Symmetric positive solutions of nonlinear boundary value problems

Abstract We study the nonlinear boundary value problem u ( 2 m ) = f ( t , u , u ′ , … , u ( 2 m − 2 ) ) , t ∈ ( 0 , 1 ) , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , i = 0 , … , m − 1 . The existence of symmetric positive solutions of the above problem is discussed. Sufficient conditions are obtained for the problem to have one, any finite number, and a countably infinite number of such solutions. Our results extend some recent work in the literature on boundary value problems of ordinary differential equations. We illustrate our results by two examples, none of which can be handled using the existing results.

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

[2]  Vishwanath B. Awati,et al.  Boundary Value Problems for Third‐Order Nonlinear Ordinary Differential Equations , 2005 .

[3]  Ravi P. Agarwal,et al.  Focal Boundary Value Problems for Differential and Difference Equations , 1998 .

[4]  D. O’Regan,et al.  Positive Solutions of Differential, Difference and Integral Equations , 1998 .

[5]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[6]  Lingju Kong,et al.  Positive solutions of boundary value problems for third-order functional difference equations , 2002 .

[7]  Johnny Henderson,et al.  Multiple solutions for 2mth order sturm—lioville boundary value problems on a measure chain , 2000 .

[8]  L. Kong,et al.  Even order nonlinear eigenvalue problems on a measure chain , 2003 .

[9]  Ravi P. Agarwal,et al.  Singular Lidstone boundary value problem with given maximal values for solutions , 2003 .

[10]  J. Baxley,et al.  Nonlinear boundary value problems with multiple solutions , 2001 .

[11]  J. Baxley,et al.  Nonlinear higher order boundary value problems with multiple positive solutions , 2003 .

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

[13]  Binggen Zhang,et al.  Existence of multiple symmetric positive solutions of higher order Lidstone problems , 2003 .

[14]  Haiyan Wang,et al.  On the existence of positive solutions of ordinary differential equations , 1994 .

[15]  R. Leggett,et al.  Multiple positive fixed points of nonlinear operators on ordered Banach spaces , 1979 .

[16]  Lingju Kong,et al.  POSITIVE SOLUTIONS OF HIGHER-ORDER BOUNDARY-VALUE PROBLEMS , 2005, Proceedings of the Edinburgh Mathematical Society.

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

[18]  Bo Yang,et al.  Boundary value problems for 2n-th order nonlinear ordinary differential equation , 2001 .

[19]  John M. Davis,et al.  General Lidstone problems : Multiplicity and symmetry of solutions , 2000 .

[20]  John R. Graef,et al.  On a nonlinear boundary value problem for fourth order equations , 1999 .

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

[22]  Multiple symmetric positive solutions of fourth-order two point boundary value problem , 2006 .

[23]  Leonard Meirovitch,et al.  Dynamics And Control Of Structures , 1990 .

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

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

[26]  K. Deimling.  Nonlinear functional analysis , 1985 .