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

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 $t \geq 0$.