Integrable solutions of highly discontinuous implicit functional-integral equations
Optimization Eruditorum, Volume 1, Issue 2, December 2024, Pages 75–87
PAOLO CUBIOTTI
Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F.Stagno d’Alcontres 31, 98166 Messina, Italy
JEN-CHIH YAO
Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan
Academy of Romanian Scientists, 50044 Bucharest, Romania
Abstract
Let \( I \) be a real compact interval. We deal with the problem of the existence of solutions \( u \in L^p(I) \) of the implicit functional-integral equation \[ f\left(t, u(t), \int_I k(t, s) u(\varphi(s)) \, ds \right) = 0, \quad \text{for a.e. }~~ t \in I, \] where \( Y \) is a closed interval, and \( f : I \times Y \times \mathbb{R} \to \mathbb{R} \), \( k : I \times I \to [0, +\infty[ \), and \( \varphi : I \to I \) are given functions. Such an equation includes, as special cases, many integral equations studied in the literature. We prove an existence result whose main peculiarity is the following: a function \( f(t, y, x) \) satisfying our assumptions can be discontinuous with respect to the third variable, even at all points \( x \in \mathbb{R} \). As regards the function \( y \mapsto f(t, y, x) \), we only require that it is continuous, that it changes its sign over \( Y \), and that it is not identically zero over any interval. No assumption of monotonicity is made on \( f \). Our result extends and improves several results in the literature. Examples and also counter-examples to possible improvements are presented.
Cite this Article as
Paolo Cubiotti and Jen-Chih Yao, Integrable solutions of highly discontinuous implicit functional-integral equations, Optimization Eruditorum, 1(2), 75–87, 2024