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

A. F. Iz{é
1990 Proceedings of the American Mathematical Society  
Consider the equation (1) x + A(t)x = -f(t, x) x(Q) = x , x £ X , a Banach sequence space with a Schauder Basis. It is proved that if /(/, 0) = 0 , A(t)(-) + f(t, •) is a positive operator and the solution operator K(t, 0)x° = x° -f¿A(s)ds -¡¿f(s, x(s))ds is compact for t > 0, then system (1) has at least one solution x(t), x(t) ^ 0 such that x(t) > 0, -x(l) < 0 , and consequently x(t) are monotone nonincreasing for t > 0 .
doi:10.1090/s0002-9939-1990-1015679-7 fatcat:ljwcljorjvdhpmyje4zl2ulk64