Open Access

Convergence theorem for finite family of lipschitzian demi-contractive semigroups

Fixed Point Theory and Applications20112011:18

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

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

Demi-contractive maps Demi-contractive semigroup Demicompact maps Fixed point

1. Introduction

Let 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 : KK 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.

A one parameter family of self mapping of K is called a nonexpansive semigroup if the following conditions are satisfied,
  1. (i)

    T(0)x = x x K;

     
  2. (ii)

    T(t + s) = T(t) T(s) t, s ≥ 0;

     
  3. (iii)

    for each x K, the mapping tT(t)x is continuos;

     
  4. (iv)

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

     
If the family satisfies conditions (i) - (iii), then it is called
  1. (a)
    pseudocontractive semigroup if for any x, y K, there exists j(x - y) J(x - y) such that
     
  2. (b)
    strictly pseudocontractive semigroup if there exists a bounded function λ : [0, ∞) → (0, ∞) and j(x - y) J(x - y) such that
     
for all x, y K;
  1. (c)
    demi-contractive semigroup if F(T(t)) ≠ t ≥ 0, there exists a bounded function λ : [0, ∞) → (0, ∞), and j(x - y) J(x - y) such that
     
for any x K and q F(T(t));
  1. (d)

    Lipschitzian semigroup if there is a bounded measurable function

     
L : [0, ∞) → (0, ∞) such that for x, y K and t ≥ 0,

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 : KK 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 T1, T2, ..., 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 [15] 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 [613], to list but a few.

In 1998, Shoiji and Takahashi [7] introduced and studied a Halpern-type scheme for common fixed point of a family of asymptotically nonexpansive semigroup in the framework of a real Hilbert space. Suzuki [8] proved that the implicit scheme defined by x, x1 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.

In 2005, Aleyner and Reich [10] proved the strong convergence of an explicit Halpern-type scheme defined by x, x1 K,
to a common fixed point of the family {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, x1 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,
  1. (i)
    ; (ii) . Assume that there exists a compact subset C of E such that t≥0T(t)(K) C and for any bounded set D K
     
Let {x n } be generated by x1 K,
(1.1)

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 2.1. Let E be a real normed linear space. Then, the following inequality holds:
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 xn+1= β n y n +(1 -β n )x n for all integers n ≥ 1 and lim sup(||yn+1- y n || - ||xn+1- x n ||) ≤ 0. Then, lim ||y n - x n || = 0.

3. Main Results

Let E be a real Banach space, and K be a nonempty, closed convex subset of E. For some fixed i , let be a Lipschitzian demi-contractive semigroup with bounded measurable function L i : [0, ∞) → (0, ∞) and bounded function λ i : [0, ∞) → (0, ∞) such that
Then, for x, y K, q F i and t ≥ 0,
and
Consider a family of Lipschitzian demi-contractive semigroups of K and let , and Clearly L < ∞ and λ > 0 and for x, y K, , t ≥ 0 and any i {1, 2, ..., N},
and
For a fixed δ (0, 1) and t ≥ 0 define a family S i (t) : KK i = 1, 2, ..., N by
(3.1)
Then, for x, y K and ,
Let then
(3.2)
Also,

Let .

Then,
(3.3)

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:
  1. (i)

    , (ii) .

     
Assume i {1,2, ..., N} for any bounded set D K the relation
(3.4)
holds. Let {x n } be a sequence generated by x1 K,
(3.5)

where T n (t n ) = T n modN (t n ).

Then,
  1. (a)

    exists for all .

     
  2. (b)

    for all i {1,2,3, ..., N}.

     
Proof. For any fixed using (3.5), we have
Thus,
(3.6)

where .

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.

From (3.6)
where, . Hence, for some m ,
Since m is arbitrary, we have
which implies
(3.7)
Next, we show that,
Let {β n } and {y n } be two sequences define by β n := δ(1 - δ)αn+1+ δ2 and . Then, using the definition of {β n } and {S n } we obtain that . Then,
Therefore,
Hence,
and by lemma 2.3,
Thus,
This implies that,
But, for i {1,2,3, ..., N},
Therefore,
Hence,
From the relation,
and condition (3.4) we get
(3.8)

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≥0T i (t)(K) C. Then, the sequence {x n } converges to some element .

Proof. By Lemma 3.1, we have .

If t≥0T 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
(3.9)
Observe that for t > 0,

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

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

. Then, similarly for t ≥ 0, we obtain

This implies that and hence q* t≥0F(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 T1, T2, ..., T N be a finite family of Lipschitzian demi-contractive self-mapping of K with positive constants λ1, λ2, ..., λ N and Lipschitz constants L1,L2, ..., L N ,

respectively. Let .

For a fixed δ (0, 1), define S n : KK by
(3.10)
Then, it follows that for x, y K and i F,

where , , and .

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 T1, T2, ..., T N : KK 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 x1 K
(3.11)

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 → ∞.

Since,

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 T1, T2, ..., T N : KK 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 T1, T2, ..., T N : KK 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 ||xn+1- q||2 ≤ (1 + σn+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* such that nn*, . Hence, for nn* and k , we have
Thus, {x n } is Cauchy and so x n q* K. We now show that q* is in F. Since , there exists m0 large enough and p* F such that for all nm0, 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

(1)
Department of Mathematical Sciences, Bayero University

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Copyright

© Ali and Ugwunnadi; licensee Springer. 2011

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