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

For each $\lambda \in {\mathbb{N}}^{\ast }$, we consider the integral equation: \int_{\lambda y} ^{\lambda x} f(t)\, d t = f(x) - f(y) for every $(x,y)\in {\mathbb R_+^2$,} where $f$ is the concatenation of two continuous functions $f_a,f_b:[0,\lambda ]\to \mathbb{R}$ along a word $u=u_0{u}_1\cdots \in \{a,b{\}}^{\mathbb{N}}$ such that $u=\sigma (u)$, where $\sigma$ is a $\lambda$-uniform substitution satisfying some combinatorial conditions. There exists some non-trivial solutions. We show in this work that the dimension of the set of solutions is at most two.