Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues
Author
Eloe, Paul W.
Neugebauer, Jeffrey T.
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.