STRONG CONVERGENCE OF APPROXIMATION FIXED POINTS FOR NONEXPANSIVE NONSELF-MAPPING

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1 STRONG CONVERGENCE OF APPROXIMATION FIXED POINTS FOR NONEXPANSIVE NONSELF-MAPPING RUDONG CHEN AND ZHICHUAN ZHU Received 17 May 2006; Accepted 22 June 2006 Let C be a closed convex subset of a uniformly smooth Banach space E, andt : C E a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that FT,and f : C C afixedcontractivemapping.fort 0,1, the implicit iterative sequence {x t } is defined by x t = Ptfx t +1 ttx t, the explicit iterative sequence {x n } is given by x n+1 = Pα n f x n +1 α n Tx n, where α n 0,1 and P is a sunny nonexpansive retraction of E onto C. Weprovethat{x t } strongly converges to a fixed point of T as t 0, and {x n } strongly converges to a fixed point of T as α n satisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu 2004 and Yisheng Song and Rudong Chen Copyright All rights reserved. 1. Introduction Let C be a nonempty closed convex subset of a Banach space E, andlett : C C be a nonexpansive mapping i.e., Tx Ty x y for all x, y C. We use FixT to denote the set of fixed points of T; thatis,fixt ={x C : x = Tx}. Recall that a selfmapping f : C C is a contraction on C if there exists a constant β 0,1 such that f x f y β x y, x, y C. 1.1 Xu see [6] defined the following two viscosity iterations for nonexpansive mappings: x t = tf x t +1 ttxt, x C, 1.2 x n+1 = α n f x n + 1 αn Txn, 1.3 where α n is a sequence in 0,1. Xu proved the strong convergence of {x t } defined by 1.2 as t 0and{x n } defined by 1.3 in both Hilbert space and uniformly smooth Banach space. International Mathematics and Mathematical Sciences Volume 2006, Article ID 16470, Pages 1 12 DOI /IJMMS/2006/16470

2 2 Strong convergence of approximation fixed points Recently, Song and Chen [2] provedifc is a closed subset of a real reflexive Banach space E which admits a weakly sequentially continuous duality mapping from E to E, and if T : C E is a nonexpansive nonself-mapping satisfying the weakly inward condition, FT φ, f : C C is a fixed contractive mapping, and P is a sunny nonexpansive retraction of E onto C, then the sequences {x t } and {x n } defined by x t = P tf x t +1 ttxt, 1.4 x n+1 = P α n f x n + 1 αn Txn 1.5 strongly converge to a fixed point of T. In this paper, we establish the strong convergence of both {x t } defined by 1.4 and {x n } defined by 1.5 for a nonexpansive nonself-mapping T in a uniformly smooth Banach space. Our results extend and improve the results in [2, 6]. 2. Preliminaries Let E be a real Banach space and let J denote the normalized duality mapping from E into 2 E given by Jx = { f E : x, f = x f, x = f } x E, 2.1 where E denotes the dual space of E and, denotes the generalized duality pairing. In the sequence, we will denote the single-valued duality mapping by j, andx n x will denote strong convergence of the sequence {x n } to x. InBanachspaceE, the following result is well known [1, 3]forallx, y E,foralljx + y Jx + y, for all jx Jx, x 2 +2 y, jx x + y 2 x 2 +2 y, jx + y. 2.2 Recall that the norm of E is said to be Gâteaux differentiable and E is said to be smooth if x + ty x lim 2.3 t 0 t exists for each x, y in its unit sphere U ={x E : x =1}. Itissaidtobeuniformly Gâteaux differentiable if, for each y U, this limit is attained uniformly for x U. Finally, the norm is said to be uniformly Fréchet differentiable and E is said to be uniformly smooth if the limit in 2.3 is attained uniformly for x, y U U. ABanachspaceE is said to be smooth if and only if J is single valued. It is also well known that if E is uniformly smooth, J is uniformly norm-to-norm continuous. These concepts may be found in [3]. If C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and D C, then a mapping P : C D is called a retraction from C to D if P 2 = P. It is easily known that a mapping P : C D is retraction, then Px = x, forallx D. A mapping P : C D is called sunny if P Px + tx Px = Px x C, 2.4

3 R. Chen and Z. Zhu 3 whenever Px + tx Px C and t>0. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D.Formoredetail, see [1, 3 5]. The following lemma is well known [3]. Lemma 2.1. Let C be a nonempty convex subset of a smooth Banach space E, D C, J : E E the normalized duality mapping of E, andp : C D a retraction. Then the following are equivalent: i x Px, jy Px 0 for all x C and y D; ii P is both sunny and nonexpansive. Let C beanonemptyconvexsubsetofabanachspacee,thenforx C, we define the inward set [4, 5]: I C x = { y E : y = x + λz x, z C and λ 0 }. 2.5 AmappingT : C E is said to be satisfying the inward condition if Tx I C x forall x C. T is also said to be satisfying the weakly inward condition if for each x C, Tx I C xi C xistheclosureofi C x. Clearly C I C x and it is not hard to show that I C x is a convex set as C is. Using above these results and definitions, we can easily show the following lemma. Lemma 2.2 [2], Lemma 1.2. Let C be a nonempty closed subset of a smooth Banach space E, let T : C E be nonexpansive nonself-mapping satisfying the weakly inward condition, and let P be a sunny nonexpansive retraction of E onto C. Then FT = FPT. Lemma 2.3 [2], Lemma 2.1. Let E be a Banach space and let C be a nonempty closed convex subset of E. SupposethatT : C E is a nonexpansive mapping such that for each fixed contractive mapping f : C C,andP is a sunny nonexpansive retraction of E onto C. For each t 0,1, {x t } is defined by 1.4. Suppose u C is a fixed point of T, then i x t f x t, jx t u 0; ii {x t } is bounded. Definition 2.4. μ is called a Banach limit if μ is a continuous linear functional on l satisfying i μe =1 = μ1, e = 1,1,1,...; ii μ n a n = μ n a n+1, for all a n a 0,a 1,... l ; iii liminf a n μa n limsup a n,foralla n a 0,a 1,... l. According to time and circumstances, we use μ n a n insteadofμa 0,a 1,... Further, we know the following result. Lemma 2.5 [3], Lemma Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gâteaux differentiable norm and let {x n } be a bounded sequence in E. Let μ be a Banach limit and u C. Then μ n x n u 2 = min y C μ n x n y 2 2.6

4 4 Strong convergence of approximation fixed points if and only if for all x C. 3. Main results μ n x u,j xn u Theorem 3.1. Let E be a uniformly smooth Banach, suppose that C is a nonempty closed convex subset of E and T : C E is a nonexpansive nonself-mapping satisfying the weakly inward condition and FT.Let f : C C be a fixed contractive mapping, and let {x t } be defined by 1.4, where P is a sunny nonexpansive retraction of E onto C. Then as t 0 {x t } converges strongly to some fixed point q of T that q is the unique solution in FT to the following variational inequality: I f q, jq u 0 u FT. 3.1 Proof. For all u FT bylemma 2.3ii, {x t } is bounded, therefore the sets {Tx t : t 0,1} and { f x t :t 0,1} are also bounded. From x t = Ptfx t +1 ttx t, we have x t PTx t = P tf x t +1 ttxt PTxt This implies that tf x t +1 ttxt Tx t = t Tx t f x t 0 ast lim t 0 x t PTx t = Assume t n 0, set x n := x tn,anddefineg : C R by gx = μ n x n x 2, x C,whereμ n is a Banach limit on l.let { K = x C : gx = min x n y 2}. 3.4 y C μ n It is easily seen that K is a nonempty closed convex bounded subset of E, since note x n Tx n 0 gtx = μ n x n Tx 2 = μ n Tx n Tx 2 μ n x n x 2 = gx. 3.5 It follows that TK K, that is, K is invariant under T. Since a uniformly smooth Banach space has the fixed point property for nonexpansive mappings, T has a fixed point, say q,ink.fromlemma2.5we get μ n x q, j xn q 0, x C. 3.6

5 R. Chen and Z. Zhu 5 For all q FT, we have tfx t +1 tq = P[tfx t +1 tq], then x t [ tf x t +1 tq ] = P [ tf ] [ ] x t +1 ttxt P tf xt +1 tq 1 t Tx t q 1 t x t q. Hence from 2.2 and the above inequality we get x t [ tf x t +1 tq ] 2 = 1 t x t q + t x t f x t 2 1 t 2 x t q 2 +2t1 t x t f x t, j xt q Therefore xt f x t, j xt q Then We get 0 x t f x t, j xt q = x t q 2 + q f q, j x t q + f q f x t, j xt q 1 β x t q 2 + q f q, j x t q x t q 2 1 f q q, j xt q β Now applying Banach limit to the above inequality, we get μ n x t q 2 1 μ n f q q, j xt q β Let x = f qin3.6, and noting 3.12, we have that is, μ n x t q 2 0, 3.13 μ n x n q 2 = and then exists a subsequence which is still denoted by {x n } such that x n q, n We have proved that for any sequence {x tn } in {x t : t 0,1}, there exists a subsequencewhichisstilldenotedby{x tn } thatconvergestosomepointq of T.Toprovethat

6 6 Strong convergence of approximation fixed points the entire net {x t } converges to q, suppose that there exists another sequence {x sk } {x t } such that x sk p, ass k 0, then we also have p FT using lim t 0 x t PTx t =0. Next we show p = q and q is the unique solution in FT to the following variational inequality: I f q, jq u u FT Since the sets {x t u} and {x t f x t } are bounded and the uniform smoothness of E implies that the duality map J is norm-to-norm uniformly continuous on bounded sets of E,foranyu FT, by x sk p s k 0, we have I f x sk I f p 0, sk 0, x sk f x sk, j xsk u I f p, jp u = x sk f x sk I f p, j xsk u I f p, j x sk u jp u I f x sk I f p x sk u + I f p, j x sk u jp u 0 assk Therefore, noting Lemma 2.3i, for any u FT, we get I f p, jp u = lim xsk f x sk, j xsk u sk 0 Similarly, we also can show I f q, jq u = xtn f x tn, j xtn u Interchange q andu to obtain I f p, jp q Interchange p andu to obtain I f q, jq p This implies that p q f p f q, jp q 0, 3.22 that is, p q 2 β p q This is a contradiction, so we must have q = p. The proof is complete.

7 From Theorem 3.1 we can get the following corollary directly. R. Chen and Z. Zhu 7 Corollary 3.2. Let E be a uniformly smooth space, suppose C is a nonempty closed convex subset of E, T : C E is a nonexpansive mapping satisfying the weakly inward condition, and FT.Let f : C C be a fixed contractive mapping from C to C. {x t } is defined by x t = tf x t +1 tptxt, 3.24 where P is a sunny nonexpansive retraction of E onto C, then x t converges strongly to some fixed point q of T as t 0 and q is the unique solution in FT to the following variational inequality: I f q, jq u u FT Lemma 3.3 [6], Lemma 2.1. Let {α n } be a sequence of nonnegative real numbers satisfying the property where {γ n } 0,1 and δ n isasequenceinr such that: i lim γ n = 0 and n=0 γ n = ; ii either n=0 δ n < + or limsup δ n /γ n 0, then lim α n = 0. α n+1 1 γ n αn + δ n n 0, 3.26 Theorem 3.4. Let E be a uniformly smooth Banach space, suppose that C is a nonempty closed convex subset of E, T : C E is a nonexpansive nonself-mapping satisfying the weakly inward condition, and FT.Let f : C C be a fixed contractive mapping, and {x n } is defined by 1.5, where P is a sunny nonexpansive retraction of E onto C, andα n 0,1 satisfies the following conditions: i α n 0,as; ii n=0 α n = ; iii either n=0 α n+1 α n < or lim α n+1 /α n = 1. Then x n converges strongly to a fixed point q of T such that q is the unique solution in FT to the following variational inequality: I f q, jq u 0 u FT Proof. Firstweshow{x n } is bounded. Take u FT, it follows that x n+1 u = P 1 α n Txn + α n f x n Pu 1 α n Txn + α n f x n u 1 α n Tx n u + α n f x n f u + f u u 1 α n x n u + α n β x n u + f u u = 1 1 βα n x n u + α n f u u max{ x n u 1, 1 β f u u }. 3.28

8 8 Strong convergence of approximation fixed points By induction, x n u { max x 0 u 1, 1 β f u u }, n 0, 3.29 and {x n } is bounded, so are {Tx n } and { f x n }.Weclaimthat Indeed we have for some appropriate constant M>0 x n+1 x n 0 asn x n+1 x n = P α n f x n + 1 αn Txn P αn 1 f x n αn 1 Txn 1 α n f x n + 1 αn Txn α n 1 f x n 1 1 αn 1 Txn 1 1 α n Txn Tx n 1 + αn α n 1 f xn 1 Txn 1 + α n f x n f xn 1 1 α n x n x n 1 [3pt]+M α n α n 1 + βα n x n x n 1 = 1 1 βα n x n x n 1 [3pt]+M α n α n By Lemma 3.3 we have x n+1 x n 0, as n. We now show that x n PTx n In fact, x n+1 PTx n = P α n f x n + 1 αn Txn PTxn α n f x n Txn This follows from 3.30that x n PTx n x n x n+1 + x n+1 PTx n x n x n+1 + α n f x n Txn asn. Let q = lim t 0 x t,where{x t } is defined in Corollary 3.2,wegetthatq is the unique solution in FT to the following variational inequality: I f q, jq u 0 u FT We next show that limsup f q q, j xn q

9 Form Corollary 3.2,letx t = tfx t +1 tptx t, indeed we can write R. Chen and Z. Zhu 9 x t x n = t f x t xn +1 t PTxt x n Noting 3.32, putting a n t = x n PTx n x n PTx n +2 x n x t 0 asn, 3.38 and using 2.2, we obtain x t x n 2 1 t 2 PTx t x n 2 +2t f xt xn, j x t x n 1 t 2 PTx t PTx n + PTx n x n 2 +2t f xt xt, j x t x n +2t x t x n 2 1 t 2 x t x n 2 +1 t 2 x n PTx n t 2 PTx n x n x t x n +2t f x t xt, j x t x n +2t x t x n 2 1+t 2 x t x n 2 + an t+2t f x t xt, j x t x n The last inequality implies From a n t 0asn we get f xt xt, j t x n x t 2 x t x n t a nt limsup f xt xt, j t x n x t M 2, 3.41 where M>0 is a constant such that M x t x n 2 for all n 0andt 0,1. By letting t 0in3.41wehave lim t 0 limsup f xt xt, j x n x t On the one hand, for all ε>0, δ 1 such that t 0,δ 1, limsup f xt xt, j ε x n x t 2. **

10 10 Strong convergence of approximation fixed points On the other hand, {x t } strongly converges to q,ast 0, the set {x t x n } is bounded, and the duality map J is norm-to-norm uniformly continuous on bounded sets of uniformly smooth space E;fromx t q t 0, we get f q q f x t xt 0, t 0, f q q, j x n q f x t xt, j x n x t = f q q, j x n q j x n x t + f q q f xt xt, j xn x t f q q j x n q j x n x t + f q q f x t xt x n x t 0, t Hence for the above ε>0, δ 2, such that for all t 0,δ 2, for all n,wehave f q q, j x n q f x t xt, j x n x t ε Therefore, we have f q q, j xn q f x t xt, j x n x t + ε Noting ** and taking δ = min{δ 1,δ 2 },forallt 0,δ, we have limsup f q q, j xn q limsup f xt xt, j ε x n x t + ε ε = ε. Since ε is arbitrary, we get limsup f q q, j xn q Finally we show x n q. Indeed x n+1 α n f x n + 1 αn q = xn+1 q α n f xn q. 3.48

11 By 2.2we have x n+1 q 2 = x n+1 α n f x n + 1 αn q + αn f xn q 2 R. Chen and Z. Zhu 11 x n+1 P α n f x n + 1 αn q 2 +2αn f xn q, j xn+1 q P [ α n f ] x n + 1 αn Txn P αn f x n + 1 αn q 2 +2α n f xn q, j xn+1 q 2 1 α n Tx n q 2 +2α n f xn f q, j xn+1 q +2α n f q q, j xn+1 q α n x n q 2 +2α n f q f x n x n+1 q +2α n f q q, j xn+1 q 2 1 α n x n q 2 + α n f q f x n 2 + x n+1 q 2 +2α n f q q, j xn+1 q. Therefore, we have 1 αn x n+1 q 2 1 α n 2 x n q 2 + α n β 2 x n q 2 +2α n f q q, j xn+1 q That is, x n+1 q β2 α n x n q + α2 n x n q 2 1 α n 1 α n + 2α n 1 α n f q q, j xn+1 q 1 γ n x n q 2 + λγ n α n β 2 γ n f q q, j xn+1 q, 3.51 where γ n = 1 β 2 /1 α n α n and λ is a constant such that λ>1/1 β 2 x n q 2. Hence, x n+1 q 2 1 γ n x n q 2 + γ n λα n + 2 f q q, j xn+1 1 β 2 q It is easily seen that γ n 0, n=1 γ n =, and noting 3.36 limsup λα n + 2 f q q, j xn+1 1 β 2 q Applying Lemma 3.3 onto 3.52, we have x n q. The proof is complete.

12 12 Strong convergence of approximation fixed points Acknowledgment This work is supported by the National Science Foundation of China, Grants and References [1] N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Analysis , no. 6, [2] Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings,journal of Mathematical Analysis and Applications , no. 1, [3] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, [4] W. Takahashi and G.-E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications , no. 3, [5] H.-K. Xu, Approximating curves of nonexpansive nonself-mappings in Banach spaces, Comptes Rendus de l Académie des Sciences. Série I. Mathématique , no. 2, [6], Viscosity approximation methods for nonexpansive mappings, Mathematical Analysis and Applications , no. 1, Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin , China address: chenrd@tjpu.edu.cn Zhichuan Zhu: Department of Mathematics, Tianjin Polytechnic University, Tianjin , China address: zhuzcnh@yahoo.com.cn

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