摘要
给出并证明了Volcrra积分方程组解的存在唯一性的一组定理,给出的充分条件,减弱了以下定理的条件,使其存在唯一性定理成为本文定理的直接推论.定理:设a,b和L为正数,a∈(0,1),c=a/L又设a)f在[0,a]上连续;b)g在U上连续,其中U={(t,s,x):0≤s≤t≤a,|x-f(t)|≤b};C)g关于x在U上满足李普希兹条件|g(t,s,x)-g(t,S,y)|≤L|x-y|.若M=max_g|g(t,s,x)|,则方程x(t)=f(t)+integral from n=0 to t g(t,s,x(s))ds在[0,T]上有唯一解,这里T=min[d,b/M,c].
This artical discusses the existence and uniquencss of solutions of the Volterra integral equations. The theorem obtained here weakens the conditions of the following theorem: Let a. b and L be positive numbers, and for some fixed a∈(0,1)define c= a/L, suppose a) f is continuous on [0, a],b) g is continuous on U ={(t, s, x,):0≤s≤t≤a and |x—f(t)|≤b},c) g satisfies a Lipschitz condition with respect to x on U|g(t,s,x)—g(t,s,y)|≤L|x—y|if (t,s,x), (t,s,y)∈ U. If M=max|g(t,s,x)|, then there is a unique solution of the equation x(t)=f(t)+integral from n=0 to t g(t,s,x(s))ds on [0, T], where r=min[a,b/M, c].
