Nonlinear Equations with Operators Satisfying Generalized Lipschitz Conditions in Scales

Abstract

By means of the contraction principle we prove existence, uniqueness and stability of solutions for nonlinear equations $u+G_0[D,tL]+L(G_1[D,u],G2[D,u])=f$ in a Banach space $E$, where $G_0,G_1,G_2$ satisfy Lipschitz conditions in scales of norms, $L$ is a bilinear operator and $D$ is a data parameter. The theory is applicable for inverse problems of memory identification and generalized convolution equations of the second kind.