Research Article
Fixed Point Results for --Contractive Mappings Including
Almost Contractions and Applications
Gonca Durmaz, Gülhan MJnak, and Ishak Altun
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey
Correspondence should be addressed to Gonca Durmaz; gncmatematik@hotmail.com
Received 29 May 2014; Accepted 9 July 2014; Published 22 July 2014
Academic Editor: Salvador Romaguera
Copyright © 2014 Gonca Durmaz 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.
In the recent paper (B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for --contractive type mappings, Nonlinear Analysis.
heory, Methods and Applications, 75 (2012), 2154-2165.), the authors introduced the concept of -admissible maps on metric
spaces. Using this new concept, they presented some nice ixed point results. Also, they gave an existence theorem for integral
equation to show the usability of their result. hen, many authors focused on this new concept and obtained a lot of ixed point
results, which are used for existence theorems. In this paper, we not only extend some of the recent results about this direction but
also generalize them. hen, we give some examples to show our results are proper extensions. Furthermore, we use our results to
obtain the existence and uniqueness result for a solution of fourth order two-point boundary value problem.
1. Introduction and Preliminaries
Fixed point theory contains many diferent ields of mathe-
matics, such as nonlinear functional analysis, mathematical
analysis, operator theory, and general topology. Historically,
the study of ixed point theory has developed in two major
branches: the irst is ixed point theory for contraction or
contraction type mappings on complete metric spaces and
the second is ixed point theory for continuous operators on
compact and convex subsets of a normed space. Recently,
there has been a lot of activities in the irst branch and several
fundamental ixed point results have been extended and
generalized by many authors in diferent directions. In this
paper, we mention some important of them and give some
new ixed point results. Also, we support our results by giving
a lot of nontrivial examples. First, we give some notations,
which will be used in this paper. Let : [0, ∞) → [0, ∞) be a
function. For convenience, we consider the following proper-
ties of this function:
(
1
) is nondecreasing,
(
2
) lim
→∞
() = 0 for all ≥0,
(
3
) () < for >0,
(
4
) (0) = 0,
(
5
) is continuous,
(
6
) is upper semicontinuous from the right,
(
7
)∑
∞
=1
() < ∞ for any >0.
In the light of the above properties, the following hold: if
1
and
2
satisied, then
3
holds. If
1
and
3
are satisied,
then
4
holds. If
3
and
5
are satisied, then
2
and
4
hold.
Denoted by Ψ the family of functions : [0,∞) →
[0, ∞) satisfying
1
and
2
, which is called comparison
functions in the literature (see [1]), by Φ the family of
functions : [0, ∞) → [0, ∞) satisfying
1
and
7
, which is
called ()-comparison functions in the literature (see [1]), and
by Υ the family of functions : [0, ∞) → [0, ∞) satisfying
3
and
6
. Now, we give some examples showing the relations
between the sets Φ, Ψ, and Υ. First, it is clear that Φ⊂Ψ.
Example 1. Let : [0, ∞) → [0, ∞) be deined by () = ,
where ∈ [0, 1), and then ∈Φ∩Υ.
Example 2. Let : [0, ∞) → [0, ∞) be deined by () =
/(1 + ), and then ∈Ψ∩Υ, but ∉Φ.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 869123, 10 pages
http://dx.doi.org/10.1155/2014/869123