TITLE:
Existence of Solutions of a Convolution Integral Equation
AUTHORS:
Henry Otoo, Benjamin Dadzie Mensah, Lewis Brew
KEYWORDS:
Volterra, Integral Equation, Convolution, Fixed Point, Existence, Noncompactness
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.5,
May
30,
2024
ABSTRACT: In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball Br belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set
M∈
B
r
the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.