Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues

It has been shown that, under suitable hypotheses, boundary value problems of the form, Ly +lambda y = f, BCy = 0 where L is a linear ordinary or partial differential operator and BC denotes a linear boundary operator, then there exists. > 0 such that integral >= 0 implies lambda y >= 0 for lambda(Upsilon) [-Alpha,Alpha] \ {0}, where y is the unique solution of Ly +.y = f, BCy = 0. So, the boundary value problem satisfies a maximum principle for lambda is an element of[-Alpha., 0) and the boundary value problem satisfies an antimaximum principle for lambda.is an element of (0,A]. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, D-alpha (0) y + ss D alpha-1 (0) y = f, BCy = 0 where D-alpha (0) is a Riemann-Liouville fractional differentiable operator of order a, 1 < a = 2, and BC denotes a linear boundary operator, then there exists B > 0 such that f = 0 implies ss D (alpha-1) (0) y >= 0 for ss.is an element of [-B, B] \ {0}, where y is the unique solution of D-alpha (0) y+ ss D alpha-1 (0) y = f, BCy = 0. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of ss D (alpha-1) (0) y. The boundary conditions are chosen so that with further analysis a sign property of ss y is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.