On the existence of a positive solution to a periodic boundary value problem for one non-linear second-order functional differential equation

Abstract

The boundary value problem is considered $$x''(t)+\rho^2x(t)+f \left (t,\left(Tx \right)(t) \right)=0,\quad 0<t<1,$$ $$x(0)=x(1),$$ $$x'(0)=x'(1),$$ where $0<\rho<\pi$, $T$ — linear positive continuous operator.

This problem was reduced to an equivalent integral equation, and using some properties of the Green's function of the operator $-d^2/dx^2$ with the corresponding periodic boundary conditions, the invariance of a completely continuous integral operator in the chosen cone of the space of continuous functions was established. Further, under power-law growth restrictions on the function $f$, relying on the well-known Krasnosel'skii theorem on the index of fixed points of an operator, the existence of at least one positive solution of the problem under consideration was proved. At the end of the article, an example of a boundary value problem for an integro-differential equation with a sublinear power right side $f$ is given, illustrating the fulfillment of sufficient conditions for the existence of at least one positive solution.