Symmetry analysis for nonlinear fractional terminal system under $w$-Hilfer fractional derivative in different weighted Banach spaces

Our objective in this study is to investigate the behavior of a nonlinear terminal fractional system under $w$-Hilfer fractional derivative in different weighted Banach spaces. We examine the system's dynamics and understand the effects of different weighted Banach spaces on the properties of solutions, including existence, uniqueness, stability, and symmetry. We derive the equivalent integral equations and employ the Schauder and Banach fixed point theorems. Additionally, we discuss three symmetric cases of the system to show how the choice of the weighted function $w(\iota )$ impacts the solutions and their symmetry properties. We study the stability of the solutions in the Ulam sense to assess the robustness and reliability of these solutions under various conditions. Finally, to understand the system's behavior, we present an illustrative example with graphs of the symmetric cases.