= Solution
A positive <fixed point> requires $\beta+u_*^\gamma=\alpha$, so it exists exactly when $\boxed{\alpha>\beta}$ and is $u_*=(\alpha-\beta)^{1/\gamma}$. Differentiating the map at it gives
$$
\boxed{F'(u_*)=1-\gamma\left(1-\frac\beta\alpha\right).}
$$
For $1<\gamma\leq2$ and $0<\beta/\alpha<1$, this <fixed-point multiplier> is strictly between $-1$ and $1$. Hence the positive <fixed point> is linearly asymptotically stable, including $\gamma=2$; positivity of $\beta$ makes the lower bound strict.
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