John M. Bownds

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We consider the linear system of integral equations (L) v(t) = ~o(t)+ Sta A(t, s)v(s) ds and its perturbation (P) u(t) = 9(t) + St a A(,, s)u(s)ds + ft p(t, s, U(,~)) ds for t _> a, where, following Strauss in [1], we assume that A(t, s) is an n x n matrix which, for some fixed to, is defined for t > s > to and satisfies f l i m f r [A(T+h,s)-A(T, s)[ ds =(More)
The basic linear, Volterra integral equation of the first kind with a weakly singular kernel is solved via a Galerkin approximation. It is shown that the approximate solution is a sum with the first term being the solution of Abel's equation and the remaining terms computable as components of the solution of an initial-value problem. The method represents a(More)
Using some basic observations from stability theory, it is shown that theclassical equationy"+a(t)y = Omust have at least one solution y(j) such that lim sup(|_y(/)l + lv'(/)l)>0 as /-»-oo. The same conclusion holds for a nonlinear perturbation of this equation provided the linearization has a stable zero equilibrium. The results may be easily and naturally(More)
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