Non‐local boundary value problems of arbitrary order

We give a new unified method of establishing the existence of multiple positive solutions for a large number of non‐linear differential equations of arbitrary order with any allowed number of non‐local boundary conditions (BCs). In particular, we are able to determine the Green's function for these problems with very little explicit calculation, which shows that studying a more general version of a problem with appropriate notation can lead to a simplification in approach. We obtain existence and non‐existence results, some of which are sharp, and give new results for both non‐local and local BCs. We illustrate the theory with a detailed account of a fourth‐order problem that models an elastic beam and also determine optimal values of constants that appear in the theory.

[1]  H. H. Schaefer,et al.  Some spectral properties of positive linear operators , 1960 .

[2]  Robert H. Martin,et al.  Nonlinear operators and differential equations in Banach spaces , 1976 .

[3]  Roger D. Nussbaum,et al.  A Periodicity Threshold Theorem for Some Nonlinear Integral Equations , 1978 .

[4]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[5]  P. Korman.  Computation of displacements for nonlinear elastic beam models using monotone iterations , 1988 .

[6]  Sotiris K. Ntouyas,et al.  Solvability of an m-Point Boundary Value Problem for Second Order Ordinary Differential Equations , 1995 .

[7]  Kunquan Lan,et al.  Positive Solutions of Semilinear Differential Equations with Singularities , 1998 .

[8]  Roger D. Nussbaum,et al.  Eigenvectors of Order‐Preserving Linear Operators , 1998 .

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

[10]  L. Erbe.  Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems , 2000 .

[11]  Kunquan Lan,et al.  Multiple Positive Solutions of Semilinear Differential Equations with Singularities , 2001 .

[12]  B. Lou.  POSITIVE SOLUTIONS FOR NONLINEAR ELASTIC BEAM MODELS , 2001 .

[13]  Bryan P. Rynne,et al.  Infinitely many solutions of superlinear fourth order boundary value problems , 2002 .

[14]  G. Karakostas,et al.  Existence of multiple positive solutions for a nonlocal boundary value problem , 2002 .

[15]  John R. Graef,et al.  A three point boundary value problem for nonlinear fourth order differential equations , 2003 .

[16]  Qingliu Yao,et al.  Positive solutions for Eigenvalue problems of fourth-order elastic beam equations , 2004, Appl. Math. Lett..

[17]  Ruyun Ma,et al.  Positive solutions for nonlinear m-point Eigenvalue problems , 2004 .

[18]  Jingxian Sun,et al.  Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations , 2004, Journal of Applied Mathematics and Computing.

[19]  Qingliu Yao,et al.  On the Positive Solutions of a Nonlinear Fourth-order Boundary Value Problem with Two Parameters , 2004 .

[20]  N. Kosmatov.  Countably Many Solutions of a Fourth Order Boundary Value Problem , 2004 .

[21]  P. Korman.  Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[23]  F. Minhós,et al.  EXISTENCE AND LOCATION RESULT FOR A FOURTH ORDER BOUNDARY VALUE PROBLEM , 2005 .

[24]  Bai,et al.  POSITIVE SOLUTIONS OF m-POINT BOUNDARY-VALUE PROBLEMS , 2005 .

[25]  Zhaocai Hao,et al.  On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems , 2005, Appl. Math. Lett..

[26]  Gennaro Infante,et al.  Loss of positivity in a nonlinear scalar heat equation , 2006 .

[27]  Zhongli Wei,et al.  The method of lower and upper solutions for fourth order singular m-point boundary value problems☆ , 2006 .

[28]  Gennaro Infante,et al.  NONLINEAR NON-LOCAL BOUNDARY-VALUE PROBLEMS AND PERTURBED HAMMERSTEIN INTEGRAL EQUATIONS , 2006, Proceedings of the Edinburgh Mathematical Society.

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

[30]  Huili Ma,et al.  Positive solution for m-point boundary value problems of fourth order , 2006 .

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

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

[33]  Alberto Cabada,et al.  Positivity and lower and upper solutions for fourth order boundary value problems , 2007 .

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

[35]  Daniel Franco,et al.  Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions , 2008, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.