Existence of Positive Solutions for Higher Order Three-Point Boundary Value Problems on Time Scales

Abstract

In this paper, by using the four functionals fixed point theorem, Avery-Henderson fixed point theorem and the five functionals fixed point theorem, respectively, we investigate the conditions for the existence of at least one, two and three positive solutions to nonlinear higher order three-point boundary value problems on time scales.

Keywords

• Boundary value problems,fixed point theorems,positive solutions,time scales

References

1. [1] Hilger, S., “Analysis on measure chains-A unified approach to continuous and discrete calculus”, Results Math., 18 (1990): 18-56.
2. [2] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
3. [3] Bohner, M. and Peterson, A. (editors), Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
4. [4] Neuberger, J. W., “The lack of self-adjointness in three point boundary value problems”,Pacific J. Math., 18 (1966): 165-168.
5. [5] Eloe, P. W. and McKelwey, J., “Positive solutions of three point boundary value problems”, Comm. Appl. Nonlinear Anal., 4 (1997): 45-54.
6. [6] Agarwal, R. P., O'Regan, D. and Yan, B., “Positive solutions for singular three-point boundary value problems”, Electron. J. Differential Equations, 2008 (2008): 1-20.
7. [7] Karaca, I. Y., “Discrete third-order three-point boundary value problem”, J. Comput. Appl. Math. 205 (2007): 458–468.
8. [8] Karaca, I. Y., “Positive solutions of an n th order three-point boundary value problem”, Rocky Mountain J. Math., 43 (2013): 205-224.

9. [9] Fan, J. and Han, F., “Existence of positive solutions to a three-point boundary value problem for second order dynamic equations with derivative on time scales”, Ann. Differential Equations, 30 (2014): 282–290.
10. [10] Guo, M., “Existence of positive solutions of p-Laplacian three-point boundary value problems on time scales”, Math. Comput. Modelling, 50 (2009): 248–253.
11. [11] Murty, K. N., Saryanayana, R. and Gopalarao, Ch., “Three-point boundary value problems on time scale dynamical systems-existence and uniqueness”, Bull. Calcutta Math. Soc., 106 (2014): 369–380.
12. [12] Prasad, K. R., Murali, P. and Rao, S. N., “Existence of multiple positive solutions to three-point boundary value problems on time scales”, Int. J. Difference Equ., 4 (2009): 219–232.
13. [13] Prasad, K. R., Sreedhar, N. and Srinivas, M. A. S., “Eigenvalue intervals for iterative systems of nonlinear three-point boundary value problems on time scales”, Nonlinear Stud., 22 (2015): 419–431.
14. [14] Sun, J., “Existence of positive solution to second-order three-point BVPs on time scales”, Bound. Value Probl,. Art. ID 685040 (2009): 1-6.
15. [15] Wang, D., “Three positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales”, Nonlinear Analysis, 68 (2008): 2172-2180.
16. [16] Yaslan, İ. and Liceli, O., “Three-point boundary value problems with delta Riemann-Liouville fractional derivative on time scales”, Fract. Differ. Calc., 6 (2016): 1–16.
17. [17] Zhao, B. and Sun, H., “Multiplicity results of positive solutions for nonlinear three-point boundary value problems on time scales”, Adv. Dyn. Syst. Appl., 4 (2009): 243–253.
18. [18] Anderson, D. R. and Avery, R. I., “An even-order three-point boundary value problem on time scales”, J. Math. Anal. Appl., 291 (2004): 514-525.
19. [19] Anderson, D. R. and Karaca, I. Y., “Higher-order three-point boundary value problem on time scales”, Comput. Math. Appl., 56 (2008): 2429-2443.
20. [20] Sang, Y., “Solvability of a higher-order three-point boundary value problem on time scales”, Abstract and Applied Analysis, Art. ID 341679 (2009): 1-16.
21. [21] Yaslan, İ., “Existence results for an even-order boundary value problem on time scales”, Nonlinear Analysis, 70 (2009): 483-491.
22. [22] Sang, Y., “Some new existence results of positive solutions to an even-order boundary value problem on time scales”, Abstract and Applied Analysis, Art. ID 314382 (2013): 1-9.
23. [23] Yaslan, İ., “Multiple positive solutions for a higher order boundary value problem on time scales”, Fixed Point Theory, 17 (2016): 201-214.
24. [24] Avery, R. I., Henderson. J. and O'Regan, D., “Four functionals fixed point theorem”, Math. Comput. Modelling, 48 (2008): 1081-1089.
25. [25] Avery, R. I. and Henderson, J., “Two positive fixed points of nonlinear operators on ordered Banach spaces”, Comm. Appl. Nonlinear Anal., 8 (2001): 27-36.
26. [26] Avery, R. I., “A generalization of the Legget-Williams fixed point theorem”, Math. Sci.Research Hot-Line, 3 (1999): 9-14.