Multiple Solutions of Generalized Multipoint Conjugate Boundary Value Problems

AbstractWe consider the boundary value problem $$y^{(n)} (t) = P(t,y),{\text{ }}t \in (0,1)$$ $$y^{(i)} (ti) = 0,{\text{ }}j = 0,...,n_i - 1,{\text{ }}i = 1,...,r,$$ where r ≥ 2, ni ≥ 1 for i = 1, ... ,r, $$\sum {_{i = 1}^r = n}$$ and 0 = t1 < t2 < ... < tr = 1. Criteria are offered for the existence of double and triple ‘positive’ (in some sense) solutions of the boundary value problem. Further investigation on the upper and lower bounds for the norms of these solutions is carried out for special cases. We also include several examples to illustrate the importance of the results obtained.

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

[2]  Izrail Moiseevich Gelfand.  Some problems in the theory of quasilinear equations , 1987 .

[3]  J. Henderson,et al.  Twin solutions for nonlinear multipoint conjugate boundary value problems , 1999 .

[4]  Ravi P. Agarwal,et al.  Eigenvalues and eigenfunctions of discrete conjugate boundary value problems , 1999 .

[5]  Seymour V. Parter,et al.  Solutions of a Differential Equation Arising in Chemical Reactor Processes , 1974 .

[6]  H. Fujita.  On the nonlinear equations $\Delta u + e^u = 0$ and $\partial v/\partial t = \Delta v + e^v$ , 1969 .

[7]  Ravi P. Agarwal,et al.  Focal Boundary Value Problems for Differential and Difference Equations , 1998 .

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

[9]  L. A. Peletier,et al.  Stabilization of solutions of a degenerate nonlinear diffusion problem , 1982 .

[10]  Ravi P. Agarwal,et al.  Difference equations and inequalities , 1992 .

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

[12]  E. N. Dancer,et al.  On the structure of solutions of an equation in catalysis theory when a parameter is large , 1980 .

[13]  L. Fox,et al.  Initial-value methods for boundary-value problems , 1987 .

[14]  Johnny Henderson,et al.  Positive solutions and nonlinear multipoint conjugate eigenvalue problems. , 1997 .

[15]  I. M. Gel'fand,et al.  Some problems in the theory of quasilinear equations , 1987 .

[16]  D. Luss.  MULTIPLE STABLE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS ARISING IN CHEMICAL REACTOR THEORY , .

[17]  G. S. S. Ludford,et al.  AN UNEXPECTED STABILITY RESULT OF THE NEAR-EXTINCTION DIFFUSION FLAME FOR NON-UNITY LEWIS NUMBERS , 1989 .

[18]  K. Schmitt,et al.  Boundary value problems for differential equations with deviating arguments , 1969 .

[19]  S. Ntouyas,et al.  Boundary value problems for differential equations with deviating arguments , 1997 .

[20]  Patricia J. Y. Wong.  Sharp inequalities for solutions of multipoint boundary value problems , 2000 .

[21]  J. Henderson,et al.  Inequalities for Solutions of Multipoint Boundary Value Problems , 1999 .

[22]  Patricia J. Y. Wong,et al.  Generalized multipoint conjugate eigenvalue problems , 2000 .

[23]  Ravi P. Agarwal,et al.  Advanced topics in difference equations , 1992 .

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

[25]  Haiyan Wang,et al.  On the existence of positive solutions of ordinary differential equations , 1994 .

[26]  S Umamaheswaram,et al.  Boundary value problems for higher order differential equations , 1975 .

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