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On existence theorems for differential equations in Banach spaces. (English) Zbl 0569.34053

Let E be a real B-space and μ a ball measure of noncompactness on E, T>0. Let A={ω:[0,T]×[0,[[0,[ be continuous functions, ω(t,·) nondecreasing, ω(t,0)=0 and such that 0 is the only continuous function on [0,T] satisfying u(t) 0 t ω(s,u(s))ds, u(0)=0}, C={ω:]0,T]×[0,[[0,[ such that for any ϵ>0 there exist δ>0 and t n 0, t n >0 and ρ n :[t n ,T][0,[ differentiable and satisfying ρ n ' (t)ω(t,ρ n (t)), ρ n (t)δt n , 0<ρ n (t)ϵ for t[t n ,T], n}·

Theorem 1: Let ω 1 :[0,T]×[0,[[0,[ be a continuous function, ω 1 (t,·) nondecreasing, ω 1 (t,0)=0 and there exist ωC such that ω 1 (t,u)ω(t,u), (t,u)]0,T]×[0,[. Then ω 1 A·

Theorem 2: If ω ¯(t,u)=sup{μ(f(t,X)):μ(X)=u,XB(x 0 ,r) ¯}, where f: [0,T]×B(x 0 ,r) ¯E is a uniformly continuous function such that μ(f(t,X))ω(t,μ(X)) for t[0,T] and for any X bounded in E, X and ω is a given Kamke function. Then ω ¯A. Theorem 3: Let f be a function bounded by A>0 and as in theorem 2 and ω is a Kamke function of C. If ATr then there exists at least one solution of x ' (t)=f(t,x(t)) for t[0,T], x(0)=x 0 .

Reviewer: G.Bottaro

MSC:
34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions