Basis properties of eigenfunctions of second-order differential operators with involution. (English) Zbl 1270.34195

Kopzhassarova, Asylzat; Sarsenbi, Abdizhakhan

Abstr. Appl. Anal. 2012, Article ID 576843, 6 p. (2012).

Consider the spectral problem \[ -u''(-x)+\alpha u''(x)=\lambda u(x),\tag{\(*\)} \] where \(\lambda\) is the spectral parameter.
The authors prove the following results.
Theorem 1. Suppose that \(\alpha^2\neq 1\). Then the eigenfunctions of \((*)\) satisfying \[ u(-1)= u(1) =0 \] form an orthonormal basis in \(L_2(-1,1)\).
Theorem 2. Assume \(\alpha^2\neq 1\). Then the eigenfunctions of \((*)\) satisfying \[ u(-1)= u(1),\;u'(-1)= u'(1)\qquad(\text{periodic case}) \] or \[ u(-1)= -u(1),\;u'(-1)= -u'(1)\qquad(\text{antiperiodic case}) \] form orthonormal basis in \(L_2(-1,1)\).

Reviewer: Klaus R. Schneider (Berlin)

MSC:

34L10	Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B09	Boundary eigenvalue problems for ordinary differential equations

Keywords:

Dirichlet; periodic; antiperiodic

References:

[1]	T. Li, Z. Han, C. Zhang, and H. Li, “Oscillation criteria for second-order superlinear neutral differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 367541, 17 pages, 2011. · Zbl 1217.34112 · doi:10.1155/2011/367541
[2]	A. Zafer, “Oscillation of second-order sublinear impulsive differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 458275, 11 pages, 2011. · Zbl 1223.34049 · doi:10.1155/2011/458275
[3]	A. M. A. El-Sayed, E. M. Hamdallah, and Kh. W. El-kadeky, “Monotonic positive solutions of nonlocal boundary value problems for a second-order functional differential equation,” Abstract and Applied Analysis, vol. 2012, Article ID 489353, 12 pages, 2012. · Zbl 1239.34077 · doi:10.1155/2012/489353
[4]	R. Cheng, “Oscillatory periodic solutions for two differential-difference equations arising in applications,” Abstract and Applied Analysis, vol. 2011, Article ID 635926, 12 pages, 2011. · Zbl 1217.34113 · doi:10.1155/2011/635926
[5]	J. D\vzurina and R. Komariková, “Asymptotic properties of third-order delay trinomial differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 730128, 10 pages, 2011. · Zbl 1210.34107 · doi:10.1155/2011/730128
[6]	C. Babbage, “An essay towards the calculus of calculus of functions,” Philosophical Transactions of the Royal Society of London, vol. 106, pp. 179-256, 1816.
[7]	J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific, Singapore, 1993. · Zbl 0874.34054
[8]	M. A. Sadybekov and A. M. Sarsenbi, “Solution of fundamental spectral problems for all the boundary value problems for a first-order differential equation with a deviating argument,” Uzbek Mathematical Journal, vol. 3, pp. 88-94, 2007 (Russian). · Zbl 1189.34124
[9]	M. A. Sadybekov and A. M. Sarsenbi, “On the notion of regularity of boundary value problems for differential equation of second order with dump argument,” Mathematical Journal, vol. 7, no. 1, article 23, 2007 (Russian). · Zbl 1189.34124
[10]	A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,” Differential Equations, vol. 46, no. 4, pp. 506-511, 2010. · Zbl 1214.34082 · doi:10.1134/S0012266110040051
[11]	A. M. Sarsenbi and A. A. Tengaeva, “On the basis properties of root functions of two generalized eigenvalue problems,” Differential Equations, vol. 48, no. 2, pp. 1-3, 2012. · Zbl 1280.34084
[12]	V. P. Mihaĭlov, “On Riesz bases in L2[0, 1],” Reports of the Academy of Sciences of the USSR, vol. 144, pp. 981-984, 1962.
[13]	G. M. Keselman, “On the unconditional convergence of eigenfunction expansions of certain differential operators,” Proceedings of Institutes of Higher Education, vol. 2, no. 39, pp. 82-93, 1964.
[14]	M. A. Naĭmark, Linear Differential Operators, Frederick Ungar, New York, NY, USA, 1968.