The results for a new modeling integral boundary value problem using Caputo-Hadamard impulsive implicit fractional differential equations with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem, Schaefer's fixed point theorem and the Banach contraction principle serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements.

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.