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Global cusp maps in differential and integral equations. (English) Zbl 0794.58005

For a Fréchet space \(E\) define the fold map \(F: R\times E\to R\times E\) by \(E(t,v)= (t^ 2,v)\) and the cusp map \(G: R^ 2\times E\to R^ 2\times E\) by \(G(s,t,v)= (s^ 3-ts,t,v)\). The fold map and the cusp map are stable singularities. After having studied abstract fold maps in Nonlinear Anal., Theory Methods Appl. 18, No. 8, 743-758 (1992; Zbl 0771.58003), the authors set themselves the problem to characterize those maps which are homeomorphically equivalent with a cusp map. Note that for our initial cusp map we have three solutions in one region, one solution in another region and two solutions on the common boundary of the two regions (except for a codimension one submanifold, where there is one solution). A map can therefore only be equivalent to the cusp map above if it exhibits similar behaviour. The paper also gives a number of applications for partial differential equations where their abstract approach leads to simplifications in the treatment.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
45G05 Singular nonlinear integral equations
35G30 Boundary value problems for nonlinear higher-order PDEs

Citations:

Zbl 0771.58003
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References:

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