The periodic problem for the second order integro-differential equations with distributed deviation

Abstract

In the paper we describe the classes of unique solvability of the Dirichlet and mixed two point boundary value problems for the second order linear integro-differential equation b u (t) = p0 (t)u(t) + p1 (t)u(1 (t)) + p(t, s)u( (s)) ds + q(t). a On the basis of the obtained and, in some sense, optimal results for the linear problems, by the a priori boundedness principle we prove the theorems of solvability and unique solvability for the second order nonlinear functional differential equations under the mentioned boundary conditions.In the paper we describe the classes of unique solvability of the Dirichlet and mixed two point boundary value problems for the second order linear integro-differential equation b u (t) = p0 (t)u(t) + p1 (t)u(1 (t)) + p(t, s)u( (s)) ds + q(t). a On the basis of the obtained and, in some sense, optimal results for the linear problems, by the a priori boundedness principle we prove the theorems of solvability and unique solvability for the second order nonlinear functional differential equations under the mentioned boundary conditions.