subseteq B$ and $f(B)\subseteq A$. 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 $(X,d)$, $f:A\cup B\to A\cup B$ be a cyclic map such that for some $x\in A$, there exists a ${\kappa }_x\in (0,1)$ such that

$$d(f^{2n}x,fy)\le {\kappa }_x\cdot d(f^{2n-1}x,y),n\in \mathbb{N},y\in A.$$

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 $(X,d)$, and let $f:A\cup B\to A\cup B$ be a cyclic orbital contraction. Then f has a fixed point in $A\cap B$.

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

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 $(X,d)$ be a metric space. We call $\psi :{\mathbb{R}}^{+}\to [0,1)$ a stronger Meir-Keeler type mapping in X if the mapping ψ satisfies the following condition:

$$\mathrm{\forall }\eta >0,\mathrm{\exists }\delta >0,\mathrm{\exists }{\gamma }_{\eta }\in [0,1),\mathrm{\forall }x,y\in X(\eta \le d(x,y)<\delta +\eta \Rightarrow \psi (d(x,y))<{\gamma }_{\eta }).$$

Definition 3 [17]

Let A and B be nonempty subsets of a metric space $(X,d)$. Suppose $f:A\cup B\to A\cup B$ is a cyclic map such that for some $x\in A$, there exists a stronger Meir-Keeler type mapping ${\psi }_x:{\mathbb{R}}^{+}\to [0,1)$ in X such that

$$d(f^{2n}x,fy)\le {\psi }_x\left(d(f^{2n-1}x,y)\right)\cdot d(f^{2n-1}x,y),$$

for all $n\in \mathbb{N}$ and $y\in A$. Then f is called a cyclic orbital stronger Meir-Keeler ${\psi }_x$-contraction.

Clearly, if $f:A\cup B\to A\cup B$ is a cyclic orbital contraction, then f is a cyclic orbital stronger Meir-Keeler ${\psi }_x$-contraction, where ${\psi }_x(t)=k_x$ for all $t\in {\mathbb{R}}^{+}$.

Theorem 3 [17]

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$, and let ${\psi }_x:{\mathbb{R}}^{+}\to [0,1)$ be a stronger Meir-Keeler type mapping in X. Suppose $f:A\cup B\to A\cup B$ is a cyclic orbital stronger Meir-Keeler ${\psi }_x$-contraction. Then $A\cap B$ is nonempty and f has a unique fixed point in $A\cap B$.

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 $(X,d)$ be a metric space, and $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$. Then ψ is called a weaker Meir-Keeler type mapping in X, if the mapping ψ satisfies the following condition:

$$\mathrm{\forall }\eta >0,\mathrm{\exists }\delta >0,\mathrm{\forall }x,y\in X(\eta \le d(x,y)<\delta +\eta \Rightarrow \mathrm{\exists }{n}_0\in \mathbb{N}{\psi }^{n_0}(d(x,y))<\eta ).$$

Definition 5 [17]

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

(${\psi }_1$) f is a weaker Meir-Keeler type mapping in X with $f(0)=0$;

(${\psi }_2$)

• (a) if ${lim}_{n\to \mathrm{\infty }}{t}_n=\gamma >0$, then ${lim}_{n\to \mathrm{\infty }}f(t_n)\le \gamma$, and

• (b) if ${lim}_{n\to \mathrm{\infty }}{t}_n=0$, then ${lim}_{n\to \mathrm{\infty }}f(t_n)=0$;

(${\psi }_3$) ${\{f^n(t)\}}_{n\in \mathbb{N}}$ is decreasing, for each $t\in {\mathbb{R}}^{+}\mathrm{\setminus }\{0\}$.

Definition 6 [17]

Let A and B be nonempty subsets of a metric space $(X,d)$. Suppose $f:A\cup B\to A\cup B$ is a cyclic map such that for some $x\in A$, there exists a ψ-mapping ${\psi }_x:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ in X such that

$$d(f^{2n}x,fy)\le {\psi }_x\left(d(f^{2n-1}x,y)\right),$$

for all $n\in \mathbb{N}$ and $y\in A$. Then f is called a cyclic orbital weaker Meir-Keeler ${\psi }_x$-contraction.

Theorem 4 [17]

Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$, and let ${\psi }_x:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a ψ-mapping in X. Suppose $f:A\cup B\to A\cup B$ is a cyclic orbital weaker Meir-Keeler ${\psi }_x$-contraction. Then $A\cap B$ is nonempty and f has a unique fixed point in $A\cap B$.

2 Fixed-point theorems (I)

In this section, we will introduce the class of generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-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 Θ the class of functions $\phi :{{\mathbb{R}}^{+}}^5\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\phi }_1$) φ is a strictly increasing, continuous function in each coordinate;

(${\phi }_2$) for all $t>0$, $\phi (t,t,t,0,2t)<t$, $\phi (t,t,t,2t,0)<t$, $\phi (t,0,0,t,t)<t$, $\phi (0,0,t,t,0)<t$, and $\phi (0,0,0,0,0)=0$.

Example 1 Let $\phi :{{\mathbb{R}}^{+}}^5\to {\mathbb{R}}^{+}$ denote

$$\phi (t_1,t_2,t_3,t_4,t_5)=\frac{2}{3}\cdot max\{t_1,t_2,t_3,\frac{1}{2}{t}_4,\frac{1}{2}{t}_5\}.$$

Then φ satisfies the above conditions $({\phi }_1)$ and $({\phi }_2)$.

We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-contraction.

Definition 7 Let A and B be nonempty subsets of a metric space $(X,d)$. Suppose $f:A\cup B\to A\cup B$ is a cyclic map such that for some $x\in A$, there exist a stronger Meir-Keeler type mapping ${\psi }_x:{\mathbb{R}}^{+}\to [0,1)$ in X and $\phi \in \mathrm{\Theta }$ such that

$$d(f^{2n}x,fy)\le {\psi }_x\left(d(f^{2n-1}x,y)\right)\cdot \theta ,$$

where

$$\theta =\phi (d(f^{2n-1}x,y),d(f^{2n-1}x,f^{2n}x),d(fy,y),d(f^{2n-1}x,fy),d(f^{2n}x,y))$$

for all $n\in \mathbb{N}$ and $y\in A$. Then f is called a generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-contraction.

Our main result is the following.

Theorem 5 Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$, and let ${\psi }_x:{\mathbb{R}}^{+}\to [0,1)$ be a stronger Meir-Keeler type mapping in X and $\phi \in \mathrm{\Theta }$. Suppose $f:A\cup B\to A\cup B$ is a generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-contraction. Then $A\cap B$ is nonempty and f has a unique fixed point in $A\cap B$.

Proof Since $f:A\cup B\to A\cup B$ is a generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-contraction and for $x\in A$, we have $f^{2n}x\in A$. Put $y=f^{2n}x$, for $n\in \mathbb{N}$. Then we have that for each $n\in \mathbb{N}$

and by the conditions of the function φ, we get

$$\theta <d(f^{2n-1}x,f^{2n}x),$$

and

$$\begin{array}{rl}d(f^{2n}x,f^{2n+1}x)& <{\psi }_x\left(d(f^{2n-1}x,f^{2n}x)\right)\cdot d(f^{2n-1}x,f^{2n}x)\\ \le d(f^{2n-1}x,f^{2n}x).\end{array}$$

(2.1)

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

and by the conditions of the function φ, we get

$$\theta <d(f^{2n}x,f^{2n+1}x),$$

and

$$\begin{array}{rl}d(f^{2n+1}x,f^{2n+2}x)& <{\psi }_x\left(d(f^{2n+1}x,f^{2n}x)\right)\cdot d(f^{2n}x,f^{2n+1}x)\\ \le d(f^{2n}x,f^{2n+1}x).\end{array}$$

(2.2)

Using inequalities (2.1) and (2.2), we deduce that $\{d(f^{n}x,f^{n+1}x)\}$ is a decreasing sequence and hence it is convergent. Let ${lim}_{n\to \mathrm{\infty }}d(f^{n}x,f^{n+1}x)=\eta$. Then there exists ${\kappa }_0\in \mathbb{N}$ and $\delta >0$ such that for all $n\ge {\kappa }_0$,

$$\eta \le d(f^{n}x,f^{n+1}x)<\eta +\delta .$$

Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ${\psi }_x$ in X, corresponding to η use, there exists ${\gamma }_{\eta }\in [0,1)$ such that

$${\psi }_x\left(d(f^{n}x,f^{n+1}x)\right)<{\gamma }_{\eta }\text{for all }n\ge {\kappa }_0.$$

(2.3)

Put $n_0=[\frac{{\kappa }_0+3}{2}]$, where $[\frac{{\kappa }_0+3}{2}]$ is the integer part of $\frac{{\kappa }_0+3}{2}$. It follows from (2.1), (2.2) and (2.3) that we deduce that for all $n\ge n_0$,

$$\begin{array}{rcl}d(f^{2n}x,f^{2n+1}x)& <& {\psi }_x\left(d(f^{2n-1}x,f^{2n}x)\right)\cdot d(f^{2n-1}x,f^{2n}x)\\ <& {\gamma }_{\eta }\cdot d(f^{2n-1}x,f^{2n}x),\end{array}$$

(2.4)

and

$$\begin{array}{rcl}d(f^{2n+1}x,f^{2n+2}x)& <& {\psi }_x\left(d(f^{2n+1}x,f^{2n}x)\right)\cdot d(f^{2n}x,f^{2n+1}x)\\ <& {\gamma }_{\eta }\cdot d(f^{2n}x,f^{2n+1}x).\end{array}$$

(2.5)

It follows from (2.4) and (2.5) that for each $n\in \mathbb{N}\cup \{0\}$

$$d(f^{2n_0+n}x,f^{2n_0+n+1}x)<{\gamma }_{\eta }^n\cdot d(f^{2n_0-1}x,f^{2n_0}x).$$

(2.6)

Since ${\gamma }_{\eta }<1$, we get

$$\underset{n\to \mathrm{\infty }}{lim}d(f^{2n_0+n}x,f^{2n_0+n+1}x)=0.$$

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

$$d(f^{2n_0+n}x,f^{2n_0+m}x)\le \sum _{i=n}^{m-1}d(f^{2n_0+i}x,f^{2n_0+i+1}x)<\frac{{\gamma }_{\eta }^{m-1}}{1-{\gamma }_{\eta }}d(f^{2n_0}x,f^{2n_0+1}x),$$

and hence $d(f^{n}x,f^{m}x)\to 0$, since $0<{\gamma }_{\eta }<1$. So, $\{f^{n}x\}$ is a Cauchy sequence. Since $(X,d)$ is a complete metric space, A and B are closed, $\{f^{n}x\}\subset A\cup B$, there exists $\nu \in A\cup B$ such that ${lim}_{n\to \mathrm{\infty }}{f}^{n}x=\nu$. Now $\{f^{2n}x\}$ is a sequence in A and $\{f^{2n+1}x\}$ is a sequence in B, and also both converge to ν. Since A and B are closed, $\nu \in A\cap B$, and so $A\cap B$ is nonempty. Next, we want to show that ν is a fixed point of f. Suppose that ν is not a fixed point of f. Then $d(\nu ,f\nu )>0$. Since ${lim}_{n\to \mathrm{\infty }}d(f^{2n-1}x,\nu )=0$ and

$$d(f^{2n}x,f\nu )\le {\psi }_x\left(d(f^{2n-1}x,\nu )\right)\cdot \theta ,$$

where

$$\theta =\phi (d(f^{2n-1}x,\nu ),d(f^{2n-1}x,f^{2n}x),d(f\nu ,\nu ),d(f^{2n-1}x,f\nu ),d(f^{2n}x,\nu )),$$

we obtain that

$$\begin{array}{rl}d(\nu ,f\nu )& =\underset{n\to \mathrm{\infty }}{lim}d(f^{2n}x,f\nu )\\ \le {\gamma }_{\eta }\cdot \phi (d(\nu ,\nu ),d(\nu ,\nu ),d(f\nu ,\nu ),d(\nu ,f\nu ),d(\nu ,\nu ))\\ \le \phi (0,0,d(\nu ,f\nu ),d(\nu ,f\nu ),0)\\ <d(\nu ,f\nu ).\end{array}$$

This leads to a contradiction. So, $d(\nu ,f\nu )=0$, that is, ν is a fixed point of f.

Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have $\nu ,\mu \in A\cap B$. Since f is a generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-contraction, we have

$$d(\nu ,\mu )=d(\nu ,f\mu )=\underset{n\to \mathrm{\infty }}{lim}d(f^{2n}x,f\mu ),$$

(2.7)

and

$$d(f^{2n}x,f\mu )\le {\psi }_x\left(d(f^{2n-1}x,\mu )\right)\cdot \theta <{\gamma }_{\eta }\cdot \theta ,$$

(2.8)

where

$$\theta =\phi (d(f^{2n-1}x,\mu ),d(f^{2n-1}x,f^{2n}x),d(f\mu ,\mu ),d(f^{2n-1}x,f\mu ),d(f^{2n}x,\mu )).$$

It follows from (2.7), (2.8) and the condition $({\phi }_2)$ of the mapping φ that

$$\begin{array}{rl}d(\nu ,\mu )& <{\gamma }_{\eta }\cdot \phi (d(\nu ,\mu ),d(\nu ,\nu ),d(f\mu ,\mu ),d(\nu ,f\mu ),d(\nu ,\mu ))\\ \le \phi (d(\nu ,\mu ),0,0,d(\nu ,\mu ),d(\nu ,\mu ))\\ <d(\nu ,\mu ).\end{array}$$

This leads to a contradiction. Therefore, $\nu =\mu$, and so ν is the unique fixed point of f. □

We give the following example to illustrate Theorem 5.

Example 2 Let $A=B=X={\mathbb{R}}^{+}$ and we define $d:X\times X\to {\mathbb{R}}^{+}$ by

$$d(x,y)=|x-y|,\text{for }x,y\in X,$$

and let $f:X\to X$ denote

$$f(x)=\{\begin{array}{cc}0,\hfill & \text{if }0\le x<1;\hfill \\ \frac{1}{16},\hfill & \text{if }x\ge 1.\hfill \end{array}$$

We next define ${\psi }_x:{\mathbb{R}}^{+}\to [0,1)$ by

$${\psi }_x(t)=\{\begin{array}{cc}\frac{1}{3},\hfill & \text{if }0\le t\le 1;\hfill \\ \frac{t}{t+1},\hfill & \text{if }t>1,\hfill \end{array}$$

and let $\phi :{{\mathbb{R}}^{+}}^5\to {\mathbb{R}}^{+}$ denote

$$\phi (t_1,t_2,t_3,t_4,t_5)=\frac{1}{2}\cdot max\{t_1,t_2,t_3,\frac{1}{2}{t}_4,\frac{1}{2}{t}_5\}.$$

Then f is a generalized cyclic orbital stronger Meir-Keeler $({\psi }_x,\phi )$-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 $({\psi }_x,\varphi )$-contraction and we study the existence and uniqueness of fixed points for such mappings.

In the sequel, we denote by Φ the class of functions $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\varphi }_1$) ϕ is lower semi-continuous, and

(${\varphi }_2$) $\varphi (0)=0$ if and only if $t=0$.

Definition 8 Let A and B be nonempty subsets of a metric space $(X,d)$. Suppose $f:A\cup B\to A\cup B$ is a cyclic map such that for some $x\in A$, there exist a ψ-mapping ${\psi }_x:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ in X and $\varphi \in \mathrm{\Phi }$ such that

$$d(f^{2n}x,fy)\le {\psi }_x\left(d(f^{2n-1}x,y)\right)-\varphi \left(d(f^{2n-1}x,y)\right),n\in \mathbb{N},y\in A.$$

(3.1)

Then f is called a generalized cyclic orbital weaker Meir-Keeler $({\psi }_x,\varphi )$-contraction.

Our second main result is the following.

Theorem 6 Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$, and let ${\psi }_x:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a ψ-mapping in X and $\varphi \in \mathrm{\Phi }$. Suppose $f:A\cup B\to A\cup B$ is a generalized cyclic orbital weaker Meir-Keeler $({\psi }_x,\varphi )$-contraction. Then $A\cap B$ is nonempty and f has a unique fixed point in $A\cap B$.

Proof Since $f:A\cup B\to A\cup B$ is a generalized cyclic orbital weaker Meir-Keeler $({\psi }_x,\varphi )$-contraction and for $x\in X$, there exist a ψ-mapping ${\psi }_x:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ in X and $\varphi \in \mathrm{\Phi }$ such that (3.1) is satisfied. Put $y=f^{2n}x$ for all $n\in \mathbb{N}$. Then we have that for each $n\in \mathbb{N}$

$$\begin{array}{rl}d(f^{2n}x,f^{2n+1}x)& \le {\psi }_x\left(d(f^{2n-1}x,f^{2n}x)\right)-\varphi \left(d(f^{2n-1}x,f^{2n}x)\right)\\ \le {\psi }_x\left(d(f^{2n-1}x,f^{2n}x)\right),\end{array}$$

and

$$\begin{array}{rl}d(f^{2n+1}x,f^{2n+2}x)& =d(f^{2n+2}x,f^{2n+1}x)\\ \le {\psi }_x\left(d(f^{2n+1}x,f^{2n}x)\right)-\varphi \left(d(f^{2n+1}x,f^{2n}x)\right)\\ \le {\psi }_x\left(d(f^{2n+1}x,f^{2n}x)\right).\end{array}$$

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

$$d(f^{n}x,f^{n+1}x)\le {\psi }_x\left(d(f^{n-1}x,f^{n}x)\right),$$

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

$$\begin{array}{rl}d(f^{n}x,f^{n+1}x)& \le {\psi }_x\left(d(f^{n-1}x,f^{n}x)\right)\\ \le {\psi }_x^2\left(d(f^{n-2}x,f^{n-1}x)\right)\\ \le \cdots \\ \le {\psi }_x^n(d(x,fx)).\end{array}$$

Since ${\{{\psi }_x^n(d(x,fx))\}}_{n\in \mathbb{N}}$ is decreasing, it must converge to some $\eta \ge 0$. We claim that $\eta =0$. On the contrary, assume that $\eta >0$. Then by the definition of weaker Meir-Keeler type mapping ${\psi }_x$ in X, there exists $\delta >0$ such that for $x,y\in X$ with $\eta \le d(x,y)<\delta +\eta$, there exists $n_0\in \mathbb{N}$ such that ${\psi }_x^{n_0}(d(x,y))<\eta$. Since ${lim}_{n\to \mathrm{\infty }}{\psi }_x^n(d(x,fx))=\eta$, there exists $m_0\in \mathbb{N}$ such that $\eta \le {\psi }_x^m(d(x,fx))<\delta +\eta$, for all $m\ge m_0$. Thus, we conclude that ${\psi }_x^{m_0+n_0}(d(x_0,x_1))<\eta$, and we get a contradiction. So, ${lim}_{n\to \mathrm{\infty }}{\psi }_x^n(d(x,fx))=0$, that is,

$$\underset{n\to \mathrm{\infty }}{lim}d(f^{n}x,f^{n+1}x)=0.$$

(3.2)

We now claim that $\{f^{n}x\}$ is a Cauchy sequence. It is sufficient to show that $\{f^{2n}x\}$ is a Cauchy sequence. Suppose $\{f^{2n}x\}$ is not Cauchy. Then there exists $\epsilon >0$ such that for all $k\in \mathbb{N}$, there are $m_k,n_k\in \mathbb{N}$ with $m_k>n_k\ge k$ satisfying:

• (i) $d(f^{2m_k}x,f^{2n_k})\ge \epsilon$, and

• (ii) $m_k$ is the smallest number greater than $n_k$ such that the condition (i) holds.

Using (3.2), we have

$$\begin{array}{rl}\epsilon & \le d(f^{2m_k}x,f^{2n_k})\le d(f^{2m_k}x,f^{2m_k-1})+d(f^{2m_k-1}x,f^{2m_k-2})+d(f^{2m_k-2}x,f^{2n_k})\\ \le d(f^{2m_k}x,f^{2m_k-1})+d(f^{2m_k-1}x,f^{2m_k-2})+\epsilon .\end{array}$$

Let $k\to \mathrm{\infty }$, we get

$$\underset{n\to \mathrm{\infty }}{lim}d(f^{2m_k}x,f^{2n_k})=\epsilon .$$

(3.3)

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

$$d(f^{2m_k}x,f^{2n_k+1})\le {\psi }_x\left(d(f^{2m_k-1}x,f^{2n_k})\right)-\varphi \left(d(f^{2m_k-1}x,f^{2n_k})\right).$$

(3.4)

Since for each $k\in \mathbb{N}$

$$d(f^{2m_k}x,f^{2n_k+1})\le d(f^{2m_k}x,f^{2n_k})+d(f^{2n_k}x,f^{2n_k+1}),$$

(3.5)

and

$$d(f^{2m_k-1}x,f^{2n_k})\le d(f^{2m_k-1}x,f^{2m_k})+d(f^{2m_k}x,f^{2n_k}),$$

(3.6)

taking $k\to \mathrm{\infty }$ and using the inequalities (3.3), (3.5) and (3.6), we get

$$\underset{n\to \mathrm{\infty }}{lim}d(f^{2m_k}x,f^{2n_k+1})=\epsilon ,$$

(3.7)

and

$$\underset{n\to \mathrm{\infty }}{lim}d(f^{2m_k-1}x,f^{2n_k})=\epsilon .$$

(3.8)

Taking into account the inequalities (3.4), (3.7) and (3.8), and by the definitions of the functions ϕ and ${\psi }_x$ , we get

$$\begin{array}{rl}\epsilon & =\underset{n\to \mathrm{\infty }}{lim}d(f^{2m_k}x,f^{2n_k+1})\\ \le \underset{n\to \mathrm{\infty }}{lim}{\psi }_x\left(d(f^{2m_k-1}x,f^{2n_k})\right)-\underset{n\to \mathrm{\infty }}{lim}\varphi \left(d(f^{2m_k-1}x,f^{2n_k})\right)\\ \le \epsilon -\varphi (\epsilon ),\end{array}$$

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

Since $(X,d)$ is a complete metric space, A and B are closed, $\{f^{n}x\}\subset A\cup B$, there exists $\nu \in A\cup B$ such that ${lim}_{n\to \mathrm{\infty }}{f}^{n}x=\nu$. Now $\{f^{2n}x\}$ is a sequence in A and $\{f^{2n+1}x\}$ is a sequence in B, and also both converge to ν. Since A and B are closed, $\nu \in A\cap B$, and so $A\cap B$ is nonempty. On the other hand, since ${lim}_{n\to \mathrm{\infty }}d(f^{2n-1}x,\nu )=0$ and

$$d(f^{2n}x,f\nu )\le {\psi }_x\left(d(f^{2n-1}x,\nu )\right)-\varphi \left(d(f^{2n-1}x,\nu )\right),$$

taking $n\to \mathrm{\infty }$,we obtain that

$$d(\nu ,f\nu )\le 0-\varphi (d(\nu ,\nu ))=0,$$

and hence $d(\nu ,f\nu )=0$, that is, ν is a fixed point of f.

Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have $\nu ,\mu \in A\cap B$. Since f is a generalized cyclic orbital weaker Meir-Keeler $({\psi }_x,\varphi )$-contraction, we have

$$d(f^{2n}x,f\mu )\le {\psi }_x\left(d(f^{2n-1}x,\mu )\right)-\varphi \left(d(f^{2n-1}x,\mu )\right).$$

Letting $n\to \mathrm{\infty }$, and by the definitions of the functions ϕ and ${\psi }_x$, we obtain that

$$d(\nu ,\mu )=d(\nu ,f\mu )=\underset{n\to \mathrm{\infty }}{lim}d(f^{2n}x,f\mu )\le d(\nu ,\mu )-\varphi (d(\nu ,\mu )),$$

which implies that $d(\nu ,\mu )=0$. Therefore, $\nu =\mu$, and so ν is the unique fixed point of f. □

We give the following example to illustrate Theorem 6.

Example 3 Let $A=B=X={\mathbb{R}}^{+}$ and we define $d:X\times X\to {\mathbb{R}}^{+}$ by

$$d(x,y)=|x-y|,\text{for }x,y\in X.$$

Define $f:X\to X$ by

$$f(x)=\{\begin{array}{cc}0,\hfill & \text{if }0\le x<1;\hfill \\ \frac{1}{16},\hfill & \text{if }x\ge 1\hfill \end{array}$$

and define $\psi ,\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ by

$${\psi }_x(t)=\frac{1}{3}t\text{and}\varphi (t)=\frac{1}{6}t\text{for }t\in {\mathbb{R}}^{+}.$$

Then f is a generalized cyclic orbital weaker Meir-Keeler $({\psi }_x,\varphi )$-contraction and 0 is the unique fixed point.