COMMON FIXED POINT THEOREMS FOR COMMUTING k-UNIFORMLY LIPSCHITZIAN MAPPINGS

We give a common fixed point existence theorem for any sequence of commuting k-uniformly Lipschitzian mappings (eventually, for k= 1 for any sequence of commuting nonexpansive mappings) defined on a bounded and complete metric space (X,d) with uniform normal structure. After that we deduce, by using the Kulesza and Lim (1996), that this result can be generalized to any family of commuting k-uniformly Lipschitzian mappings. 2000 Mathematics Subject Classification. Primary 47H10.


Introduction.
In classical theorems concerning the existence of fixed points for family of mappings, such as the Kakutani theorem [4] and its well-known generalization due to Ryll-Nardzewski [13], the mappings of the family are usually assumed to be linear, or at least to be weakly continuous and affine [11].In the nonlinear theory, a stronger geometric structure is required.In particular for a family of nonexpansive mappings, Khamsi proved in [7] that any family of nonexpansive mappings defined on a metric space (X, d) with compact and normal convexity structure Ᏺ, has a common fixed point.In his proof, Khamsi investigated the concept of 1-local retract.In this paper, we prove that any sequential family of k-uniformly Lipschitzian mappings defined on a bounded metric space with a uniform normal convexity structure Ᏺ with constant β, which contains all closed ball of (X, d), has a common fixed point provided that k 2 β < 1. Recall that any nonexpansive mapping defined on a bounded complete metric space with uniform normal structure with constant β has a nonempty fixed point set (Khamsi [6]).For more details on fixed point theory for nonexpansive and k-uniformly Lipschitzian mappings in metric spaces we refer the reader to [1,2,3].

Definitions and preliminaries.
In this work, (X, d) will be a metric space.We use B(x, r ) to denote the closed ball centered at x ∈ X with radius r > 0. For a subset A of X, we write where Ᏺ is the family of closed balls containing A. A subset A of X is said to be admissible if and only if A = cov(A).In other words, A is admissible if it is an intersection of a family of closed balls centered in X.
Definition 2.1.Let Ᏺ be a nonempty family of a subset of X.We say that Ᏺ defines a convexity structure on X if and only if it is stable by intersection.
In this work, we always assume that Ᏺ contains the balls.Also we denote by Ꮽ(X) the smallest convexity structure on X. Definition 2.2.We say that Ᏺ has the property (R) if and only if any decreasing sequence (X n ) n of nonempty bounded closed subsets of X with X n ∈ Ᏺ has a nonempty intersection.Definition 2.3.(i) We say that X has uniform normal structure if and only if r (A) ≤ βδ(A) for some 0 < β < 1 and for every A ∈ Ᏺ.
(ii) We say that Ᏺ is normal if and only if r (A) < δ(A) for every A ∈ Ᏺ.
Let us recall that a self mapping T : X → X is said to be k-uniformly Lipschitzian if there exists a k > 0 such that for every i ∈ N and every x, y in X.A 1-uniformly Lipschitzian map is called nonexpansive.For such class of mappings we recall the following most important result.
Theorem 2.4 (see [6]).Let (X, d) be a complete bounded metric space.Assume that X has uniform normal structure.Then any nonexpansive mapping defined on X has a fixed point.
In [7], Khamsi gave the definition and a characterization of a 1-local retract subset of a metric space.Definition 2.5.A subset A is said to be a k-local retract if for any family It is immediate that uniform normal structure is not hereditary.However, for 1-local retract subsets we have the following lemma.Lemma 2.6.Let (X, d) be a metric space.Suppose that Ꮽ(X) is a uniform normal convexity structure with constant The proof is based on the next lemma.Lemma 2.7 (see [7]).Let (X, d) be a metric space and A a nonempty bounded subset of X.Then (1) cov(A) = ∩ x∈X B(x, r x (A)).
(2) r x (A) = r x (cov A) for every x in X. ( Recall that (X, d) is said to have the (n, ∞) property if for any family (B i ) i∈I of closed balls of X such that ∩ i∈J B i ≠ ∅ for any finite subfamily J of I with card(J) less than n, we have ∩ i∈I B i ≠ ∅.
A metric space (X, d) is said to be convex if for all x, y in X and α ∈ [0, 1] there exists a z ∈ X such that Proof of Lemma 2.6.We assume that A is not a singleton.By (4) of Lemma 2.7, we have r from property (4) of Lemma 2.7.

Fixed points for k-uniformly Lipschitzian mappings.
In the next theorem, we obtain fixed point theorem for k-uniformly Lipschitzian mapping by utilizing the existence theorem of nonexpansive mapping [7].To our knowledge this connection has not been utilized.Moreover, Theorem 3.1 contains the result of Theorem 2.4.Theorem 3.1.Let (X, d) be a complete bounded metric space.Assume that X has a uniform normal structure with constant β < 1.Then any k-uniformly Lipschitzian mapping T : X → X has a fixed point if k 2 β < 1.
Proof.First we need the following two lemmas.

Lemma 3.2. Under the same hypothesis as Theorem 3.1, and for
Lemma 3.3.Under the same hypothesis as Theorem 3.1, and for T : X → X k-uniformly Lipschitzian, the family of all admissible subsets of (X, d ) is a uniform normal convexity structure with constant c (c ≤ k 2 β).
Proof of Lemma 3.2.(1-1) d is a metric on X. Indeed (1-1-a) For every x, y in X, we have d (x, y) = 0 is equivalent to d(T i x, T i y) = 0 for every i = 0, 1, 2,.... Specifically for i = 0, it implies that d(x, y) = 0. Since d is a metric on X, then x = y.
(1-1-b) For every i = 0, 1, 2,..., and every x, y, z in X, we have since d is a metric on X.
By passing to the supremum on i ∈ N, we obtain that (2) For every x, y in X, we have By passing in (3.9) to the infimum on z ∈ ∩ x∈X B (x, r x (A)), we get inf which implies that By Theorem 3.1, we have Fix(T ) ≠ ∅ for every k-uniformly Lipschitzian mapping T defined on a bounded complete metric space (X, d) with uniform normal convexity structure Ᏺ with constant β < 1/k 2 .Moreover, Fix(T ) is a k-local retract of X, that is, for every closed ball B(x i ,r i ) i∈I , we have Now we are able to show the following.
Theorem 3.4.Let T n : X → X; n = 0, 1, 2,... be a family of commuting k-uniformly Lipschitzian mappings.Suppose that X has a uniform normal convexity structure Ᏺ Proof of Theorem 3.4.The first part of the theorem follows immediately from Theorem 3.1.For the second part, let (B i ) i∈I be a family of closed balls centered in We have and since (T n ) n are nonexpansive mappings on (X, d ), it follows from Theorem 3.1 that The problem of whether the conclusion of Theorem 3.4 holds for any commuting family (T i ) i∈I of k-uniformly Lipschitzian mappings (k > 1) was open for several years.However, by using the result of Lim and Kulesza [8] in which they show that weak compactness and weak countably compactness are equivalent, if the metric space has normal structure, we prove the following.Theorem 3.5.Let (X, d) be a bounded complete metric space with a uniform normal convexity structure (β < 1).Then any commuting family T i : X → X, i ∈ I of k-uniformly Lipschitzian mappings has a common fixed point provided that k 2 β < 1.
Proof.Since (X, d ) has uniform normal structure with constant c (c < k 2 β), then by the well-known theorem of Khamsi [6], Ꮽ(X d ) is countably compact.
Hence by the Lim and Kulesza result, it follows that Ꮽ(X d ) is in fact compact.On the other hand, since each T i , i ∈ I is d -nonexpansive (Lemma 3.3), it follows that the result of Theorem 3.4 is a direct consequence of Khamsi's theorem in which he shows that any commuting family of nonexpansive mappings defined on a bounded metric space for which Ꮽ(X d ) is compact and normal, has a common fixed point.
We remark that the result of Theorem 3.5 was deduced from Lim and Kulesza theorem and the uniform convexity of (X, d ) (Lemma 3.3); but the problem of whether the compactness and normality of (X, d) imply the compactness and normality of ( X, d ) is still open.[10] and Kelley [5] that all Banach spaces which have the (2, ∞) property are those of form C(E), where E is a compact Stonian, for example l ∞ and L ∞ .Then by Theorem 3.5 and property (5) of Lemma 2.7, we have the following.

Corollary 4.1. The unit balls of l ∞ , L ∞ , and C(E), where E is a compact Stonian have the common fixed point property for every commuting family T
Lindenstrauss [9] has proved that l 1 has a (3, ∞) property.
Corollary 4.2.The unit ball of l 1 has the common fixed point property for every commuting family T i : X → X, i ∈ I of k-uniformly Lipschitzian mappings provided that k < 3/2.Also, we deduce from Theorem 3.5 and property (5) of Lemma 2.7, the following corollary.

Corollary 4.3. If (X, d) is a Banach space with the (n, ∞) property, and if k < n/(n − 1), then its unit ball has the common fixed point property for every commuting family T
More recently, Prus [12] has proved that all Banach spaces L p (1 < p < +∞) have uniform normal structure with constant β = (min(2 1/p , 2 1/q )) −1 , where q = p(p −1) −1 is the conjugate of p.
Hence, we have the following.
Corollary 4.4.The unit balls of L p have the common fixed point property for every commuting family T i : X → X, i ∈ I of k-uniformly Lipschitzian mappings provided that k < min(2 1/p , 2 1/q ).Now we recall the definition of the most geometrical characterization of l ∞ , L ∞ , and C(E), where E is a compact Stonian.Definition 4.5.A metric space (X, d) is said to be hyperconvex if and only if any family {B(x i ,r i ), i ∈ I} of closed balls of (X, d) such that for every i, j ∈ I, has a nonempty intersection.

Remarks.
(1) Every hyperconvex metric space is complete, and if A is an admissible subset of (X, d), then also (A, d) is a hyperconvex metric space (see [2]).
(2) Every hyperconvex space is convex.Indeed: For all x, y in X and for any α ∈ [0, 1], let u, v in X.We have 2) The hyperconvexity of (X, d) implies that Hence, for every x, y in X and for every α ∈ [0, 1], there exists a z ∈ X such that Also by Theorem 3.5 and property (5) of Lemma 2.7, we obtain the following theorem.
Theorem 4.6.Let (X, d) be a bounded hyperconvex metric space.Then any family of commuting k-uniformly Lipschitzian mappings defined on X has a common fixed point if k < √ 2.
Proof.(X, d) is a bounded hyperconvex metric space.Then from the above remarks, it is complete.Let us prove that (X, d) has the (2, ∞) property.Indeed: Let {B(x i ,r i ), i ∈ I} be a family of closed balls of (X, d), such that Then we have where x ∈ B(x i ,r i ) ∩ B(x j ,r j ).

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Proof of Lemma 3 . 3 .
Let A be an admissible subset for d , thenA = ∩ x∈X B x, r x (A) ⊂ cov(A) = ∩ x∈X B x, r x (A) .(3.7) On the other hand, it follows from the definition of d that d(z, y) ≤ d (z, y) ≤ kd(z, y) ∀z, y ∈ X. (3.8) Hence r z (A) ≤ kr z (A) ∀z ∈ X. (3.9)