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

- Journal title : Journal of the Korean Mathematical Society
- Volume 47, Issue 1, 2010, pp.135-163
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2010.47.1.135

Title & Authors

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

Liu, Yu-Ji;

Liu, Yu-Ji;

Abstract

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP 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 x(n).

Keywords

one-dimension p-Laplacian difference equation;multi-point boundary value problem;positive solution;

Language

English

Cited by

1.

2.

3.

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

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

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

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

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

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

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

9.

Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), no. 1, 193-201

10.

I. Y. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007), no. 1, 458-468

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

12.

Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006), no. 10, 1019-1023

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

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

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

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