General solution to a weakly delayed planar linear discrete system, the case of real different eigenvalues of the matrix of nondelayed terms

The paper considers a linear discrete system with a single delay where $k∈ℤ0∞:={ 0,1,…,∞ },x:ℤ0∞→ℝ2$⁠, m is a positive fixed integer, $A={ aij }i,j=12$ and the entries of matrix $B={ bij(k) }i,j=12$ are defined for every $k∈ℤ0∞$⁠. It is assumed that the system is weakly delayed and the eigenvalues of the matrix A are real and different. Analyzing and simplifying a formula for the general solution derived recently, it is shown that, for k≥m, the number of arbitrary constants in this solution can be reduced to two. Conditional stability of a given system is considered. In addition, a non-delayed planar linear discrete system is constructed such that, for k≥m and after a transformation, we get the same solutions as those of the delayed system.