@Rouben Rostamian  

Hello Rouben and thanks for your answer.

I have given a differential equation dg(x)/dx=f(x,g(x)) with boundary conditions g'(a)=0 and g'(b)=0 where a and b are specified through the equation \int_a^b g'(x)/x dx =log((1+m)/(1-n)).

m (lambda_high) and n (lambda_low) are given and it can be proved that for certain values such values a and b satisfying these conditions exist.

The difficulty is that not only the solution function $g$ is unknown, also the domain of $g$ is not given and has to be calculated.

I hope, that I could make it more clear, if not please ask again :)