Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

and Applied Analysis 3 2. Main Results We will need the following results. Lemma 10 (see [19, 20]). Let J : E → 2E ∗ be the normalized duality mapping. Then for any x, y ∈ E, one has 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2 ⟨y, j (x + y)⟩ , ∀j (x + y) ∈ J (x + y) . (13) Lemma 11 (see [13]). Let {ρ n } and {θ n } be nonnegative sequences satisfying ρ n+1 ≤ (1 − θ n ) ρ n + b n , (14) where θ n ∈ [0, 1), ∑∞ n=1 θ n = ∞, and b n = o(θ n ). Then lim n→∞ ρ n = 0. The following is our main result. Theorem 12. Let K be a nonempty closed convex subset of a real Banach space E, let S : K → K be a nonexpansive mapping, and let T : K → K be a Lipschitz strongly pseudocontractive mapping such that p ∈ F(S) ∩ F(T) = {x ∈ K : Sx = Tx = x} and condition (C3). Let {β n } be a sequence in [0, 1] satisfying (i) ∑∞ n=1 β n = ∞; (ii) lim n→∞ β n = 0. For arbitrary x 0 ∈ K, let {x n } be a sequence iteratively defined by


Introduction and Preliminaries
Let  be a real Banach space and let  be a nonempty convex subset of .Let  denote the normalized duality mapping from  to 2  * defined by  () = { * ∈  * : ⟨,  * ⟩ = ‖‖ 2 ,      *     = ‖‖} ,  ∈ , where  * denotes the dual space of  and ⟨⋅, ⋅⟩ denotes the generalized duality pairing.We will denote the single-valued duality map by .
Let  :  →  be a mapping.
Remark 4. As a consequence of a result of Kato [1], it follows from the inequality that  is pseudocontractive if and only if there exists ( − ) ∈ ( − ) such that for all ,  ∈ .
For a nonempty convex subset  of a normed space ,  :  →  is a mapping.
If   = 0 for  ≥ 1, then the Ishikawa iteration scheme becomes the Mann iteration process [4].
In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., [3]).Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [7][8][9][10][11][12][13] and the references cited therein).
Theorem 6.Let  be a compact convex subset of a Hilbert space  and let  :  →  be a Lipschitzian pseudocontractive mapping.For arbitrary  1 ∈ , let {  } be a sequence defined iteratively by where {  } and {  } are sequences satisfying Then the sequence {  } converges strongly to a fixed point of .
In [7], Chidume extended the results of Schu [12] from Hilbert spaces to the much more general class of real Banach spaces and approximate the fixed points of pseudocontractive mappings.Also, in [14], he investigated the approximation of the fixed points of strongly pseudocontractive mappings.
In [15], Zhou and Jia gave the answer of the question raised by Chidume [14] and proved the following.
If  is a real Banach space with a uniformly convex dual  * ,  is a nonempty bounded closed convex subset of , and  :  →  is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly to the unique fixed point of . In For arbitrary  0 ∈ , let {  } be a sequence iteratively defined by Then the sequence {  } converges strongly to the fixed point  * of .
For arbitrary  1 ∈ , let {  } be a sequence iteratively defined by Then the sequence {  } converges strongly to a common fixed point  of  and .
Keeping in view the importance of the implicit iteration schemes (see [17]) in this paper we establish the strong convergence theorem for the hybrid implicit -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.

Main Results
We will need the following results.
Let {  } be a sequence in [0, 1] satisfying For arbitrary  0 ∈ , let {  } be a sequence iteratively defined by Then the sequence {  } converges strongly to a common fixed point  of  and .
Proof.For strongly mappings, the existence of a fixed point follows from Deimling [21].It is shown in [15] that the set of fixed points for strongly pseudocontractions is a singleton.By (ii), since lim  → ∞   = 0, there exists  0 ∈ N such that ∀ ≥  0 , where and consequently from ( 18) and ( 19), we obtain Substituting (20) in (17) and using ( 16), we get So, from the above discussion, we can conclude that the sequence {  − } is bounded.Since  is Lipschitzian, so Also by (ii), we have For arbitrary  0 ∈ , let {  } be a sequence iteratively defined by (15).Then the sequence {  } converges strongly to a common fixed point  of  and .
Example 14.As a particular case, we may choose, for instance,   = 1/.
In 2012,Kang et al. [17]established the strong convergence for the implicit -iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.