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APPROXIMATING FIXED POINTS OF TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS YA.I.ALBER,C.E.CHIDUME,ANDH.ZEGEYE Received 10 March 2005; Revised 7 August 2005; Accepted 28 August 2005 We introduce a new class of asymptotically nonexpansive mappings and study approxi- mating methods for finding their fixed points. We deal with the Krasnosel’skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings. Copyright © 2006 Ya. I. Alber et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let K be a nonempty subset of a real linear nor med space E.LetT be a self-mapping of K.ThenT : K → K is said to be nonexpansive if Tx− Ty≤x − y, ∀x, y ∈ K. (1.1) T is said to be asymptotically nonexpansive if there exists a sequence {k n }⊂[1,∞)with k n → 1asn →∞such that for all x, y ∈ K the following inequality holds:   T n x −T n y   ≤ k n x − y, ∀n ≥ 1. (1.2) The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [18] as a generalization of the class of nonexpansive maps. They proved that if K is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self-mapping of K,thenT has a fixed point. Alber and Guerre-Delabriere have studied in [3–5] weakly contractive mappings of the class C ψ . Definit ion 1.1. An operator T is called weakly contractive of the class C ψ on a closed convex set K of the normed space E if there exists a continuous and increasing function ψ(t)definedonR + such that ψ is positive on R + \{0}, ψ(0) = 0, lim t→+∞ ψ(t) =∞and Hindaw i Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 10673, Pages 1–20 DOI 10.1155/FPTA/2006/10673 2 Total asymptotically nonexpansive mappings for all x, y ∈ K, Tx− Ty≤x − y−ψ   x − y  . (1.3) The class C ψ of weakly contractive maps contains the class of strongly contractive maps and it i s contained in the class of nonexpansive maps. In [3–5], in fact, there is also the concept of the asymptotically weakly contractive mappings of the class C ψ . Definit ion 1.2. The operator T is called asymptotically weakly contractive of the class C ψ if there exists a sequence {k n }⊂[1,∞)withk n → 1asn →∞and strictly increasing function ψ : R + → R + with ψ(0) = 0 such that for all x, y ∈ K, the following inequality holds:   T n x − T n y   ≤ k n x − y−ψ   x − y  , ∀n ≥ 1. (1.4) Bruck et al. have introduced in [11] asy mptotically nonexpansive in the intermediate sense mappings. Definit ion 1.3. An operator T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: limsup n→∞ sup x,y∈K    T n x − T n y   − x − y  ≤ 0. (1.5) Observe that if a n := sup x,y∈K    T n x − T n y   − x − y  , (1.6) then (1.5) reduces to the relation   T n x − T n y   ≤ x − y + a n , ∀x, y ∈ K. (1.7) It is known [23]thatifK is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self-mapping of K which is asymptotically nonexpan- sive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense con- tains properly the class of asymptotically nonexpansive maps (see, e.g., [22]). Iterative techniques are the main tool for approximating fixed points of nonexpansive mappings and asymptotically nonexpansive mappings, and it has been studied by various authors using Krasnosel’skii-Mann and Ishikawa schemes (see, e.g., [12, 13, 15, 20, 21, 25, 27–37]). Bose in [10]provedthatifK is a nonempty closed convex bounded subset of a uni- formly convex Banach space E satisfying Opial’s condition [26]andT : K → K is an asymptotically nonexpansive mapping, then the sequence {T n x} converges weakly to a fixed point of T provided T is asymptotically regular at x ∈ K, that is, the limit equality lim n→∞   T n x − T n+1 x   = 0 (1.8) Ya . I. Alb er et a l . 3 holds. Passty [28]andalsoXu[38] showed that the requirement of the Opial’s condition can be replaced by the Fr ´ echet differentiability of the space norm. Furthermore, Tan and Xu established in [34, 35] that the asymptotic regularity of T at a point x can be weakened to the so-called weakly asymptotic regularity of T at x, defined as follows: ω − lim n→∞  T n x − T n+1 x  = 0. (1.9) In [31, 32], Schu introduced a modified Krasnosel’skii-Mann process to approximate fixed points of asymptotically nonexpansive self-maps defined on nonempty closed con- vex and bounded subsets of a uniformly convex Banach space E.Inparticular,heproved that the iterative sequence {x n } generated by the algorithm x n+1 =  1 − α n  x n + α n T n x n , n ≥ 1, (1.10) converges weakly to some fixed point of T if the Opial’s condition holds, {k n } n≥1 ⊂ [1,∞) for all n ≥ 1, limk n = 1,  ∞ n=1 (k 2 n − 1) < ∞, {α n } n≥1 is a real sequence satisfying the in- equalities 0 < ¯ α ≤ α n ≤ α<1, n ≥ 1, for some positive constants ¯ α and α.However,Schu’s result does not apply, for instance, to L p spaces with p = 2 because none of these spaces satisfy the Opial’s condition. In [30], Rhoades obtained strong convergence theorem for asymptotically nonexpan- sive mappings in uniformly convex Banach spaces using a modified Ishikawa iteration method. Osilike and Aniagbosor proved in [27] that the results of [30–32]stillremain true without the boundedness requirement imposed on K,providedthatᏺ(T) ={x ∈ K : Tx = x} =∅.In[37], Tan and Xu extended Schu’s theorem [32]touniformlyconvex spaces with a Fr ´ echet differentiable norm. Therefore, their result covers L p spaces with 1 <p< ∞. Chang et al. [12] established convergence theorems for asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces without assuming any of the following properties: (i) E satisfies the Opial’s condition; (ii) T is asymptotically regular or weakly asymptotically regular; (iii) K is bounded. Their results improve and generalize the corresponding results of [10, 19, 28, 29, 32, 34, 35, 37, 38] and others. Recently, Kim and Kim [22] studied the strong convergence of the Krasnosel’skii- Mann and Ishikawa iterations with errors for asymptotically nonexpansive in the inter- mediate sense oper ators in Banach spaces. In all the above papers, the operator T remains a self-mapping of nonempty closed convex subset K in a uniformly convex Banach space. If, however, domain D(T)ofT is apropersubsetofE (and this is indeed the case for several applications), and T maps D( T)intoE, then the Krasnosel’skii-Mann and Ishikawa iterative processes and Schu’s modifications of type (1.10) may fail to be well-defined. More recently, C hidume et al. [14] proved the convergence theorems for asymptot- ically nonexpansive nonself-mappings in Banach spaces by having extended the corre- sponding results of [12, 27, 30]. The purpose of this paper is to introduce more general classes of asymptotically non- expansive mappings and to study approximating methods for finding their fixed points. 4 Total asymptotically nonexpansive mappings We deal with self- and nonself-mappings and the Krasnosel’skii-Mann-type iterative pro- cess (1.10). The Ishikawa iteration scheme is beyond the scope of this paper. Definit ion 1.4. AmappingT : E → E is called total asymptotically nonexpansive if there exist nonnegative real sequences {k (1) n } and {k (2) n }, n ≥ 1, with k (1) n ,k (2) n → 0asn →∞,and strictly increasing and continuous functions φ : R + → R + with φ(0) = 0suchthat   T n x − T n y   ≤ x − y + k (1) n φ   x − y  + k (2) n . (1.11) Remark 1.5. If φ(λ) = λ,then(1.11) takes the form   T n x − T n y   ≤  1+k (1) n   x − y + k (2) n . (1.12) In addition, if k (2) n = 0foralln ≥ 1, then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mapping s. If k (1) n = 0andk (2) n = 0foralln ≥ 1, then we obtain from (1.11) the class of nonexpansive mappings. Definit ion 1.6. AmappingT is called total asymptotically weakly contractive if there exist nonnegative real sequences {k (1) n } and {k (2) n }, n ≥ 1, with k (1) n ,k (2) n → 0asn →∞,and strictly increasing and continuous functions φ,ψ : R + → R + with φ(0) = ψ(0) = 0such that   T n x − T n y   ≤ x − y + k (1) n φ   x − y  − ψ   x − y  + k (2) n . (1.13) Remark 1.7. If φ(λ) = λ,then(1.13)acceptstheform   T n x − T n y   ≤  1+k (1) n   x − y−ψ   x − y  + k (2) n . (1.14) In addition, if k (2) n = 0foralln ≥ 1, then total asymptotically weakly contractive mapping coincides with the earlier known asymptotically weakly contractive mapping. If k (2) n = 0 and k (1) n = 0, then we obtain from (1.13)theclassofweaklycontractivemappings.If k (1) n ≡ 0andk (2) n ≡ a n ,wherea n := sup x,y∈K (T n x − T n y−x − y)foralln ≥ 0, then (1.13)reducesto(1.7) which has been studied as asymptotically nonexpansive mappings in the intermediate sense. The paper is organized in the following manner. In Section 2, we present characteris- tic inequalities from the standpoint of their being an important component of common theory of Banach space geometry. Section 3 is dedicated to numerical recurrent inequal- ities that are a cr ucial tool in the investigation of convergence and stability of iterative methods. In Section 4, we study the convergence of the iterative process (1.10)withto- tal asymptotically weakly contractive mappings. The next two sections deal with total asymptotically nonexpansive mappings. 2. Banach space geometry and characteristic inequalities Let E be a real uniformly convex and uniformly smooth Banach space (it is a reflexive space), and let E ∗ be a dual space with the bilinear functional of duality φ,x between Ya . I. Alb er et a l . 5 φ ∈ E ∗ and x ∈ E. We denote the norms of elements in E and E ∗ by · and · ∗ , respectively. A uniform convexity of the Banach space E means that for any given ε>0 there exists δ>0suchthatforallx, y ∈ E, x≤1, y≤1, x − y=ε the inequality x + y≤2(1 − δ) (2.1) is satisfied. The function δ E (ε) = inf  1 − 2 −1 x + y, x=1, y=1, x − y=ε  (2.2) is called to be modulus of convexity of E. A uniform smoothness of the Banach space E means that for any given ε>0there exists δ>0 such that for all x, y ∈ E, x=1, y≤δ the inequality 2 −1   x + y + x − y  − 1 ≤ εy (2.3) holds. The function ρ E (τ) = sup  2 −1   x + y + x − y  − 1, x=1, y=τ  (2.4) is called to be modulus of smoothness of E. The moduli of convexity and smoothness are the basic quantitative characteristics of a Banach space that describe its geometric properties [2, 16, 17, 24]. Let us observe that the space E is uniformly convex if and only if δ E (ε) > 0forallε>0 and it is uniformly smooth if and only if lim τ→0 τ −1 ρ E (τ) = 0. The following properties of the functions δ E (ε)andρ E (τ)areimportanttokeepin mind throughout of this paper: (i) δ E (ε) is defined on the interval [0,2], continuous and increasing on this interval, δ E (0) = 0, (ii) 0 <δ E (ε) < 1if0<ε<2, (iii) ρ E (τ) is defined on the interval [0,∞), convex, continuous and increasing on this interval, ρ E (0) = 0, (iv) the function g E (ε) = ε −1 δ E (ε) is continuous and non-decreasing on the interval [0,2], g E (0) = 0, (v) the function h E (τ) = τ −1 ρ E (τ) is continuous and non-decreasing on the interval [0, ∞), h E (0) = 0, (vi) ε 2 δ E (η) ≥ (4L) −1 η 2 δ E (ε)ifη ≥ ε>0andτ 2 ρ E (σ) ≤ Lσ 2 ρ E (τ)ifσ ≥ τ>0. Here 1 <L<1.7 is the Figiel constant. We recall that nonlinear in general operator J : E → E ∗ is called normalized duality mapping if Jx ∗ =x, Jx,x=x 2 . (2.5) It is obvious that this operator is coercive because of Jx,x x −→ ∞ as x−→∞ (2.6) 6 Total asymptotically nonexpansive mappings and monotone due to Jx− Jy,x − y≥  x−y  2 . (2.7) In addition, Jx− Jy,x − y≤  x + y  2 . (2.8) A normalized duality mapping J ∗ : E ∗ → E can be introduced by analogy. The properties of the operators J and J ∗ have been given in detail in [2]. Let us present the estimates of the normalized duality mappings used in the sequel (see [2]). Let x, y ∈ E. We denote R 1 = R 1   x, y  =  2 −1   x 2 + y 2  . (2.9) Lemma 2.1. In a uniformly convex Banach space E Jx− Jy,x − y≥2R 2 1 δ E   x − y/2R 1  . (2.10) If x≤R and y≤R, then Jx− Jy,x − y≥(2L) −1 R 2 δ E   x − y/2R  . (2.11) Lemma 2.2. In a uniformly smooth Banach space E Jx− Jy,x − y≤2R 2 1 ρ E  4x − y/R 1  . (2.12) If x≤R and y≤R, then Jx− Jy,x − y≤2LR 2 ρ E  4x − y/R  . (2.13) Next we present the upper and lower chara cteristic inequalities in E (see [2]). Lemma 2.3. Let E be uniformly convex Banach space. Then for all x, y ∈ E and for all 0 ≤ λ ≤ 1   λx +(1− λ)y   2 ≤ λx 2 +(1− λ)y 2 − 2λ(1 − λ)R 2 1 δ E   x − y/2R 1  . (2.14) If x≤R and y≤R, then   λx +(1− λ)y   2 ≤ λx 2 +(1− λ)y 2 − L −1 λ(1 − λ) R 2 δ E   x − y/2R  . (2.15) Lemma 2.4. Let E be uniformly smooth Banach space. Then for all x, y ∈ E and for all 0 ≤ λ ≤ 1   λx +(1− λ)y   2 ≥ λx 2 +(1− λ)y 2 − 8λ(1 − λ)R 2 1 ρ E  4x − y/R 1  . (2.16) Ya . I. Alb er et a l . 7 If x≤R and y≤R, then   λx +(1− λ)y   2 ≥ λx 2 +(1− λ)y 2 − 16Lλ(1 − λ)R 2 ρ E  4x − y/R  . (2.17) 3. Recur rent numerical inequalities Lemma 3.1 (see, e.g., [7]). Let {λ n } n≥1 , {κ n } n≥1 and {γ n } n≥1 be sequences of nonnegative real numbers such that for all n ≥ 1 λ n+1 ≤ (1 + κ n )λ n + γ n . (3.1) Let  ∞ 1 κ n < ∞ and  ∞ 1 γ n < ∞. Then lim n→∞ λ n exists. Lemma 3.2 [1, 8]. Let {λ k } and {γ k } be sequences of nonnegative numbers and {α k } be a sequence of positive numbers satisfying the conditions ∞  1 α n =∞,lim n→∞ γ n α n −→ 0. (3.2) Let the recursive inequality λ n+1 ≤ λ n − α n ψ  λ n  + γ n , n = 1,2, , (3.3) be given, where ψ(λ) is a continuous and nondecreasing function from R + to R + such that it is positive on R + \{0}, φ(0) = 0, lim t→∞ ψ(t) > 0. Then λ n → 0 as n →∞. We present more general statement. Lemma 3.3. Let {λ k }, {κ n } n≥1 and {γ k } be sequences of nonnegat ive numbers and {α k } be a sequence of positive numbers satisfying the conditions ∞  1 α n =∞, ∞  1 κ n < ∞, γ n α n −→ 0 as n −→ ∞ . (3.4) Let the recursive inequality λ n+1 ≤  1+κ n  λ n − α n ψ  λ n  + γ n , n = 1,2, , (3.5) be given, where ψ(λ) is the same as in Lemma 3.2. Then λ n → 0 as n →∞. Proof. We produce in ( 3.5) the following replacement: λ n = μ n Π n−1 j =1  1+κ n  . (3.6) Then μ n+1 ≤ μ n − α n  Π n−1 j =1  1+κ n   −1 ψ  μ n Π n−1 j =1  1+κ n   +  Π n−1 j =1  1+κ n   −1 γ n . (3.7) Since  ∞ 1 κ n < ∞, we conclude that there exists a constant C>0suchthat 1 ≤ Π n−1 j =1  1+κ n  ≤ C. (3.8) 8 Total asymptotically nonexpansive mappings Therefore, taking into account nondecreasing property of ψ,wehave μ n+1 ≤ μ n − α n C −1 ψ  μ n  + γ n . (3.9) Consequently, by Lemma 3.2, μ n → 0asn →∞and this implies lim n→∞ λ n = 0.  Lemma 3.4. Let {λ n } n≥1 , {κ n } n≥1 and {γ n } n≥1 be nonnegative, {α n } n≥1 be positive real numbers such that λ n+1 ≤ λ n + κ n φ  λ n  − α n ψ  λ n  + γ n , ∀n ≥ 1, (3.10) where φ,ψ : R + → R + are strictly increasing and continuous functions such that φ(0) = ψ(0) = 0.Letforalln>1 γ n α n ≤ c 1 , κ n α n ≤ c 2 , α n ≤ α<∞, (3.11) where 0 ≤ c 1 , c 2 < ∞. Assume that the equation ψ(λ) = c 1 + c 2 φ(λ) has the unique root λ ∗ on the interval (0,∞) and lim λ→∞ ψ(λ) φ(λ) >c 2 . (3.12) Then λ n ≤ max{λ 1 ,K ∗ },whereK ∗ = λ ∗ + α(c 1 + c 2 φ(λ ∗ )). In addition, if ∞  1 α n =∞, γ n + κ n α n −→ 0, (3.13) then λ n → 0 as n →∞. Proof. For each n ∈ I ={1,2, }, just one alternative can happen: either H 1 : κ n φ  λ n  − α n ψ  λ n  + γ n > 0, (3.14) or H 2 : κ n φ  λ n  − α n ψ  λ n  + γ n ≤ 0. (3.15) Denote I 1 ={n ∈ I | H 1 is tr ue} and I 2 ={n ∈ I | H 2 is tr ue}. It is clear that I 1 ∪ I 2 = I. (i) Let c 1 > 0. Since ψ(0) = 0, we see that hypothesis H 1 is valid on the interval (0,λ ∗ ) and H 2 is valid on [λ ∗ ,∞). Therefore, the following result is obtained: λ n ≤ λ ∗ , ∀n ∈ I 1 ={1,2, ,N}, λ N+1 ≤ λ N + γ N + κ N φ  λ N  ≤ λ ∗ + γ N + κ N φ(λ ∗ ) ≤ K ∗ , λ n ≤ λ N+1 ≤ K ∗ , ∀n ≥ N +2. (3.16) Thus, λ n ≤ K ∗ for all n ≥ 1. Ya . I. Alb er et a l . 9 (ii) Let c 1 = 0. This takes place if γ n = 0foralln>1. In this case, along with situ- ation described above it is possible I 2 = I and then λ n <λ 1 for all n ≥ 1. Hence, λ n ≤ max{λ 1 ,K ∗ }= ¯ C. The second assertion follows from Lemma 3.2 because λ n+1 ≤ λ n − α n ψ  λ n  + κ n φ( ¯ C)+γ n , n = 1,2, (3.17)  Lemma 3.5. Suppose that the conditions of the previous lemma are fulfilled with positive κ n for n ≥ 1, 0 <c 1 < ∞, and the equation ψ(λ) = c 1 + c 2 φ(λ) has a finite number of solutions λ (1) ∗ ,λ (2) ∗ , ,λ (l) ∗ , l ≥ 1. Then there exists a constant ¯ C>0 such that all the conclusions of Lemma 3.4 hold. Proof. It is sufficiently to consider the following two cases. (i) If there is no points of contact among λ (l) ∗ , i = 1,2, ,l,then I = I (1) 1 ∪ I (1) 2 ∪ I (2) 1 ∪ I (2) 2 ∪ I (3) 1 ∪ I (3) 2 ∪···∪I (l) 1 ∪ I (l) 2 , (3.18) where I (k) 1 ⊂ I 1 and I (k) 2 ⊂ I 2 , k = 1,2, ,l. It is not difficult to see that λ n ≤ λ ∗ on the interval I (1) 1 . Denote N (1) 1 = max{n | n ∈ I (1) 1 }.ThenN (1) 1 +1= min{n | n ∈ I (1) 2 } and this yields the inequality λ N (1) 1 +1 ≤ λ N (1) 1 + γ N (1) 1 + κ N (1) 1 φ  λ N (1) 1  ≤ λ ∗ + γ N (1) 1 + κ N (1) 1 φ(λ ∗ ) ≤ K ∗ . (3.19) By the hypothesis H 2 , for the rest n ∈ I (1) 2 ,wehaveλ n ≤ λ N (1) 1 +1 ≤ K ∗ . The same situation arrises on the intervals I (2) 1 ∪ I (2) 2 , I (3) 1 ∪ I (3) 1 , and so forth. Thus, λ n ≤ K ∗ for all n ∈ I. (ii) If some λ (i) ∗ is a point of contact, then either I i ⊂ I 2 and I i+1 ⊂ I 2 or I i ⊂ I 1 and I i+1 ⊂ I 1 . We presume, respectively, I i ∪ I i+1 ⊂ I 2 and I i ∪ I i+1 ⊂ I 1 and after this number intervals again. It is easy to verify that the proof coincides with the case (i).  Remark 3.6. Lemma 3.4 remains still valid if the equation ψ(λ) = c 1 + c 2 φ(λ) has a mani- fold of solutions on the interval (0, ∞). Lemma 3.7 (see [6]). Let {μ n }, {α n }, {β n } and {γ n } be sequences of non-negative real numbers satisfy ing the recurrence inequality μ n+1 ≤ μ n − α n β n + γ n . (3.20) Assume that ∞  n=1 α n =∞, ∞  n=1 γ n < ∞. (3.21) Then (i) there exists an infinite subsequence {β  n }⊂{β n } such that β  n ≤ 1   n j=1 α j , (3.22) and, consequently, lim n→∞ β  n = 0; 10 Total asymptotically nonexpansive mappings (ii) if lim n→∞ α n = 0 and there exists a constant κ>0 such that   β n+1 − β n   ≤ κα n (3.23) for all n ≥ 1, then lim n→∞ β n = 0. 4. Convergence analysis of the iterations (1.10) with total asymptotically weakly contractive mappings In this section, we are going to prove the strong convergence of approximations generated by the iterative process (1.10) to fixed points of the total asymptotically weakly contractive mappings T : K → K,whereK ⊆ E is a nonempty closed convex subset. In the sequal, we denote a fixed point set of T by ᏺ(T), that is, ᏺ(T): ={x ∈ K : Tx = x}. Theorem 4.1. Let E be a real linear normed space and K a nonempt y closed convex subset of E.LetT : K → K be a mapping which is total asymptotically weakly contractive. Suppose that ᏺ(T) =∅and x ∗ ∈ ᏺ(T). Starting from arbitrary x 1 ∈ K define the sequence {x n } by the iterative scheme (1.10), where {α n } n≥1 ⊂ (0,1) such that  α n =∞. Suppose that there exist constants m 1 ,m 2 > 0 such that k (1) n ≤ m 1 , k (2) n ≤ m 2 , lim λ→∞ ψ(λ) φ(λ) >m 1 (4.1) and the equation ψ(λ) = m 1 φ(λ)+m 2 has the unique root λ ∗ . Then {x n } converges strongly to x ∗ . Proof. Since K is closed convex subset of E, T : K → K and {α n } n≥1 ⊂ (0,1), we conclude that {x n }⊂K. We first show that t he sequence {x n } is bounded. From (1.10)and(1.13) one gets   x n+1 − x ∗   ≤    1 − α n  x n + α n T n x n − x ∗   ≤  1 − α n    x n − x ∗   + α n   T n x n − T n x ∗   ≤   x n − x ∗   + α n k (1) n φ    x n − x ∗    − α n ψ    x n − x ∗    + α n k (2) n . (4.2) By Lemma 3.4,weobtainthat {x n − x ∗ } is bounded, namely, x n − x ∗ ≤ ¯ C,where ¯ C = max    x 1 − x ∗   , λ ∗ + m 1 φ  λ ∗  + m 2  . (4.3) Next the convergence x n → x ∗ is shown by the relation   x n+1 − x ∗   ≤   x n − x ∗   − α n ψ    x n − x ∗    + α n k (1) n φ( ¯ C)+α n k (2) n , (4.4) applying Lemma 3.2 to the recurrent inequality (3.5)withλ n =x n − x ∗ .  [...]... construction of fixed points of asymptotically nonexpansive mappings, Journal of Mathematical Analysis and Applications 158 (1991), no 2, 407–413 , Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, [32] Bulletin of the Australian Mathematical Society 43 (1991), no 1, 153–159 [33] H F Senter and W G Dotson Jr., Approximating fixed points of nonexpansive mappings, Proceedings of. .. convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the Australian Mathematical Society 73 (1967), 591–597 [27] M O Osilike and S C Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Mathematical and Computer Modelling 32 (2000), no 10, 1181–1191 [28] G B Passty, Construction of fixed points for asymptotically nonexpansive. .. theorem for asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society 45 (1992), no 1, 25–36 , The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach [35] spaces, Proceedings of the American Mathematical Society 114 (1992), no 2, 399–404 , Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, [36] Journal of Mathematical... and D0 > 0, such that δE ( ) ≥ cδ( ), and |δ ( )| ≤ D0 for all 0 ≤ ≤ 2 Then {xn } converges strongly to a fixed point of T 16 Total asymptotically nonexpansive mappings Proof We denote F n = I − T n Since T is total asymptotically nonexpansive, one can con(1) (2) sider without loss of generality that kn ≤ c1 and kn ≤ c2 Consequently, by Lemma 2.3, ¯ ¯ if x ≤ R and y ≤ R, then 2 T nx − T n y 2 = (x... generated by the iterative process (1.10) to fixed points of the total asymptotically nonexpansive mappings T : K → K As before, we denote ᏺ(T) = {x ∈ K : Tx = x} Theorem 6.1 Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E Let T : K → K be a uniformly continuous and compact mapping which is total asymptotically nonexpansive and there exist constants M0 ,M > 0... [36] Journal of Mathematical Analysis and Applications 178 (1993), no 2, 301–308 , Fixed point iteration processes for asymptotically nonexpansive mappings, Proceedings of [37] the American Mathematical Society 122 (1994), no 3, 733–739 [38] H K Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Analysis Theory, Methods & Applications An International... convergence problems for asymptotically nonexpansive mappings, Journal of the Korean Mathematical Society 38 (2001), no 6, 1245–1260 [13] C E Chidume, Nonexpansive mappings, generalizations and iterative algorithms, to appear in Nonlinear Anal [14] C E Chidume, E U Ofoedu, and H Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, Journal of Mathematical Analysis and... asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society 35 (1972), 171–174 ´ [19] J Gornicki, Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces satisfying Opial’s condition, Journal of Mathematical Analysis and Applications 161 (1991), no 2, 440–446 [20] S Ishikawa, Fixed points by a new iteration method, Proceedings of the American... Ishikawa, Fixed points by a new iteration method, Proceedings of the American Mathematical Society 44 (1974), 147–150 , Fixed points and iteration of a nonexpansive mapping in a Banach space, Proceedings of [21] the American Mathematical Society 59 (1976), no 1, 65–71 20 Total asymptotically nonexpansive mappings [22] G E Kim and T H Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings... continuity of T yields the equality T y ∗ = y ∗ Finally, the limit of xn − y ∗ exists as n → ∞ because of Lemma 5.1 Therefore, the strong convergence of {xn } to some point of ᏺ(T) holds This accomplishes the proof Theorem 6.2 Let E be a real uniformly convex and uniformly smooth Banach space and K a nonempty closed convex subset of E Let T : K → K be a uniformly continuous and compact mapping which is total . to a fixed point of T. 16 Total asymptotically nonexpansive mappings Proof. We denote F n = I − T n .SinceT is total asymptotically nonexpansive, one can con- sider without loss of generality that. 147–150. [21] , Fixed points and iteration of a nonexpansive mapping in a Banach space, Proceedings of the American Mathematical Society 59 (1976), no. 1, 65–71. 20 Total asymptotically nonexpansive. fixed points of asymptotically nonexpansive mappings, Mathematical and Computer Modelling 32 (2000), no. 10, 1181–1191. [28] G. B. Passty, Construction of fixed points for asymptotically nonexpansive