engleski

Linear singular differntial equations in Banach space and nonrectifiable attractivity in two- dimensional linear differential systems

We study the asymptotic behaviour near t = 0 of all solutions x ∈ C 1 ((0, t0 ] ; X) of linear nonautonomous differential equation x = A(t)x, t ∈ (0, t0 ] (1) where X is an arbitrary Banach space and A : (0, t0 ] → L(X) is an operator-valued function which may be singular at t = 0. In terms of some asmyptotic behaviour of the operator norm A(t) near t = 0, the kind of singularity (resp. regularity) of equation (1) is characterized: for every x0 ∈ X and solution x of (1) such that x(t0 ) = x0 , we have x(t) X → 0 as t → 0 and x X ∈ L1 ((0, t0 ]) (resp. x X ∈ L1 ((0, t0 ])). Next, when X = R2 and equation (1) is a two-dimensional linear integrable differntial system, our previous result allows us to characterize the so-called nonrectifiable (resp. rectifiable) attractivity of zero the zero solution to the equation (1), that is x(t) R2 → 0 as t → 0, the solution’s curve Γx is a Jordan curve in R2 and length(Γx ) = ∞ (resp. length(Γx ) < ∞).

linear nonautonomous differential operator; attractivity; singualrity; zero-solution; rectifiability

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