Abstract
Hahn introduced the difference operator D-q,D-omega f(t) = (f(qt + omega) - f(t))/(t(q - 1) + omega) in 1949, where 0 < q < 1 and omega > 0 are fixed real numbers. This operator extends the classical difference operator Delta(omega) f(t) = (f(t + omega) - f (t))/omega as well as the Jackson q-difference operator D(q)f(t) = (f(qt) -f(t))/(t(q -1)).
In this paper, we study the theory of abstract linear Hahn difference equations of the form
A(0)(t)D(q,omega)(n)x(t) + A(1)(t)D(q,omega)(n-1)x(t) + ... + A(n)(t)x(t) = B(t),
where B and A(i) are mappings from an interval I into a Banach algebra X, i = 1, ..., n. We define the abstract exponential functions and the abstract trigonometric (hyperbolic) functions. We prove they are solutions of first and second order Hahn difference equations, respectively. Also, we obtain an integral equation corresponding to the second order linear Hahn difference equations which is known as the Volterra integral equation. Finally, we present the analogs of the variation of parameter technique and the annihilator method for the non-homogeneous case.