Some 'a priori' Bounds for Nonlinear Volterra Equations.

摘要

A priori bounds are obtained for the solutions, x, of the real vector Volterra equation x(t) + the integral over s=0 to t of (a(t-s)g(x(s))ds) = f(t) under several different sets of hypotheses on the prescribed functions a, g and f. In each case, the key conditions are on a and g. Thus there is a positivity assumption on the matrix a(0) and an assumption on g which, in the scalar case, is typified by (but less restrictive than) the hypothesis xg(x) > or = 0, over all x. The known procedure of reducing the problem of global existence to that of finding an a priori bound is briefly reviewed together with a relevant local existence theorem.