ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Existence of Three Positive Solutions for Nonlinear Third Order Arbitrary Two-point Boundary Value Problems

### Author(s):

Associate Professor
Department of Mathematics, Islamic University
Teacher and M. Phil. degree holder

Md Zulfikar Ali

Professor
Department of Mathematics, University of Rajshahi
Teacher and PhD degree holder

### Abstract:

In this paper, we establish the existence criteria for at least three positive solutions of an arbitrary two-point boundary value problem for nonlinear third order ordinary differential equation. The analysis of this paper is based on Leggett-William’s fixed point theorem and Krasnoselskii’s fixed point theorem. Some new existence and multiplicity results for nonlinear third order ordinary differential equation with arbitrary two-point boundary value conditions are obtained. The results of this paper extend and modify the corresponding results of several authors in literature. Some illustrative examples are given to support the analytic proof.

### Keywords

• arbitrary two-point boundary value problem
• Leggett-William’s fixed point theorem
• positive solution
• the Krasnoselskii fixed point theorem

### References:

1. D. R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl. 288 (2003), 1-14
2. D. R. Anderson, J. M. Davis, Multiple solutions and eigenvalues for three-order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), 135-157
3. D. R. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Model. 27 (1998), 49-57
4. D. R. Anderson, R. I. Avery, A. C. Peterson, Three positive solutions to a discrete focal boundary value problem, J. Comput. Appl. Math. 88 (1998), 102-118
5. R. P. Agarwal, D. O’Regan, A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. Math. Comput. 161 (2005), 433-439
6. R. P. Agarwal, D. O’Regan, P. J. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic, Boston(1999)
7. Y. Feng, S. Liu, Solvability of a third-order two-point boundary value problem, Appl. Math. Lett. 18(9) (2005), 1034-1040
8. M. Fré chet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo22 (1906), 1-74[in French].
9. Y. Guo, F. Yang, Positive solutions for third-order boundary-value problems with the integral boundary conditions and dependence on the first-order derivatives, J. Appl. Math., Hindawi Publishing Corporation Vol. 2013 (2013): Art. ID 721909, 6 pages
10. A. Guezane-Lakoud, N. Hamidane, R. Khaldi, On a third-order three-point boundary value problem, Int. J. Math. Mathematical Sciences, Hindawi Publishing Corporation Vol. 2012 (2012): Art. ID 513189, 7 pages
11. L. J. Guo, J. P. Sun, Y. H. Zhao, Existence of positive solutions for nonlinear third-order three point boundary value problems, Nonlinear Anal. 68(10) (2008), 3151-3158
12. M. Gregus, Third order linear differential equations, Int. Math. Appl. Reidel, Dordrecht: (1987)
13. T. Hu, Y. Sun, Existence and uniqueness of positive solution for third-order three-point boundary value problems, Adv. Pure Math. 4 (2014), 82-288
14. M. A. Krasnosel’skii, Positive solutions of operator equations, Noordhoff, Groningen (1964)
15. Z. Liu, S. M. Kang, J. S. Ume, Triple positive solutions of nonlinear third order boundary value problems, Taiwanese J. Math. 13(3) (2009), 955-971
16. Z. Liu, L. Debnath and S. M. Kang, Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem, Comput. Math. Appl. 55 (2008), 356-367
17. R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673-688
18. N. Nyamoradi, Existence of positive solutions for third-order boundary value Problems, The J. Math. Comput. Sci. 4(1) (2012), 8-18
19. D. J. O’Reagan, Topological transversality: applications to third order boundary value problems, SIAM J. Math. Anal. 19(1987), 630-641
20. B. Sun, J. Li, Positive solutions to a third-order boundary value problem with Riemann Stieltjes integral boundary conditions, IOP Conf. Series: Journal of Physics: Conf. Series 1039 (2018) 012004
21. Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306(2) (2005), 589-603
22. Z. Wei, Some necessary and sufficient conditions for existence of positive solutions for third order singular sublinear multi-point boundary value problems, Acta Math. Sci. 34(6) (2014), 1795-1810
23. P. J. Y. Wong, R. P. Agarwal, Multiple positive solutions of two-point right focal boundary value problems, Math. Comput. Model. 28 (1998), 41-49
24. W. Xie, H. Pang, The shooting method and integral boundary value problems of third-order differential equation, Adv. Difference Equ. 138 (2016)
25. F. Xu, F. Xu, Positive solutions of singular third-order three-point boundary value problems, Inter. J. Math. Anal. 3(31) (2009), 1539-1548
26. Q. L. Yao, Positive solutions of a weak semipositone third-order three-point boundary value problem, J. Math. Research and Exposition 30(1) (2010), 173-180
27. Q. L. Yao and Y. Feng, The existence of solution for a third-order two-point boundary value problem, Appl. Math. Lett. 15 (2002), 227-232
28. D. R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl. 288 (2003), 1-14
29. D. R. Anderson, J. M. Davis, Multiple solutions and eigenvalues for three-order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), 135-157
30. D. R. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Model. 27 (1998), 49-57
31. D. R. Anderson, R. I. Avery, A. C. Peterson, Three positive solutions to a discrete focal boundary value problem, J. Comput. Appl. Math. 88 (1998), 102-118
32. R. P. Agarwal, D. O’Regan, A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. Math. Comput. 161 (2005), 433-439
33. R. P. Agarwal, D. O’Regan, P. J. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic, Boston(1999)
34. Y. Feng, S. Liu, Solvability of a third-order two-point boundary value problem, Appl. Math. Lett. 18(9) (2005), 1034-1040
35. M. Fré chet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo22 (1906), 1-74[in French].
36. Y. Guo, F. Yang, Positive solutions for third-order boundary-value problems with the integral boundary conditions and dependence on the first-order derivatives, J. Appl. Math., Hindawi Publishing Corporation Vol. 2013 (2013): Art. ID 721909, 6 pages
37. A. Guezane-Lakoud, N. Hamidane, R. Khaldi, On a third-order three-point boundary value problem, Int. J. Math. Mathematical Sciences, Hindawi Publishing Corporation Vol. 2012 (2012): Art. ID 513189, 7 pages
38. L. J. Guo, J. P. Sun, Y. H. Zhao, Existence of positive solutions for nonlinear third-order three point boundary value problems, Nonlinear Anal. 68(10) (2008), 3151-3158
39. M. Gregus, Third order linear differential equations, Int. Math. Appl. Reidel, Dordrecht: (1987)
40. T. Hu, Y. Sun, Existence and uniqueness of positive solution for third-order three-point boundary value problems, Adv. Pure Math. 4 (2014), 82-288
41. M. A. Krasnosel’skii, Positive solutions of operator equations, Noordhoff, Groningen (1964)
42. Z. Liu, S. M. Kang, J. S. Ume, Triple positive solutions of nonlinear third order boundary value problems, Taiwanese J. Math. 13(3) (2009), 955-971
43. Z. Liu, L. Debnath and S. M. Kang, Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem, Comput. Math. Appl. 55 (2008), 356-367
44. R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673-688
45. N. Nyamoradi, Existence of positive solutions for third-order boundary value Problems, The J. Math. Comput. Sci. 4(1) (2012), 8-18
46. D. J. O’Reagan, Topological transversality: applications to third order boundary value problems, SIAM J. Math. Anal. 19(1987), 630-641
47. B. Sun, J. Li, Positive solutions to a third-order boundary value problem with Riemann Stieltjes integral boundary conditions, IOP Conf. Series: Journal of Physics: Conf. Series 1039 (2018) : (2018) 012004
48. Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306(2) (2005), 589-603
49. Z. Wei, Some necessary and sufficient conditions for existence of positive solutions for third order singular sublinear multi-point boundary value problems, Acta Math. Sci. 34(6) (2014), 1795-1810
50. P. J. Y. Wong, R. P. Agarwal, Multiple positive solutions of two-point right focal boundary value problems, Math. Comput. Model. 28 (1998), 41-49
51. W. Xie, H. Pang, The shooting method and integral boundary value problems of third-order differential equation, Adv. Difference Equ. 138 (2016)
52. F. Xu, F. Xu, Positive solutions of singular third-order three-point boundary value problems, Inter. J. Math. Anal. 3(31) (2009), 1539-1548
53. Q. L. Yao, Positive solutions of a weak semipositone third-order three-point boundary value problem, J. Math. Research and Exposition 30(1) (2010), 173-180
54. Q. L. Yao and Y. Feng, The existence of solution for a third-order two-point boundary value problem, Appl. Math. Lett. 15 (2002), 227-232