  We use $KKM$ theorem to prove the existence of a new fixed point theorem for
non-expansive mapping:Let M be a bounded closed convex subset of Hilbert space
H, and $A:M\rightarrow M$ be a non-expansive mapping, then exists a fixed point
of A in M, we also apply this Theorem to study the solution for an integral
equation,we can weak some conditions comparing with Banach's contraction
principe.
