Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 4 (2009), 1 -- 14

EXISTENCE AND NONEXISTENCE RESULTS FOR SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM

Feng Wang, Yujun Cui and Fang Zhang

Abstract. In this paper some existence and nonexistence results for positive solutions are obtained for second-order boundary value problem

-u"+Mu=f(t,u),    t∈(0,1)
with Neumann boundary conditions
u'(0)=u'(1)=0,
where M>0,  f∈C([0,1]×R+, R+). By making use of fixed point index theory in cones, some new results are obtained.

2000 Mathematics Subject Classification: 34B18; 47H10.
Keywords: Neumann BVP; Positive solutions; Cone; Fixed point index.

Full text

References

  1. R.P. Agarwal, D. O'Regan and P.J.Y. Wong, Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic Publishers, Boston, 1999. MR1680024(2000a:34046). Zbl pre01256477.

  2. A. Cabada, P. Habets and S. Lois, Monotone method for the Neumann problem with low and upper solutions in the reverse order, Appl. Math. Comput. 117 (2001) 1-14. MR1801398(2001j:34016). Zbl 1031.34021.

  3. A. Cabada, L. Sanchez. A positive operator approach to the Neumann problem for a second order differential equation, J. Math. Anal. Appl. 204 (1996) 774-785. MR1422772(97k:34020). Zbl 0871.34014.

  4. A. Cañada, J. A. Montero and S.Villegas, Liapunov-type inequalities and Neumann boundary value problems at resonance, Math. Inequal. Appl. 8 (2005) 459-475. MR2148238(2006f:34037). Zbl 1085.34014.

  5. J.F. Chu, Y.G. Sun and H. Chen, Positive solutions of Neumann problems with singularities, J. Math. Anal. Appl. 327 (2008) 1267-1272. MR2386375(2008k:34075). Zbl 1142.34315.

  6. D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988. MR0959889(89k:47084). Zbl 0661.47045.

  7. D.Q. Jiang and H. Z. Liu, Existence of positive solutions to second order Neumann boundary value problems, J. Math. Res. Exposition 20(3) (2000) 360-364. MR1787796(2001f:34044). Zbl 0963.34019.

  8. I. Rachunkova, Upper and lower solutions with inverse inequality, Ann. Polon. Math. 65 (1997), 235-244. MR1441178(92h:00002a). Zbl 0868.34014.

  9. I. Rachunkova and S. Stanek, Topological degree method in functional boundary value problems at resonance, Nonlinear Anal. TMA 27 (1996), 271-285. MR1391437(98c:34105). Zbl 0853.34062.

  10. J.P. Sun and W.T. Li, Multiple positive solutions to second-order Neumann boundary value problems, Appl. Math. Comput. 146 (2003) 187-194. MR2007778(2005a:34031). Zbl 1041.34013.

  11. J.P. Sun, W.T. Li and S. S. Cheng, Three positive solutions for second-Order Neumann boundary Value problems, Appl. Math. Lett. 17 (2004) 1079-1084. MR2087758(2005e:34064). Zbl 1061.34014.

  12. P.J.Y. Wong and R.P. Agarwal, Double positive solutions of (n,p) boundary value problems for higher order Difference equations, Comput. Math. Appl. 32 (1996), 1-21. MR1426193(98h:39003). Zbl 0873.39008.

  13. N. Yazidi, Monotone method for singular Neumann problem, Nonlinear Anal. 49 (2002), 589-602. MR1894297(2003d:34045). Zbl 1007.34019.



Feng Wang Yujun Cui
School of Mathematics and Physics, Department of Mathematics,
Jiangsu Polytechnic University, Shandong University of Science and Technology,
Changzhou, 213164, P.R. China. Qingdao, 266510, P.R. China.
e-mail: fengwang188@163.com e-mail: w.feng@yeah.net
Fang Zhang
School of Mathematics and Physics,
Jiangsu Polytechnic University,
Changzhou, 213164, P.R. China.
e-mail: fangzhang@em.jpu.cn

http://www.utgjiu.ro/math/sma