(2005-2006) Implicit Difference Equations

In my final year project, I studied the stability of singular quasi-linear difference equations (SDE). These equations are of the following form

A_n x_n+1 + B_n x_n = f_n (x_n) (n ≥ 0),

where A_n, B_n (real square matrices of order m) and the mapping f_n: R^m → R^m are given. The matrix A_n is always singular (n ≥ 0). We first defined various notions of stability of the trivial solution of such an SDE. Then, we applied the Lyapunov function to establish several necessary and sufficient conditions for the trivial solution of SDE. Our results were presented in the following paper:

K. A. Pham and S. H. Dau, "Stability of a Class of Singular Difference Equations," International Journal of Difference Equation, 1 (2), pp. 181 – 193, 2006.