Alexander Lomtatidze

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
$$ u''=f(t,u);\quad u(0)=u(\omega),\;\; u'(0)=u'(\omega), $$
where the function $f\colon[0,\omega]\times\,]0,+\infty[\,\to\bbr$ satisfies local Ca\-ra\-th\'{e}o\-do\-ry 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
$$ u''=p(t)u+q(t);\quad u(0)=u(\omega),\;\; u'(0)=u'(\omega) $$
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.

Mathematics Subject Classification: 34B16, 34B15, 34B05, 34D20, 34D09

Key words and phrases: Periodic boundary value problem, positive solution, singular equation, solvability, unique solvability, stability