EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR A \(p\)LAPLACIAN BOUNDARY VALUE PROBLEM ON TIME SCALES
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 onedimensional fourth
order \(p\)Laplacian boundary value problems on time scales. By using Krasnosel'skii's fixed point theorem of cone expansioncompression 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
 pLaplacian operator
 Positive solution
 Cone.
MSC
References

[1]
R. I. Avery, D. R. Anderson, Existence of three positive solutions to a secondorder boundary value problem on a measure chain, J. Comput. Appl. Math. , 141 (2002), 6573.

[2]
R. P. Agarwal, M. Bohner, D. ORegan, A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. , 141 (2002), 126.

[3]
R. P. Agarwal, M. Bohner, P. Rehak, Halflinear dynamic equations, Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, Kluwer Academic Publishers, Dordrecht, 1 (2003), 157.

[4]
D. R. Anderson, J. M. Davis, Multiple solutions and eigenvalues for thirdorder right focal boundary value problem, J. Math. Anal. Appl. , 267 (2002), 135157.

[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), 3845.

[6]
R. P. Agarwal, D. ORegan, Existence theorem for the onedimensional singular pLaplacian equation with sign changing nonlinearities, Appl. Math. Comput. , 143 (2003), 1538.

[7]
B. Aulbach, L. Neidhart, Integration on measure chains, in: Proc. of the Sixth Int. Conf. on Difference Equations, CRC, Boca Raton, FL, (2004), 239252.

[8]
R. P. Agarwal, D. ORegan, Nonlinear boundary value problems on time scales , Nonlinear Anal. , 44 (2001), 527535.

[9]
Z. B. Bai, The upper and lower solution method for some fourthorder boundary value problems, Nonlinear Anal. , 67 (2007), 17041709.

[10]
Z. B. Bai, Z. Du, Positive solutions for some secondorder fourpoint boundary value problems, J. Math. Anal. Appl. , 330 (2007), 3450.

[11]
Z. B. Bai, X. Q. Liang, Z. J. Du, Triple positive solutions for some secondorder boundary value problem on a measure chain, Comput. Math. Appl. , 53 (2007), 18321839.

[12]
X. Dong, Z. B. Bai , Positive solutions of fourthorder boundary value problem with variable parameters, J. Nonlinear Sci. Appl. , 1 (2008), 2130.

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

[14]
S. Hilger , Analysis on measure chainsa unified approach to continuous and discrete calculus, Results Math. , 4 (1990), 1856.

[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 fourthorder ordinary differential equations, Applicable Analysis., 59 (1995), 225231.

[17]
H. R. Sun, W. T. Li, Multiple positive solutions for pLaplacian mpoint boundary value problems on time scales, Appl. Math. Comput. , 182 (2006), 478491.

[18]
H. R. Sun, W. T. Li, Existence theory for positive solutions to onedimensional pLaplacian boundary value problems on time scales, J. Differential Equations. , 240 (2007), 217248.

[19]
J. Y. Wang, The existence of positive solutions for the onedimensional pLaplacian, Proc. Amer. Math. Soc. , 125 (1997), 22752283.

[20]
Y. Z. Zhu, J. Zhu, Several existence theorems of nonlinear mpoint boundary value problem for pLaplacian dynamic equations on time scales, J. Math. Anal. Appl. , 344 (2008), 616626.