Well-Posedness of Inverse Sturm-Liouville Problem with Fractional Derivative

Abstract

We have given the inverse nodal problem for the fractional Sturm-Liouville (S-L) problem and the stability for this problem. First of all, we have shown asymptotic forms for nodal parameters and by them, the potential function can be reconstructed with a limit of nodal parameters. We proved that this limit exists. We also gave wellposedness of the problem in the rest of study. We have basically shown that the set of potential functions satisfying the condition integral(pi)(0) q(t)d(alpha)t < infinity is homeomorphic to the space of quasi nodal sequences. Although the results given in the paper were given for the classical derivative of S-L the problem, the results here are different and more general than the previous results because they contain the fractional derivative. If alpha = 1, the results coincide with the results given for classical derivative problem.