An analogue of the Lyapunov inequality for an ordinary second-order differential equation with a fractional derivative and a variable coefficient

Abstract

This paper studies an ordinary second-order differential equation with a fractional differentiation operator in the sense of Riemann-Liouville with a variable coefficient. We use the Green’s function’s method to find a representation of the solution of the Dirichlet problem for the equation under consideration when the solvability condition is satisfied. Green’s function to the problem is constructed in terms of the fundamental solution of the equation under study and its properties are proved. The necessary integral condition for the existence of a nontrivial solution to the homogeneous Dirichlet problem, called an analogue of the Lyapunov inequality, is found.

Keywords

• fractional Riemann–Liouville integral
• fractional Riemann–Liouville derivative
• Gerasimov–Caputo fractional derivative
• Dirichlet problem
• Green’s function
• analogue of Lyapunov inequality