Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique
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We study nonoscillation/oscillation of the dynamic equation$${(rx^\Delta)}^{\Delta}(t) + p(t)x(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$where \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation$${(rx^\Delta)}^{\Delta}(t) + p(t)x^\sigma(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$which is extensively discussed on time scales.KeywordsOscillationnonoscillationdynamic equationstime scalesfixed pointmonotone operatorsequence techniqueMathematics Subject ClassificationPrimary 34C10Secondary 39A10References1.Bohner M., Erbe L., Peterson A.: Oscillation for nonlinear second order dynamic equations on a time scale. J. Math. Anal. Appl. 301, 491–507 (2005)MathSciNetCrossRefMATHGoogle Scholar2.Bohner M., Peterson A.: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Boston (2001)CrossRefMATHGoogle Scholar3.E. Braverman and B. Karpuz, Nonoscillation of second-order dynamic equations with several delays. Abstr. Appl. Anal. (2011), Art. ID 591254, 34 pp.4.Erbe L.: Oscillation criteria for second order linear equations on a time scale. Canad. Appl. Math. 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Lett. 22, 136–141 (2009)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsBaşak Karpuz1Email author1.Department of Mathematics, Faculty of Science, Tınaztepe CampusDokuz Eylül UniversityBucaTurkey About this article CrossMark Print ISSN 1661-7738 Online ISSN 1661-7746 Publisher Name Springer International Publishing 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/s11784-016-0334-8_Nonoscillation and oscillation of ", 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/s11784-016-0334-8_Nonoscillation and oscillation of ", 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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