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# Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space

Jing Quan1*, Shih-sen Chang2 and Min Liu1

Author Affiliations

1 Department of Mathematics, Yibin University, Yibin, Sichuan 644000, China

2 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

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Fixed Point Theory and Applications 2012, 2012:16  doi:10.1186/1687-1812-2012-16

 Received: 4 November 2011 Accepted: 15 February 2012 Published: 15 February 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The purpose of this article is to study the strong and weak convergence of implicit iterative sequence to a common fixed point for pseudocontractive semigroups in Banach spaces. The results presented in this article extend and improve the corresponding results of many authors.

### 1 Introduction and preliminaries

Throughout this article we assume that E is a real Banach space with norm ||·||, E* is the dual space of E; 〈·, ·〉 is the duality pairing between E and E*; C is a nonempty closed convex subset of E; ℕ denotes the natural number set; ℜ+ is the set of nonnegative real numbers; The mapping defined by

(1)

is called the normalized duality mapping. We denote a single valued normalized duality mapping by j.

Let T: C C be a nonlinear mapping; F(T) denotes the set of fixed points of mapping T, i.e., F(T) := {x C, x = Tx}. We use "→" to stand for strong convergence and "⇀" for weak convergence. For a given sequence {xn} ⊂ C, let ωw(xn) denote the weak ω-limit set.

Recall that T is said to be pseudocontractive if for all x, y C, there exists j(x - y) ∈ J(x - y) such that

(2)

T is said to be strongly pseudocontr active if there exists a constant α ∈ (0,1), such that for any x, y C, there exists j(x - y) ∈ J(x - y)

(3)

In recent years, many authors have focused on the studies about the existence and convergence of fixed points for the class of pseudocontractions. Especially in 1974, Deimling [1] proved the following existence theorem of fixed point for a continuous and strong pseudocontraction in a nonempty closed convex subset of Banach spaces.

Theorem D. Let E be a Banach space, C be a nonempty closed convex subset of E and T: C C be a continuous and strong pseudocontraction. Then T has a unique fixed point in C.

Recently, the problems of convergence of an implicit iterative algorithm to a common fixed point for a family of nonexpansive mappings or pseudocontractive mappings have been considered by several authors, see [2-5]. In 2001, Xu and Ori [2] firstly introduced an implicit iterative xn = αnxn-1 + (1 - αn)Tnxn, n ∈ ℕ, x0 C for a finite family of nonexpansive mappings and proved some weak convergence theorems to a common fixed point for a finite family of nonexpansive mappings in a Hilbert space. In 2004, Osilike [3] improved the results of Xu and Ori [2] from nonexpansive mappings to strict pseudocontractions in the framework of Hilbert spaces. In 2006, Chen et al. [4] extended the results of Osilike [3] to more general Banach spaces.

On the other hand, the convergence problems of semi-groups have been considered by many authors recently. Suzuki [6] considered the strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Xu [7] gave strong convergence theorem for contraction semigroups in Banach spaces. Chang et al. [8] proved the strong convergence theorem for nonexpansive semi-groups in Banach space. He also studied the weak convergence problems of the implicit iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces [9]. The pseudocontractive semi-groups is defined as follows.

Definition 1.1 (1) One-parameter family T: = {T(t): t ≥ 0} of mappings from C into itself is said to be a pseudo-contraction semigroup on C, if the following conditions are satisfied:

(a). T(0)x = x for each x C;

(b). T(t + s)x = T(s)T(t) for any t, s ∈ ℜ+ and x C;

(c). For any x C, the mapping t T(t)x is continuous;

(d). For all x, y C, there exists j(x - y) ∈ J(x - y) such that

(4)

(2) A pseudo-contraction semigroup of mappings from C into itself is said to be a Lipschitzian if the condition (a)-(d) and following condition (f) are satisfied.

(f) there exists a bounded measurable function L: [0, ∞) → [0, ∞) such that for any x, y C,

for any t > 0. In the sequel, we denote it by

(5)

Cho et al. [10] considered viscosity approximations with continuous strong pseudocontractions for a pseudocontraction semigroup and prove the following theorem.

Theorem Cho. Let E be a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm, and C be a nonempty closed convex subset of E. Let T(t): t ≥ 0 be a strongly continuous L-Lipschitz semigroup of pseudocontractions on C such that , where Ω is the set of common fixed points of semi-group T(t). Let f: C C be a fixed bounded, continuous and strong pseudocontraction with the coefficient α in (0,1), let αn and tn be sequences of real numbers satisfying αn ∈ (0, 1), tn > 0, and ; Let {xn} be a sequence generated in the following manner:

(6)

Assume that LIM||T(t)xn - T(t)x*|| ≤ ||xn - x*||, ∀x* K, t ≥ 0, where K := {x* C: Φ(x*) = minxC Φ(x)} with Φ(x) = LIM||xn - x||2, ∀x C. Then xn converges strongly to x* ∈ Ω which solves the following variational inequality: 〈(I - f)x*, j(x* - x)〉 ≤ 0, ∀x ∈ Ω.

Qin and Cho [11] established the theorems of weak convergence of an implicit iterative algorithm with errors for strongly continuous semigroups of Lipschitz pseudocontractions in the framework of real Banach spaces.

Theorem Q. Let E be a reflexive Banach space which satisfies Opial's condition and K a nonempty closed convex subset of E. Let be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with ; Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {xn} be a sequence generated by the following iterative process:

(7)

where {αn}, {βn}, {γn} are sequences in (0,1), {tn} is a sequence in (0, ∞) and {un} is a bounded sequence in K. Assume that the following conditions are satisfied:

(a) αn + βn + γn = 1;

(b) .

Then the sequence {xn} generated in (7) converges weakly to a common fixed point of the semigroup ;

Agarwal et al. [12] studied strongly continuous semigroups of Lipschitz pseudocontractions and proved the strong convergence theorems of fixed points in an arbitrary Banach space based on an implicit iterative algorithm.

Theorem A. Let E be an arbitrary Banach space and K a nonempty closed convex subset of E. Let be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with . Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {xn} be a sequence in

(8)

where {αn}, {βn}, {γn} are sequences in (0,1) such that αn + βn + γn = 1, {tn} is a sequence in (0, ∞) and {un} is a bounded sequence in K. Assume that , and there is a nondecreasing function f: (0, ∞) → (0, ∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0, ∞) such that, for all x C, . Then the sequence {xn} converges strongly to a common fixed point of the semigroup .

(9)

for a pseudocontraction semigroup T: = {T(t): t ≥ 0} in the framework of Banach spaces, which improves and extends the corresponding results of many author's. We need the following Lemma.

Lemma 1.1 [9]Let E be a real reflexive Banach space with Opial condition. Let C be a nonempty closed convex subset of E and T: C C be a continuous pseudocontractive mapping. Then I - T is demiclosed at zero, i.e., for any sequence {xn} ⊂ E, if xn y and ||(I - T)xn|| → 0, then (I - T)y = 0.

### 2 Main results

Theorem 2.1 Let E be a real Banach space and C be a nonempty compact convex subset of E. Let T: = {T(t): t ≥ 0}: C C be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose . Let αn and tn be sequences of real numbers satisfying tn > 0, αn ∈ [a, 1) ⊂ (0, 1) and limn→∞ αn = 1. Then the sequence {xn} defined by (9) converges strongly to a common fixed point x* F(T) in C.

Proof. We divide the proof into five steps.

(I). The sequence {xn} defined by xn = (1 - αn)xn-1 + αnT(tn)xn, n ∈ ℕ, x0 C is well defined.

In fact for all n ∈ ℕ, we define a mapping Sn as follows:

(10)

Then we have

(11)

So Sn is strongly pseudo-contraction, thus from Theorem D, there exists a point xn such that xn = (1 - αn)xn-1 + αnT(tn)xn, that is the sequence {xn} defined by xn = (1 - αn)xn-1 + αnT(tn)xn, n ∈ ℕ, x0 C is well defined.

(II). Since the common fixed-point set F(T) is nonempty let p F(T). For each p F(T), we prove that limn→∞ ||xn - p|| exists.

In fact

(12)

So we get ||xn - p|| ≤ (1 - αn)||xn-1 - p|| + αn||xn - p||, that is

This implies that the limit limn→∞ ||xn - p|| exists.

(III). We prove limn→∞ ||T(tn)xn - xn|| = 0.

The sequence {||xn - p||n∈ℕ} is bounded since limn→∞ ||xn - p|| exists, so the sequence {xn} is bounded. Since

(13)

This shows that {T(tn)xn} is bounded. In view of

and condition limn→∞ αn = 1, we have

(14)

(IV). Now we prove that for all t > 0, limn→∞ ||T(t)xn - xn|| = 0.

Since pseudocontraction semigroup T: = {T(t) : t ≥ 0} is Lipschitian, for any k ∈ ℕ,

(15)

Because limn→∞ ||T(tn)xn - xn|| = 0, so for any k ∈ ℕ,

(16)

Since

(17)

and T(·) is continuous, we have

(18)

So from

(19)

and limn→∞ ||T((k+1)tn)xn - T(ktn)xn|| = 0 as well as , we can get

(20)

(V). We prove {xn} converges strongly to an element of F(T).

Since C is a compact convex subset of E, we know there exists a subsequence , such that . So we have from limn→∞ ||T(t)xn - xn|| = 0, and

(21)

This manifests that x F(T). Because for any p F(T), limn→∞ ||xn - p|| exists, and , we have that {xn} converges strongly to an element of F(T). This completes the proof of Theorem 2.1.

Theorem 2.2 Let E be a reflexive Banach space satisfying the Opial condition and C be a nonempty closed convex subset of E. Let T: = {T(t): t ≥ 0}: C C be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose . Let αn and tn be sequences of real numbers satisfying tn > 0, αn ∈ [a, 1) ⊂ (0,1) and limn→∞ αn = 1. Then the sequence {xn} defined by xn = (1 - αn)xn-1 + αnT(tn)xn, x0 C, n ∈ ℕ, converges weakly to a common fixed point x* F(T) in C.

Proof. It can be proved as in Theorem 2.1, that for each p F(T), the limit limn→∞ ||xn - p|| exists and {T(tn)xn} is bounded, for all t > 0, limn→∞ ||T(t)xn - xn|| = 0. Since E is reflexive, C is closed and convex, {xn} is bounded, there exist a subsequence such that . For any t > 0, we have . By Lemma 1.1, x F(T(t)), ∀t > 0. Since the space E satisfies Opial condition, we see that ωw(xn) is a singleton. This completes the proof.

Remark 2.1 There is no other condition imposed on tn in the Theorems 2.1 and 2.2 except that in the definition of pseudo-contraction semigroups. So our results improve corresponding results of many authors such as [10-12], of cause extend many results in [4-8].

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.

### Acknowledgements

This work was supported by National Research Foundation of Yibin University (No.2011B07).

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