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Neural activity influences the patterning of synaptic connections and functional organization of developing sensory and motor systems, but the long-term consequences of intense neural activity such as seizures in the developing hippocampus are not adequately understood. To evaluate the possibility that abnormal neural activity during early development may(More)
The notion of strong or adjoint stability for linear ordinary differential equations is generalized to the theory of Volterra integral equations. It is found that this generalization is not unique in that equivalent definitions for differential equations lead to different stabilities for integral equations in general. Three types of stabilities arising(More)
The author considers Volterra Integral Equations of either of the two forms $$u(x) = f(x) + \int\limits_a^x {k(x - t)g(u(t))dt, a \leqslant } x \leqslant b,$$ wheref, k, andg are continuous andg satisfies a local Lipschitz condition, or $$u(x) = f(x) + \int\limits_a^x {\sum\limits_{j = 1}^m {c_j (x)g_j (t,u(t))dt} ,} $$ wheref,c j , andg j ,j=1,2,...,m, are(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)
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)
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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