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

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## 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.

### Keywords

Banach Space Nonexpansive Mapping Real Banach Space Common Fixed Point Nonempty Closed Convex Subset## 1 Introduction and preliminaries

*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 $J:E\to {2}^{{E}^{*}}$ defined by

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 {*x*_{ n }} ⊂ *C*, let *ω*_{ w }(*x*_{ n }) denote the weak *ω*-limit set.

*T*is said to be

*pseudocontractive*if for all

*x*,

*y*∈

*C*, there exists

*j*(

*x*-

*y*) ∈

*J*(

*x*-

*y*) such that

*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*)

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, 3, 4, 5]. In 2001, Xu and Ori [2] firstly introduced an implicit iterative *x*_{ n }= *α*_{ n }*x*_{n-1}+ (1 - *α*_{ n })*T*_{ n }*x*_{ n }, *n* ∈ ℕ, *x*_{0} ∈ *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* ∈ 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*

*(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*

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 G

*â*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 $\Omega \ne \mathrm{0\u0338}$, 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

*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:

Assume that *LIM*||*T*(*t*)*x*_{ n }- *T*(*t*)*x**|| ≤ ||*x*_{ n }- *x**||, ∀*x** ∈ *K*, *t* ≥ 0, where *K* := {*x** ∈ *C*: Φ(*x**) = min_{x∈C}Φ(*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 $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with $\mathfrak{F}:={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{0\u0338}$; Assume that

*sup*

_{t≥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:

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 $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$;

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 $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a strongly continuous semigroup of Lipschitz pseudocontractions from

*K*into itself with $\mathfrak{F}:={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{0\u0338}$. Assume that sup

_{t≥0}{

*L*(

*t*)} < ∞, where

*L*(

*t*) is the Lipschitz constant of the mapping

*T*(

*t*). Let {

*x*

_{ n }} be a sequence in

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*, $\text{sup}\left\{\u2225x-T\left(t\right)x\u2225:t\ge 0\right\}\ge f\left(\mathsf{\text{dist}}\left(x,\mathfrak{F}\right)\right)$. Then the sequence {*x*_{ n }} converges strongly to a common fixed point of the semigroup $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$.

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* {*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}: *C* → *C be a Lipschitian and pseudocontraction semigroup defined by Definition* 1.1 *with a bounded measurable function L*: [0, ∞) → [0, ∞). *Suppose* $F\left(T\right):={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{0\u0338}$. *Let α*_{ n }*and t*_{ n }*be sequences of real numbers satisfying t*_{ n }> 0, *α*_{ n }∈ [*a*, 1) ⊂ (0, 1) *and* lim_{n→∞}*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 })*x*_{n-1}+ *α*_{ n }*T*(*t*_{ n })*x*_{ n }, *n* ∈ ℕ, *x*_{0} ∈ *C* is well defined.

*n*∈ ℕ, we define a mapping

*S*

_{ n }as follows:

So *S*_{ n }is strongly pseudo-contraction, thus from Theorem *D*, there exists a point *x*_{ n }such that *x*_{ n }= (1 - *α*_{ n })*x*_{n-1}+ *α*_{ n }*T*(*t*_{ n })*x*_{ n }, that is the sequence {*x*_{ n }} defined by *x*_{ n }= (1 - *α*_{ n })*x*_{n-1}+ *α*_{ n }*T*(*t*_{ n })*x*_{ n }, *n* ∈ ℕ, *x*_{0} ∈ *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 lim_{n→∞}||*x*_{ n }- *p*|| exists.

*x*

_{ n }-

*p*|| ≤ (1 -

*α*

_{ n })||

*x*

_{n-1}-

*p*|| +

*α*

_{ n }||

*x*

_{ n }-

*p*||, that is

This implies that the limit lim_{n→∞}||*x*_{ n }- *p*|| exists.

(*III*). We prove lim_{n→∞}||*T*(*t*_{ n })*x*_{ n }- *x*_{ n }|| = 0.

*x*

_{ n }-

*p*||

_{n∈ℕ}} is bounded since lim

_{n→∞}||

*x*

_{ n }-

*p*|| exists, so the sequence {

*x*

_{ n }} is bounded. Since

*T*(

*t*

_{ n })

*x*

_{ n }} is bounded. In view of

_{n→∞}

*α*

_{ n }= 1, we have

(*IV*). Now we prove that for all *t* > 0, lim_{n→∞}||*T*(*t*)*x*_{ n }- *x*_{ n }|| = 0.

**T**: = {

*T*(

*t*) :

*t*≥ 0} is Lipschitian, for any

*k*∈ ℕ,

_{n→∞}||

*T*(

*t*

_{ n })

*x*

_{ n }-

*x*

_{ n }|| = 0, so for any

*k*∈ ℕ,

*T*(·) is continuous, we have

_{n→∞}||

*T*((

*k*+1)

*t*

_{ n })

*x*

_{ n }-

*T*(

*kt*

_{ n })

*x*

_{ n }|| = 0 as well as $\underset{n\to \infty}{\text{lim}}\u2225T\left(\left[\frac{t}{{t}_{n}}\right]{t}_{n}\right){x}_{n}-T\left(t\right){x}_{n}\u2225=0$, we can get

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

*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 ${\text{lim}}_{j\to \infty}\u2225T\left(t\right){x}_{{n}_{j}}-{x}_{{n}_{j}}\u2225=0$ from lim

_{n→∞}||

*T*(

*t*)

*x*

_{ n }-

*x*

_{ n }|| = 0, and

This manifests that *x* ∈ *F*(**T**). Because for any *p* ∈ *F*(**T**), lim_{n→∞}||*x*_{ n }- *p*|| exists, and ${\text{lim}}_{n\to \infty}\u2225{x}_{n}-x\u2225={\text{lim}}_{j\to \infty}\u2225{x}_{{n}_{j}}-x\u2225=0$, 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}: *C* → *C be a Lipschitian and pseudocontraction semigroup defined by Definition* 1.1 *with a bounded measurable function L*: [0, ∞) → [0, ∞). *Suppose* $F\left(T\right):={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{0\u0338}$. *Let α*_{ n }*and t*_{ n }*be sequences of real numbers satisfying t*_{ n }> 0, *α*_{ n }∈ [*a*, 1) ⊂ (0,1) *and* lim_{n→∞}*α*_{ n }= 1. *Then the sequence* {*x*_{ n }} *defined by x*_{ n }= (1 - *α*_{ n })*x*_{n-1}+ *α*_{ n }*T*(*t*_{ n })*x*_{ n }, *x*_{0} ∈ *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 lim_{n→∞}||*x*_{ n }- *p*|| exists and {*T*(*t*_{ n })*x*_{ n }} is bounded, for all *t* > 0, lim_{n→∞}||*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 ${x}_{{n}_{j}}\rightharpoonup x$. For any *t* > 0, we have ${\text{lim}}_{{n}_{j}\to \infty}\u2225T\left(t\right){x}_{{n}_{j}}-{x}_{{n}_{j}}\u2225=0$. 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* [10, 11, 12], *of cause extend many results in* [4, 5, 6, 7, 8].

## Notes

### Acknowledgements

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

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