Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line

Abstract

We are interested in the following fractional boundary value problem:Dαu(t)+atuσ=0,t∈(0,∞),limt→0⁡t2-αu(t)=0,limt→∞⁡t1-αu(t)=0, where1<α<2,σ∈(-1,1),Dαis the standard Riemann-Liouville fractional derivative, andais a nonnegative continuous function on(0,∞)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.