Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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SYMMETRIC SOLUTIONS FOR A FOURTH-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS WITH ONE-DIMENSIONAL $p$-LAPLACIAN AT RESONANCE

  • Yang, Aijun (College of Science, Zhejiang University of Technology) ;
  • Wang, Helin (College of Science, Zhejiang University of Technology)
  • Received : 2011.03.23
  • Accepted : 2011.07.15
  • Published : 2012.01.30
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Abstract

We consider the fourth-order differential equation with one-dimensional $p$-Laplacian (${\phi}_p(x^{\prime\prime}(t)))^{\prime\prime}=f(t,x(t),x^{\prime}(t),x^{\prime\prime}(t)$) a.e. $t{\in}[0,1]$, subject to the boundary conditions $x^{\prime\prime}}(0)=0$, $({\phi}_p(x^{\prime\prime}(t)))^{\prime}{\mid}_{t=0}=0$, $x(0)={\sum}_{i=1}^n{\mu}_ix({\xi}_i)$, $x(t)=x(1-t)$, $t{\in}[0,1]$, where ${\phi}_p(s)={\mid}s{\mid}^{p-2}s$, $p$ > 1, 0 < ${\xi}_1$ < ${\xi}_2$ < ${\cdots}$ < ${\xi}_n$ < $\frac{1}{2}$, ${\mu}_i{\in}\mathbb{R}$, $i=1$, 2, ${\cdots}$, $n$, ${\sum}_{i=1}^n{\mu}_i=1$ and $f:[0,1]{\times}\mathbb{R}^3{\rightarrow}\mathbb{R}$ is a $L^1$-Carath$\acute{e}$odory function with $f(t,u,v,w)=f(1-t,u,-v,w)$ for $(t,u,v,w){\in}[0,1]{\times}\mathbb{R}^3$. We obtain the existence of at least one nonconstant symmetric solution by applying an extension of Mawhin's continuation theorem due to Ge. Furthermore, an example is given to illustrate the results.

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References

  1. R. P. Agarwal and D. O'Regan, Infinite interval problems for differential, Kluwer Academic, 2001.
  2. W. Ge and J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian, Nonlinear Analysis 58 (2004), 477-488. https://doi.org/10.1016/j.na.2004.01.007
  3. P. R. Agarwal, H. S. Lu and D. O'Regan, Positive solutions for the boundary value problem $(\left|u^{{\prime}{\prime}}\right|^{p-2}u^{{\prime}{\prime}})^{{\prime}{\prime}}\;-\;{\lambda}q(t)f(u(t))\;=\;0$, Mem. Differential Equations Math. Physics 28 (2003),33-44.
  4. Z. Du, X. Lin and W. Ge, Some higher-order multi-point boundary value problem at resonance, J. Differential Equations 218 (2005), 69-90. https://doi.org/10.1016/j.jde.2005.01.005
  5. H. Pang, W. Ge and M. Tian, Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p-Laplacian, Compu. Math. Applications 56 (2008), 127-142. https://doi.org/10.1016/j.camwa.2007.11.039
  6. V. A. Il'in and E. I. Moiseev, Nonlocal boundary value problem of the second kind for a SturmCLiouville operator, Differ. Equations 23 (1987), 979-987.
  7. W. Ge, Boundary value problems for ordinary nonlinear differential equations, Science Press, Beijing, 2007.
  8. Y. Sun, Existence and multiplicity of symmetric positive solutions for three-point boundary value problem, J. Math. Anal. Applications 329 (2007), 998-1009. https://doi.org/10.1016/j.jmaa.2006.07.001
  9. Y. Sun, Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem, Nonlinear Analysis 66 (2007), 1051-1063. https://doi.org/10.1016/j.na.2006.01.004
  10. W. Feng and J. R. L.Webb, Solvability of a m-point boundary value problem with nonlinear growth, J. Math. Anal. Applications 212 (1997), 467-480. https://doi.org/10.1006/jmaa.1997.5520
  11. C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Applications 168 (1992), 540-551. https://doi.org/10.1016/0022-247X(92)90179-H
  12. C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equation, Appl. Math. Computation 89 (1998), 133-146. https://doi.org/10.1016/S0096-3003(97)81653-0
  13. C. P. Gupta, Existence theorems for a second order three-point boundary value problem, J. Math. Anal. Applications 212 (1997), 430-442. https://doi.org/10.1006/jmaa.1997.5515
  14. C. P. Gupta, Positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian, Comput. Math. Applications 42 (2001), 755-765. https://doi.org/10.1016/S0898-1221(01)00195-X
  15. S. A. Marano, A remark on a second order three-point boundary value problems, J. Math. Anal. Applications 183 (1994), 518-522. https://doi.org/10.1006/jmaa.1994.1158
  16. H. Feng, H. Lian and W. Ge, A symmetric solution of a multipoint boundary value problems with one-dimensional p-Laplacian at resonance, Nonlinear Analysis 69 (2008), 3964-3972.
  17. N. Kosmatov, Multi-point boundary value problems on time scales at resonance, J. Math. Anal. Applications 323 (2006), 253-266. https://doi.org/10.1016/j.jmaa.2005.09.082
  18. J. R. Graef and L. J. Kong, Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems, Nonlinear Analysis 68 (2008), 1529-1552. https://doi.org/10.1016/j.na.2006.12.037
  19. A. J. Yang and W. G. Ge, Existence of symmetric solutions for a fourth-order multi-point boundary value problem with a p-Laplacian at resonance, J. Appl. Math. Computation 29 (2009), 301-309. https://doi.org/10.1007/s12190-008-0131-7