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, Marcel Dekker, New York, 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), 297-312.  Google Scholar

[3]

D. R. Anderson and F. Minhós, A discrete fourth-order Lidstone problem with parameters, Appl. Math. Comput., 214 (2009), 523-533.  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), 166-182.  Google Scholar

[5]

Z. Bai, Iterative solutions for some fourth-order periodic boundary value problems, Taiwanese J. Math., 12 (2008), 1681-1690.  Google Scholar

[6]

C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations, Nonlinear Anal. 71 (2009), 53-57.  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), 518-533.  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), 935-954.  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), 459-466.  Google Scholar

[10]

D. C. Clark, A variant of the Liusternik-Schnirelman theory, Indiana Uni. Math. J., 22 (1972), 65-74.  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), 3283-3300.  Google Scholar

[12]

T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters, J. Comput. Appl. Math., 233 (2010), 2506-2520.  Google Scholar

[13]

Z. He and J. Yu, On the existence of positive solutions of fourth-order difference equations, Appl. Math. Comput., 161 (2005), 139-148.  Google Scholar

[14]

J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation, Adv. Difference Equ., 2006, Art. ID 81025, 9 pp.  Google Scholar

[15]

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

[16]

Y. Li, Positive solutions of fourth-order periodic boundary value problems, Nonlinear Anal., 54 (2003), 1069-1078.  Google Scholar

[17]

Y. Li and H. Fan, Existence of positive periodic solutions for higher-order ordinary differential equations, Comput. Math. Appl., 62 (2011), 1715-1722.  Google Scholar

[18]

R. Ma and Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations, Comput. Math. Appl., 59 (2010), 3770-3777.  Google Scholar

[19]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in "CMBS Regional Conf. Ser. Math.,'' Vol. 65, Amer. Math. Soc., Providence, RI, 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), 583-591.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, "Difference Equations and Inequalities, Theory, Methods, and Applications,'' $2^{nd}$ edition, Marcel Dekker, New York, 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), 297-312.  Google Scholar

[3]

D. R. Anderson and F. Minhós, A discrete fourth-order Lidstone problem with parameters, Appl. Math. Comput., 214 (2009), 523-533.  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), 166-182.  Google Scholar

[5]

Z. Bai, Iterative solutions for some fourth-order periodic boundary value problems, Taiwanese J. Math., 12 (2008), 1681-1690.  Google Scholar

[6]

C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations, Nonlinear Anal. 71 (2009), 53-57.  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), 518-533.  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), 935-954.  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), 459-466.  Google Scholar

[10]

D. C. Clark, A variant of the Liusternik-Schnirelman theory, Indiana Uni. Math. J., 22 (1972), 65-74.  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), 3283-3300.  Google Scholar

[12]

T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters, J. Comput. Appl. Math., 233 (2010), 2506-2520.  Google Scholar

[13]

Z. He and J. Yu, On the existence of positive solutions of fourth-order difference equations, Appl. Math. Comput., 161 (2005), 139-148.  Google Scholar

[14]

J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation, Adv. Difference Equ., 2006, Art. ID 81025, 9 pp.  Google Scholar

[15]

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

[16]

Y. Li, Positive solutions of fourth-order periodic boundary value problems, Nonlinear Anal., 54 (2003), 1069-1078.  Google Scholar

[17]

Y. Li and H. Fan, Existence of positive periodic solutions for higher-order ordinary differential equations, Comput. Math. Appl., 62 (2011), 1715-1722.  Google Scholar

[18]

R. Ma and Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations, Comput. Math. Appl., 59 (2010), 3770-3777.  Google Scholar

[19]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in "CMBS Regional Conf. Ser. Math.,'' Vol. 65, Amer. Math. Soc., Providence, RI, 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), 583-591.  Google Scholar

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