Extremal solutions for p-Laplacian fractional differential systems involving the Riemann-Liouville integral boundary conditions

where D , D , and D are the standard Riemann-Liouville fractional derivatives, I and I are the Riemann-Liouville fractional integrals, and 0 < γ < 1 < β < 2 < α < 3, ν,ω > 0, 0 < η, ξ < 1, k ∈R, f ∈ C([0, 1]×R×R,R), g ∈ C([0, 1]×R,R). The p-Laplacian operator is defined as φp(t) = |t|p–2t, p > 1, and (φp) = φq, 1 p + 1 q = 1. The study of boundary value problems in the setting of fractional calculus has received a great attention in the last decade, and a variety of results concerning the of existence of solutions, based on various analytic techniques, can be found in the literature [1–10]. In particular, much effort has been made toward the study of the existence of solutions for fractional differential equations involving p-Laplacian operators; see [11–14]. Using the