# General solution of a linear functional equation

As we know, general solution of the linear functional equation $$f(x+1)-f(x)=g(x)$$ ($$g$$ is a known function) is $$f=f_0+\phi$$, where $$f_0$$ is an its special solution and $$\phi$$ any 1-periodic function.

Now, does any one know general solution of the linear functional equation: $$rf(ax+b)+sf(cx+d)=g(x),$$ where $$a,b,c,d, r$$ and $$s$$ are real constants with $$rasc\neq0$$?

($$g$$ is a given real function, the case $$g=0$$ and the equation $$f(ax+b)=cf(x)$$ is of special interest).

Are there any papers or books regarding the problem?

• It is the same as you wrote in the first paragraph: the general solution is the sum of a single solution plus the general solution of homogeneous equation (with g=0). Sep 17, 2020 at 13:41

To describe he general solution of the homogeneous equation $$rf\phi_1+sf\circ\phi_2=0,$$ where $$\phi_1(x)=ax+b$$ and $$\phi_2(x)=cx+d$$ are affine functions, you use the fact that affine functions form a group. So there is an affine function $$\phi_3$$ such that $$\phi_2=\phi_3\circ\phi_1$$, and making the change of the variable $$t=\phi_1(x)$$ we obtain the equation $$rf(t)+sf\circ\phi_3(t)=0.$$ To further simplify this, consider two cases: a) $$\phi_3(t)=at+b,\; a\neq 1$$, and b) $$\phi_3(t)=t+b$$. In the first case, by a conjugation in the affine group, the equation is reduced to $$rg(t)+sg(at)=0.$$ In the second case, it is reduced to $$rg(t)+sg(t+1)=0.$$ The general solutions of these two standard equations must be clear.