On a system of m difference equations having exponential terms

In this paper we study the asymptotic behavior of the positive solutions of a cyclic system of the following m difference equations: {align*} x⁽ⁱ⁾ₙ₊₁&=a_ix_n⁽ⁱ⁺¹⁾+b_i x⁽ⁱ⁾ₙ₋₁e^{{-x⁽ⁱ⁺¹⁾_n}}, i=1,2,…, m-1,\\ x⁽ᵐ⁾ₙ₊₁&=a_mx⁽¹⁾_n+b_m x⁽ᵐ⁾ₙ₋₁e^{{-x⁽¹⁾_n}}, {align*} where n=0,1,…, and aᵢ,\ bᵢ, i=1,2,…,m are positive constants and the initial values x⁽ⁱ⁾₋₁, x⁽ⁱ⁾₀, i=1,2,…,m are positive numbers.