A Signed Maximum Principle for Boundary Value Problems for Riemann–Liouville Fractional Differential Equations with Analogues of Neumann or Periodic Boundary Conditions

Abstract

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for which the sufficient conditions can be verified. To show an application of the signed maximum principle, a method of upper and lower solutions coupled with monotone methods is developed to obtain sufficient conditions for the existence of a maximal solution and a minimal solution of a nonlinear boundary value problem. A specific example is provided to show that sufficient conditions for the nonlinear problem can be realized.

Keywords

• fractional boundary value problem
• signed maximum principle
• fractional Neumann boundary conditions
• fractional periodic boundary conditions