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  • Research Article
  • Open Access

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

Boundary Value Problems20072007:057481

  • Received: 18 December 2006
  • Accepted: 23 April 2007
  • Published:


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.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Iterative Technique


Authors’ Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain


  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 1987,23(7):803-811.MATHMathSciNetGoogle Scholar
  2. Il'in VA, Moiseev EI: Nonlocal boundary-value problem of the secod kind for a Sturm-Liouville operator. Differential Equations 1987,23(8):979-987.MATHMathSciNetGoogle Scholar
  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 1992,168(2):540-551. 10.1016/0022-247X(92)90179-HMATHMathSciNetView ArticleGoogle Scholar
  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 1996,124(4):1117-1126. 10.1090/S0002-9939-96-03403-XMATHMathSciNetView ArticleGoogle Scholar
  5. Ma R: Positive solutions of a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 1999, (34):1-8.Google Scholar
  6. Ma R, Castaneda N:Existence of solutions of nonlinear -point boundary-value problems. Journal of Mathematical Analysis and Applications 2001,256(2):556-567. 10.1006/jmaa.2000.7320MATHMathSciNetView ArticleGoogle Scholar
  7. Coppel WA: Disconjugacy, Lecture Notes in Mathematics. Volume 220. Springer, New York, NY, USA; 1971:iv+148.Google Scholar
  8. Eloe PW, Ahmad B:Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions. Applied Mathematics Letters 2005,18(5):521-527. 10.1016/j.aml.2004.05.009MATHMathSciNetView ArticleGoogle Scholar
  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 1997,77(6):477-479. 10.1002/zamm.19970770618MATHView ArticleGoogle Scholar
  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 2002,82(10):728-731. 10.1002/1521-4001(200210)82:10<728::AID-ZAMM728>3.0.CO;2-RMATHMathSciNetView ArticleGoogle Scholar
  11. Avery R, Henderson J:Existence of three positive pseudo-symmetric solutions for a one-dimensional Laplacian. Journal of Mathematical Analysis and Applications 2003,277(2):395-404. 10.1016/S0022-247X(02)00308-6MATHMathSciNetView ArticleGoogle Scholar
  12. Guo Y, Ge W:Three positive solutions for the one-dimensional Laplacian. Journal of Mathematical Analysis and Applications 2003,286(2):491-508. 10.1016/S0022-247X(03)00476-1MATHMathSciNetView ArticleGoogle Scholar
  13. He X, Ge W:Twin positive solutions for the one-dimensional Laplacian boundary value problems. Nonlinear Analysis 2004,56(7):975-984. 10.1016/ ArticleGoogle Scholar
  14. Li J, Shen J:Existence of three positive solutions for boundary value problems with Laplacian. Journal of Mathematical Analysis and Applications 2005,311(2):457-465. 10.1016/j.jmaa.2005.02.054MATHMathSciNetView ArticleGoogle Scholar
  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 2006,314(2):618-630. 10.1016/j.jmaa.2005.04.012MATHMathSciNetView ArticleGoogle Scholar
  16. Wang Y, Hou C:Existence of multiple positive solutions for one-dimensional Laplacian. Journal of Mathematical Analysis and Applications 2006,315(1):144-153. 10.1016/j.jmaa.2005.09.085MATHMathSciNetView ArticleGoogle Scholar
  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 2005,50(5-6):729-739. 10.1016/j.camwa.2005.04.016MATHMathSciNetView ArticleGoogle Scholar
  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.Google Scholar
  19. Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 1976,18(4):620-709. 10.1137/1018114MATHMathSciNetView ArticleGoogle Scholar
  20. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics. Volume 27. Pitman, Boston, Mass, USA; 1985:x+236.Google Scholar
  21. Nieto JJ, Jiang Y, Jurang Y: Monotone iterative method for functional-differential equations. Nonlinear Analysis 1998,32(6):741-747. 10.1016/S0362-546X(97)00524-5MATHMathSciNetView ArticleGoogle Scholar
  22. Vatsala AS, Yang J: Monotone iterative technique for semilinear elliptic systems. Boundary Value Problems 2005,2005(2):93-106. 10.1155/BVP.2005.93MATHMathSciNetView ArticleGoogle Scholar
  23. Drici Z, McRae FA, Vasundhara Devi J: Monotone iterative technique for periodic boundary value problems with causal operators. Nonlinear Analysis 2006,64(6):1271-1277. 10.1016/ ArticleGoogle Scholar
  24. West IH, Vatsala AS: Generalized monotone iterative method for initial value problems. Applied Mathematics Letters 2004,17(11):1231-1237. 10.1016/j.aml.2004.03.003MATHMathSciNetView ArticleGoogle Scholar
  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 2004,289(2):691-699. 10.1016/j.jmaa.2003.09.020MATHMathSciNetView ArticleGoogle Scholar
  26. Nieto JJ, Rodríguez-López R: Monotone method for first-order functional differential equations. Computers & Mathematics with Applications 2006,52(3-4):471-484. 10.1016/j.camwa.2006.01.012MATHMathSciNetView ArticleGoogle Scholar
  27. Ahmad B, Sivasundaram S: The monotone iterative technique for impulsive hybrid set valued integro-differential equations. Nonlinear Analysis 2006,65(12):2260-2276. 10.1016/ ArticleGoogle Scholar
  28. Nieto JJ: An abstract monotone iterative technique. Nonlinear Analysis 1997,28(12):1923-1933. 10.1016/S0362-546X(97)89710-6MATHMathSciNetView ArticleGoogle Scholar
  29. Liz E, Nieto JJ: An abstract monotone iterative method and applications. Dynamic Systems and Applications 1998,7(3):365-375.MATHMathSciNetGoogle Scholar


© B. Ahmad and J.J. Nieto 2007

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