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On discrete fourth-order boundary value problems with three parameters. (English) Zbl 1189.39006

The article deals with the following discrete nonlinear fourth-order boundary value problem

Δ 4 u(t-2)+ηΔ 2 u(t-1)-ξu(t)=λf(t,u(t)),t[a+1,b+1],
u(a)=Δ 2 u(a-1)=0,u(b+2)=Δ 2 u(b+1)=0,

where Δu(t)=u(t+1)-u(t). Using the classical critical point theory and theory of monotone operators in Banach spaces, the authors describe the intervals for λ such that the boundary value problem under consideration has no nontrivial solutions, has a unique solution, has at least one nontrivial solution, has at least two nontrivial solutions, has at least three nontrivial solutions, has at least k (k[1,b-a]) distinct pairs of nontrivial solutions.

MSC:
39A12Discrete version of topics in analysis
46N10Applications of functional analysis in optimization and programming
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
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