Three positive solutions for a generalized Laplacian boundary value problem with a parameter

Concerned is a generalized Laplacian boundary value problem with a positive parameter. First we apply the Leggett-Williams fixed point theorem to establish sufficient conditions on the existence of at least three positive solutions for the parameter belonging to an explicit interval. Then, under a little bit stronger assumptions, we show that there are at least three positive symmetric solutions for the parameter in an open interval. The obtained results are new even for Laplacian boundary value problems and they are illustrated with an example.

[1]  J. Henderson,et al.  Nonlinear eigenvalue problems for quasilinear systems , 2005 .

[2]  Weigao Ge,et al.  Multiple positive solutions for one-dimensional p-Laplacian boundary value problems , 2002, Appl. Math. Lett..

[3]  Yanping Guo,et al.  TWIN POSITIVE SYMMETRIC SOLUTIONS FOR LIDSTONE BOUNDARY VALUE PROBLEMS , 2004 .

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

[5]  Yongping Sun.  Three symmetric positive solutions for second-order nonlocal boundary value problems , 2011 .

[6]  Dingyong Bai,et al.  Positive solutions and eigenvalue regions of two-delay singular systems with a twin parameter , 2007 .

[7]  R. Ma,et al.  Positive solutions of a nonlinear three-point boundary-value problem. , 1999 .

[8]  Johnny Henderson,et al.  Positive Solutions for Nonlinear Eigenvalue Problems , 1997 .

[9]  Haiyan Wang,et al.  On the number of positive solutions of nonlinear systems , 2003 .

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

[11]  P. Ubilla,et al.  Positive solutions for the 1-dimensional generalized p-Laplacian involving a real parameter , 1998 .

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

[13]  Wan-Tong Li,et al.  Multiple positive solutions for nonlinear dynamical systems on a measure chain , 2004 .

[14]  Ravi P. Agarwal,et al.  Triple solutions to boundary value problems on time scales , 2000, Appl. Math. Lett..

[15]  Wan-Tong Li,et al.  Multiple positive solutions of a discrete difference system , 2003, Appl. Math. Comput..

[16]  Tadeusz Jankowski,et al.  Positive solutions to second order four-point boundary value problems for impulsive differential equations , 2008, Appl. Math. Comput..

[17]  Johnny Henderson,et al.  Multiple Symmetric Solutions for Discrete Lidstone Boundary Value Problems , 2001 .

[18]  Ravi P. Agarwal,et al.  A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem , 2005, Appl. Math. Comput..

[19]  Eun Kyoung Lee,et al.  A multiplicity result for generalized Laplacian systems with multiparameters , 2009 .

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

[21]  Wei-Cheng Lian,et al.  Existence of positive solutions for higher-order generalized p-Laplacian BVPs , 2000, Appl. Math. Lett..

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

[23]  Yanping Guo,et al.  Three positive solutions for a nonlinear nth-order m-point boundary value problem , 2008 .

[24]  Jiqing Qiu,et al.  Three positive solutions for higher order m-point boundary value problems , 2004 .

[25]  Wan-Tong Li,et al.  Three positive solutions of a nonlinear three-point boundary value problem , 2003 .

[26]  K. Prasad,et al.  Multiple positive solutions for nonlinear third order general three-point boundary value problems , 2008 .

[27]  Yanping Guo,et al.  Three positive solutions for second-order m-point boundary value problems , 2004, Appl. Math. Comput..

[28]  J. Henderson,et al.  An eigenvalue problem for quasilinear systems , 2007 .

[29]  Lingju Kong,et al.  Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem , 2008 .

[30]  Douglas R. Anderson,et al.  NONLINEAR TRIPLE-POINT PROBLEMS ON TIME SCALES , 2004 .

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

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

[33]  Hongzhou Wang,et al.  Existence of triple positive solutions to a class of p-Laplacian boundary value problems , 2007 .