# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford-London-New York-Paris, 1964. [2] K. Schmitt and Z. Q. Wang, ”On bifurcation from infinity for potential operators,” Differential Integral Equations, 4, No.5, 933–943 (1991). · Zbl 0736.58014 [3] A. M. Krasnosel’skii, ”Asymptotic homogeneity of hysteresis operators,” Z. Angew. Math. Mech., 76, Suppl. 2, 313–316 (1996). · Zbl 0889.47034 [4] M. A. Krasnoselskii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis, Springer-Verlag, Berlin-Heidelberg-New York-Tokio, 1984. [5] A. C. Lazer and D. E. Leach, ”Bounded perturbations of forced harmonic oscillators at resonance,” Ann. Mat. Pura Appl. (4), 82, 46–68 (1969). · Zbl 0194.12003 [6] A. M. Krasnosel’skii and J. Mawhin, ”Periodic solutions of equations with oscillating nonlinearities. Nonlinear operator theory,” Math. Comput. Modelling, 32, 1445–1455 (2000). · Zbl 0974.34042 · doi:10.1016/S0895-7177(00)00216-8 [7] P.-A. Bliman, A. M. Krasnosel’skii, M. Sorine, and A. A. Vladimirov, ”Forced oscillations in control systems with hysteresis,” Dokl. Akad. Nauk, 347, No.4, 458–461 (1996). English transl. in: Acad. Sci. Russian Dokl. Math., 53, No. 2, 312–315 (1996). · Zbl 0906.93027 [8] A. M. Krasnosel’skii, N. A. Kuznetsov, and D. I. Rachinskii, ”On resonant differential equations with unbounded non-linearities,” Z. Anal. Anwendungen, 21, No.3, 639–668 (2002). · Zbl 1025.34032 [9] P. Diamond, P. E. Kloeden, A. M. Krasnosel’skii, and A. V. Pokrovskii, ”Bifurcations at infinity for equations in spaces of vector-valued functions,” J. Austral. Math. Soc. Ser. A, 63, No.2, 263–280 (1997). · Zbl 0913.47058 · doi:10.1017/S1446788700000689 [10] A. M. Krasnosel’skii and J. Mawhin, ”The index at infinity of some twice degenerate compact vector fields,” Discrete Contin. Dynam. Systems, 1, No.2, 207–216 (1995). · Zbl 0877.47034 · doi:10.3934/dcds.1995.1.207 [11] V. I. Arnold, Arnold Problems, Springer-Verlag, Berlin-London, 2005.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.