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}\phantom{\rule{1em}{0ex}}\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}^{2{n}_{0}+n}x,{f}^{2{n}_{0}+n+1}x)<{\gamma}_{\eta}^{n}\cdot d({f}^{2{n}_{0}-1}x,{f}^{2{n}_{0}}x).

(2.6)

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

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

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

d({f}^{2{n}_{0}+n}x,{f}^{2{n}_{0}+m}x)\le \sum _{i=n}^{m-1}d({f}^{2{n}_{0}+i}x,{f}^{2{n}_{0}+i+1}x)<\frac{{\gamma}_{\eta}^{m-1}}{1-{\gamma}_{\eta}}d({f}^{2{n}_{0}}x,{f}^{2{n}_{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|,\phantom{\rule{1em}{0ex}}\text{for}x,y\in X,

and let f:X\to X denote

f(x)=\{\begin{array}{cc}0,\hfill & \text{if}0\le x1;\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}t1,\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.