Let $X$ be a Banach space. Denote for $x_0\in X$ and $r>0$ the closed ball centered at $x_0$ by $B(x_0,r)=\lbrace x\in X:\|x-x_0\|\le r\rbrace$.
Suppose $f:X\to X$ a bounded map with a fixed point $x_*$ (that is, $f(x_*)=x_*$) and assume there is an constant $r>0$ and $M\in[0,1)$ such that $\|f(x)-f(y)\|\le M\|x-y\|$ for any $x,y\in B(x_*,r)$.
Put for $\alpha\in[0,\infty)$, $\phi_\alpha(A)=\overline{\{f(x):x\in A\}}+\alpha C,$where $A,C\subset X$ are closed and bounded subsets and the overline denotes the closure. The map $A\mapsto\overline{\{f(x):x\in A\}}$ is locally Lipschitz continuous.
Let $BC[B(x_*,r)]$ denote the collection of all bounded closed subsets in $X$. Is there a $\alpha_0>0$ such that $\phi_\alpha(BC[B(x_*,r)])=BC[B(x_*,r)]$ for all $\alpha\in[0,\alpha_0]$? and does $\phi_\alpha$ have fixed point $A_*\in BC[B(x_*,r)]$ that contains $x_*$?