I am interested in solving the following equations:  

$f(x) + \int\_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = 0$  

and  

$f(x) + \int\_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = u(x)$  

when $u(x)$ is the unknown function defined on $[0^+,\infty)$ and all other functions are known and are assumed to have convenient differentiability and other reasonable properties. In particular, both lower and upper bounds $\alpha(x)$ and $\beta(x)$ are not constants.


I know that when both bounds are constant they belong to Fredholm equations, and when only one bound is constant they belong to Volterra equations, where plenty of literature work exists.


However I am not clear about the case when both bounds are variables. Can the equations defined above be conveniently translated into Fredholm or Volterra theory? If they cannot, what is the theory for these type of equations?


Thanks!



how to solve general integral equations with both variable lower and upper bounds
If $\alpha$ and $\beta$ are bounded functions, with $a\le\alpha\le\beta \le b$, then your equations are of standard type since they can be written as
$$f(x)+\int\_a^b H(x,t) u(t) dt =0 $$
where
$$ H(x,t)=K(x,t)\chi\_{[\alpha(x),\beta(x)]}(t). $$
Here $\chi\_A$ denotes the characteristic function of the set $A$.


If $\alpha,\beta$ are unbounded, then we need some thinking...


