- Research Article
- Open Access
Multiple nonnegative solutions for BVPs of fourth-order difference equations
Advances in Difference Equations volume 2006, Article number: 089585 (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.
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.
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)00192-3
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.
Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic, Dordrecht; 1997:viii+507.
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)00085-2
Hartman P: Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity. Transactions of the American Mathematical Society 1978, 246: 1–30.
Henderson J: Positive solutions for nonlinear difference equations. Nonlinear Studies 1997,4(1):29–36.
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-H
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.
Merdivenci F: Green's matrices and positive solutions of a discrete boundary value problem. Panamerican Mathematical Journal 1995,5(1):25–42.
Merdivenci F: Two positive solutions of a boundary value problem for difference equations. Journal of Difference Equations and Applications 1995,1(3):263–270. 10.1080/10236199508808026
Sun J-P: Positive solution for BVPs of fourth order difference equations. Indian Journal of Pure and Applied Mathematics 2005,36(7):361–370.
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)00354-5
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.1998.5983
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.
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)00171-0
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.
Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen; 1964:381.
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.1979.28.28046