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 | References | Similar Articles | Additional Information

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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Additional Information

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