Existence result for nonlinear fractional differential equation with \(p\)-Laplacian operator at resonance
In this paper, we consider the existence result for the nonlinear fractional differential equations with \(p\)-Laplacian operator $$\begin{aligned} {\left\{ \begin{array}{ll} \big [D_{0^+}^{\alpha } \phi _p\big ( [D_{0^+}^{\beta }u](\cdot )\big )\big ](t)=f\big (t,u(t),[D_{0^+}^{\beta }u](t)\big ), \quad t \in (0,1),\\ u(0)=0,~[D_{0^+}^{\beta }u](0)=[D_{0^+}^{\beta }u](1), \end{array}\right. } \end{aligned}$$where the \(p\)-Laplacian operator is defined as \(\phi _p(s)=|s|^{p-2}s\), \(p>1\), \(0<\alpha , \beta <1\), \(1<\alpha +\beta <2\), \(D_{0^+}^{\alpha }\) and \(D_{0^+}^{\beta }\) denote the Caputo fractional derivatives. Though Tang et al. have studied the same equations in the article as reported by Tang (JAMC 41:119131, 2013), the proof process is wrong. We point out their mistakes and give the correct proof of the existence result. The innovation of this article is that we establish a new definition and overcome the difficulties of the proof of compactness of the projector \(K_P(I-Q)N\). As applications, two examples are presented to illustrate the main results.
fractional differential equations \(p\)-laplacian pointwise equicontinuity coincidence degree theory 34a08 34b15