On the number of unbounded solution branches in a neighborhood of an asymptotic bifurcation point
This paper concerns abstract equations of the type \[ x= \lambda Ax+F_0(\lambda)+ F_1(Ax,\lambda), \quad \lambda\in\mathbb R,\;x\in B. \tag{1} \] Here, \(H\) is a Hilbert space with scalar product \(\langle\cdot,\cdot\rangle\), \(B\hookrightarrow H\) is a Banach space, \(A:H\to B\) is linear, completely continuous in \(B\) and selfadjoint in \(H\), and \(H_0:\mathbb R\to H\) and \(F_1:H\times\mathbb R\to\mathbb R\) are continuous. Further, it is supposed that \(F_1\) is asymptotically homogeneous, that \(\lambda_0\in\mathbb R\) is a characteristic value of \(A\) of multiplicity two, that \(F_0(\lambda_0)\perp\ker (I-\lambda_0A)\), and that \(F_1\) is ``coordinated with \(A\)'' (see the paper for this notion). Then each proper solution of a certain scalar equation on \(\{x\in H: x= \lambda_0Ax\), \(\|x\|=1\}\) generates a directed unbounded branch of solutions to (1). Applications to second order ODEs (forced oscillations, two-point boundary value problems) are also given.
scientific article; zbMATH DE number 5163295