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01-06-2015 | Original Research | Issue 1-2/2015

Journal of Applied Mathematics and Computing 1-2/2015

Existence result for nonlinear fractional differential equation with \(p\)-Laplacian operator at resonance

Journal:
Journal of Applied Mathematics and Computing > Issue 1-2/2015
Authors:
Lei Hu, Shuqin Zhang, Ailing Shi
Important notes
This research is supported by the Natural Science Foundation of China (11371364) and 2013 Science and Technology Research Project of Beijing Municipal Education Commission (KM201310016001).

Abstract

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 41:119–131, 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.

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