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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 719631, 13 pages doi:10.1155/2010/719631 Research Article Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi-I-Nonexpansive Mapping in Banach Space Farrukh Mukhamedov and Mansoor Saburov Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box 141, 25710 Kuantan, Malaysia Correspondence should be addressed to Farrukh Mukhamedov, far75m@yandex.ru Received 31 August 2009; Accepted 6 December 2009 Academic Editor: Mohamed A. Khamsi Copyright q 2010 F. Mukhamedov and M. Saburov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original w ork is properly cited. We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically quasi- nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space. 1. Introduction Let K be a nonempty subset of a real normed linear space X and let T : K → K be a mapping. Denote by FT the set of fixed points of T,thatis,FT{x ∈ K : Tx  x}. Throughout this paper, we always assume that FT /  ∅. Now let us recall some known definitions. Definition 1.1. A mapping T : K → K is said to be i nonexpansive, if Tx − Ty≤x − y for all x, y ∈ K; ii asymptotically nonexpansive, if there exists a sequence {λ n }⊂1, ∞ with lim n →∞ λ n  1 such that T n x − T n y≤λ n x − y for all x, y ∈ K and n ∈ N; iii quasi-nonexpansive, if Tx − p≤x − p for all x ∈ K, p ∈ FT; iv asymptotically quasi-nonexpansive, if there exists a sequence {μ n }⊂1, ∞ with lim n →∞ μ n  1 such that T n x − p≤μ n x − p for all x ∈ K, p ∈ FT and n ∈ N. 2 Fixed Point Theory and Applications Note that from the above definitions, it follows that a nonexpansive mapping must be asymptotically nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, but the converse does not hold see 1. If K is a closed nonempty subset of a Banach space and T : K → K is nonexpansive, then it is known that T may not have a fixed point unlike the case if T is a strict contraction, and even when it has, the sequence {x n } defined by x n1  Tx n the so-called Picard sequence may fail to converge to such a fixed point. In 2, 3 Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the studies of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians see for more details 1, 4. In 5 Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces. Ghosh and Debnath 6 established a necessary and sufficient condition for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banach space. The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied extensively by Goebel and Kirk 7,Liu8, Wittmann 9,Reich10, Gornicki 11,Schu 12 Shioji and Takahashi 13, and Tan and Xu 14 in the settings of Hilbert spaces and uniformly convex Banach spaces. There are many methods for approximating fixed points of a nonexpansive mapping. Xu and Ori 15 introduced implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. Recently, Sun 16 has extended an implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces. In 17 it has been studied the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces, which extends and improves the mentioned papers see also 18, 19 for applications and other methods of implicit iteration processes. There are many concepts which generalize a notion of nonexpansive mapping. One of such concepts is I-nonexpansivity of a mapping T 20. Let us recall some notions. Definition 1.2. Let T : K → K, I : K → K be two mappings of a nonempty subset K of a real normed linear space X. Then T is said to be i I-nonexpansive, if Tx − Ty≤Ix − Iy for all x, y ∈ K; ii asymptotically I-nonexpansive, if there exists a sequence {λ n }⊂1, ∞ with lim n →∞ λ n  1 such that T n x − T n y≤λ n I n x − I n y for all x, y ∈ K and n ≥ 1; iii asymptotically quasi I-nonexpansive mapping, if there exists a sequence {μ n }⊂ 1, ∞ with lim n →∞ μ n  1 such that T n x − p≤μ n I n x − p for all x ∈ K, p ∈ FT ∩ FI and n ≥ 1. Remark 1.3. If FT ∩ FI /  ∅ then an asymptotically I-nonexpansive mapping is asymptot- ically quasi-I-nonexpansive. But, there exists a nonlinear continuous asymptotically quasi I-nonexpansive mappings which is asymptotically I-nonexpansive. In 21 a weakly convergence theorem for I-asymptotically quasi-nonexpansive mapping defined in Hilbert space was proved. In 22 strong convergence of Mann iterations of I-nonexpansive mapping has been proved. Best approximation properties of Fixed Point Theory and Applications 3 I-nonexpansive mappings were investigated in 20.In23 the weak convergence of three- step Noor iterative scheme for an I-nonexpansive mapping in a Banach space has been established. Recently, in 24 the weak and strong convergence of implicit iteration process to a common fixed point of a finite family of I-asymptotically nonexpansive mappings were studied. Assume that the family consists of one I-asymptotically nonexpansive mapping T. Now let us consider an iteration method used in 24,forT, which is defined by x 1 ∈ K, x n1   1 − α n  x n  α n I n y n , y n   1 − β n  x n  β n T n x n . n ≥ 1, 1.1 where {α n } and {β n } are two sequences in 0, 1. From this formula one can easily see that the employed method, indeed, is not implicit iterative processes. The used process is some kind of modified Ishikawa iteration. Therefore, in this paper we will extend of the implicit iterative process, defined in 16, to I-asymptotically quasi-nonexpansive mapping defined on a uniformly convex Banach space. Namely, let K be a nonempty convex subset of a real Banach space X and T : K → K be an asymptotically quasi I-nonexpansive mapping, and let I : K → K be an asymptotically quasi-nonexpansive mapping. Then for given two sequences {α n } and {β n } in 0, 1 we will consider the following iteration scheme: x 0 ∈ K, x n   1 − α n  x n−1  α n T n y n , y n   1 − β n  x n  β n I n x n . n ≥ 1, 1.2 In this paper we will prove the weak and strong convergences of the implicit iterative process 1.2 to a common fixed point of T and I. All results presented here generalize and extend the corresponding main results of 15–17 in a case of one mapping. 2. Preliminaries Throughout this paper, we always assume that X is a real Banach space. We denote by FT and DT the set of fixed points and the domain of a mapping T, respectively. Recall that a Banach space X is said to satisfy Opial condition 25, if for each sequence {x n } in X, x n converging weakly to x implies that lim inf n →∞  x n − x  < lim inf n →∞   x n − y   . 2.1 for all y ∈ X with y /  x. It is well known that see 26 inequality 2.1 is equivalent to lim sup n →∞  x n − x  < lim sup n →∞   x n − y   . 2.2 4 Fixed Point Theory and Applications Definition 2.1. Let K be a closed subset of a real Banach space X and let T : K → K be a mapping. i A mapping T is said to be semiclosed demiclosed at zero, if for each bounded sequence {x n } in K, the conditions x n converges weakly to x ∈ K and Tx n converges strongly to 0 imply Tx  0. ii A mapping T is said to be semicompact, if for any bounded sequence {x n } in K such that x n − Tx n →0,n→∞, then there exists a subsequence {x n k }⊂{x n } such that x n k → x ∗ ∈ K strongly. iii T is called a uniformly L-Lipschitzian mapping, if there exists a constant L>0 such that T n x − T n y≤Lx − y for all x, y ∈ K and n ≥ 1. The following lemmas play an important role in proving our main results. Lemma 2.2 see 12. Let X be a uniformly convex Banach space and let b, c be two constants with 0 <b<c<1. Suppose that {t n } is a sequence in b, c and {x n } and {y n } are two sequences in X such that lim n →∞   t n x n   1 − t n  y n    d, lim sup n →∞  x n  ≤ d, lim sup n →∞   y n   ≤ d, 2.3 holds some d ≤ 0. Then lim n →∞ x n − y n   0. Lemma 2.3 see 14. Let {a n } and {b n } be two sequences of nonnegative real numbers with  ∞ n1 b n < ∞. If one of the following conditions is satisfied: i a n1 ≤ a n  b n ,n≥ 1, ii a n1 ≤ 1  b n a n ,n≥ 1, then the limit lim n →∞ a n exists. 3. Main Results In this section we will prove our main results. To formulate one, we need some auxiliary results. Lemma 3.1. Let X be a real Banach space and let K be a nonempty closed convex subset of X. Let T : K → K be an asymptotically quasi I-nonexpansive mapping with a sequence {λ n }⊂1, ∞ and I : K → K be an asymptotically quasi-nonexpansive mapping with a sequence {μ n }⊂1, ∞ such that F  FT ∩ FI /  ∅. Suppose A ∗  sup n α n , Λsup n λ n ≥ 1,M sup n μ n ≥ 1 and {α n } and {β n } are two sequences in 0, 1 which satisfy the following conditions: i  ∞ n1 λ n μ n − 1α n < ∞, ii A ∗ < 1/Λ 2 M 2 . If {x n } is the implicit iterative sequence defined by 1.2, then for each p ∈ F  FT ∩ FI the limit lim n →∞ x n − p exists. Fixed Point Theory and Applications 5 Proof. Since F  FT ∩ FI /  ∅, for any given p ∈ F, it follows from 1.2 that   x n − p       1 − α n   x n−1 − p   α n  T n y n − p    ≤  1 − α n    x n−1 − p    α n   T n y n − p   ≤  1 − α n    x n−1 − p    α n λ n   I n y n − p   ≤  1 − α n    x n−1 − p    α n λ n μ n   y n − p   . 3.1 Again from 1.2 we derive that   y n − p       1 − β n  x n − p   β n  I n x n − p    ≤  1 − β n    x n − p    β n μ n   x n − p   ≤  1 − β n  μ n   x n − p    β n μ n   I n x n − p   ≤ μ n   x n − p   , 3.2 which means   y n − p   ≤ μ n   x n − p   ≤ λ n μ n   x n − p   . 3.3 Then from 3.3 one finds   x n − p   ≤  1 − α n    x n−1 − p    α n λ 2 n μ 2 n   x n − p   , 3.4 and so  1 − α n λ 2 n μ 2 n    x n − p   ≤  1 − α n    x n−1 − p   . 3.5 By condition ii we have α n λ 2 n μ 2 n ≤ A ∗ Λ 2 M 2 < 1, and therefore 1 − α n λ 2 n μ 2 n ≥ 1 − A ∗ Λ 2 M 2 > 0. 3.6 Hence from 3.5 we obtain   x n − p   ≤ 1 − α n 1 − α n λ 2 n μ 2 n   x n−1 − p     1   λ 2 n μ 2 n − 1  α n 1 − α n λ 2 n μ 2 n    x n−1 − p   ≤  1   λ 2 n μ 2 n − 1  α n 1 − A ∗ Λ 2 M 2    x n−1 − p   . 3.7 6 Fixed Point Theory and Applications By putting b n λ 2 n μ 2 n − 1α n /1 − A ∗ Λ 2 M 2  the last inequality can be rewritten as f ollows:   x n − p   ≤  1  b n    x n−1 − p   . 3.8 From condition i we find ∞  n1 b n  1 1 − A ∗ Λ 2 M 2 ∞  n1  λ 2 n μ 2 n − 1  α n  1 1 − A ∗ Λ 2 M 2 ∞  n1  λ n μ n − 1  λ n μ n  1  α n ≤ ΛM  1 1 − A ∗ Λ 2 M 2 ∞  n1  λ n μ n − 1  α n < ∞. 3.9 Denoting a n  x n−1 − p in 3.8 one gets a n1 ≤  1  b n  a n , 3.10 and Lemma 2.3 implies the existence of the limit lim n →∞ a n . This means the limit lim n →∞   x n − p    d 3.11 exists, where d ≥ 0 is a constant. This completes the proof. Now we prove the following result. Theorem 3.2. Let X be a real Banach space and let K be a nonempty closed convex subset of X. Let T : K → K be a uniformly L 1 -Lipschitzian asymptotically quasi-I-nonexpansive mapping with a sequence {λ n }⊂1, ∞ and let I : K → K be a uniformly L 2 -Lipschitzian asymptotically quasi- nonexpansive mapping with a sequence {μ n }⊂1, ∞ such that F  FT ∩ FI /  ∅. Suppose A ∗  sup n α n , Λsup n λ n ≥ 1,M sup n μ n ≥ 1, and {α n } and {β n } are two sequences in 0, 1 which satisfy the following conditions: i  ∞ n1 λ n μ n − 1α n < ∞, ii A ∗ < 1/Λ 2 M 2 . Then the implicitly iterative sequence {x n } defined by 1.2 converges strongly to a common fixed point in F  FT ∩ FI /  ∅ if and only if lim inf n →∞ d  x n ,F   0. 3.12 Proof. The necessity of condition 3.12 is obvious. Let us proof the sufficiency part of theorem. Since T, I : K → K are uniformly L-Lipschitzian mappings, so T and I are continuous mappings. Therefore the sets FT and FI are closed. Hence F  FT ∩ FI is a nonempty closed set. Fixed Point Theory and Applications 7 For any given p ∈ F, we have see 3.8   x n − p   ≤  1  b n    x n−1 − p   , 3.13 here as before b n λ 2 n μ 2 n − 1α n /1 − A ∗ Λ 2 M 2  with  ∞ n1 b n < ∞. Hence, one finds d  x n ,F  ≤  1  b n  d  x n−1 ,F  . 3.14 From 3.14 due to Lemma 2.3 we obtain the existence of the limit lim n →∞ dx n ,F.By condition 3.12,onegets lim n →∞ d  x n ,F   lim inf n →∞ d  x n ,F   0. 3.15 Let us prove that the sequence {x n } converges to a common fixed point of T and I. In fact, due to 1  t ≤ expt for all t>0, and from 3.13,weobtain   x n − p   ≤ exp  b n    x n−1 − p   . 3.16 Hence, for any positive integers m, n, from 3.16 with  ∞ n1 b n < ∞ we find   x nm − p   ≤ exp  b nm    x nm−1 − p   ≤ exp  b nm  b nm−1    x nm−2 − p   ≤··· ≤ exp  nm  in1 b i    x n − p   ≤ exp  ∞  i1 b i    x n − p   , 3.17 which means t hat   x nm − p   ≤ W   x n − p   3.18 for all p ∈ F, where W  exp  ∞ i1 b i  < ∞. Since lim n →∞ dx n ,F0, then for any given ε>0, there exists a positive integer number n 0 such that d  x n 0 ,F  < ε W . 3.19 Therefore there exists p 1 ∈ F such that   x n 0 − p 1   < ε W . 3.20 8 Fixed Point Theory and Applications Consequently, for all n ≥ n 0 from 3.18 we derive   x n − p 1   ≤ W   x n 0 − p 1   <W· ε W  ε, 3.21 which means that the strong convergence of the sequence {x n } is a common fixed point p 1 of T and I. This proves the required assertion. We need one more auxiliary result. Proposition 3.3. Let X be a real uniformly convex Banach space and let K be a nonempty closed convex subset of X. Let T : K → K be a uniformly L 1 -Lipschitzian asymptotically quasi-I- nonexpansive mapping with a sequence {λ n }⊂1, ∞ and let I : K → K be a uniformly L 2 - Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence {μ n }⊂1, ∞ such that F  FT ∩ FI /  ∅. Suppose A ∗  inf n α n ,A ∗  sup n α n , Λsup n λ n ≥ 1,M sup n μ n ≥ 1 and {α n } and {β n } are two sequences in 0, 1 which satisfy the following conditions: i  ∞ n1 λ n μ n − 1α n < ∞, ii 0 <A ∗ ≤ A ∗ < 1/Λ 2 M 2 , iii 0 <B ∗  inf n β n ≤ sup n β n  B ∗ < 1. Then the implicitly iterative sequence {x n } defined by 1.2 satisfies the following: lim n →∞  x n − Tx n   0, lim n →∞  x n − Ix n   0. 3.22 Proof. First, we will prove that lim n →∞  x n − T n x n   0, lim n →∞  x n − I n x n   0. 3.23 According to Lemma 3.1 for any p ∈ F  FT ∩ FI we have lim n →∞ x n − p  d.It follows from 1.2 that   x n − p       1 − α n   x n−1 − p   α n  T n y n − p    −→ d, n −→ ∞ . 3.24 By means of asymptotically quasi-I-nonexpansivity of T and asymptotically quasi- nonexpansivity of I from 3.3 we get lim sup n →∞   T n y n − p   ≤ lim sup n →∞ λ n μ n   y n − p   ≤ lim sup n →∞ λ 2 n μ 2 n   x n − p    d. 3.25 Now using lim sup n →∞   x n−1 − p    d 3.26 Fixed Point Theory and Applications 9 with 3.25 and applying Lemma 2.2 to 3.24 one finds lim n →∞   x n−1 − T n y n    0. 3.27 Now from 1.2 and 3.27 we infer that lim n →∞  x n − x n−1   lim n →∞   α n  T n y n − x n−1     0. 3.28 On the other hand, we have   x n−1 − p   ≤   x n−1 − T n y n      T n y n − p   ≤   x n−1 − T n y n    λ n μ n   y n − p   , 3.29 which implies   x n−1 − p   −   x n−1 − T n y n   ≤ λ n μ n   y n − p   . 3.30 The last inequality with 3.3 yields that   x n−1 − p   −   x n−1 − T n y n   ≤ λ n μ n   y n − p   ≤ λ 2 n μ 2   x n − p   . 3.31 Then 3.27 and 3.24 with the Squeeze theorem imply that lim n →∞   y n − p    d. 3.32 Again from 1.2 we can see that   y n − p       1 − β n  x n − p   β n  I n x n − p    −→ d, n −→ ∞ . 3.33 From 3.11 one finds lim sup n →∞   I n x n − p   ≤ lim sup n →∞ μ n   x n − p    d. 3.34 Now applying Lemma 2.2 to 3.33 we obtain lim n →∞  x n − I n x n   0. 3.35 10 Fixed Point Theory and Applications Consider  x n − T n x n  ≤  x n − x n−1     x n−1 − T n y n      T n y n − T n x n   ≤  x n − x n−1     x n−1 − T n y n    L 1   y n − x n     x n − x n−1     x n−1 − T n y n    L 1   β n  I n x n − x n      x n − x n−1     x n−1 − T n y n    L 1 β n  I n x n − x n  . 3.36 Then from 3.27, 3.28,and3.35 we get lim n →∞  x n − T n x n   0. 3.37 Finally, from  x n − Tx n  ≤  x n − T n x n    T n x n − Tx n  ≤  x n − T n x n   L 1    T n−1 x n − x n    ≤  x n − T n x n   L 1     T n−1 x n − T n−1 x n−1        T n−1 x n−1 − x n−1      x n−1 − x n   ≤  x n − T n x n   L 1  L 1  x n − x n−1      T n−1 x n−1 − x n−1      x n−1 − x n   ≤  x n − T n x n   L 1  L 1  1   x n − x n−1   L 1    T n−1 x n−1 − x n−1    3.38 with 3.28 and 3.37 we obtain lim n →∞  x n − Tx n   0. 3.39 Analogously, one has  x n − Ix n  ≤  x n − I n x n   L 2  L 2  1   x n − x n−1   L 2    I n−1 x n−1 − x n−1    , 3.40 which with 3.28  and 3.35 implies lim n →∞  x n − Ix n   0. 3.41 Now we are ready to formulate one of main results concerning weak convergence of the sequence {x n }. 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Journal of Mathematical Analysis and Applications, vol 329, no 2, pp 759–765, 2007 22 B H Rhodes and S Temir, “Convergebce thorems for I-nonexpansive mapping, ” International Journal of Mathematics and Mathematical Sciences, vol 2006, Article ID 63435, 4 pages, 2006 23 P Kumam, W Kumethong, and N Jewwaiworn, Weak convergence theorems of three-step Noor iterative scheme for I-quasi-nonexpansive mappings in . Tan and Xu 14 in the settings of Hilbert spaces and uniformly convex Banach spaces. There are many methods for approximating fixed points of a nonexpansive mapping. Xu and Ori 15 introduced implicit. family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces. In 17 it has been studied the weak and strong convergence. original w ork is properly cited. We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically