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

John R. Graef; Lingju Kong; Min Wang

doi:10.3934/proc.2013.2013.291

Article Contents

2013, Volume 2013: 291-299. Doi: 10.3934/proc.2013.2013.291 open access

This issue Previous Article Positive solutions of nonlocal fractional boundary value problems Next Article Optimization problems for the energy integral of p-Laplace equations

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

1.
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403

Received: June 30, 2012

Revised: November 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 39A10; Secondary: 34B15.

Citation:

Related Papers

Cited by

References

[1]	R. P. Agarwal, "Difference Equations and Inequalities, Theory, Methods, and Applications,'' $2^{nd}$ edition, Marcel Dekker, New York, 2000.
[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.
[3]	D. R. Anderson and F. Minhós, A discrete fourth-order Lidstone problem with parameters, Appl. Math. Comput., 214 (2009), 523-533.
[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.
[5]	Z. Bai, Iterative solutions for some fourth-order periodic boundary value problems, Taiwanese J. Math., 12 (2008), 1681-1690.
[6]	C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations, Nonlinear Anal. 71 (2009), 53-57.
[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.
[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.
[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.
[10]	D. C. Clark, A variant of the Liusternik-Schnirelman theory, Indiana Uni. Math. J., 22 (1972), 65-74.
[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.
[12]	T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters, J. Comput. Appl. Math., 233 (2010), 2506-2520.
[13]	Z. He and J. Yu, On the existence of positive solutions of fourth-order difference equations, Appl. Math. Comput., 161 (2005), 139-148.
[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.
[15]	W. G. Kelly and A. C. Peterson, "Difference Equations, an Introduction with Applications,'' $2^{nd}$ edition, Academic Press, New York, 2001.
[16]	Y. Li, Positive solutions of fourth-order periodic boundary value problems, Nonlinear Anal., 54 (2003), 1069-1078.
[17]	Y. Li and H. Fan, Existence of positive periodic solutions for higher-order ordinary differential equations, Comput. Math. Appl., 62 (2011), 1715-1722.
[18]	R. Ma and Y. Xu, Existence of positive solution for nonlinear fourth-order difference equations, Comput. Math. Appl., 59 (2010), 3770-3777.
[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.
[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.