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Proceedings of the American Mathematical Society

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



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