数学物理学报(英文版) ›› 1996, Vol. 16 ›› Issue (S1): 15-21.
范进军
Fan Jinjun
摘要: 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].