Fixed point theorems for mappings satisfying contractive conditions of integral type
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Fixed point theorems for mappings satisfying contractive conditions of integral type Zeqing Liu1 , Yan Lu1 and Shin Min Kang2* * Correspondence: [email protected] 2 Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract Two results involving the existence, uniqueness and iterative approximations of fixed points for two contractive mappings of integral type are proved in complete metric spaces. Two nontrivial examples are included. MSC: 54H25 Keywords: contractive mappings of integral type; fixed point; complete metric space
1 Introduction In recent years, there has been increasing interest in the study of fixed points and common fixed points of mappings satisfying contractive conditions of integral type, see, for example, [–] and the references cited therein. Branciari [] introduced first the contractive mapping of integral type as follows:
d(fx,fy)
d(x,y)
ϕ(t) dt ≤ c
ϕ(t) dt,
∀x, y ∈ X,
where c ∈ (, ) is a constant, ϕ ∈ = {ϕ : ϕ : R+ → R+ satisfies that ϕ is Lebesgue inε tegrable, summable on each compact subset of R+ and ϕ(t) dt > for each ε > } and proved the existence of a fixed point for the mapping in complete metric spaces. Rhoades [] and Liu et al. [] extended Branciari’s result and obtained a few fixed point theorems for the contractive mappings of integral type below:
d(fx,fy)
M(x,y)
ϕ(t) dt ≤ c
ϕ(t) dt,
∀x, y ∈ X
and
d(fx,fy)
ϕ(t) dt ≤ α d(x, y)
d(x,y)
ϕ(t) dt,
∀x, y ∈ X,
where c ∈ (, ) is a constant, ϕ ∈ and α : R+ → [, ) is a function with lim sups→t α(s) < , ∀t > . Mongkolkeha and Kumam [] proved fixed point and common fixed point theorems for ρ-compatible mapping satisfying a generalized weak contraction of integral type in modular spaces. Sintunavarat and Kumam [, ] gave common fixed point theorems ©2013 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu et al. Fixed Point Theory and Applications 2013, 2013:267 http://www.fixedpointtheoryandapplications.com/content/2013/1/267
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for single-valued and multi-valued mappings satisfying strict general contractive conditions of integral type. Inspired and motivated by the results in [–], in this paper, we introduce two new classes of contractive mappings of integral type in complete metric spaces and study the existence, uniqueness and iterative approximations of fixed points for the mappings. The results obtained in this paper generalize and improve Theorem . in [], Theorem . in [] and Theorem in []. Two nontrivial examples are constructed.
2 Preliminaries Throughout this paper, we assume that R = (–∞, +∞), R+ = [, +∞), N denotes the set of all positive integers, N
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