Existence and Uniqueness Theorem for Voltera Equation First Order
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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