Fixed point theory for nonlinear mappings in Banach spaces and applications

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1 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 R E S E A R C H Open Access Fixed point theory for nonlinear mappings in Banach spaces and applications Atid Kangtunyakarn * * Correspondence: beawrock@hotmail.com Department of Mathematics, Faculty of Science, King Mongkut s Institute of Technology Ladkrabang, Bangkok 1050, 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 -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; -uniformly smooth 1 Introduction Let E and E be a Banach space and the dual space of E, respectively,andletc be a nonempty closed convex subset of E. Throughoutthis paper, weuse and todenote strong and weak convergence, respectively. The duality mapping J : E E is defined by Jx)={x E : x, x = x, x = x } for all x E. Definition 1.1 Let E be a Banach space. Then a function δ X :[0,] [0, 1] is said to be the modulus of convexity of E if { δ E ɛ)=inf 1 x + y }. : x 1, y 1, x y ɛ If δ E ɛ) > 0 for all ɛ 0, ], then E is uniformly convex. The function ρ E : R + R + is said to be the modulus of smoothness of E if { } x + y + x y ρ E t)=sup 1: x =1, y = t, t 0. ρ If lim E t) t 0 =0,thenE is uniformly smooth. It is well known that every uniformly t smooth Banach space is smooth and if E is smooth, then J is single-valued which is denoted by j.abanachspacee is said to be q-uniformly smooth if there exists a fixed constant c >0suchthatρ E t) ct q.ife is q-uniformly smooth, then q ande is uniformly smooth. 014 Kangtunyakarn; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page of 16 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 0, 1) such that Tx Ty, jx y) x y η I T)x I T)y 1.1) for every x, y C and for some jx y) Jx y). It is clear that 1.1) isequivalenttothe following: I T)x I T)y, jx y) η I T)x I T)y 1.) for every x, y C and for some jx y) Jx y). Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and D C, then a mapping P : C D is sunny see [1]) provided Px + tx Px))) = Px) for all x C and t 0, whenever x + tx Px)) C. A mapping P : C D is called a retraction if Px = x for all x D.Furthermore,P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. AsubsetD of C is called a sunny nonexpansive retract of C see []) if there exists a sunny nonexpansive retraction from C onto D. An operator A of C into E is said to be accretive if there exists jx y) Jx y)suchthat Ax Ay, jx y) 0, x, y C. A mapping A : C E is said to be α-inverse strongly accretive if there exist jx y) Jx y)andα >0suchthat Ax Ay, jx y) α Ax Ay, x, y C. A mapping A : C E is called γ -strongly accretive if there exist jx y) Jx y) anda constant γ >0suchthat Ax Ay, jx y) γ x y for all x, y C. In 006,Aoyama et al. [3] studied the variational inequality problem in Banach spaces. Such a problem is to find a point x C such that for some jx x ) Jx x ), Ax, j x x ) 0, x C. 1.3) The set of solutions of 1.3) in Banach spaces is denoted by SC, A), that is, SC, A)= { u C : Au, Jv u) 0, v C }. 1.4) They introduced the strong convergence theorem involving the variational inequality problem in Banach spaces as follows.

3 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 3 of 16 Theorem 1.1 Let E be a uniformly convex and -uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let Q C be a sunny nonexpansive retraction from EontoC, let α >0,and let A be an α-inverse strongly accretive operator of C into E with SC, A). Suppose x 1 = x Cand{x n } is given by x n+1 = α n x n +1 α n )Q C x n λ n Ax n ) for every n =1,,...,where {λ n } is a sequence of positive real numbers and {α n } is a sequence α in [0, 1]. If {λ n } and {α n } are chosen so that λ n [a, ] for some a >0and α K n [b, c] for some b, cwith0<b < c <1,then {x n } converges weakly to some element z of SC, A), where K is the -uniformly smoothness constant of E. Many authors have studied the variational inequality problem; see, for example, [4 8]. The variational inequality problem is an important tool for studying fixed point theory, equilibrium problems, optimization problems and partial differential equations with applications principally drawn from mechanics; see, e.g.,[9, 10]. Recently, Kangtunyakarn [11] introduced a new mapping in uniformly convex and -smooth Banach spaces to prove a strong convergence theorem for finding a common element of the set of fixed points of finite families of nonexpansive and strictly pseudocontractive mappings and two sets of solutions of variational inequality problems as follows. Theorem 1. Let C be a nonempty closed convex subset of a uniformly convex and - uniformly smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C, and let A, Bbeα and β-inverse strongly accretive mappings of C into E, respectively. Let {S i } N be a finite family of κ i-strict pseudo-contractions of C into itself, and let {T i } N be a finite family of nonexpansive mappings of C into itself with F = N FS i) N FT i) SC, A) SC, B) and κ = min{κ i : i = 1,,...,N} with K κ, where K is the -uniformly smooth constant of E. Let α j =α j 1, αj, αj 3 ) I I I, where I = [0, 1], α j 1 + αj + αj 3 =1,αj 1 0, 1], αj [0, 1] and αj 3 0, 1) for all j =1,,...,N. Let SA be the S A -mapping generated by S 1, S,...,S N, T 1, T,...,T N and α 1, α,...,α N. Let {x n } be the sequence generated by x 1, u Cand x n+1 = α n u + β n x n + γ n Q C I aa)x n + δ n Q C I bb)x n + η n S A x n, n 1, 1.5) where {α n }, {β n }, {γ n }, {δ n }, {η n } [0, 1] and α n +β n +γ n +δ n +η n =1and satisfy the following conditions: i) lim n α n =0, n=1 α n = ; ii) {γ n }, {δ n }, {η n } [c, d] 0, 1) for some c, d >0; iii) n=1 β n+1 β n, n=1 γ n+1 γ n, n=1 δ n+1 δ n, n=1 η n+1 η n, n=1 α n+1 α n < ; iv) 0<lim inf n β n lim sup n β n <1; α β v) a 0, ) and b 0, ). K K Then {x n } converges strongly to z 0 = Q F u, where Q F is the sunny nonexpansive retraction of C onto F.

4 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 4 of 16 For every i =1,,...,N,letA i : C H be a mapping. From 1.3), we introduce the combination of variational inequality problems in Banach spaces as follows: to find a point x C such that for some jx x ) Jx x ), N a i A i x, j x x ) 0 1.6) for all x C and a i is a positive real number for all i =1,,...,N with N a i =1.Theset of solutions of 1.6) in Banach spaces is denoted by SC, N a ia i ), that is, ) { N } S C, a i A i = u C : a i A i u, Jv u) 0, v C. 1.7) By using 1.6) we prove the convergence theorem for a finite family of the set of solutions of variational inequality problems and two sets of fixed points of nonlinear mappings in a Banach space. Preliminaries The following lemmas are important tools to prove our main results in the next section. Lemma.1 See [1]) Let E be a real -uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds: x + y x + y, Jx) + Ky for any x, y E. Lemma. See [13]) Let X be a uniformly convex Banach space and B r = {x X : x r}, r >0.Then there exists a continuous, strictly increasing and convex function g :[0, ] [0, ], g0) = 0 such that αx + βy + γ z α x + β y + γ z αβg x y ) for all x, y, z B r and all α, β, γ [0, 1] with α + β + γ =1. Remark.3 For every i =1,,...,N,ifx i B r 0), from Lemma.,wehave N a ix i N a i x i,wherea i [0, 1] and N a i =1. Lemma.4 See [3]) Let C be a nonempty closed convex subset of a smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all λ >0, SC, A)=F Q C I λa) ). Lemma.5 See [1]) Let r >0.If E is uniformly convex, then there exists a continuous, strictly increasing and convex function g :[0, ) [0, ), g0) = 0 such that for all x, y B r 0) = {x E : x r} and for any α [0, 1], we have αx +1 α)y α x +1 α) y α1 α)g x y ).

5 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 5 of 16 Lemma.6 See [14]) Let C be a closed and convex subset of a real uniformly smooth Banach space E, and let T : C C be a nonexpansive mapping with a nonempty fixed point FT). If {x n } C is a bounded sequence such that lim n x n Tx n =0,then there exists a unique sunny nonexpansive retraction Q FT) : C FT) such that lim sup u QFT) u, Jx n Q FT) u) 0 n for any given u C. Lemma.7 See [15]) Let {s n } be a sequence of nonnegative real numbers satisfying s n+1 1 α n )s n + δ n, n 0, where {α n } is a sequence in 0, 1) and {δ n } is a sequence such that 1) n=1 α n =, δ ) lim sup n n α n 0 or n=1 δ n <. Then lim n s n =0. Lemma.8 [11] Let C be a closed convex subset of a strictly convex Banach space E. Let T 1, T and T 3 be three nonexpansive mappings from C into itself with FT 1 ) FT ) FT 3 ). Define a mapping S by Sx = αt 1 x + βt x + γ T 3 x, x C, where α, β, γ is a constant in 0, 1) and α + β + γ =1.Then S is a nonexpansive mapping and FS)=FT 1 ) FT ) FT 3 ). Lemma.9 Let C be a nonempty closed convex subset of a real smooth Banach space E. For every i = 1,,...,N, let A i : C Ebeanα i -strongly accretive mapping with α = min,,...,n {α i } and N SC, A i). Then SC, N a ia i )= N SC, A i), where a i [0, 1] and N a i =1. Proof It is easy to see that N SC, A i) SC, N a ia i ). Let x 0 SC, N a ia i )and x N SC, A i). Then there exist jy x ) Jy x )andjy x 0 ) Jy x 0 )suchthat N a i A i x 0, jy x 0 ) 0, y C.1) and N a i A i x, j y x ) 0, y C..) From.1),.)andx 0, x C,wehave and N a i A i x 0, j x ) x 0 0.3) N a i A i x, j x 0 x ) 0..4)

6 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 6 of 16 From.3)and.4), we have N 0 a i A i x 0 a i A i x, j x ) x 0 N = a i A i x a i A i x 0, j x ) x 0 = a i Ai x A i x 0, j x ) N x 0 a i α i x x 0 a i α x x 0 = α x x 0. It implies that x = x 0,thatis,x 0 N SC, A i). Therefore SC, N a ia i ) N SC, A i). 3 Main results Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and - uniformly smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C. For every i = 1,,...,N, let A i : C Ebeα i -strongly accretive and L i -Lipschitz continuous with α = min,,...,n α i and L = max,,...,n L i. Let T : C C be a nonexpansive mapping and S : C Cbeanη-strictly pseudo-contractive mapping with K η, where K is the -uniformly smooth constant of E. Assume that F = FT) FS) N SC, A i). Let {x n } be a sequence generated by u, x 1 Cand z n = c n x n +1 c n )Sx n, y n = b n x n +1 b n )Tz n, x n+1 = α n u + β n x n + γ n Q C I λ N a ia i )y n, n 1, 3.1) where a i [0, 1] for all i =1,,...,Nand{α n }, {β n }, {γ n } [0, 1] with α n + β n + γ n =1for all n N satisfy the following conditions: i) lim n α n =0and n=1 α n = ; ii) 0<a β n, γ n, c n, b n b <1for some a, b >0, n N and N a i =1; iii) 0 λk α ; L iv) n=1 α n+1 α n, n=1 β n+1 β n, n=1 b n+1 b n, n=1 c n+1 c n <. Then {x n } converges strongly to z 0 = Q F u, where Q F is the sunny nonexpansive retraction of C onto F. Proof First, we show that N a ia i is an α L -inverse strongly monotone mapping. Let x, y C,thereexistsjx y) Jx y)and N a i A i x a i A i y, jx y) = a i Ai x A i y, jx y) a i α i x y

7 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 7 of 16 α L α i a i L A i x A i y i N a i A i x A i y α N L a i A i x a i A i y. 3.) Next, we show that Q C I λ N a ia i ) is a nonexpansive mapping. From 3.), we have ) Q C I λ a i A i x Q C I λ a i A i )y N x y λ a i A i x a i A i y) N x y λ a i A i x a i A i y, jx y) N +K λ a i A i x A i y) x y λ α N L a i A i x a i A i y N +K λ a i A i x A i y) = x y ) α λ L K λ a i A i x a i A i y x y for all x, y C.Letx F,wehave xn+1 x = α n u x ) + β n xn x ) + γ n Q C I λ α n u x + βn xn x + γn yn x ) ) a i A i y n x = α n u x + βn xn x + γn bn x n p)+1 b n ) Tz n x ) α n u x + β n x n x + γ n bn x n p +1 b n ) zn x ) = α n u x + β n x n x + γ n bn x n p +1 b n ) cn xn x ) +1 c n ) Sx n x ) ). 3.3)

8 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 8 of 16 Since S is a strictly pseudo-contractive mapping, we have cn xn x ) +1 c n ) Sx n x ) = xn x +1 c n )Sx n x n ) x n x +1 c n ) Sx n x n, j x n x ) +K 1 c n ) Sx n x n = x n x 1 c n ) I S)x n, j x n x ) +K 1 c n ) I S)xn x n x 1 c n )η I S)x n +K 1 c n ) I S)xn x n x 1 c n ) η I S)x n +K 1 c n ) I S)xn x n x 1 c n ) η K ) I S)x n xn x. 3.4) From 3.3)and3.4), we have x n+1 x α n u x + β n x n x + γ n bn x n p +1 b n ) cn xn x ) +1 c n ) Sx n x ) ) α n u x +1 α n ) x n x max { u x, x1 x }. From induction we can conclude that {x n } is bounded and so are {y n }, {z n }. Next, we show that lim n x n+1 x n =0. For every n N,wehave x n+1 x n = α nu + β n x n + γ n Q C I λ a i A i )y n α n 1 u β n 1 x n 1 γ n 1 Q C I λ ) a i A i y n 1 α n α n 1 u + β n x n x n 1 + β n β n 1 x n 1 ) ) + γ n Q C I λ a i A i y n Q C I λ a i A i y n 1 ) + γ n γ n 1 Q C I λ a i A i y n 1 α n α n 1 u + β n x n x n 1 + β n β n 1 x n 1 + γ n y n y n 1 + γ n γ n 1 Q C I λ a i A i )y n )

9 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page 9 of 16 From the definition of y n,wehave y n y n 1 = bn x n +1 b n )Tz n b n 1 x n 1 1 b n 1 )Tz n 1 b n x n x n 1 + b n b n 1 x n 1 +1 b n ) Tz n Tz n 1 + b n b n 1 Tz n 1 b n x n x n 1 + b n b n 1 x n 1 +1 b n ) z n z n 1 + b n b n 1 Tz n ) From the definition of z n,wehave z n z n 1 = cn x n +1 c n )Sx n c n 1 x n 1 1 c n 1 )Sx n 1 = c n x n x n 1 )+c n c n 1 )x n 1 +1 c n )Sx n Sx n 1 ) +c n 1 c n )Sx n 1 c n x n x n 1 )+1 c n )Sx n Sx n 1 ) + c n c n 1 x n 1 + c n c n 1 Sx n ) Since S is an η-strictly pseudo-contractive mapping, we have cn x n x n 1 )+1 c n )Sx n Sx n 1 ) = xn x n 1 1 c n ) I S)x n I S)x n 1 ) x n x n 1 1 c n ) I S)x n I S)x n 1, jx n x n 1 ) +K 1 c n ) I S)xn I S)x n 1 x n x n 1 1 c n )η I S)xn I S)x n 1 +K 1 c n ) I S)x n I S)x n 1 x n x n 1 1 c n ) η K ) I S)xn I S)x n 1 x n x n ) From 3.6), 3.7)and3.8), we have y n y n 1 b n x n x n 1 + b n b n 1 x n 1 +1 b n ) z n z n 1 + b n b n 1 Tz n 1 b n x n x n 1 + b n b n 1 x n 1 +1 b n ) c n x n x n 1 ) +1 c n )Sx n Sx n 1 ) + cn c n 1 x n 1 + c n c n 1 Sx n 1 ) + b n b n 1 Tz n 1 b n x n x n 1 + b n b n 1 x n 1

10 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page10of16 +1 b n ) x n x n 1 + c n c n 1 x n 1 + c n c n 1 Sx n 1 ) + b n b n 1 Tz n 1 x n x n 1 + b n b n 1 x n 1 + c n c n 1 x n 1 + c n c n 1 Sx n 1 + b n b n 1 Tz n ) From 3.9)and3.5), we have x n+1 x n α n α n 1 u + β n x n x n 1 + β n β n 1 x n 1 ) + γ n y n y n 1 + γ n γ n 1 Q C I λ a i A i y n 1 1 α n ) x n x n 1 + β n β n 1 x n 1 + α n α n 1 u ) + γ n γ n 1 Q C I λ a i A i y n 1 + b n b n 1 x n 1 + c n c n 1 x n 1 + c n c n 1 Sx n 1 + b n b n 1 Tz n 1 1 α n ) x n x n 1 + β n β n 1 M + α n α n 1 M + γ n γ n 1 M + b n b n 1 M + c n c n 1 M, where M = max n N { x n, u, Q C I λ N a ia i )y n, Sx n, Tz n }. Applying Lemma.7, conditions i) and iv), we have lim x n+1 x n = ) n Next, we show that lim n Q C I λ a i A i )x n x n = lim x n Tx n = lim x n Sx n =0. n n From the definition of x n,wehave xn+1 x = α n u x ) ) ) + β n xn x ) + γ n Q C I λ a i A i y n x α n u x + β n xn x + γ n Q C I λ β n γ n g 1 Q C I λ ) ) a i A i y n x n α n u x + β n xn x + γ n yn x β n γ n g 1 Q C I λ ) ) a i A i y n x n ) a i A i y n x

11 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page11of16 = α n u x + β xn n x + γ bn n xn x ) +1 b n ) Tz n x ) ) ) β n γ n g 1 Q C I λ a i A i y n x n α n u x + β n x n x + γ n bnxn x +1 b n ) zn x b n 1 b n )g xn Tz n )) ) ) β n γ n g 1 Q C I λ a i A i y n x n α n u x + β n x n x + γ n bn x n x +1 b n ) x n x 1 c n ) η K ) Sx n x n ) b n 1 b n )g xn Tz n )) ) ) β n γ n g 1 Q C I λ a i A i y n x n α n u x + β n x n x + γ xn n x γ n 1 b n )1 c n ) η K ) Sx n x n γ n b n 1 b n )g xn Tz n ) ) ) β n γ n g 1 Q C I λ a i A i y n x n α n u x + x n x γ n 1 b n )1 c n ) η K ) Sx n x n γ n b n 1 b n )g xn Tz n ) ) β n γ n g 1 Q C I λ a i A i y n x n ). It implies that γ n 1 b n )1 c n ) η K ) Sx n x n + γ n b n 1 b n )g xn Tz n ) ) ) + β n γ n g 1 Q C I λ a i A i y n x n α n u x + x n x x n+1 x α n u x + x n x + x n+1 x ) x n+1 x n. From 3.10), conditions i) and ii), we have lim Sx n x n = lim g 1 n n Q C I λ ) ) a i A i y n x n = lim g xn Tz n ) = ) n

12 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page1of16 From the properties of g 1 and g,wehave lim n Q C I λ a i A i )y n x n = lim x n Tz n =0. 3.1) n From 3.11) and the definition of z n,wehave lim n z n x n = ) From 3.1) and the definition of y n,wehave lim n y n x n = ) From 3.13)and3.14), we have lim n y n z n = ) From 3.1)and3.14), we have Then lim n Q C I λ a i A i )y n y n = ) ) ) Q C I λ a i A i )x n x n Q C I λ a i A i x n Q C I λ a i A i y n From 3.1)and3.14), we have Since ) + Q C I λ a i A i y n x n x n y n + Q C I λ a i A i )y n x n. lim n Q C I λ a i A i )x n x n = ) Tx n x n Tx n Tz n + Tz n x n x n z n + Tz n x n, from 3.1)and3.13), we have lim n Tx n x n = )

13 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page13of16 Define the mapping G : C C by Gx = αq C I λ N a ia i )x + βtx + γ Wx,whereWx = cx +1 c)sx for all x C and α, β, γ, c [0, 1] with α + β + γ = 1. We show that W is a nonexpansive mapping. Let x, y C, wehave Wx Wy = cx y)+1 c)sx Sy) = x y 1 c) I S)x I S)y ) x y 1 c) I S)x I S)y, jx y) +K 1 c) I S)x I S)y x y 1 c)η I S)x I S)y +K 1 c) I S)x I S)y x y 1 c) η K ) I S)x I S)y x y. Then W isa nonexpansivemapping. It iseasy tosee that themapping G is nonexpansive. From the definition of W,we have FS)=FW). 3.19) From 3.19) Lemmas.8,.9 and the definition of G, wehavefg) =FT) FS) N SC, A i)=f.since Gx n x n α Q C I λ a i A i )x n x n + β Tx n x n + γ Wx n x n = α Q C I λ a i A i )x n x n + β Tx n x n + γ 1 c) Sx n x n, and 3.11), 3.17)and3.18), we have lim Gx n x n =0. 3.0) n From Lemma.6,wehave lim sup u z0, jx n z 0 ) 0, 3.1) n where z 0 = Q F u. Finally, we show that the sequence {x n } converges strongly to z 0 = Q F u. From the definition of x n,wehave ) ) x n+1 z 0 = α nu z 0 )+β n x n z 0 )+γ n Q C I λ a i A i y n z 0 ) ) β nx n z 0 )+γ n Q C I λ a i A i y n z 0

14 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page14of16 +α n u z0, jx n+1 z 0 ) 1 α n ) x n z 0 +α n u z0, jx n+1 z 0 ). From Lemma.7 and condition i), we can conclude that the sequence {x n } converges strongly to z 0 = Q F u. This completes the proof. The followingcorollaryis a direct sequel of Theorem 3.1. Therefore, we omit the proof. Corollary 3. Let C be a nonempty closed convex subset of a uniformly convex and - uniformly smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C. Let A : C Ebeα-strongly accretive and L-Lipschitz continuous. Let T : C C be a nonexpansive mapping and S : C Cbeanη-strictly pseudo-contractive mapping with K η, where K is the -uniformly smooth constant of E. Assume that F = FT) FS) SC, A). Let {x n } be the sequence generated by u, x 1 Cand z n = c n x n +1 c n )Sx n, y n = b n x n +1 b n )Tz n, x n+1 = α n u + β n x n + γ n Q C I λa)y n, n 1, where {α n }, {β n }, {γ n } [0, 1] with α n + β n + γ n =1for all n N satisfy the following conditions: i) lim n α n =0and n=1 α n = ; ii) 0<a β n, γ n, c n, b n b <1, for some a, b >0, n N; iii) 0 λk α ; L iv) n=1 α n+1 α n, n=1 β n+1 β n, n=1 b n+1 b n, n=1 c n+1 c n <. Then {x n } converges strongly to z 0 = Q F u, where Q F is the sunny nonexpansive retraction of C onto F. 4 Applications Using the concepts of the S A -mapping and Theorem 3.1, we prove the strong convergence theorem for the set of fixed points of two finite families of nonlinear mappings. We need the following definition and lemma to prove our result. Definition 4.1 [11] LetC be a nonempty convex subset of a real Banach space. Let {S i } N and {T i } N be two finite families of the mappings of C into itself. For each j =1,,...,N, let α j =α j 1, αj, αj 3 ) I I I, wherei [0, 1] and αj 1 + αj + αj 3 = 1. Define the mapping S A : C C as follows: U 0 = T 1 = I, U 1 = T 1 α 1 1 S 1 U 0 + α 1 U 0 + α 1 3 I), U = T α 1 S U 1 + α U 1 + α 3 I), U 3 = T 3 α 3 1 S 3 U + α 3 U + α 3 3 I),.

15 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page15of16 U N 1 = T N 1 α N 1 1 S N 1 U N + α N 1 U N + α N 1 3 I ), S A = U N = T N α N 1 S N U N 1 + α N U N 1 + α N 3 I). This mapping is called the S A -mapping generated by S 1, S,...,S N, T 1, T,...,T N α 1, α,...,α N. and Lemma 4.1 [11] Let C be a nonempty closed convex subset of a -uniformly smooth and uniformly convex Banach space. Let {S i } N be a finite family of κ i-strict pseudocontractions of C into itself, and let {T i } N be a finite family of nonexpansive mappings of C into itself with N FS i) N FT i) and κ = min{κ i : i =1,,...,N} with K κ, where K is the -uniformly smooth constant of E. Let α j =α j 1, αj, αj 3 ) I I I, where I = [0, 1], α j 1 + αj + αj 3 =1, αj 1 0, 1], αj [0, 1] and αj 3 0, 1) for all j = 1,,...,N. Let S A be the S A -mapping generated by S 1, S,...,S N, T 1, T,...,T N and α 1, α,...,α N. Then FS A )= N FS i) N FT i) and S A is a nonexpansive mapping. Theorem 4. Let C be a nonempty closed convex subset of a uniformly convex and - uniformly smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C. For every i = 1,,...,N, let A i : C Ebeα i -strongly accretive and L i -Lipschitz continuous with α = min,,...,n α i and L = max,,...,n L i. Let S : C Cbeanη-strictly pseudo-contractive mapping with K η, where K is the -uniformly smooth constant of E. Let {S i } N be a finite family of κ i-strict pseudo-contractions of C into itself, and let {T i } N be a finite family of nonexpansive mappings of C into itself with κ = min{κ i : i =1,,...,N} with K κ. Let α j =α j 1, αj, αj 3 ) I I I, where I = [0, 1], αj 1 + αj + αj 3 =1,αj 1 0, 1], α j [0, 1] and αj 3 0, 1) for all j = 1,,...,N. Let SA be the S A -mapping generated by S 1, S,...,S N, T 1, T,...,T N and α 1, α,...,α N. Assume that F = FS) N SC, A i) N FS i) N FT i). Let {x n } be the sequence generated by u, x 1 Cand z n = c n x n +1 c n )Sx n, y n = b n x n +1 b n )S A z n, x n+1 = α n u + β n x n + γ n Q C I λ N a ia i )y n, n 1, where a i [0, 1] for all i =1,,...,Nand{α n }, {β n }, {γ n } [0, 1] with α n + β n + γ n =1for all n N and satisfy the following conditions: i) lim n α n =0and n=1 α n = ; ii) 0<a β n, γ n, c n, b n b <1for some a, b >0, n N and N a i =1; iii) 0 λk α ; L iv) n=1 α n+1 α n, n=1 β n+1 β n, n=1 b n+1 b n, n=1 c n+1 c n <. Then {x n } converges strongly to z 0 = Q F u, where Q F is the sunny nonexpansive retraction of C onto F. 4.1) Proof FromLemma 4.1and Theorem 3.1, we can reach the desired conclusion. Competing interests The author declares that they have no competing interests.

16 Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 Page16of16 Acknowledgements This research was supported by the Research Administration Division of King Mongkut s Institute of Technology Ladkrabang. Received: 16 December 013 Accepted: 1 April 014 Published: 06 May 014 References 1. Reich, S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, ). Kopecka, E, Reich, S: Nonexpansive retracts in Banach spaces. Banach Cent. Publ. 77, ) 3. Aoyama, K, Iiduka, H, Takahashi, W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 006, Article ID ).doi: /FPTA/006/ Ceng, LC, Wang, CY, Yao, JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, ) 5. Yao, Y, Liou, YC, Kang, SM: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Appl. Math. 59, ) 6. Qin, X, Kang, SM: Convergence theorems on an iterative method for variational inequality problems and fixed point problems. Bull. Malays. Math. Soc. 33, ) 7. Censor, Y, Gibali, A, Reich, S: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, ) 8. Kassay, G, Reich, S, Sabach, S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 1, ) 9. Yao, Y, Cho, YJ, Chen, R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 71, ) 10. Qin, X, Shang, M, Su, Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 48, ) 11. Kangtunyakarn, A: A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and -smooth Banach spaces. Fixed Point Theory Appl. 013,Article ID ) 1. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, ) 13. Cho, YJ, Zhou, HY, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, ) 14. Zhou, H: Convergence theorems for κ-strict pseudocontractions in -uniformly smooth Banach spaces. Nonlinear Anal. 69, ) 15. Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, ) / Cite this article as: Kangtunyakarn: Fixed point theory for nonlinear mappings in Banach spaces and applications. Fixed Point Theory and Applications 014, 014:108