A generalized-degree homotopy yielding global bifurcation results
The nonlinear eigenvalue problem \[ F(x,\lambda)\equiv Ax-\sum\limits_{j=1}^{k}\lambda^{j}B_jx-R(x,\lambda)=0,\;k\geq 1 \] is considered, where \[ F(\cdot,\lambda):X\to Y\text{ and }L(\lambda)=A-\sum\limits_{j=1}^{k}\lambda^{j}B_j \] are \(A\)-proper mappings with respect to the admissible scheme \(\Gamma=\left\{X_n,Y_n,Q_n\right\}\) for \(\lambda\) in a (possibly unbounded) open interval \(I\), \(X\) and \(Y\) are Hilbert spaces, \(X\) is embedded in \(Y\), \(A,B_j: X\to Y\) are bounded linear operators, \(X_n\) and \(Y_n\) are sequences of oriented finite-dimensional subspaces of \(X\) and \(Y\), \(Q_n: Y\to Y_n\) are continuous linear projections of \(Y\) onto \(Y_n\), \(R:X\times \mathbb{R}\to Y\) is a continuous mapping, and \(\|R(x,\lambda)\|=o(\|x\|).\) It is proved that \(\lambda_0=0\) is a global bifurcation point when \(\dim N(A)\) is odd, \(B_j\), \(j=1,\dots,k,\) are positive operators on \(N(A)\), and a standard transversality condition is satisfied. This result is applied to the periodic boundary value problem \[ -x''(t)+\lambda x(t)+\lambda^2 x'(t)+f(t, x(t),x'(t),x''(t))=0,\;x(0)=x(1),\;x'(0)=x'(1). \] Reviewer's remarks: It is noted that this result generalizes J.~F.\ Toland's results of the 1970s and 1980s, but comparisons with more recent work such as [\textit{B.~Buffoni} and \textit{J.~Toland}, ``Analytic Theory of Global Bifurcation'' (Princeton Series in Applied Mathematics, Princeton Univ.\ Press, Princeton/NJ) (2003; Zbl 1021.47044)] are absent.
scientific article; zbMATH DE number 2183185
scientific article