Leray–Schauder degree

E518467

nonlinear analysis tool topological degree topological invariant

The Leray–Schauder degree is a topological invariant that generalizes the Brouwer degree to compact perturbations of the identity in infinite-dimensional Banach spaces, providing a powerful tool for proving existence of solutions to nonlinear equations.

Try in SPARQL Jump to: Statements Referenced by

Statements (44)

Predicate	Object
instanceOf	nonlinear analysis tool ⓘ topological degree ⓘ topological invariant ⓘ
appliesTo	compact perturbations of the identity ⓘ infinite-dimensional Banach spaces ⓘ maps in Banach spaces ⓘ
assumes	Fredholm-type structure in many applications ⓘ
codomain	integers ⓘ
comparedTo	Brouwer degree NERFINISHED ⓘ
definedFor	compact maps on open bounded subsets of Banach spaces ⓘ
domain	Banach space ⓘ
field	functional analysis ⓘ nonlinear functional analysis ⓘ partial differential equations ⓘ topology ⓘ
generalizes	Brouwer degree NERFINISHED ⓘ
hasVariant	degree for Fredholm maps of index zero ⓘ degree for condensing maps ⓘ
historicalPeriod	20th century mathematics ⓘ
namedAfter	Jean Leray NERFINISHED ⓘ Julius Schauder NERFINISHED ⓘ
property	coincides with Brouwer degree in finite dimensions ⓘ homotopy invariant ⓘ integer-valued ⓘ stable under compact perturbations ⓘ
relatedTo	Leray–Schauder fixed point theorem NERFINISHED ⓘ Schauder fixed point theorem ⓘ topological degree theory ⓘ
requires	a priori bounds on solutions ⓘ compactness of the nonlinear part ⓘ
satisfies	additivity property ⓘ excision property ⓘ homotopy invariance property ⓘ normalization property ⓘ
toolFor	integral equations ⓘ nonlinear boundary value problems ⓘ nonlinear operator equations ⓘ
usedFor	boundary value problems ⓘ elliptic partial differential equations ⓘ existence of solutions of nonlinear equations ⓘ fixed point problems ⓘ
usedIn	continuation methods ⓘ degree-theoretic proofs of existence theorems ⓘ global bifurcation theory ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Schauder fixed-point theorem → relatedTo → Leray–Schauder degree ⓘ