Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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EXISTENCE OF TRIPLE POSITIVE SOLUTIONS OF A KIND OF SECOND-ORDER FOUR-POINT BVP

  • Published : 2009.01.31
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Abstract

In this paper, we considered the following four-point boundary value problem $\{{x"(t)+h(t)f(t,x(t),x'(t))=0,\;0<t<1\atop%20x'(0)=ax(\xi),\;x'(1)=bx(\eta)}\$. where $0\;<\;{\xi}\;<\;{\eta}\;<\;1,\;{\delta}\;=\;ab{\xi}\;-\;ab{\eta}\;+\;a\;-\;b\;<\;0,\;0\;<\;a\;<\;\frac{1}{\xi},\;0\;<\;b\;<\;\frac{1}{\eta}$. After the discussion of the Green function of the corresponding homogeneous system, we establish some criteria for the existence of positive solutions by using the generalized Leggett-William's fixed point theorem. The interesting point is the expression of the Green function, which is a difficulty for multi-point BVP.

Keywords

References

  1. R.P. Agarwal, D. ORegan, P.J.Y. Wong, Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic, Boston, 1999.
  2. R.P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinary differential equations with singularities, J. Math. Anal. Appl. 297(2004) 131-151. https://doi.org/10.1016/j.jmaa.2004.05.002
  3. R.I. Avery, A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42(2001) 313-322. https://doi.org/10.1016/S0898-1221(01)00156-0
  4. R.Y. MA and Bevan Thompson, Multiplicity results for second-order two-point bound ary value problems with superlinear or sublinear nonlinearities, J. Math. Anal. Appl. 303(2005) 726-735. https://doi.org/10.1016/j.jmaa.2004.09.002
  5. Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundaty value problems, Comput. Math. Appl. 48(2004) 699-707. https://doi.org/10.1016/j.camwa.2004.03.002
  6. Z.B. Bai, W.G. Ge, Y.F. Wang, Multiplicity results for some second-order four-point boundary-value problems, Nonlinear Anal. 60(2006) 491-500.
  7. Z.B. Bai,W.G. Li, W.G. Ge,Existence and multiplicity of solutions for four-point boundary value problems at resonance, Nonlinear Anal. 60(2005) 1151-1162. https://doi.org/10.1016/j.na.2004.10.013
  8. Q. Yao, Positive solutions of nonlinear second-order periodic boundary value problems, Appl. Math. Lett. 5(2007) 583-590.
  9. W. Feng, On an m-point boundary value problem, Nonlinear Anal. 30(1997) 5369-5374. https://doi.org/10.1016/S0362-546X(97)00360-X
  10. W. Feng and J.R.L. Webb, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212(1997) 467-480. https://doi.org/10.1006/jmaa.1997.5520
  11. C.P. Gupta, A Dirichlet type multi-point boundary value problem for second order ordinary differential equations, Nonlinear Anal. 26(1996) 925-931. https://doi.org/10.1016/0362-546X(94)00338-X
  12. R. Y. Ma, Positive solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 34(1998) 1-8. https://doi.org/10.1016/0022-0396(79)90015-9
  13. R.Y. Ma, Multiplicity results for a three-point boundary value problem at resonance, Non.Ana. 53(2003) 777-789. https://doi.org/10.1016/S0362-546X(03)00033-6
  14. Bing Liu, Positive solutions of a nonlinear four-point boundary value problems in Banach spaces, J. Math. Anal. Appl. 305(2005) 253-276. https://doi.org/10.1016/j.jmaa.2004.11.037
  15. Pang. H.H, Ge.W, Existence for second-order four-point boundary value problems at resonance, Anm.of D:H.Eqs. 22(2006) 343-346.