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

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.

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.

Cite this paper

nullP. Palamides and E. Papageorgiou, "Approach to a Fifth-Order Boundary Value Problem, via Sperner's Lemma,"*Applied Mathematics*, Vol. 2 No. 8, 2011, pp. 993-998. doi: 10.4236/am.2011.28137.

nullP. Palamides and E. Papageorgiou, "Approach to a Fifth-Order Boundary Value Problem, via Sperner's Lemma,"

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

[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