Existence and Continuous Dependence for a Class of Nonlinear Neutral- Differential Equations

This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form $$x'(t) = f(t, x(t), x(g(t, x)), x'(h(t, x))), x(0) = x_0,$$ where $f, g$, and $h$ are continuous functions with $g(0, x_0) =...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1971-08, Vol.29 (3), p.467-473
Main Author: Grimm, L. J.
Format: Article
Language:English
Subjects:
Cauchy problem
Continuous functions
Differential equations
Equation roots
Existence theorems
Lipschitz condition
Mathematical theorems
Uniqueness
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Summary:This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form $$x'(t) = f(t, x(t), x(g(t, x)), x'(h(t, x))), x(0) = x_0,$$ where $f, g$, and $h$ are continuous functions with $g(0, x_0) = h(0, x_0) = 0$. The existence of a continuous solution of the functional equation $z(t) = f(t, z(h(t)))$ is proved as a corollary.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1971-0287117-1