Open Access

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

Fixed Point Theory and Applications20122012:16

DOI: 10.1186/1687-1812-2012-16

Accepted: 15 February 2012

Published: 15 February 2012

## 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; denotes the natural number set; + is the set of nonnegative real numbers; The mapping $J:E\to {2}^{{E}^{*}}$ defined by
$J\left(x\right)=\left\{{f}^{*}\in {E}^{*}:〈x,{f}^{*}〉={∥x∥}^{2};∥{f}^{*}∥=∥x∥,\phantom{\rule{1em}{0ex}}x\in E\right\}$
(1)

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

Let T: CC 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 {x n } C, let ω w (x n ) 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
$〈Tx-Ty,j\left(x-y\right)〉\le {∥x-y∥}^{2};$
(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)
$〈Tx-Ty,j\left(x-y\right)\le \alpha {∥x-y∥}^{2}.$
(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: CC 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 [25]. In 2001, Xu and Ori [2] firstly introduced an implicit iterative x n = α n xn-1+ (1 - α n )T n x n , n , x0 C for a finite family of nonexpansive mappings ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ 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 + and x C;

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

(d). For all x, y C, there exists j(x - y) J(x - y) such that
$〈T\left(t\right)x-T\left(t\right)y,j\left(x-y\right)\le {∥x-y∥}^{2},\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}any\phantom{\rule{2.77695pt}{0ex}}t>0.$
(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,
$∥T\left(t\right)x-T\left(t\right)y∥\le L\left(t\right)∥x-y∥$
for any t > 0. In the sequel, we denote it by
$L=\underset{t\ge 0}{\text{sup}}L\left(t\right)<\infty$
(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 Gâ teaux 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 $\Omega \ne 0̸$, where Ω is the set of common fixed points of semi-group T(t). Let f: CC be a fixed bounded, continuous and strong pseudocontraction with the coefficient α in (0,1), let α n and t n be sequences of real numbers satisfying α n (0, 1), t n > 0, and ${\text{lim}}_{n\to \infty }{t}_{n}={\text{lim}}_{n\to \infty }\frac{{\alpha }_{n}}{{t}_{n}}=0$; Let {x n } be a sequence generated in the following manner:
${x}_{n}=\left(1-{\alpha }_{n}\right)f\left({x}_{n}\right)+{\alpha }_{n}T\left({t}_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}\forall n\ge 1.$
(6)

Assume that LIM||T(t)x n - T(t)x*|| ≤ ||x n - x*||, x* K, t ≥ 0, where K := {x* C: Φ(x*) = minxCΦ(x)} with Φ(x) = LIM||x n - x||2, x C. Then x n 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 $\mathsc{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with $\mathfrak{F}:={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne 0̸$; Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x n } be a sequence generated by the following iterative process:
${x}_{0}\in K;{x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}T\left({t}_{n}\right){x}_{n}+{\gamma }_{n}{u}_{n};\phantom{\rule{1em}{0ex}}\forall n\ge 1;$
(7)

where {α n }, {β n }, {γ n } are sequences in (0,1), {t n } is a sequence in (0, ∞) and {u n } is a bounded sequence in K. Assume that the following conditions are satisfied:

(a) α n + β n + γ n = 1;

(b) ${\text{lim}}_{n\to \infty }{t}_{n}={\text{lim}}_{n\to \infty }\frac{{\alpha }_{n}+{\gamma }_{n}}{{t}_{n}}=0$.

Then the sequence {x n } generated in (7) converges weakly to a common fixed point of the semigroup $\mathsc{T}:=\left\{T\left(t\right):t\ge 0\right\}$;

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 $\mathsc{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with $\mathfrak{F}:={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne 0̸$. Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x n } be a sequence in
${x}_{0}\in K;{x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}T\left({t}_{n}\right){x}_{n}+{\gamma }_{n}{u}_{n};\phantom{\rule{1em}{0ex}}\forall n\ge 1,$
(8)

where {α n }, {β n }, {γ n } are sequences in (0,1) such that α n + β n + γ n = 1, {t n } is a sequence in (0, ∞) and {u n } is a bounded sequence in K. Assume that $\underset{n\to \infty }{\text{lim}}{t}_{n}=\underset{n\to \infty }{\text{lim}}\frac{{\alpha }_{n}+{\gamma }_{n}}{{t}_{n}}=0$, $\underset{n\to \infty }{\text{lim}}\frac{{\gamma }_{n}}{{\alpha }_{n}+{\gamma }_{n}}<\infty$ 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, $\text{sup}\left\{∥x-T\left(t\right)x∥:t\ge 0\right\}\ge f\left(\mathsf{\text{dist}}\left(x,\mathfrak{F}\right)\right)$. Then the sequence {x n } converges strongly to a common fixed point of the semigroup $\mathsc{T}:=\left\{T\left(t\right):t\ge 0\right\}$.

${x}_{n}=\left(1-{\alpha }_{n}\right){x}_{n-1}+{\alpha }_{n}T\left({t}_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}n\in ℕ,\phantom{\rule{1em}{0ex}}{x}_{0}\in C$
(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: CC be a continuous pseudocontractive mapping. Then I - T is demiclosed at zero, i.e., for any sequence {x n } E, if x n y and ||(I - T)x n || → 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}: CC be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose $F\left(T\right):={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne 0̸$. Let α n and t n be sequences of real numbers satisfying t n > 0, α n [a, 1) (0, 1) and limn→∞α n = 1. Then the sequence {x n } 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 {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x n , n , x0 C is well defined.

In fact for all n , we define a mapping S n as follows:
${S}_{n}x=\left(1-{\alpha }_{n}\right){x}_{n-1}+{\alpha }_{n}T\left({t}_{n}\right)x,\phantom{\rule{1em}{0ex}}n\in ,\phantom{\rule{1em}{0ex}}\forall x\in C.$
(10)
Then we have
$〈{S}_{n}x-{S}_{n}y,j\left(x-y\right)〉={\alpha }_{n}〈T\left({t}_{n}\right)x-T\left({t}_{n}\right)y,\phantom{\rule{1em}{0ex}}j\left(x-y\right)〉\le {\alpha }_{n}{∥x-y∥}^{2}.$
(11)

So S n is strongly pseudo-contraction, thus from Theorem D, there exists a point x n such that x n = (1 - α n )xn-1+ α n T(t n )x n , that is the sequence {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x 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→∞||x n - p|| exists.

In fact
$\begin{array}{ll}\hfill {∥{x}_{n}-p∥}^{2}& =〈{x}_{n}-p,\phantom{\rule{1em}{0ex}}j\left({x}_{n}-p\right)〉\phantom{\rule{2em}{0ex}}\\ =〈\left(1-{\alpha }_{n}\right)\left({x}_{n-1}-p\right)+{\alpha }_{n}\left(T\left({t}_{n}\right){x}_{n}-p\right),\phantom{\rule{1em}{0ex}}j\left(x-p\right)〉\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\alpha }_{n}\right)∥{x}_{n-1}-p∥∥{x}_{n}-p∥+{\alpha }_{n}{∥{x}_{n}-p∥}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
(12)
So we get ||x n - p|| ≤ (1 - α n )||xn-1- p|| + α n ||x n - p||, that is
$∥{x}_{n}-p∥\le ∥{x}_{n-1}-p∥.$

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

(III). We prove limn→∞||T(t n )x n - x n || = 0.

The sequence {||x n - p||n} is bounded since limn→∞||x n - p|| exists, so the sequence {x n } is bounded. Since
$\begin{array}{ll}\hfill ∥T\left({t}_{n}\right){x}_{n}∥& =∥\frac{{x}_{n}-\left(1-{\alpha }_{n}\right){x}_{n-1}}{{\alpha }_{n}}∥\phantom{\rule{2em}{0ex}}\\ \le \frac{∥{x}_{n}∥}{{\alpha }_{n}}+\frac{\left(1-{\alpha }_{n}\right)∥{x}_{n-1}∥}{{\alpha }_{n}}\phantom{\rule{2em}{0ex}}\\ \le \frac{∥{x}_{n}∥}{a}+\frac{\left(1-{\alpha }_{n}\right)∥{x}_{n-1}∥}{a},\phantom{\rule{2em}{0ex}}\end{array}$
(13)
This shows that {T(t n )x n } is bounded. In view of
$∥{x}_{n}-T\left({t}_{n}\right){x}_{n}∥=∥\left(1-{\alpha }_{n}\right)\left({x}_{n-1}-T\left({t}_{n}\right){x}_{n}\right)∥=∥1-{\alpha }_{n}∥\cdot ∥{x}_{n-1}-T\left({t}_{n}\right){x}_{n}∥$
and condition limn→∞α n = 1, we have
$\underset{n\to \infty }{\text{lim}}∥T\left({t}_{n}\right){x}_{n}-{x}_{n}∥=0.$
(14)

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

Since pseudocontraction semigroup T: = {T(t) : t ≥ 0} is Lipschitian, for any k ,
$\begin{array}{l}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}∥T\left(\left(k+1\right){t}_{n}\right){x}_{n}-T\left(k{t}_{n}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥T\left(k{t}_{n}\right)T\left({t}_{n}\right){x}_{n}-T\left(k{t}_{n}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ \le L\left(k{t}_{n}\right)∥T\left({t}_{n}\right){x}_{n}-{x}_{n}∥\phantom{\rule{2em}{0ex}}\\ \le L∥T\left({t}_{n}\right){x}_{n}-{x}_{n}∥.\phantom{\rule{2em}{0ex}}\end{array}$
(15)
Because limn→∞||T(t n )x n - x n || = 0, so for any k ,
$\underset{n\to \infty }{\text{lim}}∥T\left(\left(k+1\right){t}_{n}\right){x}_{n}-T\left(k{t}_{n}\right){x}_{n}∥=0.$
(16)
Since
$\begin{array}{l}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}∥T\left(t\right){x}_{n}-T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right)T\left(t-\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ \le L∥T\left(t-\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-{x}_{n}∥\phantom{\rule{2em}{0ex}}\end{array}$
(17)
and T(·) is continuous, we have
$\underset{n\to \infty }{\text{lim}}∥T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-T\left(t\right){x}_{n}∥=0.$
(18)
So from
$\begin{array}{l}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}∥{x}_{n}-T\left(t\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ \le \sum _{k=0}^{\left[\frac{t}{{t}_{n}}\right]-1}∥T\left(\left(k+1\right){t}_{n}\right){x}_{n}-T\left(k{t}_{n}\right){x}_{n}∥+∥T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-T\left(t\right){x}_{n}∥,\phantom{\rule{2em}{0ex}}\end{array}$
(19)
and limn→∞||T((k+1)t n )x n - T(kt n )x n || = 0 as well as $\underset{n\to \infty }{\text{lim}}∥T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-T\left(t\right){x}_{n}∥=0$, we can get
$\underset{n\to \infty }{\text{lim}}∥T\left(t\right){x}_{n}-{x}_{n}∥=0.$
(20)

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

Since C is a compact convex subset of E, we know there exists a subsequence $\left\{{x}_{{n}_{j}}\right\}\subset \left\{{x}_{n}\right\}$, such that ${x}_{{n}_{j}}\to x\in C$. So we have ${\text{lim}}_{j\to \infty }∥T\left(t\right){x}_{{n}_{j}}-{x}_{{n}_{j}}∥=0$ from limn→∞||T(t)x n - x n || = 0, and
$∥x-T\left(t\right)x∥=\underset{j\to \infty }{\text{lim}}∥T\left(t\right){x}_{{n}_{j}}-{x}_{{n}_{j}}∥=0.$
(21)

This manifests that x F(T). Because for any p F(T), limn→∞||x n - p|| exists, and ${\text{lim}}_{n\to \infty }∥{x}_{n}-x∥={\text{lim}}_{j\to \infty }∥{x}_{{n}_{j}}-x∥=0$, we have that {x n } 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}: CC be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose $F\left(T\right):={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne 0̸$. Let α n and t n be sequences of real numbers satisfying t n > 0, α n [a, 1) (0,1) and limn→∞α n = 1. Then the sequence {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x n , x0 C, 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→∞||x n - p|| exists and {T(t n )x n } is bounded, for all t > 0, limn→∞||T(t)x n - x n || = 0. Since E is reflexive, C is closed and convex, {x n } is bounded, there exist a subsequence $\left\{{x}_{{n}_{j}}\right\}\subset \left\{{x}_{n}\right\}$ such that ${x}_{{n}_{j}}⇀x$. For any t > 0, we have ${\text{lim}}_{{n}_{j}\to \infty }∥T\left(t\right){x}_{{n}_{j}}-{x}_{{n}_{j}}∥=0$. By Lemma 1.1, x F(T(t)), t > 0. Since the space E satisfies Opial condition, we see that ω w (x n ) is a singleton. This completes the proof.

Remark 2.1 There is no other condition imposed on t n 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 [1012], of cause extend many results in [48].

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Department of Mathematics, Yibin University
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

## References

1. Deimling K: Zeros of accretive operators. Manuscripta Math 1974, 13: 365–374. 10.1007/BF01171148
2. Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Numer Funct Anal Optim 2001, 22: 767–773. 10.1081/NFA-100105317
3. Osilike MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J Math Anal Appl 2004, 294: 73–81. 10.1016/j.jmaa.2004.01.038
4. Chen RD, Song YS, Zhai HY: Convergence theorems for implicit iteration press for a finite family of continuous pseudocontractive mappings. J Math Anal Appl 2006, 314: 701–709. 10.1016/j.jmaa.2005.04.018
5. Zhou HY: Convergence theorems of common fixed points for a finite family of Lipschitzian pseudocontractions in Banach spaces. Nonlinear Anal 2008, 68: 2977–2983. 10.1016/j.na.2007.02.041
6. Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc Am Math Soc 2003, 131: 2133–2136. 10.1090/S0002-9939-02-06844-2
7. Xu HK: A strong convergence theorem for contraction semigruops in Banach spaces. Bull Aust Math Soc 2005, 72: 371–379. 10.1017/S000497270003519X
8. Chang SS, Yang L, Liu JA: Strong convergence theorem for nonexpansive semigroups in Banach spaces. Appl Math Mech 2007, 28: 1287–1297. 10.1007/s10483-007-1002-x
9. Zhang SS: Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semigroups in Banach spaces. Appl Math Mech (English Edition) 2009, 30(2):145–152. 10.1007/s10483-009-0202-y
10. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
11. Qin X, Cho SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl Math Lett 2010, 23: 1252–1255. 10.1016/j.aml.2010.06.008
12. Agarwal RP, Qin X, Kang SM: Strong convergence theorems for strongly continuous semigroups of pseudocontractions. Appl Math Lett 2011, 24: 1845–1848. 10.1016/j.aml.2011.05.003