**Alexander Lomtatidze**

##
Theorems on Differential Inequalities and Periodic Boundary Value Problem for
Second-Order Ordinary Differential Equations

**abstract:**

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