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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 $$\Delta\left[ p\left( t-1\right) \Delta y\left( t-1\right) \right] +q\left( t\right) y\left( t\right) +\lambda y\left( t\right) =f\left( y\left( t\right) \right) ,$$ $t=a+1,\dots,b+1,$ subject to the boundary conditions $$a_{11}y\left( a\right) +a_{12}\Delta y\left( a\right) =0,\text{ } a_{21}y\left( b+1\right) +a_{22}\Delta y\left( b+1\right) =0.$$ For bounded and continuous functions $f:\Bbb R\rightarrow\Bbb R,$ the existence and the behavior of the real valued solutions is studied using the Brouwer 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 $\varepsilon,$ 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
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##### References:
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