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Tang et al. Fixed Point Theory and Applications :00 DOI 0.86/s y R E S E A R C H Open Access Strong convergence theorems of Cesàro-type means for nonexpansive mapping in CAT0 space Jinfang

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Tang et al. Fixed Point Theory and Applications :00 DOI 0.86/s y R E S E A R C H Open Access Strong convergence theorems of Cesàro-type means for nonexpansive mapping in CAT0 space Jinfang Tang, Shih-sen Chang 2* and Jian Dong * Correspondence: 2 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 65022, China Full list of author information is available at the end of the article Abstract The iterative algorithms with Cesàro-type means for a nonexpansive mapping are proposed in CAT0 spaces. Under suitable conditions, some strong convergence theorems for the sequence generated by the algorithms to a fixed point of the nonexpansive mapping are proved. We also proved that this fixed point is also a unique solution to some kind of variational inequality. The results presented in this paper extend and improve the corresponding results of some others. Keywords: Cesàro-type means method; CAT0 space; fixed point Introduction In 975, Baillon [] first proved the following nonlinear ergodic theorem. Theorem. Suppose that C is a nonempty closed convex subset of a Hilbert space E and T : C C is a nonexpansive mapping such that FT. Then, x C, thecesàromeans T n x = n + T i x. weakly converges to a fixed point of T. In 979, Bruck [2] generalized the Baillon Cesàro s means theorem from Hilbert space to uniformly convex Banach space with Fréchet differentiable norms. In 2007, Song and Chen [3] defined the following viscosity iteration {x n } of the Cesàro means for nonexpansive mapping T: x n+ = α n f x n + α n n + T i x,.2 and they proved that the sequence {x n } converged strongly to some point in FT ina uniformly convex Banach space with weakly sequentially continuous duality mapping. In 2009, Yao et al. [4] in a Banach space introduced the following process {x n }: x n+ = α n u + β n x n + γ n Tx n, n Tang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. Tang et al. Fixed Point Theory and Applications :00 Page 2 of 3 They proved that the sequence {x n } converged strongly to a fixed point of T under suitable control conditions of parameters. In 204, Zhu and Chen [5] proposed the following iterations with Cesàro s means for nonexpansive mappings: x n+ = α n u + β n x n + γ n n + and viscosity iteration: T i x n, n 0,.4 x n+ = α n f x n +β n x n + γ n n + T i x n, n 0..5 They proved the sequences {x n } both converged strongly to a fixed point of FT inthe framework of a uniformly smooth Banach space. As is well known, an extension of a linear version usually in Banach spaces or Hilbert spaces of this well-known result to a metric space has its own importance. The above iterative methods.-.5 involve general convex combinations. If we want to extend these results from Hilbert spaces or Banach spaces to metric spaces, we need some convex structure in a metric space to investigate their convergence on a nonlinear domain. On the other hand, recently the theory and applications of CAT0 space have been studied extensively by many authors. Recall that a metric space X, d is called a CAT0 space, if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is known that any complete, simply connected Riemannian manifold having non-positive sectional curvature is a CAT0 space. Other examples of CAT0 spaces include pre-hilbert spaces see [6], R-trees see [7], Euclidean buildings see [8], the complex Hilbert ball with a hyperbolic metric see [9], and many others. A complete CAT0 space is often called a Hadamard space. A subset K of a CAT0 space X is convex if, for any x, y K, wehave[x, y] K, where[x, y] is the uniquely geodesic joining x and y. For a thorough discussion of CAT0 spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [6]. Fixed point theory in CAT0 spaces has been first studied by Kirk see [0, ]. He showed that every nonexpansive single-valued mapping defined on a bounded closed convex subset of a complete CAT0 space always has a fixed point. Motivated and inspired by the research going on in this direction, it is naturally to put forward the following. Open question Can we extend the above Cesàro nonlinear ergodic theorems for nonexpansive mapping in [ 5] from a Hilbert space or Banach space to a CAT0 space? The purpose of this paper is to give an affirmative answer to this question. For solving this problem we first propose the following new iterations with Cesàro s means for nonexpansive mappings in the setting of CAT0 spaces: x n+ = α n u β n x n γ n n + T i x n,.6 Tang et al. Fixed Point Theory and Applications :00 Page 3 of 3 and the viscosity iteration: x n+ = α n f x n β n x n γ n n + T i x n,.7 and then study the strong convergence of the iterative sequences.6 and.7. Under suitable conditions, some strong converge theorems to a fixed point of the nonexpansive mapping are proved. We also prove that this fixed point is a unique solution of some kind of variational inequality in CAT0 spaces. The results presented in this paper are new; they extend and improve corresponding previous results. Next we give some examples of iteration.6 or.7 with Cesàro s means for nonexpansive mappings to illustrate the generation of our new iterations. Example If X is a real Hilbert or Banach space, and if α n =0,β n =0, n, then the iteration.6 with Cesàro s means for nonexpansive mappings is reduced as the same iteration of Baillon []orbruck[2]. Example 2 If X is a real Banach space and n =, then the iteration.6 withcesàro s means for nonexpansive mappings is reduced to the same iteration of Yao et al. [4]. If n 2, then the iteration.6canbewritten x n+ = α n u + β n x n + γ n n + T i x n, which is a generalization of results due to Yao et al. [4]. Example 3 The iterations.6 and.7 are a generalization of the iterations of Zhu and Chen [5] from Banach space to CAT0 spaces. Example 4 Let E = R with the usual metric, T be a nonexpansive mapping defined by Tx = sin x,andf be a contractive mapping defined by f x= x 2.Letα n = 2n, β n = 4n 3 4n,and γ n =, n. Then the iteration.6 and.7 with Cesàro s means for nonexpansive 4n mappings Tx = sin x are reduced to the following iterations, respectively: x n+ = 4n 3 u + 2n 4n x n + xn + Tx n + T 2 x n + + T n x n, 4nn +.8 x n+ = 2n 2 x 4n 3 n + 4n x n + xn + Tx n + T 2 x n + + T n x n. 4nn +.9 Example 5 If X, d is a CAT0 space, n =andβ n = 0, then the iteration.7 with Cesàro s means for nonexpansive mappings is reduced to the following iteration, with Cesàro s means for nonexpansive mappings: x n+ = α n f x n γ n 2 x n γ n 2 Tx n,.0 which is a generalization of the iterations in Wangkeeree and Preechasilp [2]fromaBa- nach space to CAT0 spaces. Tang et al. Fixed Point Theory and Applications :00 Page 4 of 3 2 Preliminaries and lemmas In this paper, we write tx ty for the unique point z in the geodesic segment joining from x to y such that dx, z =tdx, y, dy, z = tdx, y. 2. Lemma 2. [3] AgeodesicspaceX, d is a CAT0 space, ifandonlyiftheinequality d 2 tx ty, z td 2 x, z+td 2 y, z t td 2 x, y 2.2 is satisfied for all x, y, z Xandt [0, ]. In particular, if x, y, z are points in a CAT0 space and t [0, ], then d tx ty, z tdx, z+tdy, z. Lemma 2.2 [6] Let X, d be a CAT0 space, p, q, r, s X, and λ [0, ]. Then d λp λq, λr λs λdp, r+ λdq, s. 2.3 By induction, we write n m= λ λ m x m := λ n x λ 2 x 2 λ n x n λ n x n. 2.4 λ n λ n λ n Lemma 2.3 [4] Let X, d be a CAT0 space, then for any sequence {λ i } n i= in [0,] satisfying n i= λ i =and for any {x i } n i= X, the following conclusions hold: d λ i x i, x i= λ i dx i, x, x X; 2.5 i= and d 2 i= λ i x i, x λ i d 2 x i, x λ λ 2 d 2 x, x 2, x X. i= Lemma 2.4 Let {x i } and {y i } be any sequences of a CAT0 space X, then for any sequence {λ i } k i= in [0, ] satisfying k i= λ i =and for any {x i } k i= X, the following inequality holds: k d λ i x i, i= k λ i y i i= k λ i dx i, y i. 2.6 i= Proof It is obvious that 2.6holdsfork =2.Supposethat2.6 holds for some k 3. Next we prove that 2.6isalsotruefork +.From2.3and2.4wehave Tang et al. Fixed Point Theory and Applications :00 Page 5 of 3 d k+ k+ λ i x i, λ i y i i= i= k k λ i λ i = d λ k+ x i λ k+ x k+, λ k+ y i λ k+ y k+ λ k+ λ k+ λ k+ d λ k+ i= k k i= k+ = λ i dx i, y i. i= i= λ i λ k+ x i, i= i= k λ i y i + λ k+ dx k+, y k+ λ k+ λ i λ k+ dx i, y i +λ k+ dx k+, y k+ This implies that 2.6holds. Berg and Nikolaev [5] introduced the concept of quasilinearization as follows. Let us denote a pair a, b X X by ab and call it a vector. Then quasilinearization is defined as a map, :X X X X R defined by ab, cd = 2 d 2 a, d+d 2 b, c d 2 a, c d 2 b, d a, b, c, d X. 2.7 It is easy to seen that ab, cd = cd, ab, ab, cd = ba, cd, and ax, cd + xb, cd = ab, cd for all a, b, c, d X.WesaythatX satisfies the Cauchy-Schwarz inequality if ab, cd da, bdc, d 2.8 for all a, b, c, d X. Itiswellknown[5] that a geodesically connected metric space is a CAT0 space if and only if it satisfies the Cauchy-Schwarz inequality. Let C be a nonempty closed convex subset of a complete CAT0 space X. Themetric projection P C : X C is defined by u = P C x du, x=inf { dy, x:y C }, x X. Lemma 2.5 [6] Let C be a nonempty convex subset of a complete CAT0 space X, x X, and u C. Then u = P C x if and only if yu, ux 0, y C. Lemma 2.6 [2] LetCbeaclosedconvexsubsetofacompleteCAT0 space X, and let T : C C be a nonexpansive mapping with FT. Let f be a contraction on C with coefficient α .For each t [0, ], let {x t } be the net defined by x t = tf x t ttx t. 2.9 Tang et al. Fixed Point Theory and Applications :00 Page 6 of 3 Then lim t 0 x t = x, some point FT which is the unique solution of the following variational inequality: xf x, x x 0, x FT. 2.0 Lemma 2.7 [2] Let X be a complete CAT0 space. Then, for any u, x, y X, the following inequality holds: d 2 x, u d 2 y, u+2 xy, xu. Lemma 2.8 [2] Let X be a complete CAT0 space. For any t [0, ] and u, v X, let u t = tu tv. Then, for any x, y X, i ut x, ut y t ux, ut y + t vx, ut y; ii ut x, uy t ux, uy + t vx, uy, and ut x, vy t ux, vy + t vx, vy. Lemma 2.9 [7] Let {a n } be a sequence of non-negative real numbers satisfying the property a n+ α n a n + α n β n, n 0, where {α n } 0, and {β n } R such that i n=0 α n = ; ii lim sup n β n 0 or n=0 α nβ n . Then {a n } converges to zero as n. 3 Approximative iterative algorithms Throughout this section, we assume that C is a nonempty and closed convex subset of a complete CAT0 space X,andT : C C is a nonexpansive mappings with FT.Let f be a contraction on C with coefficient k 0,. Suppose {α n }, {β n },and{γ n } are three real sequences in 0, satisfying i α n + β n + γ n =, for all n 0; ii lim n α n =0and n=0 α n = ; iii lim n γ n =0. In the following, we first present two important results. The first one is to prove the mapping S := n n+ T i : C Cisnonexpansive: In fact, for any x, y C, from Lemma 2.4 we get dsx, Sy=d dx, y, n + T i x, n n + d T i x, T i y n + T i y i.e., S is nonexpansive. The second one is to prove the following mapping T t,n : C C is contractive: T t,n x = α nt f x tγ n n + T i x. Tang et al. Fixed Point Theory and Applications :00 Page 7 of 3 In fact, the mapping T t,n can be written as T t,n x = λ n f x λ n n + T i x, x C, where λ n = α nt γ n +tβ n. Therefore this kind of mappings has just a similar form to 2.9. Hence for any x, y C, from Lemma 2.4,wehave d T t,n x, T t,n y = d λ n f x λ n n n + T i x, λ n f y λ n λ n d f x, f y + λ n d λ n kdx, y+ λ n n + d T i x, T i y λ n kdx, y+ λ n dx, y = λ n k dx, y, n + T i x, n n n + T i y n + T i y i.e., T t,n is a contractive mapping. Hence, it has a unique fixed point denote by z t,n, i.e., z t,n = λ n f z t,n λ n n + T i z t,n. From Lemma 2.6,foreachgivenn, we have lim z t,n = x n F t 0 n + T i, 3. and it is the unique solution of the following variational inequality: x n f x n, xxn 0, x F Next we prove that for each n FT=F n + T i. 3.2 n + T i. 3.3 If fact, if x FT, it is obvious that x F n n+ T i, for each n. This implies that FT F n n+ T i foreachn. By using induction, now we prove that F n + T i FT foreachn. 3.4 Tang et al. Fixed Point Theory and Applications :00 Page 8 of 3 Indeed, if n =,andx F 2 T i, i.e., x = 2 x 2 Tx,thenwehave 2 x 2 x = 2 x Tx Since the geodesic segment joining any two points in CAT0 space is unique, it follows from 3.5thatx = Tx, i.e., x FT. If 3.4istrueforsomen 2, now we prove that 3.4isalsotrueforn +. Indeed, if x F n+ n+2 T i, i.e., n+ x = n +2 T i x. 3.6 It is easy to see that 3.6canberewrittenas n + n +2 x n +2 x = n + { x n +2 n + Tx n + T n } x n + n +2 T n+ x. 3.7 By the same reasoning showing that the geodesic segment joining any two points in CAT0 space is unique, from 3.7wehave n + n +2 x = n + { x n +2 n + Tx n + T n } x n + This implies that and n +2 x = n +2 T n+ x. x = x n + Tx n + T n x n + and x = T n+ x. 3.8 By the assumption of induction, from 3.8wehavex = Tx.Theconclusion3.4isproved. Therefore the conclusion 3.3isalsoproved. By using 3.3, 3.canbewrittenas lim z t,n = x n FT, 3.9 t 0 and x n is the unique solution of the following variational inequality: x n f x n, xxn 0, x FT. 3.0 Next we prove that for any positive integers m, n, x n = x m where x n is the limit in 3.9. Therefore 3.9canbewrittenas lim t 0 z t,n = x n = x FT, 3. where x is some point, and it is the unique solution of the following variational inequality: xf x, x x 0, x FT. 3.2 Tang et al. Fixed Point Theory and Applications :00 Page 9 of 3 In fact, it follows from 3.0that x n f x n, xxn 0, x m f x m, yxm 0, x FT, y FT. Taking x = x m in the first inequality and y = x n in the second inequality and then adding up the resultant two inequalities, we have 0 x n f x n, x m x n x m f x m, x m x n x n f x m, x m x n + f x m f x n, x m x n x m x n, x m x n x n f x m, x m x n = f x m f x n, x m x n xm x n, x m x n d f x m, f x n dx m, x n d 2 x m, x n kd 2 x m, x n d 2 x m, x n =k d 2 x m, x n. Since k 0,, this implies that dx m, x n =0,i.e., x m = x n, m, n. The conclusion is proved. We are now in a position to propose the iterative algorithms with Cesàro s means for a nonexpansive mapping in a complete CAT0 space X, and prove that the sequence generated by the algorithms converges strongly to a fixed point of the nonexpansive mapping which is also a unique solution of some kind of variational inequality. Theorem 3. Let C be a closed convex subset of a complete CAT0 space X, and T : C C be a nonexpansive mapping with FT. Let f be a contraction on C with coefficient k 0, 2. Suppose x 0 C and the sequence {x n } is given by x n+ = α n f x n β n x n γ n n + T i x n, 3.3 where T 0 = I, {α n }, {β n }, and {γ n } are three real sequences in 0, satisfying conditions i- iii. Then {x n } converges strongly to x suchthat x = P FT f x, which is equivalent to the following variational inequality: xf x, x x 0, x FT. 3.4 Proof We first show that the sequence {x n } is bounded. For any p FT, we have dx n+, p=d α n f x n β n x n γ n α n d f x n, p + β n dx n, p+γ n d n + T i x n, p n + T i x n, p α n d f xn, f p + d f p, p + β n dx n, p+γ n n + d T i x n, T i p Tang et al. Fixed Point Theory and Applications :00 Page 0 of 3 By induction, we have α n kdx n, p+α n d f p, p + β n dx n, p+γ n dx n, p = α n k dx n, p+α n k { max dx n, p, k d f p, p }. { dx n, p max dx 0, p, k d f p, p } k d f p, p for all n 0. Hence {x n } is bounded, and so are {f x n } and {T i x n }. Let {z t,n } be a sequence in C such that z t,n = α nt f z t,n tγ n n + T i z t,n. It follows from 3.that{z t,n } convergesstronglytoafixedpoint x FT, which is also a unique solution of the variational inequality 3.4. Now we claim that lim sup f x x, xn+ x n It follows from Lemma 2.8 and Lemma 2.4 that d 2 z t,n, x n+ = z t,n x n+, z t,n x n+ α nt f z t,n x n+, z t,n x n+ + tγ n n = α nt [ f z t,n f x, z t,n x n+ + f x x, z t,n x n+ + xzt,n, z t,n x n+ + z t,n x n+, z t,n x n+ ] + tγ n [ n n n + T i z t,n ] n + n + T i x n+ x n+, z t,n x n+ n + T i x n+, z t,n x n+ α nt [ kdzt,n, xdz t,n, x n+ + f x x, z t,n x n+ n + T i z t,n x n+, z t,n x n+ + d x, z t,n dz t,n, x n+ +d 2 z t,n, x n+ ] [ + tγ n n d n + T i z t,n, n + T i x n+ dz t,n, x n+ Tang et al. Fixed Point Theory and Applications :00 Page of 3 where + d ] n + T i x n+, x n+ dz t,n, x n+ α nt [ kdzt,n, xdz t,n, x n+ + f x x, z t,n x n+ + d x, z t,n dz t,n, x n+ +d 2 z t,n, x n+ ] [ + tγ n d 2 z t,n, x n+ + d ] T i x n+, x n+ dzt,n, x n+ n + α nt [ Mkdzt,n, x+ f x x, z t,n x n+ + Md x, zt,n ] + d 2 z t,n, x n+ + tγ nmn, M sup { dz t,n, x n+, n 0, 0 t }, N sup { d T i x n+, x n+, n 0, i 0 }. This implies that f x x, x n+ z t,n + kmdzt,n, x+ tγ nmn α n t. 3.6 Taking the upper limit as n first, and then taking the upper limit as t 0, we get Since lim sup t 0 lim sup f x x, x n+ z t,n n f x x, xn+ x = f x x, x n+ z t,n + f x x, zt,n x f x x, x n+ z t,n + d f x, x dzt,n, x. Thus, by taking the upper limit as n first, and then taking the upper limit as t 0, it follows from z t,n x and 3.7that Hence lim sup t 0 lim sup f x x, xn+ x 0. n lim sup f x x, xn+ x 0. n Finally, we prove that x n x as n.infact,foranyn 0, let { u n := α n x α n y n, y n := β n α n x n γ n α n n n+ T i x n. 3.8 Tang et al. Fixed Point Theory and Applications :00 Page 2 of 3 FromLemma 2.7 andlemma 2.8 we have d 2 x n+, x d 2 u n, x+2 x n+ u n, xn+ x α n 2 d 2 y n, x+2 [ α n f x n u n, xn+ x + α n yn u n, xn+ x ] α n 2 d 2 x n, x+2 [ αn 2 f x n x, xn+ x + α n α n f x n y n, xn+ x + α n α n yn x, xn+ x + α n 2 yn y n, xn+ x ] = α n 2 d 2 x n, x+2 [ αn 2 f x n x, xn+ x + α n α n f x n x, xn+ x ] = α n 2 d 2 x n, x+2α n f x n x, xn+ x = α n 2 d 2 x n, x+2α n f x n f x, xn+ x +2α n f x x, xn+ x α n 2 d 2 x n, x+2α n kdx n, xdx n+, x+2α n f x x, xn+ x α n 2 d 2 x n, x+α n k d 2 x n, x+d 2 x n+, x +2α n f x x, xn+ x. This implies that d 2 x n+, x 2 kα n + αn 2 d 2 x n, x+ α n k = α n2 2k α n α n k Then it follows that d 2 x n, x+ 2α n f x x, xn+ x α n k 2α n f x x, xn+ x. α n k d 2 x n+, x α n d 2 x n, x+α n β n, where α n = α n2 2k α n, β n α n k = 2 f x x, xn+ x. 2 2k α n Applying Lemma 2.9,wecanconcludethatx n x as n. This completes the proof. Letting f x n u for all n N, the following theorem can be obtained from Theorem 3. immediately. Theorem 3.2 Let C be a closed convex subset of a complete CAT0 space X, and let T : C C be a nonexpansive mapping satisfying FT. For given x 0 Carbitrarilyand fixed point u C, the sequence {x n } is given by x n+ = α n u β n x n γ n n + T i x n, 3.9 Tang et al. Fixed Point Theory and Applications :00 Page 3 of 3 where T 0 = I, {α n }, {β n }, and {γ n } are three real sequences in 0, satisfying conditions i-iii. Then {x n } converges strongly to x suchthat x = P FT u, which is equivalent to the following variational inequality: xu, x x 0, x FT Remark 3.3 Theorem 3. and Theorem 3.2 are two new results. The proofs are simple and different from many others which extend the Cesàro nonlinear ergodic theorems for nonexpansive mappings in Baillon [], Bruck [2], Song and Chen [3], Zhu and Chen [5], Yao et al. [4] from Hilbert space or Banach space to CAT0 spaces. Theorem 3. is also a generalization of the results by Wangkeeree and Preechasilp [2]. Competing interests The authors declare that they have no competing interests. Authors contributions All the authors contributed equally to the writing of the present article. They also read and approved the final paper. Author details Department of Mathematics, Yibin University,

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