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**Md Asaduzzaman**

Associate Professor

Department of Mathematics, Islamic University

Kushtia-7003, Bangladesh

Teacher and M. Phil. degree holder

**Md Zulfikar Ali**

Professor

Department of Mathematics, University of Rajshahi

Rajshahi-6205, Bangladesh

Teacher and PhD degree holder

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.

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

- D. R. Anderson,
*Green’s function for a third-order generalized right focal problem*, J. Math. Anal. Appl.**288**(2003), 1-14 - 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 - D. R. Anderson,
*Multiple positive solutions for a three-point boundary value problem*, Math. Comput. Model.**27**(1998), 49-57 - 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 - 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 - R. P. Agarwal, D. O’Regan, P. J. Wong,
*Positive solutions of differential, difference and integral equations*, Kluwer Academic, Boston(1999) - Y. Feng, S. Liu,
*Solvability of a third-order two-point boundary value problem*, Appl. Math. Lett.**18(9)**(2005), 1034-1040 - M. Fré chet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo
**22**(1906), 1-74[in French]. - 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 - 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 - 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 - M. Gregus,
*Third order linear differential equations*, Int. Math. Appl. Reidel, Dordrecht: (1987) - 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 - M. A. Krasnosel’skii,
*Positive solutions of operator equations*, Noordhoff, Groningen (1964) - 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 - 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 - 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 - N. Nyamoradi,
*Existence of positive solutions for third-order boundary value Problems*, The J. Math. Comput. Sci.**4(1)**(2012), 8-18 - D. J. O’Reagan,
*Topological transversality: applications to third order boundary value problems*, SIAM J. Math. Anal.**19**(1987), 630-641 - 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 - Y. Sun,
*Positive solutions of singular third-order three-point boundary value problem*, J. Math. Anal. Appl.**306(2)**(2005), 589-603 - 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 - 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 - W. Xie, H. Pang,
*The shooting method and integral boundary value problems of third-order differential equation*, Adv. Difference Equ.**138**(2016) - F. Xu, F. Xu,
*Positive solutions of singular third-order three-point boundary value problems*, Inter. J. Math. Anal.**3(31)**(2009), 1539-1548 - 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 - 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 - D. R. Anderson,
*Green’s function for a third-order generalized right focal problem*, J. Math. Anal. Appl.**288**(2003), 1-14 - 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 - D. R. Anderson,
*Multiple positive solutions for a three-point boundary value problem*, Math. Comput. Model.**27**(1998), 49-57 - 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 - 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 - R. P. Agarwal, D. O’Regan, P. J. Wong,
*Positive solutions of differential, difference and integral equations*, Kluwer Academic, Boston(1999) - Y. Feng, S. Liu,
*Solvability of a third-order two-point boundary value problem*, Appl. Math. Lett.**18(9)**(2005), 1034-1040 - M. Fré chet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo
**22**(1906), 1-74[in French]. - 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 - 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 - 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 - M. Gregus,
*Third order linear differential equations*, Int. Math. Appl. Reidel, Dordrecht: (1987) - 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 - M. A. Krasnosel’skii,
*Positive solutions of operator equations*, Noordhoff, Groningen (1964) - 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 - 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 - 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 - N. Nyamoradi,
*Existence of positive solutions for third-order boundary value Problems*, The J. Math. Comput. Sci.**4(1)**(2012), 8-18 - D. J. O’Reagan,
*Topological transversality: applications to third order boundary value problems*, SIAM J. Math. Anal.**19**(1987), 630-641 - 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 - Y. Sun,
*Positive solutions of singular third-order three-point boundary value problem*, J. Math. Anal. Appl.**306(2)**(2005), 589-603 - 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 - 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 - W. Xie, H. Pang,
*The shooting method and integral boundary value problems of third-order differential equation*, Adv. Difference Equ.**138**(2016) - F. Xu, F. Xu,
*Positive solutions of singular third-order three-point boundary value problems*, Inter. J. Math. Anal.**3(31)**(2009), 1539-1548 - 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 - 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