# Differentiability of Inverse Operators

• Simon Y. Serovajsky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

## Abstract

The Inverse Function Theorem is a mighty tool of the local nonlinear analysis. It guarantees the existence of the inverse function and its differentiability. However the first property is sometimes not used. It is true, for example, for the optimal control theory and the inverse problems of mathematical physics. The inverse operator can be interpreted as a control-state mapping here. Its existence is a corollary of the state equation properties, and the differentiability of the inverse operator is used for the differentiation of the minimizing functional or the discrepancy. We establish a differentiability criterion of the inverse operator. Moreover, we prove a property which can be interpreted as a weak form of the operator differentiability. The Dirichlet problem for a nonlinear elliptic equation is considered as an example.

## Keywords

Inverse function theorem Operator derivative Extended derivative Nonlinear elliptic equation Necessary conditions of optimality

## Mathematics Subject Classification

58C20 46T20 35J60 49K20

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