On Nodal Properties for Some Nonlinear Conformable Fractional Differential Equations

Abstract

A class of conformable fractional differential equations \[ D_x^{\alpha} D_x^{\alpha} u(x) + \omega(x) f(u(x)) = 0 \quad \textrm{on $(0,1)$} \] is considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function $\omega$. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.