# Solve non-linear equation in R

I need to solve the following equation for $$x$$ in [0, 1]. Assume $$0<\alpha<1$$ and $$0<\lambda$$.

$$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$

Would very much appreciate any kind of help!

$$y(x,\lambda,\alpha)=2\lambda(\alpha+1)x^\alpha+(1-x)^{\alpha+1}-\lambda(1+x)^{\alpha+1}$$
be the function we are finding roots of, writing $$y=y(x)=y(x,\lambda,\alpha)$$ for shorthand. Then notice our boundary cases are
$$y(0)=1-\lambda\implies\begin{cases}y(0)\le0,&\lambda\ge1\\y(0)>0,&\lambda<1\end{cases}$$ $$y(1)=2\lambda(\alpha+1-2^\alpha)\ge0$$
which means that when $$\lambda\ge1$$, the uniroot function finds the root with guaranteed convergence because $$y(0)\le0\le y(1)$$ has a root in-between.
And although I haven't proven it, graphing strongly suggests that there are no solutions for $$0<\lambda<1$$, so there is no point in root-finding there.