International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales

  • K.R. Prasad1*
  • MD. Khuddush1
  1. Andhra University

* Corresponding author

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.

How to Cite

K. R. Prasad, M. 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 (2017), 198–210. https://doi.org/10.28924/2291-8639-15-2017-198

BibTeX
@article{Prasad2017,
  author  = {Prasad, K.R. and Khuddush, MD.},
  title   = {Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales},
  journal = {Int. J. Anal. Appl.},
  volume  = {15},
  year    = {2017},
  pages   = {198--210},
  doi     = {10.28924/2291-8639-15-2017-198}
}

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