On the difference equation xn+1=axn−l+bxn−k+f(xn−l,xn−k) $x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} )$ - Mahmoud A. E. Abdelrahman (Advances in Difference Equations, 2018)
Abstract In this paper, we study the asymptotic behavior of the solutions of a new class of difference equations xn+1=axn−l+bxn−k+f(xn−l,xn−k), $$x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} ), $$ where l and k are nonnegative integers, a and b are nonnegative real numbers, the initial values x−s,x−s+1,…,x0 $x_{-s}, x_{-s+1},…, x_{0}$ are positive real numbers, s=max{l,k} $s=\max\{l,k\}$, and f(u,v):(0,∞)2→(0,∞) $f (u,v ): ( 0,∞ ) ^{2}→ ( 0,∞ ) $ is a continuous and homogeneous real function of degree zero. We consider the stability, boundedness, and periodicity of the solutions of this equation which is the most general form of linear difference equations. Thus, the results in this paper apply to several other equations that are special cases of the studied equation. Moreover, we present a new method to study periodic solutions of period two.