A global continuation theorem for obtaining eigenvalues and bifurcation points. (English) Zbl 0665.35010

This paper proves the existence of a connected branch of solutions [\(\lambda\),u,\(\tau\) ] of the equation \(u- T(\lambda)u+H_{\tau}(\lambda,u)=0\), \(u\in Banach\) space with the additional norm condition \(\| u\|^ 2=\delta \tau\) containing at least one \([\lambda_{\tau},u_{\tau},\tau]\) for any \(\tau \in <0,1)\), starting at \([\lambda_ 0,0,0]\) and lying in a suitable subinterval \(J_ 0\) of J (the interval of \(\lambda)\). Here \(\lambda\in J\) is a bifurcation parameter, \(\tau\in (0,1)\) is an additional parameter, \(\delta >0\) is fixed.
Reviewer: J.H.Tian


35B32 Bifurcations in context of PDEs
39B52 Functional equations for functions with more general domains and/or ranges
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B60 Continuation and prolongation of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
34G20 Nonlinear differential equations in abstract spaces
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