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Solving an integral for an unknown function

Dear All,

I wonder if anyone could let me know if there is a general solution to the following problem in Mathematica.

Suppose the functions g[u,w] and h[w] are known. I want to find a function f[u] such that


In addition I know that 




Is it possible to solve the above for the unknown function f[u]? Of course there might be several solutions but I would be happy to find one.

Many thanks in advance for any suggestions. 


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