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


Author: A. F. Izé
Journal: Proc. Amer. Math. Soc. 110 (1990), 77-84
MSC: Primary 34G20; Secondary 34C35, 58D25
MathSciNet review: 1015679
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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$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1015679-7
Keywords: Differential equations, Banach spaces, infinite dimensional spaces, positive solutions, operator equations, strongly positive, solid cone, egress points, strict egress points, trajectory, orbit, consequent operator, left shadow, process, retract, infinitesimal generator, classical solution
Article copyright: © Copyright 1990 American Mathematical Society