Continuous solutions for a class boundary value problem with mixed $\psi$-Riemann-Liouville tempered fractional derivatives

We study a class of nonlinear boundary value problems driven by mixed $\psi$-Riemann-Liouville tempered fractional derivatives of order $\alpha \in \left(\frac{1}{2},1\right)$ under homogeneous Dirichlet boundary conditions. The analysis is performed in a suitable fractional Sobolev-type space associated with $\psi$-tempered operators. We first establish sharp mapping properties for the $\psi$-Riemann-Liouville tempered fractional integral, proving its boundedness from $L^p((a,b),{\psi }^{\prime })$ into Hölder spaces $H^{\alpha -\frac{1}{2}}[a,b]$ with explicit constants depending on the incomplete gamma function. These estimates provide the functional framework required to handle the corresponding nonlocal problem. Using variational methods, we associate the problem with a $C^1$ energy functional on ${\mathbb{H}}_{\psi ,0}^{\alpha ,\sigma }(a,b)$ and prove compactness via the Palais–Smale condition under optimal growth assumptions. We then establish a precise variational structure yielding: (ⅰ) existence of nontrivial solutions via minimization arguments, (ⅱ) multiplicity results based on a mountain pass geometry, and (ⅲ) existence of infinitely many solutions by means of symmetric critical point theory. Our results extend the current theory of fractional equations with respect to another function by incorporating tempered effects and explicit operator estimates, thus providing a unified and flexible framework for a broad class of nonlocal problems.