I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with convexity.
Although Vidit Nanda has addressed Lifchitz fixed point theorem that replace the convexity asumption with contractibility, however I wonder if we still can relax contractibility or acyclic set to a weaker condition?



In the absence of convex images, one typically relies on algebraic topology as you have guessed. If your setvalued map has a reasonably nice domain and contractible images, then you can easily string together two results: Theorem: [Contractible Carrier] Let $K$ be a locally finite simplicial complex and $T$ a topological space. Assume the existence of a map $C$ from simplices of $K$ to subsets of $T$ so that for each simplex $\sigma \in K$ the image $C(\sigma) \subset T$ is contractible, and for each face relation $\sigma < \tau$ you have $C(\sigma) \subset C(\tau)$. Then, (a) there exists a continuous map $f:K \to T$ carried by $C$, i.e., $f(\sigma) \subset C(\sigma)$ for each $\sigma \in K$, and (b) any two maps carried by $C$ are homotopic. This theorem is pretty much folklore at this point, see for instance Chapter II Theorem 9.2 in Lundell and Weingram's topology of CW complexes. There is an inductive process underlying (a): you can actually construct maps $f:K \to T$ carried by $C$ by inducting on skeleta of $K$. Things are easier in your case since $T = K$ as well. Once you have any continuous $f:K \to K$ carried by $C$, compute its Lefschetz number, hope it is nonzero, and use the Lefschetz fixed point theorem. 


There is the EilenbergMontgomery fixed point theorem. You can find it and many other results in Advanced Fixed Point Theory for Economics by Andy McLennan. 

