THEOREMS ON DIFFERENTIAL INEQUALITIES AND PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

The aim of the present article is to get efficient conditions for the solvability of the periodic boundary value problem u '' = f(t,u); u(0) = u(w), u'(0) = u'(w), where the function f : [0, w] x] 0, +infinity [-> R satisfies local Caratheodory conditions, i.e., it may have "singularity" for u = 0. For this purpose, first the technique of differential inequalities is developed and the question on existence and uniqueness of a positive solution of the linear problem u '' = p(t,u) + q(t); u(0) = u(w), u ''(0) = u'(w) is studied. A systematic application of the above-mentioned technique enables one to derive sufficient and in certain cases also necessary conditions for the solvability of the nonlinear problem considered.