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On ordinary linear differential equations in J . (English) Zbl 0838.34075
Let J be a countably infinite set and let J be the FrĂ©chet space of all mappings x:J written as x=(x j ) jJ endowed with the product topology. Let A be a row-finite matrix, f([0,T], J ) and y 0 J . Consider the initial value problem y ' (t)=Ay(t)+f(t), y(0)=y 0 J . The author proves that if σ(A) is at most countable, then for every f([0,T], J ) and every y 0 J there are infinitely many y 1 ([0,T], J ) solving the problem above. (σ(A) denotes the spectrum of A.) This and other theorems are used in what follows to investigate row-finite initial value problems of second-order and the uniqueness of positive solutions for initial value problems of the considered form in the case that σ(A) is at most countable for special sets of matrices.
Reviewer: R.Manthey (Jena)
MSC:
34G10Linear ODE in abstract spaces