Strong solutions for differential equations in abstract spaces
نویسنده
چکیده
Let (E,F) be a locally convex space. We denote the bounded elements of E by Eb : ={x ∈ E : ‖x‖F = sup ∈F (x)<∞}. In this paper, we prove that if BEb is relatively compact with respect to the F topology and f : I × Eb → Eb is a measurable family of F-continuous maps then for each x0 ∈ Eb there exists a norm-differentiable, (i.e. differentiable with respect to the ‖ · ‖F norm) local solution to the initial valued problem ut (t)= f (t, u(t)), u(t0)= x0. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy–Littlewood maximal operator. © 2004 Elsevier Inc. All rights reserved. MSC: primary 45N05; 45G10; 65L05; secondary 46A03; 46B10; 46B50
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