International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 549364
Research Article
Open Access

Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of Strict Pseudocontractions in Banach Spaces

Phayap Katchang, 

Phayap Katchang

Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

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Wiyada Kumam, 

Wiyada Kumam

Department of Mathematics, Faculty of Science, King Mongkut′s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand kmutt.ac.th

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Usa Humphries, 

Usa Humphries

Department of Mathematics, Faculty of Science, King Mongkut′s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand kmutt.ac.th

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Poom Kumam, 

Corresponding Author

Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut′s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand kmutt.ac.th

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First published: 10 May 2011
https://doi.org/10.1155/2011/549364
Academic Editor: Vittorio Colao

Abstract

We introduce a new modified Ishikawa iterative process and a new W-mapping for comput- ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.

1. Introduction

Let E be a real Banach space with norm ∥·∥ and C a nonempty closed convex subset of E. Let E* be the dual space of E, and let 〈·, ·〉 denotes the generalized duality pairing between E and E*. For q > 1, the generalized duality mapping is defined by
(1.1)
for all x ∈ E. In particular, if q = 2, the mapping J = J2 is called the normalized duality mapping and Jq(x) = ∥x∥q−2J2(x) for x ≠ 0. It is well known that if E is smooth, then Jq is single-valued, which is denoted by jq.
A mapping T : C → C is called nonexpansive if
(1.2)
We use F(T) to denote the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}.
T is said to be a λ-strict pseudocontraction in the terminology of Browder and Petryshyn [1] if there exists a constant λ > 0 and for some jq(x − y) ∈ Jq(x − y) such that
(1.3)
T is said to be a strong pseudocontraction if there exists k ∈ (0,1) such that
(1.4)

Remark 1.1 (see [2].)Let T be a λ-strict pseudocontraction in a Banach space. Let x ∈ C and p ∈ F(T). Then,

(1.5)

Recall that a self mapping f : C → C is contraction on C if there exists a constant α ∈ (0,1) and x, y ∈ C such that
(1.6)
We use ΠC to denote the collection of all contractions on C. That is, ΠC = {f∣f : C → C  a  contraction}. Note that each f ∈ ΠC has a unique fixed point in C.
Very recently, Cai and Hu [3] also proved the strong convergence theorem in Banach spaces. They considered the following iterative algorithm:
(1.7)
where Ti is a non-self-λi-strictly pseudocontraction, f is a contraction, and A is a strongly positive linear bounded operator.
Dong et al. [2] proved the sequence {xn} converges strongly in Banach spaces under certain appropriate assumptions and used the Wn mapping defined by (1.11). Let the sequences {xn} be generated by
(1.8)
On the other hand, Katchang and Kumam [4, 5] introduced the following new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces; let the sequences {xn} be generated by
(1.9)
where f is a contraction, A is a strongly positive linear bounded self-adjoint operator, and Wn mapping (see [6, 7]) is defined by
(1.10)
where T1, T2, … is an infinite family of nonexpansive mappings of C into itself and λ1, λ2, … is real numbers such that 0 ≤ λn ≤ 1 for every n ∈ ℕ. In 2010, Cho [8] considered and proved the strong convergence of the implicit iterative process for an infinite family of strict pseudocontractions in an arbitrary real Banach space.
In this paper, motivated and inspired by Cai and Hu [9], we consider the mapping Wn defined by
(1.11)
where t1, t2, … are real numbers such that 0 ≤ tn ≤ 1.  Tn,k = θn,kSk + (1 − θn,k)I, where Sk is a λk-strict pseudocontraction of C into itself and θn,k ∈ (0, μ], μ = min {1, {qλ/Cq} 1/q−1}, where λ = inf  λk for all k ∈ ℕ. By Lemma 2.3, we know that Tn,k is a nonexpansive mapping, and therefore, Wn is a nonexpansive mapping. We note that the W-mapping (1.10) is a special case of a W-mapping (1.11) when θn,k = θk is constant for all n ≥ 1.
Throughout this paper, we will assume that inf λi > 0, 0 < tn ≤ b < 1 for all n ∈ ℕ and {θn,k} satisfies
  • (H1)

    for all k ∈ ℕ,

  • (H2)

    |θn+1,k − θn,k | ≤ an for all n ∈ ℕ and 1 ≤ k ≤ n, where {an} satisfies .

The hypothesis (H2) secures the existence of for all k ∈ ℕ. Set for all k ∈ ℕ. Furthermore, we assume that
  • (H3)

    θ1,k > 0 for all k ∈ ℕ.

It is obvious that θ1,k satisfy (H1). Using condition (H3), from Tn,k = θn,kSk + (1 − θn,k)I, we define mappings T1,kx : = lim n→∞Tn,kx = θ1,kSkx + (1 − θ1,k)x for all x ∈ C.

Our results improve and extend the recent ones announced by Cai and Hu [3], Dong et al. [2], Katchang and Kumam [4, 5], and many others.

2. Preliminaries

Recall that U = {x ∈ E : ∥x∥ = 1}. A Banach space E is said to be uniformly convex if, for any ϵ ∈ (0,2], there exists δ > 0 such that for any x, y ∈ U, ∥x − y∥≥ϵ implies ∥(x + y)/2∥≤1 − δ. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [10]). A Banach space E is said to be smooth if the limit lim t→0(∥x + ty∥−∥x∥)/t exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U.

In a smooth Banach space, we define an operator A as strongly positive if there exists a constant with the property
(2.1)
where I is the identity mapping and J is the normalized duality mapping.
If C and D are nonempty subsets of a Banach space E such that C is a nonempty closed convex and D ⊂ C, then a mapping Q : C → D is sunny [11, 12] provided that Q(x + t(x − Q(x))) = Q(x) for all x ∈ C and t ≥ 0 whenever x + t(x − Q(x)) ∈ C. A mapping Q : C → C is called a retraction if Q2 = Q. If a mapping Q : C → C is a retraction, then Qz = z for all z in the range of Q. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [11, 12]: if E is a smooth Banach space, then Q : C → D is a sunny nonexpansive retraction if and only if there holds the inequality
(2.2)

We need the following lemmas for proving our main results.

Lemma 2.1 (see [13].)In a Banach space E, the following holds:

(2.3)
where j(x + y) ∈ J(x + y).

Lemma 2.2 (see [14].)Let E be a real q-uniformly smooth Banach space, then there exists a constant Cq > 0 such that

(2.4)
In particular, if E be a real 2-uniformly smooth Banach space with the best smooth constant K, then the following inequality holds:
(2.5)

The relation between the λ-strict pseudocontraction and the nonexpansive mapping can be obtained from the following lemma.

Lemma 2.3 (see [15].)Let C be a nonempty convex subset of a real q-uniformly smooth Banach space E and S : C → C a λ-strict pseudocontraction. For α ∈ (0,1), one defines Tx = (1 − α)x + αSx. Then, as α ∈ (0, μ], μ = min {1, {qλ/Cq} 1/q−1}, T : C → C is nonexpansive such that F(T) = F(S), where Cq is the constant in Lemma 2.2.

Concerning Wn, we have the following lemmas, which are important to prove the main results.

Lemma 2.4 (see [2].)Let C be a nonempty closed convex subset of a q-uniformly smooth and strictly convex Banach space E. Let Si, i = 1,2, …, be a λi-strict pseudocontraction from C into itself such that , and let inf λi > 0. Let tn, n = 1,2, …, be real numbers such that 0 < tn ≤ b < 1 for any n ≥ 1. Assume that the sequence {θn,k} satisfies (H1) and (H2). Then, for every x ∈ C and k ∈ ℕ, the limit lim n→∞Un,kx exists.

Using Lemma 2.4, we define the mappings U1,k and W : C → C as follows:
(2.6)
for all x ∈ C. Such W is called the W-mapping generated by S1, S2, …,   t1, t2, … and θn,k, for  all  n ∈ ℕ and 1 ≤ k ≤ n.

Lemma 2.5 (see [2].)Let {xn} be a bounded sequence in a q-uniformly smooth and strictly convex Banach space E. Under the assumptions of Lemma 2.4, it holds

(2.7)

Lemma 2.6 (see [2].)Let C be a nonempty closed convex subset of a q-uniformly smooth and strictly convex Banach space E. Let Si, i = 1,2, …, be a λi-strict pseudocontraction from C into itself such that , and let inf λi > 0. Let tn, n = 1,2, …, be real numbers such that 0 < tn ≤ b < 1 for any n ≥ 1. Assume that the sequence {θn,k} satisfies (H1)–(H3). Then, .

Lemma 2.7 (see [16].)Assume that {an} is a sequence of nonnegative real numbers such that

(2.8)
where {αn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that
  • (i)

    ,

  • (ii)

    limsup n→∞δn/αn ≤ 0 or .

Then, lim n→∞an = 0.

Lemma 2.8 (see [17].)Let {xn} and {yn} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0 < liminf n→∞βn ≤ limsup n→∞βn < 1. Suppose that xn+1 = (1 − βn)yn + βnxn for all integers n ≥ 0 and limsup n→∞(∥yn+1 − yn∥−∥xn+1 − xn∥) ≤ 0. Then, lim n→∞∥yn − xn∥ = 0.

Lemma 2.9 (see [3].)Assume that A is a strong positive linear bounded operator on a smooth Banach space E with coefficient and 0 < ρ ≤ ∥A∥−1. Then, .

3. Main Results

In this section, we prove a strong convergence theorem.

Theorem 3.1. Let E be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. Let A be a strongly positive linear bounded operator on E with coefficient such that , and let f be a contraction of C into itself with coefficient α ∈ (0,1). Let Si,   i = 1,2, …, be λi-strict pseudocontractions from C into itself such that and inf λi > 0. Assume that the sequences {αn}, {βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:

  • (i)

    ; and lim n→∞αn = 0,

  • (ii)

    0 < liminf n→∞ βn ≤ limsup n→∞ βn < 1,

  • (iii)

    lim n→∞ | γn+1 − γn | = 0,

  • (iv)

    lim n→∞ | δn+1 − δn | = 0,

  • (v)

    δn(1 + γn) − 2γn > a for some a ∈ (0,1),

and the sequence {θn,k} satisfies (H1)–(H3). Then, the sequence {xn} generated by
(3.1)
converges strongly to , which solves the following variational inequality:
(3.2)

Proof. By (i), we may assume, without loss of generality, that αn ≤ (1 − βn)∥A∥−1 for all n. Since A is a strongly positive bounded linear operator on E and by (2.1), we have

(3.3)
Observe that
(3.4)
This shows that (1 − βn)I − αnA is positive. It follows that
(3.5)

First, we show that {xn} is bounded. Let . By the definition of {zn}, {yn}, and {xn}, we have

(3.6)
and from this, we have
(3.7)

It follows that

(3.8)
By induction on n, we obtain for every n ≥ 0 and x0 ∈ C, then {xn} is bounded. So, {yn}, {zn}, {Ayn}, {Wnxn}, {Wnzn}, and {f(xn)} are also bounded.

Next, we claim that ∥xn+1 − xn∥→0 as n → ∞. Let x ∈ C and . Fix k ∈ ℕ for any n ∈ ℕ with n ≥ k, and since Tn,k and Un,k are nonexpansive, we have ∥Tn,kx − p∥≤∥x − p∥ and ∥Un,kx − p∥≤∥x − p∥, respectively. From (1.5), it follows that . We can set

(3.9)

From (1.11), we have

(3.10)
for all n ≥ 0. Similarly, we also have ∥Wn+1zn − Wnzn∥≤(bn+1 + an(b/(1 − b)))M2 for all n ≥ 0. We compute that
(3.11)
and
(3.12)
where . Observe that we put ln = (xn+1 − βnxn)/1 − βn, then
(3.13)
Now, we have
(3.14)
Therefore, we have
(3.15)
From the conditions (i)–(iv), (H2), 0 < b < 1 and the boundedness of {xn}, {f(xn)}, {Ayn}, {Wnxn}, and {Wnzn}, we obtain
(3.16)
It follows from Lemma 2.8 that lim n→∞∥ln − xn∥ = 0. Noting (3.13), we see that
(3.17)
as n → ∞. Therefore, we have
(3.18)
We also have ∥yn+1 − yn∥→0 and ∥zn+1 − zn∥→0 as n → ∞. Observing that
(3.19)
it follows that
(3.20)
By the conditions (i), (ii), (3.18), and the boundedness of {xn}, {f(xn)}, and {Ayn}, we obtain
(3.21)
Consider
(3.22)
It follows that
(3.23)
This implies that
(3.24)
From the condition (v) and (3.21), we get
(3.25)
On the other hand,
(3.26)
From the boundedness of {xn} and using (2.7), we have ∥Wxn − Wnxn∥→0 as n → ∞. It follows that
(3.27)

Next, we prove that

(3.28)
where x* = lim t→0xt with xt being the fixed point of contraction x ↦ tγf(x)+(1 − tA)Wx. Noticing that xt solves the fixed point equation xt = tγf(xt)+(1 − tA)Wxt, it follows that
(3.29)
It follows from Lemma 2.1 that
(3.30)
where
(3.31)
Since A is linearly strong and positive and using (2.1), we have
(3.32)
Substituting (3.32) in (3.30), we have
(3.33)
Letting n → ∞ in (3.33) and noting (3.31) yield that
(3.34)
where M3 > 0 is a constant such that for all t ∈ (0,1) and n ≥ 0. Taking t → 0 from (3.34), we have
(3.35)
On the other hand, we have
(3.36)
which implies that
(3.37)

Noticing that J is norm-to-norm uniformly continuous on bounded subsets of C, it follows from (3.35) that

(3.38)

Therefore, we obtain that (3.28) holds.

Finally, we prove that xn → x* as n → ∞. Now, from Lemma 2.1, we have

(3.39)
and consequently,
(3.40)
where M4 is an appropriate constant such that M4 ≥ sup n≥0∥xn − x*∥2. Setting and , then we have
(3.41)
By (3.28), (i) and applying Lemma 2.7 to (3.41), we have xn → x* as n → ∞. This completes the proof.

Corollary 3.2. Let E be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. Let A be a strongly positive linear bounded operator on E with coefficient such that , and let f be a contraction of C into itself with coefficient α ∈ (0,1). Let Si,   i = 1,2, …, be λi-strict pseudocontractions from C into itself such that and inf λi > 0. Assume that the sequences {αn}, {βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:

  • (i)

    ; and lim n→∞ αn = 0,

  • (ii)

    0 < liminf n→∞ βn ≤ limsup n→∞ βn < 1,

  • (iii)

    lim n→∞ | γn+1 − γn | = 0,

  • (iv)

    lim n→∞ | δn+1 − δn | = 0,

  • (v)

    δn(1 + γn) − 2γn > a for some a ∈ (0,1),

and the sequence {θn} satisfies (H1). Then, the sequence {xn} generated by
(3.42)
converges strongly to , which solves the following variational inequality:
(3.43)

Corollary 3.3. Let E be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C + C ⊂ C. Let A be a strongly positive linear bounded operator on E with coefficient such that , and let f be a contraction of C into itself with coefficient α ∈ (0,1). Let Si,   i = 1,2, …, be a nonexpansive mapping from C into itself such that and inf λi > 0. Assume that the sequences {αn}, {βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:

  • (i)

    ; and lim n→∞ αn = 0,

  • (ii)

    0 < liminf n→∞ βn ≤ limsup n→∞ βn < 1,

  • (iii)

    lim n→∞ | γn+1 − γn | = 0,

  • (iv)

    lim n→∞ | δn+1 − δn | = 0,

  • (v)

    δn(1 + γn) − 2γn > a for some a ∈ (0,1).

Then, the sequence {xn} generated by
(3.44)
converges strongly to , which solves the following variational inequality:
(3.45)

Remark 3.4. Theorem 3.1, Corollaries 3.2, and 3.3, improve and extend the corresponding results of Cai and Hu [3], Dong et al. [2], and Katchang and Kumam [4, 5] in the following senses.

  • (i)

    For the mappings, we extend the mappings from an infinite family of nonexpansive mappings to an infinite family of strict pseudocontraction mappings.

  • (ii)

    For the algorithms, we propose new modified Ishikawa iterative algorithms, which are different from the ones given in [2–5] and others.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT) and the Faculty of Science KMUTT for financial support. Furthermore, they also would like to thank the National Research University Project of Thailand′s Office of the Higher Education Commission for financial support (NRU-CSEC Project no. 54000267). Finally, they are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work.

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