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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymptotic behavior of solutions of nonlinear functional differential equations in Banach space


Author: John R. Haddock
Journal: Trans. Amer. Math. Soc. 231 (1977), 83-92
MSC: Primary 34G05; Secondary 34K20
DOI: https://doi.org/10.1090/S0002-9947-1977-0442404-9
MathSciNet review: 0442404
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Abstract: Let X be a Banach space and let $ C = C([ - r,0],X)$ denote the space of continuous functions from $ [ - r,0]$ to X. In this paper the problem of convergence in norm of solutions of the nonlinear functional differential equation $ \dot x = F(t,{x_t})$ is considered where $ F:[0,\infty ) \times C \to X$. As a special case of the main theorem, stability results are given for the equation $ \dot x(t) = f(t,x(t)) + g(t,{x_t})$, where $ - f(t, \cdot ) - \alpha (t)I$ satisfies certain accretive type conditions and $ g(t, \cdot )$ is Lipschitzian with Lipschitz constant $ \beta (t)$ closely related to $ \alpha (t)$.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0442404-9
Keywords: Banach space, nonlinear differential equation, accretive operator, stability
Article copyright: © Copyright 1977 American Mathematical Society