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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0255881-2
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\vert \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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DOI: https://doi.org/10.1090/S0002-9939-1970-0255881-2
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