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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

${{\Delta }}^{4}u\left(t-2\right)+\eta {{\Delta }}^{2}u\left(t-1\right)-\xi u\left(t\right)=\lambda f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in ℤ\left[a+1,b+1\right],$
$u\left(a\right)={{\Delta }}^{2}u\left(a-1\right)=0,\phantom{\rule{2.em}{0ex}}u\left(b+2\right)={{\Delta }}^{2}u\left(b+1\right)=0,$

where ${\Delta }u\left(t\right)=u\left(t+1\right)-u\left(t\right)$. Using the classical critical point theory and theory of monotone operators in Banach spaces, the authors describe the intervals for $\lambda$ 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 $k$ ($k\in ℤ\left[1,b-a\right]$) distinct pairs of nontrivial solutions.

##### MSC:
 39A12 Discrete version of topics in analysis 46N10 Applications of functional analysis in optimization and programming 39A10 Additive difference equations 34B15 Nonlinear boundary value problems for ODE
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