A New Hybrid Algorithm for λ-Strict Asymptotically Pseudocontractions in 2-Uniformly Smooth Banach Spaces

and Applied Analysis 3 Theorem 1.3. Let E be a real q-uniformly smooth Banach space which is also uniformly convex, let C be a nonempty closed convex subset of E, let T : C → C be a (λ, {kn})-strictly asymptotically pseudocontractive mapping such that ∑∞ n 1 k 2 n − 1 < ∞, and let F T / ∅. Let {αn} ⊂ 0, 1 be a real sequence satisfying the following condition: 0 < a ≤ αn ≤ b < q 1 − k 2cq 1 L − q−2 , ∀n ≥ 1. 1.10 Let {xn} be the sequence defined by 1.9 . Then, {xn} converges weakly to a fixed point of T . On the other hand, by using the metric projection, Nakajo and Takahashi 22 introduced the following iterative algorithms for the nonexpansive mapping T in the framework of Hilbert spaces: x0 x ∈ C, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥∥z − yn ∥∥} ≤ ‖z − xn‖, Qn {z ∈ C : 〈xn − z, x − xn〉 ≥ 0}, xn 1 PCn∩Qnx, n 0, 1, 2, . . . , 1.11 where {αn} ⊂ 0, α , α ∈ 0, 1 and PCn∩Qn is the metric projection from a Hilbert spaceH onto Cn ∩Qn. They proved that {xn} generated by 1.11 converges strongly to a fixed point of T . In 2006, Xu 23 extended Nakajo and Takahashi’s theorem to Banach spaces by using the generalized projection. In 2008, Matsushita and Takahashi 24 presented the following iterative algorithms for the nonexpansive mapping T in the framework of Banach spaces: x0 x ∈ C, Cn co{z ∈ C : ‖z − Tz‖ ≤ tn‖xn − Txn‖}, Dn {z ∈ C : 〈xn − z, J x − xn 〉 ≥ 0}, xn 1 PCn∩Dnx, n 0, 1, 2, . . . , 1.12 where coC denotes the convex closure of the set C, J is normalized duality mapping, {tn} is a sequence in 0, 1 with tn → 0, and PCn∩Dn is the metric projection from E onto Cn ∩Dn. Then, they proved that {xn} generated by 1.12 converges strongly to a fixed point of nonexpansive mapping T . Recently, Dehghan 25 introduced the following hybrid projection algorithm for an asymptotically nonexpansive mapping T in the framework of Banach spaces: x0 x ∈ C, C0 D0 C, Cn co{z ∈ Cn−1 : ‖z − Tnz‖ ≤ tn‖xn − Txn‖}, Dn {z ∈ Dn−1 : 〈xn − z, J x − xn 〉 ≥ 0}, xn 1 PCn∩Dnx, n 1, 2, . . . , 1.13 4 Abstract and Applied Analysis where coC denotes the convex closure of the set C, {tn} is a sequence in 0, 1 with tn → 0, and PCn∩Dn is the metric projection from E onto Cn ∩Dn. Then, he proved that {xn} generated by 1.13 converges strongly to a fixed point of an asymptotically nonexpansive mappings T . Motivated by the research work going on in this direction, the purpose of this paper is to introduce the following iteration for finding a fixed point of λ, {kn} -strict asymptotically pseudocontraction in a uniformly convex and 2-uniformly smooth Banach spaces: x0 x ∈ C, C0 C, Cn co{z ∈ Cn−1 : ‖z − Tnz‖ ≤ tn‖xn − Txn‖}, xn 1 PCnx, n 1, 2, . . . , 1.14 where coC denotes the convex closure of the set C, {tn} is a sequence in 0, 1 with tn → 0, and PCn is the metric projection from E onto Cn. Under suitable conditions some strong convergence theorem for the sequence {xn} defined by 1.14 to converge a fixed point of an asymptotically λ-strictly pseudocontraction. The result presented in the paper extends and improves the main results of Matsushita and Takahashi 24 , Dehghan 25 , Kang and Wang 26 , and others. 2. Preliminaries In this section, we recall the well-known concepts and results which will be needed to prove our main results. Throughout this paper, we assume that E is a real Banach space and C is a nonempty subset of E. When {xn} is a sequence in E, we denote strong convergence of {xn} to x ∈ E by xn → x and weak convergence by xn ⇀ x. A Banach space E is said to be strictly convex if ‖x y‖/2 < 1 for all x, y ∈ U {z ∈ E : ‖z‖ 1} with x / y. E is said to be uniformly convex if for each > 0 there is a δ > 0 such that for x, y ∈ E with ||x||, ||y|| ≤ 1 and ||x − y|| ≥ , ||x y|| ≤ 2 1 − δ holds. The modulus of convexity of E is defined by δE inf { 1 − ∥∥∥ x y 2 ∥∥∥ : ‖x‖, ∥∥y∥∥ ≤ 1,∥∥x − y∥∥ ≥ } , 2.1 E is said to be smooth if the limit lim t→ 0 ∥∥x ty∥∥ − ‖x‖ t 2.2 exists for all x, y ∈ U. The modulus of smoothness of E is defined by ρE t sup { 1 2 (∥∥x y∥∥ ∥∥x − y∥∥) − 1 : ‖x‖ ≤ 1,∥∥y∥∥ ≤ t } . 2.3 A Banach space E is said to be uniformly smooth if ρE t /t → 0 as t → 0. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρE t ≤ ct. Abstract and Applied Analysis 5 If E is a reflexive, strictly convex and smooth Banach space, then for any x ∈ E, there exists a unique point x0 ∈ C such that ‖x0 − x‖ min y∈C ∥∥y − x∥∥. 2.4and Applied Analysis 5 If E is a reflexive, strictly convex and smooth Banach space, then for any x ∈ E, there exists a unique point x0 ∈ C such that ‖x0 − x‖ min y∈C ∥∥y − x∥∥. 2.4 The mapping PC : E → C defined by PCx x0 is called the metric projection from E onto C. Let x ∈ E and u ∈ C. Then it is known that u PCx if and only if 〈 u − y, J x − u 〉 ≥ 0, ∀y ∈ C. 2.5 For the details on the metric projection, refer to 27–30 . In the sequel, we make use of the following lemmas for our main results. Lemma 2.1 see 31 . Let E be a real Banach space, C a nonempty subset of E, and T : C → C a (λ, {kn})-strictly asymptotically pseudocontractive mapping. Then T is uniformly L-Lipschitzian. Lemma 2.2 see 32 . Let E be a real 2-uniformly smooth Banach spaces with the best smooth constant K. Then the following inequality holds: ∥∥x y∥∥2 ≤ ‖x‖ 2〈y, J x 〉 2∥∥Ky∥∥2, 2.6 for any x, y ∈ E. Lemma 2.3 demiclosed principle 21 . Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let C be a nonempty closed convex subset of E and T : C → C a (λ, {kn})strictly asymptotically pseudocontractive mapping with a nonempty fixed point set. Then I − T is demiclosed at zero, where I is the identical mapping. Lemma 2.4 see 33 . Let C be a closed convex subset of a uniformly convex Banach space. Then for each r > 0, there exists a strictly increasing convex continuous function γ : 0,∞ → 0,∞ such that γ 0 0 and γ ⎛ ⎝ ∥∥∥∥∥ T ⎛ ⎝ m ∑