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

• Abdussalam A Bojeldain

  Mathematics Department, Faculty of Science, Omar Al_Mukhtar University, El_Beida, Libya.

  Author

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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