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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?

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See also mathoverflow.net/questions/143696/… –  András Bátkai Oct 2 '13 at 13:10
    
@ Andras , that question which is post by me is just concern about the convexity of a set valued map. however this question does not assume any convex or contractible or other feature for the $G(\mathbf x)$. The fixed point is actually to be concluded by only homotopy property. –  behrad mahboobi Oct 2 '13 at 13:17
    
Sorry for my comment which was too short. I just wanted to add context to your question here. –  András Bátkai Oct 2 '13 at 13:19

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