Implicit functional-integral equations associated with unbounded discontinuous functions
Applicable Nonlinear Analysis, Volume 3, Issue 2, August 2026, Pages: 93–105
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 and Academy of Romanian Scientists, 50044 Bucharest, Romania
Abstract
Let \(I=[0,1]\). We deal with the existence of solutions \(u\in L^p(I)\) of the implicit functional-integral equation $$ \psi(u(t))=f\Big(t,\int_I k(t,s)\,u(\varphi(s))\,ds\Big)\quad\hbox{for a.e.}\quad t\in I, $$ where \(Y\subseteq{\bf R}\) is a closed interval, \(p\in]1,+\infty[\,\), and \(\psi:Y\to{\bf R}\), \(f:I\times {\bf R}\to{\bf R}\), \(k:I\times I\to [0,+\infty[\) and \(\varphi:I\to I\) are given functions. We prove an existence result where the function \(f\) can be discontinuous, with respect to the second variable, even at all points \(x\in{\bf R}\). Our result improve in several aspects a very recent result in the field. In particular, we impose a linear growth condition for the function \(\psi^{-1}(f(t,\cdot))\), meaningfully weaker than boundedness condition which was previously imposed. As regards the function \(\psi\), we only require that it is continuous and non-constant on intervals.
Cite this Article as
Paolo Cubiotti and Jen-Chih Yao, Implicit functional-integral equations associated with unbounded discontinuous functions, Applicable Nonlinear Analysis, 3(2), 93–105, 2026