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Title: Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions (English)
Author: Bereanu, Cristian
Author: Mawhin, Jean
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 131
Issue: 2
Year: 2006
Pages: 145-160
Summary lang: English
Category: math
Summary: We use Brouwer degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above. (English)
Keyword: nonlinear difference equations
Keyword: Ambrosetti-Prodi problem
Keyword: Brouwer degree
MSC: 39A11
MSC: 47H11
MSC: 47N20
idZBL: Zbl 1110.39003
idMR: MR2242841
Date available: 2009-09-24T22:25:06Z
Last updated: 2012-06-18
Stable URL:
Reference: [1] C. Bereanu, J. Mawhin: Existence and multiplicity results for periodic solutions of nonlinear difference equations.(to appear). MR 2243830
Reference: [2] R. E. Gaines: Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations.SIAM J. Numer. Anal. 11 (1974), 411–434. Zbl 0279.65068, MR 0383757
Reference: [3] J. Henderson, H. B. Thompson: Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations.J. Differ. Equ. Appl. 7 (2001), 297–321. MR 1923625
Reference: [4] J. Henderson, H. B. Thomson: Existence of multiple solutions for second-order discrete boundary value problems.Comput. Math. Appl. 43 (2002), 1239–1248. MR 1906350
Reference: [5] R. Chiappinelli, J. Mawhin, R. Nugari: Generalized Ambrosetti-Prodi conditions for nonlinear two-point boundary value problems.J. Differ. Equations 69 (1987), 422–434. MR 0903395
Reference: [6] K. Deimling: Nonlinear Functional Analysis.Springer, Berlin, 1985. Zbl 0559.47040, MR 0787404
Reference: [7] M. Lees: A boundary value problem for nonlinear ordinary differential equations.J. Math. Mech. 10 (1961), 423–430. Zbl 0099.06902, MR 0167672
Reference: [8] M. Lees: Discrete methods for nonlinear two-point boundary value problems.Numerical Solutions of Partial Differential Equations, Bramble ed., Academic Press, New York, 1966, pp. 59–72. Zbl 0148.39206, MR 0202323
Reference: [9] M. Lees, M. H. Schultz: A Leray-Schauder principle for A-compact mappings and the numerical solution of non-linear two-point boundary value problems.Numerical Solutions of Nonlinear Differential Equations, Greenspan ed., Wiley, New York, 1966, pp. 167–179. MR 0209924
Reference: [10] J. Mawhin: Topological Degree Methods in Nonlinear Boundary Value Problems.CBMS Series No. 40, American Math. Soc., Providence, 1979. Zbl 0414.34025, MR 0525202
Reference: [11] J. Mawhin: Boundary value problems with nonlinearities having infinite jumps.Comment. Math. Univ. Carol. 25 (1984), 401–414. Zbl 0562.34010, MR 0775560
Reference: [12] J. Mawhin: Points fixes, points critiques et problèmes aux limites.Sémin. Math. Sup. No. 92, Presses Univ. Montréal, Montréal (1985). Zbl 0561.34001, MR 0789982
Reference: [13] J. Mawhin: Ambrosetti-Prodi type results in nonlinear boundary value problems.Differential equations and mathematical physics. Lect. Notes in Math. 1285, Springer, Berlin, 1987, pp. 290–313. Zbl 0651.34014, MR 0921281
Reference: [14] J. Mawhin: A simple approach to Brouwer degree based on differential forms.Advanced Nonlinear Studies 4 (2004), 535–548. Zbl 1082.47052, MR 2100911
Reference: [15] J. Mawhin, H. B. Thompson, E. Tonkes: Uniqueness for boundary value problems for second order finite difference equations.J. Differ. Equations Appl. 10 (2004), 749–757. MR 2069640
Reference: [16] H. B. Thompson: Existence of multiple solutions for finite difference approximations to second-order boundary value problems.Nonlinear Anal. 53 (2003), 97–110. Zbl 1019.65054, MR 1992406
Reference: [17] H. B. Thompson, C. C. Tisdell: The nonexistence of spurious solutions to discrete, two-point boundary value problems.Appl. Math. Lett. 16 (2003), 79–84. MR 1938194


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