Existence and global asymptotic behavior of positive solutions for nonlinear fractional Dirichlet problems on the half-line
Summary: We are interested in the following fractional boundary value problem: \(D^\alpha u(t) + a \left(t\right) u^\sigma = 0\), \(t \in(0, \infty)\), \(\lim_{t \rightarrow 0} t^{2 - \alpha} u(t) = 0\), \(\lim_{t \rightarrow \infty} t^{1 - \alpha} u(t) = 0\), where \(1 < \alpha < 2\), \(\sigma \in(- 1,1)\), \(D^\alpha\) is the standard Riemann-Liouville fractional derivative, and \(a\) is a nonnegative continuous function on \((0, \infty)\) satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.
scientific article; zbMATH DE number 7022577