On the eigenfunction expansions associated with semilinear Sturm-Liouville-type problems. (English) Zbl 1162.34363

Summary: We consider semilinear second-order ordinary differential equations, mainly autonomous, in the form \(-u{^{\prime\prime}}=f(u)+\lambda u\), supplied with different sets of standard boundary conditions. Here \(\lambda\) is a real constant or it plays the role of a spectral parameter. Mainly, we study problems in the interval (0,1). It is shown that in this case each problem that we deal with has an infinite sequence of solutions or eigenfunctions. Our aim in the present article is to review recent results on basis properties of sequences of these solutions or eigenfunctions. In a number of cases, it is proved that such a system is a basis in \(L_{2}\) (in addition, a Riesz or Bari basis). In addition, we briefly consider a problem for the half-line \((0,\infty )\). In this case, the spectrum of the problem fills a half-line and an analog of the expansions into the Fourier integral is obtained. The proofs are mainly based on the Bari theorem and, in addition, on our general result on sufficient conditions for a sequence of functions to be a Riesz basis in \(L_{2}\).


34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34L30 Nonlinear ordinary differential operators
34L99 Ordinary differential operators
42C15 General harmonic expansions, frames
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[1] Bari, N. K., On bases in Hilbert space, Dokl. Akad. Nauk SSSR, 54, 5, 383-386 (1946), (in Russian)
[2] Bari, N. K., Biorthogonal systems and bases in Hilbert space, Moskovskogo Gos. Universiteta Učenye Zapiski, 148, Matematika, 4, 69-107 (1951), (in Russian)
[3] Brown, K. J., A completeness theorem for a nonlinear problem, Proc. Edinburgh Math. Soc., 19, 2, 169-172 (1974) · Zbl 0292.34014
[4] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill Book Company, Inc.: McGraw-Hill Book Company, Inc. New York, Toronto, London · Zbl 0042.32602
[5] Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340 (1971) · Zbl 0219.46015
[6] Gohberg, I. Ts.; Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators (1969), AMS: AMS Providence, RI, (English translation) · Zbl 0181.13504
[8] Makhmudov, A. P., On the completeness of eigenelements of some nonlinear operator equations, Dokl. Akad. Nauk SSSR, 263, 1, 23-27 (1982), (in Russian) · Zbl 0527.34020
[9] Makhmudov, A. P., Basics of Nonlinear Spectral Analysis (1984), Azarbaijan State Univ.: Azarbaijan State Univ. Baku, (in Russian) · Zbl 0609.34024
[10] Makhmudov, A. P., Nonlinear Eigenvalue Problems and Bifurcation Theory (1996), ALM Publ.: ALM Publ. Baku, (in Russian)
[11] Makin, A. S.; Thompson, H. B., Convergence of eigenfunction expansions corresponding to nonlinear Sturm-Liouville operators, Electron. J. Differential Equations, 2004, 87, 1-10 (2004) · Zbl 1078.34532
[12] Pohozaev, S. I., On an approach to nonlinear equations, Dokl. Akad. Nauk SSSR, 247, 6, 1327-1331 (1979), (in Russian)
[13] Pohozaev, S. I., A method of fibering for solving nonlinear boundary-value problems, Trudy Matem. Inst. imeni V.A. Steklova, 192, 146-163 (1990), (in Russian) · Zbl 0734.35036
[14] Rabinowitz, P. H., Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J., 23, 8, 729-754 (1974) · Zbl 0278.35040
[15] Rabinowitz, P. H., (Minimax Methods in Critical Point Theory and Applications to Differential Equations. Minimax Methods in Critical Point Theory and Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., vol. 65 (1986), AMS: AMS Providence) · Zbl 0609.58002
[16] Zhidkov, P. E., Completeness of systems of eigenfunctions for the Sturm-Liouville operator with potential depending on the spectral parameter and for one non-linear problem, Sb. Math., 188, 7, 1071-1084 (1997) · Zbl 0959.34018
[19] Zhidkov, P. E., Riesz basis property of the system of eigenfunctions for a non-linear problem of Sturm-Liouville type, Sb. Math., 191, 3, 359-368 (2000) · Zbl 0961.34072
[20] Zhidkov, P. E., Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems, Electron. J. Differential Equations, 2000, 28, 1-13 (2000) · Zbl 0945.34066
[21] Zhidkov, P. E., Sufficient conditions for functions to form Riesz bases in \(L_2\) and applications to nonlinear boundary-value problems, Electron. J. Differential Equations, 2001, 74, 1-10 (2001) · Zbl 1009.34077
[22] Zhidkov, P. E., On the property of being a basis for a denumerable set of solutions of a nonlinear Schrödinger-type boundary-value problem, Nonlinear Anal.: TMA, 43, 4, 471-483 (2001) · Zbl 0972.34016
[23] Zhidkov, P. E., On the property of being a Bari basis for system of eigenfunctions of a nonlinear integrodifferential equation, Differ. Uravn., 38, 9, 1183-1189 (2002), (in Russian)
[24] Zhidkov, P. E., An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation, Nonlinear Anal.: TMA, 52, 3, 737-754 (2003) · Zbl 1025.34087
[25] Zhidkov, P. E., (Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math., vol. 1756 (2001), Springer-Verlag: Springer-Verlag Heidelberg) · Zbl 0987.35001
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