Green's functions for discrete \(m\)th-order problems (Q1943765)

This paper concerns the \(m\)th-order linear ordinary difference equation \(a^m_iu_{i+m} +\dots + a^2_iu_{i+2} + a^1_iu_{i+1} + a^0_iu_i = f_i,\) \(i = \overline{0, n-m}\), where \(0 \leq n\in \mathbb{N}\), \(m = \overline{0, n}\), and \(a^m_i\), \(a^0_i\not= 0\). The expression of the solution to the equation is given under additional conditions: \(\langle L_1, u\rangle= g_1\in \mathbb{K},\dots, \langle L_m, u\rangle= g_m\in \mathbb{K}\), where \(L_1,\dots,L_m\) are linearly independent functionals. Moreover, a discrete Green function is defined, and the Green function is constructed for the equation. Applications to two problems with nonlocal boundary conditions are also given.