Volume 20, Issue 3

Abstract. The idea of the construction of Boehmians was initiated by the concept of regular operators. The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it just gives the field of quotients. In this article we consider two spaces of Boehmians. The strong space of Boehmians is continuously viewed in the space of general Boehmians. The Hartley transform is extended and obtained as a well-defined continuous mapping with respect to the convergence for which certain theorems have been proved. The article is ended up in defining the inverse Hartley transform and discussing some of its properties in detail.

Abstract. The one-dimensional full von Kármán beam is stabilized using the dissipative effect through heat conduction acting on the longitudinal component and the frictional damping acting on the transversal component of the beam.

Abstract. The purpose of the present research is to investigate the mixed Dirichlet–Neumann boundary value problems for the Helmholtz equation in a 2D domain with finite number of non-cuspidal angular points on the boundary. Using localization, the problem is reduced in [Proc. A. Razmadze Math. Inst. 162 (2013), 37–44] to model problems in plane angles of magnitude , . In the present paper, we apply the potential method and reduce the model mixed BVP (with Dirichlet–Neumann conditions on the boundary) to an equivalent boundary integral equation (BIE) of Mellin convolution type. Applying the recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev–Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava and by R. Duduchava, the unique solvability criteria (Fredholm criteria) of the above mentioned mixed BVP are obtained in classical finite energy space and also in non-classical Bessel potential spaces when . The problem has been tackled before only for angular domains of magnitude for rational angles .

Abstract. This paper is devoted to the study of the existence of a solution to the Dirichlet boundary value problem for the second-order differential equation where the functions satisfy the local Carathéodory conditions and may have singularities both in the time (for and ) and the phase (for and ) variables. Sufficient conditions for the solvability of the above-mentioned problem are established.

Abstract. Our purpose here is to show that an application of the theory of regular variation (in the sense of Karamata) gives the possibility of determining the existence and precise asymptotic behavior of positive solutions of the third-order nonlinear differential equation , where and are positive constants satisfying , is a continuous regularly varying function and , .

Abstract. It is shown that well-orderings are either negligible or nonmeasurable with respect to the completions of σ-finite product measures. Several consequences of this fact are discussed in the light of some classical problems formulated by Hilbert, Lebesgue and Luzin.

Abstract. Two-weight norm estimates for Hardy-type transforms, maximal functions and singular integrals in variable exponent Lebesgue spaces defined, generally speaking, on spaces of homogeneous type (quasi-metric measure spaces with doubling measure) are established. The derived conditions are written in terms of norms and are simultaneously necessary and sufficient for appropriate inequalities mainly in the case when weights are of radial type. An appropriate example of a pair of non-Muckenhoupt weights governing the two-weight estimate for maximal and Calderón–Zygmund operators is also constructed.

Abstract. New oscillation and nonoscillation criteria are established for the two-dimensional system , , where are locally integrable functions, for a.e. , and .

Abstract. The theorems of the optimal element existence are proved for a two-stage nonlinear optimal problem with constant delays in phase coordinates and with general boundary conditions and a functional. An element implies the collection of initial and switching moments, an initial function and vector, delay parameters, a control function and a final moment.