Fractional BVPs with strong time singularities and the limit properties of their solutions

摘要

In the first part, we investigate the singular BVP d/(dt)c~D~αu + (a/t)~cD~αu = Hu, u(0) = A, u(1) = B, ~cD~αu(t)|_(t=0) = 0, where H is a continuous operator, α ∈ (0, 1) and a < 0. Here, ~cD denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems d/(dt)~cD~(αn)u+(a/t)cD~(αn)u = f(t, u, ~cD~(βn)u), u(0) = A, u(1) = B, ~cD~(αn)u(t)|_(t=0) = 0, where a < 0, 0 < β_n ≤ α_n < 1, lim_(n→∞) β_n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.