A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

In this paper, we study the system of third-order difference equations {equation*} xₙ₊₁=a+{a₁}{yₙ}+{a₂}{yₙ₋₁}+{a₃}{yₙ₋₂}% , yₙ₊₁=b+{b₁}{xₙ}+{b₂}{xₙ₋₁}+{b₃}{% xₙ₋₂}, n∈ N₀, {equation*}% where the parameters a, aᵢ, b, bᵢ, $i=1,2,3$, and the initial values x₋ⱼ, y₋ⱼ, $j=0,1,2$, are positive real numbers. We first prove a general convergence theorem. By applying this convergence theorem to the system, we show that positive equilibrium is a global attractor. We also study the local asymptotic stability of the equilibrium and show that it is globally asymptotically stable. Finally, we study the invariant set of solutions.