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http://dx.doi.org/10.4134/JKMS.2010.47.1.135

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)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.1, 2010 , pp. 135-163 More about this Journal
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
one-dimension p-Laplacian difference equation; multi-point boundary value problem; positive solution;
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1 I. Y. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007), no. 1, 458-468   DOI   ScienceOn
2 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   DOI
3 Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006), no. 10, 1019-1023   DOI   ScienceOn
4 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   DOI   ScienceOn
5 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   DOI   ScienceOn
6 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   DOI   ScienceOn
7 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   DOI   ScienceOn
8 J. Yu and Z. Guo, Boundary value problems of discrete generalized Emden-Fowler equation, Sci. China Ser. A 49 (2006), no. 10, 1303-1314   DOI   ScienceOn
9 G. Zhang and R. Medina, Three-point boundary value problems for difference equations, Comput. Math. Appl. 48 (2004), no. 12, 1791-1799   DOI   ScienceOn
10 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   DOI   ScienceOn
11 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   DOI   ScienceOn
12 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   DOI   ScienceOn
13 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   DOI   ScienceOn
14 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
15 X. Cai and J. Yu, Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2006), no. 2, 649-661   DOI   ScienceOn
16 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   DOI   ScienceOn
17 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   DOI   ScienceOn
18 Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), no. 1, 193-201   DOI   ScienceOn