International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 (2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES G. S. SALUJA Abstract. In this paper, we study hybrid fixed point of a modified two-step iteration process with errors for a pair of asymptotically quasi-nonexpansive mapping and asymptotically quasi-nonexpansive mapping in the intermediate sense in the framework of Banach spaces. Also we establish some strong con- vergence theorems and a weak convergence theorem for the iteration scheme and mappings. The results presented in this paper extend, improve and gen- eralize some previous work from the existing literature. 1. Introduction Let K be a nonempty subset of a real Banach space E. Let T : K → K be a mapping, then we denote the set of all fixed points of T by F(T). The set of com- mon fixed points of two mappings S and T will be denoted by F = F(S) ∩F(T). A mapping T : K → K is said to be: (1) nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖ for all x,y ∈ K; (2) quasi-nonexpansive if F(T) 6= ∅ and ‖Tx−p‖ ≤ ‖x−p‖ for all x ∈ K and p ∈ F(T); (3) asymptotically nonexpansive if there exists a sequence {kn} ∈ [1,∞) with limn→∞kn = 1 and ‖Tnx−Tny‖ ≤ kn‖x−y‖ for all x, y ∈ K and n ≥ 1; 2010 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. asymptotically quasi-nonexpansive mapping; asymptotically quasi- nonexpansive mapping in the intermediate sense; modified two-step iteration process with errors; hybrid fixed point; strong convergence; weak convergence; Banach space. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 69 70 SALUJA (4) asymptotically quasi-nonexpansive if F(T) 6= ∅ and there exists a sequence {kn}∈ [1,∞) such that limn→∞kn = 1 and ‖Tnx−p‖ ≤ kn‖x−p‖ for all x ∈ K, p ∈ F(T) and n ≥ 1; (5) uniformly L-Lipschitzian if there exists a constant L > 0 such that ‖Tnx−Tny‖ ≤ L‖x−y‖ for all x, y ∈ K and n ≥ 1; (6) uniformly quasi-Lipschitzian if there exists L ∈ [1, +∞) such that ‖Tnx−p‖ ≤ L‖x−p‖ for all x ∈ K, p ∈ F(T) and n ≥ 1. Remark 1.1. It is clear that every nonexpansive mapping is asymptotically non- expansive and every asymptotically nonexpansive is uniformly Lipschitzian. Also, if F(T) 6= ∅, then a nonexpansive mapping is a quasi-nonexpansive mapping, an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive, a uniformly L-Lipschitzian mapping must be uniformly quasi-Lipschitzian and an asymptotically quasi-nonexpansive mapping must be uniformly quasi-Lipschitzian mapping with L = supn≥1{kn}≥ 1 but the converse is not true in general. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] as a generalization of the class of nonexpansive mappings. Recall also that a mapping T : K → K is said to be asymptotically quasi-nonexpansive in the intermediate sense [18] provided that T is uniformly continuous and lim sup n→∞ sup x∈K, p∈F(T) ( ‖Tnx−p‖−‖x−p‖ ) ≤ 0. From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense. But the converse does not hold as the following example: Example 1.2. Let X = R be a normed linear space and K = [0, 1]. For each x ∈ K, we define T(x) = { kx, if x 6= 0, 0, if x = 0, where 0 < k < 1. Then |Tnx−Tny| = kn|x−y| ≤ |x−y| for all x,y ∈ K and n ∈ N. Thus T is an asymptotically nonexpansive mapping with constant sequence {1} and lim sup n→∞ {|Tnx−Tny|− |x−y|} = lim sup n→∞ {kn‖x−y‖−‖x−y‖} ≤ 0, CONVERGENCE OF HYBRID FIXED POINT 71 because limn→∞k n = 0 as 0 < k < 1, for all x,y ∈ K, n ∈ N and T is continuous. Hence T is an asymptotically nonexpansive mapping in the intermediate sense. Example 1.3. Let X = R, K = [−1 π , 1 π ] and |k| < 1. For each x ∈ K, define T(x) = { k xsin(1/x), if x 6= 0, 0, if x = 0. Then T is an asymptotically nonexpansive mapping in the intermediate sense but it is not asymptotically nonexpansive mapping. Since 1972, many authors have studied weak and strong convergence problem of the iterative sequences (with errors) for asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces (see, e.g., [5, 6, 8, 10, 11, 12, 13, 17] and refer- ences therein). In 2007, Agarwal et al. [1] introduced the following iteration process: x1 = x ∈ K, 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 contrac- tions. The above process deals with one mapping only. The case of two mappings in iterative processes has also remained under study since Das and Debate [3] gave and studied a two mappings process. Later on, many authors, for example Khan and Takahashi [8], Shahzad and Udomene [14] and Takahashi and Tamura [16] have studied the two mappings case of iterative schemes for different types of mappings. Recently, Khan et al. [7] studied the modified two-step iteration process for two mappings as follows: x1 = x ∈ K, xn+1 = (1 −αn)Tnxn + αnSnyn, yn = (1 −βn)xn + βnTnxn, n ≥ 1,(1.2) where {αn} and {βn} are sequences in (0, 1). They established weak and strong convergence theorems in the setting of real Banach spaces. Inspired and motivated by [1, 7] and many others, in this paper we introduce the following iteration scheme for a pair of asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense. The pro- posed iteration scheme is as follows. Definition 1.4. Let S : K → K be an asymptotically quasi-nonexpansive mapping and T : K → K be an asymptotically quasi-nonexpansive mapping in the interme- diate sense on a closed convex subset K of a real Banach space E with K +K ⊂ K. 72 SALUJA Let {xn} be the sequence defined as follows: x1 = x ∈ K, xn+1 = (1 −αn)Tnxn + αnSnyn + un, yn = (1 −βn)xn + βnTnxn + vn, n ≥ 1,(1.3) where {αn} and {βn} are sequences in (0, 1) and {un} and {vn} are two sequences in K. The iteration scheme (1.3) is called modified two-step iteration process with errors for a pair of above said mappings. The aim of this paper is to establish some strong convergence theorems and a weak convergence theorem for newly proposed iteration scheme (1.3) in the frame- work of real Banach spaces. The results presented in this paper extend, improve and generalize some previous work from the existing literature. 2. Preliminaries For the sake of convenience, we restate the following concepts. A mapping T : K → K is said to be demiclosed at zero, if for any sequence {xn} in K, the condition xn converges weakly to x ∈ K and Txn converges strongly to 0 imply Tx = 0. A mapping T : K → K is said to be semi-compact [2] if for any bounded sequence {xn} in K such that ‖xn −Txn‖ → 0 as n → ∞, then there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ K strongly. We say that a Banach space E satisfies the Opial’s condition [9] 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 [9]. Now, we state the following useful lemma to prove our main results. Lemma 2.1. (See [15]) Let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of non- negative 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 con- vergence theorem of the iteration scheme (1.3) for a pair of asymptotically quasi- nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense in the framework of real Banach spaces. CONVERGENCE OF HYBRID FIXED POINT 73 Theorem 3.1. Let E be a real Banach space and K be a nonempty closed convex subset of E with K+K ⊂ K. Let S : K → K be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn − 1) < ∞ and T : K → K be uniformly L-Lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞ and∑∞ n=1 ‖vn‖ < ∞. Put Dn = max { sup x∈K, q∈F ( ‖Tnx−q‖−‖x−q‖ ) ∨ 0 : n ≥ 1 } (3.1) such that ∑∞ n=1 Dn < ∞. Then {xn} converges to a hybrid fixed point of the map- pings S and T if and only if lim infn→∞d(xn,F) = 0, where d(x,F) = infp∈F d(x,p). Proof. The necessity is obvious. Thus we only prove the sufficiency. Let q ∈ F. Then from (1.3) and (3.1), we have ‖yn −q‖ = ‖(1 −βn)xn + βnTnxn + vn −q‖ ≤ (1 −βn)‖xn −q‖ + βn‖Tnxn −q‖ + ‖vn‖ ≤ (1 −βn)‖xn −q‖ + βn[‖xn −q‖ + Dn] + ‖vn‖ = (1 −βn)‖xn −q‖ + βn‖xn −q‖ + βnDn + ‖vn‖ ≤ ‖xn −q‖ + Dn + ‖vn‖.(3.2) Again using (1.3), (3.1) and (3.2), we obtain ‖xn+1 −q‖ = ‖(1 −αn)Tnxn + αnSnyn + un −q‖ ≤ (1 −αn)‖Tnxn −q‖ + αn‖Snyn −q‖ + ‖un‖ ≤ (1 −αn)[‖xn −q‖ + Dn] + αnkn‖yn −q‖ + ‖un‖ ≤ (1 −αn)‖xn −q‖ + αnkn[‖xn −q‖ + Dn + ‖vn‖] +(1 −αn)Dn + ‖un‖ ≤ [1 + (kn − 1)αn]‖xn −q‖ + [1 + (kn − 1)αn]Dn +‖un‖ + kn‖vn‖ = (1 + tn)‖xn −q‖ + θn(3.3) where tn = (kn − 1)αn and θn = [1 + (kn − 1)αn]Dn + ‖un‖ + kn‖vn‖. Since by hypothesis of the theorem ∑∞ n=1(kn−1)αn < ∞, ∑∞ n=1 Dn < ∞, ∑∞ n=1 ‖un‖ < ∞ and ∑∞ n=1 ‖vn‖ < ∞, it follows that ∑∞ n=1 tn < ∞ and ∑∞ n=1 θn < ∞. Hence by Lemma 2.1, we know that the limit limn→∞‖xn − q‖ exists. Also from (3.3), we obtain d(xn+1,F) ≤ (1 + tn) d(xn,F) + θn(3.4) for all n ≥ 1. From Lemma 2.1 and (3.4), we know that limn→∞d(xn,F) exists. Since lim infn→∞d(xn,F) = 0, we have that limn→∞d(xn,F) = 0. 74 SALUJA 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 (3.3), we have ‖xn+m −q‖ ≤ (1 + tn+m−1)‖xn+m−1 −q‖ + θn+m−1 ≤ etn+m−1‖xn+m−1 −q‖ + θn+m−1 ≤ etn+m−1 [etn+m−2‖xn+m−2 −q‖ + θn+m−2] + θn+m−1 ≤ e(tn+m−1+tn+m−2)‖xn+m−2 −q‖ +e(tn+m−1+tn+m−2)[θn+m−2 + θn+m−1] ≤ . . . ≤ e (∑n+m−1 k=n tk ) ‖xn −q‖ + e (∑n+m−1 k=n tk ) n+m−1∑ k=n θk ≤ e (∑∞ n=1 tk ) ‖xn −q‖ + e (∑∞ n=1 tk ) n+m−1∑ k=n θk = R‖xn −q‖ + R n+m−1∑ k=n θk(3.5) where R = e (∑∞ n=1 tk ) < ∞. Since lim n→∞ d(xn,F) = 0, ∞∑ n=1 θn < ∞(3.6) for any given ε > 0, there exists a positive integer n1 such that d(xn,F) < ε 4(R + 1) , ∞∑ k=n θk < ε 2R ∀n ≥ n1.(3.7) Hence, there exists q1 ∈ F such that ‖xn −q1‖ < ε 2(R + 1) ∀n ≥ n1.(3.8) Consequently, for any n ≥ n1 and m ≥ 1, from (3.5), we have ‖xn+m −xn‖ ≤ ‖xn+m −q1‖ + ‖xn −q1‖ ≤ R‖xn −q1‖ + R n+m−1∑ k=n θk + ‖xn −q1‖ ≤ (R + 1)‖xn −q1‖ + R n+m−1∑ k=n θk < (R + 1). ε 2(R + 1) + R. ε 2R = ε.(3.9) This implies that {xn} is a Cauchy sequence in E and so is convergent since E is complete. Let limn→∞xn = q ∗. Then q∗ ∈ K. It remains to show that q∗ ∈ F. Let ε1 > 0 be given. Then there exists a natural number n2 such that ‖xn −q∗‖ < ε1 2(L + 1) , ∀n ≥ n2.(3.10) CONVERGENCE OF HYBRID FIXED POINT 75 Since limn→∞d(xn,F) = 0, there must exists a natural number n3 ≥ n2 such that for all n ≥ n3, we have d(xn,F) < ε1 3(L + 1) (3.11) and in particular, we have d(xn3,F) < ε1 3(L + 1) .(3.12) Therefore, there exists z∗ ∈ F such that ‖xn3 −z ∗‖ < ε1 2(L + 1) .(3.13) Consequently, we have ‖Tq∗ −q∗‖ = ‖Tq∗ −z∗ + z∗ −xn3 + xn3 −q ∗‖ ≤ ‖Tq∗ −z∗‖ + ‖z∗ −xn3‖ + ‖xn3 −q ∗‖ ≤ L‖q∗ −z∗‖ + ‖z∗ −xn3‖ + ‖xn3 −q ∗‖ ≤ L‖q∗ −xn3 + xn3 −z ∗‖ + ‖z∗ −xn3‖ +‖xn3 −q ∗‖ ≤ L[‖q∗ −xn3‖ + ‖xn3 −z ∗‖] + ‖z∗ −xn3‖ +‖xn3 −q ∗‖ ≤ (L + 1)‖q∗ −xn3‖ + (L + 1)‖z ∗ −xn3‖ < (L + 1). ε1 2(L + 1) + (L + 1). ε1 2(L + 1) < ε1.(3.14) This implies that q∗ ∈ F(T). Similarly, we can show that q∗ ∈ F(S). Since S is asymptotically quasi-nonexpansive mapping, so it is uniformly quasi-Lipschitzian with L = supn≥1{kn} ≥ 1 (by remark 1.1). So, as above we can show that q∗ ∈ F(S). Thus q∗ ∈ F = F(S) ∩ F(T), that is, q∗ is a hybrid fixed point of the mappings S and T. This completes the proof. � Theorem 3.2. Let E be a real Banach space and K be a nonempty closed convex subset of E with K+K ⊂ K. Let S : K → K be a asymptotically quasi-nonexpansive mapping with sequence {kn}∈ [1,∞) such that ∑∞ n=1(kn−1) < ∞ and T : K → K be uniformly L-Lipschitzian asymptotically quasi-nonexpansive mapping in the in- termediate sense such that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞,∑∞ n=1 ‖vn‖ < ∞ and Dn be taken as in Theorem 3.1. Suppose that the mappings S and T satisfy the following conditions: (C1) limn→∞‖xn −Sxn‖ = 0 and limn→∞‖xn −Txn‖ = 0, (C2) there exists a constant A > 0 such that{ a1 ‖xn −Sxn‖ + a2 ‖xn −Txn‖ } ≥ Ad(xn,F) for all n ≥ 1, where a1 and a2 are two nonnegative real numbers such that a1 +a2 = 1. 76 SALUJA Then {xn} converges strongly to a hybrid fixed point of the mappings S and T . Proof. From conditions (C1) and (C2), we have limn→∞d(xn,F) = 0, it follows as in the proof of Theorem 3.1, that {xn} must converges strongly to a hybrid fixed point of the mappings S and T. This completes the proof. � Theorem 3.3. Let E be a real Banach space and K be a nonempty closed convex subset of E with K+K ⊂ K. Let S : K → K be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn − 1) < ∞ and T : K → K be uniformly L-Lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that F = F(S)∩F(T) 6= ∅. Let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). Let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞, ∑∞ n=1 ‖vn‖ < ∞ and Dn be taken as in Theorem 3.1. Suppose limn→∞‖xn−Sxn‖ = 0 and limn→∞‖xn−Txn‖ = 0. If at least one of the mappings S and T is semi-compact, then the sequence {xn} converges strongly to a hybrid fixed point of the mappings S and T . Proof. Without loss of generality, we may assume that T is semi-compact. By Theorem 3.1, {xn} is bounded and by assumption of the theorem limn→∞ ‖xn − Sxn‖ = 0 and limn→∞‖xn − Txn‖ = 0. This means that there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ K as nk →∞. Now again by the hypothesis of the theorem, we find ‖x∗ −Tx∗‖ = lim nk→∞ ‖xnk −Txnk‖ = 0 and ‖x∗ −Sx∗‖ = lim nk→∞ ‖xnk −Sxnk‖ = 0. This shows that x∗ ∈ F. According to Theorem 3.1, the limit limn→∞‖xn −x∗‖ exists. Then lim n→∞ ‖xn −x∗‖ = lim nk→∞ ‖xnk −x ∗‖ = 0, which means that {xn} converges to x∗ ∈ F, that is, the sequence {xn} converges strongly to a hybrid fixed point of the mappings S and T . This completes the proof. � Theorem 3.4. Let E be a real Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E with K + K ⊂ K. Let S : K → K be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn−1) < ∞ and T : K → K be uniformly L-Lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that F = F(S)∩F(T) 6= ∅. Let {αn} and {βn} be sequences in [δ, 1−δ] for some δ ∈ (0, 1). Let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn−1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞, ∑∞ n=1 ‖vn‖ < ∞ and Dn be taken as in Theorem 3.1. Suppose that S and T have a hybrid fixed point, I − S and I − T are demiclosed at zero and {xn} is an approximating hybrid fixed point sequence for S and T , that is, limn→∞‖xn − Sxn‖ = 0 and limn→∞‖xn −Txn‖ = 0. Then {xn} converges weakly to a hybrid fixed point of the mappings S and T . Proof. Let p be a hybrid fixed point of S and T. Then limn→∞‖xn − p‖ 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 CONVERGENCE OF HYBRID FIXED POINT 77 {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(S) ∩F(T). Again in the same fashion, we can prove that v ∈ 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→∞ ‖xnj −u‖ = lim n→∞ ‖xn −u‖. This is a contradiction. Hence u = v ∈ F . Thus {xn} converges weakly to a hybrid fixed point of the mappings S and T . This completes the proof. � Remark 3.5. Our results extend, improve and generalize many known results from the existing literature (see, e.g., [1], [3], [4], [6]-[8], [10]-[14] and many others). 4. Conclusion The results proved in this paper of the iteration scheme (1.3) for a pair of asymp- totically quasi-nonexpansive and asymptotically quasi-nonexpansive mappings in the intermediate sense which is wider than that the class of nonexpansive, quasi- nonexpansive and asymptotically nonexpansive mappings. Thus our results are good improvement and extension of some corresponding previous work from the existing literature. References [1] R.P. Agarwal, Donal O’Regan and D.R. 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