Convergence theorem for finite family of lipschitzian demi-contractive semigroups

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

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 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,

• (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) 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));

• (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 : 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 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 [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.

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,

• (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 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 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 ∈ Fⁱ 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) : K → K 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:

• (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,

• (a) exists for all .

• (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 ∈ ℕ,

Since m ∈ ℕ 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)α_n+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 : K → K 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 : 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 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 : 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 T₁, T₂, ..., 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

Thus, {x _n } is Cauchy and so x _n → q* ∈ K. We now show that q* is in F. Since , there exists m₀ ∈ ℕ large enough and p* ∈ F such that for all n ≥ m₀, and . Hence,

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