Abstract

This paper is devoted to the construction and analysis of solution operators for a broad class of fractional-order systems. Both coupled systems with a memory-decoupled structure and fully coupled systems are considered. Using Laplace transform techniques and matrix-valued operator methods, explicit representations of the associated operator families are derived. The developed framework extends classical fractional resolvent theory to distributed-order and fully coupled systems, highlighting the role of coupling in shaping the structure of solution operators. These operators provide a natural setting for the fractional Duhamel principle and thus play a central role in the analysis of nonhomogeneous problems. Finally, several examples are presented to illustrate the theory and demonstrate the construction of solution operators for representative distributed-order and fully coupled systems.

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