Solutions for Linear Non-Homogeneous Set-Valued Differential Equations with Memory under Generalized Derivatives
Applicable Nonlinear Analysis, Volume 3, Issue 3, December 2026, Pages: 148–163
Yonglin Liu
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, China
Abstract
We study linear non-homogeneous set-valued differential equations in \(\mathbb{R}^n\) with memory effects under generalized derivatives. The memory term is described by a convolution-type radius function, which leads to equivalent set-valued Volterra integral equations. Under suitable structural assumptions on the coefficient matrix, we derive explicit solution representations for the integral equation associated with the Hukuhara derivative, which yield globally defined solutions with monotone non-decreasing diameters. In contrast, for the Plotnikov--Skripnik derivative and the Bede--Gal derivative, we obtain ball-valued solutions that are typically only locally defined in time and have monotone decreasing diameters. Several examples are included to illustrate qualitative differences in the solution behavior induced by different notions of generalized differentiability.
Cite this Article as
Yonglin Liu, Solutions for Linear Non-Homogeneous Set-Valued Differential Equations with Memory under Generalized Derivatives, Applicable Nonlinear Analysis, 3(3), 148–163, 2026