In this paper, we consider the eigenvalue problem for the equation of a vibrating beam with a spectral parameter contained in the boundary conditions. The general characteristics of the location of eigenvalues on the real axis are studied, the multiplicities of eigenvalues are found, and the oscillatory properties of the eigenfunctions of this spectral problem are investigated. Moreover, asymptotic formulas for the eigenvalues are obtained and sufficient conditions are established for the subsystems of root functions to form a basis in Lebesgue spaces.

In this paper is proved the law of large numbers for the Markov random walks, discribed by the first-order autoregressive process (AR(1)).

This paper is devoted to the study of a nonlinear boundary value problem for the Sturm-Liouville equation with a parameter contained both in the equation and in the boundary condition. We show the existence of solutions to this problem with a fixed number of simple nodal zeros.

In this work we study the second order differential operator with integral boundary conditions. Under weaker than previously known conditions on the functions asymptotic formulas for eigenvalues and Eigen functions are found functions, an estimate of the resolvent was obtained and theorem on completeness and minimality of eigenfunctions in some subspace of space codimension 2.

In this paper we study the behaviour of the sequence of scalar ( real or complex) numbers satisfying the relation , where is a fixed sequence of scalar numbers. Such kind of sequences arise in problems of analysis, fixed point theory, dynamical sistems, theory of chaos and ets. For example, investigating the spectra of triple and more than triple band triangle operator-matrices arise above mentioned sequences which required to study the behaviour of these sequences. From the point of application, the proved results and formulas in the literature for the spectra of the operator-matrices look like very complicated. In this work for the eliminating of indicated flaws we apply new approach, where the formulas for the spectra for the generalized difference operator-matrices describe circular domains, and also we study problems describing the theory of natural processes.

Using congruence schemes, we present a characterization of compatible reflexive relations for k-majority algebras (i.e. for algebras having k-ary near unanimity operation among its term operations, k≥3). For algebras in congruence n-permutable varieties we show that every binary (n-1)-pretransitive compatible relation is symmetric and we obtain some consequences for special types of relations.

We study the classical solution of the nonlinear inverse boundary value problem for for pseudo hyperbolic equation of the fourth order The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. To solve the considered problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to the auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is made, as a result, a conclusion is made about the solvability of the original inverse problem.

A mixed problem for nonlinear hyperbolic equations with nonlinear acoustic transmission condition is considered. The theorem on existence and uniqueness of solutions for this problem is proved by the semigroup method.

In this paper we consider the bundle of type tensor frames over a smooth manifold, define the horizontal and complete lifts of symmetric affine connection from a given manifold to this bundle. Also we investigate the properties of the geodesic lines corresponding to the complete lift of an affine connection and determine the relations between Sasaki metric and lifted connections on the bundle of type tensor frames.

In this paper we consider global bifurcation from infinity in nonlinearizable Dirac problem with a spectral parameter contained in both boundary conditions. We prove the existence of two families of unbounded components of the set of nontrivial solutions to this problem, which bifurcate from asymptotic intervals and contained in classes of vector- functions possessing oscillatory properties of the eigenvector-functions of the corresponding linear Dirac problem in the neighborhood of these intervals.