International Journal of Analysis and Applications ISSN 2291-8639 Volume 10, Number 1 (2016), 24-39 http://www.etamaths.com ON WEAK AND STRONG CONVERGENCE THEOREMS OF MODIFIED SP-ITERATION SCHEME FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS G. S. SALUJA∗ Abstract. In this paper, we study modified SP -iteration scheme for three total asymptotically nonexpansive mappings and also establish some weak and strong convergence theorems for mentioned mappings and scheme to converge to common fixed points in the framework of Banach spaces. Our results extend and generalize the previous works from the current existing literature. 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 three mappings T1, T2 and T3 will be denoted by F = ∩3i=1F(Ti). Definition 1.1. Let T : C → C be a mapping. Then (1) T is said to be nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖(1.1) for all x, y ∈ C. (2) T is said to be asymptotically nonexpansive if there exists a positive sequence hn ∈ [1,∞) with limn→∞hn = 1 such that ‖Tnx−Tny‖ ≤ hn ‖x−y‖(1.2) for all x, y ∈ C and n ≥ 1. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [6] as a generalization of the class of nonexpansive mappings. They proved that if C is a nonempty closed con- vex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on C, then has a fixed point. T is said to be asymptotically noneexpansive in the intermediate sense if it is continuous and the following inequality holds: lim sup n→∞ sup x,y∈C ( ‖Tnx−Tny‖−‖x−y‖ ) ≤ 0.(1.3) Observe that if we define cn = lim sup n→∞ sup x,y∈C ( ‖Tnx−Tny‖−‖x−y‖ ) and νn = max{0,cn}, then νn → 0 as n →∞. It follows that (1.3) is reduced to ‖Tnx−Tny‖ ≤ ‖x−y‖ + νn,(1.4) 2010 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. Total asymptotically nonexpansive mapping; modified SP -iteration scheme; common fixed point; strong convergence; weak convergence; Banach space. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 24 MODIFIED SP -ITERATION SCHEME 25 for all x, y ∈ C and n ≥ 1. The class of mappings which are asymptotically nonexpansive in the intermediate sense was intro- duced by Bruck, Kuczumow and Reich [3]. It is known [10] that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is asymptotically nonexpansive in the in- termediate sense mapping, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate contains properly the class of asymptoti- cally nonexpansive mappings. In 2006, Albert et al. [2] introduced the notion of total asymptotically nonexpansive mappings. Definition 1.2. ([2]) The mapping T is said to be total asymptotically nonexpansive if ‖Tnx−Tny‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(1.5) for all x, y ∈ C and n ≥ 1, where {µn} and {νn} are nonnegative real sequences such that µn → 0 and νn → 0 as n →∞ and a strictly increasing continuous function ψ : [0,∞) → [0,∞) with ψ(0) = 0. From the definition, we see that the class of total asymptotically nonexpansive mappings include the class of asymptotically nonexpansive mappings as a special case; see also [4] for more details. Remark 1.3. From the above definition, it is clear that each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with νn = 0, µn = kn − 1 for all n ≥ 1, ψ(t) = t, t ≥ 0. (1) Mann iteration [12]: Chose x1 ∈ C and define xn+1 = (1 −αn)xn + αnTxn, n ≥ 1,(1.6) where {αn} is a sequence in (0,1). (2) Ishikawa iteration [9]: Chose x1 ∈ C and define yn = (1 −βn)xn + βnTxn xn+1 = (1 −αn)xn + αnTyn, n ≥ 1,(1.7) where {αn} and {βn} are sequences in (0,1). (3) S-iteration [1]: Chose x1 ∈ C and define yn = (1 −βn)xn + βnTxn xn+1 = (1 −αn)Txn + αnTyn, n ≥ 1,(1.8) where {αn} and {βn} are sequences in (0,1). Note that (1.8) is independent of (1.7) (and hence (1.6)). Agarwal, O’Regan and Sahu [1] showed that their process independent of those of Mann and Ishikawa and converges faster than both of these (see [[1], Proposition 3.1]). (4) Modified S-iteration [1]: Chose x1 ∈ C and define yn = (1 −βn)xn + βnTnxn xn+1 = (1 −αn)Tnxn + αnTnyn, n ≥ 1,(1.9) where {αn} and {βn} are sequences in (0,1). (5) Noor iteration [13]: Chose x1 ∈ C and define zn = (1 −γn)xn + γnTxn yn = (1 −βn)xn + βnTzn xn+1 = (1 −αn)xn + αnTyn, n ≥ 1,(1.10) where {αn}, {βn} and {γn} are sequences in [0,1]. 26 SALUJA (6) Modified Noor iteration [21]: Chose x1 ∈ C and define zn = (1 −γn)xn + γnTnxn yn = (1 −βn)xn + βnTnzn xn+1 = (1 −αn)xn + αnTnyn, n ≥ 1,(1.11) where {αn}, {βn} and {γn} are sequences in [0,1]. Recently, Phuengrattana and Suantai [16] introduced the following iteration scheme. (7) SP-iteration [16]: Chose x1 ∈ C and define zn = (1 −γn)xn + γnTxn yn = (1 −βn)zn + βnTzn xn+1 = (1 −αn)yn + αnTyn, n ≥ 1,(1.12) where {αn}, {βn} and {γn} are sequences in [0,1]. Inspired and motivated by [16], we modify iteration scheme (1.12) for three total asymptotically nonexpansive self mappings of C as follows: (8) Modified SP-iteration: Chose x1 ∈ C and define zn = (1 −γn)xn + γnTn3 xn yn = (1 −βn)zn + βnTn2 zn xn+1 = (1 −αn)yn + αnTn1 yn, n ≥ 1,(1.13) where {αn}, {βn} and {γn} are sequences in [0,1]. Remark 1.4. If we take Tn1 = T n 2 = T n 3 = T for all n ≥ 1, then (1.13) reduces to the SP-iteration scheme (1.12). The three-step iterative approximation problems were studied extensively by Noor [13, 14], Glowin- sky and Le Tallec [7], and Haubruge et al [8]. It has been shown [7] that three-step iterative scheme gives better numerical results than the two step and one step approximate iterations. Thus we conclude that three step scheme plays an important and significant role in solving various problems, which arise in pure and applied sciences. The purpose of this paper is to study modified SP-iteration scheme (1.13) and establish some strong and weak convergence theorems for total asymptotically nonexpansive mappings in the setting of Ba- nach spaces. Our results extend and generalize the previous works from the current existing literature. 2. Preliminaries For the sake of convenience, we restate the following definitions and lemmas. Let E be a Banach space with its dimension greater than or equal to 2. The modulus of convexity of E is the function δE(ε) : (0, 2] → [0, 1] defined by δE(ε) = inf { 1 −‖ 1 2 (x + y)‖ : ‖x‖ = 1, ‖y‖ = 1, ε = ‖x−y‖ } . A Banach space E is uniformly convex if and only if δE(ε) > 0 for all ε ∈ (0, 2]. We recall the following: Let S = {x ∈ E : ‖x‖ = 1} and let E∗ be the dual of E, that is, the space of all continuous linear functionals f on E. Definition 2.1. (i) Opial condition: The space E has Opial condition [15] if for any sequence {xn} in E, xn converges to x weakly it follows that lim supn→∞‖xn −x‖ < lim supn→∞‖xn −y‖ for all y ∈ E with y 6= x. Examples of Banach spaces satisfying Opial condition are Hilbert spaces and all spaces MODIFIED SP -ITERATION SCHEME 27 lp(1 < p < ∞). On the other hand, Lp[0, 2π] with 1 < p 6= 2 fail to satisfy Opial condition. (ii) A mapping T : C → C is said to be demiclosed at zero, if for any sequence {xn} in K, the condition xn converges weakly to x ∈ C and Txn converges strongly to 0 imply Tx = 0. (iii) A Banach space E has the Kadec-Klee property [19] if for every sequence {xn} in E, xn → x weakly and ‖xn‖→‖x‖ it follows that ‖xn −x‖→ 0. Definition 2.2. Condition (A): The mapping T : C → E with F(T) 6= ∅ is said to satisfy condition (A) [18] if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(t) > 0 for all t ∈ (0,∞) such that ‖x−Tx‖≥ f(d(x,F(T))) for all x ∈ C, where d(x,F(T)) = inf{‖x−p‖ : p ∈ F(T)}. Now, we modify Condition (A) for three mappings. Definition 2.3. Condition (B): Three mappings T1, T2, T3 : C → C are said to satisfy condition (B) if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(t) > 0 for all t ∈ (0,∞) such that a1 ‖x−T1x‖ + a2 ‖x−T2x‖ + a3 T3x ≥ f(d(x,F)) for all x ∈ C, where d(x,F) = inf{‖x−p‖ : p ∈ F = ∩3i=1F(Ti)}, where a1, a2 and a3 are nonnegative real numbers such that a1 + a2 + a3 = 1. Note that condition (B) reduces to condition (A) when T1 = T2 = T3 = T and hence is more general than the demicompactness of T1, T2 and T3 [18]. A mapping T : C → C is called: (1) demicompact if any bounded sequence {xn} in C such that {xn − Txn} converges has a convergent subsequence; (2) semicompact (or hemicompact) if any bounded sequence {xn} in C such that {xn −Txn}→ 0 as n →∞ has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general. Senter and Dotson [18] have approximated fixed points of a nonexpansive mapping T by Mann iterates whereas Maiti and Ghosh [11] and Tan and Xu [20] have approximated the fixed points using Ishikawa iterates under the condition (A) of [18]. Tan and Xu [20] pointed out that condition (A) is weaker than the compactness of C. We shall use condition (B) instead of compactness of C to study the strong convergence of {xn} defined by iteration scheme (1.13). Lemma 2.4. (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 (i) limn→∞αn exists; (ii) In particular, if {αn}∞n=1 has a subsequence which converges strongly to zero, then limn→∞αn = 0. Lemma 2.5. (See [17]) Let E be a uniformly convex Banach space and 0 < α ≤ tn ≤ β < 1 for all n ∈ N. Suppose further that {xn} and {yn} are sequences of E such that lim supn→∞‖xn‖ ≤ a, lim supn→∞‖yn‖≤ a and limn→∞‖tnxn+(1−tn)yn‖ = a hold for some a ≥ 0. Then limn→∞‖xn−yn‖ = 0. Lemma 2.6. (See [19]) Let E be a real reflexive Banach space with its dual E∗ has the Kadec-Klee property. Let {xn} be a bounded sequence in E and p, q ∈ ww(xn) (where ww(xn) denotes the set of all weak subsequential limits of {xn}). Suppose limn→∞‖txn + (1 − t)p − q‖ exists for all t ∈ [0, 1]. Then p = q. Lemma 2.7. (See [19]) Let K be a nonempty convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing continuous convex function φ: [0,∞) → [0,∞) with φ(0) = 0 such that for each Lipschitzian mapping T : C → C with the Lipschitz constant L, ‖tTx + (1 − t)Ty −T(tx + (1 − t)y)‖≤ Lφ−1 ( ‖x−y‖− 1 L ‖Tx−Ty‖ ) for all x, y ∈ K and all t ∈ [0, 1]. 28 SALUJA Proposition 2.8. Let C be a nonempty subset of a Banach space E and T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings. Then there exist nonnegative real sequences {µn} and {νn} in [0,∞) with µn → 0 and νn → 0 as n → ∞ and a strictly increasing continuous function ψ : R+ → R+ with ψ(0) = 0 such that ‖Tn1 x−T n 1 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.1) ‖Tn2 x−T n 2 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.2) and ‖Tn3 x−T n 3 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.3) for all x, y ∈ C and n ≥ 1. Proof. Since T1, T2, T3 : C → C are three total asymptotically nonexpansive mappings, there exist nonnegative real sequences {µn1}, {µn2}, {µn3}, {νn1}, {νn2} and {νn3} in [0,∞) with µn1,µn2,µn3 → 0 and νn1,νn2,νn3 → 0 as n → ∞ and strictly increasing continuous functions ψ1,ψ2,ψ3 : R+ → R+ with ψi(0) = 0 for i = 1, 2, 3 such that ‖Tn1 x−T n 1 y‖ ≤ ‖x−y‖ + µn1ψ1(‖x−y‖) + νn1,(2.4) ‖Tn2 x−T n 2 y‖ ≤ ‖x−y‖ + µn2ψ2(‖x−y‖) + νn2,(2.5) and ‖Tn3 x−T n 3 y‖ ≤ ‖x−y‖ + µn3ψ3(‖x−y‖) + νn3,(2.6) for all x, y ∈ C and n ≥ 1. Setting µn = max{µn1, µn2, µn3}, νn = max{νn1, νn2, νn3} and ψ(r) = max{ψi(r), for i = 1, 2, 3 and for r ≥ 0}, then we get that there exist nonnegative real sequences {µn} and {νn} with µn → 0 and νn → 0 as n →∞ and strictly increasing continuous function ψ : R+ → R+ with ψ(0) = 0 such that ‖Tn1 x−T n 1 y‖ ≤ ‖x−y‖ + µn1ψ1(‖x−y‖) + νn1 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, ‖Tn2 x−T n 2 y‖ ≤ ‖x−y‖ + µn2ψ2(‖x−y‖) + νn2 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, and ‖Tn3 x−T n 3 y‖ ≤ ‖x−y‖ + µn3ψ3(‖x−y‖) + νn3 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, for all x, y ∈ C and n ≥ 1. � 3. Strong Convergence Theorems In this section, we prove some strong convergence theorems for three total asymptotically nonex- pansive mappings in the framework of real Banach spaces. First, we shall need the following lemmas. Lemma 3.1. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. MODIFIED SP -ITERATION SCHEME 29 Then limn→∞‖xn −p‖ and limn→∞d(xn,F) both exist for all p ∈ F . Proof. Let p ∈ F. Then from (1.13), we have ‖zn −p‖ = ‖(1 −γn)xn + γnTn3 xn −p‖ ≤ (1 −γn)‖xn −p‖ + γn‖Tn3 xn −p‖ ≤ (1 −γn)‖xn −p‖ + γn[‖xn −p‖ +µnψ(‖xn −p‖) + νn] ≤ (1 −γn)‖xn −p‖ + γn[‖xn −p‖ +µnM‖xn −p‖ + νn] ≤ ‖xn −p‖ + µnM‖xn −p‖ + νn = (1 + µnM)‖xn −p‖ + νn.(3.1) Again from (1.13) and (3.1), we have ‖yn −p‖ = ‖(1 −βn)zn + βnTn2 zn −p‖ ≤ (1 −βn)‖zn −p‖ + βn‖Tn2 zn −p‖ ≤ (1 −βn)‖zn −p‖ + βn[‖zn −p‖ +µnψ(‖zn −p‖) + νn] ≤ (1 −βn)‖zn −p‖ + βn[‖zn −p‖ +µnM‖zn −p‖ + νn] ≤ ‖zn −p‖ + µnM‖zn −p‖ + νn = (1 + µnM)‖zn −p‖ + νn ≤ (1 + µnM)[(1 + µnM)‖xn −p‖ + νn] + νn ≤ (1 + µnM)2‖xn −p‖ + (2 + µnM)νn.(3.2) Finally, using (1.13) and (3.2), we have ‖xn+1 −p‖ = ‖(1 −αn)yn + αnTn1 yn −p‖ ≤ (1 −αn)‖yn −p‖ + αn‖Tn1 yn −p‖ ≤ (1 −αn)‖yn −p‖ + αn[‖yn −p‖ +µnψ(‖yn −p‖) + νn] ≤ (1 −αn)‖yn −p‖ + αn[‖yn −p‖ +µnM‖yn −p‖ + νn] ≤ ‖yn −p‖ + µnM‖yn −p‖ + νn = (1 + µnM)‖yn −p‖ + νn ≤ (1 + µnM)[(1 + µnM)2‖xn −p‖ +(2 + µnM)νn] + νn ≤ (1 + µnM)3‖xn −p‖ + (1 + µnM) × (2 + µnM)νn + νn ≤ (1 + µnQ1)‖xn −p‖ + νnQ2(3.3) for some Q1,Q2 > 0. For any p ∈ F, from (3.3), we obtain the following inequality d(xn+1,F) ≤ (1 + µnQ1)d(xn,F) + νnQ2.(3.4) Since ∑∞ n=1 µn < ∞ and ∑∞ n=1 νn < ∞, therefore applying Lemma 2.4(i) in (3.3) and (3.4), we have limn→∞‖xn −p‖ and limn→∞d(xn,F) both exist. This completes the proof. � Lemma 3.2. Let E be a uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let 30 SALUJA {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. Then limn→∞‖xn −Tixn‖ = 0 for i = 1, 2, 3. Proof. By Lemma 3.1, limn→∞‖xn−p‖ exists for all p ∈ F , so we can assume that limn→∞‖xn−p‖ = c. Then c > 0 otherwise there is nothing to prove. Now (3.1) and (3.2) implies that lim sup n→∞ ‖zn −p‖ ≤ c,(3.5) and lim sup n→∞ ‖yn −p‖ ≤ c.(3.6) Also ‖Tn1 yn −p‖ ≤ ‖yn −p‖ + µnψ(‖yn −p‖) + νn ≤ ‖yn −p‖ + µnM‖yn −p‖ + νn = (1 + µnM)‖yn −p‖ + νn, and so lim sup n→∞ ‖Tn1 yn −p‖ ≤ c.(3.7) Since c = ‖xn+1 −p‖ = ‖(1 −αn)(yn −p) + αn(Tn1 yn −p)‖. It follows from Lemma 2.5 that lim n→∞ ‖Tn1 yn −yn‖ = 0.(3.8) Again note that ‖Tn3 xn −p‖ ≤ ‖xn −p‖ + µnψ(‖xn −p‖) + νn ≤ ‖xn −p‖ + µnM‖xn −p‖ + νn = (1 + µnM)‖xn −p‖ + νn, ‖Tn2 zn −p‖ ≤ ‖zn −p‖ + µnψ(‖zn −p‖) + νn ≤ ‖zn −p‖ + µnM‖zn −p‖ + νn = (1 + µnM)‖zn −p‖ + νn. Hence, from above inequalities, we obtain lim sup n→∞ ‖Tn3 xn −p‖ ≤ c,(3.9) and lim sup n→∞ ‖Tn2 zn −p‖ ≤ c.(3.10) Further, note that ‖yn −p‖ ≤ ‖yn −Tn1 yn‖ + ‖T n 1 yn −p‖ ≤ ‖yn −Tn1 yn‖ + ‖yn −p‖ + µnψ(‖yn −p‖) + νn ≤ ‖yn −Tn1 yn‖ + ‖yn −p‖ + µnM‖yn −p‖ + νn ≤ ‖yn −Tn1 yn‖ + (1 + µnM)‖yn −p‖ + νn. MODIFIED SP -ITERATION SCHEME 31 It follows from (3.6) and (3.8) that c ≤ lim inf n→∞ ‖yn −p‖.(3.11) From (3.6) and (3.11), we get lim n→∞ ‖yn −p‖ = c.(3.12) Now, we have c = lim n→∞ ‖yn −p‖ = ‖(1 −βn)(zn −p) + βn(Tn2 zn −p)‖.(3.13) It follows from (3.5), (3.10) and Lemma 2.5 that lim n→∞ ‖Tn2 zn −zn‖ = 0.(3.14) Again note that ‖zn −p‖ ≤ ‖zn −Tn2 zn‖ + ‖T n 2 zn −p‖ ≤ ‖zn −Tn2 zn‖ + ‖zn −p‖ + µnψ(‖zn −p‖) + νn ≤ ‖zn −Tn2 zn‖ + ‖zn −p‖ + µnM‖zn −p‖ + νn ≤ ‖zn −Tn2 zn‖ + (1 + µnM)‖zn −p‖ + νn. It follows from (3.5) and (3.14) that c ≤ lim inf n→∞ ‖zn −p‖.(3.15) From (3.5) and (3.15), we get lim n→∞ ‖zn −p‖ = c.(3.16) Now, we see that c = lim n→∞ ‖zn −p‖ = ‖(1 −γn)(xn −p) + γn(Tn3 xn −p)‖.(3.17) It follows from Lemma 2.5 that lim n→∞ ‖Tn3 xn −xn‖ = 0.(3.18) Again note that ‖xn −zn‖ = γn‖xn −Tn3 xn‖ ≤ (1 −δ)‖xn −Tn3 xn‖.(3.19) Using (3.18) in (3.19), we get lim n→∞ ‖xn −zn‖ = 0.(3.20) Further, note that ‖xn −yn‖ = βn‖zn −Tn2 zn‖ ≤ (1 − δ)‖zn −Tn2 zn‖.(3.21) Using (3.14) in (3.21), we get lim n→∞ ‖xn −yn‖ = 0.(3.22) Note that ‖xn −Tn2 zn‖ ≤ ‖xn −zn‖ + ‖zn −T n 2 zn‖.(3.23) Using (3.14) and (3.20) in (3.23), we get lim n→∞ ‖xn −Tn2 zn‖ = 0.(3.24) 32 SALUJA Hence ‖xn −Tn2 xn‖ ≤ ‖xn −T n 2 zn‖ + ‖T n 2 zn −T n 2 xn‖ ≤ ‖xn −Tn2 zn‖ + ‖zn −xn‖ + µnψ(‖zn −xn‖) + νn ≤ ‖xn −Tn2 zn‖ + ‖zn −xn‖ + µnM‖zn −xn‖ + νn = ‖xn −Tn2 zn‖ + (1 + µnM)‖zn −xn‖ + νn.(3.25) Using (3.20) and (3.24) in (3.25), we get lim n→∞ ‖xn −Tn2 xn‖ = 0.(3.26) Again notice that ‖xn −Tn1 yn‖ ≤ ‖xn −yn‖ + ‖yn −T n 1 yn‖.(3.27) Using (3.8) and (3.22) in (3.27), we get lim n→∞ ‖xn −Tn1 yn‖ = 0.(3.28) Hence ‖xn −Tn1 xn‖ ≤ ‖xn −T n 1 yn‖ + ‖T n 1 xn −T n 1 yn‖ ≤ ‖xn −Tn1 yn‖ + ‖xn −yn‖ + µnψ(‖xn −yn‖) + νn ≤ ‖xn −Tn1 yn‖ + ‖xn −yn‖ + µnM‖xn −yn‖ + νn = ‖xn −Tn1 yn‖ + (1 + µnM)‖xn −yn‖ + νn.(3.29) Using (3.22) and (3.28) in (3.29), we get lim n→∞ ‖xn −Tn1 xn‖ = 0.(3.30) By the definitions of xn+1, we have ‖xn −xn+1‖ ≤ ‖xn −yn‖ + ‖Tn1 yn −yn‖.(3.31) Using (3.8) and (3.22) in (3.31), we get lim n→∞ ‖xn −xn+1‖ = 0.(3.32) By (3.30), (3.31) and uniform continuity of T1, we have ‖xn −T1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −Tn+11 xn+1‖ +‖Tn+11 xn+1 −T n+1 1 xn‖ + ‖T n+1 1 xn −T1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −Tn+11 xn+1‖ + ‖xn+1 −xn‖ +µn+1ψ(‖xn+1 −xn‖) + νn+1 + ‖Tn+11 xn −T1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −Tn+11 xn+1‖ + ‖xn+1 −xn‖ +µn+1M‖xn+1 −xn‖ + νn+1 + ‖Tn+11 xn −T1xn‖ = (2 + µn+1M)‖xn −xn+1‖ + ‖xn+1 −Tn+11 xn+1‖ +‖Tn+11 xn −T1xn‖ + νn+1 → 0 as n →∞.(3.33) Similarly, we can prove that ‖xn −T2xn‖ = 0 and ‖xn −T3xn‖ = 0.(3.34) This completes the proof. � Theorem 3.3. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) is closed. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. MODIFIED SP -ITERATION SCHEME 33 Then {xn} converges strongly to a common fixed point of the mappings T1, T2 and T3 if and only if lim infn→∞d(xn,F) = 0, where d(x,F) = inf{‖x−p‖ : p ∈ F}. Proof. The necessity is obvious. Indeed, if xn → q ∈ F as n →∞, then d(xn,F) = inf q∈F d(xn,q) ≤‖xn −q‖→ 0 (n →∞). This shows that lim infn→∞d(xn,F) = 0. Conversely, suppose that lim infn→∞d(xn,F) = 0. By Lemma 3.1, we have that limn→∞d(xn,F) exists. Further, by assumption lim infn→∞d(xn,F) = 0, from (3.4) and Lemma 2.4(ii), we conclude that limn→∞d(xn,F) = 0. Next, we show that {xn} is a Cauchy sequence. From (3.3), we know that ‖xn+1 −p‖ ≤ (1 + µnQ1)‖xn −p‖ + νnQ2 = (1 + dn)‖xn −p‖ + Q2νn,(3.35) where dn = Q1µn and for some Q1,Q2 > 0. Since ∑∞ n=1 µn < ∞, it follows that ∑∞ n=1 dn < ∞. Since 1 + x ≤ ex for all x ≥ 0, therefore from (3.35), we have ‖xn+m −p‖ ≤ (1 + dn+m−1)‖xn+m−1 −p‖ + Q2νn+m−1 ≤ edn+m−1‖xn+m−1 −p‖ + Q2νn+m−1 ≤ e[dn+m−1+dn+m−2]‖xn+m−2 −p‖ + edn+m−1Q2νn+m−2 +Q2νn+m−1 ≤ e[dn+m−1+dn+m−2]‖xn+m−2 −p‖ + edn+m−1Q2[νn+m−2 +νn+m−1] ... ≤ ( e ∑n+m−1 j=n dj ) ‖xn −p‖ + ( e ∑n+m−1 j=n dj ) Q2 n+m−1∑ j=n νj ≤ ( e ∑∞ j=1 dj ) ‖xn −p‖ + ( e ∑∞ j=1 dj ) Q2 n+m−1∑ j=n νj ≤ Q3 ‖xn −p‖ + Q2Q3 n+m−1∑ j=n νj(3.36) for all natural numbers m,n, where Q3 = e ∑∞ j=1 dj < ∞. Now, given ε > 0, since limn→∞d(xn,F) = 0 and ∑∞ n=1 νn < ∞, there exists a natural number n1 > 0 such that for all n ≥ n1, d(xn,F) < ε8Q3 and ∑∞ j=1 νj < ε 4Q2Q3 . So, we get d(xn1,F) < ε 4Q3 and ∑∞ j=n1 νj < ε 4Q2Q3 . This means that there exists a p1 ∈ F such that ‖xn1 −p1‖ ≤ ε 4Q3 . Hence, 34 SALUJA for all integers n ≥ n1 and m ≥ 1, we obtain from (3.36) that ‖xn+m −xn‖ ≤ ‖xn+m −p1‖ + ‖xn −p1‖ ≤ Q3 ‖xn1 −p1‖ + Q2Q3 n+m−1∑ j=n1 νj +Q3 ‖xn1 −p1‖ + Q2Q3 n+m−1∑ j=n1 νj = 2 ( Q3 ‖xn1 −p1‖ + Q2Q3 n+m−1∑ j=n1 νj ) ≤ 2 ( Q3 ‖xn1 −p1‖ + Q2Q3 ∞∑ j=n1 νj ) < 2 ( Q3. ε 4Q3 + Q2Q3. ε 4Q2Q3 ) = ε. This proves that {xn} is a Cauchy sequence in C. Thus, the completeness of E implies that {xn} must be convergent. Assume that limn→∞xn = z. We will prove that z is a common fixed point of T1, T2 and T3, that is, we will show that z ∈ F = ∩3i=1F(Ti). Since C is closed, therefore z ∈ C. Next, we show that z ∈ F. Now limn→∞d(xn,F) = 0 gives that d(z,F) = 0. Since F is closed, z ∈ F. Thus, z is a common fixed point of the mappings T1, T2 and T3. This completes the proof. � We deduce the following result as corollary from Theorem 3.3 as follows. Corollary 3.4. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) is closed. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. Then {xn} converges strongly to a point p ∈ F if and only if there exists some subsequence {xnj} of {xn} which converges to p ∈ F . Theorem 3.5. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. Then lim infn→∞d(xn,F) = lim supn→∞d(xn,F) = 0 if {xn} converges to a unique point in F . Proof. Let p ∈ F. Since {xn} converges to p, limn→∞d(xn,p) = 0. So, for a given ε > 0, there exists n1 ∈ N such that d(xn,p) < ε for all n ≥ n1. Taking the infimum over p ∈ F(S,T), we obtain that d(xn,F) < ε for all n ≥ n1. This means that limn→∞d(xn,F) = 0. Thus we obtain that lim inf n→∞ d(xn,F) = lim sup n→∞ d(xn,F) = 0. This completes the proof. � As an application of Theorem 3.3, we establish some strong convergence results as follows. MODIFIED SP -ITERATION SCHEME 35 Theorem 3.6. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. If one of the mappings in {Ti : i = 1, 2, 3} is demicompact, then {xn} converges strongly to a common fixed point of the mappings T1, T2 and T3. Proof. Without loss of generality, we can assume that T1 is demicompact. It follows from (3.33) in Lemma 3.2 that limn→∞‖xn − T1xn‖ = 0 and {xn} is bounded, by demicompactness of T1, there exists a subsequence {xnk} of {xn} that converges strongly to some q ∈ C as k →∞. From (3.33) in Lemma 3.2 we have lim k→∞ ‖xnk −T1xnk‖ = ‖q −T1q‖ = 0. This implies that q ∈ F(T1). Similarly, we can prove that q ∈ F(T2) and q ∈ F(T3). Thus, we obtain that q ∈ F = ∩3i=1F(Ti). It follows from Lemma 3.1 and Theorem 3.3 that {xn} must converges strongly to a common fixed point of the mappings T1, T2 and T3. This completes the proof. � Theorem 3.7. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. If T1, T2 and T3 satisfy condition (B), then {xn} converges strongly to a common fixed point of the mappings T1, T2 and T3. Proof. By Lemma 3.2, we know that lim n→∞ ‖xn −Tixn‖ = 0, for i = 1, 2, 3.(3.37) From condition (B) and (3.37), we get f(d(xn,F) ≤ a1.‖xn −T1xn‖ + a2.‖xn −T2xn‖ + a3.‖xn −T3xn‖ = 0, that is, f(d(xn,F) = 0. Since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f(0) = 0, f(t) > 0 for all t ∈ (0,∞), therefore we obtain lim n→∞ d(xn,F) = 0. Now all the conditions of Theorem 3.3 are satisfied, therefore by its conclusion {xn} converges strongly to a common fixed point of the mappings T1, T2 and T3. This completes the proof. � 4. Weak Convergence Theorems In this section, we prove some weak convergence theorems of iteration scheme (1.13) for three total asymptotically nonexpansive mappings in a uniformly convex Banach space such that either it satisfies the Opial property or its dual space has the Kadec-Klee property (KK-property). Theorem 4.1. Let E be a uniformly convex Banach space satisfying Opial’s condition and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1−δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. If the mappings I −Ti for all i = 1, 2, 3, where I denotes the identity mapping, are demiclosed at zero, then {xn} converges weakly to a common fixed point of the mappings T1, T2 and T3. 36 SALUJA Proof. Let q ∈ F, from Lemma 3.1 the sequence {‖xn −q‖} is convergent and hence bounded. Since E is uniformly convex, every bounded subset of E is weakly compact. Thus there exists a subsequence {xnk}⊂{xn} such that {xnk} converges weakly to q ∗ ∈ C. From Lemma 3.2, we have lim k→∞ ‖xnk −T1xnk‖ = 0, lim k→∞ ‖xnk −T2xnk‖ = 0, lim k→∞ ‖xnk −T3xnk‖ = 0. Since the mappings I − Ti for all i = 1, 2, 3 are demiclosed at zero, therefore Tiq∗ = q∗ for all i = 1, 2, 3, which means q∗ ∈ F . Finally, let us prove that {xn} converges weakly to q∗. Suppose on contrary that there is a subsequence {xnj} ⊂ {xn} such that {xnj} converges weakly to p∗ ∈ C and q∗ 6= p∗. Then by the same method as given above, we can also prove that p∗ ∈ F . From Lemma 3.1 the limits limn→∞‖xn − q∗‖ and limn→∞‖xn −p∗‖ exist. By virtue of the Opial condition of E, we obtain lim n→∞ ‖xn −q∗‖ = lim nk→∞ ‖xnk −q ∗‖ < lim nk→∞ ‖xnk −p ∗‖ = lim n→∞ ‖xn −p∗‖ = lim nj→∞ ‖xnj −p ∗‖ < lim nj→∞ ‖xnj −q ∗‖ = lim n→∞ ‖xn −q∗‖ which is a contradiction, so q∗ = p∗. Thus {xn} converges weakly to a common fixed point of the mappings T1, T2 and T3. This completes the proof. � Lemma 4.2. Under the conditions of Lemma 3.2 and for any p, q ∈ F , limn→∞‖txn + (1 − t)p−q‖ exists for all t ∈ [0, 1]. Proof. By Lemma 3.1, limn→∞‖xn −z‖ exists for all z ∈ F and therefore {xn} is bounded. Letting an(t) = ‖txn + (1 − t)p−q‖ for all t ∈ [0, 1]. Then limn→∞an(0) = ‖p− q‖ and limn→∞an(1) = ‖xn − q‖ exists by Lemma 3.1. It, therefore, remains to prove the Lemma 4.2 for t ∈ (0, 1). For all x ∈ C, we define the mapping Wn : C → C by: Un(x) = (1 −γn)x + γnTn3 x Vn(x) = (1 −βn)Un(x) + βnTn2 Un(x) and Wn(x) = (1 −αn)Vn(x) + αnTn1 Vn(x)). Then it follows that zn = Unxn, yn = Vnxn, xn+1 = Wnxn and Wnp = p for all p ∈ F. Now from (3.1), (3.2) and (3.3) of Lemma 3.1, we see that ‖Un(x) −Un(y)‖ ≤ (1 + µnM)‖x−y‖ + νn ‖Vn(x) −Vn(y)‖ ≤ (1 + µnM)2‖x−y‖ + (2 + µnM)νn and ‖Wn(x) −Wn(y)‖ ≤ (1 + µnQ1)‖x−y‖ + Q2νn = Kn ‖x−y‖ + Q2νn,(4.1) for some Q1,Q2 > 0 and for all x,y ∈ C, where Kn = 1 + µnQ1 with ∑∞ n=1 νn < ∞ and Kn → 1 as n →∞. Setting Hn, m = Wn+m−1Wn+m−2 . . .Wn, m ≥ 1(4.2) MODIFIED SP -ITERATION SCHEME 37 and bn, m = ‖Hn, m(txn + (1 − t)p) − (tHn, mxn + (1 − t)Hn,mq)‖. From (4.1) and (4.2), we have ‖Hn, m(x) −Hn, m(y)‖ = ‖Wn+m−1Wn+m−2 . . .Wn(x) −Wn+m−1Wn+m−2 . . .Wn(y)‖ ≤ Kn+m−1‖Wn+m−2 . . .Wn(x) −Wn+m−2 . . .Wn(y)‖ +Q2νn+m−1 ≤ Kn+m−1Kn+m−2‖Wn+m−3 . . .Wn(x) −Wn+m−3 . . .Wn(y)‖ +Q2νn+m−1 + Q2νn+m−2 ... ≤ (n+m−1∏ j=n Kj ) ‖x−y‖ + Q2 n+m−1∑ j=n νj = Mn‖x−y‖ + Q2 n+m−1∑ j=n νj(4.3) for all x,y ∈ C, where Mn = ∏n+m−1 j=n Kj and Hn, mxn = xn+m, Hn, mp = p for all p ∈ F. Thus an+m(t) = ‖txn+m + (1 − t)p−q‖ ≤ bn, m + ‖Hn, m(txn + (1 − t)p) −q‖ ≤ bn, m + Mnan(t) + Q2 n+m−1∑ j=n νj ≤ bn, m + Mnan(t) + Q2 ∞∑ j=1 νj.(4.4) By using [ [5], Theorem 2.3], we have bn,m ≤ ϕ−1(‖xn −u‖−‖Hn,mxn −Hn,mu‖) ≤ ϕ−1(‖xn −u‖−‖xn+m −u + u−Hn,mu‖) ≤ ϕ−1(‖xn −u‖− (‖xn+m −u‖−‖Hn,mu−u‖)) and so the sequence {bn,m} converges uniformly to 0, i.e., bn,m → 0 as n →∞. Since limn→∞Mn = 1, Q2 > 0 and νj → 0 as j →∞, therefore from (4.4), we have lim sup n→∞ an(t) ≤ lim n,m→∞ bn,m + lim inf n→∞ an(t) + 0 = lim inf n→∞ an(t). This shows that limn→∞an(t) exists, that is, limn→∞‖txn + (1− t)p−q‖ exists for all t ∈ [0, 1]. This completes the proof. � Theorem 4.3. Let E be a real uniformly convex Banach space such that its dual E∗ has the Kadec-Klee property and C be a nonempty closed convex subset of E. Let T1, T2, T3 : C → C be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and F = ∩3i=1F(Ti) 6= ∅. Let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1−δ] for all n ∈ N and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant M > 0 such that ψ(t) ≤ M t, t ≥ 0. If the mappings I −Ti for all i = 1, 2, 3, where I denotes the identity mapping, are demiclosed at zero, then {xn} converges weakly to a common fixed point of the mappings T1, T2 and T3. Proof. By Lemma 3.1, {xn} is bounded and since E is reflexive, there exists a subsequence {xnj} of {xn} which converges weakly to some p ∈ C. By Lemma 3.2, we have lim j→∞ ‖xnj −Tixnj‖ = 0 for all i = 1, 2, 3. 38 SALUJA Since by hypothesis the mappings I −Ti for all i = 1, 2, 3 are demiclosed at zero, therefore Tip = p for all i = 1, 2, 3, which means p ∈ F. Now, we show that {xn} converges weakly to p. Suppose {xni} is another subsequence of {xn} converges weakly to some q ∈ C. By the same method as above, we have q ∈ F and p, q ∈ ww(xn). By Lemma 4.2, the limit lim n→∞ ‖txn + (1 − t)p−q‖ exists for all t ∈ [0, 1] and so p = q by Lemma 2.6. Thus, the sequence {xn} converges weakly to p ∈ F. This completes the proof. � Example 4.4. Let E be the real line with the usual norm |.|, C = [0,∞). Assume that T1(x) = x, T2(x) = x 3 and T3(x) = sin x for all x ∈ C. Let φ be the strictly increasing continuous function such that φ: R+ → R+ with φ(0) = 0. Let {µn}n≥1 and {νn}n≥1 be two nonnegative real sequences defined by µn = 1 n2 and νn = 1 n3 for all n ≥ 1 with µn → 0 and νn → 0 as n → ∞. Then T1, T2 and T3 are total asymptotically nonexpansive mappings with common fixed point 0, that is, F = F(T1) ∩F(T2) ∩T3 = {0}. 5. Conclusion In this paper, we establish some weak and strong convergence theorems for modified SP iteration scheme for three total asymptotically nonexpansive mappings in the framework of real Banach spaces. The results presented in this paper extend and generalize several results from the current existing literature to the case of more general class of mappings, spaces and iteration schemes considered in this paper. References [1] R. P. Agarwal, Donal O’Regan, D. R. 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Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178(1993), 301–308. [21] B. L. Xu, M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267(2002), 444–453. Department of Mathematics, Govt. Nagarjuna P.G. College of Science, Raipur - 492010 (C.G.), India ∗Corresponding author: saluja1963@gmail.com