Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

This paper is concerned with the one-dimensional p-Laplacian multi-point boundary value problem on time scales T: (jp(uD))\nabla + h(t)f(u) = 0, t \in [0,T]T, subject to multi-point boundary conditions u(0) - B0(\sumi=1m-2ai uD(xi)) = 0, uD(T) = 0, or uD(0) = 0, u(T) + B1(\sumi=1m-2biuD(x'i)) = 0, where jp(u) is p-Laplacian operator, i.e., jp(u = |u|p-2u, p>1, xi,x'i\in [0,T]T, m \geq 3 and satisfy 0 \leq x1 < x2 < ... < xm-2 < r(T), s(0) < x'1 < x'2 < ... < x'm-2 \leq T, ai, bi\in [0,\infty) (i=1,2,..., m-2). Some new sufficient conditions are obtained for the existence of at least one positive solution by using Krasnosel'skii's fixed-point theorem and new sufficient conditions are obtained for the existence of twin, triple or arbitrary odd positive solutions by using generalized Avery and Henderson fixed-point theorem and Avery-Peterson fixed-point theorem. Our results include and extend some known results. As applications, two examples are given to illustrate the main results and their differences. These results are new even for the special cases of continuous and discrete equations, as well as in the general time scale setting.

Anahtar Kelimeler:

Time scales; boundary value problem; positive solutions; p-Laplacian; fixed-point theorem

Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

Keywords:

Time scales; boundary value problem; positive solutions; p-Laplacian; fixed-point theorem,

PDF

___

Agarwal, R.P., Bohner, M. and Li, W.T.: Nonoscilation and oscillation theory for functional differential equations, Pure Appl. Math. S. 267, Marcel Dekker, 2004.
Agarwal, R.P., Bohner,M. and Rehak, P.: Half—linear dynamic equations, Nonlinear Anal. Appl.: to V. Lakshmikan— tham on his 80th birthday, vol. 1, Kluwer Acad. Publ. Dordrecht, 1-57 (2003).
Anderson, D.R.: Solutions to second-order three-point problems on time scales, J. Differ. Equations Appl. 8, 673-688 (2002).
Aulbach, B. and Neidhart, L.: Integration on measure chains, In proceedings of the sixth international conference on difference equations, CRC, Boca Raton, FL, 239-252 (2004).
He, Z. and Li L.: Triple positive solutions for the one—dimensional p—Laplacian dynamic equations on time scales,
Math. Comput. Modelling 45, 68-79 (2007).
Hilger, S.: Ein MaŞkettenkalkiil mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universitat Würzburg, (1988).
Hong, S.: Triple positive solutions of three—point boundary value problem for p—Laplacian dynamic equations on
time scales, J. Comput. Appl. Math. 206, 967-976 (2007).
J amieson, V. and Spedding, V.: Taming nature’s numbers, the global science and technology weekly, New Scientist,
, 28—31 (2003). Jones M.A., Song,B. and Thomas, D.M.: Controlling wound healing through debridement, Math. Comput. Mod— elling 40, 1057—1064 (2004). Krasnosel’skii, M.: Positive solutions of operator equations, Noordhoff, Groningen (1964). Lakshmikantham, V., Sivasundaram, S. and Kaymakcalan, B.: Dynamic systems on measure chains, Kluwer Academic Publishers, Boston, 1996.
Leggett, R. and Williams, L.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces,
Indiana Univ. Math. J. 28, 673—688 (1979).
Li, C. and Ge, W.: Positive solutions of one-dimensional p—Laplacian singular Sturm—Liouville boundary value
problems, Math. Appl. 15, 13—17 (2002).
Liu, Y. and Ge, W.: Multiple positive solutions to a three-point boundary value problems with p—Laplacian, J.
Math. Anal. Appl. 277, 293—302 (2003).
Liu, Y. and Ge, W.: Twin positive solutions of boundary value problems for Şnite difference equations with p—
Laplacian, J. Math. Anal. Appl. 278, 551-561 (2003).
Su, Y.H.: Multiple positive pseudo-symmetric solutions of p—Laplacian dynamic equations on time scales, Math.
Comput. Modelling 49, 1664—1681 (2009).
Su, Y.H., Li, W.T.: Existence of positive solutions to a singular p—Laplacian dynamic equations With sign changing
nonlinearity, Acta. Math. Sinica, Chinese Series, 52, 181—196 (2009).
Su,Y.H., Li,W.T. and Sun, H.R.: Positive solutions of singular p—Laplacian BVPs with sign changing nonlinearity
on time scales, Math. Comput. Modelling 48, 845-858 (2008).
Wang, D.B.: Triple positive solutions of three-point boundary value problems for p—Laplacian dynamic equations
on time scales, Nonlinear Anal. 68, 2172—2180 (2008).
Wang, H.: Positive periodic solutions of functional differential equations, J. Differential Equations 202, 354-366 (2004).
Wang, J .: The existence of positive solutions for the one—dimensional p—Laplacian, Proc. Amer. Math. Soc. 125, 2275—2283 (1997).
Yaslan, I.: Existence of positive solutions for nonlinear three-point problems on time scales, J . Comput. Appl. Math. 206, 888-897 (2007).
You-Hui SU Received: 09.04.2009
Mathematics and Physical Sciences Technology,
Xuzhou Institute of Technology,
Xuzhou, Jiangsu, 221111, CHINA
e—mail: suyouhui©xzit.edu.cn