Abstract. In this paper we introduce the notions of generalized d− derivations, (, )− generalized module maps and (, )− generalized dynamics on Modules. We show that if \\\' : R ! B(M) is a − dynamics on Banach algebra A with the infinitesimal generator d and T : R ! B(M) is a (, )− generalized dynamics on Banach A− bimodule M with the infinitesimal generator , then  is a generalized d− derivation. Using the computation formula for n(ax) and the concept of − one parameter groups on M, we prove the converse under some more restrictions on .

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