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∈FS∩FT=x∈K:Sx=Tx=x and x-Sy≤Sx-Syandx-Ty≤Tx-Ty for all x,y∈K. Let βn be a sequence in 0,1 satisfying (i) ∑n=1∞βn=∞; (ii) limn→∞βn=0. For arbitrary x0∈K, let xn be a sequence iteratively defined by xn=Syn,yn=1-βnxn-1+βnTxn,n≥1. Then the sequence xn converges strongly to a common fixed point p of S and T.

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∈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 L>1 such that
(2)∥Tx-Ty∥≤L∥x-y∥
for all x,y∈K.

Definition 2.

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

Definition 3.

The mapping T is said to be pseudocontractive if
(4)∥x-y∥≤∥x-y+t((I-T)x-(I-T)y)∥
for all x,y∈K and t>0.

Remark 4.

As a consequence of a result of Kato [1], it follows from the inequality that T is pseudocontractive if and only if there exists j(x-y)∈J(x-y) such that
(5)〈Tx-Ty,j(x-y)〉≤∥x-y∥2
for all x,y∈K.

Definition 5.

The mapping T is said to be strongly pseudocontractive if there exists a constant t>1 such that
(6)∥x-y∥≤∥(1+r)(x-y)-rt(Tx-Ty)∥
for all x,y∈K and r>0. Or equivalently (see [2]) one has for 0<k<1(7)〈Tx-Ty,j(x-y)〉≤k∥x-y∥2
for all x,y∈K.

For a nonempty convex subset K of a normed space E, T:K→K is a mapping.

(I) The sequence {xn}, defined by, for arbitrary x1∈K,
(8)xn+1=(1-an)xn+anTyn,yn=(1-bn)xn+bnTxn,n≥1,
where {an} and {bn} are sequences in [0,1], is known as the Ishikawa iteration process [3].

If bn=0 for n≥1, then the Ishikawa iteration scheme becomes the Mann iteration process [4].

(S) The sequence {xn}, defined by, for arbitrary x1∈K,
(9)xn+1=Tyn,yn=(1-bn)xn+bnTxn,n≥1,
where {bn} is a sequence in [0,1], is known as the S-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., [7–13] and the references cited therein).

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

Theorem 6.

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
(10)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=1∞αnβn=∞.

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

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 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 to the unique fixed point of T.

In [16], Liu et al. introduced the following condition.

Remark 7.

Let S,T:K→K be two mappings. The mappings S and T are said to satisfy condition (C1) if
(C1)∥x-Ty∥≤∥Sx-Ty∥
for all x,y∈K.

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

Theorem 8.

Let K be a compact convex subset of a real Hilbert space H and let T:K→K be a Lipschitzian hemicontractive mapping satisfying
(C2)∥x-Ty∥≤∥Tx-Ty∥
for all x,y∈K. Let {βn} be a sequence in [0,1] satisfying

∑n=1∞βn=∞;

∑n=1∞βn2<∞.

For arbitrary x0∈K, let {xn} be a sequence iteratively defined by
(11)xn=Tyn,yn=(1-βn)xn-1+βnTxn,n≥1.
Then the sequence {xn} converges strongly to the fixed point x* of T.

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

Theorem 9.

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
(C3)∥x-Sy∥≤∥Sx-Sy∥,∥x-Ty∥≤∥Tx-Ty∥
for all x,y∈K. Let {βn} be a sequence in [0,1] satisfying

∑n=1∞βn=∞;

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 to a common fixed point p of S and T.

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 S-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 [<xref ref-type="bibr" rid="B19">19</xref>, <xref ref-type="bibr" rid="B20">20</xref>]).

Let J:E→2E* be the normalized duality mapping. Then for any x,y∈E, one has
(13)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,wwwwwww∀j(x+y)∈J(x+y).

Lemma 11 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let {ρn} and {θn} be nonnegative sequences satisfying
(14)ρn+1≤(1-θn)ρn+bn,
where θn∈[0,1), ∑n=1∞θn=∞, and bn=o(θn). Then limn→∞ρn=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 {βn} be a sequence in [0,1] satisfying

∑n=1∞βn=∞;

limn→∞βn=0.

For arbitrary x0∈K, let {xn} be a sequence iteratively defined by
(15)xn=Syn,yn=(1-βn)xn-1+βnTxn,n≥1.
Then the sequence {xn} converges strongly to a common fixed point p of S and T.

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 limn→∞βn=0, there exists n0∈N such that ∀n≥n0,
(16)βn≤min{14k,1-k2(1+L)(1+2L)},
where k<1/2 and L is a Lipschitz constant of T. Consider
(17)∥xn-p∥2=〈xn-p,j(xn-p)〉=〈Syn-p,j(xn-p)〉=〈Txn-p,j(xn-p)〉+〈Syn-Txn,j(xn-p)〉≤k∥xn-p∥2+∥Syn-Txn∥∥xn-p∥,
where
(18)∥Syn-Txn∥≤∥Syn-Tyn∥+∥Tyn-Txn∥≤∥xn-Syn∥+∥xn-Tyn∥+∥Tyn-Txn∥≤∥Sxn-Syn∥+∥Txn-Tyn∥+∥Tyn-Txn∥=∥Sxn-Syn∥+2∥Txn-Tyn∥≤(1+2L)∥xn-yn∥,(19)∥xn-yn∥≤∥xn-xn-1∥+∥xn-1-yn∥=∥Syn-xn-1∥+∥xn-1-yn∥≤∥Sxn-1-Syn∥+∥xn-1-yn∥≤2∥xn-1-yn∥=2βn∥xn-1-Txn∥≤2βn(∥xn-1-p∥+∥p-Txn∥)≤2βn(∥xn-1-p∥+L∥xn-p∥),
and consequently from (18) and (19), we obtain
(20)∥Syn-Txn∥≤2(1+2L)βn∥xn-1-p∥+2L(1+2L)βn∥xn-p∥.
Substituting (20) in (17) and using (16), we get
(21)∥xn-p∥≤2(1+2L)βn1-k-2L(1+2L)βn∥xn-1-p∥≤∥xn-1-p∥forn≥n0.
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
(22)∥xn-1-yn∥=βn∥xn-1-Txn∥≤M1βn⟶0
as n→∞, which implies that {xn-1-yn} is bounded, so let M2=supn≥1∥xn-1-yn∥+M1. Further
(23)∥yn-p∥≤∥yn-xn-1∥+∥xn-1-p∥≤M2,
which implies that {yn-p} is bounded. Therefore {Tyn-p} is also bounded.

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

Denote M=M1+M2+M3. Obviously M<∞.

Now, from (15), for all n≥1, we obtain
(25)∥xn-p∥2=∥Syn-p∥2≤∥yn-p∥2,
and by Lemma 10,
(26)∥yn-p∥2=∥(1-βn)xn-1+βnTxn-p∥2=∥(1-βn)(xn-1-p)+βn(Txn-p)∥2≤(1-βn)2∥xn-1-p∥2+2βn〈Txn-p,j(yn-p)〉=(1-βn)2∥xn-1-p∥2+2βn〈Tyn-p,j(yn-p)〉+2βn〈Txn-Tyn,j(yn-p)〉≤(1-βn)2∥xn-1-p∥2+2kβn∥yn-p∥2+2βn∥Txn-Tyn∥∥yn-p∥≤(1-βn)2∥xn-1-p∥2+2kβn∥yn-p∥2+2MLβn∥xn-yn∥,∀j(yn-p)∈J(yn-p),
which implies that
(27)∥yn-p∥2≤(1-βn)21-2kβn∥xn-1-p∥2+2MLβn1-2kβn∥xn-yn∥≤(1-βn)∥xn-1-p∥2+4MLβn∥xn-yn∥forn≥n0.
Because of (16), we have (1-βn)/(1-2kβn)≤1 and 1/(1-2kβn)≤2. Also, by (ii) and (19), ∥xn-yn∥≤2M(1+L)βn→0 as n→∞.

Hence (25) and (27) give
(28)∥xn-p∥2≤(1-βn)∥xn-1-p∥2+4MLβn∥xn-yn∥.

For all n≥1, put
(29)ρn=∥xn-1-p∥,θn=βn,bn=4MLβn∥xn-yn∥;
then according to Lemma 11, we obtain from (28) that
(30)limn→∞∥xn-p∥=0.
This completes the proof.

Corollary 13.

Let K be a nonempty closed convex subset of a real Hilbert space H, 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 {βn} be a sequence in [0,1] satisfying conditions (i) and (ii) in Theorem 12.

For arbitrary x0∈K, let {xn} be a sequence iteratively defined by (15). Then the sequence {xn} converges strongly to a common fixed point p of S and T.

Example 14.

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

Remark 15.

(1) Condition (C2) is due to Kang et al. [17] and condition (C1) with S=T becomes condition (C2).

(2) Condition (C3) is due to Kang et al. [18] and condition (C3) with S=T becomes condition (C2).

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.

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