Publication

Bifurcation at isolated singular points of the Hadamard derivative

Charles Stuart
2014
Journal Articles

Abstract

For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F(lambda, u) = 0, where F : R x X -> Y with F(lambda, 0) = 0 for all lambda is an element of R. The focus is on the situation where F(lambda, center dot) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Frechet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on R-N, where Frechet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.

Official source

https://infoscience.epfl.ch/entities/publication/810a2ebe-3e7c-4896-87bc-685321eb0696

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Charles Stuart
2014
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/810a2ebe-3e7c-4896-87bc-685321eb0696

About this result

Related concepts (32)

Gateaux derivative

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.

Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces.

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.

Related publications (32)

Bifurcation at isolated singular points of the Hadamard derivative

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