Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions

Abstract The nonlinear n th-order singular nonlocal boundary value problem { u ( n ) ( t ) + λ a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where ∫ 0 1 u ( s ) d A ( s ) is given by a Riemann–Stieltjes integral with a signed measure, a may be singular at t = 0 and/or t = 1 , f ( t , x ) may also have singularity at x = 0 . The existence of positive solutions is obtained by means of the fixed point index theory in cones.

[1]  J. Webb.  Uniqueness of the principal eigenvalue in nonlocal boundary value problems , 2007 .

[2]  K. Q. Lan,et al.  Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems , 2004, Appl. Math. Comput..

[3]  Yanping Guo,et al.  The existence of countably many positive solutions for some nonlinear nth order m-point boundary value problems , 2009, J. Comput. Appl. Math..

[4]  Lingju Kong,et al.  Higher order boundary value problems with nonhomogeneous boundary conditions , 2010 .

[5]  J. Webb,et al.  Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type , 2006 .

[6]  J. Webb.  Nonlocal conjugate type boundary value problems of higher order , 2009 .

[7]  Yonghong Wu,et al.  Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions , 2009, Appl. Math. Comput..

[8]  Jingbao Yang,et al.  Positive solutions of nth order m-point boundary value problem , 2008, Appl. Math. Comput..

[9]  J. Webb,et al.  Positive solutions of nonlocal boundary value problems involving integral conditions , 2008 .

[10]  Gennaro Infante,et al.  Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach , 2006 .

[11]  Weigao Ge,et al.  Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p-Laplacian , 2008, Computers and Mathematics with Applications.

[12]  Lishan Liu,et al.  Positive solutions of fourth-order nonlinear singular Sturm-Liouville eigenvalue problems , 2007 .

[13]  John R. Graef,et al.  Positive solutions to a multi-point higher order boundary value problem , 2006 .

[14]  Lishan Liu,et al.  Positive Solutions for Nonlinearth-Order Singular Nonlocal Boundary Value Problems , 2007 .

[15]  John R. Graef,et al.  A class of nth-order BVPs with nonlocal conditions , 2009, Comput. Math. Appl..

[16]  Wei Dong,et al.  Green's function and positive solutions of nth order m-point boundary value problem , 2006, Appl. Math. Comput..

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

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

[19]  Weigao Ge,et al.  Existence results for a class of nth order m-point boundary value problems in Banach spaces , 2009, Appl. Math. Lett..

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

[21]  Yonghong Wu,et al.  Existence of positive solutions for singular higher-order differential equations , 2008 .

[22]  Kunquan Lan,et al.  Multiple eigenvalues for singular Hammerstein integral equations with applications to boundary value problems , 2006 .

[23]  Weihua Jiang.  Multiple positive solutions for nth-order m-point boundary value problems with all derivatives , 2008 .

[24]  Bashir Ahmad,et al.  Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions , 2005, Appl. Math. Lett..

[25]  L. Kong,et al.  Existence of Solutions for a Higher Order Multi-Point Boundary Value Problem , 2009 .