A Multiplicity Results for a Singular Problem Involving a Riemann-Liouville Fractional Derivative

Abstract

In this work, we investigate the following nonlinear singular problem with Riemann-Liouville Fractional Derivative (Pλ)    −tDα 1 |0Dα t (u(t))| p−2 0Dα t u(t) = g(t) uγ (t) + λf(t, u(t)) t ∈ (0, T); u(0) = u(T) = 0, where λ is a positive parameter, 2 < q < p < r, 1 2 < α ≤ 1, 0 < γ < 1, g ∈ C([0, 1]) and f ∈ C([0, T] × R, R). Under appropriate assumptions on the function f, we employ the method of the Nehari manifold combined with the fibering maps in order to show the existence of λ0 such that for all λ ∈ (0, λ0) the problem (Pλ) has at least two positive solutions. Keywords: Nonlinear singular fractional differential equation, Riemann-Liouville Fractional Derivative, Fibreging maps, Nehari Manifold, Existence of solutions.