On Perturbed Discrete Boundary Value Problems |
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Authors: | Debra L Etheridge Jesu´s Rodriguez |
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Institution: | 1. Department of Mathematics , University of North Carolina , Chapel Hill, Chapel Hill, NC, 27599, USA;2. Department of Mathematics , North Carolina State University , Box 8205, Raleigh, NC, 27695-8205, USA |
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Abstract: | In this paper, we study nonlinear discrete boundary value problems of the form x ( t +1)= A ( t ) x ( t )+ h ( t )+ k f ( t , x ( t ), k ) subject to Bx (0)+ Dx ( J )= u + k g ( x (0), x ( J ), k ) where k is a "small" parameter. Our main concern is the case of resonance, that is, the situation where the associated linear homogeneous boundary value problem x ( t +1)= A ( t ) x ( t ), Bx (0)+ Dx ( J )=0 admits nontrivial solutions. We establish conditions for the solvability of the nonlinear boundary value problem when k is "small". We also establish qualitative properties of these solutions. |
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Keywords: | Difference Equations Boundary Value Problems Resonance Implicit Function Theorem |
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