Second-Order Self-Adjoint Differential Equations Using a Conformable   Proportional Derivative
  In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Several reduction-of-order theorems are given. Solutions of the conformable second-order self-adjoint equation are then shown to be related to corresponding solutions of a first-order Riccati equation and a related quadratic functional and a conformable Picone identity. The first part of the study is concluded with a comprehensive roundabout theorem relating key equivalences among all these results. Subsequently, we establish a Lyapunov inequality, factorizations of the second-order equation, and conclude with a section on boundary value problems and Green's functions. 
Douglas R. Anderson
math.CA