How to solve the delay algebraic equation $xf(x) + \alpha f(x - {x_0}) - \alpha f(x + {x_0}) = 0$?

Question

In the process of solving a problem, I am faced with the problem of finding a non-zero function $f:\mathbb{R} \to \mathbb{R}$ which satisfies the equation $$xf(x) + \alpha f(x - {x_0}) - \alpha f(x + {x_0}) = 0$$ for a known $x_0$ and $\alpha$. Unfortunately all I could find searching online is the topic of delay differential equation which seems to be more general than my question. Could anyone help me with some references, keywords or hints? Thanks in advance.

Answer 1

Let $f(x)=\int_a^be^{xs}K(s)~ds$ ,

Then $x\int_a^be^{xs}K(s)~ds+\alpha\int_a^be^{(x-x_0)s}K(s)~ds-\alpha\int_a^be^{(x+x_0)s}K(s)~ds=0$

$\int_a^bK(s)~d(e^{xs})+\alpha\int_a^be^{-x_0s}e^{xs}K(s)~ds-\alpha\int_a^be^{x_0s}e^{xs}K(s)~ds=0$

$[e^{xs}K(s)]_a^b-\int_a^be^{xs}~d(K(s))+\alpha\int_a^be^{-x_0s}e^{xs}K(s)~ds-\alpha\int_a^be^{x_0s}e^{xs}K(s)~ds=0$

$[e^{xs}K(s)]_a^b-\int_a^be^{xs}K'(s)~ds+\alpha\int_a^be^{-x_0s}e^{xs}K(s)~ds-\alpha\int_a^be^{x_0s}e^{xs}K(s)~ds=0$

$[e^{xs}K(s)]_a^b-\int_a^b(K'(s)+\alpha(e^{x_0s}-e^{-x_0s})K(s))e^{xs}~ds=0$

$\therefore K'(s)+\alpha(e^{x_0s}-e^{-x_0s})K(s)=0$

$K'(s)=\alpha(e^{-x_0s}-e^{x_0s})K(s)$

$\dfrac{K'(s)}{K(s)}=-2\alpha\sinh x_0s$

$\int\dfrac{K'(s)}{K(s)}~ds=-2\alpha\int\sinh x_0s~ds$

$\ln K(s)=-\dfrac{2\alpha\cosh x_0s}{x_0}+c_1$

$K(s)=ce^{-\frac{2\alpha\cosh x_0s}{x_0}}$

$\therefore f(x)=\int_a^bce^{xs-\frac{2\alpha\cosh x_0s}{x_0}}~ds$

But since the above procedure in fact suitable for any complex number $s$ ,

$\therefore f_n(x)=\int_{a_n}^{b_n}c_ne^{xk_nt-\frac{2\alpha\cosh x_0k_nt}{x_0}}~d(k_nt)=k_nc_n\int_{a_n}^{b_n}e^{k_nxt-\frac{2\alpha\cosh k_nx_0t}{x_0}}~dt$

For some $x$-independent real number choices of $a_n$ and $b_n$ and $x$-independent complex number choices of $k_n$ such that:

$\lim\limits_{t\to a_n}e^{k_nxt-\frac{2\alpha\cosh k_nx_0t}{x_0}}=\lim\limits_{t\to b_n}e^{k_nxt-\frac{2\alpha\cosh k_nx_0t}{x_0}}$

$\int_{a_n}^{b_n}e^{k_nxt-\frac{2\alpha\cosh k_nx_0t}{x_0}}~dt$ converges

One of the choice involving

$\int_{-\infty}^\infty e^{\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt$

$=\int_{-\infty}^0e^{\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt+\int_0^\infty e^{\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt$

$=\int_\infty^0e^{-\frac{xt}{x_0}-\frac{2\alpha\cosh(-t)}{x_0}}~d(-t)+\int_0^\infty e^{\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt$

$=\int_0^\infty e^{-\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt+\int_0^\infty e^{\frac{xt}{x_0}-\frac{2\alpha\cosh t}{x_0}}~dt$

$\propto\int_0^\infty e^{-\frac{2\alpha\cosh t}{x_0}}\cosh\dfrac{xt}{x_0}~dt$

$\propto K_\frac{x}{x_0}\left(\dfrac{2\alpha}{x_0}\right)$

You will find that this functional equation if fact is very similar to that of modified Bessel functions

In fact the general solution can consider as $f(x)=\Theta_1(x)I_\frac{x}{x_0}\left(\dfrac{2\alpha}{x_0}\right)+\Theta_2(x)K_\frac{x}{x_0}\left(\dfrac{2\alpha}{x_0}\right)$, where $\Theta_1(x)$ and $\Theta_2(x)$ are arbitrary periodic functions with period $|x_0|$ .

Answer 2

For fixed $x$, it is a difference equation with known solution methods. This defines your function (a two-dimensional set of possibilities) on one of the cosets of $x_0\mathbb Z$. It is defined independently on each coset. (Of course, such a definition is likely not continuous. But the problem does not require continuity.)

added

what could have been done if the delay was $x_0$ but the advance was $y_0$ (in the case that these didn't have a common factor, e.g. $\sqrt{2}$ and $\pi$)

Incommensurable intervals: that is harder. Now the cosets, although still countable, are dense in $\mathbb R$. So (with no requirement of continuity) you can still solve on each coset separately. And the cosets are group-isomorphic to $\mathbb Z^2$. So now you need the theory of two-parameter difference equations.

a plug

Suppose we do require continuity, or measurability, and integrability or boundedness. I have a paper with Rosenblatt on this...

G. A. Edgar & J. M. Rosemblatt, "Difference equations over locally compact abelian groups." Trans. Amer. Math. Soc. 253 (1979) 273--289. MR 80i:39001