Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator
In this paper, existence of positive periodic solutions is considered for differential equations of the form \[[x(t)-cx(t-\tau(t))]''+a(t)x(t)=f(t, x(t-\tau(t)))+e(t),\] where $c$ is a constant, $a(t) \geq 0$ and $\tau, a, e$ are real valued $\omega$-periodic functions, $f(t, u)$ is $\omega$-periodic in $t$ and with a singularity at $u=0$: $\lim_{u\to 0^+}f(t, u)=+\infty$ (or $\lim_{u\to 0^+}f(t, u)=-\infty$) uniformly in $t$. By using a fixed-point theorem of Krasnoselskii, the existence of positive $\omega$-periodic solutions is proved under various sufficient conditions. There are many results in literature on existence of periodic solutions for delay differential equations, neutral differential equations, equations with a difference operator, and differential equations with a singularity term. But it seems that this is the first paper on periodic solutions of differential equations combining a difference operator with a singularity term.
scientific article; zbMATH DE number 7075056
scientific article