Eigenschaften der Lösungsmenge eines Systems von Volterra-Integralgleichungen

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∞R^m, 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),...,u_n(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.