Multiple Solutions and Eigenvalues for Third-Order Right Focal Boundary Value Problems

Abstract We are concerned with the right focal boundary value problem x‴(t) = f(t, x(t)), t 1 ≤ t ≤ t 3 , x(t 1 ) = x′(t 2 ) = x″(t 3 ) = 0, and the associated eigenvalue problem x‴(t) = λa(t)f(x(t)) with the same boundary conditions. Under various assumptions on f, a, and λ we establish intervals of the parameter λ which yield the existence of a positive solution of the eigenvalue problem. By placing certain restrictions on the nonlinearity, we prove the existence of at least one, at least two, at least three, and infinitely many positive solutions of the boundary value problem by applying some known fixed-point theorems as well as some recent generalizations of these fixed-point theorems.

[1]  Uniqueness of solutions of right focal problems for third order differential equations , 1984 .

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

[3]  P. Habets,et al.  EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEMS OF 2D ORDER ODE , 1999 .

[4]  Douglas R. Anderson,et al.  Multiple rositive solutions for a three-point boundary value problem , 1998 .

[5]  John M. Davis,et al.  Eigenvalue intervlas for nonlinear right focal problems , 2000 .

[6]  J. Henderson.  TRIPLE POSITIVE SYMMETRIC SOLUTIONS FOR A LIDSTONE BOUNDARY VALUE PROBLEM , 1999 .

[7]  F. Sadyrbaev,et al.  Multiple solutions of a third order boundary value problem , 1992 .

[8]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

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

[10]  T. Ouyang,et al.  Exact Multiplicity Results for Two Classes of Periodic Equations , 1995 .

[11]  Twin solutions to singular boundary value problems , 2000 .

[12]  D. Anderson.  Positivity of green's function for an n-point right focal boundary value problem on measure chains , 2000 .

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

[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]  Multiple positive fixed points of weakly inward mappings , 1990 .

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

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

[19]  P. J. McKenna,et al.  ON MULTIPLE SOLUTIONS OF A NONLINEAR DIRICHLET PROBLEM , 1980 .

[20]  D. Anderson.  Multiple Positive Solutions for a Three-Point Boundary Value Problem , 2003 .

[21]  On multiple solutions of a nonlinear Dirichlet problem , 1997 .

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

[23]  On a Criterion for the Existence of at Least Four Solutions of Functional Boundary Value Problems , 1997 .

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

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

[26]  Ravi P. Agarwal,et al.  Positive solutions and eigenvalues of conjugate boundary value problems , 1999, Proceedings of the Edinburgh Mathematical Society.