Set-contractive mappings and difference equations in Banach spaces

In this paper, we consider the difference equation on an arbitrary Banach space (X, • x), Δ(qnΔxn + fn(xn) = 0, where {qn} is a positive sequence and fn is X-valued. We shall give conditions so that for a given x ε X, there exists a solution of this equation asymptotically equal to x.