An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces
Alexander Davidovich Ioffe, Efim Schwartzman · Proceedings of the American Mathematical Society · 1997
The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations A x + φ λ ( x ) = λ x Ax + \varphi _\lambda (x) = \lambda x with φ \varphi continuous jointly in ( λ , x ) (\lambda ,x) and φ λ ( ⋅ ) \varphi _\lambda (\cdot ) of class C 1 , 1 C^{1,1} . We also prove a bifurcation theorem for critical points of the function g λ ( x ) g_{\lambda }(x) which is just continuous and changes at x = 0 x=0 an isolated minimum (in x x ) to isolated maximum when λ \lambda passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.