On solutions of a class of tempered fractional boundary value problem in tempered fractional Sobolev space

In this paper, we investigate tempered fractional boundary value problems (TFBVPs) with and without oscillating terms and establish the existence of solutions in the tempered fractional Sobolev space for fractional order μ ∈ (0, 1). Using variational principles and critical point theory, we derive new criteria ensuring the existence of at least one or infinitely many solutions. The problem is formulated in terms of an associated energy functional, whose critical points correspond to solutions of the TFBVPs. Additionally, we explore the connection between these problems and eigenvalue formulations, demonstrating how oscillating terms influence the spectral properties and solution structure. Our results contribute to the broader theory of fractional differential equations by providing insights into their variational structure and spectral behavior.