International Journal of Analysis and Applications ISSN 2291-8639 Volume 6, Number 1 (2014), 89-96 http://www.etamaths.com CONVERGENCE TO COMMON FIXED POINT FOR NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES G. S. SALUJA Abstract. The purpose of this paper is to study modified S-iteration process to converge to common fixed point for two nearly asymptotically nonexpansive mappings in the framework of Banach spaces. Also we establish some strong convergence theorems and a weak convergence theorem for said mappings and iteration scheme under appropriate conditions. 1. Introduction Let C be a nonempty subset of a Banach space E and T : C → C a nonlinear mapping. We denote the set of all fixed points of T by F(T). The set of common fixed points of two mappings S and T will be denoted by F = F(S) ∩F(T). The mapping T is said to be Lipschitzian [1, 16] if for each n ∈ N, there exists a constant kn > 0 such that ‖Tnx−Tny‖ ≤ kn ‖x−y‖ for all x, y ∈ C. A Lipschitzian mapping T is said to be uniformly k-Lipschitzian if kn = k for all n ∈ N and asymptotically nonexpansive [4] if kn ≥ 1 for all n ∈ N with limn→∞kn = 1. It is easy to observe that every nonexpansive mapping T (i.e., ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C) is asymptotically nonexpansive with constant sequence {1} and every asymptotically nonexpansive mapping is uniformly k-Lipschitzian with k = supn∈N kn. The asymptotic fixed point theory has a fundamental role in nonlinear func- tional analysis (see, [2]). The theory has been studied by many authors (see, e.g., [6], [7], [10], [12], [21]) for various classes of nonlinear mappings (e.g., Lipschitzian, uniformly k-Lipschitzian and non-Lipschitzian mappings). A branch of this theo- ry related to asymptotically nonexpansive mappings has been developed by many authors (see, e.g., [4], [5], [9], [11], [12], [14], [15], [17]-[19]) in Banach spaces with 2010 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. Nearly asymptotically nonexpansive mapping, modified S-iteration process, common fixed point, strong convergence, weak convergence, Banach space. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 89 90 G. S. SALUJA suitable geometrical structure. Fix a sequence {an} ⊂ [0,∞) with limn→∞an = 0, then according to Agarwal et al. [1], T is said to be nearly Lipschitzian with respect to {an} if for each n ∈ N, there exist constants kn ≥ 0 such that ‖Tnx − Tny‖ ≤ kn(‖x − y‖ + an) for all x, y ∈ C. The infimum of constants kn for which the above inequality holds is denoted by η(Tn) and is called nearly Lipschitz constant. A nearly Lipschitzian mapping T with sequence {an,η(Tn)} is said to be nearly asymptotically nonexpansive if η(Tn) ≥ 1 for all n ∈ N and limn→∞η(Tn) = 1 and nearly uniformly k-Lipschitzian if η(Tn) ≤ k for all n ∈ N. In 2007, Agarwal et al. [1] introduced the following iteration process: x1 = x ∈ C, xn+1 = (1 −αn)Tnxn + αnTnyn, yn = (1 −βn)xn + βnTnxn, n ≥ 1(1.1) where {αn} and {βn} are sequences in (0, 1). They showed that this process con- verge at a rate same as that of Picard iteration and faster than Mann for con- tractions and also they established some weak convergence theorems using suitable conditions in the framework of uniformly convex Banach space. We modify iteration scheme (1.1) for two nonlinear mappings. Let C be a nonempty subset of a Banach space E and S, T : C → C be two nearly asymptotically nonexpansive mappings. For given x1 = x ∈ C, the iterative sequence {xn} defined as follows: x1 = x ∈ C, xn+1 = (1 −αn)Tnxn + αnSnyn, yn = (1 −βn)xn + βnTnxn, n ≥ 1(1.2) where {αn} and {βn} are sequences in (0, 1). The iteration scheme (1.2) is called modified S-iteration scheme for two nonlinear mappings. If we put S = T, then iteration scheme (1.2) reduces to S-iteration scheme (1.1). The aim of this paper is to establish some strong convergence theorems and a weak convergence theorem of modified S-iteration scheme (1.2) for two nearly asymptotically nonexpansive mappings in the framework of Banach spaces. 2. Preliminaries For the sake of convenience, we restate the following concepts. A mapping T : C → C is said to be demiclosed at zero, if for any sequence {xn} in C, the condition xn converges weakly to x ∈ C and Txn converges strongly to 0 imply Tx = 0. NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 91 A mapping T : C → C is said to be semi-compact [3] if for any bounded sequence {xn} in C such that ‖xn −Txn‖ → 0 as n → ∞, then there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ C strongly. We say that a Banach space E satisfies the Opial’s condition [13] if for each sequence {xn} in E weakly convergent to a point x and for all y 6= x lim inf n→∞ ‖xn −x‖ < lim inf n→∞ ‖xn −y‖. The examples of Banach spaces which satisfy the Opial’s condition are Hilbert spaces and all Lp[0, 2π] with 1 < p 6= 2 fail to satisfy Opial’s condition [13]. Now, we state the following useful lemma to prove our main results. Lemma 2.1. (See [20]) Let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of nonnegative numbers satisfying the inequality αn+1 ≤ (1 + βn)αn + rn, ∀n ≥ 1. If ∑∞ n=1 βn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞αn exists. 3. Main Results In this section, we prove some strong convergence theorems and a weak conver- gence theorem for two nearly asymptotically nonexpansive mappings in the frame- work of Banach spaces. Theorem 3.1. Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T : C → C be two nearly asymptotically nonexpansive map- pings with sequences {a′n,η(Sn)}, {a′′n,η(Tn)} and F = F(S) ∩F(T) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)η(Tn)−1 ) < ∞. Let {xn} be the mod- ified S-iteration scheme defined by (1.2). Then {xn} converges to a common fixed point of the mappings S and T if and only if lim infn→∞d(xn,F) = 0. Proof. The necessity is obvious. Thus we only prove the sufficiency. Let q ∈ F. For the sake of convenience, set Anx = (1 −βn)x + βnTnx and Gnx = (1 −αn)Tnx + αnSnAnx. Then yn = Anxn and xn+1 = Gnxn. Moreover, it is clear that q is a fixed point of Gn for all n. Let η = supn∈N η(S n) ∨ supn∈N η(Tn) and an = max{a′n,a′′n} for all n. 92 G. S. SALUJA Consider ‖Anx−Any‖ = ‖((1 −βn)x + βnTnx) − ((1 −βn)y + βnTny)‖ = ‖(1 −βn)(x−y) + βn(Tnx−Tny)‖ ≤ (1 −βn)‖x−y‖ + βnη(Tn)(‖x−y‖ + a′′n) ≤ (1 −βn)‖x−y‖ + βnη(Tn)‖x−y‖ + βnanη(Tn) ≤ (1 −βn)η(Tn)‖x−y‖ + βnη(Tn)‖x−y‖ +βnanη(T n) ≤ η(Tn)‖x−y‖ + anη(Tn).(3.1) Choosing x = xn and y = q, we get ‖yn −q‖ ≤ η(Tn)‖xn −q‖ + anη(Tn).(3.2) Now, consider ‖Gnx−Gny‖ = ‖((1 −αn)Tnx + αnSnAnx) − ((1 −αn)Tny + αnSnAny)‖ = ‖(1 −αn)(Tnx−Tny) + αn(SnAnx−SnAny)‖ ≤ (1 −αn)η(Tn)(‖x−y‖ + a′′n) + αnη(S n)(‖Anx−Any‖ + a′n) ≤ (1 −αn)η(Tn)(‖x−y‖ + an) + αnη(Sn)(‖Anx−Any‖ + an) ≤ (1 −αn)η(Tn)‖x−y‖ + αnη(Sn)‖Anx−Any‖ +(1 −αn)anη(Tn) + αnanη(Sn).(3.3) Now using (3.1) in (3.3), we get ‖Gnx−Gny‖ ≤ (1 −αn)η(Tn)‖x−y‖ + αnη(Sn)[η(Tn)‖x−y‖ +anη(T n)] + (1 −αn)anη(Tn) + αnanη(Sn) ≤ (1 −αn)η(Tn)η(Sn)‖x−y‖ + αnη(Tn)η(Sn)‖x−y‖ +(1 −αn + 2αn)anη(Tn)η(Sn) ≤ η(Tn)η(Sn)‖x−y‖ + 2anη(Tn)η(Sn) ≤ [ 1 + ( η(Tn)η(Sn) − 1 )] ‖x−y‖ + 2anη2 = (1 + Pn)‖x−y‖ + Qn,(3.4) where Pn = ( η(Tn)η(Sn)−1 ) and Qn = 2anη 2. Since by hypothesis ∑∞ n=1 ( η(Sn)η(Tn)− 1 ) < ∞ and ∑∞ n=1 an < ∞. It follows that ∑∞ n=1 Pn < ∞ and ∑∞ n=1 Qn < ∞. Choosing x = xn and y = q in (3.4), we get ‖xn+1 −q‖ = ‖Gnxn −q‖≤ (1 + Pn)‖xn −q‖ + Qn.(3.5) Applying Lemma 2.1 in (3.5), we have limn→∞‖xn −q‖ exists. NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 93 Next, we shall prove that {xn} is a Cauchy sequence. Since 1 +x ≤ ex for x ≥ 0, therefore, for any m,n ≥ 1 and for given q ∈ F, from (3.5), we have ‖xn+m −q‖ ≤ (1 + Pn+m−1)‖xn+m−1 −q‖ + Qn+m−1 ≤ ePn+m−1‖xn+m−1 −q‖ + Qn+m−1 ≤ ePn+m−1 [ePn+m−2‖xn+m−2 −q‖ + Qn+m−2] + Qn+m−1 ≤ e(Pn+m−1+Pn+m−2)‖xn+m−2 −q‖ +e(Pn+m−1+Pn+m−2)[Qn+m−2 + Qn+m−1] ≤ . . . ≤ e (∑n+m−1 k=n Pk ) ‖xn −q‖ + e (∑n+m−1 k=n Pk ) n+m−1∑ k=n Qk ≤ e (∑∞ n=1 Pn ) ‖xn −q‖ + e (∑∞ n=1 Pn ) n+m−1∑ k=n Qk = K‖xn −q‖ + K n+m−1∑ k=n Qk(3.6) where K = e (∑∞ n=1 Pn ) < ∞. Since lim n→∞ d(xn,F) = 0, ∞∑ n=1 Qn < ∞(3.7) for any given ε > 0, there exists a positive integer n1 such that d(xn,F) < ε 4(K + 1) , n+m−1∑ k=n Qk < ε 2K ∀n ≥ n1.(3.8) Hence, there exists q1 ∈ F such that ‖xn −q1‖ < ε 2(K + 1) ∀n ≥ n1.(3.9) Consequently, for any n ≥ n1 and m ≥ 1, from (3.6), we have ‖xn+m −xn‖ ≤ ‖xn+m −q1‖ + ‖xn −q1‖ ≤ K‖xn −q1‖ + K n+m−1∑ k=n Qk + ‖xn −q1‖ ≤ (K + 1)‖xn −q1‖ + K n+m−1∑ k=n Qk < (K + 1) ε 2(K + 1) + K ε 2K = ε.(3.10) This implies that {xn} is a Cauchy sequence in E and so is convergent since E is complete. Assume that limn→∞xn = q ∗. Since C is closed, therefore q∗ ∈ C. Next, we show that q∗ ∈ F . Now limn→∞d(xn,F) = 0 gives that d(q∗,F) = 0. Since F is closed, q∗ ∈ F. Thus {xn} converges strongly to a common fixed point of the mappings S and T. This completes the proof. 94 G. S. SALUJA Theorem 3.2. Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T : C → C be two nearly asymptotically nonexpansive map- pings with sequences {a′n,η(Sn)}, {a′′n,η(Tn)} and F = F(S) ∩F(T) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)η(Tn) − 1 ) < ∞. Let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). Let {xn} be the modified S-iteration scheme defined by (1.2). If either S is semi-compact and limn→∞‖xn−Sxn‖ = 0 or T is semi-compact and limn→∞‖xn −Txn‖ = 0, then the sequence {xn} converge strongly to a point of F. proof. Suppose that T is semi-compact and limn→∞‖xn − Txn‖ = 0. Then there exists a subsequence {xnj} of {xn} such that xnj → q ∈ C. Also, we have limj→∞‖xnj −Txnj‖ = 0 and we make use of the fact that every nearly asymptot- ically nonexpansive mapping is nearly k-Lipschitzian. Hence, we have ‖q −Tq‖ ≤ ‖q −xnj‖ + ‖xnj −Txnj‖ + ‖Txnj −Tq‖ ≤ (1 + k)‖q −xnj‖ + ‖xnj −Txnj‖→ 0. Thus q ∈ F . By (3.5), ‖xn+1 −q‖≤ (1 + Pn)‖xn −q‖ + Qn. Since by hypothesis ∑∞ n=1 Pn < ∞ and ∑∞ n=1 Qn < ∞, by Lemma 2.2, limn→∞‖xn− q‖ exists and xnj → q ∈ F gives that xn → q ∈ F . This shows that {xn} converges strongly to a point of F. This completes the proof. As an application of Theorem 3.1, we establish another strong convergence result as follows. Theorem 3.3. Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T : C → C be two nearly asymptotically nonexpansive map- pings with sequences {a′n,η(Sn)}, {a′′n,η(Tn)} and F = F(S) ∩F(T) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)η(Tn) − 1 ) < ∞. Let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). Let {xn} be the modified S-iteration scheme defined by (1.2). If S and T satisfy the following conditions: (i) limn→∞‖xn −Sxn‖ = 0 and limn→∞‖xn −Txn‖ = 0. (ii) There exists a constant A > 0 such that[ a1‖xn −Sxn‖ + a2‖xn −Txn‖ ] ≥ Ad(xn,F) where a1 and a2 are two non-negative real numbers such that a1 + a2 = 1. Then the sequence {xn} converge strongly to a point of F. proof. From conditions (i) and (ii), we have limn→∞d(xn,F) = 0, it follows as in the proof of Theorem 3.1, that {xn} must converge strongly to a point of F. This completes the proof. NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 95 Theorem 3.4. Let E be a Banach space satisfying Opial’s condition and C be a nonempty closed convex subset of E. Let S, T : C → C be two nearly asymptotical- ly nonexpansive mappings with sequences {a′n,η(Sn)}, {a′′n,η(Tn)} and F = F(S)∩ F(T) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)η(Tn)−1 ) < ∞. Let {αn} and {βn} be sequences in [δ, 1 −δ] for some δ ∈ (0, 1). Let {xn} be the mod- ified S-iteration scheme defined by (1.2). Suppose that S and T have a common fixed point, I −S and I −T are demiclosed at zero and {xn} is an approximating common fixed point sequence for S and T , that is, limn→∞‖xn − Sxn‖ = 0 and limn→∞‖xn −Txn‖ = 0. Then {xn} converges weakly to a common fixed point of S and T. Proof: Let q be a common fixed point of S and T . Then limn→∞‖xn−q‖ exists as proved in Theorem 3.1. We prove that {xn} has a unique weak subsequential limit in F = F(S) ∩ F(T). For, let u and v be weak limits of the subsequences {xni} and {xnj} of {xn}, respectively. By hypothesis of the theorem, we know that limn→∞‖xn −Sxn‖ = 0 and I −S is demiclosed at zero, therefore we obtain Su = u. Similarly, Tu = u. Thus u ∈ F = F(S)∩F(T). Again in the same fashion, we can prove that v ∈ F = F(S) ∩F(T). Next, we prove the uniqueness. To this end, if u and v are distinct then by Opial’s condition, lim n→∞ ‖xn −u‖ = lim ni→∞ ‖xni −u‖ < lim ni→∞ ‖xni −v‖ = lim n→∞ ‖xn −v‖ = lim nj→∞ ‖xnj −v‖ < lim nj→∞ ‖xni −u‖ = lim n→∞ ‖xn −u‖. This is a contradiction. Hence u = v ∈ F. Thus {xn} converges weakly to a com- mon fixed point of the mappings S and T. This completes the proof. Remark 3.1. Our results extend and generalize the corresponding results of [8], [14], [15], [17], [18], [20] and many others from the existing literature to the case of modified S-iteration scheme and more general class of nonexpansive and asymptotically nonexpansive mappings considered in this paper. References [1] R.P. Agarwal, Donal O’Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Nonlinear Convex Anal. 8(1)(2007), 61-79. [2] F.E. Browder, Nonlinear operators and nonlinear equations of evolution, Proc. Amer. Math. Symp. Pure Math. XVII, Amer. Math. Soc., Providence, 1976. [3] C.E. Chidume and B. Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330(2007), 377-387. 96 G. S. SALUJA [4] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpan- sive mappings, Proc. Amer. Math. Soc. 35(1)(1972), 171-174. [5] J. 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