Abstract: For a cluster of maximal monotone mappings, Hausdorff continuity of themselves and their domain were constructed. The multi-valued mappings were transformed into single-valued mappings by means of their Yosida approximation, the homotopy invariance of Brouwer degree was used to approach its topological degree with the degree of its Yosida approximation, and the homotopy invariance of the degree for the cluster of maximal monotone mappings was obtained. Besides, some basic  properties of the topological degree for the cluster of maximal monotone mappings as defined above were obtained. Similarly, under some additional assumptions, a theorem was derived about the homotopy invariance of the degree for the sum of two maximal monotone mappings.

Key words: maximal monotone mapping; topological degree; the homotopy invariance; continuity; Yosida approximation