# Convergence theorem for finite family of lipschitzian demi-contractive semigroups

- Bashir Ali
^{1}Email author and - Godwin Chidi Ugwunnadi
^{1}

**2011**:18

https://doi.org/10.1186/1687-1812-2011-18

© Ali and Ugwunnadi; licensee Springer. 2011

**Received: **6 March 2011

**Accepted: **23 July 2011

**Published: **23 July 2011

## Abstract

Let *E* be a real Banach space and *K* be a nonempty, closed, and convex subset of *E*. Let
be a finite family of Lipschitzian demi-contractive semigroups of *K*, with sequences of bounded measurable functions *L*_{
i
} : [0, ∞) → (0, ∞) and bounded functions *λ*_{
i
} : [0, ∞) → (0, ∞), respectively, where
, *i* = 1,2, ..., *N*. Strong convergence theorem for common fixed point for finite family
is proved in a real Banch space. As an application, a new convergence theorem for finite family of Lipschitzian demi-contractive maps is also proved.

**Mathematics subject classification (2000)** 47H09, 47J25

## Keywords

## 1. Introduction

*E*be a real Banach space and

*E** be the dual space of

*E*. The normalized duality mapping is defined by,

*x*∈

*E*,

where 〈., .〉 denotes the normalized duality pairing. For any *x* ∈ *E*, an element of

*Jx* is denoted by *j*(*x*).

Let *K* be a nonempty, closed and convex subset of *E*. Let *T* : *K* → *K* be a map, a point *x* ∈ *K* is called a fixed point of *T* if *Tx* = *x*, and the set of all fixed points of *T* is denoted by *F*(*T*). The mapping *T* is called *L*-Lipschitzian or simply Lipschitz if ∃*L >* 0, such that ||*Tx* -*Ty*|| ≤ *L*||*x* - *y*|| ∀*x*, *y* ∈ *K* and if *L* = 1, then the map *T* is called *nonexpansive*.

*K*is called a

*nonexpansive semigroup*if the following conditions are satisfied,

- (i)
*T*(0)*x*=*x*∀*x*∈*K*; - (ii)
*T*(*t*+*s*) =*T*(*t*) ∘*T*(*s*) ∀*t*,*s*≥ 0; - (iii)
for each

*x*∈*K*, the mapping*t*→*T*(*t*)*x*is continuos; - (iv)
for

*x*,*y*∈*K*and*t*≥ 0, ||*T*(*t*)*x*-*T*(*t*)*y*|| ≤ ||*x*-*y*||.

*x*,

*y*∈

*K*;

- (c)

*x*∈

*K*and

*q*∈

*F*(

*T*(

*t*));

- (d)
*Lipschitzian semigroup*if there is a bounded measurable function

It is known that every strictly pseudocontractive semigroup is Lipschitzian, and every strictly pseudocontractive semigroup with fixed point is demi-contractive semi-group.

Let *E* be a real Banach space and let *K* be a nonempty closed convex subset of *E*. A mapping *T* : *K* → *K* is *demicompact* if for every bounded sequence {*x*_{
n
}} in *K* such that {*xn* - *Tx*_{
n
}} converges, and there exists a subsequence, say
of {*x*_{
n
}} that converges strongly to some *x** in *K*. *T* is said to be demi-contractive if *F*(*T*) ≠ ∅, and there exists *λ >* 0 such that 〈*Tx*- *q*, *j*(*x* - *q*)〉 ≤ ||*x* - *q*||^{2} - λ||*x* - *Tx*||^{2} ∀ *x* ∈ *K*, *q* ∈ *F*(*T*) and *j*(*x* - *q*) ∈ *J*(*x* - *q*).

Let *T*_{1}, *T*_{2}, ..., *T*_{
N
} be a family of self-mappings of *K* such that
. Then, the family is said to satisfy condition
if there exists a nondecreasing function *f* : [0, ∞) → [0, ∞) with *f* (0) = 0 and *f* (*r*) *>* 0 ∀ *r* ∈ (0, ∞) such that *f* (*d*(*x*, *F*)) ≤ ||*x* - *T*_{
s
}*x*|| for some *s* in {1, 2, ..., *N*} and for all *x* ∈ *K*, where *d*(*x*, *F*) = *inf* {||*x* - *q*|| : *q* ∈ *F*}.

Existence theorems for family of nonexpansive mappings are proved in [1–5] and actually many others. Recently, Suzuki [6] proved the equivalence between the fixed point property for nonexpansive mappings and that of the nonexpansive semi-groups.

Both implicit and explicit, Mann, Ishikawa, and Halpern-type schemes were studied for approximation of common fixed points of family of nonexpansive semigroups and their generalizations in various spaces; see, for example [6–13], to list but a few.

*x*,

*x*

_{1}∈

*K*,

converges strongly to a common fixed point of the family of nonexpansive semigroup in a real Hilbert space. Xu [9] extended the result of Suzuki to a more general real uniformly convex Banach space having a weakly sequentially continuous duality mapping.

*x*,

*x*

_{1}∈

*K*,

*T*(

*t*) :

*t*≥ 0} of nonexpansive semigroup in a reflexive Banach space with uniformly Gatéuax differentiable norm. Recently, Zhang et al. [11] introduced and studied a composite iterative scheme defined by

*x*,

*x*

_{1}∈

*K*,

Those authors proved strong convergence of the sequence {*x*_{
n
}} to a common fixed point of the family {*T*(*t*) : *t* ≥ 0} of nonexpansive semigroup.

Very recently, Chang et al. [12] proved a strong convergence theorem which extended and improved the results in [10, 9] and some others. They proved the following theorem.

**Theorem 1.1**.

*Chang et al.*[12]

*Let K be a nonempty, closed, and convex subset of a real Banach space E: Let*

*be a Lipschitzian demi-contractive semigroup of K with bounded measurable function L*: [0, ∞) → (0, ∞)

*and bounded function λ*: [0, ∞) → (0, ∞)

*such that*

*Let*{

*t*

_{ n }}

*be an increasing sequence in*[0, ∞)

*and*{

*α*

_{ n }}

*be a sequence in (0,1) satisfying the following conditions*,

- (i)

*Then, the sequence* {*x*_{
n
}} *converges strongly to some element in F*.

The purpose in this article is to prove a strong convergence theorem for common fixed point for finite families of demi-contractive semigroups in a real Banach space. As application, we also prove convergence theorem for finite family of demi-contractive mappings. Our theorems generalize and improve several recent results. For instance, Theorem 1.1, which generalized, extended and improved several recent results, is a special case of our Theorem.

## 2. Preliminaries

We shall make use of the following lemmas.

**Lemma 2.2**. (Xu [14])

*Let*{

*a*

_{ n }} and {

*b*

_{ n }}

*be sequences of nonnegative real numbers satisfying the inequality*

*If*
, *then*
*exists. If in addition* {*a*_{
n
}} *has a subsequence which converges strongly to zero, then*
.

**Lemma 2.3**. (Suzuki [15]) *Let* {*x*_{
n
}} *and* {*y*_{
n
}} *be bounded sequences in a Banach space E and let* {*β*_{
n
}} *be a sequence in* [0, 1] *with* 0 *<* lim *inf β*_{
n
} ≤ lim *supβ*_{
n
} *<* 1. *Suppose x*_{n+1}= *β*_{
n
}*y*_{
n
} +(1 -*β*_{
n
})*x*_{
n
} *for all integers n* ≥ 1 *and* lim *sup*(||*y*_{n+1}- *y*_{
n
}|| - ||*x*_{n+1}- *x*_{
n
}||) ≤ 0. *Then*, lim ||*y*_{
n
} - *x*_{
n
}|| = 0.

## 3. Main Results

*E*be a real Banach space, and

*K*be a nonempty, closed convex subset of

*E*. For some fixed

*i*∈ ℕ, let be a Lipschitzian demi-contractive semigroup with bounded measurable function

*L*

_{ i }: [0, ∞) → (0, ∞) and bounded function

*λ*

_{ i }: [0, ∞) → (0, ∞) such that

*K*and let , and Clearly

*L <*∞ and

*λ >*0 and for

*x*,

*y*∈

*K*, ,

*t*≥ 0 and any

*i*∈ {1, 2, ...,

*N*},

Hence, for each *i* ∈ {1, 2, ... *N*}, *S*_{
i
} is Lipschitz with Lipschitz constant
.

**Lemma 3.1**.

*Let E be a real Banach space and K be a nonempty closed convex subset of E. Let*

*be a finite family of Lipschitzian demi-contractive semigroups of K with sequences of bounded measurable functions L*

_{ i }: [0, ∞) → (0, ∞)

*and bounded functions λ*

_{ i }: [0, ∞) → (0, ∞)

*i*= 1, 2, ...,

*N such that for each i*= 1, 2, ...,

*N*,

*Let*, {

*t*

_{ n }}

*be an increasing sequence in*[0, ∞)

*and*{

*α*

_{ n }}

*be a sequence in (0,1) satisfying the following conditions:*

- (i)

*where T*_{
n
}(*t*_{
n
}) = *T*_{
n modN
} (*t*_{
n
}).

Since , by lemma 2.2, it follows that exists.

Hence, {*x*_{
n
}} is bounded, which implies that {*T*_{
n
}(*t*_{
n
})*x*_{
n
}} and {*S*_{
n
}(*t*_{
n
})*x*_{
n
}} are also bounded.

*β*

_{ n }} and {

*y*

_{ n }} be two sequences define by

*β*

_{ n }:=

*δ*(1 - δ)

*α*

_{n+1}+

*δ*

^{2}and . Then, using the definition of {

*β*

_{ n }} and {

*S*

_{ n }} we obtain that . Then,

It follows from (3.8) that . This completes the proof. □

**Theorem 3.2**. *Let E*, *K*,
, {*α*_{
n
}}, {*t*_{
n
}},
*and* {*x*_{
n
}} *be as in lemma 3.1. Assume that, for at least one i* ∈ {1, 2, ..., *N*}, *there exists a compact subset C of E such that* ∪_{t≥0}*T*_{
i
}(*t*)(*K*) ⊂ *C*. *Then, the sequence* {*x*_{
n
}} *converges to some element*
.

*Proof*. By Lemma 3.1, we have
.

_{t≥0}

*T*

_{ s }(

*t*)(

*K*) ⊂

*C*for some compact subet

*C*of

*E*and some

*s*∈ {1, 2, ...,

*N*}, then there exists a subsequence , of {

*x*

_{ n }} and

*q** ∈

*K*, such that

From the above we have
. Using (3.9) and the fact that *Ts* is Lipschitzian, we get *q** ∈ ∩_{t≥0}*F*(*T*_{
s
}(*t*)).

Now, for any *l* ∈ {1,2, ...,*N* }, since
, there exists a subsequence
of
such that

This implies that
and hence *q** ∈ ∩_{t≥0}*F*(*T*_{
l
}(*t*)). Since *l* ∈ {1, 2, ... *N*} is arbitrarily chosen, we have
. As the limit
exists, the conclusion of the theorem follows immediately and this completes the proof. □

*Remark* 3.3. Observe that considering a single one-parameter family of demi-contractive semigroup in Theorem 3.2, we obtain the conclusion of Theorem 1.1.

Let *T*_{1}, *T*_{2}, ..., *T*_{
N
} be a finite family of Lipschitzian demi-contractive self-mapping of *K* with positive constants *λ*_{1}, λ_{2}, ..., *λ*_{
N
} and Lipschitz constants *L*_{1},*L*_{2}, ..., *L*_{
N
} ,

The following Theorem is a consequence of Lemma 3.1.

**Theorem 3.4**.

*Let E, K and*{

*α*

_{ n }}

*be as in Lemma 3.1. Let T*

_{1},

*T*

_{2}, ...,

*T*

_{ N }:

*K*→

*K be Lipschitzian demi-contractive mappings with T*

_{ s }

*demicompact for at least one s*∈ {1, 2, ...,

*N*}.

*Let*{

*x*

_{ n }]

*be a sequence generated by x*

_{1}∈

*K*

*where T*_{
n
} = *T*_{
n
} _{
modN
} . *Then*, {*x*_{
n
}} *converges strongly to a common fixed point of the family*
.

*Proof*. Following the line of proof of lemma 3.1 we immediately obtain

*qk*exists for any

*q*∈

*F*and , ∀

*i*∈ {1,2, ...

*N*}. Let be a subsequence of {

*x*

_{ n }} such that

Therefore
and, by demicompactness of *T*_{
s
}, there exists a subsequence
of
and *q** ∈ *K*, such that
as *j* → ∞.

we obtain *q** ∈ *F*. But,
exists, thus *x*_{
n
} → *q** ∈ *F* and this completes the proof. □

The following corollaries follow from Theorem 3.4

**Corollary 3.5**. *Let E*, *K and* {*α*_{
n
}} *be as in Theorem 3.4. Let T*_{1}, *T*_{2}, ..., *T*_{
N
} : *K* → *K be Lipschitzian demi-contractive mappings. Suppose there exists a compact subset C in E such that*
. *Let* {*x*_{
n
}} *be defined by* (3.11). *Then*, {*x*_{
n
}} *converges strongly to a common fixed point of the family*
.

**Corollary 3.6**. *Let E; K and* {*α*_{
n
}} *be as in Theorem 3.4. Let T*_{1}, *T*_{2}, ..., *T*_{
N
} : *K* → *K be Lipschitzian demi-contractive mappings satisfying condition*
. *Let* {*x*_{
n
}} *be defined by* (3.11). *Then*, {*x*_{
n
}} *converges strongly to a common fixed point of the family*
.

*Proof*. Following the line of proof of lemma 3.1, we obtain for all

*i*∈ {1, 2, 3, ...,

*N*} and ||

*x*

_{n+1}-

*q*||

^{2}≤ (1 +

*σ*

_{n+1}) ||

*x*

_{ n }-

*q*||

^{2}, where . Since , by lemma 2.2 exists and consequently exists. Let be a subsequence of {

*x*

_{ n }} such that . Then, by using condition , there exists

*s*∈ {1, 2, ...,

*N*} such that and, using the property of

*f*, we get that , and since the limit exists we have that . We next show that {

*x*

_{ n }} is Cauchy. Let

*ε*> 0 be given, then there exists

*p** ∈

*F*and

*n** ∈ ℕ such that ∀

*n*≥

*n**, . Hence, for

*n*≥

*n** and

*k*∈ ℕ, we have

*x*

_{ n }} is Cauchy and so

*x*

_{ n }→

*q** ∈ K. We now show that

*q** is in

*F*. Since , there exists

*m*

_{0}∈ ℕ large enough and

*p** ∈

*F*such that for all

*n*≥

*m*

_{0}, and . Hence,

Thus, *q** ∈ *F*(*T*_{
l
}) and since *l* ∈ {1, 2, ..., *N*} is arbitrary, we have *q** ∈ *F*. This completes the proof. □

## Declarations

### Acknowledgements

This study was conducted when the first author was visiting the AbdusSalam International Center for Theoretical Physics Trieste Italy as an Associate, and the hospitality and financial support provided by the centre is gratefully acknowledged.

## Authors’ Affiliations

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