The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.

This motivates us to consider the following "weak fixed point property" by replacing the identity map with another continuous map:

**A weak fixed point property $"P"$:**

A topological space $X$ satisfies $P$ if there is a continuous map $F:X\to X$ such that for every continuous map $g:X\to X$ there exist a point $x_{0}$ such that $g(x_{0})=F(x_{0})$. That is: the graph of all maps intersect the graph of $F$.

We search for some examples (among topological space in particular manifolds) such that a space satisfies $P$ but does not satisfy the usual fixed point property. In particular do spheres, real or complex projective space satisfy $P$?