 Research
 Open Access
 Published:
MeirKeeler ɑcontractive fixed and common fixed point theorems
Fixed Point Theory and Applicationsvolume 2013, Article number: 19 (2013)
The Erratum to this article has been published in Fixed Point Theory and Applications 2013 2013:110
Abstract
Generalized MeirKeeler αcontractive functions and pairs are introduced and their fixed and common fixed point theorems are obtained. Also, the socalled generalized MeirKeeler αfcontractive maps commuting with f are introduced and their coincidence and common fixed point theorems are investigated. New sufficient conditions different from those in (Samet et al. in Nonlinear Anal. 75:21542165, 2012) are used. An application to the coupled fixed point is established as well. An example is given to show that the αMeirKeeler generalization is real.
AMS Subject Classification: 47H10, 54H25.
1 Introduction
Fixed point theory is of wide and endless applications in many fields of engineering and science. Its core, the Banach contraction principle, has attracted many researchers who tried to generalize it in different aspects. Some dealt with the contractive condition itself, of worth mentioning MeirKeeler contractive type [1–4], some extended it to more generalized metrictype spaces [5–11] and others applied to common [12], coupled and tripled versions (see [13, 14] and the references therein). In 1969 Meir and Keeler [15] established a fixed point theorem in a metric space $(X,d)$ for mappings satisfying the following condition, called the MeirKeeler type contractive condition:
In 1978 Maiti and Pal [16] generalized a fixed point for maps satisfying the following condition:
Later in 1981, Park and Rhoades in [3] established fixed point theorems for a pair of mappings f, g satisfying a contractive condition that can be reduced to the following generalization of (2) when $f=g$.
In this article we develop the fixed point theorems for αcontractive type maps introduced recently in [17] (for the αψcontractive multivalued case, see [18]) to MeirKeeler versions and hence generalize the results obtained in [3] and the references therein. Then, we apply part of our results to the coupled case on the basis of AminiHarandi [19].
2 Fixed and common fixed point theorems for generalized MeirKeeler αcontractive maps and pairs
The first part of the following definition was introduced in [17].
Definition 1 Let $f,g:X\to X$ be selfmappings of a set X and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping, then the mapping f is called αadmissible if
and the pair $(f,g)$ is called αadmissible if
Example 2 Let $X=\mathbb{R}$ and
Then the pair $({x}^{1/2},{x}^{1/3})$ is αadmissible but the pair $({x}^{1/2},x+1)$ is not αadmissible.
Definition 3 Let $(X,d)$ be a metric space and $f:X\to X$ be a selfmapping, $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. Then f is called MeirKeeler αcontractive if, given an $\u03f5>0$, there exists a $\delta >0$ such that
Definition 4 Let $(X,d)$ be a metric space and $f:X\to X$ be a selfmapping, $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. Then f is called generalized MeirKeeler αcontractive if, given an $\u03f5>0$, there exists a $\delta >0$ such that
where
Definition 5 Let $(X,d)$ be a metric space and $f,g:X\to X$ be selfmappings, $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. Then the pair $(f,g)$ is called generalized MeirKeeler αcontractive if, given an $\u03f5>0$, there exists a $\delta >0$ such that
where
We write ${M}_{f}(x,y)={M}_{(f,f)}(x,y)$.
Clearly, f is generalized MeirKeeler αcontractive if and only if $(f,f)$ is generalized MeirKeeler αcontractive.
Definition 6 Let X be any set, ${x}_{0}\in X$ and f, g be selfmaps of X. Define ${x}_{2n+1}=f{x}_{2n}$ and ${x}_{2n+2}=g{x}_{2n}$, $n=0,1,2,\dots $ . Then $\{{x}_{n}\}$ is called the $(f,g)$orbit of ${x}_{0}$. If d is a metric on X, then $(X,d)$ is called $(f,g)$orbitally complete if every Cauchy sequence in the $(f,g)$orbit of ${x}_{0}$ is convergent and the map f or g is called orbitally continuous if it is continuous on the orbit.
The proof of the following lemma is immediate.
Lemma 7 Let $f,g:X\to X$ be selfmappings of a set X, $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping and $\{{x}_{n}\}$ be the $(f,g)$orbit of ${x}_{0}$ with $\alpha ({x}_{0},f{x}_{0})\ge 1$. If the pair $(f,g)$ is αadmissible, then $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n=0,1,2,\dots $ .
Theorem 8 Let $(X,d)$ be an $(f,g)$orbitally complete metric space, where f, g are selfmappings of X. Also, let $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. Assume the following:

1.
$(f,g)$ is αadmissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},f{x}_{0})\ge 1$;

2.
the pair $(f,g)$ is generalized MeirKeeler αcontractive.
Then the sequence ${d}_{n}=d({x}_{n},{x}_{n+1})$ is monotone decreasing. If, moreover, we assume that

3.
on the $(f,g)$orbit of ${x}_{0}$, we have $\alpha ({x}_{n},{x}_{j})\ge 1$ for all n even and $j>n$ odd and that f and g are continuous on the $(f,g)$orbit of ${x}_{0}$.
Then either (1) f or g has a fixed point in the $(f,g)$orbit $\{{x}_{n}\}$ of ${x}_{0}$ or (2) f and g have a common fixed point p and $lim{x}_{n}=p$. If, moreover, we assume that the following condition (H) holds: If $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all n and ${x}_{n}\to x$ implies $\alpha ({x}_{n},x)\ge 1$ for all n, then uniqueness of the fixed point is obtained.
Proof Define ${d}_{n}=d({x}_{n},{x}_{n+1})$ for $n=0,1,2,\dots $ . If ${d}_{n}=0$ for some even integer n, then f has a fixed point. If ${d}_{n}=0$ for some odd integer n, then g has a fixed point. Hence, we may assume that ${d}_{n}\ne 0$ for each n. The fact that the pair $(f,g)$ is generalized MeirKeeler αcontractive implies that
Note that assumption (3) implies that $\alpha ({x}_{0},f{x}_{0})\ge 1$. Hence, since $(f,g)$ is αadmissible, then Lemma 7 implies that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n=0,1,2,\dots $ and hence by (5), we have
whence ${d}_{2n}<{d}_{2n1}$. □
Similarly, it can be shown that ${d}_{2n+1}<{d}_{2n}$. Thus, $\{{d}_{n}\}$ is monotone decreasing in n and converges to a limit, say ϱ.
Suppose $\varrho >0$. Then, for each $\delta >0$, there exists a positive integer $N=N(\delta )$ such that $\varrho \le {d}_{N}=d({x}_{N},{x}_{N+1})<\varrho +\delta $, where N can be chosen even. Thus, from assumption (1) and Lemma 7, we have ${d}_{N+1}\le \alpha ({x}_{N},{x}_{N+1})d(f{x}_{N},g{x}_{N+1})<\varrho $, a contradiction. Therefore, $\varrho =0$. To show that $\{{x}_{n}\}$ is Cauchy, we assume the contrary. Thus, there exists an ${\u03f5}^{\mathrm{\prime}}>0$ such that for each integer N, there exist integers $m>n>N$ such that $d({x}_{m},{x}_{n})\ge {\u03f5}^{\mathrm{\prime}}$. Define ϵ by ${\u03f5}^{\mathrm{\prime}}=2\u03f5$. Choose a number δ, $0<\delta <\u03f5$, for which (4) is satisfied. Since $\varrho =0$, there exists an integer $N=N(\delta )$ such that ${d}_{i}<\frac{\delta}{6}$ for $i\ge N$. With this choice of N, pick integers $m>n>N$ such that
in which it is clear that $mn>6$. Otherwise, $d({x}_{m},{x}_{n})\le {\sum}_{i=0}^{5}{d}_{i+n}<\delta <\delta +\u03f5$, contradicting (7). Without loss of generality, we may assume that n is even since from (7) it follows that $d({x}_{m},{x}_{n+1})>\u03f5+\frac{\delta}{3}$. From (7) there exists the smallest odd integer $j>n$ such that
Hence, $d({x}_{n},{x}_{j2})<\u03f5+\frac{\delta}{3}$, and so $d({x}_{n},{x}_{j})\le d({x}_{n},{x}_{j2})+{d}_{j1}+{d}_{j}<\u03f5+\frac{\delta}{3}+2(\frac{\delta}{6})=\u03f5+\frac{2\delta}{3}$. Therefore, we have
so that, by (7) and assumption (3), $d({x}_{n+1},{x}_{j+1})\le \alpha ({x}_{n},{x}_{j})d({x}_{n+1},{x}_{j+1})<\u03f5$. Then we have
This contradicts the choice of j in (8). Therefore, $\{{x}_{n}\}$ is Cauchy.
Since X is $(f,g)$orbitally complete, $\{{x}_{n}\}$ converges to some point $p\in X$. Since f and g are orbitally continuous, then p is a common fixed point of f and g. To prove uniqueness, assume p is the common fixed point obtained as ${x}_{n}\to p$ and q is another common fixed point. Then (5) and the condition (H) yield
If we let $n\to \mathrm{\infty}$, then we reach $d(p,q)<d(p,q)$, which implies that $p=q$.
Corollary 9 Let $(X,d)$ be an forbitally complete metric space, where f is a selfmapping of X. Also, let $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. Assume the following:

1.
f is αadmissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},f{x}_{0})\ge 1$;

2.
f is generalized MeirKeeler αcontractive.
Then the sequence ${d}_{n}=d({x}_{n},{x}_{n+1})$ is monotone decreasing. If, moreover, we assume that

3.
on the forbit of ${x}_{0}$, we have $\alpha ({x}_{n},{x}_{j})\ge 1$ for all n even and $j>n$ odd.
Then either (1) f has a fixed point in the forbit $\{{x}_{n}\}$ of ${x}_{0}$ or (2) f has a fixed point p and $lim{x}_{n}=p$. If, moreover, we assume that the following condition (H) holds: If $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all n and ${x}_{n}\to x$, then $\alpha ({x}_{n},x)\ge 1$ for all n, then uniqueness of the fixed point is obtained.
Since generalized MeirKeeler αcontractions are MeirKeeler αcontractions, then Corollary 9 is valid also for MeirKeeler αcontractions. In the following example, the existence and uniqueness of the fixed point cannot be proved in the category of MeirKeeler contractions, but can be proved by means of Corollary 9.
Example 10 Let $X=[0,2]$ with the absolute value metric $d(x,y)=xy$. Define $f:X\to X$ by
Then, for $\u03f5=\frac{1}{2}$, $x=\frac{1}{4}$ and any $\delta >0$, we have $\frac{1}{2}\le \frac{1}{4}y<\delta +\frac{1}{2}$ implies $y\in [\frac{1}{2},2]$ and hence $d(fx,fy)=d(0,\frac{3}{2})=\frac{3}{2}>\u03f5$. Hence, f is not a MeirKeeler contraction. However, f is a MeirKeeler αcontraction, where
Indeed, for $0<\u03f5<1$ (the case $\u03f5\ge 1$ is trivial, since $fxfy\le 1$), let $\delta =(1\u03f5)$, then $\u03f5\le \alpha (x,y)d(x,y)<\delta +\u03f5=1$ implies that $x,y\in [\frac{1}{2},2]$ and hence $d(fx,fy)=\frac{3}{2}\frac{3}{2}=0<\u03f5$. Also, notice that f is continuous on the orbit of ${x}_{0}=1$ and that $\alpha ({x}_{n},{x}_{j})\ge 1$ for all n, j. Clearly, $p=\frac{3}{2}$ is the unique fixed point.
Remark 11 Note that the admissibility condition (1) in Theorem 8 is not enough to proceed to guarantee the existence of the fixed point. However, such an admissibility condition was used in obtaining the main result in Theorem 2.2 of [17].
3 Generalized MeirKeeler αfcontractive fixed points
Definition 12 Let f be a continuous selfmap of a metric space $(X,d)$, ${C}_{f}=\{g:g:X\to X,\text{such that}fg=gf\text{and}gX\subseteq fX\}$, the sequence $\{f{x}_{n}\}$ defined by $f{x}_{n+1}=g{x}_{n}$, $n=0,1,2,\dots $ , with the understanding that if $f{x}_{n}=f{x}_{n+1}$ for some n, then $f{x}_{n+j}=f{x}_{n}$ for each $j\ge 0$ is called the fiteration of ${x}_{0}$ under g.
Definition 13 Let f be a selfmap of a metric space $(X,d)$ and $g\in {C}_{f}$. Then g is called a MeirKeeler αfcontractive map if for each $\u03f5>0$, there exists a $\delta >0$ such that for all $x,y\in X$,
Definition 14 Let f be a selfmap of a metric space $(X,d)$ and $g\in {C}_{f}$. Then g is called a generalized MeirKeeler αfcontractive map if for each $\u03f5>0$, there exists a $\delta >0$ such that for all $x,y\in X$,
where ${M}_{g}(f)(x,y)=max\{d(fx,fy),d(fx,gx),d(fy,gy),\frac{d(fx,gy)+d(fy,gx)}{2}\}$.
Lemma 15 Let f, g be continuous selfmaps of a metric space $(X,d)$ such that $g\in {C}_{f}$. Assume g is a generalized MeirKeeler αfcontractive map such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all n. Then $inf\{d(f{x}_{n},f{x}_{n+1}):n=0,1,2,\dots \}=0$.
Proof Let $\sigma =inf\{d(f{x}_{n},f{x}_{n+1}):n=0,1,2,\dots \}$ and $\sigma >0$. From the definition of the fiteration of ${x}_{0}$ under g and from the fact that g is a generalized MeirKeeler αfcontractive map, for each n, we have
Hence, $d(f{x}_{n+1},f{x}_{n+2})<d(f{x}_{n},f{x}_{n+1})$ and $\{d(f{x}_{n},f{x}_{n+1})\}$ is monotone decreasing so that $\sigma ={lim}_{n\to \mathrm{\infty}}d(f{x}_{n},f{x}_{n+1})$. From the assumption that g is a MeirKeeler αfcontractive map, for $\u03f5=\sigma $, find $\delta >0$ such that (10) is satisfied. For the chosen δ, pick N so that $\sigma \le d(f{x}_{n},f{x}_{n+1})<\sigma +\delta $. Noting that for $x={x}_{n}$ and $y={x}_{n+1}$, ${M}_{g}(f)(x,y)=d(f{x}_{n},f{x}_{n+1})$, we by (10) conclude that $d(g{x}_{n},g{x}_{n+1})\le \alpha ({x}_{n},{x}_{n+1})d(g{x}_{n},g{x}_{n+1})<\sigma $. But $d(g{x}_{n},g{x}_{n+1})=d(f{x}_{n+1},f{x}_{n+2})<\sigma $, a contradiction. □
Theorem 16 Let f, g be continuous selfmaps of a metric space $(X,d)$ such that $g\in {C}_{f}$. Assume $\alpha ({x}_{n},{x}_{m})\ge 1$ for all $m>n$. If g is a generalized MeirKeeler αfcontractive map such that α satisfies the condition (fH): If $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{m})\ge 1$ for all $m>n$ and $f{x}_{n}\to z$, then $\alpha (f{x}_{n},z)\ge 1$ and $\alpha (f{x}_{n},fz)\ge 1$ for all n. Then f and g have a unique common fixed point.
Proof Let ${x}_{0}\in X$ for which its fiteration under g satisfies the assumptions of the theorem. The proof will be divided into four steps.

Step 1: By Lemma 15, $inf\{d(f{x}_{n},f{x}_{n+1}):n=0,1,2,\dots \}=0$.

Step 2: We find a coincidence point for f and g. That is to find a $z\in X$ such that $fz=gz$. If there exists an n such that $d(f{x}_{n},f{x}_{n+1})=0$, then $f{x}_{n+1}=g{x}_{n}=f{x}_{n}$, and we are finished. Hence, we may assume that $d(f{x}_{n},f{x}_{n+1})\ne 0$ for each n. We claim to show that $\{f{x}_{n}\}$ is Cauchy. Suppose not. Then there exists an $\u03f5>0$ and a subsequence $\{f{x}_{{n}_{i}}\}$ of $\{f{x}_{n}\}$ such that $d(f{x}_{{n}_{i}},f{x}_{{n}_{i+1}})>2\u03f5$. From (10), there exists a δ satisfying $0<\delta <\u03f5$ for which (10) is true. Since ${lim}_{n\to \mathrm{\infty}}d(f{x}_{n},f{x}_{n+1})$=0, there exists an N such that
$$d(f{x}_{m},f{x}_{m+1})<\frac{\delta}{6}\phantom{\rule{1em}{0ex}}\text{for all}mN.$$Let ${n}_{i}\ge N$. We will show that there exists an integer j satisfying ${n}_{i}<j<{n}_{i+1}$ such that
$$\u03f5+\frac{\delta}{3}\le d(f{x}_{{n}_{i}},f{x}_{j})<\u03f5+\frac{2\delta}{3}.$$(11)First of all, there exist values of j such that $d(f{x}_{{n}_{i}},f{x}_{j})\ge \u03f5+\frac{\delta}{3}$. For example, choose $j={n}_{i+1}$. The inequality is also true for $j={n}_{i+1}1$. If not, then $d(f{x}_{{n}_{i}},f{x}_{j})<\u03f5+\frac{\delta}{3}$ and hence
$$\begin{array}{rcl}d(f{x}_{{n}_{i}},f{x}_{{n}_{i+1}})& \le & d(f{x}_{{n}_{i}},f{x}_{{n}_{i+1}}1)+d(f{x}_{{n}_{i+1}}1,f{x}_{{n}_{i+1}})\\ <& \u03f5+\frac{\delta}{3}+\frac{\delta}{6}<2\u03f5,\end{array}$$a contradiction. There are also values of j such that $d(f{x}_{{n}_{i}},f{x}_{j})<\u03f5+\frac{\delta}{3}$. For example, choose $j={n}_{i}+1$ and $j={n}_{i}+2$. Pick j to be the smallest integer greater than ${n}_{i}$ such that $d(f{x}_{{n}_{i}},f{x}_{j})\ge \u03f5+\frac{\delta}{3}$. Then $d(f{x}_{{n}_{i}},f{x}_{i}1)<\u03f5+\frac{\delta}{3}$, and hence
$$d(f{x}_{{n}_{i}},f{x}_{j})\le d(f{x}_{{n}_{i}},f{x}_{j}1)+d(f{x}_{j}1,f{x}_{j})<\u03f5+\frac{\delta}{3}+\frac{\delta}{6}<\u03f5+\frac{2\delta}{3}.$$Thus (11) is established. Now, note that
$$\u03f5+\frac{\delta}{3}\le d(f{x}_{{n}_{i}},f{x}_{j})\le max\{d(f{x}_{{n}_{i}},f{x}_{j}),d(f{x}_{{n}_{i}},g{x}_{{n}_{i}}),d(f{x}_{j},g{x}_{j}),\frac{d(f{x}_{{n}_{i}},g{x}_{j})+d(f{x}_{j},g{x}_{{n}_{i}})}{2}\}.$$Then from the choice of j and the fact that $f{x}_{{n}_{i}}+1=g{x}_{{n}_{i}}$, $f{x}_{j}+1=g{x}_{j}$, we reach
$$\u03f5\le d(f{x}_{{n}_{i}},f{x}_{j})<\delta +\u03f5.$$Hence,
$$d(f{x}_{{n}_{i}+1},f{x}_{j+1})=d(g{x}_{{n}_{i}},g{x}_{j})\le \alpha ({x}_{{n}_{i}},{x}_{j})d(g{x}_{{n}_{i}},g{x}_{j})<\u03f5.$$On the other hand,
$$\begin{array}{rcl}d(f{x}_{{n}_{i}},f{x}_{j})& \le & d(f{x}_{{n}_{i}},{f}_{{n}_{i}+1})+d({f}_{{n}_{i}+1},f{x}_{j+1})+d(f{x}_{j+1},f{x}_{j})\\ <& \frac{\delta}{6}+\u03f5+\frac{\delta}{6}=\u03f5+\frac{\delta}{3},\end{array}$$contradicting (11). Therefore, $\{f{x}_{n}\}$ is Cauchy hence convergent to $z\in X$. Since $ff{x}_{n}=fg{x}_{n1}=gf{x}_{n1}$, the continuity of f and g implies that $fz=gz$.

Step 3: We show that $\eta =fz=gz$ is a common fixed point for f and g. Assume $f\eta \ne \eta $, then ${f}^{2}z\ne fz$ and by the help of the (fH) condition, we have
$$\begin{array}{rcl}d(\eta ,f\eta )& =& d(gz,fgz)=d(gz,gfz)\\ \le & d(gz,gf{x}_{n})+d(gf{x}_{n},gfz)\\ \le & \alpha (f{x}_{n},z)d(gz,gf{x}_{n})+\alpha (f{x}_{n},fz)d(gf{x}_{n},gfz)\\ <& max\{d(fz,ff{x}_{n}),d(fz,gz),d(ff{x}_{n},gf{x}_{n}),\frac{d(fz,gf{x}_{n})+d(ff{x}_{n},gfz)}{2}\}\\ +max\{d(ff{x}_{n},ffz),d(ff{x}_{n},gf{x}_{n}),d(ffz,gfz),\frac{d(ff{x}_{n},gfz)+d(ffz,gf{x}_{n})}{2}\}.\end{array}$$If we let $n\to \mathrm{\infty}$ above and use continuity and commutativity of f and g, then we reach $d(\eta ,f\eta )<d(\eta ,f\eta )$ and hence $f\eta =\eta $. Moreover, $g\eta =gfz=f\eta =\eta $.

Step 4: Uniqueness of the common fixed point. Assume $\eta =fz=gz$ is our common fixed point for f and g where $f{x}_{n}\to z$ and ω is another common fixed point. Then, by the (fH) condition, we have
$$\begin{array}{rcl}d(\eta ,\omega )& =& d(g\eta ,\omega )\le d(g\eta ,gf{x}_{n})+d(gf{x}_{n},\omega )\\ \le & \alpha (\eta ,f{x}_{n})d(g\eta ,gf{x}_{n})+d(gf{x}_{n},\omega )\\ <& max\{d(f\eta ,ff{x}_{n}),d(f\eta ,g\eta ),d(ff{x}_{n},gf{x}_{n}),\frac{d(f\eta ,gf{x}_{n})+d(ff{x}_{n},g\eta )}{2}\}.\end{array}$$If we let $n\to \mathrm{\infty}$ above and use the continuity of f and g, we conclude that $d(\eta ,\omega )<d(\eta ,\omega )$ and hence $\eta =\omega $.
□
Remark 17 Theorem 16 has been proved for commuting maps. It would be interesting to extend it for weakly commuting and compatible mappings and so forth. For example, can we extend the results in [20–22] to αtype contractions?
4 Application to coupled αMeirKeeler fixed points
Let $F:X\times X\to X$ be a mapping. We say that $(x,y)\in X\times X$ is a coupled fixed point of F if $F(x,y)=x$ and $F(y,x)=y$. If we define $T:X\times X\to X\times X$ by $T(x,y)=(F(x,y),F(y,x))$, then clearly $(x,y)$ is a coupled fixed point of F if and only if $(x,y)$ is a fixed point of T. If $({x}_{0},{y}_{0})\in X\times X$, then the Forbit of $({x}_{0},{y}_{0})$ means the orbit $\{({x}_{n},{y}_{n}):n=0,1,2,\dots \}$, where $({x}_{n+1},{y}_{n+1})=T({x}_{n},{y}_{n})$.
If $(X,d)$ is a metric space, then $\rho :X\times X\to \mathbb{R}$ defined by $\rho ((x,y),(u,v))=d(x,u)+d(y,v)$ is a metric on $X\times X$.
Theorem 18 Let $(X,d)$ be a complete metric space and $F:X\times X\to X$ be a continuous mapping. Also, let $\alpha :{X}^{2}\times {X}^{2}\to [0,\mathrm{\infty})$ be a mapping. Assume the following:

1.
For all $(x,y),(u,v)\in X\times X$, we have
$$\alpha ((x,y),(u,v))\ge 1\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}\alpha ((F(x,y),F(y,x)),(F(u,v),F(v,u)))\ge 1.$$Also, assume there exists $({x}_{0},{y}_{0})\in X\times X$ such that $\alpha ((F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})),({y}_{0},{x}_{0}))\ge 1$ and $\alpha (({x}_{0},{y}_{0}),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})))\ge 1$;

2.
For each $\u03f5>0$, there exists $\delta >0$ such that
$$\u03f5\le \frac{1}{2}[d(x,u)+d(y,v)]<\delta +\u03f5\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}\alpha ((x,y),(u,v))d(F(x,y),F(u,v))<\u03f5.$$Then the sequence ${\rho}_{n}=\rho (({x}_{n},{y}_{n}),({x}_{n+1},{y}_{n+1}))$ is monotone decreasing. If, moreover, we assume that

3.
on the Forbit of $({x}_{0},{y}_{0})$, we have $\alpha (({x}_{n},{y}_{n}),({x}_{j},{y}_{j}))\ge 1$ and $\alpha (({y}_{j},{x}_{j}),({y}_{n},{x}_{n}))\ge 1$ for all n, j.
Then either (1) F has a coupled fixed point in the Forbit $\{({x}_{n},{y}_{n})\}$ of $({x}_{0},{y}_{0})$ or (2) F has a coupled fixed point $(p,q)$ and $lim\rho ({x}_{n},{y}_{n})=(p,q)$. If, moreover, we assume that the following condition (H) holds: If $\{({x}_{n},{y}_{n})\}$ is a sequence in $X\times X$ such that $\alpha (({x}_{n},{y}_{n}),({x}_{n+1},{y}_{n+1}))\ge 1$ for all n and $d({x}_{n},x)\to 0$, $d({y}_{n},y)\to 0$, then $\alpha (({x}_{n},{y}_{n}),(x,y))\ge 1$ and $\alpha ((y,x),({y}_{n},{x}_{n}))\ge 1$ for all n, then uniqueness of the coupled fixed point is obtained.
Proof The proof will follow by applying Corollary 9, with $f=T$ as above, to the metric space $(X\times X,\rho )$. The controlling function will be $\beta :{X}^{2}\times {X}^{2}\to [0,\mathrm{\infty})$ given by
In fact, if $\u03f5>0$ is given, then by assumption (2), find ${\delta}^{\mathrm{\prime}}>0$ such that
Let $\delta =2{\delta}^{\mathrm{\prime}}$ and assume $\u03f5\le \rho ((x,y),(u,v))<\delta +\u03f5$. Then
and
Hence,
and
which leads to
□
References
 1.
Aydi H, Karapınar E: A MeirKeeler common type fixed point theorem on partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 26
 2.
Aydi H, Karapınar E: New MeirKeeler type tripled fixed point theorems on ordered partial metric spaces. Math. Probl. Eng. 2012., 2012: Article ID 409872
 3.
Park S, Rhoades BE: MeirKeeler type contractive conditions. Math. Jpn. 1981, 26(1):13–20.
 4.
Abdeljawad T, Karapınar E, Aydi H: A new MeirKeeler type coupled fixed point on ordered partial metric spaces. Math. Probl. Eng. 2012., 2012: Article ID 327273. doi:10.1155/2012/327273
 5.
Matthews, SG: Partial metric topology. Research report 212, Dept. of Computer Science, University of Warwick (1992)
 6.
Matthews SG: Partial metric topology.Annals of the New York Academy of Sciences 728. General Topology and Its Applications. Proc. 8th Summer Conf., Queen’s College (1992) 1994, 183–197.
 7.
Abdeljawad T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 2011, 54(11–12):2923–2927. 10.1016/j.mcm.2011.07.013
 8.
Abdeljawad T, Murthy PP, Taş K: A Gregus type common fixed point theorem of setvalued mappings in cone metric spaces. J. Comput. Anal. Appl. 2011, 13(4):622–628.
 9.
Abdeljawad T, Alzabut JO, Mukheimer E, Zaidan Y: Banach contraction principle for cyclical mappings on partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 154
 10.
Abdeljawad T, Karapınar E, Taş K: Common fixed point theorems in cone Banach spaces. Hacet. J. Math. Stat. 2011, 40(2):211–217.
 11.
Shatanawi W, Nashine HK: A generalization of Banach’s contraction principle for nonlinear contraction in a partial metric space. J. Nonlinear Sci. Appl. 2012, 5: 37–43.
 12.
Abdeljawad T, Karapınar E, Taş K: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 2011, 24(11):1900–1904. 10.1016/j.aml.2011.05.014
 13.
Abdeljawad T: Coupled fixed point theorems for partially contractive type mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 148
 14.
Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012. doi:10.1016/j.mcm.2011.08.042
 15.
Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022247X(69)900316
 16.
Maiti M, Pal TK: Generalizations of two fixed point theorems. Bull. Calcutta Math. Soc. 1978, 70: 57–61.
 17.
Samet B, Vetro C, Verto P: Fixed point theorems for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
 18.
Asl JH, Rezapour S, Shahzad N: On fixed points of α  ψ contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212 (26 November 2012)
 19.
AminiHarandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with applications to initial value problems. Math. Comput. Model. 2011. doi:10.1016/j.mcm.2011.12.006
 20.
Sessa S: On a week commutativity condition of mappings in fixed point considerations. Publ. Inst. Math. (Belgr.) 1982, 32(46):149–153.
 21.
Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9(4):333–338.
 22.
Cho YJ, Murthy PP, Jungck J: A common fixed point theorem of Meir and Keeler type. Int. J. Math. Math. Sci. 1993, 16(4):669–674. 10.1155/S0161171293000833
Author information
Additional information
Competing interests
The author declares that he has no competing interests.
An erratum to this article is available at http://dx.doi.org/10.1186/168718122013110.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Accepted
Published
DOI
Keywords
 orbitally complete metric space
 common fixed point
 MeirKeeler αcontractive
 αadmissible mapping
 coupled fixed point