Hindawi Publishing Corporation
Journal of Function Spaces and Applications
Volume 2013, Article ID 143686, 9 pages
http://dx.doi.org/10.1155/2013/143686
Research Article
Suzuki-Type Fixed Point Results in Metric-Like Spaces
Nabiollah Shobkolaei,
1
Shaban Sedghi,
2
Jamal Rezaei Roshan,
2
and Nawab Hussain
3
1
Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran
2
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
3
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Nawab Hussain; nhusain@kau.edu.sa
Received 16 May 2013; Accepted 17 July 2013
Academic Editor: Pei De Liu
Copyright © 2013 Nabiollah Shobkolaei et al. his is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We demonstrate a fundamental lemma for the convergence of sequences in metric-like spaces, and by using it we prove some
Suzuki-type ixed point results in the setup of metric-like spaces. As an immediate consequence of our results we obtain certain
recent results in partial metric spaces as corollaries. Finally, three examples are presented to verify the efectiveness and applicability
of our main results.
1. Introduction
here are a lot of generalizations of Banach ixed-point prin-
ciple in the literature. So far several authors have studied
the problem of existence and uniqueness of a ixed point for
mappings satisfying diferent contractive conditions (e.g., [1–
20]). In 2008, Suzuki introduced an interesting generalization
of Banach ixed-point principle. his interesting ixed-point
result is as follows.
heorem 1 (see [19]). Let (,) be a complete metric space,
and let be a mapping on . Deine a nonincreasing function
from [0,1] into [1/2,1] by
()=
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
1, 0≤≤
√
5−1
2
1−
2
,
√
5−1
2
≤≤
1
√
2
1
1+
,
1
√
2
≤<1.
(1)
Assume that there exists ∈[0,1], such that
()(,)≤ (,)⇒(,)≤(,), (2)
for all ,∈, then there exists a unique ixed-point of .
Moreover, lim
→∞
= for all ∈.
Suzuki proved also the following version of Edelstein’s
ixed point theorem.
heorem 2. Let (,) be a compact metric space. Let :→
be a self-map, satisfying for all ,∈, ̸ = the condition
1
2
(,)≤ (,)⇒(,)<(,). (3)
hen has a unique ixed point in .
his theorem was generalized in [3].
In addition to the above results, Kikkawa and Suzuki [8]
provided a Kannan type version of the theorems mentioned
before. In [14], Chatterjea type version is provided. Popescu
[15] presented a Ciri´ c type version. Recently, Kikkawa and
Suzuki also provided multivalued versions which can be
found in [9, 10].
Very recently Hussain et al. [4] have extended Suzuki’s
heorems 1 and 2, as well as Popescu’s results from [15] to the
case of metric type spaces and cone metric type spaces (see
also [5–7, 11]).
he aim of this paper is to generalize the above-men-
tioned results. Indeed we prove a ixed point theorem in the
set up of metric-like spaces and derive certain new results as
corollaries. Finally, three examples are presented to verify the
efectiveness and applicability of our main results.