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 i, for 1 i q, with the same eigenvalue P, such that the general solution ψ to (*) is given byψ(ξ)=(c1+ 1(ξ)) (c2+ 2(ξ)) (cq+ q(ξ)).Here ci [−∞,∞) are arbitrary quantities and denotes the maximum operator. In many cases q=1 so the solution ψ is unique up to an additive constant.