We study Fréchet differentiable stable operators in real Banachspaces. We present the theory of linear and nonlinear stableoperators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for aclass of nonlinear expanding operators. Copyright © 2006 Vadim Azhmyakov.