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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FIRST ORDER DIFFERENTIAL EQUATIONS CONTD
FUNCTIONAL DIFFERENTIAL EQUATIONS
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absolutely continuous Assume Banach space Carathèodory conditions classical abstract solution conditions of Theorem Consider the initial continuous function continuous in ro continuously differentiable contradiction Corollary defined and continuous differential equation differential inequality Example exists a Lyapunov exp ſ extended solution f(ro finite function f(r Further hence Hilbert space hypothesis implies initial value problem interval ro Lebesgue Lemma Let f(r Let the function lim-o lime—ot Lipschitz Uniqueness Theorem Lyapunov function V(r maximal solution mean value theorem nondecreasing nonincreasing ordinary differential equations p(ro point ro positive number proof of Theorem r e ro ro-H ro-Ha rols satisfies the Carathèodory satisfies the conditions satisfies the inequality satisfy the Lipschitz sequence ſly solution in ro solution z(z solutions of 1.1.1 solutions y(r sufficiently small Suppose that f(r Suppose y(r unique solution v(ro value problem 1.1.1 waſ y(ro y(tr yo(r zero
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Page 308 - Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano. Mem. Coll. Sci. Univ. Kyoto. Ser. A, 24 (1942), 2128.
Page 305 - MA Krasnosel'skii and SG Krein, On a class of uniqueness theorems for the equation У