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MULTIPLE SOLUTIONS OF PERIODIC BOUNDARY VALUE PROBLEMS FOR FIRST-ORDER DIFFERENCE EQUATIONS

Part of: Difference equations Difference and functional equations

Published online by Cambridge University Press:  01 August 2008

DA-BIN WANG
DA-BIN WANG*
Affiliation:
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China (email: wangdb@lut.cn)
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Abstract

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In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

Keywords

periodic boundary value problempositive solutionfixed point theoremdifference equationcone

MSC classification

Secondary: 39A10: Difference equations, additive
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 1 , August 2008 , pp. 1 - 11
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Agarwal, R. P., Difference Equations and Inequalities (Marcel Dekker, New York, 1992).Google Scholar
[2]Agarwal, R. P., Bohner, M. and Wong, P. J. Y., ‘Eigenvalues and eigenfunctions of discrete conjugate boundary value problems’, Comput. Math. Appl. 38(3–4) (1999), 159–183.Google Scholar
[3]Agarwal, R. P. and Henderson, J., ‘Positive solutions and nonlinear eigenvalue problems for third-order difference equations’, Comput. Math. Appl. 36(10–12) (1998), 347–355.Google Scholar
[4]Agarwal, R. P. and O’Regan, D., ‘Multiple solutions for higher-order difference equations’, Comput. Math. Appl. 37 (1999), 39–48.Google Scholar
[5]Agarwal, R. P. and Wong, F. H., ‘Existence of positive solutions for higher order difference equations’, Appl. Math. Lett. 10(5) (1997), 67–74.Google Scholar
[6]Agarwal, R. P. and Wong, P. J. Y., Advanced Topics in Difference Equations (Kluwer Academic, Dordrecht, 1997).Google Scholar
[7]Anderson, D. R., ‘Discrete third-order three-point right-focal boundary value problems’, Comput. Math. Appl. 45 (2003), 861–871.Google Scholar
[8]Anderson, D. R. and Avery, R. I., ‘Multiple positive solutions to a third-order discrete focal boundary value problem’, Comput. Math. Appl. 42 (2001), 333–340.Google Scholar
[9]Atici, F. M. and Cabada, A., ‘Existence and uniqueness results for discrete second-order periodic boundary value problems’, Comput. Math. Appl. 45 (2003), 1417–1427.Google Scholar
[10]Atici, F. M. and Guseinov, G. Sh., ‘Positive periodic solutions for nonlinear difference equations with periodic cofficients’, J. Math. Anal. Appl. 232 (1999), 166–182.Google Scholar
[11]Avery, R. I., Chyan, C. J. and Henderson, J., ‘Twin solutions of boundary value problems for ordinary differential equations and finite difference equations’, Comput. Math. Appl. 42 (2001), 695–704.Google Scholar
[12]Chyan, C. J., Henderson, J. and Lo, H. C., ‘Existence of triple solutions of discrete (n,p) boundary value problems’, Appl. Math. Lett. 14 (2001), 347–352.Google Scholar
[13]Eloe, P. W., ‘A generalization of concavity for finite differences’, J. Math. Anal. Appl. 36(10–12) (1998), 109–113.Google Scholar
[14]Guo, D. and Lakshmikantham, V., Nonlinear Problems in Abstract Cones (Academic Press, New York, 1988).Google Scholar
[15]He, Z. M., ‘On the existence of positive solutions of p-Laplacian difference equations’, J. Comput. Appl. Math. 161 (2003), 193–201.Google Scholar
[16]He, X. M. and Ge, W. G., ‘Triple solutions for second-order three-point boundary value problems’, J. Math. Anal. Appl. 268 (2002), 256–265.CrossRefGoogle Scholar
[17]He, X. M., Ge, W. G. and Peng, M. S., ‘Multiple positive solutions for one-dimensional p-Laplacian boundary value problems’, Appl. Math. Lett. 15 (2002), 937–943.Google Scholar
[18]He, Z. M. and Yu, J. S., ‘On the existence of positive solutions of fourth-order difference equations’, Appl. Math. Comput. 161 (2005), 139–148.Google Scholar
[19]Henderson, J., ‘Positive solutions for nonlinear difference equations’, Nonlinear Stud. 4(1) (1997), 29–36.Google Scholar
[20]Henderson, J. and Wong, P. J. Y., ‘Positive solutions for a system of nonpositive difference equations’, Aequationes Math. 62 (2001), 249–261.Google Scholar
[21]Kelley, W. G. and Peterson, A. C., Difference Equations (Academic Press, Boston, 1991).Google Scholar
[22]Lakshmikantham, V. and Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications (Academic Press, New York, 1988).Google Scholar
[23]Leggett, R. W. and Williams, L. R., ‘Multiple positive fixed points of nonlinear operators on ordered Banach spaces’, Indiana Univ. Math. J. 28 (1979), 673–688.Google Scholar
[24]Liu, Y. S., ‘Multiple solutions of periodic boundary value problems for first-order differential equations’, Comput. Math. Appl. 54 (2007), 1–8.Google Scholar
[25]Sun, J. P., ‘Positive solution for first-order discrete periodic boundary value problem’, Appl. Math. Lett. 19 (2006), 1244–1248.Google Scholar
[26]Sun, J. P. and Li, W. T., ‘Multiple positive solutions of a discrete difference system’, Appl. Math. Comput. 143 (2003), 213–221.CrossRefGoogle Scholar
[27]Wang, Y. and Shi, Y. M., ‘Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions’, J. Math. Anal. Appl. 309 (2005), 56–69.CrossRefGoogle Scholar
[28]Wong, P. J. Y., ‘Positive solutions of difference equations with two-point right focal boundary conditions’, J. Math. Anal. Appl. 224 (1998), 34–58.Google Scholar
[29]Wong, P. J. Y. and Agarwal, R. P., ‘Double positive solutions of (n,p) boundary value problems for higher order difference equations’, Comput. Math. Appl. 32 (1996), 1–21.Google Scholar
[30]Wong, P. J. Y. and Agarwal, R. P., ‘Existence theorems for a system of difference equations with (n,p)-type conditions’, Appl. Math. Comput. 123 (2001), 389–407.Google Scholar
[31]Wong, P. J. Y. and Agarwal, R. P., ‘Further results on fixed-sign solutions for a system of higher-order difference equations’, Comput. Math. Appl. 42(3–5) (2001), 497–514.Google Scholar