An analytical solution of the integral equation $ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz $

Asked Mar 24 '24 at 08:13

Active Mar 25 '24 at 07:13

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How to solve for the f(x) in the integral equation?

$$ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz \\ \alpha \in (1,2), C_2\text{ is a positive constant.} \\ \text{What is the definition of the function } f(x) \text{ in the given integral equation?} $$

I have tried using Taylor series method to attempt to obtain some numerical solutions, as well as attempting to use iterative methods to derive some results, but without success.

I am unable to figure out on how to solve it.

Can someone please give some hints?

edited Mar 25 '24 at 07:13

asked Mar 24 '24 at 08:13

YuanLan

I'm assuming you want to look for nonzero solutions, otherwise $f \equiv 0$ is one if you just wanted existence. – Bruno B Mar 25 '24 at 08:06
Does this equation have an analytical solution or other numerical solutions? The problem is from a Russian textbook from 1964. – YuanLan Mar 26 '24 at 15:17

An analytical solution of the integral equation $ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz $

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