Volume 5 | Number 2 | Year 2014 | Article Id. IJMTT-V5P526 | DOI : https://doi.org/10.14445/22315373/IJMTT-V5P526
In this paper we present existential and uniqueness results for BVPs associated with 4th order nonlinear singular interface problems on Time Scales. We discuss these results using the classical fixed point theorems of Banach and Schauder.
[1] F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8 Academic Press, New York-London, 1964.
[2] M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications, Birkhuser Boston, Inc., Boston, MA, 2001.
[3] C. Allan Boyles, Acoustic Waveguides, Applications to Oceanic Sciences, John- Wiley and Sons, 1978.
[4] C. T. Fulton, Parametrizations of Titchmarshs m()-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63.
[5] P. K. Ghosh, The Mathematics of Waves and Vibrations, MacMillan, 1975.
[6] D. B. Hinton and J. K. Shaw, Titchmarsh-Weyl theory for Hamiltonian sys- tems. Spectral theory of dierential operators (Birmingham, Ala., 1981), North- Holland Math. Stud., 219-231, 55, North-Holland, Amsterdam- New York, 1981.
[7] John Von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionalop- eratoren, Math. Ann. 102(1929), 49{131.
[8] H. G. Kaper, M. Kwong, and A. Zettl, Anton Characterizations of the Friedrichs extensions of singular Sturm-Liouville expressions, SIAM J. Math. Anal. 17 (1986), 772-777.
[9] K. Kodaria, On ordinary dierential equations of any even order and the corre- sponding eigenfunction expansions, Amer. J. Math. 72 (1950), 502-544.
[10] B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary dierential operators, Translated from the Russian by Amiel Feinstein. Translations of Mathematical Monographs, Vol. 39. American Mathematical Society, Providence, R.I., 1975.
[11] G. L. Lamb, Elements of Soliton Theory, JohnWiley and Sons, 1980.
[12] F. Merdivenci Atici and G. Sh. Guseinov, On Greens functions and posi- tive solutions for boundary value problems on time scales, J. Comput. Appl. Math.141(2002), 75-99.
[13] M. A. Naimark, Linear dierential operators. Part II: Linear dierential opera- tors in Hilbert space, With additional material by the author, and a supplement by V. . Ljance. Translated from the Russian by E. R. Dawson. English transla- tion edited by W. N. Everitt Frederick Ungar Publishing Co., New York 1968.
[14] K. Noda, Optical Fiber Transmission, Studies in Telecommunications, North- Holland, 1986.
[15] Pallav Kumar Baruah and Dibya Jyothi Das, Singular Sturm Liouville problems with an interface, Int. J. Math. Sci. 3 (2004), 323-340.
[16] Pallav Kumar Baruah and D. K. K. Vamsi, IVPs for Singular Interface Prob- lems, to appear.
[17] Pallav Kumar Baruah and Dasu Krishna Kiran Vamsi, Oscillation Theory for a Pair of Second Order Dynamic Equations with a Singular Interface, Electron. J. Dierential Equations 43 (2008), 1{7.
[18] Pallav Kumar Baruah and M. Venkatesulu, Characterization of the resolvent of a dierential operator generated by a pair of singular ordinary dierential expressions satisfying certain matching interface conditions, Int. J. Mod. Math. 1 (2006), 31{47.
[19] Pallav Kumar Baruah and M. Venkatesulu, Deciency indices of a dierential operator satisfying certain matching interface conditions, Electron. J. Dieren- tial Equations 38 (2005), 1{9.
[20] Pallav Kumar Baruah and M. Venkatesulu, Number of linearly independent square integrable solutions of a pair of ordinary dierential equations satisfying certain matching interface conditions, to appear.
[21] Pallav Kumar Baruah and M. Venkatesulu, Self adjoint boundary value prob- lems associated with a pair of singular ordinary dierential expressions with interface spatial conditions, submitted.
[22] Pallav Kumar Baruah and M. Venkatesulu, Spectrum of pair of ordinary dif- ferential operators with a matching interface conditions, to appear.
[23] A. Pleijel, Generalized Weyl circles, Lecture Notes in Math., Vol. 415, Springer, Berlin, 1974, 211-226.
[24] E. C. Titchmarsh, Eigenfunction expansions associated with second-order dif- ferential equations, Part I, Second Edition Clarendon Press, Oxford, 1962.
[25] D. K. K. Vamsi, IVPs for a Pair of Dynamic Equations with Matching Interface Conditions, Sri Sathya Sai Institute of Higher Learning, March 2006.
[26] M. Venkatesulu and Pallav Kumar Baruah, A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations-I, J. Appl. Math. Stoch. Anal. 14 (2001), 205-214.
[27] M. Venkatesulu and Pallav Kumar Baruah, A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations-II, J. Appl. Math. Stoch. Anal. 15 (2002), 197-203.
[28] M. Venkatesulu and T. Gnana Bhaskar, Computation of Greens matrices for boundary value problems associated with a pair of mixed linear regular ordinary dierential operators, Int. J. Math. Math. Sci. 18 (1995), 789-797.
[29] M. Venkatesulu and T.Gnana Bhaskar, Fundamental systems and solutions of nonhomogeneous equations for a pair of mixed linear ordinary dierential equations, J. Aust. Math. Soc.(Series A), 49 (1990), 161-173.
[30] M. Venkatesulu and T.Gnana Bhaskar, Selfadjoint boundary value problems associated with a pair of mixed linear ordinary dierential equations, J. Math. Anal. Appl. 144 (1989), 322-341.
[31] M. Venkatesulu and Pallav Kumar Baruah, Solutions of initial value problems for a pair of linear rst order ordinary dierential systems with interface spatial conditions, J. Appl. Math. Stoch. Anal. 9 (1996), 303-314.
[32] M. Venkatesulu and T.Gnana Bhaskar, Solutions of initial value problems asso- ciated with a pair of mixed linear ordinary dierential equations, J. Math. Anal. Appl. 148 (1990), 63-78.
[33] D.K.K.Vamsi, A study of pair of Dynamic equations with singular interface, M.Phil. Thesis, Sri Sathya Sai Institute of Higher Learning, 2008.
[34] D.K.K.Vamsi and Pallav Kumar Baruah, Existential Results for Nonlinear Sin- gular Interface Problems Involving Second Order Nonlinear Dynamic Equations Using Picards Iterative Technique, The Journal of Nonlinear Analysis and Ap- plications, Vol. 4, No. 3 , (2011), 200-209.
[35] D.K.K.Vamsi and Pallav Kumar Baruah, Disconjugacy(D) and Non- Oscillation(N) Domains for Nonlinear Singular Interface Problems on Semi In- nite Time Scales, ISRN Mathematical Analysis, Vol. 2011, 23.
[36] D.K.K.Vamsi and Pallav Kumar Baruah, Existential Results for Nonlinear Sturm-Liouville Singular Interface Problems, International Journal of Mathe-matics Research, Vol. 4, No. 1, (2012), 1-14.
[37] D.K.K.Vamsi and Pallav Kumar Baruah, Existential of Positive Solutions and Eigenvalue Intervals for Nonlinear Sturm Liouville Problems with a Singular In- terface, Electronic Journal of Dierential Equations, Vol. (2012), No. 53, (2012), 1-12.
[38] D.K.K.Vamsi and Pallav Kumar Baruah, Existential of Multiple Positive So- lutions Localization of Eigen Values for a Nonlocal Boundary Value Problem Involving a Pair of Second Order Dynamic Equations with a Singular Inter- face, Advances in Theoretical and Applied Mathematics, Vol. 7, No. 3, (2012), 281-294.
[39] D.K.K.Vamsi and Pallav Kumar Baruah, Existential of Multiple Positive Solu- tions Localization of Eigen Values for Nonlinear Sturm Liouville Problems with a Singular Interface using the Green's Matrix, International Journal of Mathe- matical Analysis, Vol. 6, No. 39, (2012), 1891-1910.
[40] D.K.K.Vamsi and Pallav Kumar Baruah, Existential and Uniqueness Results for IVPs associated with Nonlinear Singular Interface Problems on Time Scales, International Journal of Nonlinear Analysis and Applications, Vol. 2012.
[41] D.K.K.Vamsi and Pallav Kumar Baruah, Existential and Uniqueness Results for BVPs associated with Nonlinear Singular Interface Problems on Time Scales, Advanced Dynamical Systems and Applications, Vol. 6, No. 2, (2011), 271-290.
[42] H. Weyl, Uber gewhnliche Dierentialgleichungen mit Singularitten und die zugehrigen Entwicklungen willkrlicher Funktionen (German), Math. Ann. 68 (1910), 220-269.
[43] Chi-Teh Wang, Applied Elasticity, McGraw Hill, 1953.
D. K. K. Vamsi , I. Aditya , KNVSD. Dwarakanath , P. K. Baruah, "Existential and Uniqueness Results for Boundary Value Problems associated with Non-linear Singular Interface Problems on Time Scales using Fixed Point Theorems," International Journal of Mathematics Trends and Technology (IJMTT), vol. 5, no. 2, pp. 112-155, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V5P526