Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces

and Applied Analysis 3 Lemma 1.1 see 21 . let X be a uniformly convex Banach space. Let b and c be two constants with 0 < b < c < 1. Suppose that {tn} is a sequence in b, c . Let {xn} and {yn} be two sequences in X such that lim sup n→∞ ‖xn‖ ≤ d, lim sup n→∞ ∥ yn ∥ ∥ ≤ d, lim n→∞ ∥ tnxn 1 − tn yn ∥ ∥ d 1.8 hold for some d ≥ 0, then limn→∞‖xn − yn‖ 0. Lemma 1.2 see 26 . Let {an}, {bn}, and {cn} be three nonnegative sequences satisfying the following condition: an 1 ≤ 1 bn an cn, ∀n ≥ n0, 1.9 where n0 is some nonnegative integer, ∑∞ n 1 bn < ∞ and ∑∞ n 1 cn < ∞. Then the limit limn→∞an exists. 2. Main Results Lemma 2.1. Let X be a real Banach space and K a nonempty closed and convex subset of X. Let T : K → K be a asymptotically I-nonexpansive in the intermediate sense and I : K → K a asymptotically nonexpansive in the intermediate sense. Assume that F : F T ∩ F I / ∅. Let σn max{0, supx,y∈k ‖Tnx−Tny‖−‖Inx−Iny‖ } and σn max{0, supx,y∈k ‖Tnx−Tny‖−‖x−y‖ }. Let {αn}, {βn}, {γn}, {α̂n}, {β̂n}, {γ̂n} be six real number sequences in 0, 1 . Let {xn} be a sequence generated in the following iterative process: x1 ∈ C, yn α̂nxn β̂nIxn γ̂nvn, xn 1 αnxn βnTyn γnun, n ≥ 1, 2.1 where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied: a αn βn γn α̂n β̂n γ̂n 1; b ∑∞ n 1 σn < ∞, ∑∞ n 1 σn < ∞; c ∑∞ n 1 γn < ∞, ∑∞ n 1 γ̂n < ∞. Then limn→∞‖xn − p‖ exists for all p ∈ F. 4 Abstract and Applied Analysis Proof. Letting p ∈ F, we see that ∥ yn − p ∥ ∥ ∥ ∥ ∥ ( 1 − β̂n − γ̂n ) xn β̂nIxn γ̂nvn − p ∥ ∥ ∥ ≤ ( 1 − β̂n − γ̂n )∥ xn − p ∥ ∥ β̂n ∥ ∥Ixn − p ∥ ∥ γ̂n ∥ vn − p ∥ ∥ ≤ ( 1 − β̂n − γ̂n )∥ xn − p ∥ ∥ β̂n (∥ xn − p ∥ ∥ σn ) γ̂n ∥ vn − p ∥ ∥ ( 1 − γ̂n )∥ xn − p ∥ ∥ γ̂n ∥ vn − p ∥ ∥ β̂nσn ≤ ∥∥xn − p ∥ ∥ γ̂n ∥ vn − p ∥ ∥ β̂nσn, 2.2 ∥ xn 1 − p ∥ ∥ ∥ ∥ ( 1 − βn − γn ) xn βnTyn γnun − p ∥ ∥ ≤ (1 − βn − γn )∥ xn − p ∥ ∥ βn ∥ ∥Tyn − p ∥ ∥ γn ∥ un − p ∥ ∥ ( 1 − βn − γn )∥ xn − p ∥ ∥ βn ∥ ∥Tyn − Tp ∥ ∥ γn ∥ un − p ∥ ∥ ≤ (1 − βn − γn )∥ xn − p ∥ ∥ βn (∥ ∥Iyn − Ip ∥ ∥ σn ) γn ∥ un − p ∥ ∥ ( 1 − βn − γn )∥ xn − p ∥ ∥ βn ∥ ∥Iyn − Ip ∥ ∥ βnσn γn ∥ un − p ∥ ∥ ≤ (1 − βn − γn )∥ xn − p ∥ ∥ βn (∥ yn − p ∥ ∥ σn ) βnσn γn ∥ un − p ∥ ∥ ( 1 − βn − γn )∥ xn − p ∥ ∥ βn ∥ yn − p ∥ ∥ γn ∥ un − p ∥ ∥ βn σn σn ≤ (1 − βn )∥ xn − p ∥ ∥ βn ∥ yn − p ∥ ∥ γn ∥ un − p ∥ ∥ βn σn σn . 2.3 Substituting 2.2 into 2.3 ,we obtain that ∥ xn 1 − p ∥ ∥ ≤ (1 − βn )∥ xn − p ∥ ∥ βn (∥ xn − p ∥ ∥ γ̂n ∥ vn − p ∥ ∥ β̂nσn ) γn ∥ un − p ∥ ∥ βn σn σn ∥ xn − p ∥ ∥ [ βnγ̂n ∥ vn − p ∥ ∥ γn ∥ un − p ∥ ∥ βnσn βnσn ( 1 β̂n )] . 2.4 Let an ‖xn − p‖, bn 0, and cn βnγ̂n ∥ vn − p ∥ ∥ γn ∥ un − p ∥ ∥ βnσn βnσn ( 1 β̂n ) . 2.5 It follows from 2.4 that an 1 ≤ an cn. 2.6 In view of the restrictions b and c , we see that ∑∞ n 1 cn < ∞. We can easily conclude the desired conclusion with the aid of Lemma 1.2. This completes the proof of Lemma 2.1. Abstract and Applied Analysis 5 Theorem 2.2. Let X be a real Banach space and K a nonempty closed and convex subset of X. Let T : K → K be a asymptotically I-nonexpansive in the intermediate sense and I : K → K a asymptotically nonexpansive in the intermediate sense. Assume that F : F T ∩ F I / ∅. Let σn max{0, supx,y∈k ‖Tnx−Tny‖−‖Inx−Iny‖ } and σn max{0, supx,y∈k ‖Tnx−Tny‖−‖x−y‖ }. Let {αn}, {βn}, {γn}, {α̂n}, {β̂n}, {γ̂n} be six real number sequences in 0, 1 . Let {xn} be a sequence generated in the following iterative process:and Applied Analysis 5 Theorem 2.2. Let X be a real Banach space and K a nonempty closed and convex subset of X. Let T : K → K be a asymptotically I-nonexpansive in the intermediate sense and I : K → K a asymptotically nonexpansive in the intermediate sense. Assume that F : F T ∩ F I / ∅. Let σn max{0, supx,y∈k ‖Tnx−Tny‖−‖Inx−Iny‖ } and σn max{0, supx,y∈k ‖Tnx−Tny‖−‖x−y‖ }. Let {αn}, {βn}, {γn}, {α̂n}, {β̂n}, {γ̂n} be six real number sequences in 0, 1 . Let {xn} be a sequence generated in the following iterative process: x1 ∈ C, yn α̂nxn β̂nIxn γ̂nvn, xn 1 αnxn βnTyn γnun, n ≥ 1, 2.7 where {un} and {vn} be two bounded sequences in K. Assume that the following restrictions are satisfied: a αn βn γn α̂n β̂n γ̂n 1; b ∑∞ n 1 σn < ∞, ∑∞ n 1 σn < ∞; c ∑∞ n 1 γn < ∞, ∑∞ n 1 γ̂n < ∞. If both T and I are continuous, then the sequence {xn} strongly converges to a common fixed point of T and I if and only if lim inf n→∞ d xn, F 0. 2.8 Proof. The necessity is obvious. Next, we prove the sufficiency part of the theorem. Note that continuity of T and I implies that the set F T and F I are closed. It follows from 2.6 that ∥ xn 1 − p ∥ ∥ ≤ ∥∥xn − p ∥ ∥ cn. 2.9 This implies in turn that d xn 1, F ≤ d xn, F cn. 2.10 Now applying Lemma 1.2 to 2.10 , we obtain the existence of the limit limn→∞d xn, F . By condition 2.8 , we have lim n→∞ d xn, F lim inf n→∞ d xn, F 0. 2.11 6 Abstract and Applied Analysis Next we prove that the sequence {xn} is a Cauchy sequence in K. For any positive integers n, m, from 2.9 it follows that ∥ xn m − p ∥ ∥ ≤ ∥∥xn m−1 − p ∥ ∥ cn m−1 ≤ (∥∥xn m−2 − p ∥ ∥ cn m−2 ) cn m−1 ≤ · · · ≤ ∥∥xn − p ∥ ∥ n m−1 ∑