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

For each $\lambda \in \mathbb N^*$, we consider the integral equation: \[ \int_{\lambda y} ^{\lambda x} f(t)\, d t = f(x) - f(y) \mbox{ for every $(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two continuous functions $f_a,f_b:[0,\lambda] \rightarrow {\mathbb R}$ along a word $u= u_0u_1\cdots\in\{a,b\}^{\mathbb N}$ 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.