Open Access

Multiple nonnegative solutions for BVPs of fourth-order difference equations

Advances in Difference Equations20062006:089585

DOI: 10.1155/ADE/2006/89585

Received: 31 March 2006

Accepted: 18 September 2006

Published: 7 December 2006


First, existence criteria for at least three nonnegative solutions to the following boundary value problem of fourth-order difference equation Δ4x(t-2) = a(t)f(x(t)),t [2, T], x(0) = x(T + 2) = 0, Δ2x(0) = Δ2x(T) = 0 are established by using the well-known Leggett-Williams fixed point theorem, and then, for arbitrary positive integer m, existence results for at least 2m-1 nonnegative solutions are obtained.


Authors’ Affiliations

Department of Applied Mathematics, Lanzhou University of Technology


  1. Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York; 2000:xvi+971.Google Scholar
  2. Agarwal RP, Bohner M, Wong PJY: Eigenvalues and eigenfunctions of discrete conjugate boundary value problems. Computers & Mathematics with Applications 1999,38(3–4):159–183. 10.1016/S0898-1221(99)00192-3MathSciNetView ArticleMATHGoogle Scholar
  3. Agarwal RP, Henderson J: Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Computers & Mathematics with Applications 1998,36(10–12):347–355.MathSciNetView ArticleMATHGoogle Scholar
  4. Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic, Dordrecht; 1997:viii+507.View ArticleGoogle Scholar
  5. Agarwal RP, Wong F-H: Existence of positive solutions for higher order difference equations. Applied Mathematics Letters 1997,10(5):67–74. 10.1016/S0893-9659(97)00085-2MathSciNetView ArticleMATHGoogle Scholar
  6. Hartman P: Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity. Transactions of the American Mathematical Society 1978, 246: 1–30.MathSciNetMATHGoogle Scholar
  7. Henderson J: Positive solutions for nonlinear difference equations. Nonlinear Studies 1997,4(1):29–36.MathSciNetMATHGoogle Scholar
  8. Henderson J, Wong PJY: On multiple solutions of a system of m discrete boundary value problems. Zeitschrift für Angewandte Mathematik und Mechanik 2001,81(4):273–279. 10.1002/1521-4001(200104)81:4<273::AID-ZAMM273>3.0.CO;2-HMathSciNetView ArticleMATHGoogle Scholar
  9. Liang R-J, Zhao Y-H, Sun J-P: A new theorem of existence to fourth-order boundary value problem. International Journal of Differential Equations and Applications 2003,7(3):257–262.MathSciNetMATHGoogle Scholar
  10. Merdivenci F: Green's matrices and positive solutions of a discrete boundary value problem. Panamerican Mathematical Journal 1995,5(1):25–42.MathSciNetMATHGoogle Scholar
  11. Merdivenci F: Two positive solutions of a boundary value problem for difference equations. Journal of Difference Equations and Applications 1995,1(3):263–270. 10.1080/10236199508808026MathSciNetView ArticleMATHGoogle Scholar
  12. Sun J-P: Positive solution for BVPs of fourth order difference equations. Indian Journal of Pure and Applied Mathematics 2005,36(7):361–370.MathSciNetMATHGoogle Scholar
  13. Sun J-P, Li W-T: Multiple positive solutions of a discrete difference system. Applied Mathematics and Computation 2003,143(2–3):213–221. 10.1016/S0096-3003(02)00354-5MathSciNetView ArticleMATHGoogle Scholar
  14. Wong PJY: Positive solutions of difference equations with two-point right focal boundary conditions. Journal of Mathematical Analysis and Applications 1998,224(1):34–58. 10.1006/jmaa.1998.5983MathSciNetView ArticleMATHGoogle Scholar
  15. Wong PJY, Agarwal RP: Further results on fixed-sign solutions for a system of higher-order difference equations. Computers & Mathematics with Applications 2001,42(3–5):497–514.MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang B, Kong L, Sun Y, Deng X: Existence of positive solutions for BVPs of fourth-order difference equations. Applied Mathematics and Computation 2002,131(2–3):583–591. 10.1016/S0096-3003(01)00171-0MathSciNetView ArticleMATHGoogle Scholar
  17. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Massachusetts; 1988:viii+275.MATHGoogle Scholar
  18. Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen; 1964:381.Google Scholar
  19. Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana University Mathematics Journal 1979,28(4):673–688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar


© Jian-Ping Sun 2006

This article is published under license to BioMed Central Ltd. 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.