Existence theory for single and multiple solutions to singular positone discrete Dirichlet boundary value problems to the one-dimension p -Laplacian

Daqing Jiang; Li Li Zhang; Donal O'Regan; Ravi P. Agarwal

Archivum Mathematicum (2004)

  • Volume: 040, Issue: 4, page 367-381
  • ISSN: 0044-8753

Abstract

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In this paper we establish the existence of single and multiple solutions to the positone discrete Dirichlet boundary value problem Δ [ φ ( Δ u ( t - 1 ) ) ] + q ( t ) f ( t , u ( t ) ) = 0 , t ∈ { 1 , 2 , ⋯ , T } u ( 0 ) = u ( T + 1 ) = 0 , where φ ( s ) = | s | p - 2 s , p > 1 and our nonlinear term f ( t , u ) may be singular at u = 0 .

How to cite

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Jiang, Daqing, et al. "Existence theory for single and multiple solutions to singular positone discrete Dirichlet boundary value problems to the one-dimension $p$-Laplacian." Archivum Mathematicum 040.4 (2004): 367-381. <http://eudml.org/doc/249291>.

@article{Jiang2004,
abstract = {In this paper we establish the existence of single and multiple solutions to the positone discrete Dirichlet boundary value problem \[ \left\lbrace \begin\{array\}\{l\} \Delta \big [\phi (\Delta u(t-1))\big ]+ q(t) f(t,u(t))=0\,,\quad t\in \lbrace 1,2,\dots ,T\rbrace \\[3pt] u(0)=u(T+1)=0\,, \end\{array\} \right. \] where $\phi (s) = |s|^\{p-2\}s$, $p>1$ and our nonlinear term $f(t,u)$ may be singular at $u=0$.},
author = {Jiang, Daqing, Zhang, Li Li, O'Regan, Donal, Agarwal, Ravi P.},
journal = {Archivum Mathematicum},
keywords = {multiple solutions; singular; existence; discrete boundary value problem; discrete Dirichlet boundary value problem; one-dimensional -Laplacian; positive solution; Leray-Schauder alternative; cone fixed point theorem},
language = {eng},
number = {4},
pages = {367-381},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence theory for single and multiple solutions to singular positone discrete Dirichlet boundary value problems to the one-dimension $p$-Laplacian},
url = {http://eudml.org/doc/249291},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Jiang, Daqing
AU - Zhang, Li Li
AU - O'Regan, Donal
AU - Agarwal, Ravi P.
TI - Existence theory for single and multiple solutions to singular positone discrete Dirichlet boundary value problems to the one-dimension $p$-Laplacian
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 4
SP - 367
EP - 381
AB - In this paper we establish the existence of single and multiple solutions to the positone discrete Dirichlet boundary value problem \[ \left\lbrace \begin{array}{l} \Delta \big [\phi (\Delta u(t-1))\big ]+ q(t) f(t,u(t))=0\,,\quad t\in \lbrace 1,2,\dots ,T\rbrace \\[3pt] u(0)=u(T+1)=0\,, \end{array} \right. \] where $\phi (s) = |s|^{p-2}s$, $p>1$ and our nonlinear term $f(t,u)$ may be singular at $u=0$.
LA - eng
KW - multiple solutions; singular; existence; discrete boundary value problem; discrete Dirichlet boundary value problem; one-dimensional -Laplacian; positive solution; Leray-Schauder alternative; cone fixed point theorem
UR - http://eudml.org/doc/249291
ER -

References

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  10. Henderson J., Singular boundary value problems for difference equations, Dynam. Systems Appl. (1992), 271–282. (1992) Zbl0761.39002MR1182649
  11. Henderson J., Singular boundary value problems for higher order difference equations, In Proceedings of the First World Congress on Nonlinear Analysis, (Edited by V. Lakshmikantham), Walter de Gruyter, 1994, 1139–1150. (1994) MR1389147
  12. Jiang D. Q., Multiple positive solutions to singular boundary value problems for superlinear higher-order ODEs, Comput. Math. Appl. 40 (2000), 249–259. Zbl0976.34019MR1763623
  13. Jiang D. Q., Pang P. Y. H., Agarwal R. P., Upper and lower solutions method and a superlinear singular discrete boundary value problem, Dynam. Systems Appl., to appear. MR2370156
  14. O’Regan D., Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic, Dordrecht, 1997. (1997) Zbl1077.34505MR1449397

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