Existence theory of fractional order three-dimensional differential system at resonance

This paper deals with three-dimensional differential system of nonlinear fractional order problem

$$\begin{array}{r}{D}_{0^{+}}^{\alpha }\upsilon (\varrho )=f(\varrho ,\omega (\varrho ),{\omega }^{\mathrm{\prime }}(\varrho ),{\omega }^{\mathrm{\prime }\mathrm{\prime }}(\varrho ),...,{\omega }^{(n-1)}(\varrho )),\varrho \in (0,1),\\ D_{0^{+}}^{\beta }\nu (\varrho )=g(\varrho ,\upsilon (\varrho ),{\upsilon }^{\mathrm{\prime }}(\varrho ),{\upsilon }^{\mathrm{\prime }\mathrm{\prime }}(\varrho ),...,{\upsilon }^{(n-1)}(\varrho )),\varrho \in (0,1),\\ D_{0^{+}}^{\gamma }\omega (\varrho )=h(\varrho ,\nu (\varrho ),{\nu }^{\mathrm{\prime }}(\varrho ),{\nu }^{\mathrm{\prime }\mathrm{\prime }}(\varrho ),...,{\nu }^{(n-1)}(\varrho )),\varrho \in (0,1),\end{array}$$

with the boundary conditions,

$$\begin{array}{r}\upsilon (0)={\upsilon }^{\mathrm{\prime }}(0)=...={\upsilon }^{(n-2)}(0)=0,{\upsilon }^{(n-1)}(0)={\upsilon }^{(n-1)}(1),\\ \nu (0)={\nu }^{\mathrm{\prime }}(0)=...={\nu }^{(n-2)}(0)=0,{\nu }^{(n-1)}(0)={\nu }^{(n-1)}(1),\\ \omega (0)={\omega }^{\mathrm{\prime }}(0)=...={\omega }^{(n-2)}(0)=0,{\omega }^{(n-1)}(0)={\omega }^{(n-1)}(1),\end{array}$$

where $D_{0^{+}}^{\alpha },D_{0^{+}}^{\beta },D_{0^{+}}^{\gamma }$ are the standard Caputo fractional derivative, $n-1<\alpha ,\beta ,\gamma \le n,n\ge 2$ and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.