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Banaś, J. (1997) Applications of Measures of Weak Noncompactness and Some Classes of Operators in the Theory of Functional Equations in the Lebesgue Space. Nonlinear Analysis: Theory, Methods & Applications, 30, 3283-3293.
https://doi.org/10.1016/S0362-546X(96)00157-5

has been cited by the following article:

• 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.