Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

and Applied Analysis 3 where {an}, {bn}, {αn} are appropriate sequences in 0, 1 . The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied e.g., see Khan et al. 6 , Plubtieng and Wangkeeree 7 , Fukhar-ud-din and Khan 5 , Petrot 13 , and Suantai 14 . It has been shown in 15 that three-step method performs better than two-step and one-step methods for solving variational inequalities. In 2001, Khan and Takahashi 16 have approximated common fixed points of two asymptotically nonexpansive mappings by the modified Ishikawa iteration. Jeong and Kim 17 have approximated common fixed points of two asymptotically nonexpansive mappings. Plubtieng et al. 18 , in 2006, modified Noor iterations with errors and have approximated common fixed points of three asymptotically nonexpansive mappings. Shahzad and Udomene 10 established convergence theorems for the modified Ishikawa iteration process of to asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. Plubtieng and Wangkeeree 7 , in 2006, established strong convergence theorems of the modified multistep Noor iterations with errors for an asymptotically quasinonexpansivemapping and asymptotically nonexpansivemapping in the intermediate sense. Very recently, Khan et al. 6 , in 2008, established convergence theorems for the modified multistep Noor iterations process of finite family of asymptotically quasinonexpansive mappings to a common fixed point of the mappings. For rerated results with errors terms, we refer to 5–7, 17–21 . Inspired and motivated by these facts, we introduce a new iteration process for a finite family of {Ti : i 1, 2, . . . , k} of generalized asymptotically quasi-nonexpansive mappings as follows. Let Ti : C → C i 1, 2, . . . , k be mappings and F : ⋂k i 1F Ti . For a given x1 ∈ C, and a fixed k ∈ N N denote the set of all positive integers , compute the iterative sequences {xn} and {yin} by xn 1 ykn αknT k y k−1 n βknxn γknukn, y k−1 n α k−1 nT k−1y k−2 n β k−1 nxn γ k−1 nu k−1 n, .. y3n α3nT 3 y2n β3nxn γ3nu3n, y2n α2nT 2 y1n β2nxn γ2nu2n, y1n α1nT 1 y0n β1nxn γ1nu1n, 1.7 where y0n xn and {u1n}, {u2n}, . . . , {ukn} are bounded sequences in C with {αin}, {βin}, and {γin} are appropriate real sequences in 0, 1 such that αin βin γin 1 for all i 1, 2, . . . , k and all n. Our iteration includes and extends the Mann iteration 1.5 , three-step iteration by Xu and Noor 1.6 , the multistep Noor iterations with errors by Plubtieng and Wangkeeree 7 , and the iteration defined by Khan et al. 6 simultaneously. The purpose of this paper is to establish several strong convergence theorems of the iterative scheme 1.7 for a finite family of generalized asymptotically quasi-nonexpansive mappings when one mapping Ti satisfies a condition which is weaker than demicompactness and we also weak convergence theorem for a finite family of generalized asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space satisfying Opial’s property. Our results generalize and improve the corresponding ones announced by Khan et al. 6 , Fukhar-ud-din and Khan 5 , and many others. 4 Abstract and Applied Analysis 2. Preliminaries In the sequel, the following lemmas are needed to prove our main results. A mapping T with domain D T and range R T in X is said to be demiclosed at 0 if whenever {xn} is a sequence inD T such that {xn} converges weakly to x ∈ D T and {Txn} converging strongly to 0, we have Tx 0. A Banach spaceX is said to satisfyOpial’s property if for each x inX and each sequence {xn}weakly convergent to x, the following condition holds for x / y: lim inf n→∞ ‖xn − x‖ < lim inf n→∞ ∥ xn − y ∥ ∥. 2.1 It is well known that all Hilbert spaces and lp 1 < p < ∞ spaces have Opial’s property while Lp spaces p / 2 have not. A family {Ti : i 1, 2, . . . , k} of self-mappings of C with F : ⋂k i 1F Ti / Ø is said to satisfy the following conditions. 1 Condition A 22 . If there is a nondecreasing function f : 0,∞ → 0,∞ with f 0 0 and f t > 0 for all t ∈ 0,∞ such that 1/k∑i 1 ‖x − Tix‖ ≥ f d x, F for all x ∈ C, where d x, F inf {‖x − p‖ : p ∈ F}. 2 Condition B 22 . If there is a nondecreasing function f : 0,∞ → 0,∞ with f 0 0 and f t > 0 for all t ∈ 0,∞ such that max1≤i≤k {‖x − Tix‖} ≥ f d x, F for all x ∈ C. 3 Condition C 22 . If there is a nondecreasing function f : 0,∞ → 0,∞ with f 0 0 and f t > 0 for all t ∈ 0,∞ such that ‖x − Tlx‖ ≥ f d x, F for all x ∈ C and for at least one Tl, l 1, 2, . . . , k. Note that B and C are equivalent, condition B reduces to condition I when all but one of Ti’s are identities, and in addition, it also condition A . It is well known that every continuous and demicompact mapping must satisfy condition I see 4 . Since every completely continuous T : C → C is continuous and demicompact so that it satisfies condition I . Thus we will use condition C instead of the demicompactness and complete continuity of a family {Ti : i 1, 2, . . . , k}. Lemma 2.1 see 8, Lemma 1 . Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality an 1 ≤ 1 δn an bn, ∀n 1, 2, . . . . 2.2 If ∑∞ n 1 δn < ∞ and ∑∞ n 1 bn < ∞, then i limn→∞ an exists; ii limn→∞ an 0 whenever lim infn→∞ an 0. Lemma 2.2 see 7, Lemma 3.1 . Let X be a uniformly convex Banach space, {xn}, {yn} ⊂ X, real numbers a ≥ 0, α, β ∈ 0, 1 , and let {αn} be a real sequence number which satisfies i 0 < α ≤ αn ≤ β < 1, for all n ≥ n0 and for some n0 ∈ N; ii lim supn→∞ ‖xn‖ ≤ a and lim supn→∞ ‖yn‖ ≤ a; iii limn→∞ ‖αnxn 1 − αn yn‖ a. Then limn→∞ ‖xn − yn‖ 0. Abstract and Applied Analysis 5and Applied Analysis 5 Lemma 2.3 see 14, Lemma 2.7 . Let X be a Banach space which satisfies Opial’s property and let {xn} be a sequence in X. Let u, v ∈ X be such that limn→∞ ‖xn − u‖ and limn→∞ ‖xn − v‖ exist. If {xnk} and {xmk} are subsequences of {xn} which converge weakly to u and v, respectively, then u v. 3. Convergence Theorems in Banach Spaces Our first result is the strong convergence theorems of the iterative scheme 1.7 for a finite family of generalized asymptotically quasi-nonexpansive mappings in a Banach space. In order to prove our main results, the following lemma is needed. Lemma 3.1. Let X be a Banach space and C a nonempty closed and convex subset of X, and {Ti : i 1, 2, . . . , k} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequences {bin}, {cin} ⊂ 0,∞ such that ∑∞ n 1 bin < ∞ and ∑∞ n 1 cin < ∞ for all i 1, 2, . . . , k. Assume that F / Ø and ∑∞ n 1 γin < ∞ for each i 1, 2, . . . , k. For a given x1 ∈ C, let the sequences {xn} and {yin} be defined by 1.7 . Then a there exist sequences {vn} and {ein} in 0,∞ such that ∑∞ n 1 vn < ∞, ∑∞ n 1 ein < ∞, and ‖yin − p‖ ≤ 1 vn ‖xn − p‖ ein, for all i 1, 2, . . . , k and all p ∈ F; b limn→∞ ‖xn − p‖ exists for all p ∈ F; c there exist constant M > 0 and {si} in 0,∞ such that ∑∞ i 1 si < ∞ and ‖xn m − p‖ ≤ M‖xn − p‖ ∑∞ i n si for all p ∈ F and n,m ∈ N. Proof. a Let p ∈ F, vn max1≤i≤k {bin} and dn max1≤i≤k {cin} for all n. Since ∑∞ n 1 bin < ∞ and ∑∞ n 1 cin < ∞, for all i 1, 2, . . . , k, therefore ∑∞ n 1 vn < ∞ and ∑∞ n 1 dn < ∞. For each n ≥ 1, we note that ∥ y1n − p ∥ ∥ ∥ α1nT n 1 y0n β1nxn γ1nu1n − p ∥ ∥ ≤ α1n ∥ ∥T 1 xn − p ∥ ∥ β1n ∥ xn − p ∥ ∥ γ1n ∥ u1n − p ∥ ∥ ≤ α1n 1 b1n ∥ xn − p ∥ ∥ α1nc1n β1n ∥ xn − p ∥ ∥ γ1n ∥ u1n − p ∥ ∥ ≤ α1n 1 vn ∥ xn − p ∥ ∥ α1ndn β1n 1 vn ∥ xn − p ∥ ∥ γ1n ∥ u1n − p ∥ ∥ ≤ 1 vn ∥ xn − p ∥ ∥ e1n, 3.1 where e1n α1ndn γ1n‖u1n − p‖. Since {u1n} is bounded, ∑∞ n 1 γ1n < ∞ and ∑∞ n 1 dn < ∞, we obtain that ∑∞ n 1 e1n < ∞. It follows from 3.1 that ‖y2n − p‖ ≤ α2n ∥ ∥T 2 y1n − p ∥ ∥ β2n ∥ xn − p ∥ ∥ γ2n ∥ u2n − p ∥ ∥ ≤ α2n 1 vn ∥ y1n − p ∥ ∥ α2ndn β2n ∥ xn − p ∥ ∥ γ2n ∥ u2n − p ∥ ∥ ≤ α2n 1 vn ( 1 vn ∥ xn − p ∥ ∥ e1n ) α2ndn β2n 1 vn 2 ∥ xn − p ∥ ∥ γ2n ∥ u2n − p ∥ ∥ ( α2n β2n ) 1 vn 2 ∥ xn − p ∥ ∥ α2n 1 vn e1n α2ndn γ2n ∥ u2n − p ∥ ∥ ≤ 1 vn 2 ∥ xn − p ∥ ∥ e2n, 3.2 6 Abstract and Applied Analysis where e2n α2n 1 vn e1n α2ndn γ2n‖u2n −p‖. Since {u2n}, {vn} are bounded, ∑∞ n 1 e1n < ∞, ∑∞ n 1 dn < ∞, and ∑∞ n 1 γ2n < ∞, it follows that ∑∞ n 1 e2n < ∞. Moreover, we see that ∥ y3n − p ∥ ∥ ≤ α3n ∥ ∥T 3 y2n − p ∥ ∥ β3n ∥ xn − p ∥ ∥ γ3n ∥ u3n − p ∥ ∥ ≤ α3n 1 vn ∥ y2n − p ∥ ∥ α3ndn β3n ∥ xn − p ∥ ∥ γ3n ∥ u3n − p ∥ ∥ ≤ α3n 1 vn ( 1 vn 2 ∥ xn − p ∥ ∥ e2n ) α3ndn β3n 1 vn 3 ∥ xn − p ∥ ∥ γ3n ∥ u3n − p ∥ ∥ ( α3n β3n ) 1 vn 3 ∥ xn − p ∥ ∥ α3n 1 vn e2n α3ndn γ3n ∥ u3n − p ∥ ∥ ≤ 1 vn 3 ∥ xn − p ∥ ∥ e3n, 3.3 where e3n α3n 1 vn e2n α3ndn γ3n‖u3n − p‖. Since {u3n}, {vn} are bounded, ∑∞ n 1 e2n < ∞, ∑n 1 dn < ∞, and ∑∞ n 1 γ3n < ∞, it follows that ∑∞ n 1 e3n < ∞. By continuing the above method, there are nonnegative real sequences {ein} in 0,∞ such that ∑∞ n 1 ein < ∞ and ∥ yin − p ∥ ∥ ≤ 1 vn i ∥ xn − p ∥ ∥ ein ∀i 1, 2, . . . , k. 3.4 This completes the proof of a . b From part a , for the case i k, we have ∥ xn 1 − p ∥ ∥ ≤ 1 vn k ∥ xn − p ∥ ∥ ekn, ∀n, p ∈ F. 3.5 It follows from Lemma 2.1 i that limn→∞ ‖xn − p‖ exists, for all p ∈ F. c If t ≥ 0, then 1 t ≤ e and so, 1 t k ≤ e, for k 1, 2, . . . . Thus, from 3.5 , it follows that ∥ xn m − p ∥ ∥ ≤ 1 vn m−1 k ∥ xn m−1 − p ∥ ∥ ek n m−1 ≤ exp {kvn m−1} ∥ xn m−1 − p ∥ ∥ ek n m−1