Strong Convergence for Hybrid S-Iteration Scheme

We establish a strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.


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.
Definition 3. The mapping is said to be pseudocontractive if for all , ∈ , there exists ( − ) ∈ ( − ) such that Definition 4. The mapping is said to be strongly pseudocontractive if for all , ∈ , there exists ∈ (0, 1) such that Let be a nonempty convex subset of a normed space .
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., [1]). Results which had 2 Journal of Applied Mathematics been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [5][6][7][8][9][10] and the references cited therein).

Theorem 5.
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 at a fixed point of .
In [6], Chidume extended the results of Schu [9] from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of (strongly) pseudocontractive mappings.
In [11], Zhou and Jia gave the more general answer of the question raised by Chidume [5] 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 at the unique fixed point of .
In this paper, we establish the strong convergence for the hybrid -iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. We also improve the result of Zhou and Jia [11].

Main Results
We will need the following lemmas.

Theorem 8. Let be a nonempty closed convex subset of a real Banach space , let : → be nonexpansive, and let :
→ be Lipschitz strongly pseudocontractive mappings such that ∈ ( ) ∩ ( ) = { ∈ : = = } and For arbitrary 1 ∈ , let { } be a sequence iteratively defined by +1 = , Then the sequence { } converges strongly at the common fixed point of and .
Proof. For strongly pseudocontractive mappings, the existence of a fixed point follows from Delmling [13]. It is shown in [11] that the set of fixed points for strongly pseudocontractions is a singleton. By (v), since lim → ∞ = 0, there exists 0 ∈ N such that for all ≥ 0 , where < 1/2. Consider which implies that Journal of Applied Mathematics 3 and consequently from (16), we obtain Substituting (18) in (15) and using (13), we get So, from the above discussion, we can conclude that the which implies that { − } is bounded. Therefore, { − } is also bounded. Set Denote = 1 + 2 + 3 . Obviously, < ∞. Now from (12) for all ≥ 1, we obtain and by Lemma 6, we get which implies that then according to Lemma 7, we obtain from (26) that This completes the proof.

Corollary 9.
Let be a nonempty closed convex subset of a real Hilbert space , let : → be nonexpansive, and let : → be Lipschitz strongly pseudocontractive mappings such that ∈ ( ) ∩ ( ) and the condition (C). Let { } be a sequence in [0, 1] satisfying the conditions (iv) and (v).
For arbitrary 1 ∈ , let { } be a sequence iteratively defined by (12). Then the sequence { } converges strongly at the common fixed point of and .
Example 10. As a particular case, we may choose, for instance, = 1/ .

Remark 11.
(1) The condition (C) is not new and it is due to Liu et al. [14].
(2) We prove our results for a hybrid iteration scheme, which is simple in comparison to the previously known iteration schemes.