## AN EXISTENCE THEOREM FOR A NONLINEAR INTEGRAL EQUATION OF URYSOHN TYPE IN L^P(R^N)

## W. G. El-Sayed, M. M. El-Borai, M. M. A. Metwali, and N. I. Shemais

DOI: https://doi.org/10.37418/amsj.9.11.107

The aim of this paper is investigating and solvability of the nonlinear integral Equation due to Urysohn, in the space of $~p^{th}~$ Lebesgue integrable functions on $~\mathbb{R^N},~(L^p (\mathbb{R^N}))$. The Urysohn integral equations are enjoying interest among mathematicians, physicists and engineers. We try to assume the sufficient conditions under which

the existence theorem of the given integral equation can be proved. The main tool is using Dabo fixed point theorem via a certain measure of noncompactness introduced by Aghajani et. Al \cite{AgOrHa} in the space $~L^p (\mathbb{R^N})$, as an application to prove the desired existence theorem of our Urysohn integral equation. At the end of this paper, we introduce an example that ensure the importance of the hypothesis that assumed in our existence theorem.

**Keywords: **Ursohn integral equation, Existence, measure of noncompactness, Darbo’s fixed point theorem, Fixed point.