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KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE
范进军
数学物理学报(英文版). 1996 (S1):
15-21.
In this paper. we consider the following Cauchy problem of first order nonlinear integro-differential equation of Volterra type ẋ=H(t,x,Tx),x(t0)=x0 (1) where (Tx)(t)=∫t0 tK(t,s)x(s)ds,x∈C[J,Ω] (2) K∈C[J×J,R],|K(t,s)| ≤ k(k> 0).(t,s)∈J×J,H∈C[J×Ω×Ω1,E].J=[t0.t0+a](a>0).Ω=B(0,N)={x∈E:||x||<N}(N>0).Ω1=B(0.kaN).x0∈Ω,E being real Banach space. The problem (1) is equivalent to the following equation x(t)=x0+∫t0 tH(s,r(s),(Tx)(s))ds (3) We study the toplogical properties of the problem (3)-continuum. The obtained result is the extention of the results of the ordinary differential equation and the reference[1].
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