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During the course of a calculation I was doing for my research, I derived a delay integral equation of the form

[tex]g(x) = \int_0^1 dy K(y,x)g(x-y)[/tex]

where K(x,y) is a known, but somewhat ugly, kernel that has a [itex](1-y^2)^{-1/2}[/itex] singularity, but is integrable such that [itex]\int_0^1 dy K(y,x) = 1[/itex]. K is really a probability density function [itex]\rho(t;C)[/itex], where t is the variable and C is a parameter, and [itex]K(y,x) = \rho(y;a + (x-y)b)[/itex] . This also has the property that [itex]\rho(0;C) = 0[/itex].

g(x) is also interpretable as a probability distribution, so it should have the normalization [itex]\int_0^\infty dx g(x) = 1[/itex]

Although solving this equation exactly would be fantastic, what I would really like is to obtain the large x asymptotics of this equation. (g = 0 or 1 are obviously solutions of the integral equation, but fail to satisfy the normalization condition).

I tried assuming a functional relation [itex]g(x - y) = g(x)/K(y,x)[/itex], which solves the integral equation and leaves me with a functional equation to solve. Setting y = 0 in the equation, however, demonstrates that this can't be a solution because K(0,x) vanishes.

I have so far been unable to find any good references that might help find the asymptotic solution to this. Most of the delay integral equation papers I've found are either searching for forced periodic solutions, specific nonlinear integrands or very mathematical papers that show existence, uniqueness, stability, etc, but don't give me any clue on how I might solve it.

Does anyone here know of any good references that might help me solve for at least the large x asymptotics of such an equation?

Thanks.

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# Asymptotics of a linear delay integral equation

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