Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EIGENVALUE PROBLEMS FOR p-LAPLACIAN DYNAMIC EQUATIONS ON TIME SCALES

  • Guo, Mingzhou (SCHOOL OF MATHEMATICS AND STATISTICS LANZHOU UNIVERSITY) ;
  • Sun, Hong-Rui (SCHOOL OF MATHEMATICS AND STATISTICS LANZHOU UNIVERSITY)
  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we are concerned with the following eigenvalue problems of m-point boundary value problem for p-Laplacian dynamic equation on time scales $(\varphi_p(u^{\Delta}(t)))^\nabla+{\lambda}h(t)f(u(t))=0,\;t\in(0,T)$, $u(0)=0,\varphi_p(u^{\Delta}(T))=\sum\limits_{i=1}^{m-2}a_i\varphi_p(u^{\Delta}(\xi_i))$, where $\varphi_p(u)=|u|^{p-2}$u, p > 1 and $\lambda$ > 0 is a real parameter. Under certain assumptions, some new results on existence of one or two positive solution and nonexistence are obtained for $\lambda$ evaluated in different intervals. Our work develop and improve many known results in the literature even for the continual case. In doing so the usual restriction that $f_0=lim_{u{\rightarrow}0}+f(u)/\varphi_p(u)$ and $f_\infty = lim_{u{\rightarrow}{\infty}}f(u)/\varphi_p({u})$ exist is removed. As an applications, an example is given to illustrate the main results obtained.

Keywords

References

  1. R. P. Agarwal, H. Lu, and D. O'Regan, Eigenvalues and the one-dimensional p- Laplacian, J. Math. Anal. Appl. 266 (2002), no. 2, 383–400 https://doi.org/10.1006/jmaa.2001.7742
  2. D. R. Anderson, Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl. 8 (2002), no. 8, 673–688 https://doi.org/10.1080/1023619021000000717
  3. D. R. Anderson, R. Avery, and J. Henderson, Existence of solutions for a one dimensional p-Laplacian on time-scales, J. Difference Equ. Appl. 10 (2004), no. 10, 889–896 https://doi.org/10.1080/10236190410001731416
  4. F. M. Atici, D. C. Biles, and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling 43 (2006), no. 7-8, 718–726 https://doi.org/10.1016/j.mcm.2005.08.014
  5. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001
  6. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003
  7. J. J. DaCunha, J. M. Davis, and P. K. Singh, Existence results for singular three point boundary value problems on time scales, J. Math. Anal. Appl. 295 (2004), no. 2, 378-391 https://doi.org/10.1016/j.jmaa.2004.02.049
  8. L. Erbe, A. Peterson, and R. Mathsen, Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain, J. Comput. Appl. Math. 113 (2000), no. 1-2, 365–380 https://doi.org/10.1016/S0377-0427(99)00267-8
  9. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic press, San Diego, 1988
  10. S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18–56
  11. M. A. Jones, B. Song, and D. M. Thomas, Controlling wound healing through debridement, Math. Comput. Modelling 40 (2004), no. 9-10, 1057–1064 https://doi.org/10.1016/j.mcm.2003.09.041
  12. E. R. Kaufmann and Y. N. Raffoul, Eigenvalue problems for a three-point boundaryvalue problem on a time scale, Electron. J. Qual. Theory Differ. Equ. 2004 (2004), no. 2, 10 pp
  13. M. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd. Groningen, 1964
  14. V. Spedding, Taming nature's numbers, New Scientist 179 (2003), no. 2404, 28–31
  15. H. R. Sun and W. T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004), no. 2, 508–524 https://doi.org/10.1016/j.jmaa.2004.03.079
  16. H. R. Sun and W. T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica (Chin. Ser.) 49 (2006), no. 2, 369–380
  17. H. R. Sun and W. T. Li, Positive solutions for p-Laplacian m-point boundary value problems on time scales, Tainwanese J. Math. 12 (2008), no. 1, 93–115
  18. H. R. Sun and W. T. Li, Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales, Appl. Math. Comput. 182 (2006), no. 1, 478–491 https://doi.org/10.1016/j.amc.2006.04.009
  19. D. M. Thomas, L. Vandemuelebroeke, and K. Yamaguchi, A mathematical evolution model for phytoremediation of metals, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 2, 411-422 https://doi.org/10.3934/dcdsb.2005.5.411