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Section
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Mathematics
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Title
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On the question of extended convexity of Green operator
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Author(-s)
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Islamov G.G.a
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Affiliations
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Udmurt State Universitya
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Abstract
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Let $Q$ be a differential operator of order $m-1$, $2\leqslant m \leqslant n$, for which $(a, b)$ is the interval of nonoscillation, and the Green's operator $G: L[a, b]\to W^n[a, b]$ of boundary value problem $Lx = f$, $l_i(x) = 0$, $i = 1, \ldots,n$ has the property of generalized convexity: $QGP>0$ for some linear homeomorphism $P$ of Lebesgue space $L[a,b].$ Under some conditions, we prove, that the perturbed boundary value problem $Lx = PVQx + f$, $l_i(x) = 0$, $i = 1,\ldots,n$ is also uniquely solvable in the Sobolev space $W ^ n [a, b]$ and the Green's operator $\widehat G$ inherits the property of $G$, that is $Q \widehat GP> 0.$
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Keywords
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Green‘s operator, extended convexity
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UDC
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517.929
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MSC
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34K06, 34K10
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DOI
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10.20537/vm120103
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Received
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1 February 2012
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Language
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Russian
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Citation
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Islamov G.G. On the question of extended convexity of Green operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 1, pp. 26-31.
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References
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- Azbelev N.V., Rakhmatullina L.F., Maksimov V.P. Metody sovremennoi teorii lineinykh funktsional’no-differentsial’nykh uravnenii (Methods of modern theory of linear functional–differential equations), Izhevsk: Regular and Chaotic Dynamics, 2000, 300 p.
- Islamov G.G. Estimation of the minimal rank of finite–dimensional perturbations of Green’s operators, Differ. Uravn., 1989, vol. 25, no. 9, pp. 1046–1052.
- Islamov G.G. Some applications of the theory of abstract functional-differential equation. I, Differ. Uravn., 1989, vol. 25, no. 11, pp. 1309–1317.
- Islamov G.G. Some applications of the theory of abstract functional-differential equation. II, Differ. Uravn., 1990, vol. 26, no. 2, pp. 167–173.
- Islamov G.G. The solvability criterion for equations with boundary inequalities, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., Izhevsk, 1994, no. 2, pp. 3–24.
- Azbelev N.V., Rakhmatullina L.F., Tsalyuk Z.B. A note on the positivity of inverse operators, Uch. Zap. Udmurt. Gos. Ped. Inst., 1958, no. 12, pp. 47–49.
- Vulikh B.Z. Vvedenie v teoriyu poluuporyadochennykh prostranstv (Introduction to the theory of partially ordered spaces), Moscow: Fizmatgiz, 1961, 408 p.
- Islamov G.G. On the existence of positive solutions of equations with retarded argument, Proceedings of the Third All-Union Conference on Theory and Applications of Differential Equations with Deviating Argument, Chernovtsy State University, 1972, pp. 95–97.
- Karlin S., Studden W.J. Tchebycheff systems: with applications in analysis and statistics, Interscience Publishers, 1966.
- Islamov G.G. On the question of an upper estimation of the spectral radius, Vestn. Udmurt. Univ.,1992, no. 1, pp. 82–86.
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