Global bounded solution of the chemotaxis attraction repulsion Cauchy problem with the nonlinear signal production in $\mathbb{R}^{N}$

In this paper, we consider the following attraction repulsion chemotaxis model with nonlinear signal term: \begin{align*} &u_{t}=\nabla \cdot(\nabla u-\xi_{1} u \nabla v +\xi_{2} u \nabla w), \quad &0=\Delta v -\lambda_{1}v +f_{1}(u), \quad &0=\Delta w -\lambda_{2}w +f_{2}(u), \quad x \in \mathbb{R}^{N}, t>0, \end{align*} where $\xi_{1},\xi_{2},\lambda_{1},\lambda_{2}$ are for some positive constants, and \begin{equation*} f_{1} \in C^{1}([0,\infty)) \; \text{satisfying} \; 0 \leqslant f_{1}(s) \leqslant c_{1}s^{l}, \; \forall s \geqslant 0 \ \text{and} \ l> 0, \end{equation*} \begin{equation*} f_{2} \in C^{1}([0,\infty))\; \text{satisfying} \; 0 \leqslant f_{2}(s) \leqslant c_{2}s^{m}, \; \forall s \geqslant 0 \ \text{and} \ m> 0. \end{equation*} We will show that this problem has a unique global bounded solution when $ l>\frac{2}{N}, l