Open Access Research Article

The Monotone Iterative Technique for Three-Point Second-Order Integrodifferential Boundary Value Problems with -Laplacian

Bashir Ahmad1* and Juan J Nieto2

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain

For all author emails, please log on.

Boundary Value Problems 2007, 2007:057481 doi:10.1155/2007/57481


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/057481


Received:18 December 2006
Revisions received:1 February 2007
Accepted:23 April 2007
Published:5 June 2007

© 2007 Ahmad and Nieto

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A monotone iterative technique is applied to prove the existence of the extremal positive pseudosymmetric solutions for a three-point second-order -Laplacian integrodifferential boundary value problem.

References

  1. Il'in, VA, Moiseev, EI: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations. 23(7), 803–811 (1987)

  2. Il'in, VA, Moiseev, EI: Nonlocal boundary-value problem of the secod kind for a Sturm-Liouville operator. Differential Equations. 23(8), 979–987 (1987)

  3. Gupta, CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. Journal of Mathematical Analysis and Applications. 168(2), 540–551 (1992). Publisher Full Text OpenURL

  4. Lian, WC, Wong, FH, Yeh, CC: On the existence of positive solutions of nonlinear second order differential equations. Proceedings of the American Mathematical Society. 124(4), 1117–1126 (1996). Publisher Full Text OpenURL

  5. Ma, R: Positive solutions of a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations.(34), 1–8 (1999)

  6. Ma, R, Castaneda, N: Existence of solutions of nonlinear -point boundary-value problems. Journal of Mathematical Analysis and Applications. 256(2), 556–567 (2001). Publisher Full Text OpenURL

  7. Coppel, WA: Disconjugacy, Lecture Notes in Mathematics,p. iv+148. Springer, New York, NY, USA (1971)

  8. Eloe, PW, Ahmad, B: Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions. Applied Mathematics Letters. 18(5), 521–527 (2005). Publisher Full Text OpenURL

  9. Wang, J-Y, Zheng, D-W: On the existence of positive solutions to a three-point boundary value problem for the one-dimensional Laplacian. Zeitschrift für Angewandte Mathematik und Mechanik. 77(6), 477–479 (1997). PubMed Abstract | Publisher Full Text OpenURL

  10. He, X, Ge, W: A remark on some three-point boundary value problems for the one-dimensional Laplacian. Zeitschrift für Angewandte Mathematik und Mechanik. 82(10), 728–731 (2002). PubMed Abstract | Publisher Full Text OpenURL

  11. Avery, R, Henderson, J: Existence of three positive pseudo-symmetric solutions for a one-dimensional Laplacian. Journal of Mathematical Analysis and Applications. 277(2), 395–404 (2003). Publisher Full Text OpenURL

  12. Guo, Y, Ge, W: Three positive solutions for the one-dimensional Laplacian. Journal of Mathematical Analysis and Applications. 286(2), 491–508 (2003). Publisher Full Text OpenURL

  13. He, X, Ge, W: Twin positive solutions for the one-dimensional Laplacian boundary value problems. Nonlinear Analysis. 56(7), 975–984 (2004). Publisher Full Text OpenURL

  14. Li, J, Shen, J: Existence of three positive solutions for boundary value problems with Laplacian. Journal of Mathematical Analysis and Applications. 311(2), 457–465 (2005). Publisher Full Text OpenURL

  15. Wang, Z, Zhang, J: Positive solutions for one-dimensional Laplacian boundary value problems with dependence on the first order derivative. Journal of Mathematical Analysis and Applications. 314(2), 618–630 (2006). Publisher Full Text OpenURL

  16. Wang, Y, Hou, C: Existence of multiple positive solutions for one-dimensional Laplacian. Journal of Mathematical Analysis and Applications. 315(1), 144–153 (2006). Publisher Full Text OpenURL

  17. Ma, D-X, Du, Z-J, Ge, W-G: Existence and iteration of monotone positive solutions for multipoint boundary value problem with Laplacian operator. Computers & Mathematics with Applications. 50(5-6), 729–739 (2005). PubMed Abstract | Publisher Full Text OpenURL

  18. Ma, D-X, Ge, W: Existence and iteration of positive pseudo-symmetric solutions for a three-point second order Laplacian BVP. Applied Mathematics Letters (2007)

  19. Amann, H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review. 18(4), 620–709 (1976). Publisher Full Text OpenURL

  20. Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics,p. x+236. Pitman, Boston, Mass, USA (1985)

  21. Nieto, JJ, Jiang, Y, Jurang, Y: Monotone iterative method for functional-differential equations. Nonlinear Analysis. 32(6), 741–747 (1998). Publisher Full Text OpenURL

  22. Vatsala, AS, Yang, J: Monotone iterative technique for semilinear elliptic systems. Boundary Value Problems. 2005(2), 93–106 (2005). Publisher Full Text OpenURL

  23. Drici, Z, McRae, FA, Vasundhara Devi, J: Monotone iterative technique for periodic boundary value problems with causal operators. Nonlinear Analysis. 64(6), 1271–1277 (2006). Publisher Full Text OpenURL

  24. West, IH, Vatsala, AS: Generalized monotone iterative method for initial value problems. Applied Mathematics Letters. 17(11), 1231–1237 (2004). Publisher Full Text OpenURL

  25. Jiang, D, Nieto, JJ, Zuo, W: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations. Journal of Mathematical Analysis and Applications. 289(2), 691–699 (2004). Publisher Full Text OpenURL

  26. Nieto, JJ, Rodríguez-López, R: Monotone method for first-order functional differential equations. Computers & Mathematics with Applications. 52(3-4), 471–484 (2006). PubMed Abstract | Publisher Full Text OpenURL

  27. Ahmad, B, Sivasundaram, S: The monotone iterative technique for impulsive hybrid set valued integro-differential equations. Nonlinear Analysis. 65(12), 2260–2276 (2006). Publisher Full Text OpenURL

  28. Nieto, JJ: An abstract monotone iterative technique. Nonlinear Analysis. 28(12), 1923–1933 (1997). Publisher Full Text OpenURL

  29. Liz, E, Nieto, JJ: An abstract monotone iterative method and applications. Dynamic Systems and Applications. 7(3), 365–375 (1998)