On the nonexistence of positive solutions of integro-differential inequalities (Q1360216)

Sufficient conditions for the nonexistence of positive solutions to the integro-differential inequalities \[ y'(t)+ \int^t_0 K(t-s)f(y(s))ds\leq 0,\quad t\in[T,\infty),\tag{E\(_1\)} \] and \[ y''(t)- \int^t_0 K(t-s)f(y(s))ds\geq 0,\quad t\in[T,\infty),\tag{E\(_2\)} \] are given in the form \[ -\lambda+\theta \int^\infty_0 e^{\lambda s}K(s)ds>0\quad\text{and} \quad-\lambda^2+\theta \int^\infty_0 e^{\lambda s}K(s)ds>0\quad(\text{for }\lambda>0), \] respectively, where (C1) \(K\in C(R_+,R_+)\) is not identically zero near the origin, (C2) \(f\in C(R_0,R_0)\) with \(f(y)/y\geq \theta>0\) in \((0,y_0)\); here \(\theta\), \(y_0\) are positive constants, \(R_+= [0,\infty)\) and \(R_0=(0,\infty)\). These conditions have been proved by \textit{Ch. G. Philos} and \textit{Y. G. Sficas} [Appl. Anal. 36, No. 3/4, 189-210 (1990; Zbl 0672.45007)] and \textit{G. Ladas}, \textit{Ch. G. Philos} and \textit{Y. G. Sficas} [Differ. Integral Equa. 4, No. 5, 1113-1120 (1991; Zbl 0742.45003)] under different assumptions. Namely, for the special case of \((\text{E}_2)\) when \(f(y)=y\), and for \((\text{E}_1)\) with (C2) replaced by \(f\in C(R,R)\) satisfying \(yf(y)>0\) for \(y\neq 0\) and \(\theta:=\inf_{y>0} {f(y)\over y}> 0\); here \(R=(-\infty,\infty)\). By an example, the authors show that the function \(f\) in \((\text{E}_1)\) cannot be a superlinear function.