Nan Yan, Chengbo Zhai, The existence and uniqueness of solutions to a new fractional differential system with p-Laplacian operators, Vol. 2021 (2021), Article ID 7, pp. 1-12.

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DOI: 10.23952/cot.2021.7

Received March 2, 2021; Accepted May 21, 2021; Published June 25, 2021

Abstract. In this paper, we focus on the following fractional differential system with p-Laplacian operators
$D_{0^{+}}^{\beta_{1}}(\varphi_{p}(D_{0^{+}}^{\alpha_{1}}u(t)))+f_{1}(t,u(t),v(t))=0, 0\leq t\leq1,$
$D_{0^{+}}^{\beta_{2}}(\varphi_{p}(D_{0^{+}}^{\alpha_{2}}v(t)))+f_{2}(t,u(t),v(t))=0, 0\leq t\leq1,$
$u(0)=u(1)=u'(0)=D_{0^{+}}^{\alpha_{1}-2}u(0)=D_{0^{+}}^{\alpha_{1}}u(0)=0,$
$v(0)=v(1)=v'(0)=D_{0^{+}}^{\alpha_{2}-2}v(0)=D_{0^{+}}^{\alpha_{2}}v(0)=0,$
where $3\textless\alpha_i\leq 4,$ $0\textless\beta_i\leq 1,$ $f_i\in C([0,1]\times [0,\infty)\times [0,\infty), [0,\infty)),$ $D_{0^+}^{\beta_i}$ and $D_{0^+}^{\alpha_i}$ are the standard Riemann-Liouville fractional derivatives, $i=1,2$ , and $\varphi_p(s)$ is the p-Laplacian operator defined by $\varphi_p(s)=|s|^{p-2}s,$ and $\varphi_p^{-1}(s)=\varphi_q(s)$ with $\frac{1}{p}+\frac{1}{q}=1,$ $p>1.$ The existence and uniqueness solutions are obtained via the Leray-Schauder nonlinear alternative and Banach’s fixed point theorem. Finally, an example is given to verify the effectiveness and applicability of our main results.

How to Cite this Article:
Nan Yan, Chengbo Zhai, The existence and uniqueness of solutions to a new fractional differential system with p-Laplacian operators, Communications in Optimizaton Theory, Vol. 2021 (2021), Article ID 7, pp. 1-12.