Positive solutions and eigenvalues of conjugate boundary value problems

We consider the following boundary value problem where λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

[1]  G. Gustafson.  A Green's function convergence principle, with applications to computation and norm estimates , 1976 .

[2]  A. Peterson,et al.  Bounds for positive solutions for a focal boundary value problem , 1998 .

[3]  R. Agarwal,et al.  Eigenvalues of boundary value problems for higher order di erential equations , 1996 .

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

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

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

[7]  M. Kwong,et al.  Semilinear elliptic problems in annular domains , 1989 .

[8]  Shouchuan Hu,et al.  Multiple Positive Solutions of Some Boundary Value Problems , 1994 .

[9]  Ravi P. Agarwal,et al.  On the existence of positive solutions of higher order difference equations , 1997 .

[10]  Donal O'Regan,et al.  Theory of singular boundary value problems , 1994 .

[11]  J. Henderson.  Singular boundary value problems for higher order difference equations , 1996 .

[12]  Haiyan Wang,et al.  On the existence of positive solutions for semilinear elliptic equations in the annulus , 1994 .

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

[14]  A. M. Fink,et al.  Eigenvalues of generalized Gelfand models , 1993 .

[15]  Ferhan Merdivenci,et al.  Two positive solutions of a boundary value problem for difference equations , 1995 .

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

[17]  Paul Waltman,et al.  Singular nonlinear boundary value problems for second order ordinary di erential equations , 1989 .

[18]  R. Agarwal,et al.  Extension of continuous and discrete inequalities due to Eloe and Henderson , 1998 .

[19]  Ravi P. Agarwal,et al.  On the existence of solutions of singular boundary value problems for higher order difference equations , 1997 .

[20]  P. J. Wong,et al.  Eigenvalue characterization for (n · p) boundary-value problems , 1998, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[21]  Boundary Value Problems for an $n$th Order Linear Difference Equation , 1984 .

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

[23]  J. Henderson,et al.  Positive solutions for ( n −1,1) conjugate boundary value problems , 1997 .

[24]  Patricia J. Y. Wong,et al.  Existence of solutions for singular boundary problems for higher order differential equations , 1995 .

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

[26]  Ravi P. Agarwal,et al.  Double positive solutions of (n, p) boundary value problems for higher order difference equations , 1996 .

[27]  R. Agarwal,et al.  ON THE EIGENVALUES OF BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER DIFFERENCE EQUATIONS , 1998 .

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

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

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

[31]  Ravi P. Agarwal,et al.  Boundary value problems for higher order differential equations , 1986 .

[32]  C. Bandle,et al.  Nonlinear elliptic problems in annular domains , 1987 .

[33]  X. Garaizar.  Existence of Positive Radial Solutions for Semilinear Elliptic Equations in the Annulus , 1987 .

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

[35]  Darrel Hankerson,et al.  Comparison of eigenvalues for focal point problems for $n$th order difference equations , 1990, Differential and Integral Equations.