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Completeness theorem for the dissipative Sturm-Liouville operator on bounded time scales
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In this paper we consider a second-order Sturm-Liouville operator of the form $$l(y): = - [p(t)y^\Delta (t)]^\nabla + q(t)y(t)$$ on bounded time scales. In this study, we construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint and other extensions of the dissipative Sturm-Liouville operators in terms of boundary conditions. Using Krein’s theorem, we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales.Key wordsTime scalesSturm-Liouville operatorΔ-differentiabledissipative operatorcompleteness of the system of eigenvectors and associated vectorsKrein theoremboundary value spaceReferences1.R. P. Agarwal, M. Bohner and W. T. Li, Nonoscillation and oscillation theory for functional differential equations, Pure Appl. Math., Dekker, Florida, (2004).Google Scholar2.D. R. Anderson, G. Sh. Guseinov and J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194(2) (2006), 309–342.MathSciNetCrossRefMATHGoogle Scholar3.F. Merdivenci Atici and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141(1-2) (2002), 75–99.MathSciNetCrossRefMATHGoogle Scholar4.E. Bairamov and E. Ugurlu, The determinants of dissipative Sturm–Liouville operators with transmission conditions, Math. Comput. Model, 53 (2011), 805–813.MathSciNetCrossRefMATHGoogle Scholar5.E. Bairamov and E. Ugurlu, Krein’s theorems for a dissipative boundary value transmission problem, Complex Anal. Oper. Theory, 7(4) (2013), 831–842.MathSciNetCrossRefMATHGoogle Scholar6.M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhäuser, Boston, (2001).CrossRefMATHGoogle Scholar7.M. Bohner and A. Peterson, (Eds.), Advances in dynamic equations on time scales, Birkhäuser, Boston, (2003).CrossRefMATHGoogle Scholar8.J. Eckhardt and G. Teschl, Sturm-Liouville operators with measure-valued coefficients, J. Anal. Math., 120 (2013), 151–224.MathSciNetCrossRefMATHGoogle Scholar9.I. C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Amer. Math. Soc., Providence, (1969).Google Scholar10.M. L. Gorbachuk and V. I. Gorbachuk, Boundary value problems for operator differential equations, Naukova Dumka, Kiev, (1984); English transl. Birkhauser Verlag, (1991).MATHGoogle Scholar11.G. Guseinov, Completeness theorem for the dissipative Sturm-Liouville operator, Doga-Tr. J. Math., 17 (1993), 48–54.MathSciNetMATHGoogle Scholar12.G. Guseinov and H. Tuncay, The determinants of perturbation connected with a dissipative Sturm-Liouville operator, J. Math. Anal. Appl., 194 (1995), 39–49.MathSciNetCrossRefMATHGoogle Scholar13.G. Guseinov, Completeness of the eigenvectors of a dissipative second order difference operator: dedicated to Lynn Erbe on the occasion of his 65th birthday, In honor of Professor Lynn Erbe, J. Difference Equ. Appl., 8(4) (2002), 321–331.MathSciNetCrossRefGoogle Scholar14.G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29(4) (2005), 365–380.MathSciNetMATHGoogle Scholar15.S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56.MathSciNetCrossRefMATHGoogle Scholar16.A. Huseynov, Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010), 379–392.MathSciNetMATHGoogle Scholar17.A. N. Kochubei, Extensions of symmetric operators and symmetric binary relations, Mat. Zametki, 17 (1975), 41–48; English transl. in Math. Notes, 17 (1975), 25-28.MathSciNetGoogle Scholar18.M. G. Krein, On the indeterminate case of the Sturm-Liouville boundary problem in the interval (0,8); 1Ş, Izv. Akad. Nauk SSSR Ser. Mat., 16(4) (1952), 293–324 (in Russian).MathSciNetGoogle Scholar19.V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic systems on measure chains, Kluwer Academic Publishers, Dordrecht, (1996).CrossRefMATHGoogle Scholar20.M. A. Naimark, Linear differential operators, 2nd edn., (1968), Nauka, Moscow, English transl. of 1st. edn., New York, 1,2, (1969).MATHGoogle Scholar21.B. P. Rynne, L 2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007), 1217–1236.MathSciNetCrossRefMATHGoogle ScholarCopyright information© The Indian National Science Academy 2016Authors and AffiliationsHüseyin Tuna1Email author1.Department of MathematicsMehmet Akif Ersoy UniversityBurdurTurkey About this article CrossMark Print ISSN 0019-5588 Online ISSN 0975-7465 Publisher Name Indian National Science Academy About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s13226-016-0196-1_Completeness theorem for the dissi", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s13226-016-0196-1_Completeness theorem for the dissi", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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