Fixed point theory for nonlinear mappings in Banach spaces and applications
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RESEARCH
Open Access
Fixed point theory for nonlinear mappings in Banach spaces and applications Atid Kangtunyakarn* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
Abstract The purpose of this research is to study a finite family of the set of solutions of variational inequality problems and to prove a convergence theorem for the set of such problems and the sets of fixed points of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. We also prove a fixed point theorem for finite families of nonexpansive and strictly pseudo-contractive mappings in the last section. Keywords: nonexpansive mapping; strictly pseudo-contractive mapping; inverse-strongly monotone; variational inequality problems; uniformly convex; 2-uniformly smooth
1 Introduction Let E and E∗ be a Banach space and the dual space of E, respectively, and let C be a nonempty closed convex subset of E. Throughout this paper, we use ‘→’ and ‘’ to denote ∗ strong and weak convergence, respectively. The duality mapping J : E → E is defined by J(x) = {x∗ ∈ E∗ : x, x∗ = x , x = x∗ } for all x ∈ E. Definition . Let E be a Banach space. Then a function δX : [, ] → [, ] is said to be the modulus of convexity of E if x + y : x ≤ , y ≤ , x – y ≥ . δE () = inf – If δE () > for all ∈ (, ], then E is uniformly convex. The function ρE : R+ → R+ is said to be the modulus of smoothness of E if ρE (t) = sup
x + y + x – y – : x = , y = t ,
t ≥ .
If limt→ ρEt(t) = , then E is uniformly smooth. It is well known that every uniformly smooth Banach space is smooth and if E is smooth, then J is single-valued which is denoted by j. A Banach space E is said to be q-uniformly smooth if there exists a fixed constant c > such that ρE (t) ≤ ct q . If E is q-uniformly smooth, then q ≤ and E is uniformly smooth. © 2014 Kangtunyakarn; 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.
Kangtunyakarn Fixed Point Theory and Applications 2014, 2014:108 http://www.fixedpointtheoryandapplications.com/content/2014/1/108
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A mapping T : C → C is called nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C. T is called η-strictly pseudo-contractive if there exists a constant η ∈ (, ) such that Tx – Ty, j(x – y) ≤ x – y – η(I – T)x – (I – T)y
(.)
for every x, y ∈ C and for some j(x – y) ∈ J(x – y). It is clear that (.) is equivalent to the following: (I – T)x – (I – T)y, j(x – y) ≥ η(I – T)x – (I – T)y
(.)
for every x, y ∈ C and for some j(x – y) ∈ J(x – y). Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and D ⊂ C, then
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