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I have the following conjecture to show some sort of a continuous selection of fixed points in a correspondence:

Let $S$ and $\Theta $ be non-empty, compact and convex subsets of some Euclidean space $\mathbb{R}^{n}$. Let $\Gamma :$ $S\times \Theta \rightarrow 2^{S}$ be a set-valued function on $S\times \Theta $ with a closed graph and the property that $\Gamma \left( x,\theta \right) $ is non-empty and convex for all $x$ $\in $ $S$ and all $\theta \in \Theta $. Moreover, for all $x\in S$, $\Gamma \left( x,\cdot \right) $ is lower hemicontinuous with respect to $\Theta $. Then, for all $\theta \in \Theta $, there exist a non empty, upper-hemicontinuous, convex-valued correspondence $x^{\ast }:\Theta \rightarrow S$ such that $x\in \Gamma \left( x,\tilde{\theta}\right) $ for all $x\in x^{\ast }\left( \tilde{\theta}\right) $ and all $\tilde{\theta}$ in a neighborhood of $\theta $.

The proof of the non-empty and upper-hemicontinuity part is easy. For all $% \tilde{\theta}$ in a neighborhood of $\theta $ apply the Kakutani Fixed Point Theorem and set $x^{\ast }\left( \tilde{\theta}\right) $ equal to the fixed point. Then, consider a sequence $\left \{ x^{k},\theta ^{k}\right \} \rightarrow \left( x,\theta \right) $ such that $x^{k}\in x^{\ast }\left( \theta ^{k}\right) $. It follows that $x^{k}\in \Gamma \left( x^{k},\theta ^{k}\right) $ for all $k$. Because $\Gamma $ has a closed graph, $x\in \Gamma \left( x,\theta \right) $.

The challenge is to use lower hemicontinuity of $\Gamma $ with respect to $% \theta $ to show the existence of a convex-valued correspondence. Are you aware of any result that proves this conjecture? Can you think of a counterexample?

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