Stability and absolute stability of a three-point difference scheme

Consider a three-point difference scheme −h−2Δ(2)yn + qn(h)yn = fn(h), n ε Z = {0, ±1, ±2, …}, where h ε (0, h0], h0 is a given positive number, Δ(2)yn = yn+1 + yn−1, f(h) = {fn(h)}n ε Z ε L∞(h), L∞(h) = {f(h) : f(h) L∞(h) < ∞}, f(h) L∞(h) = supnεZ fn(h) . We assume a unique a priori requirement 0 <- qn(h) < ∞ for any n ε Z and h ε (0, h0]. The main results are a criterion of stability and absolute stability of the difference scheme (1) in the space L∞(h).