Title: Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales
Author(s): K.R. Prasad, MD. Khuddush
Pages: 198-210
Cite as:
K.R. Prasad, MD. Khuddush, Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales, Int. J. Anal. Appl., 15 (2) (2017), 198-210.

Abstract


In this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using Hölder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.

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References


  1. R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications. Result Math, 35(1999), 3–22.

  2. R. P. Agarwal, V. Otero-Espinar, K. Perera and D.R. Vivero, Basic properties of Sobolev’s spaces on time scales. Advan. Diff. Eqns., 2006(2006), Art. ID 038121.

  3. D. R. Anderson, I. Y. Karaca, Higher-order three-point boundary value problem on time scales. Comput. Math. Appl.56(2008), 2429–2443.

  4. G. A. Anastassiou, Intelligent mathematics: computational analysis. Vol. 5. Heidelberg: Springer, 2011.

  5. M. Bohner and H. Luo, Singular second-order multipoint dynamic boundary value problems with mixed derivatives, Adv. Diff. Eqns., 2006(2006), Art. ID 054989.

  6. M. Bohner, and A. Peterson, Dynamic equations on time scales: An introduction with applications. Birkhauser, Boston, (2001).

  7. M. Bohner, and A. Peterson, Advances in Dynamic Equations on Time Scales. Birkhauser, Boston, (2003).

  8. E. Cetin, F. S. Topal, Existence results for solutions of integral boundary value problems on time scales, Abstr. Appl. Anal., 2013(2013), Art. ID 708734.

  9. F. T. Fen, and I. Y. Karaca, Existence of positive solutions for nonlinear second-order impulsive boundary value problems on time scales, Med. J. Math., 13(2016), 191-204.

  10. M. Feng, Existence of symmetric positive solutions for boundary value problem with integral boundary conditions, Appl. Math. Lett., 24(2011), 1419-1427.

  11. J. M. Gallardo, Second-order differential operators with integral boundary conditions and generation of analytic semigroups, Rocky Mountain J. Math., 30(2000), No. 4, 1265-1292.

  12. C. S. Goodrich, Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale, Comment. Math. Univ. Carol., 54(2013), No. 4, 509-525.

  13. C. S. Goodrich, On a first-order semipositone boundary value problem on a time scale, Appl. Anal. Disc. Math., 8 (2014), 269-287.

  14. D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Academic press, New York, 1988.

  15. Y. Guo, Y. Liu and Y. Liang, Positive solutions for the third order boundary value problems with the second derivatives, Bound. value probl., 2012(2012), Art. ID 34.

  16. G. S. Guseinov, Integration on time scales. J. Math. Anal. App., 285(2003), No. 1, 107–127.

  17. N. A. Hamal and F. Yoruk, Symmetric positive solutions of fourth order integral BVP for an increasing homeomor- phism and homomorphism with sign-changing nonlinearity on time scales, Comput. Math. Appl., 59(2010), No. 11, 3603-3611.

  18. M. Hu and L. Wang, Triple positive solutions for an impulsive dynamic equation with integral boundary condition on time scales, Inter. J. App. Math. Stat., 31(2013), No. 1, 67-78.

  19. I. Y. Karaca, Positive solutions for boundary value problems of second-order functional dynamic equations on time scales, Adv. Difference Equ. 21(2009) Art. ID 829735.

  20. I. Y. Karaca, F. Tokmak, Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales, J. Ineq. App., 2013(2013), No. 1, 1-12.

  21. G. L. Karakostas and P. Ch. Tsamatos, Multiple positive solutions of some fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 30(2002), No. 30, 1-17.

  22. R. A. Khan, the generalized method of quasi-linearization and nonlinear boundary value problems with integral boundary conditions, Electron. J. Qual. Theory Differential Equations, 19(2003), No. 15, 1-15.

  23. V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer, Dordrecht, (1996).

  24. Y. Li and L. Sun, Infinite many positive solutions for nonlinear first-order BVPs with integral boundary conditions on time scales, Top. Meth. Nonl. Anal., 41(2013), No. 2, 305-321.

  25. Y. Li and L. Wang, Multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales, Adv. Diff. Eqns., 2015(2015), Art. ID 90.

  26. Y. Li and T. Zhang, Multiple Positive Solutions for Second-Order p-Laplacian Dynamic Equations with Integral Boundary Conditions, Boun. Val. Prob., 2011(2010), Art. ID 19.

  27. A. Lomtatidze and L. Malaguti, On a nonlocal boundary-value problems for second order nonlinear singular differ- ential equations, Georgian Math. J., 7(2000), 133-154.

  28. A. D. Oguz and F. S. Topal, Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales, J. Appl. Anal and Comp., 6(2016), No. 2, 531–542.

  29. U. M. Ozkan and M. Z. Sarikaya and H. Yildirim, Extensions of certain integral inequalities on time scales. Appl. Math. Let., 21(2008), No. 10, 993–1000.

  30. B. P. Rynne, L2 spaces and boundary value problems on time-scales. J. Math. Anal. App., 328(2007), No. 2, 1217-1236.

  31. N. Sreedhar, V. V. R. R. B. Raju and Y. Narasimhulu, Existence of positive solutions for higher order boundary value problems with integral boundary conditions on time scales. J. Nonlinear Funct. Anal., 2017(2017), Article ID 5, 1-13.

  32. P . Thiramanus, and T. Jessada, Positive solutions of m-point integral boundary value problems for second-order p-Laplacian dynamic equations on time scales, Adv. Diff. Eqns., 2013(2013), Art. ID 206.

  33. S. P. Timoshenko and J. M. Gere, Theory of elastic stability, McGraw-Hill, New York(1961).

  34. S. G. Topal and A. Denk, Existence of symmetric positive solutions for a semipositone problem on time scales, Hacet. J. Math. Stat., 45(2016), No. 1, 23–31.

  35. P. A. Williams, Unifying fractional calculus with time scales

  36. [Ph.D. thesis], University of Melbourne, (2012).

  37. X. Zhang and M. Feng, W. Ge, Existence of solutions of boundary value problems with integral boundary conditions for second order impulsive integro-differential equations in Banach spaces, Comput. Appl. Math., 233(2010), 1915- 1926.