Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 017, 386 394 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT0 Spaces Liang-cai Zhao a, Shih-sen Chang b,, Lin Wang c, Gang Wang c a College of Mathematics, Yibin University, Yibin, Sichuan, 644007, China. b Center for General Education, China Medical University, Taichung, 4040, Taiwan. c College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 6501, China. Communicated by X. Qin Abstract The purpose of this paper is to introduce the implicit midpoint rule of nonexpansive mappings in CAT0 spaces. The strong convergence of this method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit of the sequence generated by the implicit midpoint rule solves an additional variational inequality. Applications to nonlinear Volterra integral equations and nonlinear variational inclusion problem are included. The results presented in the paper extend and improve some recent results announced in the current literature. c 017 all rights reserved. Keywords: Viscosity, implicit midpoint rule, nonexpansive mapping, projection, variational inequality, CAT0 space. 010 MSC: 47H09, 47J5. 1. Introduction The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differential equations and differential algebraic equations. For related works, please refer to [ 5, 15, 17, 18, 1]. For the ordinary differential equation x t = ft, x0 = x 0, 1.1 the implicit midpoint rule generates a sequence {x n } by the recursion procedure x n+1 = x n + hf x n + x n+1, n 0, 1. where h > 0 is a stepsize. It is known that if f : R N R N is Lipschitz continuous and sufficiently smooth, then the sequence {x n } generated by 1. converges to the exact solution of 1.1 as h 0 uniformly over t [0, T] for any fixed T > 0. Corresponding author Email address: changss013@163.com Shih-sen Chang doi:10.436/jnsa.010.0.05 Received 016-08-5
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 387 Based on the above fact, Alghamdi et al. [] presented the following semi-implicit midpoint rule for nonexpansive mappings in the setting of Hilbert space H: x n+1 = 1 α n x n + α n T x n + x n+1, n 0, 1.3 where α n 0, 1 and T : H H is a nonexpansive mapping. They proved the weak convergence of 1.3 under some additional conditions on {α n }. Recently, Xu et al. [] and Yao et al. [3] in a Hilbert spaces presented the following viscosity implicit midpoint rule for nonexpansive mappings: x n+1 = α n fx n + 1 α n T x n + x n+1, n 0, where α n 0, 1 and f is a contraction. Under suitable conditions and by using a very complicated method, the authors proved that the sequence {x n } converges strongly to a fixed point of T, which is also the unique solution of the following variational inequality I fq, x q 0, x FixT. Very recently, Zhao et al. [5] presented the following viscosity implicit midpoint rule for an asymptotically nonexpansive mapping T in a Hilbert space: x n+1 = α n fx n + 1 α n T n x n + x n+1, n 0, and under suitable conditions some strong convergence theorems to a fixed point of T are proved. On the other hand, 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 geodetically 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, R-trees [8, 16], Euclidean buildings [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, we have [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 [8]. Motivated and inspired by the research going on in this direction, it is natural to put forward the following. Open Question: Can we establish the viscosity implicit midpoint rule for nonexpansive mapping in CAT0 and generalize the main results in [, 3] to CAT0 spaces? The purpose of this paper is to give an affirmative answer to the above open question. In our paper we introduce and consider the following semi-implicit algorithm which is called the viscosity implicit midpoint rule in CAT0: x n+1 = α n fx n 1 α n T, n 0. 1.4 Under suitable conditions, some strong converge theorems to a fixed point of the nonexpansive mapping in CAT0 space are proved. Moreover, it is shown that the limit of the sequence {x n } generated by 1.4 solves an additional variational inequality. As applications, we shall utilize the results presented in the paper to study the existence problems of solutions of nonlinear variational inclusion problem, and nonlinear Volterra integral equations. The results presented in the paper also extend and improve the main results in Xu [], Yao et al. [3], and others.
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 388. Preliminaries In this paper, we write 1 tx ty for the unique point z in the geodesic segment joining from x to y such that dz, x = tdx, y, and dz, y = 1 tdx, y. The following lemmas play an important role in our paper. Lemma.1 [13]. Let X be a CAT0 space, x, y, z X and t [0, 1]. Then i dtx 1 ty, z tdx, z + 1 tdy, z; ii d tx 1 ty, z td x, z + 1 td y, z t1 td x, y. Lemma. [6]. Let X be a CAT0 space, p, q, r, s X and λ [0, 1]. Then dλp 1 λq, λr 1 λs λdp, r + 1 λdq, s. Berg and Nikolaev [7] 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 = 1 d a, d + d b, c d a, c d b, d, a, b, c, d X. It is easy to see that ab, cd = cd, ab, ab, cd = ba, cd and ax, cd + xb, cd = ab, cd for all a, b, c, d X. We say that X satisfies the Cauchy-Schwartz inequality if ab, cd da, bdc, d for all a, b, c, d X. It is well-known [7] that a geodesically connected metric space is a CAT0 space if and only if it satisfies the Cauchy-Schwartz inequality. Let C be a nonempty closed convex subset of a complete CAT0 space X. The metric projection P C : X C is defined by u = P C x du, x = inf{dy, x : y C}, x X. Lemma.3 [11]. 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 u is a solution of the following variational inequality i.e., u satisfies the following inequality: yu, ux 0, y C, d x, y d y, u d u, x 0, y C. Lemma.4 [14]. Every bounded sequence in a complete CAT0 space always has a -convergent subsequence. Lemma.5 [1]. Let X be a complete CAT0 space, {x n } be a sequence in X and x X. Then {x n } -converges to x if and only if lim sup xx n, xy 0 for all y X. Lemma.6 [1]. Let X be a complete CAT0 space. Then for all u, x, y X, the following inequality holds d x, u d y, u + xy, xu. Lemma.7 [19]. Let X be a complete CAT0 space. For any t [0, 1] and u, v X, let u t = tu 1 tv. Then, for all x, y X,
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 389 i ut x, u t y t ux, u t y + 1 t vx, u t y ; ii ut x, uy t ux, uy + 1 t vx, uy and ut x, vy t ux, vy + 1 t vx, vy. Lemma.8 [0]. Let {a n } be a sequence of nonnegative real numbers satisfying a n+1 1 γ n a n + δ n, n 0, where {γ n } is a sequence in 0, 1 and {δ n } is a sequence in R such that 1 n=1 γ n = ; δ lim sup n γ n 0 or n=1 δ n <. Then lim a n = 0. 3. Main results Theorem 3.1. Let C be a closed convex subset of a complete CAT0 space X, and T : C C be a nonexpansive mapping with FixT. Let f be a contraction on C with coefficient k [0, 1, and for the arbitrary initial point x 0 C, let {x n } be generated by x n+1 = α n fx n 1 α n T, n 0, where {α n } 0, 1 satisfies the following conditions: i lim α n = 0; ii n=0 α n = ; iii n=0 α α n α n+1 <, or lim n+1 α n = 1. Then the sequence {x n } converges strongly to x = P FixT f x, which is a fixed point of T and it is also a solution of the following variational inequality: xf x, x x 0, x FixT, i.e., x satisfies the following inequality equation: Proof. We divide the proof into five steps. d f x, x d x, x d f x, x 0, x FixT. Step 1. We prove that {x n } is bounded. To see this, we take p FixT to deduce that dx n+1, p = d α n fx n 1 α n T, p α n dfx n, p + 1 α n d T, p α n dfx n, fp + dfp, p + 1 α n d T, p α n kdx n, p + α n dfp, p + 1 α n d, p It then follows that α n kdx n, p + α n dfp, p + 1 α n dx n, p + dx n+1, p. dx n+1, p 1 + k 1α n dx n, p + α n dfp, p,
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 390 and, moreover By induction we readily obtain dx n+1, p 1 + k 1α n dx n, p + α n dfp, p 1 + α n = 1 1 kα n dx n, p + 1 kα n 1 dfp, p 1 k { } 1 max dx n, p, dfp, p. 1 k dx n, p max{dx 0, p, 1 dfp, p}, 1 k for all n 0. Hence {x n } is bounded, and so are {fx n } and {T x n x n+1 }. Step. We show that lim dx n+1, x n = 0. Observe that x dx n+1, x n = d α n fx n 1 α n T, α n 1 fx n 1 1 α n 1 T d α n fx n 1 α n T x n x n+1, α n fx n 1 α n T x + d α n fx n 1 α n T x, α n fx n 1 1 α n T x + d α n fx n 1 1 α n T x, α n 1 fx n 1 1 α n 1 T x 1 α n d T x n x n+1, T x + α n dfx n, fx n 1 + α n α n 1 d fx n 1, T x 1 α n d, x + α n kdx n, x n 1 + α n α n 1 M 1 α n [ dxn+1, x n + dx n, x n 1 ] + α n kdx n, x n 1 + α n α n 1 M. Here M > 0 is a constant such that { M sup d fx n 1, T x }, n 0. It turns out that Consequently, we arrive at dx n+1, x n 1 + k 1α n dx n, x n 1 + M α n α n 1. dx n+1, x n 1 + kα n α n dx n, x n 1 + M α n α n 1 = + kα n α n dx n, x n 1 + M α n α n 1 1 + α n = 1 1 kα n dx n, x n 1 + M α n α n 1.
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 391 Since {α n } 0, 1, then <, 1 1+α n > 1, 1 1 kα n 1+α n < 1 1 kα n. We have dx n+1, x n 1 1 kα n dx n, x n 1 + M α n α n 1. 3.1 By virtue of the conditions ii and iii, we can apply Lemma.8 to 3.1 to obtain lim dx n+1, x n = 0. Step 3. We show that lim dx n, Tx n = 0. In fact, we have dx n, Tx n dx n, x n+1 + d x n+1, T x n x n+1 + d T x n x n+1, Tx n dx n, x n+1 + α n d fx n, T x n x n+1 + d, x n dx n, x n+1 + α n d fx n, T x n x n+1 + 1 dx n, x n+1 Step 4. Now we prove 3 dx n, x n+1 + α n M 0 as n. lim sup f x x, x n x 0. Since {x n } is bounded, there exists a subsequence {x nj } of {x n } such that -converges to x and Since {x nj } -converges to x, by Lemma.5, we have This together with 3. shows that lim sup f x x, x n x = lim sup f x x, x nj x. 3. lim sup f x x, x nj x 0. lim sup f x x, x n x = lim sup f x x, x nj x 0. 3.3 Step 5. Finally, we prove that x n x FixT as n. For any n N, we set z n = α n x 1 α n T x n x n+1. It follows from Lemma.6 and Lemma.7 that d x n+1, x d z n, x + x n+1 z n, x 1 α n d T x n x n+1, x + 1 α n T x n x n+1 z n, ] x n+1 x [ 1 α n d, x + + α n 1 α n fx n T x n x n+1, x n+1 x + 1 α n T x n x n+1 T x n x n+1 1 α n d, x + α n 1 α n [ + α n fx n z n, x α n α n fx n x, x + α n 1 α n ] x n+1 x, [ + α n fx n x, x fx n T x n x n+1, x n+1 x T x n x n+1 x, x n+1 x + α n 1 α n T x n x n+1 x, ] x n+1 x
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 39 1 α n d, x 1 α n d, x + [α n fx n x, x + α n 1 α n fx n x, x + α n fx n x, x [ 1 1 α n d x n, x + 1 d x n+1, x 1 ] 4 d x n, x n+1 + α n fx n f x, x n+1 x + α n f x x, x 1 α n [d x n, x + d x n+1, x] + α n kdx n, xdx n+1, x + α n f x x, x 1 α n [d x n, x + d x n+1, x] + α n k[d x n, x + d x n+1, x] + α n f x x, x 1 αn + α n k [d x n, x + d x n+1, x] + α n f x x, x 1 1 kα n [d x n, x + d x n+1, x] + α nm f x x, x. Here M 1 > 0 is a constant such that It follows that d x n+1, x 1 1 kα n d x n, x + 1 + 1 kα n 1 1 kα n 1 + 1 kα n M 1 sup{d x n, x, n 0}. d x n, x + 1 Since 1 + 1 kα n < k, 1+1 kα n > 1 k, we have d x n+1, x 1 1 kα n k 1 1 kα n k α n 1 + 1 kα n M 1 + α n 1 + 1 kα n M 1 + α n 1 + 1 kα n M 1 + d x n, x + d x n, x + α nm 1 + 4α n f x x, x 1 + 1 kα n 4α n f x x, x. 1 + 1 kα n 4α n f x x, x 1 + 1 kα n 4α n f x x, x. 1 + 1 kα n Take γ n = 1 kα n k, δ n = α 4α nm 1 + n 1+1 kα n f x x, x. It follows from conditions i, ii, and 3.3 that {γ n } 0, 1, n=1 γ n = and δ n k lim sup = lim sup α n M 1 + f x x, x 0. γ n 1 k 1 + 1 kα n From Lemma.8 we have that x n x as n. This completes the proof. Remark 3.. Since every Hilbert space is a complete CAT0 space, Theorem 3.1 is an improvement and generalization of the main results in Xu et al. [] and Yao et al. [3]. The following result can be obtained from Theorem 3.1 immediately. Theorem 3.3. Let C be a nonempty closed and convex subset of a real Hilbert space H, and let T : C C be a nonexpansive mapping with FixT. Let f be a contraction on C with coefficient k [0, 1, and for the arbitrary initial point x 0 C, let {x n } be the sequence generated by xn + x n+1 x n+1 = α n fx n + 1 α n T, n 0, 3.4
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 393 where {α n } 0, 1 satisfies the conditions i, ii, and iii in Theorem 3.1. Then the sequence {x n } defined by 3.4 converges strongly to x such that x = P FixT f x which is equivalent to the following variational inequality: 4. Applications x f x, x x 0, x FixT. 4.1. Application to nonlinear variational inclusion problem Let H be a real Hilbert space, M : H H be a multi-valued maximal monotone mapping. Then, the resolvent mapping J M λ : H H associated with M, is defined by J M λ x := I + λm 1 x, x H, for some λ > 0, where I stands for the identity operator on H. We note that for all λ > 0 the resolvent operator J M λ is a single-valued nonexpansive mapping. The so-called monotone variational inclusion problem in short, MVIP [10] is to find x H such that 0 Mx. 4.1 From the definition of resolvent mapping J M λ, it is easy to know that MVIP 4.1 is equivalent to find x H such that x FixJ M λ for some λ > 0. For any given function x 0 H, define a sequence by x n+1 = α n fx n + 1 α n J M λ From Theorem 3.3 we have the following. xn + x n+1, n 0. 4. Theorem 4.1. Let M and J M λ be the same as above. Let f : H H be a contraction. Let {x n} be the sequence defined by 4.. If the sequence {α n } 0, 1 satisfies the conditions i, ii, and iii in Theorem 3.1 and FixJ M λ, then {x n } converges strongly to the solution of monotone variational inclusion 4.1, which is also a solution of the following variational inequality: x f x, x x 0, x FixJ M λ. 4.. Application to nonlinear Volterra integral equations Let us consider the following nonlinear Volterra integral equation xt = gt + t 0 Ft, s, xsds, t [0, 1], 4.3 where g is a continuous function on [0, 1] and F : [0, 1] [0, 1] R R is continuous and satisfies the following condition. Ft, s, x Ft, s, y x y, t, s [0, 1], x, y R. Then equation 4.3 has at least one solution in L [0, 1] see, for example, [4]. Define a mapping T : L [0, 1] L [0, 1] by Txt = gt + t 0 Ft, s, xsds, t [0, 1]. It is easy to see that T is a nonexpansive mapping. This means that finding the solution of integral equation 4.3 is reduced to find a fixed point of the nonexpansive mapping T in L [0, 1]. For any given function x 0 L [0, 1], define a sequence of functions {x n } in L [0, 1] by xn + x n+1 x n+1 = α n fx n + 1 α n T, n 0. 4.4 From Theorem 3.3 we have the following.
L.-C. Zhao, S.-S. Chang, L. Wang, G. Wang, J. Nonlinear Sci. Appl., 10 017, 386 394 394 Theorem 4.. Let F, g, T, L [0, 1] be the same as above. Let f be a contraction on L [0, 1] with coefficient k [0, 1. Let {x n } be the sequence defined by 4.4. If the sequence {α n } 0, 1 satisfies the conditions i, ii, and iii in Theorem 3.1, then {x n } converges strongly in L [0, 1] to the solution of integral equation 4.3 which is also a solution of the following variational inequality: x f x, x x 0, x FixT. Acknowledgment The authors would like to express their thanks to the referee and the editors for their helpful comments and advices. This work was supported by Scientific Research Fund of SiChuan Provincial Education Department No.14ZA07, This work was also supported by the National Natural Science Foundation of China Grant No.11361070, and the Natural Science Foundation of China Medical University, Taiwan. References [1] B. Ahmadi Kakavandi, Weak topologies in complete CAT0 metric spaces, Proc. Amer. Math. Soc., 141 013, 109 1039..5 [] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 014 014, 9 pages. 1, 1 [3] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 1996, 769 806. [4] W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 1989, 469 499. [5] G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 1983, 373 398. 1 [6] S. Banach, Metric spaces of nonpositive curvature, Springer, New York, 1999.. [7] I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 008, 195 18. [8] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999. 1 [9] K. S. Brown, Buildings, Springer-Verlag, New York, 1989. 1 [10] S. Y. Cho, W.-L. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 013 013, 14 pages. 4.1 [11] H. Dehghan, J. Rooin, A characterization of metric projection in CAT0 spaces, In: International Conference on Functional Equation, Geometric Functions and Applications ICFGA 01, 10-1th May 01, pp. 41-43. Payame Noor University, Tabriz, 01..3 [1] S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 007, 35 45..6 [13] S. Dhompongsa, B. Panyanak, On -convergence theorems in CAT0 spaces, Comput. Math. Appl., 56 008, 57 579..1 [14] W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 008, 3689 3696..4 [15] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 41 000, 46 55. 1 [16] S. Saejung, Halpern s iteration in CAT0 spaces, Fixed Point Theory Appl., 010 010, 13 pages. 1 [17] C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 1993, 1 10. 1 [18] S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 00, 37 33. 1 [19] R. Wangkeeree, P. Preechasilp, Viscosity approximation methods for nonexpansive mappings in CAT0 spaces, J. Inequal. Appl., 013 013, 15 pages..7 [0] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 00, 40 56..8 [1] H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 98 004, 79 91. 1 [] H.-K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 015 015, 1 pages. 1, 1, 3. [3] Y.-H. Yao, N. Shahzad, N.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 015 015, 15 pages. 1, 1, 3. [4] S.-S. Zhang, Integral equations, Chongqing press, Chongqing, 1984. 4. [5] L.-C. Zhao, S.-S. Chang, C.-F. Wen, Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces, J. Nonlinear Sci. Appl., 9 016, 4478 4488. 1