Qualitative Analysis of Non-Local Fractional Sturm–Liouville–Langevin Differential Equation and Inclusion With Time Delays

Abstract

We introduce time delays into the fractional Sturm–Liouville–Langevin framework to model hereditary effects and reaction lags in both single-valued and multivalued formulations. The problems are formulated using the generalized 𝜅-Caputo fractional derivative, and we establish existence and uniqueness results. For the singlevalued case, uniqueness is first obtained by applying the Banach fixed-point theorem with a 𝜅-Bielecki norm. The effectiveness of this norm lies in its capability to relax the strong sufficient hypothesis commonly imposed in the application of Banach’s fixed point theorem under the classical supremum norm — specifically, the contraction constant condition — by treating it in a more flexible and efficient manner through the use of this norm. In a special case, by subdividing the time interval and applying Burton’s progressive contraction method, we obtain uniqueness under the relaxed Lipschitz assumption, thus extending beyond the strict hypothesis required by Banach’s theorem — specifically, the contraction Lipschitz condition —. For the multivalued case, the Leray–Schauder nonlinear alternative is employed, which significantly broadens the existence theory by accommodating noncompact operators under 𝐿1 -Carathéodory conditions. Finally, illustrative examples validate the theoretical results.