Volume 18, Issue 2

Abstract. We consider short mappings from a bounded subset of a Euclidean space into that space, that is, mappings which do not increase distances between points. By Kirszbraun's theorem any such mapping can be extended to the entire space to be short again. In general, the extension is not unique. We show that there are single-valued extension operators continuous in the supremum norm. The multivalued extension operator is lower semicontinuous.

Abstract. In this paper, we study the Chebyshev polynomial approximation of entire solutions of Helmholtz equations in in Banach spaces ( space, Hardy space and Bergman space). Some bounds on generalized order of entire solutions of Helmholtz equations of slow growth have been obtained in terms of the coefficients and approximation errors using function theoretic methods.

Abstract. In this paper, we answer the question: What are the greatest value and least value such that the double inequality holds for any and all with ? Here, , and denote the arithmetic, geometric and p -th one-parameter means of a and b , respectively.

Abstract. In this paper, we investigate the existence, uniqueness and other properties of solutions of fractional semilinear evolution equations in Banach spaces. The results are obtained by using fractional calculus, the well-known Banach fixed point theorem coupled with Bielecki type norm and the integral inequality established by E. Hernandez.

Abstract. In this paper, we investigate the existence of mild solutions on a compact interval to second-order impulsive neutral functional differential inclusions in Banach spaces. The results are obtained by using the theory of continuous cosine families and a fixed point theorem due to Dhage.

Abstract. Let K be a closed convex cone with nonempty interior in a real Banach space. The main objective of this paper is to show that if is a regular sine family of continuous additive set-valued functions associated with , then are of the form

Abstract. In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372–379].

Abstract. The classical Lebesgue density theorem says that almost each point of a measurable set A is a density point of A . It is well known that the density point of a measurable set A can be described in terms of the convergence in measure of a sequence of characteristic functions of sets similar to A . In this note it is shown that in the Lebesgue density theorem the convergence in measure cannot be replaced by the convergence almost everywhere.

Abstract. Certain aspects of Lagrangian and Hamiltonian dynamics are investigated within the framework of an extended complex phase space approach, characterized by the transformation and by the presence of the imaginary time variable in the complex variable in a local chart. It is an application of imaginary time which is essential in connecting quantum mechanics with statistical physics. We argue that the novel complexified approach enhances the system of dynamical equations obtained in the sense that the new derived equations appear as certain combinations of former equations. Furthermore, it was shown that the physics we experience in the real time is somewhat different from what is experienced in imaginary region. Further consequences are discussed in some details.

Abstract. In 1935, E. Marczewski defined the families and where is the family of perfect sets and X is a Polish space. We say that the pair (where is the algebra of subsets of and the ideal of sets) has MB-representation if there exists a family such that and , where and are constructed analogously to and . We will use two theorems published in [J. Appl. Anal. 9 (2003), 275–286] and [Bull. Pol. Acad. Sci. Math. 53 (2005), 239–250], to prove the theorem which is stated in the title.