DOI QR코드

DOI QR Code

POSITIVE SOLUTIONS FOR NONLINEAR m-POINT BVP WITH SIGN CHANGING NONLINEARITY ON TIME SCALES

  • HAN, WEI (Department of Mathematics, North University of China) ;
  • REN, DENGYUN (Department of Mathematics, North University of China)
  • 투고 : 2017.02.27
  • 심사 : 2017.07.20
  • 발행 : 2017.09.30

초록

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for nonlinear m-point boundary value problem for the following second-order dynamic equations on time scales $$u^{{\Delta}{\nabla}}(t)+a(t)f(t,u(t))=0,\;t{\in}(0,T),\;{\beta}u(0)-{\gamma}u^{\Delta}(0)=0,\;u(T)={\sum_{i=1}^{m-2}}\;a_iu({\xi}_i),\;m{\geq}3$$, where $a(t){\in}C_{ld}((0,T),\;[0,+{\infty}))$, $f{\in}C([0,T]{\times}[0,+{\infty}),\;(-{\infty},+{\infty}))$, the nonlinear term f is allowed to change sign. We obtain several existence theorems of positive solutions for the above boundary value problems. In particular, our criteria generalize and improve some known results [15] and the obtained conditions are different from related literature [14]. As an application, an example to demonstrate our results is given.

키워드

참고문헌

  1. R.P. Agarwal, D. O'Regan, Nonlinear boundary value Problems on time scales, Nonl. Anal. 44, 2001, 527-535. https://doi.org/10.1016/S0362-546X(99)00290-4
  2. 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, 2003, p. 1-57.
  3. D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Differ. Equ. Appl. 8 (2002), 673-688. https://doi.org/10.1080/1023619021000000717
  4. F.M. Atici, G.Sh. Gnseinov, On Green'n functions and positive solutions for boundary value problems on time scales, J. Comput. Anal. Math. 141 (2002), 75-99. https://doi.org/10.1016/S0377-0427(01)00437-X
  5. B. Aulbach, L. Neidhart, Integration on measure chain, in: proc. of the Sixth Int. Conf. on Difference Equations, CRC, BocaRaton, Fl, 2004, pp. 239-252.
  6. M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications , Birkhauser, Boston, Cambridge, MA, 2001.
  7. M. Bohner, A. Peterson, Advances in Dynamic Equations on time scales, Birkhauser Boston, Cambridge, MA, 2003.
  8. E.R. Kaufmann, Positive solutions of a three-point boundary value problem on a time scale, Eletron. J. Diff. Equ. 82 (2003), 1-11.
  9. D. Guo, V. Lakshmikanthan, Nonlinear problems in Abstract Cones, Academic Press, San Diego, 1988.
  10. J.P. Sun, W.T. Li, Existence and nonexistence of positive solutions for second-order time scale systems, Nonlinear Anal. 68 (2008), 3107-3114. https://doi.org/10.1016/j.na.2007.03.003
  11. K.Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. 63 (2001), 690-704. https://doi.org/10.1112/S002461070100206X
  12. H. Luo, Q.Z. Ma, Positive solutions to a generalized second-order three-point boundary value problem on time scales, Eletron. J. Differen. Equ. 17 (2005), 1-14.
  13. Y.H. Su, W. T. Li, H. R. Sun, Positive solutions of singular p-Laplacian BVPs with sign changing nonlinearity on time scales, Mathematical and Computer Modelling, 48 (2008), 845-858. https://doi.org/10.1016/j.mcm.2007.11.008
  14. H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004), 508-524. https://doi.org/10.1016/j.jmaa.2004.03.079
  15. H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Mathematica Sinica 49 (2006), 369-380(in Chinese).
  16. C.Wang, R.P.Agarwal, Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales, Commun Nonlinear Sci Numer Simulat 36 (2016), 238-251. https://doi.org/10.1016/j.cnsns.2015.12.003