Abstract. In this article, we consider a map $A\colon M\rightarrow X$ and a multivalued map $B\colon M\rightarrow CB(X)$ where $M$ is a closed convex subset of a Banach space $X$. We give sufficient conditions for the existence of a fixed point $x_0\in M$ of the multivalued operator $A+B$ satisfying $Ax_0+Bx_0=\{x_0\} $. This result includes the well-known Krasnoselskii's fixed point theorem for the sum of two nonlinear single valued operators.

AMS Subject Classification (1991): 54H25, 47H10

Keyword(s): fixed point, Hausdorff metric, multivalued contraction, measure of noncompactness

Received October 18, 2007, and in final form March 12, 2008. (Registered under 6052/2009.)