I need to prove that the following equation has a unique solution ${\forall x \in [0,1]}$:
${f\left(x\right) = \int_{0}^{x}{ \frac{1}{(1+s)(1+ f\left(s\right) ^{2}) } \,ds} }$
I am trying to prove that ${F(f\left(s\right)) ={ \frac{1}{(1+s)(1+ f\left(s\right) ^{2}) }} }$ is Lipschitz continuous with constant < 1 which I believe would be enough for the proof. I am getting nowhere with even showing it's Lipschitz so I would appreciate any help on how to approach this. Thanks.