Constructive proof for existence of nonlinear two-point boundary value problems

Owing to the importance of differential equations in physics, the existence of solutions for differential equations has been paid much attention. In this paper, the existence of solution are obtained for the nonlinear second order two-point boundary value problem in the reproducing kernel space. Under certain assumptions on right-hand side, we propose constructive proof for the existence result, and a method is presented to obtain the exact solution expressed by the form of series. This paper is a extension of previous paper [Wei Jiang, Minggen Cui, The exact solution and stability analysis for integral equation of third or first kind with singular kernel, Appl. Math. Comput. 202 (2) (2008) 666-674], which extends a method of solving linear problems to present method for solving nonlinear problems.

[1]  Tanya G. Melton,et al.  Generalized quasilinearization method and higher order of convergence for second-order boundary value problems , 2006 .

[2]  Yuji Liu.  Studies on Neumann-type boundary value problems for second-order nonlinear p-Laplacian-like functional differential equations☆ , 2009 .

[3]  Bashir Ahmad,et al.  The bellman-kalaba-lakshmikantham quasilinearization method for Neumann problems , 2001 .

[4]  Aydin Tiryaki,et al.  More on the explicit solutions for a second-order nonlinear boundary value problem , 2008, Appl. Math. Lett..

[5]  Christopher C. Tisdell,et al.  Existence of solutions to first-order periodic boundary value problems , 2006 .

[6]  Pablo Amster,et al.  Existence of solutions to boundary value problems for dynamic systems on time scales , 2005 .

[7]  Elsayed M. E. Elbarbary,et al.  Chebyshev finite difference approximation for the boundary value problems , 2003, Appl. Math. Comput..

[8]  Bashir Ahmad,et al.  Generalized quasilinearization method for mixed boundary value problems , 2002, Appl. Math. Comput..

[9]  Weigao Ge,et al.  Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator , 2003 .

[10]  Rahmat Ali Khan.  The generalized quasilinearization technique for a second order differential equation with separated boundary conditions , 2006, Math. Comput. Model..

[11]  A. Dang Quang,et al.  Iterative method for solving a nonlinear boundary value problem , 2007, Appl. Math. Comput..

[12]  Edward H. Twizell,et al.  Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems , 2002, Appl. Math. Lett..

[13]  Zhong Chen,et al.  The exact solution of a linear integral equation with weakly singular kernel , 2008 .

[14]  Wei-Cheng Lian,et al.  Existence of solutions for nonlinear BVPs of second-order differential equations on measure chains , 2001 .

[15]  N. Aykut,et al.  Existence of positive solutions forboundary value problems of second-order functional difference equations , 2004 .

[16]  S. Ha.  A nonlinear shooting method for two-point boundary value problems , 2001 .

[17]  Wei Jiang,et al.  The exact solution and stability analysis for integral equation of third or first kind with singular kernel , 2008, Appl. Math. Comput..

[18]  Bashir Ahmad,et al.  Generalized quasilinearization method for a second order three point boundary-value problem with nonlinear boundary conditions , 2002 .

[19]  Guozhu Gao,et al.  Existence, uniqueness and approximation of classical solutions to nonlinear two-point boundary value problems , 2002 .

[20]  Zhan Zhou,et al.  Existence of nontrivial solutions for discrete nonlinear two point boundary value problems , 2006, Appl. Math. Comput..

[21]  Peixuan Weng,et al.  Existence and multiple solutions for a second-order difference boundary value problem via critical point theory , 2007 .

[22]  Johnny Henderson,et al.  Existence of Multiple Solutions for Second Order Boundary Value Problems , 2000 .

[23]  Fuyi Li,et al.  Existence of solutions to a class of nonlinear second order two-point boundary value problems , 2005 .

[24]  Minggen Cui,et al.  Nonlinear Numerical Analysis in Reproducing Kernel Space , 2009 .

[25]  Qingkai Kong,et al.  Existence and nonexistence of solutions of second-order nonlinear boundary value problems , 2007 .

[26]  Johnny Henderson,et al.  On the existence and uniqueness of solutions to boundary value problems on time scales , 2004 .

[27]  Johnny Henderson,et al.  Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and solvability , 2007 .

[28]  Patrick Habets,et al.  A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions , 2001, Appl. Math. Comput..