Existence and Uniqueness Theorem for Voltera Equation First Order


  • Abd El-Salam Bo-Geldain Dep. of Mach, Faculty of Science, Omar Al-Mukhtar University, Libya.




In this paper I introduced two theorems for the local existence of a unique solution, one for Nonlinear Volterra Integro-Differential Equation in the additive form , and the other for the general form ; where , with the i.c.  and .

The proof is done by proving that the following operator:

, with , is a contraction mapping in the metric space:

 where  and ; noting that  is a subset of the Banach space  given by:

, each of u and w is a continuous function in its arguments, and  is equipped with the following weighted norm which is known as bielecki's type norm:

 for any finite numbers  and  such that  is the Lipschitz's coefficient of  and  is the Lipschitz's coefficient of  By using the above mentioned norm, I concluded that  is contractive on the following form:

Which reveals that the contraction coefficient  is in (0, 1) ; whence the existence of a unique solution is guaranteed globally if we can guarantee that . Indeed we got this result for all the examples.



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How to Cite

Bo-Geldain, A. E.-S. (2008). Existence and Uniqueness Theorem for Voltera Equation First Order. Al-Mukhtar Journal of Sciences, 18(1), 140–148. https://doi.org/10.54172/mjsc.v18i1.800



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