On the solvability of the periodic problem for systems of linear functional differential equations with regular operators

Abstract

Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators we get effective necessary and sufficient conditions for the unique solvability of the periodic problem. We consider some classes of two-dimensional systems of first order linear func- tional differential equations with regular operators. General necessary and sufficient conditions for the solvability of the periodic problem for such classes are obtained. These conditions mean that some function on a set in a finite-dimensional space is positive (this functions is quadratic with respect to all variables). Moreover, in terms of norms of the operators appearing in the functional differential system, we get the necessary and sufficient conditions for the unique solvability of the periodic problem for one case of two-dimensional system with monotonic operators. It is found there exist two domains of parameters corresponding to the unique solvability. These result do not have analogues for systems. Non-improvable results for periodic problem are known only for cyclic first order functional differential systems (32). Necessary and sufficient conditions for the unique solvability of two-dimensional