LAUSR

Given a set $$T\subset (0, +\infty )$$T⊂(0,+∞), a function $$c:T\rightarrow \mathbb R$$c:T→R and a real number p we study continuous solutions $$\varphi $$φ of the simultaneous equations $$\begin{aligned} \varphi (tx)=\varphi (x)+c(t)x^p, \qquad t \in T. \end{aligned}$$φ(tx)=φ(x)+c(t)xp,t∈T.Here $$\varphi $$φ is defined on an interval $$I\subset (0, +\infty )$$I⊂(0,+∞), so the equations are postulated on a restricted domain: for any fixed $$t \in T$$t∈T we assume that $$x \in I$$x∈I is such that $$tx \in I$$tx∈I. In the case when T is large in a sense, we determine the form of $$\varphi $$φ on a non-trivial subinterval of I. The research is a continuation of that of “non-restricted”, where $$I=(0,+\infty )$$I=(0,+∞), made in Jarczyk (Ann Univ Sci Budapest Sect Comp 40:353–362, 2013).