Definition 1.2.

We prove that if X i , i = 1 , 2 , … , are Banach spaces that are weak* uniformly rotund, then their l p product space 1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> ( p > 1 ) is weak* uniformly rotund, and for any weak or weak* uniformly rotund Banach space, its quotient space is also weak or weak* uniformly rotund, respectively.

1. Definitions and preliminaries.In this note, X and Y denote Banach spaces and X * and Y * denote the conjugate spaces of X and Y , respectively.Let A ⊂ X be a closed subset and X/A denote the quotient space.We use S(X) for the unit sphere in X and P lp (X i ) for the l p product space.We refer to [1,3] for the following definitions and notations.For more recent treatment, one may see, for example, [2].Definition 1.1.A Banach space X is UR A , where A is a nonempty subset of X * , if and only if for any pair of sequences {x n } and {y n } in S(X), if ) , where Q : X → X * * is the canonical embedding.

Some results on the weak
].We now choose a subsequence with the diagonal method, without loss of generality, still use {n} as the index such that for each i, we have lim for each i, by the lemma, we have Suppose that P lp (X i ) is not W * UR, then there exist sequences {x n } ∈ S(P lp (X i )), {y n } ∈ S(P lp (X i )), x n + y n → 2, but x n − y n does not converge (w * ) to θ.So, there must be an a = (a 1 ,a 2 ,...,a i ,...) in P lq (Y i ), with a i ∈ Y i , such that |(x n − y n )(a)| does not converge to 0. Therefore, there exist > 0 and a subsequence of {n} (for simplicity, we still use {n}) such that |(x n − y n )(a)| > , which implies that one can find an integer m, sufficiently large, so that m i=1 Let (n k ) be the subsequence of {n} such that (2.1) holds.By (2.2), we have 3), we have a contradiction 0 > /2.The proof is complete.
Here, x = π(x), where π : X → X/A.Now, for each n, take x n ∈ xn and y n ∈ ỹn , 1 π is w * -w * continuous.So, we must have xn − ỹn w * → θ.That contradicts the above, and the proof is complete.Theorem 2.4.Suppose that A is a closed subspace of X and X is WUR (Definition 1.2), then X/A is WUR.

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: From the definition, we clearly have the following corollary.The Banach space X is W * UR if and only if for any pair of sequences {x n } and {y n } in X, if x n − y n → 0, { y n } is bounded, and }∈P lp (X i ), x n +y n →2.Using the properties of l p norm and Minkowski inequality, one can see, for each i, that there exists a subsequence of {n}, {n i k }, such that lim k→∞ x * and weak uniform rotundity.