Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation

Abstract

For each λ∈N⁎, we consider the integral equation:∫λyλxf(t)dt=f(x)−f(y) for every (x,y)∈R+2, where f is the concatenation of two continuous functions fa,fb:[0,λ]→R along a word u=u0u1⋯∈{a,b}N such that u=σ(u), where σ is a λ-uniform substitution satisfying some combinatorial conditions.There exists some non-trivial solutions ([1]). We show in this work that the dimension of the set of solutions is at most two.