Slant Differentiability and Semismooth Methods for Operator Equations
by
Zuhair Nashed
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Let X and Y be Banach spaces, and F: D ⊂ X → Y be a continuous mapping on an open domain D. The following concepts of slant differentiability and slanting function were introduced in [1]. A function F : D ⊂ X → Y is said to be slantly differentiable at x ∈ D if there exist a mapping f^○ : D → L (X, Y) such that the family f^○(x+h) of bounded linear operators is uniformly bounded in the operator norm for h sufficiently small and

_h 0F(x+h) - F(x) -f^0(x+h)h=0.

The function f^○ is called a slanting function for F at x. A function F: D ⊂ X → Y is said to be slantly differentiable in an open domain D₀ ⊂ D if there exists a mapping f⁰ : D → L(X, Y) such that f^○ is a slanting function for F at every point x ∈ D₀. In this case, f^○ is called a slanting function for F in D₀.

In this talk I will discuss operator-theoretic aspects of slant differentiability and related variants. Application to Newton-like methods, optimal control theory, and nonlinear ill-posed problems will be indicated. A unifying framework for semismooth analysis will be sketched and compared with the setting in [2] for smooth analysis.

[1] X. Chen, Z. Nashed, and L.Qi, smoothig methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal. 38 (2000), no. 4, 1200-1216.

[2] M. Z. Nashed, Differentiability adn related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, in L.B. Rall, ed., "Nonlinear Functional Analysis and Applications", Academic Press, New York, 1971, pp. 103-309.

Date received: May 2, 2007