KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE

数学物理学报(英文版) ›› 1996, Vol. 16 ›› Issue (S1): 15-21.

KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE

范进军

Department of Mathematics. Shandong Normal University. Jinan 250014. China

收稿日期:1992-12-05 出版日期:1996-12-31 发布日期:1996-12-31

KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE

Fan Jinjun

Department of Mathematics. Shandong Normal University. Jinan 250014. China

Received:1992-12-05 Online:1996-12-31 Published:1996-12-31

摘要/Abstract

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].

关键词: Banach space, Measures of noncompact

Abstract: 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].

Key words: Banach space, Measures of noncompact

范进军. KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE[J]. 数学物理学报(英文版), 1996, 16(S1): 15-21.

Fan Jinjun. KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE[J]. Acta mathematica scientia,Series B, 1996, 16(S1): 15-21.

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KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE

KNESER'S THEOREM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE IN A BANACH SPACE

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