Let $u(x)$ be a continuous function on $[0,1]$ and satisfies $$ u(x)=1+\lambda\int_x^1 u(y)u(y-x)dy $$ prove that $\lambda\leq 1/2$.
My try: My idea is to use Laplace transform. If the 2nd term of RHS is a kind of convolution, then we have $$ \hat{u}(w)=1/w+\lambda\hat{u}^2(w) $$ Then we consider this equation as a quadratic equation w.r.t $\hat{u}(w)$. So the discriminant gives $\lambda\leq 1/4$. However, this is not the answer we want and 2nd term of RHS is actually not a convolution. Thanks in advance.