The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

• DOI: 10.1007/s12190-019-01264-z
• Publication Date: 2019-04-25T06:05:32Z
• Authors: Xiaosong Tang; Xinchang Wang; Zhiwei Wang; Peichang Ouyang
• Keywords: Operator (biology); Fractional Laplacian; Degree (music)

In this paper, we consider the following mixed fractional resonant boundary value problem with p(t)-Laplacian operator $$\begin{aligned} \left\{ \begin{array}{lll} ^{C} D^{\beta }_{0^{+}}\varphi _{p(t)}(D^{\alpha }_{0^{+}}u(t))=f(t, u(t), D^{\alpha }_{0^{+}}u(t)),~~t\in [0, T],\\ t^{1-\alpha }u(t)\mid _{t=0}=0,~~D^{\alpha }_{0^{+}}u(0)=D^{\alpha }_{0^{+}}u(T), \end{array}\right. \end{aligned}$$ where $$^{C} D^{\beta }_{0^{+}}$$ is Caputo fractional derivative, $$D^{\alpha }_{0^{+}}$$ is Riemann–Liouville fractional derivative, $$\varphi _{p(t)}$$ is p(t)-Laplacian operator, $$p(t)>1$$ , $$p(t)\in C^{1}[0, T]$$ with $$p(0)=p(T)$$ . Under the appropriate conditions of the nonlinear term, the existence of solutions for the above mixed fractional resonant boundary value problem is obtained by using the continuation theorem of coincidence degree theory, which enrich the existing literatures. In addition, an example is included to demonstrate the main result.