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

Abstract

<p>In this article we consider the two-dimensional boundary-value problem</p> <pre> d_m,n = d_m-1,n + ƒ_nd_m-1,n-1, 1 ≤ n < m,<br> d_m,0 = α_m, d_m,m = b_m, m ∈ ℕ,</pre> <p>where α_m, b_m, m ∈ ℕ and ƒ_n, 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 ƒ_n, 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.</p>