Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces

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1 Available online at J. Nonlinear Sci. Appl , Research Article Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces Liang-Cai Zhao a, Shih-Sen Chang b,, Ching-Feng Wen c a College of Mathematics, Yibin University, Yibin, Sichuan, , P. R. China. b Center for General Education, China Medical University, Taichung, 4040, Taiwan. c Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan. Communicated by P. Kumam Abstract The purpose of this paper is to introduce the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to nonlinear variational inclusion problem, nonlinear Volterra integral equations, variational inequality problem and hierarchical minimization problems are included. The results presented in the paper extend and improve some recent results announced in the current literature. c 016 All rights reserved. Keywords: Viscosity, implicit midpoint rule, asymptotically nonexpansive mapping, projection, variational inequality. 010 MSC: 47J5, 47H Introduction Let H be a Hilbert space, T : H H be a nonexpansive mapping and f : H H be a contraction. The viscosity approximation method for nonexpansive mapping in Hilbert spaces was introduced by Moudafi [9], following the ideas of Attouch []. Refinements in Hilbert spaces and extensions to Banach spaces were obtained by Xu [15]. Corresponding author address: changss013@163.com Shih-Sen Chang Received

2 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , The explicit viscosity method for nonexpansive mappings generates a sequence {x n } through the iteration process: x n+1 = α n fx n + 1 α n T x n, n 0, 1.1 where I is the identity of H. It is well known [9, 15] that under certain conditions, the sequence {x n } converges in norm to a fixed point q of T which also solves the variational inequality I fq, x q 0, x F T, 1. where F T is the set of fixed points of T. The implicit midpoint rule is one of the powerful methods for solving ordinary differential equations; see [3, 4, 7, 10, 11, 13] and the references cited therein. For instance, consider the initial value problem for the differential equation y t = fyt with the initial condition y0 = y 0, where f is a continuous function from R d to R d. The implicit midpoint rule is that which generates a sequence {y n } via the relation 1 h y n+1 y n = f yn+1 + y n The implicit midpoint rule has been extended [1] to nonexpansive mappings, which generates a sequence {x n } by the implicit procedure: xn + x n+1 x n+1 = 1 t n x n + t n T, n Recently, Xu et al [16] in a Hilbert spaces introduced the following process: xn + x n+1 x n+1 = α n fx n + 1 α n T, n 0, 1.4 where T is a nonexpansive mapping. They proved that the sequence {x n } converges strongly to a fixed point of T, which, in addition, also solves the variational inequality 1.. Motivated and inspired by the research going on in this direction. The purpose of this paper is to introduce the viscosity implicit midpoint rule for asymptotically nonexpansive mapping in Hilbert space. More precisely, we consider the following implicit iterative algorithm: x n+1 = α n fx n + 1 α n T n xn + x n+1, n Under suitable conditions, some strong converge theorems to a fixed point of the asymptotically nonexpansive mapping are proved. Also, it is shown that the limit solves an additional variational inequality. Applications to nonlinear variational inclusion problem, nonlinear Volterra integral equations, variational inequality problem and hierarchical minimization problems are included. The results presented in the paper extend and improve some recent results announced in the current literature.. preliminaries In the sequel, we always assume that H is a real Hilbert space and C is a nonempty, closed, and convex subset of H. The nearest point projection from H onto C, P C, is defined by. P C x := arg min z C x z, x H..1 Namely, P C x is the only point in C that minimizes the objective x z over z C. Note that P C x is characterized as follows: P C x C and x P C x, z P C x 0 for all z C..

3 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , Recall that a mapping T : C C is said to be nonexpansive if T x T y x y, x, y C. Recall that a mapping T : C C is said to be asymptotically nonexpansive if there exists a sequence {k n } [1, + with lim k n = 1 such that T n x T n y k n x y, x, y C, n 1. The set of fixed points of T is denoted by FixT, that is, FixT = {x C : T x = x}. Note that if T : C C is an asymptotically nonexpansive mapping, then FixT is always closed and convex. Further if, in addition, C is bounded, then FixT is nonempty. The demiclosedness principle of asymptotically nonexpansive mappings is quite helpful in verifying the weak convergence of an algorithm to a fixed point of a asymptotically nonexpansive mapping. Lemma.1 [6]. Demiclosedness principle. Let H be a real Hilbert space, C be a nonempty closed and convex subset of H, and T : C C be a asymptotically nonexpansive mapping with F ixt. If {x n } is a sequence in C such that i {x n } weakly converges to x and ii I T x n converges strongly to 0, then x = T x. The following lemmas play an important role in our paper. Lemma. [8]. Let H be a real Hilbert space. x, y H and t [0, 1]. Then tx + 1 ty t x + 1 t y t1 t x y. It is easy to prove that the following lemma holds: Lemma.3. Let H be a Hilbert space. Then for all u, x, y H, the following inequality holds x u y u + x y, x u. Lemma.4 [14]. Let {a n } be a sequence of nonnegative real numbers satisfying a n+1 1 γ n a n + δ n, n 0,.3 where {γ n } is a sequence in 0, 1 and {δ n } is a sequence in R such that 1 n=1 γ n = ; lim sup δ n γn Then lim a n = 0. 0 or n=1 δ n < ; 3. Main results Theorem 3.1. Let C be a nonempty closed and convex subset of a Hilbert space H, and T : C C be a asymptotically nonexpansive mapping with a sequence {k n } [1, +, lim k n = 1 and F T. Let f be a contraction on C with coefficient α [0, 1. For an arbitrary initial point x 0 C, let {x n } be the sequence generated by x n+1 = α n fx n + 1 α n T n xn + x n+1, n where {α n } 0, 1 satisfies the following conditions: i lim α n = 0;

4 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , ii n=0 α n = ; iii lim k n 1 α n = 0; If the following condition is satisfied iv lim T n x n x n = 0; then the sequence {x n } converges strongly to x = P F T f x, which solves the following variational inequality: Proof. We divided the proof into six steps. Step 1. We prove that {x n } is bounded. I fq, x q 0, x F T. In fact, for any p F T, we have x n+1 p = α nfx n + 1 α n T n xn + x n+1 p 1 α n T n xn + x n+1 p + α n fx n p x n + x n+1 1 α n k n p + α n fx n fp + fp p 1 α nk n After simplifying, it follows that x n p + x n+1 p + α n α x n p + fp p. that is, 1 1 α nk n x n+1 p 1 α nk n + αα n x n p + α n fp p, 1 k n 1 + k n α n x n+1 p 1 + k n 1 α n k n + αα n x n p + α n fp p 1 + εα n α n k n + αα n x n p + α n fp p = 1 k n α εα n x n p + α n fp p 1 1 α εα n x n p + α n fp p. Also by condition iii, for any given positive number ε, 0 < ε < 1 α, there exists a sufficient large positive integer n 0, such that for any n n 0, we have kn 1 εα n, and k n 1 k n + 1 k n 1 k n 1 εα n. 3.3 Since {k n } [1, + and k n 1 εα n for all n n 0, then we have 3. 1 k n 1 + k n α n 1 εα n + k n α n = 1 + k n εα n εα n. 3.4 Substituting 3.4 into 3., after simplifying we have x n+1 p 1 1 α εα n α n x n p + fp p εα n εα n

5 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , By induction we readily obtain 1 1 α εα n α n x n p + fp p εα n εα n 1 1 α εα n x n p + 1 α εα n 1 fp p εα n εα n 1 α ε { } 1 max x n p, fp p, n n 0. 1 α ε x n p max { x 0 p, 1 1 α ε Hence {x n } is bounded, and so are {fx n }, {T n x n } and Step. } fp p, n n 0. { T n xn+x n+1 We show that lim x n+1 x n = 0. Observe that x n+1 x n x n+1 T n x n + T n x n x n = α nfx n + 1 α n T n xn + x n+1 1 α n T n xn + x n+1 1 α n k n x n + x n+1 Here M > 0 is a constant such that It turns out that }. T n x n + T n x n x n T n x n + α n fx n T n x n + T n x n x n x n + α n fx n T n x n + T n x n x n 1 α nk n x n+1 x n + α n fx n T n x n + T n x n x n 1 α nk n x n+1 x n + α n M + T n x n x n. M sup { fx n T n x n, n 1}. 1 1 α nk n x n+1 x n α n M + T n x n x n. By αnkn = 1 kn 1+knαn 1+1 εαn for all n n 0. Consequently, we arrive at x n+1 x n By virtue of the conditions i and iv, we have α n 1 M + T n x n x n εα n εα n lim x n+1 x n = Step 3. We show that lim x n T x n = 0. In fact, since x n T n 1 x n = α n 1fx n α n 1 T n 1 xn 1 + x n T n 1 x n α n 1 fx n 1 T n 1 x n + 1 α n 1 T n 1 xn 1 + x n T n 1 x n α n 1 fx n 1 T n 1 x n x n 1 + x n + 1 α n 1 k n 1 x n α n 1 fx n 1 T n 1 x n + 1 α n 1k n 1 x n 1 x n,

6 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , by condition i and 3.5, we have lim x n T n 1 x n = Therefore x n T x n x n T n x n + T n x n T x n x n T n x n + k 1 T n 1 x n x n 0. Step 4. From Step 3 and Lemma.1, it is a straightforward consequence that the following weak ω limit set of {x n }: is contained in F ixt. Step 5. ω w x n = {x H : there exists a subsequence of {x n } weakly converging to x} Now we prove lim sup fq q, x n q 0, 3.7 where q F ixt is the unique fixed point of the contraction P F ixt f, that is, q = P F ixt fq. As a matter of fact, since {x n } is bounded, there exists a subsequence {x nj } of {x n } such that {x nj } converges weakly to a point p and moreover lim sup fq q, x n q = lim fq q, x nj q. 3.8 j Since p F ixt, by using., 3.7, 3.8 and by virtue of Step 4, we can conclude that lim sup fq q, x n q = fq q, p q Step 6. Finally, we prove that x n q F T as n. Indeed, for any n 1, we set z n = α n q + 1 α n T n xn+x n+1.3 that. It follows from Lemma. and Lemma x n+1 q z n q + x n+1 z n, x n+1 q 1 α n T n x n + x n+1 q + x n+1 z n, x n+1 q 1 α n kn x n + x n+1 q + α n fx n q, x n+1 q = 1 α n kn x n + x n+1 q + α n fx n fq, x n+1 q + α n fq q, x n+1 q 1 α n kn x n + x n+1 q + α n fq q, x n+1 q 1 α n kn x n + x n+1 q + α n fx n fq x n+1 q + α n α x n q x n+1 q + α n fq q, x n+1 q 1 1 α n kn x n q + 1 x n+1 q 1 4 x n+1 x n + α n α x n q + x n+1 q + α n fq q, x n+1 q

7 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , αn kn Here M 1 > 0 is a constant such that xn + α n α q + x n+1 q + α n fq q, x n+1 q 1 α nkn + α n α xn q + x n+1 q + α nm 1 + α n fq q, x n+1 q. M 1 sup { k n x n q, n 1 }. It follows from 3.3 that for all n n α nkn + α n α x n+1 q 1 α nkn + α n α x n q + α nm 1 + α n fq q, x n+1 q = 1 + k n 1 kn αα n x n q + α nm 1 + α n fq q, x n+1 q 1 + εα n 1 αα n x n q + α nm 1 + α n fq q, x n+1 q = 1 1 ε αα n x n q + α nm 1 + α n fq q, x n+1 q. Since consequently, we arrive at 1 1 α nk n + αα n = 1 k n 1 + k n αα n 1 εα n + 1 αα n = ε αα n, n n 0, x n+1 q 1 1 ε αα n x n q αn + M ε αα n ε αα n 4α n + fq q, x n+1 q ε αα n = 1 41 ε αα n x n q αn + M ε αα n ε αα n 4α n + fq q, x n+1 q ε αα n α n Now, take γ n = 41 ε ααn 1+1 ε αα n, δ n = 1+1 ε αα n M ε αα n fq q, x n+1 q. It follows from conditions i, ii and 3.7 that {γ n } 0, 1, n=1 γ n = and α n M 1 + fq q, x n+1 q 0. lim sup δ n γ n = lim sup 1 1 ε α From Lemma.4 we have that x n q as n. This completes the proof. 4α n

8 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , Remark 3.. Since every nonexpansive mapping is an asymptotically nonexpansive mapping, Theorem 3.1 is an improvement and generalization of the main results in Alghamdi et al. [1] and Xu et al. [16]. 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 F ixt. 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, 3.10 where {α n } 0, 1 satisfies the conditions: i, ii and iii in Theorem 3.1. Then the sequence {x n } defined by 3.10 converges strongly to q such that q = P F ixt fq which is also a solution of the following variational inequality: q fq, x q 0, x F ixt. Proof. It suffices to prove that the following condition is satisfied: lim x n T x n = In fact, by the same method as given in [16] we can prove that x n x n+1 0 as n. Therefore we have x n T x n = α xn 1 + x n n 1fx n α n 1 T T x n α n 1 fx n 1 T x n + 1 α n 1 x n 1 x n α n 1 M + 1 α n 1 x n 1 x n 0 as n, where M = sup n fx n 1 T x n. This completes the proof of Theorem Applications 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 Jλ M : H H associated with M, is defined by J M λ J M λ x := I + λm 1 x, x H 4.1 for some λ > 0, where I stands identity operator on H. We note that for all λ > 0 the resolvent operator is a single-valued nonexpansive mapping. The so-called monotone variational inclusion problem in short, MVIP is to find x H such that 0 Mx. 4. From the definition of resolvent mapping Jλ M, it is easy to know that MVIP 4. is equivalent to find x H such that x F ixjλ M for some λ > 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

9 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , Theorem 4.1. Let M, Jλ M be the same as above. Let f : H H be a contraction. 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 and F ixjλ M, then {x n} converges strongly to the solution of monotone variational inclusion 4., which is also a solution of the following variational inequality: x f x, x x 0, x F ixj M λ. 4.. Application to nonlinear Volterra integral equations Let us consider the following nonlinear Volterra integral equation xt = gt + t 0 F t, s, xs ds, t [0, 1], 4.5 where g is a continuous function on [0, 1] and F : [0, 1] [0, 1] R R is continuous and satisfies the following condition F t, s, x F t, s, y x y, t, s [0, 1] x, y R. Define a mapping T : L [0, 1] L [0, 1] by T xt = gt + t 0 F t, s, xsds, t [0, 1]. 4.6 It is easy to see that T is a nonexpansive mapping. This means that to find the solution of integral equation 4.5 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 From Theorem 3.3 we have the following, 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 4.7. If the sequence {α n } 0, 1 satisfies the conditions: i, ii and iii in Theorem 3.1 and F ixt. Then {x n } converges strongly in L [0, 1] to the solution of integral equation 4.5 which is also a solution of the following variational inequality: 4.3. Application to variational inequalities Consider the variational inequality VI x f x, x x 0, x F ixt. Ax, x x 0, x C, 4.8 where A is a single-valued monotone operator in Hilbert space H and C is a closed convex subset of H with C doma. An example of 4.8 is the constrained minimization problem min ϕx, 4.9 x C where ϕ : H R is a proper convex and lower-semicontinuous function. If ϕ is Fréchet differentiable, then the minimization problem 4.9 is equivalently reformulated as 4.8 with A = ϕ. Notice that the VI 4.8 is equivalent to the fixed point problem, for any λ > 0, T x = x, T x := P C I λax If A is Lipschitzian and strongly monotone, then, for λ > 0 small enough, T is a contraction and its unique fixed point is also the unique solution of the VI 4.8. However, if A is not strongly monotone, T is no longer a contraction, in general. In this case we must deal with nonexpansive mappings for solving the VI 4.8. More precisely, we assume

10 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , A1 A is L Lipschitzian for some L > 0, that is, Ax Ay L x y, x, y H. A A is µ inverse strongly monotone µ ism for some µ > 0, namely, Ax Ay, x y µ Ax Ay, x, y H. Note that if ϕ is L Lipschtzian, then ϕ is 1 L ism. Under the conditions A1 and A, it is well known [5] that the operator T = P C I λa is nonexpansive provided 0 < λ < µ. Applying Theorem 3.3 we can get the following result: Theorem 4.3. Assume the VI 4.8 is solvable. Assume also A satisfies A1 and A, and 0 < λ < µ. Let f : C C be a contraction. Define a sequence {x n } by the viscosity implicit midpoint rule: xn + x n+1 x n+1 = α n fx n + 1 α n P C I λa, n 0. In addition, assume {α n } satisfies the conditions i iii in Theorem 3.1. Then {x n } converges in norm to a solution x of the VI 4.8 which is also a solution to the VI I fx, x x 0, x A Application to hierarchical minimization We next consider a hierarchical minimization problem see [1] and references cited therein. Let ϕ 0, ϕ 1 : H R be a lower semicontinuous convex function. Consider the following hierarchical minimization problem: min ϕ 1 x, S 0 := arg min ϕ 0x. 4.1 x S 0 x H Here we always assume that S 0 is nonempty. Let S := arg min x S0 ϕ 1 x and assume S. Assume ϕ 0 and ϕ 1 are differentiable and their gradients satisfy the Lipschitz continuity conditions: ϕ 0 x ϕ 0 y L 0 x y, ϕ 1 x ϕ 1 y L 1 x y Note that the condition 4.13 implies that ϕ i is 1 L i ismi = 0, 1. Now let T 0 = I γ 0 ϕ 0, T 1 = I γ 1 ϕ 1, where γ 0 > 0 and γ 1 > 0. Note that T i is nonexpansive [5] if 0 < γ i < L i i = 0, 1. Also, it is easily seen that S 0 = F ixt 0. The optimality condition for x S 0 to be a solution of the hierarchical minimization 4.1 is the VI: x S 0, ϕ 1 x, x x 0, x S This is the VI 4.8 with C = S 0 and A = ϕ 1. From Theorem 3.3 we have the following result. Theorem 4.4. Assume the hierarchical minimization problem 4.1 is solvable. contraction. Define a sequence {x n } by the viscosity implicit midpoint rule: xn + x n+1 x n+1 = α n fx n + 1 α n P S0 I λ ϕ 1. Let f : C C be a In addition, assume {α n } satisfies the conditions i iii in Theorem 3.1. If the condition 4.13 is satisfied and 0 < γ i < L i i = 0, 1, then {x n } converges in norm to a solution x of the VI 4.14 that is, a solution of hierarchical minimization problem 4.1 which also solves the VI I fx, x x 0, x S.

11 L.-C. Zhao, S.-S. Chang, C.-F. Wen, J. Nonlinear Sci. Appl , Remark 4.5. As we have observed that Theorem 3.1 can be viewed as an extension of the main result in [1, 16]. It remains an open question whether Theorem 3.1 holds without the condition iv, that is, we have the following: Open Question Let C be a nonempty closed and convex subset of a Hilbert space H, and T : C C be a asymptotically nonexpansive mapping with a sequence {k n } [1, +, lim k n = 1 and F T. Let f be a contraction on C with coefficient α [0, 1. For an arbitrary initial point x 0 C, let {x n } be the sequence generated by 3.1. If the sequence {α n } 0, 1 satisfies the conditions i-iii in Theorem 3.1, does the conclusion of Theorem 3.1 hold? Acknowledgment This work was supported by Scientific Research Fund of SiChuan Provincial Education Department No.14ZA07, 16ZA0333, This work was also supported by the National Natural Science Foundation of China Grant No , and the Natural Science Foundation of China Medical University, Taiwan. References [1] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., , 9 pages. 1, 3., 4.5 [] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., , [3] 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., , [4] G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., , [5] I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, , , 4.4 [6] S. S. Chang, Y. J. Cho, H. Zhou, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc., , [7] P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., , [8] K. Geobel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, [9] A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., , , 1 [10] C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., , [11] S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 00, [1] P. Sunthrayuth, Y. J. Cho, P. Kumam, Viscosity approximation methods for zeros of accretive operators and fixed point problems in Banach spaces, Bull. Malays. Math. Sci. Soc., , [13] M. Van Veldhuxzen, Asymptotic expansions of the global error for the implicit midpoint rule stiff case, Computing, , [14] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 00, [15] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., , , 1 [16] 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., , 1 pages. 1, 3., 3, 4.5