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

Abstract

Abstract In the first part, we investigate the singular BVP $$\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u$$, u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $$\mathcal{H}$$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D 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 $$\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)$$, u(0) = A, u(1) = B, $$\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0$$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ 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.