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EXISTENCE RESULTS FOR POSITIVE SOLUTIONS OF NON-HOMOGENEOUS BVPS FOR SECOND ORDER DIFFERENCE EQUATIONS WITH ONE-DIMENSIONAL p-LAPLACIAN

  • Liu, Yu-Ji (Department of Mathematics Guangdong University of Business Studies)
  • Published : 2010.01.01

Abstract

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP $\LARGE\left\{{{\;{\Delta}[{\phi}({\Delta}x(n))]\;+\;f(n,\;x(n\;+\;1),{\Delta}x(n),{\Delta}x(n + 1))\;=\;0,\;n\;{\in}\;[0,N],}}\\{\;{x(0)-{\sum}^m_{i=1}{\alpha}_ix(n_i) = A,}}\\{\;{x(N+2)-\;{\sum}^m_{i=1}{\beta}_ix(n_i)\;=\;B.}}\right.$ The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem are established by using a new fixed point theorem obtained in [5]. An example is presented to illustrate the main result. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operator ${\Delta}$x(n).

Keywords

References

  1. D. Anderson, Discrete third-order three-point right-focal boundary value problems, Advances in difference equations, IV. Comput. Math. Appl. 45 (2003), no. 6-9, 861-871 https://doi.org/10.1016/S0898-1221(03)80157-8
  2. D. Anderson and R. I. Avery, Multiple positive solutions to a third-order discrete focal boundary value problem, Comput. Math. Appl. 42 (2001), no. 3-5, 333-340 https://doi.org/10.1016/S0898-1221(01)00158-4
  3. R. I. Avery and A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (2001), no. 3-5, 313-322 https://doi.org/10.1016/S0898-1221(01)00156-0
  4. N. Aykut, Existence of positive solutions for boundary value problems of second-order functional difference equations, Comput. Math. Appl. 48 (2004), no. 3-4, 517-527 https://doi.org/10.1016/j.camwa.2003.10.007
  5. Z. Bai and W. Ge, Existence of three positive solutions for a one-dimensional p-Laplacian, Acta Math. Sinica (Chin. Ser.) 49 (2006), no. 5, 1045-1052
  6. X. Cai and J. Yu, Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2006), no. 2, 649-661 https://doi.org/10.1016/j.jmaa.2005.07.029
  7. W. Cheung, J. Ren, P. J. Y. Wong, and D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl. 330 (2007), no. 2, 900-915 https://doi.org/10.1016/j.jmaa.2006.08.034
  8. J. R. Graef and J. Henderson, Double solutions of boundary value problems for 2mthorder differential equations and difference equations, Comput. Math. Appl. 45 (2003), no. 6-9, 873-885 https://doi.org/10.1016/S0898-1221(03)00063-4
  9. Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), no. 1, 193-201 https://doi.org/10.1016/j.cam.2003.08.004
  10. I. Y. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007), no. 1, 458-468 https://doi.org/10.1016/j.cam.2006.05.030
  11. R. Leggett and L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673-688 https://doi.org/10.1512/iumj.1979.28.28046
  12. Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006), no. 10, 1019-1023 https://doi.org/10.1016/j.aml.2005.10.020
  13. H. Pang, H. Feng, and W. Ge, Multiple positive solutions of quasi-linear boundary value problems for finite difference equations, Appl. Math. Comput. 197 (2008), no. 1, 451-456 https://doi.org/10.1016/j.amc.2007.06.027
  14. P. J. Y.Wong and R. P. Agarwal, Existence theorems for a system of difference equations with (n, p)-type conditions, Appl. Math. Comput. 123 (2001), no. 3, 389-407 https://doi.org/10.1016/S0096-3003(00)00078-3
  15. P. J. Y. Wong and L. Xie, Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl. 45 (2003) no. 6-9, 1445-1460 https://doi.org/10.1016/S0898-1221(03)00102-0
  16. C. Yang and P. Weng, Green functions and positive solutions for boundary value problems of third-order difference equations, Comput. Math. Appl. 54 (2007), no. 4, 567-578 https://doi.org/10.1016/j.camwa.2007.01.032
  17. J. Yu and Z. Guo, Boundary value problems of discrete generalized Emden-Fowler equation, Sci. China Ser. A 49 (2006), no. 10, 1303-1314 https://doi.org/10.1007/s11425-006-1999-z
  18. G. Zhang and R. Medina, Three-point boundary value problems for difference equations, Comput. Math. Appl. 48 (2004), no. 12, 1791-1799 https://doi.org/10.1016/j.camwa.2004.09.002

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