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**On six solutions for \(m\)-point differential equations system with two coupled parallel sub-super solutions.**
*(English)*
Zbl 1241.34022

Summary: Assuming the existence of two coupled parallel sub-super solutions, the existence of at least six solutions for a class of second-order \(m\)-point boundary value problems is obtained by using the fixed point index theory. An example to illustrate our result is given.

### MSC:

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

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\textit{J. Liu} et al., J. Appl. Math. 2012, Article ID 359251, 15 p. (2012; Zbl 1241.34022)

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### References:

[1] | M. Moshinsky, “Sobre los problems de condiciones a la frontiera en una dimension de caracteristicas discontinuas,” Boletín de la Sociedad Matemática Mexicana, vol. 7, pp. 1-25, 1950. |

[2] | S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, NY, USA, 1961. |

[3] | V. A. Il’in and E. I. Moiseev, “onlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects,” Journal of Difference Equations, vol. 23, pp. 803-810, 1987. · Zbl 0668.34025 |

[4] | C. P. Gupta, “Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 540-551, 1992. · Zbl 0763.34009 |

[5] | C. P. Gupta, “A generalized multi-point boundary value problem for second order ordinary differential equations,” Applied Mathematics and Computation, vol. 89, no. 1-3, pp. 133-146, 1998. · Zbl 0910.34032 |

[6] | N. Aykut Hamal and F. Yoruk, “Positive solutions of nonlinear m-point boundary value problems on time scales,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 92-105, 2009. · Zbl 1185.34143 |

[7] | G. Infante and P. Pietramala, “Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 1301-1310, 2009. · Zbl 1169.45001 |

[8] | R. A. Khan and J. R. L. Webb, “Existence of at least three solutions of a second-order three-point boundary value problem,” Nonlinear Analysis, vol. 64, no. 6, pp. 1356-1366, 2006. · Zbl 1101.34005 |

[9] | R. Ma, “Existence of solutions of nonlinear m-point boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556-567, 2001. · Zbl 0988.34009 |

[10] | R. Ma and D. O’Regan, “Solvability of singular second order m-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 124-134, 2005. · Zbl 1062.34018 |

[11] | F. Xu, Z. Chen, and F. Xu, “Multiple positive solutions for nonlinear second-order m-point boundary-value problems with sign changing nonlinearities,” Electronic Journal of Differential Equations, pp. 1-12, 2008. · Zbl 1171.34015 |

[12] | Z. Yang, “Positive solutions to a system of second-order nonlocal boundary value problems,” Nonlinear Analysis, vol. 62, no. 7, pp. 1251-1265, 2005. · Zbl 1089.34022 |

[13] | X. Xian, “Three solutions for three-point boundary value problems,” Nonlinear Analysis, vol. 62, no. 6, pp. 1053-1066, 2005. · Zbl 1076.34011 |

[14] | Z. Zhang and J. Wang, “The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 147, no. 1, pp. 41-52, 2002. · Zbl 1019.34021 |

[15] | J. Sun and K. Zhang, “On the number of fixed points of nonlinear operators and applications,” Journal of Systems Science and Complexity, vol. 16, no. 2, pp. 229-235, 2003. · Zbl 1131.47308 |

[16] | J. Sun and Y. Cui, “Multiple solutions for nonlinear operators and applications,” Nonlinear Analysis, vol. 66, no. 9, pp. 1999-2015, 2007. · Zbl 1120.47050 |

[17] | X. Xian, “Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions,” Nonlinear Analysis, vol. 69, no. 7, pp. 2251-2266, 2008. · Zbl 1165.47044 |

[18] | K. M. Zhang and J. X. Sun, “Multiple solutions for superlinear operator equations in Banach spaces and applications,” Acta Mathematica Sinica, vol. 48, no. 1, pp. 99-108, 2005 (Chinese). · Zbl 1117.47304 |

[19] | X. Xu, D. O’Regan, and J. Sun, “Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition,” Mathematical and Computer Modelling, vol. 45, no. 1-2, pp. 189-200, 2007. · Zbl 1140.34009 |

[20] | Z. Yang, “Existence and nonexistence results for positive solutions of an integral boundary value problem,” Nonlinear Analysis, vol. 65, no. 8, pp. 1489-1511, 2006. · Zbl 1104.34017 |

[21] | J. R. L. Webb and K. Q. Lan, “Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type,” Topological Methods in Nonlinear Analysis, vol. 27, no. 1, pp. 91-115, 2006. · Zbl 1146.34020 |

[22] | R. Ma and D. O’Regan, “Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues,” Nonlinear Analysis, vol. 64, no. 7, pp. 1562-1577, 2006. · Zbl 1101.34006 |

[23] | B. P. Rynne, “Spectral properties and nodal solutions for second-order, m-point, boundary value problems,” Nonlinear Analysis, vol. 67, no. 12, pp. 3318-3327, 2007. · Zbl 1142.34010 |

[24] | L. Kong, Q. Kong, and J. S. W. Wong, “Nodal solutions of multi-point boundary value problems,” Nonlinear Analysis, vol. 72, no. 1, pp. 382-389, 2010. · Zbl 1195.34028 |

[25] | J. R. L. Webb, “Remarks on a non-local boundary value problem,” Nonlinear Analysis, vol. 72, no. 2, pp. 1075-1077, 2010. · Zbl 1186.34029 |

[26] | K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. · Zbl 0559.47040 |

[27] | D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988. · Zbl 0661.47045 |

[28] | D. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, China, 2nd edition, 2001. |

[29] | J. X. Sun, “Nontrivial solutions of superlinear Hammerstein integral equations and their applications,” Chinese Annals of Mathematics. Series A, vol. 7, no. 5, pp. 528-535, 1986 (Chinese). · Zbl 0633.45006 |

[30] | H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620-709, 1976. · Zbl 0345.47044 |

[31] | X.-l. Han, “Positive solutions of a three-point boundary value problem,” Journal of Mathematical Research and Exposition, vol. 27, no. 3, pp. 497-504, 2007. · Zbl 1144.34311 |

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