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*〉=∥x∥2,∥f*∥=∥x∥},∀x,y∈E,
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:K→K be a mapping.

Definition 1.

The mapping T is said to be Lipschitzian if there exists a constant L>1 such that
(2)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈K.

Definition 2.

The mapping T is said to be nonexpansive if
(3)∥Tx-Ty∥≤∥x-y∥,∀x,y∈K.

Definition 3.

The mapping T is said to be pseudocontractive if for all x,y∈K, there exists j(x-y)∈J(x-y) such that
(4)〈Tx-Ty,j(x-y)〉≤∥x-y∥2.

Definition 4.

The mapping T is said to be strongly pseudocontractive if for all x,y∈K, there exists k∈(0,1) such that
(5)〈Tx-Ty,j(x-y)〉≤k∥x-y∥2.

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

The sequence {xn} defined by, for arbitrary x1∈K,
(6)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1,
where {αn} and {βn} are sequences in [0,1], is known as the Ishikawa iteration process [1]. If βn=0 for n≥1, then the Ishikawa iteration process becomes the Mann iteration process [2].

The sequence {xn} defined by, for arbitrary x1∈K,
(7)xn+1=Tyn,yn=(1-βn)xn+βnTxn,n≥1,
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., [1]). 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., [5–10] and the references cited therein).

In 1974, Ishikawa [1] proved the following result.

Theorem 5.

Let K be a compact convex subset of a Hilbert space H and let T:K→K be a Lipschitzian pseudocontractive mapping. For arbitrary x1∈K, let {xn} be a sequence defined iteratively by
(8)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1,
where {αn} and {βn} are sequences satisfying

0≤αn≤βn≤1,

limn→∞βn=0,

∑n≥1αnβn=∞.

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

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 X is a real Banach space with a uniformly convex dual X*, K is a nonempty bounded closed convex subset of X, and T:K→K 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 [11].

2. Main Results

We will need the following lemmas.

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

Let J:E→2E be the normalized duality mapping. Then for any x,y∈E, one has
(9)∥x+y∥2≤∥x∥2+2〈y,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],∑n≥1θn=∞, and ωn=o(θn). Then
(11)limn→∞ρn=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:K→K be nonexpansive, and let T:K→K be Lipschitz strongly pseudocontractive mappings such that p∈F(S)∩F(T)={x∈K:Sx=Tx=x} and
(C)∥x-Sy∥≤∥Sx-Sy∥,∀x,y∈K,∥x-Ty∥≤∥Tx-Ty∥,∀x,y∈K.
Let {βn} be a sequence in [0,1] satisfying

∑n≥1βn=∞,

limn→∞βn=0.

For arbitrary x1∈K, let {xn} be a sequence iteratively defined by
(12)xn+1=Syn,yn=(1-βn)xn+βnTxn,n≥1.

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 [13]. It is shown in [11] that the set of fixed points for strongly pseudocontractions is a singleton.

By (v), since limn→∞βn=0, there exists n0∈ℕ such that for all n≥n0,
(13)βn≤min{14k,1-k(1+L)(1+3L)},
where k<1/2. Consider
(14)∥xn+1-p∥2=〈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)〉≤k∥xn+1-p∥2+∥Syn-Txn+1∥∥xn+1-p∥,
which implies that
(15)∥xn+1-p∥≤11-k∥Syn-Txn+1∥,
where
(16)∥Syn-Txn+1∥≤∥Syn-Tyn∥+∥Tyn-Txn+1∥≤∥xn-Syn∥+∥xn-Tyn∥+∥Tyn-Txn+1∥≤∥Sxn-Syn∥+∥Txn-Tyn∥+∥Tyn-Txn+1∥≤∥Sxn-Syn∥+L(∥xn-yn∥+∥yn-xn+1∥),(17)∥yn-xn+1∥≤∥yn-xn∥+∥xn-xn+1∥=∥yn-xn∥+∥xn-Syn∥≤∥yn-xn∥+∥Sxn-Syn∥,
and consequently from (16), we obtain
(18)∥Syn-Txn+1∥≤(1+L)∥Sxn-Syn∥+2L∥xn-yn∥≤(1+3L)∥xn-yn∥=(1+3L)βn∥xn-Txn∥≤(1+L)(1+3L)βn∥xn-p∥.

Substituting (18) in (15) and using (13), we get
(19)∥xn+1-p∥≤(1+L)(1+3L)1-kβn∥xn-p∥≤∥xn-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=supn≥1∥xn-p∥+supn≥1∥Txn-p∥. Also by (ii), we have
(20)∥xn-yn∥=βn∥xn-Txn∥≤M1βn→0
as n→∞, implying that {xn-yn} is bounded, so let M2=supn≥1∥xn-yn∥+M1. Further,
(21)∥yn-p∥≤∥yn-xn∥+∥xn-p∥≤M2,
which implies that {yn-p} is bounded. Therefore, {Tyn-p} is also bounded.

Set
(22)M3=supn≥1∥yn-p∥+supn≥1∥Tyn-p∥.

Denote M=M1+M2+M3. Obviously, M<∞.

Now from (12) for all n≥1, we obtain
(23)∥xn+1-p∥2=∥Syn-p∥2≤∥yn-p∥2,
and by Lemma 6, we get
(24)∥yn-p∥2=∥(1-βn)xn+βnTxn-p∥2=∥(1-βn)(xn-p)+βn(Txn-p)∥2≤(1-βn)2∥xn-p∥2+2βn〈Txn-p,j(yn-p)〉=(1-βn)2∥xn-p∥2+2βn〈Tyn-p,j(yn-p)〉m+2βn〈Txn-Tyn,j(yn-p)〉≤(1-βn)2∥xn-p∥2+2kβn∥yn-p∥2m+2βn∥Txn-Tyn∥∥yn-p∥≤(1-βn)2∥xn-p∥2+2kβn∥yn-p∥2m+2MLβn∥xn-yn∥,
which implies that
(25)∥yn-p∥2≤(1-βn)21-2kβn∥xn-p∥2+2MLβn1-2kβn∥xn-yn∥≤(1-βn)∥xn-p∥2+4MLβn∥xn-yn∥
because by (13), we have ((1-βn)/(1-2kβn))≤1 and (1/(1-2kβn))≤2. Hence, (23) gives us
(26)∥xn+1-p∥2≤(1-βn)∥xn-p∥2+4MLβn∥xn-yn∥.

For all n≥1, put
(27)ρn=∥xn-p∥,θn=βn,ωn=4MLβn∥xn-yn∥,
then according to Lemma 7, we obtain from (26) that
(28)limn→∞∥xn-p∥=0.

This completes the proof.

Corollary 9.

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

For arbitrary x1∈K, 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. [14].

(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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