×

On the existence of positive solutions of fourth-order difference equations. (English) Zbl 1068.39008

Let \(T\geq 1\) fixed, \(f\in C(\mathbb{R}^+,\mathbb{R}^+)\), \(a:\{1,2,\dots, T+1\}\subset\mathbb{Z}\to \mathbb{R}^+\) and \(\lim_{x\to 0,x> 0}\,x^{-1} f(x)\) and \(\lim_{x\to+\infty}\, x^{-1} f(x)\) be limits that exist.
The authors determine eigenvalues \(\lambda\) for which there exist positive solutions \(u\) of the fourth-order difference equation \[ \Delta^4 u(t- 2)- \lambda a(t) f(u(t))= 0,\quad t\in \{2,3,\dots, T+ 2\}\subset\mathbb{Z} \] satisfying the boundary conditions \[ u(0)= \Delta^2 u(0)= u(T+ 2)= \Delta^2 u(T)= 0 \] or \[ u(0)= \Delta^2 u(0)= \Delta u(t+ 1)= \Delta^3 u(T- 1)= 0 \] by means of Krasnosel’skij’s fixed point theorem.
Reviewer: D. M. Bors (Iaşi)

MSC:

39A11 Stability of difference equations (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P.; Henderson, J., Positive solutions and nonlinear eigenvalue problems for third-order difference equations, Comp. math. appl, 36, 11-12, 347-355, (1998) · Zbl 0933.39003
[2] Henderson, J., Positive solutions for nonlinear difference equations, Nonlinear stud, 4, 1, 29-36, (1997) · Zbl 0883.39002
[3] Merdivenci, F., Two positive solutions of a boundary value problem for difference equations, J. diff. eqns. appl, 1, 263-270, (1995) · Zbl 0854.39001
[4] Agarwal, R.P.; O’Regan, D., A fixed-point approach for nonlinear discrete boundary value problems, Comp. math. appl, 36, 10-12, 115-121, (1998) · Zbl 0933.39004
[5] Henderson, J.; Wang, H.Y., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl, 208, 252-259, (1997) · Zbl 0876.34023
[6] Zhang, B.G.; Kong, L.J.; Sun, Y.J.; Deng, X.H., Existence of positive solutions for BVPs of fourth-order difference equations, Appl. math. comput, 131, 583-591, (2002) · Zbl 1025.39006
[7] Ma, R.Y.; Wang, H.Y., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal, 59, 225-231, (1995) · Zbl 0841.34019
[8] Graef, J.R.; Yang, B., On a nonlinear boundary value problem for fourth order equations, Appl. anal, 72, 439-448, (1999) · Zbl 1031.34017
[9] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhooff Groningen · Zbl 0121.10604
[10] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.