Solvability for a nonlinear third-order three-point boundary value problem

Year 2018, Volume: 1 Issue: 2, 125 - 131, 26.06.2018

Djourdem Habib Slimane Benaicha

https://doi.org/10.32323/ujma.400179

Cited By: 3

Abstract

In this article, the existence of positive solutions for a nonlinear third-order three-point boundary value problem with integral condition is investigated. By using Leray-Schauder fixed point theorem, sufficient conditions for the existence of at least one positive solution are obtained. Illustrative examples are also presented to show the applicability of our results.

Keywords

Positive solutions, nonlocal integral boundary value problem, Concavity, Leray-Schauder fixed point theorem

References

• [1] A. Granas, J. Dugundji, Fixed point theory, Springer-Verlag, New York, (2003).
• [2] A. Guezane-Lakoud and A. Frioui, Existence of solutions of a nonlinear third order boundary value problem, Fixed Point Theory, 13(2012), 501-506.
• [3] A. Lomtatidze and L. Malaguti, ”On a nonlocal boundary value problem for second order nonlinear singular differential equations,” Georgian Mathematical Journal, vol. 7, no. 1, pp. 133–154, 2000.
• [4] A. P. Palamides, Nikolaos M. Stavrakakis, Existence and uniqueness of a positive solution for a third-order three-point boundary-value problem, Electron. J. Differential Equations 155 (2010), 1-12.
• [5] A. Rezaiguia, S. Kelaiaia, Existence of a positive solution for a third-order three point boundary value problem. Matematicki Vesnik, 68(2016), 12–25.
• [6] B.W. Niu, J.P. Sun and Q. Y. Ren, Two positive solutions of third-order BVP with integral boundary condition and sign-changing green’s function, Volume 2015, Article ID 491423, 8 pages.
• [7] F. Haddouchi, S. Benaicha; Multiple positive solutions for a nonlinear three-point integral boundary-value problem. Int. J. Open Problems Compt. Math. 8(2015), 29-42.
• [8] F. Haddouchi, S. Benaicha, Positive solutions of a nonlinear three-point eigenvalue problem with integral boundary conditions, Romanian Journal of Mathematics and Computer Science. 5(2015), 202-213.
• [9] G. L. Karakostas and P. Ch. Tsamatos, “Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, ”Electronic Journal of Differential Equations, vol. 2002, no. 30, pp. 1–17, 2002.
• [10] H. Djourdem, S. Benaicha, Existence of positive solutions for a nonlinear three-point boundary value problem with integral boundary conditions. Acta Math. Univ. Comenianae, (2018)(to appear).
• [11] J. M. Gallardo, “Second-order differential operators with integral boundary conditions and generation of analytic semigroups,” Te Rocky Mountain Journal of Mathematics. 30(2000), 1265–1291.
• [12] J.R.Graef, B.Yang: Positive solutions of a third order nonlocal boundary value problem. Discrete Contin. Dyn. Syst. Ser. (2008), 89–97.
• [13] J. P. Sun and H. B. Li, Monotone positive solution of nonlinear third-order BVP with integral boundary conditions, Boundary Value Problems, Volume 2010, Article ID 874959, 12 pages.
• [14] L. J. Gao, J. P. Sun, Positive solutions of a third-Order three-Point BVP with sign-changing green’s function. Mathematical Problems in Engineering, Volume 2014, Article ID 406815, 6 pages.
• [15] M.R. Grossinho, F. Minhos, Existence result for some third order separated boundary value problems, Nonlinear Anal. 47 (2001), 2407–2418.
• [16] N. Bouteraa, S. Benaicha, Triple Positive solutions of higher-order nonlinear boundary value problems. Journal of Computer Science and Computational Mathematics, Volume 7, Issue 2, June 2017, 25-31.
• [17] S. Benaicha and F. Hadouchi, Positive solutions of a nonlinear fourth-order integral boundary value problem, Annals of West University of Timisoara - Mathematics and Computer Science, 54, (2016), 73- 86.
• [18] S. H. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem. J. Math. Anal. Appl. 323(2006), 413–425.
• [19] X. Liu, D. Ma, The existence of positive solution for a third-order two-point boundary value problem with integral boundary conditions, Scientific Journal of Mathematics Research. 4, February 2014, 1–7.
• [20] Y. Sun, Existence of triple positive solutions for a third-order three-point boundary value problem. Journal of Computational and Applied Mathematics 221 (2008) 194–201.