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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 281070, 13 pages doi:10.1155/2010/281070 Research Article Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense Jing Zhao 1, 2 and Songnian He 1, 2 1 College of Science, Civil Aviation University of China, Tianjin 300300, China 2 Tianjin Key Laboratory For Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Jing Zhao, zhaojing200103@163.com Received 23 June 2010; Accepted 19 October 2010 Academic Editor: W. A. Kirk Copyright q 2010 J. Zhao and S. He. 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. We study the convergence of Ishikawa iteration process for the class of asymptotically κ-strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others. 1. Introduction and Preliminaries Throughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm ·.and → denote weak and strong convergence, respectively. ω w x n denotes the weak ω-limit set of {x n },thatis,ω w x n {x ∈ H : ∃x n j x}.LetC be a nonempty closed convex subset of H. It is well known that for every point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that x − P C x ≤ x − y , 1.1 for all y ∈ C. P C is called the metric projection of H onto C. P C is a nonexpansive mapping of H onto C and satisfies x − y, P C x − P C y ≥ P C x − P C y 2 , ∀x, y ∈ H. 1.2 2 Fixed Point Theory and Applications Let T : C → C be a mapping. In this paper, we denote the fixed point set of T by FT. Recall that T is said to be uniformly L-Lipschitzian if there exists a constant L>0, such that T n x − T n y ≤ L x − y , ∀x, y ∈ C, ∀n ≥ 1. 1.3 T is said to be nonexpansive if Tx − Ty ≤ x − y , ∀x, y ∈ C. 1.4 T is said to be asymptotically nonexpansive if there exists a sequence {k n } in 1, ∞ with lim n →∞ k n 1, such that T n x − T n y ≤ k n x − y , ∀x, y ∈ C, ∀n ≥ 1. 1.5 The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk 1 as a generalization of the class of nonexpansive mappings. T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: lim sup n →∞ sup x,y∈C T n x − T n y − x − y ≤ 0. 1.6 Observe that if we define τ n max 0, sup x,y∈C T n x − T n y − x − y , 1.7 then τ n → 0asn →∞. It follows that 1.6 is reduced to T n x − T n y ≤ x − y τ n , ∀x, y ∈ C, ∀n ≥ 1. 1.8 The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. 2. It is known 3 that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is asymptotically nonexpansive 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 contains properly the class of asymptotically nonexpansive mappings. Recall that T is said to be a κ-strict pseudocontraction if there exists a constant κ ∈ 0, 1, such that Tx − Ty 2 ≤ x − y 2 κ I − Tx − I − Ty 2 , ∀x, y ∈ C. 1.9 Fixed Point Theory and Applications 3 T is said to be an asymptotically κ-strict pseudocontraction with sequence {γ n } if there exist a constant κ ∈ 0, 1 and a sequence {γ n }⊂0, ∞ with γ n → 0asn →∞, such that T n x − T n y 2 ≤ 1 γ n x − y 2 κ I − T n x − I −T n y 2 , ∀x, y ∈ C, n ≥ 1. 1.10 The class of asymptotically κ-strict pseudocontractions was introduced by Qihou 4 in 1996 see also 5.KimandXu6 studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically κ-strict pseudocontractive mapping with sequence {γ n } is a uniformly L-Lipschitzian mapping with L sup{κ 1 1 − κγ n /1 κ : n ∈ N}. Recently, Sahu et al. 7 introduced a class of new mappings: asymptotically κ- strict pseudocontractive mappings in the intermediate sense. Recall that T is said to be an asymptotically κ-strict pseudocontraction in the intermediate sense with sequence {γ n } if there exist a constant κ ∈ 0, 1 and a sequence {γ n }⊂0, ∞ with γ n → 0asn →∞, such that lim sup n →∞ sup x,y∈C T n x − T n y 2 − 1 γ n x − y 2 − κ I − T n x − I − T n y 2 ≤ 0. 1.11 Throughout this paper, we assume that c n max 0, sup x,y∈C T n x − T n y 2 − 1 γ n x − y 2 − κ I − T n x − I − T n y 2 . 1.12 It follows that c n → 0asn →∞and 1.11 is reduced to the relation T n x − T n y 2 ≤ 1 γ n x − y 2 κ I − T n x − I −T n y 2 c n , ∀x, y ∈ C. 1.13 They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see 7 for more details. In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of asymptotically κ-strict pseudocontractive mappings in the intermediate sense. In order to prove our main results, we also need the following lemmas. Lemma 1.1 see 8, 9. Let {δ n }, {β n }, and {γ n } be three sequences of nonnegative numbers satisfying the recursive inequality δ n1 ≤ β n δ n γ n , ∀n ≥ 1. 1.14 If β n ≥ 1, ∞ n1 β n − 1 < ∞ and ∞ n1 γ n < ∞,thenlim n →∞ δ n exists. 4 Fixed Point Theory and Applications Lemma 1.2 see 10. Let {x n } be a bounded sequence in a reflexive Banach space X.Ifω w x n {x},thenx n x. Lemma 1.3 see 11. Let C be a nonempty closed convex subset of a real Hilbert space H.Given x ∈ H and z ∈ C,thenz P C x if and only if x − z, y − z≤0, for all y ∈ C. Lemma 1.4 see 11. For a real Hilbert space H, the following identities hold: i x −y 2 x 2 −y 2 − 2x − y, y, for all x, y ∈ H, ii tx 1 − ty 2 tx 2 1−ty 2 −t1−tx −y 2 , for all t ∈ 0, 1, for all x, y ∈ H; iii (Opial condition) If {x n } is a sequence in H weakly convergent to z,then lim sup n →∞ x n − y 2 lim sup n →∞ x n − z 2 z − y 2 , ∀y ∈ H. 1.15 Lemma 1.5 see 7. Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ n }.Then T n x − T n y ≤ 1 1 − κ κ x − y 1 1 − κ γ n x − y 2 1 − κ c n , ∀x, y ∈ C, ∀n ∈ N. 1.16 Lemma 1.6. Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ n }.Letn ∈ N.Ifγ n < 1, then T n x − T n y ≤ 1 1 − κ κ √ 2 − κ x − y √ c n , ∀x, y ∈ C. 1.17 Proof. If γ n < 1, for x, y ∈ C,weobtainfromLemma 1.5 that T n x − T n y ≤ 1 1 − κ κ x − y 1 1 − κ γ n x − y 2 1 − κ c n ≤ 1 1 − κ κ x − y 2 − κ x − y 2 c n ≤ 1 1 − κ κ x − y √ 2 − κ x − y √ c n 2 1 1 − κ κ √ 2 − κ x − y √ c n . 1.18 Lemma 1.7 see 7. Let C be a nonempty subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ n }.Let{x n } be a sequence in C such that x n − x n1 →0 and x n − T n x n →0 as n →∞, then x n − Tx n →0 as n →∞. Fixed Point Theory and Applications 5 Lemma 1.8 see 7,Proposition3.1. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense. Then I − T is demiclosed at zero in the sense that if {x n } is a sequence in C such that x n x∈ C and lim sup m →∞ lim sup n →∞ x n − T m x n 0,thenI − Tx 0. Lemma 1.9 see 7. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense. Then FT is closed and convex. 2. Main Results Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ n } such that FT / ∅.Let{x n } ∞ n1 be a sequence in C generated by the following Ishikawa iterative process: x 1 ∈ C, y n β n T n x n 1 − β n x n , x n1 α n T n y n 1 − α n x n , ∀n ≥ 1, 2.1 where {α n } and {β n } are sequences in 0, 1. Assume that the following restrictions are satisfied: i ∞ n1 α n c n < ∞ and ∞ n1 1 γ n 2 − 1 < ∞, ii 0 <a≤ α n ≤ β n ≤ b for some a>0 and b ∈ 0, −1 − κ 2 1 − κ 4 2κ √ 2 − κ 2 1 − κ 2 /2κ √ 2 − κ 2 . Then the sequence {x n } given by 2.1 converges weakly to an element of FT. Proof. Let p ∈ FT.From1.13 and Lemma 1.4,weseethat y n − p 2 β n T n x n − p1 − β n x n − p 2 β n T n x n − p 2 1 − β n x n − p 2 − β n 1 − β n x n − T n x n 2 ≤ β n 1 γ n x n − p 2 κ x n − T n x n 2 c n 1 − β n x n − p 2 − β n 1 − β n x n − T n x n 2 ≤ 1 γ n x n − p 2 − β n 1 − β n − κ x n − T n x n 2 β n c n . 2.2 6 Fixed Point Theory and Applications Without loss of generality, we may assume that γ n < 1 for all n ∈ N. Since x n − y n 2 x n − β n T n x n − 1 − β n x n 2 β 2 n x n − T n x n 2 , 2.3 it follows from Lemma 1.6 that y n − T n y n 2 β n T n x n − T n y n 1 − β n x n − T n y n 2 β n T n x n − T n y n 2 1 − β n x n − T n y n 2 − β n 1 − β n x n − T n x n 2 ≤ β n 1 − κ 2 κ √ 2 − κ x n − y n √ c n 2 1 − β n x n − T n y n 2 − β n 1 − β n x n − T n x n 2 ≤ 2β 3 n κ √ 2 − κ 1 − κ 2 x n − T n x n 2 2β n c n 1 − κ 2 1 − β n x n − T n y n 2 − β n 1 − β n x n − T n x n 2 . 2.4 By 2.2 and 2.4,weobtainthat T n y n − p 2 ≤ 1 γ n y n − p 2 κ y n − T n y n 2 c n ≤ 1 γ n 2 x n − p 2 − β n 1 γ n 1 − β n − κ x n − T n x n 2 β n 1 γ n c n 2κβ 3 n κ √ 2 − κ 1 − κ 2 x n − T n x n 2 2κβ n c n 1 − κ 2 κ 1 − β n x n − T n y n 2 − κβ n 1 − β n x n − T n x n 2 c n 1 γ n 2 x n − p 2 − β n ⎡ ⎣ 1 γ n 1 − β n − κ − 2κβ 2 n κ √ 2 − κ 1 − κ 2 κ 1 − β n ⎤ ⎦ × x n − T n x n 2 κ 1 − β n x n − T n y n 2 c n M 1 , 2.5 Fixed Point Theory and Applications 7 where M 1 sup n≥1 {β n 1 γ n 2κβ n /1 − κ 2 1}. It follows from 2.5 and α n ≤ β n that x n1 − p 2 α n T n y n − p1 − α n x n − p 2 α n T n y n − p 2 1 − α n x n − p 2 − α n 1 − α n T n y n − x n 2 ≤ α n 1 γ n 2 x n − p 2 − α n β n ⎡ ⎣ 1 γ n 1 − β n − κ − 2κβ 2 n κ √ 2 − κ 1 − κ 2 κ 1 − β n ⎤ ⎦ × x n − T n x n 2 α n κ 1 − β n x n − T n y n 2 α n c n M 1 1 − α n x n − p 2 − α n 1 − α n T n y n − x n 2 ≤ 1 γ n 2 x n − p 2 − α n β n ⎡ ⎣ 1 γ n 1 − β n − κγ n − 2κβ 2 n κ √ 2 − κ 1 − κ 2 − κβ n ⎤ ⎦ × x n − T n x n 2 − α n 1 − α n − κ 1 − β n x n − T n y n 2 α n c n M 1 ≤ 1 γ n 2 x n − p 2 − α n β n ⎡ ⎣ 1 γ n 1 − β n − κγ n − 2κβ 2 n κ √ 2 − κ 1 − κ 2 − κβ n ⎤ ⎦ × x n − T n x n 2 α n c n M 1 . 2.6 From the condition ii and γ n → 0, we see that there exists n 0 such that 1 γ n 1 − β n − κγ n − 2κβ 2 n κ √ 2 − κ 1 − κ 2 − κβ n ≥ 1 − β n − κγ n − 2β 2 n κ √ 2 − κ 1 − κ 2 − κβ n ≥ 1 − 2β n − κγ n − 2β 2 n κ √ 2 − κ 1 − κ 2 ≥ 1 − 2b − 2b 2 κ √ 2 − κ 1 − κ 2 − κγ n ≥ 1 2 ⎛ ⎝ 1 − 2b − 2b 2 κ √ 2 − κ 1 − κ 2 ⎞ ⎠ > 0, ∀n ≥ n 0 . 2.7 8 Fixed Point Theory and Applications By 2.6, we have x n1 − p 2 ≤ 1 γ n 2 x n − p 2 α n c n M 1 , ∀n ≥ n 0 . 2.8 In view of Lemma 1.1 and the condition i, we obtain that lim n →∞ x n − p exists. For any n ≥ n 0 ,itiseasytoseefrom2.6 and 2.7 that a 2 2 ⎛ ⎝ 1 − 2b − 2b 2 κ √ 2 − κ 1 − κ 2 ⎞ ⎠ x n − T n x n 2 ≤ 1 γ n 2 x n − p 2 − x n1 − p 2 α n c n M 1 , 2.9 which implies that lim n →∞ x n − T n x n 0. 2.10 Note that x n1 − x n α n T n y n − x n ≤ α n T n y n − T n x n α n T n x n − x n ≤ α n 1 − κ κ √ 2 − κ x n − y n √ c n α n T n x n − x n α n β n 1 − κ κ √ 2 − κ x n − T n x n α n √ c n 1 − κ α n T n x n − x n . 2.11 From 2.10, we have lim n →∞ x n1 − x n 0. 2.12 Since T is uniformly continuous, we obtain from 2.10, 2.12 and Lemma 1.7 that lim n →∞ x n − Tx n 0. 2.13 By the boundedness of {x n }, there exist a subsequence {x n k } of {x n } such that x n k x. Observe that T is uniformly continuous and x n − Tx n →0asn →∞, for any m ∈ N we have x n − T m x n →0asn →∞.FromLemma 1.8,weseethatx ∈ FT. To complete the proof, it suffices to show that ω w {x n } consists of exactly one point, namely, x. Suppose there exists another subsequence {x n j } of {x n } such that {x n j } converges Fixed Point Theory and Applications 9 weakly to some z ∈ C and z / x. As in the case of x, we can also see that z ∈ FT. It follows that lim n →∞ x n −x and lim n →∞ x n −z exist. Since H satisfies the Opial condition, we have lim n →∞ x n − x lim k →∞ x n k − x < lim k →∞ x n k − z lim n →∞ x n − z , lim n →∞ x n − z lim j →∞ x n j − z < lim j →∞ x n j − x lim n →∞ x n − x , 2.14 which is a contradiction. We see x z and hence ω w {x n } is a singleton. Thus, {x n } converges weakly to x by Lemma 1.2. Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping with sequence {γ n } such that FT / ∅.Let{x n } ∞ n1 be a sequence in C generated by the following Ishikawa iterative process: x 1 ∈ C, y n β n T n x n 1 − β n x n , x n1 α n T n y n 1 − α n x n , ∀n ≥ 1, 2.15 where {α n } and {β n } are sequences in 0, 1. Assume that the following restrictions are satisfied: i ∞ n1 1 γ n 2 − 1 < ∞, ii 0 <a≤ α n ≤ β n ≤ b for some a>0 and b ∈ 0, −1 − κ 2 1 − κ 4 2κ √ 2 − κ 2 1 − κ 2 /2κ √ 2 − κ 2 . Then the sequence {x n } given by 2.15 converges weakly to an element of FT. Next, we modify Ishikawa iterative process to get a strong convergence theorem. Theorem 2.3. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {γ n } such that FT / ∅ and bounded. Let {α n } and {β n } are sequences in 0, 1.Let {x n } ∞ n1 be a sequence in C generated by the modified Ishikawa iterative process: x 1 ∈ C, y n β n T n x n 1 − β n x n , z n α n T n y n 1 − α n x n , C n z ∈ C : z n − z 2 ≤ x n − z 2 θ n − ρ n x n − T n x n 2 , Q n { z ∈ C : x n − z, x 1 − x n ≥ 0 } , x n1 P C n ∩Q n x 1 , 2.16 where θ n α n c n M 1 2γ n γ 2 n Δ n , M 1 sup n≥1 {β n 1 γ n 2κβ n /1 − κ 2 1}, Δ n sup{x n − z 2 : z ∈ FT} < ∞ and ρ n α n β n 1−2β n −κγ n −2β 2 n κ √ 2 − κ/1 −κ 2 for each 10 Fixed Point Theory and Applications n ≥ 1. Assume that the control sequences {α n } and {β n } are chosen such that 0 <a≤ α n ≤ β n ≤ b for some a>0 and b ∈ 0, −1 − κ 2 1 − κ 4 2κ √ 2 − κ 2 1 − κ 2 /2κ √ 2 − κ 2 .Then the sequence {x n } given by 2.16 converges strongly to an element of FT. Proof. We break the proof into six steps. Step 1 C n ∩Q n is closed and convex for each n ≥ 1. It is obvious that Q n is closed and convex and C n is closed for each n ≥ 1. Note that the defining inequality in C n is equivalent to the inequality 2 x n − z n ,z ≤ x n 2 − z n 2 θ n − ρ n x n − T n x n 2 , 2.17 it is easy to see that C n is convex for each n ≥ 1. Hence, C n ∩Q n is closed and convex for each n ≥ 1. Step 2 FT ⊂ C n ∩ Q n for each n ≥ 1.Letp ∈ FT. Following 2.6, 2.7 and the algorithm 2.16, we have z n − p 2 ≤ 1 γ n 2 x n − p 2 − α n β n ⎡ ⎣ 1 γ n 1 − β n − κγ n − 2κβ 2 n κ √ 2 − κ 1 − κ 2 − κβ n ⎤ ⎦ × x n − T n x n 2 α n c n M 1 ≤ 1 γ n 2 x n − p 2 − α n β n ⎡ ⎣ 1 − 2β n − 2β 2 n κ √ 2 − κ 1 − κ 2 − κγ n ⎤ ⎦ × x n − T n x n 2 α n c n M 1 x n − p 2 − ρ n x n − T n x n 2 α n c n M 1 2γ n γ 2 n x n − p 2 ≤ x n − p 2 − ρ n x n − T n x n 2 θ n , 2.18 where θ n α n c n M 1 2γ n γ 2 n Δ n , M 1 sup n≥1 {β n 1 γ n 2κβ n /1 − κ 2 1}, Δ n sup{x n − z 2 : z ∈ FT} < ∞ and ρ n α n β n 1 − 2β n − κγ n − 2β 2 n κ √ 2 − κ/1 − κ 2 for each n ≥ 1. Hence p ∈ C n for each n ≥ 1. Next, we show that FT ⊂ Q n for each n ≥ 1. We prove this by induction. For n 1, we have FT ⊂ C Q 1 . Assume that FT ⊂ Q n for some n>1. Since x n1 is the projection of x 1 onto C n ∩ Q n , we have x n1 − z, x 1 − x n1 ≥ 0, ∀z ∈ C n ∩ Q n . 2.19 [...]... 10, pp 3502–3511, 2009 8 M O Osilike and S C Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, ” Mathematical and Computer Modelling, vol 32, no 10, pp 1181–1191, 2000 9 K.-K Tan and H.-K Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol 114, no 2,... 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This completes the proof Acknowledgments This research is supported by Fundamental Research Funds for the Central Universities ZXH2009D021 and supported by the Science Research Foundation Program in Civil Aviation University of China no 09CAUC-S05 as well References 1 K Goebel and W A Kirk, “A fixed point theorem for asymptotically nonexpansive mappings, ” Proceedings of the American Mathematical Society,... 399–404, 1992 10 R P Agarwal, D O’Regan, and D R Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, ” Journal of Nonlinear and Convex Analysis, vol 8, no 1, pp 61–79, 2007 11 G Marino and H.-K Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 329, no 1, pp 336–346,...Fixed Point Theory and Applications 11 By the induction consumption, we know that F T ⊂ Cn ∩ Qn In particular, for any p ∈ F T we have xn 1 − p, x1 − xn ≥ 0 1 2.20 This implies that p ∈ Qn 1 That is, F T ⊂ Qn 1 By the principle of mathematical induction, we get F T ⊂ Qn and hence F T ⊂ Cn ∩ Qn for all n ≥ 1 This means that the iteration algorithm 2.16 is well defined Step 3 limn → ∞ xn −x1 exists and. .. {xn } is bounded In view of 2.16 , we see that xn and xn 1 PCn ∩Qn x1 ∈ Qn It follows that xn − x1 ≤ xn 1 − x1 PQn x1 2.21 for each n ≥ 1 We, therefore, obtain that the sequence { xn − x1 } is nondecreasing Noticing that F T ⊂ Qn and xn PQn x1 , we have x1 − xn ≤ x1 − p , ∀p ∈ F T 2.22 This shows that the sequence { xn − x1 } is bounded Therefore, the limit of { xn − x1 } exists and {xn } is bounded... Point Theory and Applications αn T n yn Combing 2.25 and 2.26 and noting zn α2 T n yn − xn n 2 1 − αn xn , we obtain that 2 αn T n yn − xn , xn − xn ≤ θn − ρn xn − T n xn 2 1 2.27 From the assumption and 2.7 , we see that there exists n0 ∈ N such that 2 1 − 2βn − κγn − 2βn √ κ 2 2−κ 1−κ ⎛ 1⎝ 1 − 2b − 2b2 ≥ 2 κ √ 2 2−κ 1−κ ⎞ ⎠ > 0, 2.28 ∀n ≥ n0 For any n ≥ n0 , it follows from the definition of ρn and. .. Fixed Point Theory and Applications 13 This implies that x1 − PF T lim x1 − xni n→∞ x1 x1 − x , x1 − PF T x1 2.33 2.34 Hence x PF T x1 by the uniqueness of the nearest point projection of x1 onto F T Since {x} and {xni } is an arbitrary weakly convergent subsequence, it follows that ωw {xn } x It is easy to see as 2.34 that x1 − xn → x1 − x Since H has the Kadec-Klee hence xn property, we obtain that . nonexpansive 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 contains properly the. class of new mappings: asymptotically κ- strict pseudocontractive mappings in the intermediate sense. Recall that T is said to be an asymptotically κ -strict pseudocontraction in the intermediate. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 281070, 13 pages doi:10.1155/2010/281070 Research Article Weak and Strong Convergence Theorems for Asymptotically
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