Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems

Abstract We study the nonlinear boundary value problem with multi-point boundary condition ( | u ″ | p − 1 u ″ ) ″ = f ( t , u , u ′ , u ″ ) , t ∈ ( 0 , 1 ) , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = ∑ j = 1 m a i j u ( 2 i ) ( t j ) , i = 0 , 1 . Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions of this problem using fixed point theorems on cones. Applications of our results to the special case where f is a power function of u and its derivatives are also discussed. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our results extend some recent work in the literature on boundary value problems for ordinary differential equations.

[1]  W. Ames,et al.  Nonlinear problems in abstract cones , 1988 .

[2]  Lokenath Debnath,et al.  A necessary and sufficient condition for the existence of positive solutions of singular boundary value problems , 2005, Appl. Math. Lett..

[3]  John R. Graef,et al.  Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations , 2002 .

[4]  R. Ma,et al.  Positive solutions of singular sublinear fourth-order boundary value problems , 2005 .

[5]  Shaozhu Chen,et al.  Positive solutions of fourth-order superlinear singular boundary value problems , 2002, Bulletin of the Australian Mathematical Society.

[6]  Lingju Kong,et al.  Erratum Journal of Difference Equations and Applications, 2003 Vol. 9(1), pp. 121–133: Positive Solutions of Nonlinear m-point Boundary Value Problems on a Measure Chain* , 2003 .

[7]  Johnny Henderson,et al.  Multiple symmetric positive solutions for a second order boundary value problem , 2000 .

[8]  Lingju Kong,et al.  Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems , 2007 .

[9]  Zhongli Wei,et al.  A necessary and sufficient condition for 2nth-order singular super-linear m-point boundary value problems☆ , 2007 .

[10]  Zhongli Wei,et al.  Existence of positive solutions for 2nth-order singular sublinear boundary value problems☆ , 2005 .

[11]  Johnny Henderson,et al.  Three symmetric positive solutions for a second-order boundary value problem , 2000, Appl. Math. Lett..

[12]  L. Kong,et al.  Positive Solutions of Nonlinear m-point Boundary Value Problems on a Measure Chain , 2003 .

[13]  John M. Davis,et al.  Triple positive solutions and dependence on higher order derivatives , 1999 .

[14]  Ravi P. Agarwal,et al.  Unbounded solutions for singular boundary value problems on the semi-infinite interval , 2006 .

[15]  Shaozhu Chen,et al.  Positive solutions of even higher-order singular superlinear boundary value problems☆ , 2003 .

[16]  John M. Davis,et al.  Multiplicity of Positive Solutions for Higher Order Sturm-Liouville Problems , 2001 .

[17]  Lingju Kong,et al.  Symmetric positive solutions of nonlinear boundary value problems , 2007 .