±39ff±1ffi ß Ω χ Vol.39, No.1 2010fl2fl ADVANCES IN MATHEMATICS Feb., 2010 The Equivalence of the Convergence of Four Kinds of Iterations for a Finite Family of Uniformly Asymptotically ø-pseudocontractive Mappings WANG Xuewu (School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, Shandong, 264005, P. R. China) Abstract: In this paper, we prove the equivalences of the strong convergence each other the modified Mann iterative process with errors, the modified Ishikawa iterative process with errors, the implicit iterative process with errors and the composite implicit iterative process with errors for a finite family of uniformly asymptotically ø-pseudocontractive mappings in arbitrary real Banach space. The results presented in this paper generalize and improve the corresponding results of Roades, Soltuz, and Huang. Key words: uniformly asymptotically ø-pseudocontractive mapping; composite implicit iterative process; implicit iterative process; Ishikawa iterative process; Mann iterative process; equivalence MR(2000) Subject Classification: 47H05; 47H10; 47H15 / CLC number: O177.91 Document code: A Article ID: 1000-0917(2010)01-0079-09 0 Introduction Throughout this paper, we assume that X is a real Banach space, X is the dual space of X. Let J denote the normalized duality mapping form X into 2 X given by J(x) ={f X : x, f = x 2 = f 2 } for all x X, where, denotes the generalized duality pairing. In the sequel, we shall denote the single-valued duality mapping by j. Let T 1,T 2,,T N be N self-mappings of E X and suppose that F (T )= N i=1 F (T i) Ø, the set of common fixed points of T i,i =1, 2,,N. Hereafter, we will denote the index set {1, 2,,N} by I. Mann in [1], Ishikwaw in [2], Liu in [3] and Xu in [4] introduced Mann iteration, Ishikwaw iteration and modified Mann and Ishikawa iteration with errors, respectively. Xu and Ori in [5], Sun in [6], Chang in [7] and Gu in [8] introduced an implicit iterative process, an implicit iterative process with errors and a composite implicit iterative process with errors for a finite family of mappings, respectively. In this paper, we introduced a modified Mann and Ishikawa iterative processes with errors for a finite family of mappings {T 1,T 2,,T N } as follows : For Received date: 2007-12-12. E-mail: wangxuewuxx@163.com
80 ο Ψ ffi 39ff any given x 0 E the sequence {x n } defined by x n+1 =(1 α n γ n )x n + α n T k n y n + γ n u n,y n =(1 β n δ n )x n + β n T k n x n + δ n v n,n 0 (1) is called the modified Ishikawa iteration sequence with errors for a finite family of mappings, where {α n }, {γ n },{β n }, {δ n } are four sequences in [0, 1] satisfying the conditions α n + γ n 1 and β n + δ n 1 for all n 1, and T k = T k(mod N), n =(k 1)N + i, i I = {1, 2,,N}. In particular, if β n = δ n =0foralln 0, then {z n } defined by z 0 X, z n+1 =(1 α n γ n )z n + α n T k n z n + γ n w n, n 0 (2) is called the modified Mann iteration sequence with errors for a finite family of mappings. we also introduced a new composite implicit iterative process for a finite family of mappings as follows: x n =(1 α n γ n )x n 1 + α n Tn k y n + γ n u n, n 1; y n =(1 β n δ n )x n + β n T n k x n + δ n v n, n 1, (3) where n =(k 1)N + i, i I, T n = T n(mod N), {α n }, {β n }, {γ n}, {δ n } are four real sequences in [0, 1] satisfying α n + γ n 1andβ n + δ n 1 for all n 1, {u n}, {v n} are two sequences in E and x 0 is a given point. This sequence {x n } defined by (3) is called the composite implicit iterative sequence with errors for a finite family of mappings. Especially, if β n =0,δ n =0foralln 1, we have z n =(1 α n γ n )z n 1 + α n T k nz n + γ n w n, n 1, (4) where n =(k 1)N +i, i I, T n = T n(mod N), {α n }, {γ n } are two real sequences in [0, 1] satisfying α n + γ n 1andforalln 1, {w n} is a sequence in E and x 0 is a initial point. Huang and Bu in [9] have proved the equivalence of the convergence between the modified Mann iteration process with errors and the modified Ishikawa iteration process with errors for strongly successively pseudocontractive mapping in uniformly smooth Banach space. Rhoades and Soltuz in [11, 12] have proved the equivalence of the convergence between the original Ishikawa iterative sequence and original Mann iterative sequence under some strict conditions. Huang in [13] have established the equivalence theorems of the convergence between the modified Mann iteration process with errors and the modified Ishikawa iteration process with errors for strongly successively ø-pseudocontractives mapping without Lipschitzian assumptions in uniformly smooth Banach space. The main purpose of this paper is to study the equivalences of the convergence each other the modified Mann iterative process with errors, the modified Ishikawa iterative process with errors, the implicit iterative process with errors and the composite implicit iterative process with errors for a finite family of uniformly asymptotically ø-pseudocontractive mappings in an arbitrary real Banach space. Our results extend and improve the corresponding results presented recently in [9, 11 13] and some others.
1ffi»ο Ξ The Equivalence of the Convergence of Four Kinds of Iterations 81 1 Preliminaries For the sake of convenience, we first recall some definitions and conclusions. Definition 1.1 A mapping T with domain D(T ) and range R(T )inx is called asymptotically ø-pseudocontractive mapping, if for all x, y D(T ), there exist j(x y) J(x y) and a strictly increasing function ø : [0, + ) [0, + ) withø(0)=0, such that T n x T n y, j(x y) k n x y 2 ø( x y ), (5) where k n [1, + ) with lim n k n =1. Definition 1.2 A finite family of mappings {T 1,T 2,,T N } be said to be uniformly asymptotically ø-pseudocontractive mappings if for all x, y D(T )= N i=1 D(T i), there exist j(x y) J(x y) and a strictly increasing function ø : [0, + ) [0, + ) withø(0)=0, such that Tn k x T n k y, j(x y) k n x y 2 ø( x y ), (6) for all x, y X, n, k 1andj(x y) J(x y), where k n [1, + ) with lim n k n =1. Definition 1.3 A finite family of mappings {T 1,T 2,,T N } be said to be uniformly Lipschitzian, if for any x, y D(T )= N i=1 D(T i) and nonnegative integers n, thereexistsa constant L>0 such that T k nx T k n y L x y, (7) where n =(k 1)N + i, i I, T n = T n(mod N). Ramark If every T i,i I be uniformly Lipschitzian, then a finite family of mappings {T 1,T 2,,T N } be uniformly Lipschitzian. Lemma 1.1 [4] Let X be a real Banach space, and let j : X 2 X be the normalized duality mapping. Then for any x, y X we have x + y 2 x 2 +2 y, j(x + y). (8) Lemma 1.2 [14] Let {a n }, {b n }, {δ n } are three sequences of nonnegative real numbers, if there exists n 0, for all n>n 0 such that a n+1 (1 + δ n )a n + b n, (9) where n=1 δ n < and n=1 b n <. Then (i) lim n a n exists; (ii) lim n a n = 0 whenever lim inf n a n =0. Lemma 1.3 [15] Let {a n } be a sequences of positive real numbers such that n=1 a n =. Suppose that n=1 a nb n < where b n > 0 for all n 1. Then lim inf n b n =0. Lemma 1.4 Let ø : [0, ) [0, ) be a strictly increasing function with ø(0) = 0 and let {a n }, {b n }, {c n } are nonnegative real sequences such that lim n b n =0, b n =, n=1 c n <. (10) n=1
82 ο Ψ ffi 39ff Suppose that there exists an integer N 1 > 0 such that for all n N 1 a 2 n+1 a 2 n 2b n ø(a n+1 )+c n. (11) Then lim n a n =0. Proof Since the inequality (11) implies that a 2 n+1 a 2 n + c n. By (10) and Lemma 1.2, we have that lim n a n exists. By virtue of (11) and (10), we obtain that n=n 0 2b n ø(a n+1 ) a 2 n 0 + n=n 0 c n <, n 0 N 1. From n=1 b n = and Lemma 1.3 we have lim inf n ø(a n+1 ) = 0. In view of the properties of ø, it is obvious that lim inf n a n+1 = 0. By virtue of the conclusion (ii) in Lemma 1.2, we have lim n a n =0. 2 Main Results and Proofs Theorem 2.1 Let X be an arbitrary real Banach space and let {T 1,T 2,,T N } : X X be uniformly asymptotically ø-pseudocontractive mappings with bounded rang, and be uniformly Lipschitzian with L>0. The sequences {x n } and {z n } be defined by (1) and (2), respectively, where the sequences {u n },{v n },{w n } are bounded. If the sequences {α n },{γ n }, {β n }, {δ n } [0, 1] satisfy the following conditions: (a) n=1 γ n < ; (b) n=1 α n =, n=1 α2 n < ; (c) n=1 α nβ n <, n=1 α nδ n <, n=1 α n(k n 1) <. Then for any initial point z 0,x 0 X, the following two assertions are equivalent: (i) The modified Mann iteration sequence with errors (2) converges to common fixed x F (T ); (ii) The modified Ishikawa iteration sequence with errors (1) converges to common fixed x F (T ). Proof If the modified Ishikawa iteration sequence with errors (1) converges to common fixed x F (T ), setting β n = δ n =0, n 0, then we can get that the modified Mann iteration sequence with errors (2) converges to common fixed x F (T ). Next we will prove the result (i) (ii). Since T k n X, {x n }, {y n }, {z n }, {u n }, {v n }, {w n } are bounded, we set M =sup{ z n y n, z n x n, u n x n, v n x n, w n u n, n w n z n, Tn k x n x n, Tn k y n x n, Tn k z n z n }, (12)
1ffi»ο Ξ The Equivalence of the Convergence of Four Kinds of Iterations 83 obviously, M<. Form (1), (2), (12) and Lemma 1.1 we have z n+1 x n+1 2 (1 α n γ n )(z n x n )+α n (Tn k z n Tn k y n)+γ n (w n u n ) 2 (1 α n γ n ) 2 z n x n 2 +2 α n (Tn k z n Tn k y n )+γ n (w n u n ),j(z n+1 x n+1 ) (1 α n ) 2 z n x n 2 +2α n (Tn k z n+1 Tn k x n+1),j(z n+1 x n+1 ) +2α n Tn k z n Tn k y n (Tn k z n+1 Tn k x n+1 ),j(z n+1 x n+1 ) +2γ n w n u n,j(z n+1 x n+1 ) (1 α n ) 2 z n x n 2 +2α n k n z n+1 x n+1 2 2α n ø( z n+1 x n+1 ) +2α n L( z n+1 z n + x n+1 y n ) z n+1 x n+1 +2γ n w n u n z n+1 x n+1 (1 α n ) 2 z n x n 2 +2α n k n z n+1 x n+1 2 2α n ø( z n+1 x n+1 ) +2α n Mσ n +2γ n M 2, (13) where σ n = z n+1 z n + x n+1 y n. In view of (1) and (2), we have σ n = x n+1 y n + z n+1 z n α n (Tn k y n x n )+γ n (u n x n ) β n (Tn k x n x n ) δ n (v n x n ) + α n (Tn k z n z n )+γ n (w n z n ) 2M(α n + γ n + β n + δ n ). (14) From (1) and (2), we have z n+1 x n+1 = z n x n (α n + γ n )(z n x n )+α n (T k n z n T k n y n)+γ n (w n u n ), which implies that z n+1 x n+1 2 ( z n x n +(α n + γ n ) z n x n + α n Tnz k n Tn k y n + γ n w n u n ) 2 [ z n x n +((L +1)α n +2γ n )M] 2 z n x n 2 +((L +1)α n +2γ n )(2L +7)M 2. (15) Substituting (14) and (15) into (13) and simplifying, we obtain that z n+1 x n+1 2 z n x n 2 2α n ø( z n+1 x n+1 )+2α n (k n 1)M 2 +2α n Mσ n + α 2 nm 2 +2α n k n ((L +1)α n +2γ n )(2L +7)M 2 +2γ n M 2. (16) Suppose that a n = z n x n,b n = α n,c n =2α n (k n 1)M 2 +2α n Mσ n + α 2 n M 2 +2α n k n [(L + 1)α n +2γ n ](2L +7)M 2 +2γ n M 2. From (a), (b) and (c), we have that n=1 b n = and n=1 c n <. By virtue of Lemma 1.4 we obtain that lim n a n = 0. Hence lim n z n x n =0. Since the modified Mann iteration sequence with errors (2) converges to common fixed x F (T ), that is, lim n z n x = 0. From the inequality 0 x n x z n x + x n z n, we have lim n x n x = 0. This completes the proof of Theorem 2.1.
84 ο Ψ ffi 39ff Theorem 2.2 Let X be an arbitrary real Banach space and let {T 1,T 2,,T N } : X X be uniformly asymptotically ø-pseudocontractive mappings with bounded rang, and be uniformly Lipschitzian with L>0. The sequences {x n} and {z n} be defined by (3) and (4), respectively, where the sequences {u n }, {v n }, {w n } are bounded. If the sequences {α n}, {γ n }, {β n }, {δ n } [0, 1] satisfy the following conditions: (a) n=1 γ n < ; (b) n=1 α n =, n=1 α2 n < ; (c) n=1 α nβ n <, n=1 α nδ n <, n=1 α n(k n 1) <. Then for any initial point z 0,x 0 X, the following two assertions are equivalent: (iii) The implicit iteration sequence with errors (4) converges to common fixed x F (T ); (iv) The composite implicit iteration sequence with errors (3) converges to common fixed x F (T ). Proof If the composite implicit iteration sequences with errors (3) converges to common fixed x F (T ), taking β n = δ n =0, n 1, then we can get that implicit iteration sequence with errors (4) converges to common fixed x F (T ). Next we will prove the result (iii) (iv). Since Tn kx, {x n }, {y n }, {z n }, {u n }, {v n }, {w n } are bounded, we set M =sup{ Tnx k n x n, z n x n, z n y n, w n u n, w n v n, x n v n }, (17) n it is obvious that M<. Form (3), (4) and (17) we have z n x n 2 = (1 α n γ n )(z n 1 x n 1 )+α n(tn k z n T n k y n )+γ n(w n u n ) (1 α n γ n ) 2 z n 1 x n 1 2 +2 α n (Tn k z n Tn k y n)+γ n (w n u n),j(z n x n) (1 α n ) 2 z n 1 x n 1 2 +2α n (Tn k z n T n k x n ),j(z n x n ) +2α n (Tn k z n Tn k y n (Tn k z n Tn k x n),j(z n x n) +2γ n w n u n,j(z n x n) (1 α n ) 2 z n 1 x n 1 2 +2α n k n z n x n 2 2α n ø( z n x n ) +2α n Tn k x n Tn k y n z n x n +2γ n w n u n z n x n (1 α n ) 2 z n 1 x n 1 2 +2α n k n z n x n 2 2α n ø( z n x n ) +2α n LMσ n +2γ n M 2, (18) where σ n = x n y n. Byvirtueof(3),wehave It follows from (3) and (4) that σ n = x n y n = β n (T n k x n x n )+δ n (v n x n ) β n T nx k n x n + δ n v n x n (19) (β n + δ n )M. z n x n 2 [ z n 1 x n 1 +(α n + γ n ) z n 1 x n 1 + α n T k nz n T k n y n + γ n w n u n ] 2 z n 1 x n 1 2 +[(L +1)α n +2γ n ](2L +7)M 2. (20)
1ffi»ο Ξ The Equivalence of the Convergence of Four Kinds of Iterations 85 Substituting (19) and (20) into (18) and simplifying, we have z n x n 2 z n 1 x n 1 2 2α n ø( z n x n )+2α n (k n 1)M 2 +2LMα n σ n + α 2 n M 2 +2α n k n [(L +1)α n +2γ n ](2L +7)M 2 +2γ n M 2. (21) Suppose that a n = z n x n,b n = α n,c n =2α n (k n 1)M 2 +2LMα n σ n + α 2 n M 2 +2α n k n [(L + 1)α n +2γ n ](2L +7)M 2 +2γ n M 2. By (a), (b) and (c) we have n=1 b n = and n=1 c n <. By virtue of Lemma 1.4 we obtain that lim n a n =0. Hence lim n z n x n =0. Since implicit iteration sequence with errors (4) converges to common fixed x F (T ), that is, lim n z n x = 0. From the inequality 0 x n x z n x + x n z n,wehave lim n x n x = 0. This completes the proof of Theorem 2.2. Theorem 2.3 Let X be an arbitrary real Banach space and let {T 1,T 2,,T N } : X X be uniformly asymptotically ø-pseudocontractive mappings with bounded rang, and be uniformly Lipschitzian with L>0. The sequences {z n } and {z n } be defined by (2) and (4), respectively, where the sequences {w n }, {w n} are bounded. The sequences {α n },{γ n } [0, 1] satisfy the following conditions: (a) n=1 γ n < ; (b) n=1 α n =, n=1 α2 n <, n=1 α n(k n 1) <. Then for any initial point z 0,z 0 X, the following two assertions are equivalent: (v) The implicit iteration sequence with errors (4) converges to common fixed x F (T ); (vi) The modified Mann iteration sequence with errors (2) converges to common fixed x F (T ). Proof Since Tn kx, {z n}, {z n }, {w n}, {w n } are bounded, we set M =sup{ Tnz k n z n, z n+1 z n, w n w n, z n w n, z n z n }, (22) n obviously, M<. From (2), (4) and (22), we have z n+1 z n 2 (1 α n γ n )(z n z n 1 )+α n(t k n z n T k n z n )+γ n(w n w n ) 2 (1 α n γ n ) 2 z n z n 1 2 +2 α n (Tn k z n Tn k z n)+γ n (w n w n),j(z n+1 z n) (1 α n ) 2 z n z n 1 2 +2α n (Tn k z n+1 Tn k z n ),j(z n+1 z n ) +2α n Tn k z n Tn k z n (Tn k z n+1 Tn k z n),j(z n+1 z n)) +2γ n w n w n,j(z n+1 z n) (1 α n ) 2 z n z n 1 2 +2α n k n z n+1 z n 2 2α n ø( z n+1 z n ) +2α n LMσ n +2γ n M 2, (23) where σ n = z n+1 z n.byvirtueof(2),wehave By (2) and (4) we have that σ n = z n+1 z n α n T k n z n z n + γ n z n w n (α n + γ n )M. (24) z n+1 z n =(1 α n γ n )(z n z n 1)+α n (Tn k z n Tn k z n)+γ n (w n w n) =(z n z n 1 ) (α n + γ n )(z n z n 1 )+α n(tn k z n Tn k z n )+γ n(w n w n ).
86 ο Ψ ffi 39ff Therefore we have z n+1 z n 2 z n z n 1 2 +[(L +1)α n +2γ n ](2L +7)M 2. (25) Substituting (25) and (24) into (23) and simplifying, we obtain that z n+1 z n 2 z n z n 1 2 2α n ø( z n+1 z n )+2α n (k n 1)M +2α n k n [(L +1)α n +2γ n ](2L +7)M 2 +2α n LMσ n +2γ n M 2. (26) Let a n = z n+1 z n,b n = α n,c n =2α n (k n 1)M +2α n k n [(L +1)α n +2γ n ](2L +7)M 2 + 2α n LMσ n +2γ n M 2. By (a) and (b) we have n=1 b n =, n=1 c n <. By virtue of Lemma 1.4 we obtain that lim n a n = 0. That is, lim n z n+1 z n =0. If implicit iteration sequence with errors (4) converges to common fixed z F (T ), that is, lim n z n z = 0. From the inequality 0 z n+1 z z n z + z n+1 z n we have lim n z n+1 z = 0. Conversely, if the Mann iteration sequence with errors (2) converges to common fixed z F (T ), that is, lim n z n+1 z = 0. From the inequality 0 z n z z n+1 x + z n+1 z n,wehavelim n z n z = 0. This completes the proof of Theorem 2.3. Theorem 2.4 Let X be an arbitrary real Banach space and let {T 1,T 2,,T N } : X X be uniformly asymptotically ø-pseudocontractive mappings with bounded rang, and be uniformly Lipschitzian with L>0. The sequences {x n } and {x n} be defined by (1) and (3), respectively, where the sequences {u n }, {v n }, {w n }, {u n }, {v n }, {w n } are bounded. If the sequences {α n}, {γ n }, {β n }, {δ n }, {β n}, {δ n} [0, 1] satisfy the following conditions: (a) n=1 γ n < ; (b) n=1 α n =, n=1 α2 n < ; (c) n=1 α nβ n <, n=1 α nδ n <, n=1 α n(k n 1) <. (d) n=1 α nβ n <, n=1 α nδ n <. Then for any initial point z 0,x 0 X, the following two assertions are equivalent: (vii) The modified Ishikawa iteration sequence with errors (1) converges to common fixed x F (T ); (viii) The composite implicit iteration sequence with errors (3) converges to common fixed x F (T ). Proof By virtue of the conclusions of Theorem 2.1, Theorem 2.2 and Theorem 2.3, we obtain that (i) (ii), (iii) (iv) and (v) (vi). Again from (i) and (vii), (ii) and (v), (iii) and (v), (iv) and (vi) are the same, respectively. Therefore, we have that (viii) (vii). This completes the proof of Theorem 2.4. References [1] Ishikawa, S., Fixed point and iteration of a nonexpansive mapping in a Banach spaces, Proc. Amer. Math. Soc., 1976, 73: 65-71. [2] Mann, W.R., Mean value methods in iteration, Proc. Amer. Math. Soc., 1953, 4: 506-510.
1ffi»ο Ξ The Equivalence of the Convergence of Four Kinds of Iterations 87 [3] Liu L.S., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mapping in Banach spaces, J. Math. Anal. Appl., 1995, 194: 114-125. [4] Xu Y.G., Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., 1998, 224: 91-101. [5] Xu H.K., Ori, R.G., An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 2001, 22: 767-773. [6] Sun Z.H., Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings, J. Math. Anal. Appl., 2003, 286: 351-358. [7] Chang S.S., Tan K.K., et al, On the convergence of implicit iteration process with error for a finit family of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 2006, 313: 273-283. [8] Gu F., The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings, J. Math. Anal. Appl., 2007, 294: 766-776. [9] Huang Z.Y., Bu F.W., The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl., 2007, 325: 586-595. [10] Agarwal, R.P., Cho, Y.J., Stability of iterative procedures with errors approximating common fixed points for a couple of qusi-contrative mapping in q-uniformly smooth Banach spaces, J. Math. Anal. Appl., 2002, 272: 435-447. [11] Roades, B.E., Soltuz, S.M., The equivalence between the convergence of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl., 20024, 289: 266-278. [12] Roades, B.E., Soltuz, S.M., The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal., 2004, 58: 219-228. [13] Huang Z.Y., Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively ø-pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl., 2007, 329: 935-947. [14] Tan K.K., Xu H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process, J. Math. Anal. Appl., 1993, 178: 301-308. [15] Ofoedu, E.U., Strong convergence theorem for uniformly L-Lipschitzan asymptotically pseudocontractive mapping in real Banach space, J. Math. Anal. Appl., 2006, 321: 722-728. &:91>7<)* ø- 05/82".=$!4,-+"#'3?A@ (ffl Φ οfi οffνμfiοοfi ρ ffl 264005) ;6Λ Bev[Zt^nG Banach XR_GKsrgxmwSU ø- dlbqigwf CGku Mann IFNE WfCGku Ishikwaw IFNE WfCGo`IFNEOWfC GPDo`IFNEaYGHQj TMcLOJUZ Roades, Soltuz O Huang H]Ghp TM %(ψλ mwsv ø- dlbqiπpdo`ifneπo`ifneπ Ishikwaw IFN EΠ Mann IFNEΠHQ