We obtain a representation of a solution of the Cauchy problem for a linear inhomogeneous differential equation with constant coefficients and pure delay. We use special matrix functions called a delayed matrix sine and a delayed matrix cosine. They have the form of matrix polynomials of degree dependent on the value of delay.

Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space.

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The paper deals with a system of difference equations, where coefficients depend on Markov chains. The functional equations for particular density and the moment equations for the system are derived and used in the investigation of solvability and stability. An application of the results is shown how to solve various economic problems. c 2013 Mathematical Institute, Slovak Academy of Sciences. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 34K50, 60H10, 60H30, 65C30. K e y w o r d s: stochastic systems, Markov chain, moment equations, stability.

Exponential solutions of equatioṅAsymptotic behaviour of solutions of first-order differential equation with two deviating arguments of the forṁis discussed for t → ∞. A criterion for representing solutions in exponential form is proved. As consequences, inequalities for such solutions are given. Connections with known results are discussed and a sufficient condition for existence of unbounded solutions, generalizing previous ones, is derived. An illustrative example is considered, too.

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