Vector solutions of implicit functional-integral equations with highly discontinuous operators
Applicable Nonlinear Analysis, Volume 2, Issue 1, April 2025, Pages: 10–21
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 \(I \subseteq \mathbb{R}\) be a compact interval. In this paper we prove an existence result for solutions \(u \in L^p(I, \mathbb{R}^n)\), with \(p \in ]1, +\infty]\), of the implicit functional-integral equation \[ f\big(t, u(t), \int_I \xi(t, s)\, u(\varphi(s))\, ds\big) = 0 \quad \text{for almost every} \quad t \in I, \] where \(f: I \times S \times \mathbb{R}^n \to \mathbb{R}\), \(\varphi: I \to I\) and \(\xi: I \times I \to [0, +\infty[\) are given functions, and \(S \subseteq \mathbb{R}^n\) is a suitable closed, connected, and locally connected set. The main peculiarity of our result is the regularity assumption on \(f\) with respect to the third variable, considerably weaker than the usual continuity required in the literature. A function \(f\) satisfying the assumptions of our result can be discontinuous with respect to the third variable, even at each point \(x \in \mathbb{R}^n\). Our result extends a very recent result proved in the scalar case \(n = 1\). Such an extension is not trivial and requires more articulated assumptions, together with a more articulated and delicate technical construction.
Cite this Article as
Paolo Cubiotti and Jen-Chih Yao, Vector solutions of implicit functional-integral equations with highly discontinuous operators,, Applicable Nonlinear Analysis, 2(1), 10–21, 2025