On the Existence of A Unique Solution for Nonlinear Ordinary Differential Equations of Order m

Authors

  • Abdussalam A. Bojeldain Mathematics Department, Faculty of Science, Omar Al_Mukhtar University, El_Beida, Libya.

DOI:

https://doi.org/10.54172/mjsc.v30i1.44

Keywords:

Nonlinear Ordinary Differential Equation of Order m, Banach Space of Bounded Functions x(t)∈C^m (R), Lipschitz Condition, Contraction Mapping Theorem, Existence of a Unique Solution Globally.

Abstract

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)

Where  for  ,and  are finite real numbers.

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Published

2015-06-30

How to Cite

Bojeldain, A. A. . (2015). On the Existence of A Unique Solution for Nonlinear Ordinary Differential Equations of Order m. Al-Mukhtar Journal of Sciences, 30(1), 10–17. https://doi.org/10.54172/mjsc.v30i1.44

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