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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Perron integral and existence and uniqueness theorems for a first order nonlinear differential equation

Author: Manoug N. Manougian
Journal: Proc. Amer. Math. Soc. 25 (1970), 34-38
MSC: Primary 34.04
MathSciNet review: 0255881
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Abstract: The Perron integral is used to establish an existence and uniqueness theorem concerning the initial value problem $y’(t) = f(t,y((t))$, and $y({t_0}) = \alpha$, for $t$ on the interval $I = \{ t|0 \leqq t \leqq 1\}$. The existence and uniqueness of the solution is obtained by use of a generalized Lipschitz condition, and a Picard sequence which is equiabsolutely continuous on $I$. Also, we prove a theorem on the uniqueness of solution by a generalization of Gronwall’s inequality.

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Keywords: Initial value problem, Lebesgue integral, Perron integral, bounded variation, Picard sequence, locally absolutely continuous, equicontinuous, equiabsolutely continuous, Cauchy-Euler meth[ill]d, Gronwall inequality
Article copyright: © Copyright 1970 American Mathematical Society