Vol 64, No 4 (2018): Contemporary Problems in Mathematics and Physics

We study the difference splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coefficients with lower terms. A discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution.

In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the field of online/real-time algorithms, e.g., [20, 21, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any finite number of iterations, before moving, in a serial way, to the next block. The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the fields of fixed point theory and variational inequalities, such as [34]. Numerical experiments for linear and non-linear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme, e.g., to online/real-time scenarios.

In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.

This work is a continuation of our work [12] where we considered linear spaces in the following two situations: a real space admits a multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with a multiplication by complex scalars. We studied there also how they manifest themselves when the initial space possesses additional structures: topology, norm, inner product, as well as what happens with linear operators acting between such spaces. Changing the linearities of the linear spaces unmasks some very subtle properties which are not so obvious when the set of scalars is not changed. In the present work, we follow the same idea considering now Bochner and Pettis integrals for functions ranged in real and complex Banach and Hilbert spaces. Finally, this leads to the study of strong and weak random elements with values in real and complex Banach and Hilbert spaces, in particular, some properties of their expectations.

In this paper, we establish the relation between one-sided ball potentials by means of radially singular operators in a spherical layer. Moreover, we construct new Chern-type one-sided ball potentials