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Convergence theorem for finite family of lipschitzian demicontractive semigroups
Fixed Point Theory and Applications volume 2011, Article number: 18 (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 demicontractive 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 demicontractive maps is also proved.
Mathematics subject classification (2000) 47H09, 47J25
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 : 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 LLipschitzian or simply Lipschitz if ∃L > 0, such that Tx Ty ≤ Lx  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,

(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.
If the family satisfies conditions (i)  (iii), then it is called

(a)
pseudocontractive semigroup if for any x, y ∈ K, there exists j(x  y) ∈ J(x  y) such that

(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;

(c)
demicontractive 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));

(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 demicontractive semigroup.
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 demicontractive if F(T) ≠ ∅, and there exists λ > 0 such that 〈Tx q, j(x  q)〉 ≤ x  q^{2}  λx  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 selfmappings 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 semigroups.
Both implicit and explicit, Mann, Ishikawa, and Halperntype 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.
In 1998, Shoiji and Takahashi [7] introduced and studied a Halperntype 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, 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.
In 2005, Aleyner and Reich [10] proved the strong convergence of an explicit Halperntype scheme defined by x, x_{1} ∈ 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, 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 demicontractive 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)
; (ii) . Assume that there exists a compact subset C of E such that ∪_{t≥0}T(t)(K) ⊂ C and for any bounded set D ⊂ K
Let {x_{ n }} be generated by x_{1} ∈ K,
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 demicontractive semigroups in a real Banach space. As application, we also prove convergence theorem for finite family of demicontractive 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 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
Let 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 demicontractive 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 demicontractive 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) : K → K i = 1, 2, ..., N by
Then, for x, y ∈ K and ,
Let then
Also,
Let .
Then,
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 demicontractive 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)
, (ii) .
Assume ∀ i ∈ {1,2, ..., N} for any bounded set D ⊂ K the relation
holds. Let {x_{ n }} be a sequence generated by x_{1} ∈ K,
where T_{ n }(t_{ n }) = T_{ n modN } (t_{ n }).
Then,

(a)
exists for all .

(b)
for all i ∈ {1,2,3, ..., N}.
Proof. For any fixed using (3.5), we have
Thus,
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 ∈ ℕ,
Since m ∈ ℕ is arbitrary, we have
which implies
Next, we show that,
Let {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,
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
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 .
If ∪_{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
Observe that for t > 0,
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
. Then, similarly for t ≥ 0, we obtain
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 oneparameter family of demicontractive 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 demicontractive selfmapping of K with positive constants λ_{1}, λ_{2}, ..., λ_{ N } and Lipschitz constants L_{1},L_{2}, ..., L_{ N } ,
respectively. Let .
For a fixed δ ∈ (0, 1), define S_{ n } : K → K by
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 T_{1}, T_{2}, ..., T_{ N } : K → K be Lipschitzian demicontractive 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 → ∞.
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 T_{1}, T_{2}, ..., T_{ N } : K → K be Lipschitzian demicontractive 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 demicontractive 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
Thus, {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. □
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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.
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Authors' contributions
BA conceived the study, GCU carried out the computations for Theorem 3.4. BA Modified Theorem 3.4 to obtain Theorem 3.2. Both authors read and approved the final manuscript.
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Ali, B., Ugwunnadi, G.C. Convergence theorem for finite family of lipschitzian demicontractive semigroups. Fixed Point Theory Appl 2011, 18 (2011). https://doi.org/10.1186/16871812201118
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DOI: https://doi.org/10.1186/16871812201118
Keywords
 Demicontractive maps
 Demicontractive semigroup
 Demicompact maps
 Fixed point