Solutions of a class of multiplicatively advanced differential equations

Abstract The multiplicatively advanced differential equations (MADEs) of form f ( n ) ( t ) = α f ( β t ) with α ≠ 0 , β > 1 are studied along with a class of their solutions of type f μ , λ ( t ) defined on [ 0 , ∞ ) . For λ ∈ Q + , μ ∈ R , the solutions f μ , λ ( t ) are extended to ( − ∞ , ∞ ) in a non-unique manner to obtain Schwartz wavelet solutions F μ , λ ( t ) of the original MADE, with all moments of F μ , λ ( t ) vanishing. Examples are studied in detail. The Fourier transform of each F μ , λ ( t ) is computed and, in a number of examples, is related to the Jacobi theta function. Additional conditions sufficient for the uniqueness of certain MADE initial value problems are given. Conditions for decay and non-decay at −∞ are obtained. Decay rates at ±∞ in terms of familiar functions are established.