# Volume 70, № 6, 2018

### Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 727-738

We establish new Lyapunov-type inequalities for two classes of nonlinear systems with homogeneous Dirichlet boundary conditions, which generalize and improve some results known from the literature.

### Spectral properties of nonself-adjoint nonlocal boundary-value problems for the operator of differentiation of even order

Baranetskij Ya. O., Kalenyuk P. I., Kolyasa L. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 739-751

We study spectral properties of an essentially nonself-adjoint problem generated by nonlocal multipoint conditions for the operator of differentiation of order 2n and analyze the cases of regular and irregular Birkhoff boundary conditions. A system of root functions of the problem and elements of biorthogonal systems are constructed. We also establish sufficient conditions under which these systems are complete and form a Riesz basis under certain additional assumptions.

### Weighted pseudoinversion with indefinite weights

Galba E. F., Khimich A. N., Sergienko I. V., Vareniuk N. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 752-772

We present the definition of weighted pseudoinverse matrices with nonsingular indefinite weights and study these matrices. The theorems on existence and uniqueness for these matrices are proved. Weighted pseudoinverse matrices with indefinite weights are represented in terms of the coefficients of characteristic polynomials of symmetrizable matrices. The decompositions of weighted pseudoinverse matrices into matrix power series and products and their limit representations are obtained. We also propose regularized iterative methods for the determination of these matrices.

### Hedging of the European option with nonsmooth payment function

Glonti O. A., Purtukhiya O. G.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 773-787

We consider onе type of European option in the case of the Black – Scholes financial market model whose payment function is a certain combination of binary and Asian options. The corresponding hedging scheme is analyzed.We deduce the formula for the Clark stochastic integral representation of the corresponding Wiener functional with integrand represented in the explicit form.

### The exponential twice continuously differentiable $ B$-spline algorithm for Burgers’ equation

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 788-800

The exponential twice continuously differentiable $ B$-spline functions known from the literature as the exponential are used to set up the collocation method for finding solutions of the Burgers’ equation. The effect of the exponential cubic $ B$-splines in the collocation method is sought by studying the text problems.

### One method for the investigation of the fundamental solution of the Cauchy problem for parabolic systems

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 801-811

A recursive method for the investigation of the fundamental solution of the Cauchy problem for parabolic Shilov systems with time-dependent coefficients is proposed. It is based on the general formula for the solution of linear inhomogeneous systems of differential equations of the first order and does not require the use of the genus of the analyzed system.

### A version of the projection-iterative method for the solution of Fredholm integral equations of the second kind

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 812-822

We discuss the problem of application of Alpert’s multiwavelets to the solution of Fredholm integral equations by the projection-iterative method.

### Lagrange stability and instability of nonregular semilinear differential-algebraic equations and applications

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 823-847

We consider an nonregular (singular) semilinear differential-algebraic equation $$\frac d{dt} [Ax] + Bx = f(t, x)$$ and prove the theorems on Lagrange stability and instability. The theorems give sufficient conditions for the existence, uniqueness, and boundedness of a global solution of the Cauchy problem for the semilinear differential-algebraic equation and sufficient conditions for the existence and uniqueness of the solution with finite escape time for the analyzed Cauchy problem (this solution is defined on a finite interval and unbounded). The proposed theorems do not contain constraints similar to the global Lipschitz condition. This enables us to use them for solving more general classes of applied problems. Two mathematical models of radioengineering filters with nonlinear elements are studied as applications.

### On the lacunary $(A, ϕ)$ -statistical convergence of double sequences

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 848-858

We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong $(A, ϕ)$-convergence with respect to a modulus function. In addition, we also study some relationships between double lacunary strong $(A, ϕ)$-convergence with respect to a modulus and double lacunary statistical convergence.

### Properties of the logical consequence operation and its relationship with the independence of propositional logic

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 857-864

We investigate the properties of the logical consequence operation and the characteristic features of independent sets of formulas. Further, we apply these results to propositional logic. Finally, we show under what conditions the results of addition of a formula to independent sets of formulas and the union of two independent sets of formulas are also independent, by using the operation of logical consequence, i.e., we establish a relationship between the logical consequence and the preservation of independence in propositional logic.