Sharp estimates on the solutions to combined fractional boundary value problems on the half-line

We prove the existence and the uniqueness of a positive solution to the following combined fractional boundary value problem on the half-line {Dαu(t)+a1(t)uσ1+a2(t)uσ2=0,t∈(0,∞),1 α 2limt→0t2−αu(t)=0,limt→∞t1−αu(t)=0,where D α is the standard Riemann{Liouville fractional derivative, σ 1 ; σ 2 ∈ ( − 1 ; 1 ) , and a 1 ; a 2 are non-negative continuous functions on ( 0 , ∞ ), which may be singular at t = 0 and satisfying some convenient assumptions related to the Karamata regular variation theory. We also give sharp estimates on such solution