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Disconjugacy and nonoscillation domains for nonlinear singular interface problems on semi-infinite time scales. (English) Zbl 1243.34138
Summary: We define and discuss the disconjugacy $$(\mathcal D)$$ and nonoscillation $$(\mathcal N)$$ domains for a pair of dynamic equations along with matching interface conditions on the semi-infinite time scale $$[0, c]_{\mathbb T} \cup [\sigma (c), \infty]_{\mathbb T}$$. We show that these domains are closed and convex subsets of the parameter space $$\mathbb R^{n+m}$$. The theory developed is used to discuss the oscillatory behavior of initial and boundary value problems associated with interface problems in the fields of applied elasticity, acoustic wave guides in ocean, and transverse vibrations in strings.
##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] A. B. Mingarelli, “A survey of the regular weighted Sturm-Liouville problem-the non-definite case,” in Proceedings of the Workshop on Applied Differential Equations, Tsinghua University, Beiing, China, 1985. · Zbl 0624.34021 [2] R. A. Moore, “The least eigenvalue of Hill’s equation,” Journal d’Analyse Mathématique, vol. 5, pp. 183-196, 1956/57. · Zbl 0077.08703 · doi:10.1007/BF02937345 [3] L. Markus and R. A. Moore, “Oscillation and disconjugacy for linear differential equations with almost periodic coefficients,” Acta Mathematica, vol. 96, pp. 99-123, 1956. · Zbl 0071.08302 · doi:10.1007/BF02392359 [4] A. B. Mingarelli and S. G. Halvorsen, Nonoscillation Domains of Differential Equations with Two Parameters, vol. 1338, Springer, Berlin, Germany, 1988. · Zbl 0657.34035 · doi:10.1007/BFb0080637 [5] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics. Resultate der Mathematik, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153 [6] R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1-26, 2002. · Zbl 1020.39008 · doi:10.1016/S0377-0427(01)00432-0 [7] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1008.37032 · doi:10.1090/S0002-9939-02-06614-5 [8] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 1107.34304 · doi:10.1016/S0898-1221(01)00189-4 [9] M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 4, pp. 1239-1254, 2004. · Zbl 1075.34028 · doi:10.1216/rmjm/1181069797 [10] E. Akin-Bohner and J. Hoffacker, “Oscillation properties of an Emden-Fowler type equation on discrete time scales,” Journal of Difference Equations and Applications, vol. 9, no. 6, pp. 603-612, 2003. · Zbl 1038.39009 · doi:10.1080/1023619021000053575 [11] L. Erbe, A. Peterson, and S. H. Saker, “Kamenev-type oscillation criteria for second-order linear delay dynamic equations,” Dynamic Systems and Applications, vol. 15, no. 1, pp. 65-78, 2006. · Zbl 1104.34026 [12] L. Erbe and A. Peterson, “An oscillation result for a nonlinear dynamic equation on a time scale,” The Canadian Applied Mathematics Quarterly, vol. 11, no. 2, pp. 143-157, 2003. · Zbl 1086.39004 [13] O. Doand D. Marek, “Half-linear dynamic equations with mixed derivatives,” Electronic Journal of Differential Equations, no. 90, p. 118, 2005. · Zbl 1092.39004 · emis:journals/EJDE/Volumes/2005/90/abstr.html [14] S. H. Saker, “Oscillation of second-order forced nonlinear dynamic equations on time scales,” Electronic Journal of Qualitative Theory of Differential Equations, no. 23, p. 17, 2005. · Zbl 1097.34027 · emis:journals/EJQTDE/2005/200523.html · eudml:126473 [15] C. Allan Boyles, Acoustic Waveguides, Applications to Oceanic Sciences, John Wiley and Sons, 1978. [16] P. K. Ghosh, The Mathematics of Waves and Vibrations, MacMillan, India, Delhi, 1975. [17] G. L. Lamb, Jr., Elements of Soliton Theory, John Wiley & Sons, New York, NY, USA, 1980. · Zbl 0445.35001 [18] K. Noda, Optical Fibre Transmission, vol. 6 of Studies in Telecommuniations, North Holland, Amsterdam, The Netherlands, 1986. [19] Wang, Applied Elasticity, MacGraw Hill, 1953. · Zbl 0053.43902 [20] P. K. Baruah and D. J. Das, “Study of a pair of singular Sturm-Liouville equations for an interface problem,” International Journal of Mathematical Sciences, vol. 3, no. 2, pp. 323-340, 2004. · Zbl 1266.34034 [21] P. K. Baruah and M. Venkatesulu, “Deficiency indices of a differential operator satisfying certain matching interface conditions,” Electronic Journal of Differential Equations, vol. 2005, no. 38, p. 19, 2005. · Zbl 1075.34087 · emis:journals/EJDE/Volumes/2005/38/abstr.html · eudml:125296 [22] P. K. Baruah and M. Venkatesulu, “Number of linearly independent square integrable solutions of a pair of ordinary differential equations satisfying certain matching interface conditions,” International Journal of Mathematics and Analysis, vol. 3, no. 2-3, pp. 131-144, 2006. [23] M. Venkatesulu and P. K. Baruah, “A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations-I,” Journal of Applied Mathematics and Stochastic Analysis, vol. 14, no. 2, pp. 205-214, 2001. · Zbl 0992.34020 · doi:10.1155/S1048953301000168 · eudml:49340 [24] M. Venkatesulu and P. K. Baruah, “A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations-II,” Journal of Applied Mathematics and Stochastic Analysis, vol. 15, no. 2, pp. 197-203, 2002. · Zbl 1025.34026 · doi:10.1155/S1048953302000163 · eudml:49340 [25] T. G. Bhaskar and M. Venkatesulu, “Computation of Green’s matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 4, pp. 789-797, 1995. · Zbl 0835.34024 · doi:10.1155/S0161171295001013 · www.hindawi.com · eudml:47380 [26] P. K. Baruah and D. K. K. Vamsi, “Oscillation theory for a pair of second order dynamic equations with a singular interface,” Electronic Journal of Differential Equations, no. 43, pp. 1-7, 2008. · Zbl 1165.39002 · emis:journals/EJDE/Volumes/2008/43/abstr.html · eudml:130344 [27] P. K. Baruah and D. K. K. Vamsi, “Green matrix for a pair of dynamic equations with singular interface,” International Journal of Modern Mathematics, vol. 4, no. 2, pp. 135-152, 2009. · Zbl 1179.34105 · www.ijmm.dixiewpublishing.com [28] P. K. Baruah and D. K. K. Vamsi, “IVPs for singular interface problems,” Advances in Dynamical Systems and Applications, vol. 3, no. 2, pp. 209-227, 2008. [29] D. K. K. Vamsi and P. K. Baruah, “Bounds for solutions of nonlinear singular interface problems on time scales using a monotone iterative method,” Electronic Journal of Differential Equations, vol. 2010, no. 109, pp. 1-9, 2010. · Zbl 1200.34114 · emis:journals/EJDE/Volumes/2010/109/abstr.html · eudml:224855 [30] D. K. K. Vamsi and P. K. Baruah, “Existential results for nonlinear singular interface problems involving second order nonlinear dynamic equations using picards iterative technique,” Accepted for publication, The Journal of Nonlinear Science and Applications. · Zbl 1301.34116
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