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Zbl 1076.39016
Rodriguez, Jesús
Nonlinear discrete Sturm-Liouville problems.
(English)
[J] J. Math. Anal. Appl. 308, No. 1, 380-391 (2005). ISSN 0022-247X

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 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. \par 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.
[N. C. Apreutesei (Iaşi)]
MSC 2000:
*39A12 Discrete version of topics in analysis
34L15 Estimation of eigenvalues for OD operators

Keywords: discrete boundary value problems; Brower fixed point theorem; implicit function theorem; eigenvalue

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