On Existence Results for Nonlinear Fractional Differential Equations Involving the p-Laplacian at Resonance

Abstract

In this paper, we study the existence result for the nonlinear fractional differential equations with p-Laplacian operator $$\left\{\begin{array}{ll}D_{0^+}^{\beta} \phi_p( D_{0^+}^{\alpha} u(t))=f(t,u(t),D_{0^+}^{\alpha}u(t)), \quad t\in(0,1),\\ D_{0^+}^{\alpha}u(0)=D_{0^+}^{\alpha}u(1)=0,\end{array}\right.$$ where the p-Laplacian operator is defined as $${\phi_p(s) = |s|^{p-2}s,p > 1, \,\,{\rm and}\,\, \phi_q(s) = \phi_p^{-1}(s), \frac{1}{p}+\frac{1}{q} = 1;\, 0 < \alpha, \beta < 1, 1 < \alpha + \beta < 2 \,\,{\rm and}\,\, D_{0^+}^{\alpha}, D_{0^+}^{\beta}}$$ denote the Caputo fractional derivatives, and $${f : [0,1] \times \mathbb{R}^2\rightarrow \mathbb{R}}$$ is continuous. Though Chen et al. have studied the same equations in their article, the proof process is not rigorous. We point out the mistakes and give a correct proof of the existence result. The innovation of this article is that we introduce a new definition to weaken the conditions of Arzela–Ascoli theorem and overcome the difficulties of the proof of compactness of the projector K P (I − Q)N. As applications, an example is presented to illustrate the main results.