On a Class of Multilinear Operator Equations

Abstract

By means of contraction principle in a Banach space $E$ with a scale of norms $\Vert \cdot {\Vert }_{\sigma }(\sigma \ge 0)$ existence, uniqueness and stability of solutions are proved for a general class of operator equations $u+G_{0}u+G_{1}u=g$ including multilinear ones where $G_0,G_1\in (E\to E)$ are some operators. The theorems are applicable to equations with operators of generalized convolution type.