Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(H,K)$ between $(F,D)$ and $(G,E)$ as follow: \begin{array}{rl} H_t(\mathbf{x}):&K(t)\rightarrow\rightarrow K(t),t\in[0,1]\\ K(t):&Y\rightarrow\rightarrow Y\\ K(0)=&D, \quad K(1)=E\\ H_0(\mathbf x)=&F(\mathbf x), \quad H_1(\mathbf x)=G(\mathbf x) \end{array}

If $F$ and $H_t$ has fixed point over the image sets $D,K(t)$ for $t\in[0,1)$ (and not on the boundaries of their image sets), can we say that $G$ has fixed point over its image set? If the answer is negative, then is we may need to If the above statement is true, is it possible to relax the condition of no fixed point on the boundary sets?