Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line

Abstract

Abstract Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem: { D α u ( t ) = u ( t ) φ ( t , u ( t ) ) , t ∈ ( 0 , ∞ ) , 1 < α ≤ 2 , lim t → 0 t 2 − α u ( t ) = a , lim t → ∞ t 1 − α u ( t ) = b , $$\begin{equation*} \left\{ \begin{array}{l} D^{\alpha} u(t)=u(t)\varphi (t, u(t)),\ \ t\in (0,\infty),\ 1\lt \alpha \leq 2, \\ \underset{t\rightarrow 0}{\lim} t^{2-\alpha} u(t)=a,\ \underset{ t\rightarrow \infty} {\lim} t^{1-\alpha} u(t)=b, \end{array} \right. \end{equation*}$$ where D α is the standard Riemann-Liouville fractional derivative, a, b are nonnegative constants such that a + b > 0 and φ(t, s) is a nonnegative continuous function that is required to satisfy an appropriate condition related to a class 𝜅 α satisfying suitable integrability condition. We also give a global behavior of such solution.