JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 705814 10.1155/2013/705814 705814 Research Article Strong Convergence for Hybrid S-Iteration Scheme 0000-0002-5294-4447 Kang Shin Min 1 Rafiq Arif 2 0000-0002-5028-0640 Kwun Young Chel 3 Sahu D. R. 1 Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea gnu.ac.kr 2 School of CS and Mathematics Hajvery University 43-52 Industrial Area Gulberg-III Lahore 54660 Pakistan hup.edu.pk 3 Department of Mathematics Dong-A University Pusan 614-714 Republic of Korea donga.ac.kr 2013 11 3 2013 2013 19 11 2012 04 02 2013 2013 Copyright © 2013 Shin Min Kang et al. 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 establish a strong convergence for the hybrid S-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.

1. Introduction and Preliminaries

Let E be a real Banach space and let K be a nonempty convex subset of E. Let J denote the normalized duality mapping from E to 2E* defined by (1)J(x)={f*E*:x,f*=x2,f*=x},x,yE, where E* denotes the dual space of E and ·,· denotes the generalized duality pairing. We will denote the single-valued duality map by j.

Let T:KK be a mapping.

Definition 1.

The mapping T is said to be Lipschitzian if there exists a constant L>1 such that (2)Tx-TyLx-y,x,yK.

Definition 2.

The mapping T is said to be nonexpansive if (3)Tx-Tyx-y,      x,yK.

Definition 3.

The mapping T is said to be pseudocontractive if for all x,yK, there exists j(x-y)J(x-y) such that (4)Tx-Ty,j(x-y)x-y2.

Definition 4.

The mapping T is said to be strongly pseudocontractive if for all x,yK, there exists k(0,1) such that (5)Tx-Ty,j(x-y)kx-y2.

Let K be a nonempty convex subset C of a normed space E.

The sequence {xn} defined by, for arbitrary x1K, (6)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n1, where {αn} and {βn} are sequences in [0,1], is known as the Ishikawa iteration process . If βn=0 for n1, then the Ishikawa iteration process becomes the Mann iteration process .

The sequence {xn} defined by, for arbitrary x1K, (7)xn+1=Tyn,yn=(1-βn)xn+βnTxn,n1, where {βn} is a sequence in [0,1], is known as the S-iteration process [3, 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., ). 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.,  and the references cited therein).

In 1974, Ishikawa  proved the following result.

Theorem 5.

Let K be a compact convex subset of a Hilbert space H and let T:KK be a Lipschitzian pseudocontractive mapping. For arbitrary x1K, let {xn} be a sequence defined iteratively by (8)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n1, where {αn} and {βn} are sequences satisfying

0αnβn1,

limnβn=0,

n1αnβn=.

Then the sequence {xn} converges strongly at a fixed point of T.

In , Chidume extended the results of Schu  from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of (strongly) pseudocontractive mappings.

In , Zhou and Jia gave the more general answer of the question raised by Chidume  and proved the following.

If X is a real Banach space with a uniformly convex dual X*, K is a nonempty bounded closed convex subset of X, and T:KK is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly at the unique fixed point of T.

In this paper, we establish the strong convergence for the hybrid S-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. We also improve the result of Zhou and Jia .

2. Main Results

We will need the following lemmas.

Lemma 6 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let J:E2E be the normalized duality mapping. Then for any x,yE, one has (9)x+y2x2+2y,j(x+y),j(x+y)J(x+y).

Lemma 7 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

Let {ρn} be nonnegative sequence satisfying (10)ρn+1(1-θn)ρn+ωn, where θn[0,1],n1θn=, and ωn=o(θn). Then (11)limnρn=0.

The following is our main result.

Theorem 8.

Let K be a nonempty closed convex subset of a real Banach space E, let S:KK be nonexpansive, and let T:KK be Lipschitz strongly pseudocontractive mappings such that pF(S)F(T)={xK:Sx=Tx=x} and (C)x-SySx-Sy,x,yK,x-TyTx-Ty,x,yK. Let {βn} be a sequence in [0,1] satisfying

n1βn=,

limnβn=0.

For arbitrary x1K, let {xn} be a sequence iteratively defined by (12)xn+1=Syn,yn=(1-βn)xn+βnTxn,n1.

Then the sequence {xn} converges strongly at the common fixed point p of S and T.

Proof.

For strongly pseudocontractive mappings, the existence of a fixed point follows from Delmling . It is shown in  that the set of fixed points for strongly pseudocontractions is a singleton.

By (v), since limnβn=0, there exists n0 such that for all nn0, (13)βnmin{14k,1-k(1+L)(1+3L)}, where k<1/2. Consider (14)xn+1-p2=xn+1-p,j(xn+1-p)=Syn-p,j(xn+1-p)=Txn+1-p,j(xn+1-p)22+Syn-Txn+1,j(xn+1-p)kxn+1-p2+Syn-Txn+1xn+1-p, which implies that (15)xn+1-p11-kSyn-Txn+1, where (16)Syn-Txn+1Syn-Tyn+Tyn-Txn+1xn-Syn+xn-Tyn+Tyn-Txn+1Sxn-Syn+Txn-Tyn+Tyn-Txn+1Sxn-Syn+L(xn-yn+yn-xn+1),(17)yn-xn+1yn-xn+xn-xn+1=yn-xn+xn-Synyn-xn+Sxn-Syn, and consequently from (16), we obtain (18)Syn-Txn+1(1+L)Sxn-Syn+2Lxn-yn(1+3L)xn-yn=(1+3L)βnxn-Txn(1+L)(1+3L)βnxn-p.

Substituting (18) in (15) and using (13), we get (19)xn+1-p(1+L)(1+3L)1-kβnxn-pxn-p.

So, from the above discussion, we can conclude that the sequence {xn-p} is bounded. Since T is Lipschitzian, so {Txn-p} is also bounded. Let M1=supn1xn-p+supn1Txn-p. Also by (ii), we have (20)xn-yn=βnxn-TxnM1βn0 as n, implying that {xn-yn} is bounded, so let M2=supn1xn-yn+M1. Further, (21)yn-pyn-xn+xn-pM2, which implies that {yn-p} is bounded. Therefore, {Tyn-p} is also bounded.

Set (22)M3=supn1yn-p+supn1Tyn-p.

Denote M=M1+M2+M3. Obviously, M<.

Now from (12) for all n1, we obtain (23)xn+1-p2=Syn-p2yn-p2, and by Lemma 6, we get (24)yn-p2=(1-βn)xn+βnTxn-p2=(1-βn)(xn-p)+βn(Txn-p)2(1-βn)2xn-p2+2βnTxn-p,j(yn-p)=(1-βn)2xn-p2+2βnTyn-p,j(yn-p)m+2βnTxn-Tyn,j(yn-p)(1-βn)2xn-p2+2kβnyn-p2m+2βnTxn-Tynyn-p(1-βn)2xn-p2+2kβnyn-p2m+2MLβnxn-yn, which implies that (25)yn-p2(1-βn)21-2kβnxn-p2+2MLβn1-2kβnxn-yn(1-βn)xn-p2+4MLβnxn-yn because by (13), we have ((1-βn)/(1-2kβn))1 and (1/(1-2kβn))2. Hence, (23) gives us (26)xn+1-p2(1-βn)xn-p2+4MLβnxn-yn.

For all n1, put (27)ρn=xn-p,θn=βn,ωn=4MLβnxn-yn, then according to Lemma 7, we obtain from (26) that (28)limnxn-p=0.

This completes the proof.

Corollary 9.

Let K be a nonempty closed convex subset of a real Hilbert space H, let S:KK be nonexpansive, and let T:KK be Lipschitz strongly pseudocontractive mappings such that pF(S)F(T) and the condition (C). Let {βn} be a sequence in [0,1] satisfying the conditions (iv) and (v).

For arbitrary x1K, let {xn} be a sequence iteratively defined by (12). Then the sequence {xn} converges strongly at the common fixed point p of S and T.

Example 10.

As a particular case, we may choose, for instance, βn=1/n.

Remark 11.

(1) The condition (C) is not new and it is due to Liu et al. .

(2) We prove our results for a hybrid iteration scheme, which is simple in comparison to the previously known iteration schemes.

Acknowledgment

This study was supported by research funds from Dong-A University.

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