Existence of positive solutions for BVPs of fourth-order difference equations. (English) Zbl 1025.39006

The authors consider the boundary value problem (BVP) \[ \Delta^4x(t-2)= \lambda a(t)f\bigl(t,x(t) \bigr), \]
\[ x(0)=x(T+2)= \Delta^2x(0)= \Delta^2x(T)= 0,\;2\leq t\leq T\text{ and }T\geq 6. \] Under the conditions \(f\) increasing in \(x\) and continuous in both arguments, \(f(t,0)>0\) and \(a(t)\geq 0\) there exists a \(\lambda_0\geq 0\) such that the BVP has at least one positive solution for \(0\leq\lambda \leq\lambda_0\). If, moreover, \(f(t,x)\geq dx\) for some \(d>0\) then there is no solution for \(\lambda> \lambda_0\).
Reviewer’s remark: The case where \(a(t)\) vanishes identically must be excluded in the last assertion.


39A11 Stability of difference equations (MSC2000)
Full Text: DOI


[1] Agarwal, R.P.; Wong, P.J.Y., Advanced topics in difference equations, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0914.39005
[2] Wei, Z., Positive solution of singular boundary value problems of fourth order differential equations, Acta math. sinica, 42, 4, 715-722, (1999) · Zbl 1022.34018
[3] J. Henderson, Positive solutions for nonlinear difference equations, Nonlinear Stud. 4 (1) (1997) · Zbl 0883.39002
[4] Merdivenci, F., Two positive solutions of a boundary value problem for difference equations, J. difference equations appl., 1, 263-270, (1995) · Zbl 0854.39001
[5] Merdivenci, F., Positive solutions for focal point problem for 2nth order difference equations, Pan. amer. math. J., 5, 25-42, (1995)
[6] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press Orlando, FL · Zbl 0661.47045
[7] O’Regan, D., Solvability of some fourth (and high) order singular boundary value problems, J. math. anal. appl., 161, 78-116, (1991) · Zbl 0795.34018
[8] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964, MR31:6107
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.