Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations

0460328 - MÚ 2017 RIV GE eng J - Journal Article
Lomtatidze, Alexander
Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations.
Memoirs on Differential Equations and Mathematical Physics. Roč. 67, č. 1 (2016), s. 1-129. ISSN 1512-0015
Institutional support: RVO:67985840
Keywords : periodic boundary value problem * positive solution * singular equation
OECD category: Applied mathematics
http://rmi.tsu.ge/jeomj/memoirs/vol67/abs67-1.htm

The aim of the present article is to get efficient conditions for the solvability of the periodic boundary value problém $$ u''=f(t,u);quad u(0)=u(omega),;; u'(0)=u'(omega), $$ where the function $fcolon[0,omega]times,]0,+infty[,tobbr$ 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 problém $$ u''=p(t)u+q(t);quad u(0)=u(omega),;; u'(0)=u'(omega) $$ 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.
Permanent Link: http://hdl.handle.net/11104/0260435