Global asymptotic behavior of the difference equation

In this work, we study the boundedness and the global asymptotic behavior of the solutions of the difference equation y n + 1 = α ⋅ e − ( n y n + ( n − k ) y n − k ) β + n y n + ( n − k ) y n − k , n = 0 , 1 , … , where α and β are positive real numbers, k ∈ { 1 , 2 , … } and the initial conditions y − k , … , y − 1 , y 0 are arbitrary numbers.