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## Archive of Issues

Russia Izhevsk
Year
2012
Issue
1
Pages
26-31
 Section Mathematics Title On the question of extended convexity of Green operator Author(-s) Islamov G.G.a Affiliations Udmurt State Universitya Abstract 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 $Q \widehat GP> 0.$ Keywords Green‘s operator, extended convexity UDC 517.929 MSC 34K06, 34K10 DOI 10.20537/vm120103 Received 1 February 2012 Language Russian Citation 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. References 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. Full text