Consider the homogeneous functional equations
\begin{equation*} au(x+s)+bu(x+r)=0,\text{ }x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion . \end{equation*}
It is known that for $a=1,b=-1$ and $u$ is periodic that $u$ is constant if and only if $\frac{s}{r}\in %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion .$ For example, The sum of two continuous periodic functions is periodic if and only if the ratio of their periods is rational? or A real continuous periodic function with two incommensurate periods is constant.
My question: What can happens if $a,b$ are arbitrary coefficients with $\frac{s}{r}\notin %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion $.? Can we prove the non-existence of such a function $u$ under a suitable conditions on $a$ and $b$?.
If we look for a solution of the exponential form $u(x)=Ce^{\lambda x},$ then we find that under a suitable assumption on $a$ and $b$ the solution exists if and only if $\frac{s}{r}\in %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion .$ However, we don't know if this form of solution is the unique one.
Is there any reference that treats this case or any ideas?.
Thank you.
If $a=b=0$ then $u$ can be anything. Otherwise, without loss of generality let $b \ne 0$ and define
$$v(x) \equiv u(x+s),$$ $$c \equiv -a/b,$$ $$ q \equiv r - s.$$
Then, the original equation becomes
$$v(x+q) = c v(x).$$
Then, we could specify the value of $v(x)$ for $0 \le x < q$ and otherwise obtain the solution
$$v(x) = v(x- nq)c^n,$$
where $n$ is an integer chosen so that $0 \le x - n q < q$. Then, we just undo my redefinitions to obtain $u(x)$.