Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations
World Scientific, 1993 - Mathematics - 312 pages
This monograph aims to fill a void by making available a source book which first systematically describes all the available uniqueness and nonuniqueness criteria for ordinary differential equations, and compares and contrasts the merits of these criteria, and second, discusses open problems and offers some directions towards possible solutions.
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absolutely continuous Banach space classical abstract solution conditions of Theorem Consider the initial constant continuous and nonnegative continuous for x0 continuous function continuous in S+ continuously differentiable contradiction Corollary defined and continuous differential inequality Example exists a Lyapunov extended solution finite following result function f(x,y function y(x Further hence Henstock integral hypothesis implies initial value problem interval x0 Lebesgue Lebesgue integrable Lemma Let f(x,y Let the function Let y(x limsup Lipschitz condition 1.2.1 Lipschitz Uniqueness Theorem Lyapunov function Lyapunov function V(x,y Math maximal solution mean value theorem nondecreasing nonincreasing ordinary differential equations positive number proof of Theorem satisfies the conditions satisfies the inequality satisfies the Lipschitz sequence solution in 0,a solutions of 1.1.1 sufficiently small Suppose that y(x Suppose y(x unique solution value problem 1.1.1 XQ,XO XQ,XQ y(xi y(Xo zero
Page 309 - Conditions for the existence and uniqueness of a solution of the Cauchy problem for a system of ordinary differential equations with hysteresis nonlinearities', Dokl.