2
votes
2answers
70 views

solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$ |f(t,x)|\leq \phi(t)|x|. $$ Suppose ...
3
votes
0answers
194 views

Solving an integral equation.

Let $K(x,y)=(x+y)e^{-y^2/2-xy}$. I need a constructive way (not simply verifying it is a solution) to show that $f(x)=e^{-x^2/2}$ is the solution to the integral equation: ...
5
votes
1answer
143 views

Integral equation and existence: $g(x)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$

I'd like to know how one would go about showing that the following function, $f$, that is almost everywhere positive exists: $$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$ ...