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On the number of unbounded solution branches in a neighborhood of an asymptotic bifurcation point. (English. Russian original) Zbl 1128.47055
Funct. Anal. Appl. 39, No. 3, 194-206 (2005); translation from Funkts. Anal. Prilozh. 39, No. 3, 37-53 (2005).
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.

##### MSC:
 47J15 Abstract bifurcation theory involving nonlinear operators 34C23 Bifurcation theory for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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