# EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR A $p$-LAPLACIAN BOUNDARY VALUE PROBLEM ON TIME SCALES

Volume 3, Issue 1, pp 32-38 Publication Date: February 13, 2010
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### Authors

YUANYUAN PANG - College of Information Science and Engineering, Shandong University of Science and Technology, Qing Dao, 266510, P. R. China. ZHANBING BAI - College of Information Science and Engineering, Shandong University of Science and Technology, Qing Dao, 266510, P. R. China.

### Abstract

In this paper, we study the solvability of one-dimensional fourth- order $p$-Laplacian boundary value problems on time scales. By using Krasnosel'skii's fixed point theorem of cone expansion-compression type, some existence and multiplicity results of positive solution have been required according to different growth condition of nonlinear form f at zero and at infinity.

### Keywords

• Time scales
• p-Laplacian operator
• Positive solution
• Cone.

•  34B15
•  34B18

### References

• [1] R. I. Avery, D. R. Anderson, Existence of three positive solutions to a second-order boundary value problem on a measure chain, J. Comput. Appl. Math. , 141 (2002), 65-73.

• [2] R. P. Agarwal, M. Bohner, D. ORegan, A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. , 141 (2002), 1-26.

• [3] R. P. Agarwal, M. Bohner, P. Rehak, Half-linear dynamic equations, Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, Kluwer Academic Publishers, Dordrecht, 1 (2003), 1-57.

• [4] D. R. Anderson, J. M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problem, J. Math. Anal. Appl. , 267 (2002), 135-157.

• [5] G. A. Afrouzi, M. K. Moghaddam, J. Mohammadpour, M. Zameni, On the positive and negetive solutions of Laplacian BVP with Neumann boundary conditions, J. Nonlinear Sci. Appl. , 2 (2009), 38-45.

• [6] R. P. Agarwal, D. ORegan, Existence theorem for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities, Appl. Math. Comput. , 143 (2003), 15-38.

• [7] B. Aulbach, L. Neidhart, Integration on measure chains, in: Proc. of the Sixth Int. Conf. on Difference Equations, CRC, Boca Raton, FL, (2004), 239-252.

• [8] R. P. Agarwal, D. ORegan, Nonlinear boundary value problems on time scales , Nonlinear Anal. , 44 (2001), 527-535.

• [9] Z. B. Bai, The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal. , 67 (2007), 1704-1709.

• [10] Z. B. Bai, Z. Du, Positive solutions for some second-order four-point boundary value problems, J. Math. Anal. Appl. , 330 (2007), 34-50.

• [11] Z. B. Bai, X. Q. Liang, Z. J. Du, Triple positive solutions for some second-order boundary value problem on a measure chain, Comput. Math. Appl. , 53 (2007), 1832-1839.

• [12] X. Dong, Z. B. Bai , Positive solutions of fourth-order boundary value problem with variable parameters, J. Nonlinear Sci. Appl. , 1 (2008), 21-30.

• [13] M. Bohner, A. Peterson, An Introduction with Applications, Dynamic Equations on Time Scales, Birkhäuser, Boston (2001)

• [14] S. Hilger , Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. , 4 (1990), 18-56.

• [15] M. A. Krasnosel'skii , Positive Solutions of Operator Equations, Noordhoff, Gronignen (1964)

• [16] R. Y. Ma, H. Y. Wang , On the existence of positive solutions of fourth-order ordinary differential equations, Applicable Analysis., 59 (1995), 225-231.

• [17] H. R. Sun, W. T. Li, Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales, Appl. Math. Comput. , 182 (2006), 478-491.

• [18] H. R. Sun, W. T. Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, J. Differential Equations. , 240 (2007), 217-248.

• [19] J. Y. Wang, The existence of positive solutions for the one-dimensional p-Laplacian, Proc. Amer. Math. Soc. , 125 (1997), 2275-2283.

• [20] Y. Z. Zhu, J. Zhu, Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, J. Math. Anal. Appl. , 344 (2008), 616-626.