American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 291-299. doi: 10.3934/proc.2013.2013.291

Existence of multiple solutions to a discrete fourth order periodic boundary value problem

John R. Graef 1, , Lingju Kong 1, and  Min Wang 1,

1. 

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States

Received  July 2012 Revised  December 2012 Published  November 2013

Sufficient conditions are obtained for the existence of multiple solutions to the discrete fourth order periodic boundary value problem \begin{equation*} \begin{array}{l} \Delta^4 u(t-2)-\Delta(p(t-1)\Delta u(t-1))+q(t) u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb{Z}},\\ \Delta^iu(-1)=\Delta^iu(N-1),\quad i=0, 1,2, 3. \end{array} \end{equation*} Our analysis is mainly based on the variational method and critical point theory. One example is included to illustrate the result.
Keywords: fourth order, variational methods., critical points, Discrete boundary value problem, multiple solutions.
Mathematics Subject Classification: Primary: 39A10; Secondary: 34B1.
Citation: John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291
References:
[1]

R. P. Agarwal, "Difference Equations and Inequalities, Theory, Methods, and Applications,'', $2^{nd}$ edition, (2000).   Google Scholar

[2]

D. R. Anderson and R. I. Avery, Existence of a periodic solution for continuous and discrete periodic second-order equations with variable potentials,, J. Appl. Math. Comput., 37 (2011), 297.   Google Scholar

[3]

D. R. Anderson and F. Minhós, A discrete fourth-order Lidstone problem with parameters,, Appl. Math. Comput., 214 (2009), 523.   Google Scholar

[4]

F. M. Atici and G. Sh. Guseinov, Positive periodic solutions for nonlinear difference equations with periodic coefficients,, J. Math. Anal. Appl., 232 (1999), 166.   Google Scholar

[5]

Z. Bai, Iterative solutions for some fourth-order periodic boundary value problems,, Taiwanese J. Math., 12 (2008), 1681.   Google Scholar

[6]

C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations,, Nonlinear Anal. 71 (2009), 71 (2009), 53.   Google Scholar

[7]

A. Cabada and N. Dimitrov, Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities,, J. Math. Anal. Appl., 371 (2010), 518.   Google Scholar

[8]

A. Cabada and J. B. Ferreiro, Existence of positive solutions for nth-order periodic difference equations,, J. Difference Equ. Appl., 17 (2011), 935.   Google Scholar

[9]

X. Cai and Z. Guo, Existence of solutions of nonlinear fourth order discrete boundary value problem,, J. Difference Equ. Appl. 12 (2006), 12 (2006), 459.   Google Scholar

[10]

D. C. Clark, A variant of the Liusternik-Schnirelman theory,, Indiana Uni. Math. J., 22 (1972), 65.   Google Scholar

[11]

M. Conti, S. Terracini and G. Verzini, Infinitely many solutions to fourth order superlinear periodic problems,, Trans. Amer. Math. Soc., 356 (2004), 3283.   Google Scholar

[12]

T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters,, J. Comput. Appl. Math., 233 (2010), 2506.   Google Scholar

[13]

Z. He and J. Yu, On the existence of positive solutions of fourth-order difference equations,, Appl. Math. Comput., 161 (2005), 139.   Google Scholar

[14]

J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation,, Adv. Difference Equ., (2006).   Google Scholar

[15]

W. G. Kelly and A. C. Peterson, "Difference Equations, an Introduction with Applications,'', $2^{nd}$ edition, (2001).   Google Scholar

[16]

Y. Li, Positive solutions of fourth-order periodic boundary value problems,, Nonlinear Anal., 54 (2003), 1069.   Google Scholar

[17]

Y. Li and H. Fan, Existence of positive periodic solutions for higher-order ordinary differential equations,, Comput. Math. Appl., 62 (2011), 1715.   Google Scholar

[18]

R. Ma and Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations,, Comput. Math. Appl., 59 (2010), 3770.   Google Scholar

[19]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, in, 65 (1986).   Google Scholar

[20]

B. Zhang, L. Kong, Y. Sun and X. Deng, Existence of positive solutions for BVPs of fourth-order difference equation,, Appl. Math. Comput., 131 (2002), 583.   Google Scholar

show all references

References:
[1]

R. P. Agarwal, "Difference Equations and Inequalities, Theory, Methods, and Applications,'', $2^{nd}$ edition, (2000).   Google Scholar

[2]

D. R. Anderson and R. I. Avery, Existence of a periodic solution for continuous and discrete periodic second-order equations with variable potentials,, J. Appl. Math. Comput., 37 (2011), 297.   Google Scholar

[3]

D. R. Anderson and F. Minhós, A discrete fourth-order Lidstone problem with parameters,, Appl. Math. Comput., 214 (2009), 523.   Google Scholar

[4]

F. M. Atici and G. Sh. Guseinov, Positive periodic solutions for nonlinear difference equations with periodic coefficients,, J. Math. Anal. Appl., 232 (1999), 166.   Google Scholar

[5]

Z. Bai, Iterative solutions for some fourth-order periodic boundary value problems,, Taiwanese J. Math., 12 (2008), 1681.   Google Scholar

[6]

C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations,, Nonlinear Anal. 71 (2009), 71 (2009), 53.   Google Scholar

[7]

A. Cabada and N. Dimitrov, Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities,, J. Math. Anal. Appl., 371 (2010), 518.   Google Scholar

[8]

A. Cabada and J. B. Ferreiro, Existence of positive solutions for nth-order periodic difference equations,, J. Difference Equ. Appl., 17 (2011), 935.   Google Scholar

[9]

X. Cai and Z. Guo, Existence of solutions of nonlinear fourth order discrete boundary value problem,, J. Difference Equ. Appl. 12 (2006), 12 (2006), 459.   Google Scholar

[10]

D. C. Clark, A variant of the Liusternik-Schnirelman theory,, Indiana Uni. Math. J., 22 (1972), 65.   Google Scholar

[11]

M. Conti, S. Terracini and G. Verzini, Infinitely many solutions to fourth order superlinear periodic problems,, Trans. Amer. Math. Soc., 356 (2004), 3283.   Google Scholar

[12]

T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters,, J. Comput. Appl. Math., 233 (2010), 2506.   Google Scholar

[13]

Z. He and J. Yu, On the existence of positive solutions of fourth-order difference equations,, Appl. Math. Comput., 161 (2005), 139.   Google Scholar

[14]

J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation,, Adv. Difference Equ., (2006).   Google Scholar

[15]

W. G. Kelly and A. C. Peterson, "Difference Equations, an Introduction with Applications,'', $2^{nd}$ edition, (2001).   Google Scholar

[16]

Y. Li, Positive solutions of fourth-order periodic boundary value problems,, Nonlinear Anal., 54 (2003), 1069.   Google Scholar

[17]

Y. Li and H. Fan, Existence of positive periodic solutions for higher-order ordinary differential equations,, Comput. Math. Appl., 62 (2011), 1715.   Google Scholar

[18]

R. Ma and Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations,, Comput. Math. Appl., 59 (2010), 3770.   Google Scholar

[19]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, in, 65 (1986).   Google Scholar

[20]

B. Zhang, L. Kong, Y. Sun and X. Deng, Existence of positive solutions for BVPs of fourth-order difference equation,, Appl. Math. Comput., 131 (2002), 583.   Google Scholar

[1]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269
[2]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[3]

Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615
[4]

Feliz Minhós, T. Gyulov, A. I. Santos. Existence and location result for a fourth order boundary value problem. Conference Publications, 2005, 2005 (Special) : 662-671. doi: 10.3934/proc.2005.2005.662
[5]

John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84
[6]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[7]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[8]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276
[9]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89
[10]

Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032
[11]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[12]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
[13]

Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure & Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785
[14]

Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624
[15]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
[16]

Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245
[17]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[18]

Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure & Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251
[19]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489
[20]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084

 Impact Factor: 

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences