Approach to a Fifth-Order Boundary Value Problem, via Sperner's Lemma

Panos K. Palamides, Evgenia H. Papageorgiou

.

DOI: 10.4236/am.2011.28137                

Abstract

We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.

Keywords

Fifth-Order Differential Equation, Vector Field, Kneser’s Theorem, Sperner’s Lemma

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. H. Erbe and H. Wang, “On Existence of Positive Solutions of Ordinary Differential Equations,” Proceedings of the American Mathematical Society, Vol. 120, 1994, pp. 743-748. doi:10.1090/S0002-9939-1994-1204373-9
[2]	D. R. Anderson and J. M. Davis, “Multiple Solutions and Eigenvalues for Third-Order Right Focal Boundary-Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 267, No. 1, 2002, pp. 135-157. doi:10.1006/jmaa.2001.7756
[3]	Z. Bai and H. Wang, “On Positive Solutions of Some Nonlinear Fourth-Order Beam Equations,” Journal of Mathematical Analysis and Applications, Vol. 270, No. 2, 2002, pp. 357-368. doi:10.1016/S0022-247X(02)00071-9
[4]	J. R. Graef and B. Yang, “On a Nonlinear Boundary-Value Problem for Fourth Order Equations,” Applied Analysis, Vol. 72, 1999, pp. 439-448. doi:10.1080/00036819908840751
[5]	J. R. Graef and B. Yang, “Positive Solutions to a Multi-Point Higher Order Boundary-Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 316, No. 2, 2006, pp. 409-421. doi:10.1016/j.jmaa.2005.04.049
[6]	J. R. Graef, J. Henderson and B. Yang, “Positive Solutions of a Nonlinear Higher Order Boundary-Value Problem,” Electronic Journal of Differential Equations, Vol. 2007, No. 45, 2007, pp. 1-10.
[7]	P. S. Kelevedjievand P. K. Palamides and N. I. Polivanov “Another Understanding of Fourth-Order Four-Point Boundary-Value Problems,” Electronic Journal of Differential Equations, Vol. 2008, No. 47, 2008, pp. 1-15.
[8]	Z. Hao, L. Liu and L. Debnath, “A Necessary and Sufficiently Condition for the Existence of Positive Solution of Fourth-Order Singular Boundary-Value Problems,” Applied Mathematics Letters, Vol. 16, No. 3, 2003, pp. 279-285. doi:10.1016/S0893-9659(03)80044-7
[9]	J. Ge and C. Bai, “Solvability of a Four-Point Boundary- Value Problem for Fourth-Order Ordinary,” Electronic Journal of Differential Equations, Vol. 2007, No. 123, 2007, pp. 1-9.
[10]	G. G. Doronin and N. A. Larkin, “Boundary Value Problems for the Stationary Kawahara Equation,” Nonlinear Analysis, Vol. 69, No. 5-6, 2008, pp. 1655-1665. doi:10.1016/j.na.2007.07.005
[11]	S. N. Odda, “Existence Solution for 5th Order Differential Equations under Some Conditions,” Applied Mathematics, Vol. 1, No. 4, 2010, pp. 279-282. doi:10.4236/am.2010.14035
[12]	M. El-shahed and S. Al-Mezel, “Positive Solutions for Boundary Value Problems of Fifth-Order Differential Equations,” International Mathematical Forum, Vol. 4, No. 33, 2009, pp. 1635-1640.
[13]	M. A. Noor and S. T. Mohyud-Din, “A New Approach to Fifth-Order Boundary Value Problems,” International Journal of Nonlinear Science, Vol. 7, No. 2, 2009, pp. 143-148.
[14]	M. A. Noor and S. T. Mohyud-Din, “Variational Iteration Method for Fifth-Order Boundary Value Problems Using He’s Polynomials,” Hindawi Publishing Corporation Mathematical Problems in Engineering, Vol. 2008, 2008, Article ID 954794.
[15]	A. Granas and J. Dugundji, “Fixed Point Theory,” Springer-Verlag, New York, 2003.
[16]	W. A. Copel, “Stability and Asymptotic Behavior of Differential Equations,” Heath & Co. Boston, Boston, 1965.
[17]	P. K. Palamides, G. Infante and P. Pietramala, “Nontrivial Solutions of a Nonlinear Heat Flow Problem via Sperner’s Lemma,” Applied Mathematics Letters, Vol. 22, No. 9, 2009, pp. 1444-1450. doi:10.1016/j.aml.2009.03.014