Perturbations for linear difference equations
The authors study the following boundary value problem \[ \varepsilon y_{k+2} + a_k y_{k+1} + b_k y_k = f_k \quad (0\leq k\leq N-2),\quad y_0=\alpha,\quad y_N=\beta, \] where \(a_k,b_k\), and \(f_k\) are given sequences of real numbers, \(\varepsilon\) is a small parameter, and \(\alpha,\beta\) are real constants. They prove that a unique solution of the above problem exists which is given by \(y_k(\varepsilon)=\sum\limits_{j=0}^\infty \varepsilon^j y_k^{(j)}\) for \(0\leq k\leq N\), and satisfies \[ \left| y_k(\varepsilon)-\sum_{j=0}^n \varepsilon^j y_k^{(j)}\right| \leq C{(| \varepsilon| /\varepsilon_0)^{n+1}\over 1-| \varepsilon| /\varepsilon_0}, \] where the constants \(C\) and \(\varepsilon_0\) depend only on the sequences \(a_k\) and \(b_k\), and \(y_k^{(0)}\) and \(y_k^{(j)}\) are the solutions of \[ a_ky_{k+1}^{(0)}+b_ky_k^{(0)}=f_k\quad (0\leq k\leq N-2),\quad y_0^{(0)}=\alpha,\quad y_N^{(0)}=\beta \] and \[ a_ky_{k+1}^{(j)}+b_ky_k^{(j)}=-y_{k+2}^{(j-1)}\quad (0\leq k\leq N-2),\quad y_0^{(j)}=0,\quad y_N^{(j)}=0, \] respectively.
scientific article; zbMATH DE number 2159771