Vol 212, No 3 (2016)

We consider a multipoint problem with data lying on the characteristics of a system of hyperbolic equations of the second order with two independent variables. We introduce new unknown functions and reduce the problem to an equivalent problem formed by a family of multipoint problems for systems of ordinary differential equations and functional relations. We establish conditions for the existence of a unique solution in terms of the coefficients of multipoint problems.

We establish sufficient conditions for the existence of periodic solutions of a system of nonlinear functional-differential equations with small parameter and linear deviations of the argument and study the properties of these solutions.

We establish conditions required for the generalized invertibility of Fredholm integral operators with degenerate kernels in Banach spaces. New results on the generalized inversion of Fredholm integral operators with degenerate kernels in Banach spaces are obtained. A detailed analysis of an example is also presented.

We construct an asymptotic expansion for the fundamental system of solutions of a linear singularly perturbed system of m-order differential equations with degenerate principal matrix at the higher-order derivatives. The case where the corresponding characteristic polynomial has a multiple spectrum is investigated. It is proved that, in this case, the asymptotic expansions are constructed in the fractional powers of the small parameter. The recurrence formulas for the coefficients of these expansions are obtained.

We introduce a new class of almost periodic operators and establish the conditions of existence of almost periodic solutions of nonlinear discrete equations. These solutions can be not almost periodic in Bochner’s sense.