Solvability of boundary-value problems for a linear partial difference equation

Abstract

In this article we consider the two-dimensional boundary-value problem dm,n = dm−1,n + fndm−1,n−1,1 ≤n < m, dm,0 = am,dm,m = bm, m ∈ ℕ, where am, bm, m ∈ ℕ and fn,n ∈ ℕ,are complex sequences. Employing recently introduced method of half-lines, it is shown that the boundary-value problem is solvable, by finding an explicit formula for its solution on the domain, the, so called, combinatorial domain. The problem is solved for each complex sequence fn, n ∈ N, that is, even if some of its members are equal to zero. The main result here extends a recent result in the topic.