In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation invariant curves from the perspective of normal form theory. A computer simulation is used to illustrate the analytical results.

The boundedness nature and persistence, global and local behavior, and rate of convergence of positive solutions of a second-order system of exponential difference equations, is investigated in this work. Where the parameters $A,B,C,\alpha ,\beta ,\gamma ,\delta ,\eta$, and $\xi$are constants that are positive, and the initials $U_{-1},U_0,V_{-1},V_0,W_{-1}$, and $W_0$ are non-negative real numbers. Some examples are provided to support our theoretical results.

We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: ${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}$, ${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations.

In the setting of $ 2 $-uniformly convex Banach spaces, we prove the existence of solutions for a variant of the implicit state-dependent convex sweeping processes. Our approach is based on a differential equation associated with the generalized projection operator.