Existence of Multiple Positive Solutions for *n*^{th} Order Two-Point Boundary Value Problems on Time Scales

Affiliation(s)

Department of Applied Mathematics Andhra University, Visakhapatnam, India.

Department of Mathematics, VITAM College of Engineering, Visakhapatnam, India.

Department of Applied Mathematics Andhra University, Visakhapatnam, India.

Department of Mathematics, VITAM College of Engineering, Visakhapatnam, India.

ABSTRACT

We consider the*n*^{th} order nonlinear differential equation on time scales

We consider the

subject to the right focal type two-point boundary conditions

We establish a criterion for the existence of at least one positive solution by utilizing Krasnosel’skii fixed point theorem. And then, we establish the existence of at least three positive solutions by utilizing Leggett-Williams fixed point theorem.

Cite this paper

K. Prasad, P. Murali and N. Suryanarayana, "Existence of Multiple Positive Solutions for*n*^{th} Order Two-Point Boundary Value Problems on Time Scales," *Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 70-77. doi: 10.4236/apm.2013.31009.

K. Prasad, P. Murali and N. Suryanarayana, "Existence of Multiple Positive Solutions for

References

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[2] J. P. Keener, “Principles of Applied Mathematics,” Addison-Wesley, Redwood City, 1988.

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[7] P. W. Eloe and J. Henderson, “Positive Solutions and Nonlinear (k,n-k) Conjugate Eigenvalue Problems,” Journal of Differential Equations Dynamical Systems, Vol. 6, 1998, pp. 309-317.

[8] F. M. Atici and G. Sh. Guseinov, “Positive Periodic Solutions for Nonlinear Difference Equations with Periodic Coefficients,” Journal of Mathematical Analysis and Applications, Vol. 232, No. 1, 1999, pp. 166-182. doi:10.1006/jmaa.1998.6257

[9] D. R. Anderson and R. I. Avery, “Multiple Positive Solutions to a Third-Order Discrete Focal Boundary Value Problem,” Journal of Computers and Mathematics with Applications, Vol. 42, No. 3-5, 2001, pp. 333-340. doi:10.1016/S0898-1221(01)00158-4

[10] R. I. Avery and A. C. Peterson, “Multiple Positive Solutions of a Discrete Second Order Conjugate Problem,” Panamerican Mathematical Journal, Vol. 8, No. 3, 1998, pp. 1-12.

[11] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Positive Solutions of Differential, Difference and Integral Equations,” Kluwer Academic Publishers, Dordrecht, 1999.

[12] K. Deimling, “Nonlinear Functional Analysis,” Springer, New York, 1985. doi:10.1007/978-3-662-00547-7

[13] M. Gregus, “Third Order Linear Differential Equations and Mathematical Applications,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3715-4

[14] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” Acadamic Press, San Diego, 1988.

[15] J. Henderson and S. K. Ntouyas, “Positive Solutions for System of nth Order Three-Point Nonlocal Boundary Value Problems,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 18, No. 5, 2007, pp. 1-12.

[16] B. Hopkins and N. Kosmator, “Third Order Boundary Value Problem with Sign-Changing Solution,” Nonlinear Analysis, Vol. 67, No. 1, 2007, pp. 126-137. doi:10.1016/j.na.2006.05.003

[17] S. Li, “Positive Solutions of Nonlinear Singular Third Order Two-Point Boundary Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 413-425. doi:10.1016/j.jmaa.2005.10.037

[18] M. El-Shehed, “Positive Solutions of Boundary Value Problems for n^{th} Order Differential Equations,” Electronic Journal of Qualitative Theory of Differential Equations, No. 1, 2008, pp. 1-9.

[19] K. R. Prasad and P. Murali, “Eigenvalue Intervals for nth Order Differential Equations on Time Scales,” International Journal of Pure and Applied Mathematics, Vol. 44, No. 5, 2008, pp. 737-753.

[20] R. W. Leggett and L. R. Williams, “Multiple Positive Fixed Points of Nonlinear Operator on Order Banach Spaces,” Indiana University Mathematics Journal, Vol. 28, No. 4, 1979, pp. 673-688. doi:10.1512/iumj.1979.28.28046

[1] G. H. Meyer, “Initial Value Methods for Boundary Value Problems,” Journal of Applied Mathematics and Computing, Vol. 158, No. 1, 2004, pp. 345-351.

[2] J. P. Keener, “Principles of Applied Mathematics,” Addison-Wesley, Redwood City, 1988.

[3] P. C. Wayner, Y. K. Kao and L. V. Lacroin, “The Interlimne Heat Transfer Coefficient of an Eveporating Wetting Film,” International Journal of Heat and Mass Transfer, Vol. 19, No. 5, 1976, pp. 487-492. doi:10.1016/0017-9310(76)90161-7

[4] M. Bohner and A. C. Peterson, “Dynamic Equations on Time Scales, an Introduction with Applications,” Birkhauser, Boston, 2001.

[5] L. H. Erbe and H. Wang, “On the Existence of Positive Solutions of Ordinary Differential Equations,” Proceedings of the American Mathematical Society, Vol. 120, No. 3, 1994, pp. 743-748. doi:10.1090/S0002-9939-1994-1204373-9

[6] P. W. Eloe and J. Henderson, “Positive Solutions for (n-1,1) Conjugate Boundary Value Problems,” Nonlinear Analysis, Vol. 28, No. 10, 1997, pp. 1669-1680. doi:10.1016/0362-546X(95)00238-Q

[7] P. W. Eloe and J. Henderson, “Positive Solutions and Nonlinear (k,n-k) Conjugate Eigenvalue Problems,” Journal of Differential Equations Dynamical Systems, Vol. 6, 1998, pp. 309-317.

[8] F. M. Atici and G. Sh. Guseinov, “Positive Periodic Solutions for Nonlinear Difference Equations with Periodic Coefficients,” Journal of Mathematical Analysis and Applications, Vol. 232, No. 1, 1999, pp. 166-182. doi:10.1006/jmaa.1998.6257

[9] D. R. Anderson and R. I. Avery, “Multiple Positive Solutions to a Third-Order Discrete Focal Boundary Value Problem,” Journal of Computers and Mathematics with Applications, Vol. 42, No. 3-5, 2001, pp. 333-340. doi:10.1016/S0898-1221(01)00158-4

[10] R. I. Avery and A. C. Peterson, “Multiple Positive Solutions of a Discrete Second Order Conjugate Problem,” Panamerican Mathematical Journal, Vol. 8, No. 3, 1998, pp. 1-12.

[11] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Positive Solutions of Differential, Difference and Integral Equations,” Kluwer Academic Publishers, Dordrecht, 1999.

[12] K. Deimling, “Nonlinear Functional Analysis,” Springer, New York, 1985. doi:10.1007/978-3-662-00547-7

[13] M. Gregus, “Third Order Linear Differential Equations and Mathematical Applications,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3715-4

[14] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” Acadamic Press, San Diego, 1988.

[15] J. Henderson and S. K. Ntouyas, “Positive Solutions for System of nth Order Three-Point Nonlocal Boundary Value Problems,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 18, No. 5, 2007, pp. 1-12.

[16] B. Hopkins and N. Kosmator, “Third Order Boundary Value Problem with Sign-Changing Solution,” Nonlinear Analysis, Vol. 67, No. 1, 2007, pp. 126-137. doi:10.1016/j.na.2006.05.003

[17] S. Li, “Positive Solutions of Nonlinear Singular Third Order Two-Point Boundary Value Problem,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 413-425. doi:10.1016/j.jmaa.2005.10.037

[18] M. El-Shehed, “Positive Solutions of Boundary Value Problems for n

[19] K. R. Prasad and P. Murali, “Eigenvalue Intervals for nth Order Differential Equations on Time Scales,” International Journal of Pure and Applied Mathematics, Vol. 44, No. 5, 2008, pp. 737-753.

[20] R. W. Leggett and L. R. Williams, “Multiple Positive Fixed Points of Nonlinear Operator on Order Banach Spaces,” Indiana University Mathematics Journal, Vol. 28, No. 4, 1979, pp. 673-688. doi:10.1512/iumj.1979.28.28046