Positive and Negative Results Concerning Maximum Principles and Principal Eigenvalues for Fractional Problems

ABSTRACT

Fractional differential equations (FDEs) of order between 1 and 2 are studied when equal to a non‐negative integrable function , for the Riemann‐Liouville (R‐L) and Caputo fractional derivatives, each with three sets of boundary conditions: Dirichlet, right and left focal. It is shown that solutions of the FDEs are solutions of integral equations, and there is some regularity of the solution. Possible maximum principles are first studied, the R‐L derivative case has been previously studied, but the right and left focal boundary conditions are treated here for the first time. It is proved, with a different method than previously, that the maximum principle holds for the R‐L FDE with Dirichlet BCs. It is also proved to hold for the R‐L FDE with the right focal boundary conditions. For the Caputo differential problems, in both cases, a maximum principle may fail. It is shown that failure occurs with a large class of functions , which can be strictly positive. For the left focal boundary conditions, it is shown that a maximum principle does hold in the Caputo case but not in the R‐L case. Existence of the principal eigenvalue is then studied. This is proved to exist for R‐L FDEs with Dirichlet and right focal boundary conditions, and for the Caputo case with left focal boundary conditions. A new method is used, simpler than previous papers, proving some inequalities which show that the corresponding integral operators are ‐positive in the cone of non‐negative functions in the space of continuous functions.