Multiple positive solutions of singular and nonsingular discrete problems via variational methods. (English) Zbl 1070.39005

Agarwal, Ravi P.; Perera, Kanishka; O’Regan, Donal

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58, No. 1-2, 69-73 (2004).

The authors use the critical point theory to obtain the existence of multiple positive solutions of the discrete boundary value problem \[ \Delta ^{2}y(k-1)+f(k,y(k))=0,\;k\in \{1,...,T\},\quad y(0)=0=y(T+1), \] where \(T\) is a positive integer, \(\Delta y(k)=y(k+1)-y(k)\) is the forward difference operator and \(f\in C(\{1,...,T\}\times [ 0,\infty );\mathbb{R}) \) is a given function with some specific properties.

Reviewer: N. C. Apreutesei (Iaşi)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

discrete boundary value problem; multiple solutions; variational methods; critical point theory; positive solutions

References:

[1]	Agarwal, R.P., Difference equations and inequalities, Monographs and textbooks in pure and applied mathematics, Vol. 228, (2000), Marcel Dekker Inc New York · Zbl 1006.00501
[2]	Agarwal, R.P.; Regan, D.O.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Dordrecht
[3]	P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, Vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. · Zbl 0609.58002