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Nonlinear discrete Sturm-Liouville problems. (English) Zbl 1076.39016

The paper is devoted to discrete boundary value problems of the form

${\Delta }\left[p\left(t-1\right){\Delta }y\left(t-1\right)\right]+q\left(t\right)y\left(t\right)+\lambda y\left(t\right)=f\left(y\left(t\right)\right),$

$t=a+1,\cdots ,b+1,$ subject to the boundary conditions

${a}_{11}y\left(a\right)+{a}_{12}{\Delta }y\left(a\right)=0,\phantom{\rule{4.pt}{0ex}}{a}_{21}y\left(b+1\right)+{a}_{22}{\Delta }y\left(b+1\right)=0·$

For bounded and continuous functions $f:ℝ\to ℝ,$ the existence and the behavior of the real valued solutions is studied using the Brower Fixed Point Theorem. Here $\lambda$ is an eigenvalue of the linear problem ($f=0$), so one supposes there exists a nontrivial solution of the associated linear boundary value problem.

If one multiplies $f$ by a “small” parameter $\epsilon ,$ one gives conditions which ensure the solvability of the problem. The Implicit Function Theorem is used to obtain criteria for the existence and for the qualitative behavior of the solutions.

MSC:
 39A12 Discrete version of topics in analysis 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators