Regular and irregular solutions of a system of functional equations (Q2505169)

The authors consider the system of functional equations \[ \begin{cases} f \left(\frac x2\right)=a_0f(x)+g_0(x),\;x\in[0,1)\\ f\left(\frac{x+1}{2} \right)=a_1f(x)+g_1 (x),\;x\in[0,1]\end{cases}\tag{F} \] with \(a_0,a_1\in \mathbb{R}\), \(|a_0|<1\), \(|a_1|<1\), \(g_0:[0,2]\to\mathbb{R}\), \(g_1:[0,1]\to \mathbb{R}\) for an unknown function \(f:[0,1]\to \mathbb{R}\). Systems of this type have been used by the first author to characterize nowhere differentiable functions. [cf. \textit{R. Girgensohn}, ibid. 47, No. 1, 89--99 (1994; Zbl 0791.26005)]. \textit{K. Kawamura} [J. Math. Kyoto Univ. 42, No. 2, 255--286 (2002; Zbl 1048.28004)] has considered the system \[ f(x)=\frac {a_0}{2}f(2x)+bu,\;0\leq x\leq\frac 12\quad f(x)=\frac c2 f(2x-1)+d(1-x),\;\frac 12\leq x\leq 1 \] with \(a,b,c,d\in\{-1,1\}\). This system were further investigated by \textit{H.-H. Kairies} [On a problem of K. Kawamura, Mathematik-Bericht 2003/07, TU Clausthal (2003)]. The authors continue the investigation of Kawamura's equations by setting these equations into a much more general framework. In this manner they obtain the properties of Kawamura's equations from general theorems on equations of type (F). They give the general solutions of (F). They also prove the uniqueness of bounded solutions and the uniqueness of Lebesgue integrable solutions.