An Existence Result for Neumann Type Boundary Value Problems for Second Order Nonlinear Functional Differential Equation

Liu, Yuji

  • Received : 2006.10.26
  • Published : 2008.12.31


New sufficient conditions for the existence of at least one solution of Neumann type boundary value problems for second order nonlinear differential equations $$\array{\{{p(t)\phi(x'(t)))'=f(t,x(t),\;x(\tau_1(t)),\;{\cdots},\;x(\tau_m(t))),\;t\in[0,T],\\x'(0)=0,\;x'(T)=0,}\,}$$, are established.


solutions;second order functional differential equation;Neumann boundary value problem;fixed-point theorem;growth condition


  1. R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer, Dordrecht, 1998.
  2. R. P. Agarwal, D.O'Regan, P. J. Y.Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999.
  3. B. Xuan, Z. Chen, On the singular one dimensional p−Laplacian-like equation with Neumann boundary conditions, Ann. of Diff. Equs., 16(2000), 369-380.
  4. R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer, Berlin, 1977.
  5. A. Cabada, P. Habets, S. Lois, Monotone method for Neumann problem with lower and upper solutions in the inverse order, Applied Math. Comput., 117(2001), 1-14.
  6. A. Granas, R. B. Guenther, J. W. Lee, Topological transversality II: Applications in the Neumann problem for y" = f(t, y, y'), Pacific J. Math., 104(1983), 95-109.
  7. A. Granas, Z. E. A. Guennoun, Quelques results dans la theorie de Bernstein Caratheodory de equations y" = f(t, y, y'), C. R. Acad. Sc. Paris, t. 306(1988), 703-706.
  8. Z. E. A. Guennoun, Existence de solutions au sens de Caratheodory pour le probleme de Neumann, Can. J. Math. Vol.43, 5(1991), 998-1009.
  9. A. Boucherif, N. Al-Malki, Solvability of Neumann boundary value problems with Caratheodory nonlinearities, Electronic J. of Diff. Equs., 2004(2004), 1-7.
  10. S. Atslaga, Multiplicity results for the Neumann boundary value problem, Math. Modelling and Applications, 2006, to appear.
  11. P. Girg, Neumann and periodic boundary value problems for qusilinear ordinary differential equations with a nonlinearity in the derivative, Electronic J. of Diff. Equs., 2000(2000), 63:1-28.
  12. A. Canada, P. Drabek, On similinear problems with nonlinearities depending only on derivatives, SIAM J. Math. Anal., 27(1996), 543-557.
  13. E. N. Dancer, On the ranges of certain damped nonlinear differential equation, Ann. Math. Pura. Appl., 119(1979), 281-295.
  14. J. Mawhin, Some remarks on similinear problems at resonance where the nonlinearity depends only on the derivatives, Acta Math. Inform. Univ. Ostraviensis, 2(1994), 61-69.