Existence and approximations of fixed points for contractive mappings of integral type
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RESEARCH
Open Access
Existence and approximations of fixed points for contractive mappings of integral type Zeqing Liu1 , Xiaozhu Li1 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 The existence, uniqueness, and iterative approximations of fixed points for four classes of contractive mappings of integral type in complete metric spaces are established. The results presented in this paper generalize indeed several results of Branciari (J. Math. Math. Sci. 29(9):531-536, 2002), Rhoades (Int. J. Math. Math. Sci. 2003(63):4007-4013, 2003) and Liu et al. (Fixed Point Theory Appl. 2011:64, 2011). Four illustrative examples with uncountably many points are also included. MSC: 54H25 Keywords: contractive mappings of integral type; fixed point; complete metric space
1 Introduction Over the past decade the researchers [–] introduced a lot of contractive mappings of integral type and discussed the existence of fixed points and common fixed points for these mappings in metric spaces and modular spaces, respectively. Branciari [] was the first to study the existence of fixed points for the contractive mapping of integral type and proved the following result, which extends the Banach fixed point theorem. Theorem . ([]) Let f be a mapping from a complete metric space (X, d) into itself satisfying
d(fx,fy)
d(x,y)
ϕ(t) dt ≤ c
ϕ(t) dt,
∀x, y ∈ X,
where c ∈ (, ) is a constant and ϕ ∈ = {ϕ : ϕ : R+ → R+ is Lebesgue integrable, ε summable on each compact subset of R+ and ϕ(t) dt > for each ε > }. Then f has a unique fixed point a ∈ X such that limn→∞ f n x = a for each x ∈ X. Rhoades [] and Liu et al. [] extended the result of Branciari and proved the following fixed point theorems. Theorem . ([]) Let f be a mapping from a complete metric space (X, d) into itself satisfying
d(fx,fy)
max{d(x,y),d(x,fx),d(y,fy), [d(x,fy)+d(y,fx)]}
ϕ(t) dt ≤ c
ϕ(t) dt,
∀x, y ∈ X,
© 2014 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 2014, 2014:138 http://www.fixedpointtheoryandapplications.com/content/2014/1/138
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where c ∈ (, ) is a constant and ϕ ∈ . Then f has a unique fixed point a ∈ X such that limn→∞ f n x = a for each x ∈ X. Theorem . ([]) Let f be a mapping from a complete metric space (X, d) into itself satisfying
d(fx,fy)
max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}
ϕ(t) dt ≤ c
ϕ(t) dt,
∀x, y ∈ X,
where c ∈ (, ) is a constant and ϕ ∈ . Assume that f has a bounded orbit at some point x ∈ X. Then f has a unique fixed point a ∈ X such that limn→∞ f n x = a. Theorem . ([]) Let f be
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