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

Abstract

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