Volume 5, Issue 1

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem d x ( t ) = dA ( t ) · f ( t , x ( t )), x i ( t i ) = ϕ i ( x ) ( i = 1, . . . , n ), where , A : [ a , b ] → R n×n is a matrix-function with bounded variation components, f : [ a ; b ] × R n → R n is a vector-function belonging to the Carathéodory class corresponding to A ; t 1 , . . . , t n ∈ [ a , b ] and ϕ 1 , . . . , ϕ n are the continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.

Lord–Shulman's system of partial differential equations of generalized thermoelasticity [Green and Lindsay, J. Elasticity 2: 1–7, 1972] is considered, in which the finite velocity of heat propagation is taken into account by introducing a relaxation time constant. General aspects of the theory of boundary value and initial-boundary value problems and representation of solutions by series and quadratures are considered using the method of a potential.

A real sequence { x k } is said to be ( * )-monotone with respect to a sequence { p k } and a positive integer σ if x k > 0 and (–1) n Δ n ( x k – p k x k–σ ) ≥ 0 for n ≥ 0. This paper is concerned with the existence of (*)-monotone solutions of a neutral difference equation. Existence criteria are derived by means of a comparison theorem and by establishing explicit existence criteria for positive and/or bounded solutions of a majorant recurrence relation.

The systems of bases are constructed for the spaces of cusp forms S k (Γ 0 (3), χ) ( k ≥6), S k (Γ 0 (7), χ) ( k ≥3) and S k (Γ 0 (11), χ) ( k ≥3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .

The paper deals with the Darboux problem for the equation D xy z ( x, y ) = f ( x, y, z ( x , y )) where z ( x , y ) is a function defined by z ( x , y )( t, s ) = z ( x + t , y + s ), ( t, s ) ∈ [– a 0, 0] × [– b 0, 0]. We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.

A general formula is derived for the number of representations r ( n ; f ) of a natural number n by diagonal quadratic forms f with five variables of level 16. For f belonging to one-class series, r ( n ; f ) coincides with the sum of a singular series, while in the case of a many-class series an additional term is required, for which the generalized theta-function introduced by T. V. Vepkhvadze [Vepkhvadze, Acta Arithmetica 53: 433–990] is used.