Continuous Transformations of Differential Equations with Delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer’s transformation z(t) = g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of the nth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions aer cobstructed. Let I = [a, b), J = [c, d) be intervals, where b, d may be infinite. Further, let t = f1(x, y), z = f2(x, y) where (x, y) ∈ I ×R and denote by F = (f1, f2) a pointwise transformation of I × R into U ⊂ R2. In this article we shall study transformations F of a linear homogeneous differential equation of the nth order with m delays