Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval.