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A note on recent fixed point results involving g-quasicontractive type mappings in partially ordered metric spaces
- Erdal Karapınar^{1, 2}Email author and
- Bessem Samet^{3}
https://doi.org/10.1186/1687-1812-2014-126
© Karapinar and Samet; licensee Springer. 2014
- Received: 5 March 2014
- Accepted: 13 May 2014
- Published: 23 May 2014
Abstract
In this note, we establish the equivalence between recent fixed point theorems involving quasicontractive type mappings in metric spaces endowed with a partial order.
MSC:47H10.
Keywords
- g-quasicontraction
- coincidence point
- fixed point
- partial order
- metric space
1 Introduction
for some constant $\lambda \in (0,1)$. If $g=i{d}_{X}$ (the identity map on X), then f is said to be an ordered quasicontraction.
In [1], the authors established the following result.
- (i)
$fX\subseteq gX$;
- (ii)
gX is complete;
- (iii)
f is g-nondecreasing, i.e., $gx\u2aafgy\u27f9fx\u2aaffy$;
- (iv)
f is an ordered g-quasicontraction;
- (v)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aaff{x}_{0}$;
- (vi)
if $\{g{x}_{n}\}$ is a nondecreasing sequence (w.r.t. ⪯) that converges to some $gz\in gX$, then $g{x}_{n}\u2aafgz$ for each $n\in \mathbb{N}$.
Then f and g have a coincidence point, i.e., there exists $z\in X$ such that $fz=gz$.
Taking $g=i{d}_{X}$ in Theorem 1.1, we obtain immediately the following result.
- (iii)
f is nondecreasing, i.e., $x\u2aafy\u27f9fx\u2aaffy$;
- (iv)
f is an ordered quasicontraction;
- (v)
there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$;
- (vi)
if $\{{x}_{n}\}$ is a nondecreasing sequence (w.r.t. ⪯) that converges to some $z\in X$, then ${x}_{n}\u2aafz$ for each $n\in \mathbb{N}$.
Then f has a fixed point.
Let us denote by Ψ the set of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:
(${\mathrm{\Psi}}_{1}$) ψ is nondecreasing;
(${\mathrm{\Psi}}_{2}$) ψ is subadditive, i.e., $\psi (s+t)\le \psi (s)+\psi (t)$, for every $s,t\ge 0$;
(${\mathrm{\Psi}}_{3}$) ψ is continuous;
(${\mathrm{\Psi}}_{4}$) $\psi (t)=0\u27fat=0$.
In [3], the authors established the following result.
- (i)
$fX\subseteq gX$;
- (ii)
gX is complete;
- (iii)
f is g-nondecreasing;
- (iv)there exists $\psi \in \mathrm{\Psi}$ such that$\begin{array}{rl}\psi (d(fx,fy))\le & \lambda max\{\psi (d(gx,gy)),\psi (d(gx,fx)),\psi (d(gy,fy)),\\ \psi (d(gx,fy)),\psi (d(gy,fx))\}\end{array}$
- (v)
there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aaff{x}_{0}$;
- (vi)
if $\{g{x}_{n}\}$ is a nondecreasing sequence that converges to some $gz\in gX$, then $g{x}_{n}\u2aafgz$ for each $n\in \mathbb{N}$.
Then f and g have a coincidence point.
The aim of this note is to prove that Theorems 1.1, 1.2 and 1.3 are equivalent.
2 Main result
Our main result in this note is the following.
Proof We consider three steps in the proof.
⋄ Step 1. Theorem 1.2 ⟹ Theorem 1.1.
From condition (v) of Theorem 1.1, there exists ${x}_{0}\in X$ such that $g{x}_{0}\u2aaff{x}_{0}$. Let ${u}_{0}=g{x}_{0}\in gE$, we have ${u}_{0}\u2aafT{u}_{0}$. Finally, from condition (iv) of Theorem 1.1, if $\{{u}_{n}\}\subset gE$ is a nondecreasing sequence that converges to some $u\in gE$, then ${u}_{n}\u2aafu$ for each $n\in \mathbb{N}$. Thus we proved that T satisfies all the conditions of Theorem 1.2. Then we deduce that T has a fixed point ${u}^{\ast}\in gE$. This means that there exists some ${x}^{\ast}\in X$ such that $f{x}^{\ast}=T(g{x}^{\ast})=g{x}^{\ast}$, that is, ${x}^{\ast}\in X$ is a coincidence point of f and g.
⋄ Step 2. Theorem 1.1 ⟹ Theorem 1.3.
for all $x,y\in X$ such that $gy\u2aafgx$. Now, applying Theorem 1.1 with the metric space $(X,{d}_{\psi})$, we obtain the result of Theorem 1.3.
⋄ Step 3. Theorem 1.3 ⟹ Theorem 1.2.
Taking $g=i{d}_{X}$ and $\psi (t)=t$ in Theorem 1.3, we obtain immediately the result of Theorem 1.2. □
Declarations
Acknowledgements
The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.
Authors’ Affiliations
References
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