ISSN:
1432-1785
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract By a well known theorem of H. Kneser [4] the set U of all solutions of the initial value problem $$u' = f(x,u)forx\varepsilon [0,a],u(0) = u_o $$ has the following property: If f is continuous and bounded then U(x0)={u(x0): u∈U} is a continuum (i.e. a compact and connected subset) for every x0∈[0,a]. In the present paper we claim to extend this theorem to a system of Volterra integral equations in several variables of the form x∈B∞Rm, ν=1,...,n that had been investigated in [8]. In fact we shall prove that U is a continuum of the Banach space Cn(B) of all ‘vector functions’ u(x)=(u1(x),...,un(x)), continuous on B. It is an immediate consequence from this that U(x0) is a continuum of Rn. These results will be established by the help of a suitable modification of a method used by M. Müller [5] to prove Kneser's theorem. Especially, we obtain new theorems for some initial value problems for hyperbolic equations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01190276
