Abstract

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 all . Let be a sequence in satisfying (i) ; (ii) For arbitrary , let be a sequence iteratively defined by Then the sequence converges strongly to a common fixed point of and .

1. 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 defined by 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 1. The mapping is said to be Lipschitzian if there exists such that for all .

Definition 2. The mapping is said to be nonexpansive if for all .

Definition 3. The mapping is said to be pseudocontractive if for all and .

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 .

Definition 5. The mapping is said to be strongly pseudocontractive if there exists a constant such that for all and . Or equivalently (see [2]) one has for for all .

For a nonempty convex subset of a normed space , is a mapping.

(I) The sequence , defined by, for arbitrary , where and are sequences in , is known as the Ishikawa iteration process [3].

If for , then the Ishikawa iteration scheme becomes the Mann iteration process [4].

(S) The sequence , defined by, for arbitrary , where is a sequence in , is known as the -iteration process [5, 6].

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., [713] and the references cited therein).

In 1974, Ishikawa [3] proved the following result.

Theorem 6. Let be a compact convex subset of a Hilbert space and let be a Lipschitzian pseudocontractive mapping. For arbitrary , let be a sequence defined iteratively by where and are sequences satisfying(i);(ii);(iii).
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 [16], Liu et al. introduced the following condition.

Remark 7. Let be two mappings. The mappings and are said to satisfy condition if for all .

In 2012, Kang et al. [17] established the strong convergence for the implicit -iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

Theorem 8. Let be a compact convex subset of a real Hilbert space and let be a Lipschitzian hemicontractive mapping satisfying for all . Let be a sequence in satisfying(i);(ii).
For arbitrary , let be a sequence iteratively defined by Then the sequence converges strongly to the fixed point of .

In 2013, Kang et al. [18] proved the following result.

Theorem 9. Let be a nonempty closed convex subset of a real Banach space , let be a nonexpansive mapping, and let be a Lipschitz strongly pseudocontractive mapping such that and for all . Let be a sequence in satisfying(i); (ii).
For arbitrary , 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.

2. Main Results

We will need the following results.

Lemma 10 (see [19, 20]). Let be the normalized duality mapping. Then for any , one has

Lemma 11 (see [13]). Let and be nonnegative sequences satisfying where , , and . Then .

The following is our main result.

Theorem 12. Let be a nonempty closed convex subset of a real Banach space , let be a nonexpansive mapping, and let be a Lipschitz strongly pseudocontractive mapping such that and condition .
Let be a sequence in satisfying(i);(ii).
For arbitrary , let be a sequence iteratively defined by Then the sequence converges strongly to a common fixed point of and .

Proof. For strongly pseudocontractive 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 , there exists such that , where and is a Lipschitz constant of . Consider 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 is also bounded. Let . Also by (ii), we have as , which implies that is bounded, so let . Further which implies that is bounded. Therefore is also bounded.
Set
Denote . Obviously .
Now, from (15), for all , we obtain and by Lemma 10, which implies that Because of (16), we have and . Also, by (ii) and (19), as .
Hence (25) and (27) give
For all , put then according to Lemma 11, we obtain from (28) that This completes the proof.

Corollary 13. Let be a nonempty closed convex subset of a real Hilbert space , let be a nonexpansive mapping, and let be a Lipschitz strongly pseudocontractive mapping such that and condition . Let be a sequence in satisfying conditions (i) and (ii) in Theorem 12.
For arbitrary , 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, .

Remark 15. Condition is due to Kang et al. [17] and condition with becomes condition .
(2) Condition is due to Kang et al. [18] and condition with becomes condition .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and all referees for their valuable comments and suggestions for improving the paper. This study was supported by research funds from Dong-A University.