A basis theorem for a class of max-plus eigenproblems

We study the max-plus equation where H:[0, M]→(-∞, ∞) and γ:[0, M]→[0, M] are given functions. The function ψ:[0, M]→[-∞, ∞) and the quantity P are unknown, and are, respectively, an eigenfunction and additive eigenvalue. Eigensolutions ψ are known to describe the asymptotics of certain solutions of singularly perturbed differential equations with state dependent time lags. Under general conditions we prove the existence of a finite set (a basis) of eigensolutions, for 1≤i≤q, with the same eigenvalue P, such that the general solution ψ to (*) is given by Here cⁱ ∈[-∞, ∞) are arbitrary quantities and v denotes the maximum operator. In many cases q = 1 so the solution ψ is unique up to an additive constant.