Abstract
In this work, we investigate the condition of the given interval which ensures the existence and uniqueness of solutions for two-point boundary value problems within conformable-type local fractional derivative. The method of analysis is obtained by the principle of contraction mapping. Furthermore, benefiting from calculating the integral of the Green's function, we are able to improve a recent result by obtaining a sharper lower bound for an eigenvalue problem. Two examples are presented to clarify the obtained results. Finally, we present an open problem for the interested reader.