On the Existence of A Unique Solution for Nonlinear Ordinary Differential Equations of Order m
DOI:
https://doi.org/10.54172/mjsc.v30i1.44Keywords:
Nonlinear Ordinary Differential Equation of Order m, Banach Space of Bounded Functions , Lipschitz Condition, Contraction Mapping Theorem, Existence of a Unique Solution GloballyAbstract
In this work I state and prove a theorem for local existence of a unique solution for the Nonlinear Ordinary Differential Equations (NODE):
(1) of order m; where m is a positive integer; having the initial conditions:
(2) Since the (NODE) (1) with the initial conditions (2) is equivalent to the Integral Equation:
(3)We denote the right hand side (r.h.s.) of (3) by the nonlinear operator; then prove that this operator is contractive in a metric space E subset of the Banach space B of the class of continuous bounded functions defined by:
(4) and B is equipped with the weighted norm:
(5) which is known as Bielescki's type norm. , are finite real numbers, where is the Lipschitz coefficient of the r.h.s. of (1) in B1(a subset of the Banach space B given by (4)) defined by:
(6) , and are finite real numbers.
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