×

zbMATH — the first resource for mathematics

Correct and self-adjoint problems for biquadratic operators. (English) Zbl 1402.47002
J. Math. Sci., New York 179, No. 6, 714-725 (2011) and Zap. Nauchn. Semin. POMI 381, 145-162 (2011).
Summary: In this paper, we continue a series of previous articles and present a simple method of proving the correctness and self-adjointness of operators of the form \(B^{4}\) corresponding to some boundary value problems. We also give representations for the unique solutions of these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used.

MSC:
47B25 Linear symmetric and selfadjoint operators (unbounded)
34K10 Boundary value problems for functional-differential equations
Software:
DERIVE; Mathematica
PDF BibTeX Cite
Full Text: DOI
References:
[1] M. G. Krein, ”The theory of self-adjoint extensions of semi-bounded Hermitian operators and its applications,” Mat. Sb., 20, No. 3, 431–495 (1947). · Zbl 0029.14103
[2] M. I. Vishik. ”On general boundary value problems for elliptical differential equations,” Trudy Moskov. Mat. Obshch., 1, 187–246 (1952).
[3] E. A. Coddington, ”Self-adjoint subspace extensions of nondensely defined symmetric operators,” Adv. Math., 14, 309–332 (1974). · Zbl 0307.47028
[4] E. A. Coddington and A. Dijksma, ”Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces,” J. Differential Equations, 20, 473–526 (1976). · Zbl 0306.34023
[5] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht (1991). · Zbl 0751.47025
[6] A. N. Kochubei, ”Extensions of positive definite symmetric operators,” Dokl. Akad. Nauk Ukraln. SSR, Ser. A. 3, 168–171 (1979).
[7] A. A. Dezin, General Aspects of Boundary Value Problems [in Russian], Nauka. Moscow (1980). · Zbl 0494.35084
[8] B. K. Kokebaev, M. Otelbaev, and A. N. Shynybekov, ”About restrictions and extensions of operators,” Dokl. Akad. Nauk SSSR, 271, No. 6, 1307–1310 (1983). · Zbl 0568.35093
[9] R. O. Oinarov and I. N. Parasidis, ”Correct extensions of operators with finite defect in Banach spaces,” Izv. Akad. Nauk Kazakh. SSR, 5. 42–46 (1988).
[10] I. N. Parasidis and P. C. Tsekrekos, ”Correct self-adjoint and positive extensions of nondensely defined symmetric operators,” Abstract and Applied Analysis, 7, 767–790 (2005). · Zbl 1104.47026
[11] I. N. Parasidis and P. C. Tsekrekos. ”Some quadratic correct extensions of minimal operators in Banach spaces,” Operators and Matrices, 4, No. 2, 225–243 (2010). · Zbl 1192.47005
[12] I. N Parasidis and P. C. Tsekrekos, ”Correct and self–adjoint problems for quadratic operators,” Eurasian Math. J., 1, No. 2, 122–135 (2010). · Zbl 1233.47010
[13] I. N. Parasidis, P. C. Tsekrekos, and Th. G. Lokkas, ”Correct and self-adjoint problems with cubic operators.” J. Math. Sci., 166, No. 2, 420–427 (2010). · Zbl 1288.47024
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.