An infinite-dimensional extension of theorems of Hartman and Wintner on monotone positive solutions of ordinary differential equations

Abstract:

Consider the equation (1) $\dot x + A\left ( t \right )x = - f\left ( {t,x} \right )\;x\left ( 0 \right ) = {x^0},{x^0} \in X$, a Banach sequence space with a Schauder Basis. It is proved that if $f\left ( {t,0} \right ) = 0,A\left ( t \right )\left ( \cdot \right ) + f\left ( {t, \cdot } \right )$ is a positive operator and the solution operator $K\left ( {t,0} \right ){x^0} = {x^0} - \int _0^t {A\left ( s \right )ds - \int _0^t {f\left ( {s,x\left ( s \right )} \right )ds} }$ is compact for $t > 0$, then system (1) has at least one solution $x\left ( t \right ),x\left ( t \right )\not \equiv 0$ such that $x\left ( t \right ) \geq 0, - \dot x\left ( t \right ) \leq 0$, and consequently $x\left ( t \right )$ are monotone nonincreasing for $t \geq 0$.