RT Journal Article
T1 Reflection equation with piecewise constant arguments
A1 Cabada Fernández, Alberto
A1 Cambeses Franco, Paula
K1 Green’s function
K1 Equation with reflection
K1 Piecewise constant arguments
K1 Constant sign solutions
K1 Nonlinear problem
K1 Monotone method
K1 Nonlocal problems
AB In this work, we study nonlocal differential equations with  particular focus on those with reflection in their argument and piecewise constant dependence. The approach entails deriving the explicit expression of the solution to the linear problem by constructing the corresponding Green's function, as well as developing a novel formula to delineate the set of parameters involved in the analyzed equations for which the Green's function exhibits a constant sign. Furthermore, we demonstrate the existence of solutions for nonlinear problems through the utilisation of diverse results derived from fixed-point theory.The aforementioned methodology is specifically applied to the linear problem with periodic conditions $v'(t) + mv(-t) + Mv([t]) = h(t)$ for $t \in [-T,T]$, proving several existence results for the associated nonlinear problem and precisely delimiting the region where the Green's function $H_{m,M}$ has a constant sign. % when $T \in (0,1]$.The equations studied have the potential to be applied in fields such as biomedicine or quantum mechanics. Furthermore, this work represents a significant advance, as it is the first time that equations with involution and piecewise constant arguments have been studied together.
PB Elsevier
SN 1878-7274
YR 2026
FD 2026-02-01
LK https://hdl.handle.net/10347/44902
UL https://hdl.handle.net/10347/44902
LA eng
NO Commun Nonlinear Sci Numer Simulat 153 (2026) 109488
NO The authors were partially supported by Grant PID2020-113275GB-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” of the “European Union”, and by Xunta de Galicia (Spain), project ED431C 2023/12. The second author would like to express her gratitude to the Spanish Ministry of Science, Innovation and Universities for financial support (Grant reference FPU 23/02202).
DS Minerva
RD 26 abr 2026