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Positive solutions and eigenvalues of conjugate boundary value problems

Published online by Cambridge University Press:  20 January 2009

Ravi P. Agarwal ,
Martin Bohner  and
Patricia J. Y. Wong
Ravi P. Agarwal
Affiliation:
National University of Singapore, Department of Mathematics, 10 Kent Ridge Crescent, Singapore119260 E-mail address: matravip@leonis.nus.sg martin@saturn.sdsu.edu
Martin Bohner
Affiliation:
National University of Singapore, Department of Mathematics, 10 Kent Ridge Crescent, Singapore119260 E-mail address: matravip@leonis.nus.sg martin@saturn.sdsu.edu
Patricia J. Y. Wong
Affiliation:
Nanyang Technological University, Division of Mathematics, 469 Bukit Timah Road, Singapore, 259756 E-mail address: wongjyp@nievax.nie.ac.sg
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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 2 , June 1999 , pp. 349 - 374
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations (World Scientific, Singapore, 1986).CrossRefGoogle Scholar
2.Agarwal, R. P. and Wong, P. J. Y., Existence of solutions for singular boundary value problems for higher order differential equations. Rend. Sem. Mat. Fis. Milano 65 (1995), 249–264.CrossRefGoogle Scholar
3.Agarwal, R. P. and Wong, P. J. Y., Advanced Topics in Difference Equations (Kjuwer Academic Publishers, Dordrecht, 1997).CrossRefGoogle Scholar
4.Aronson, D., Crandall, M. G. and Peletier, L. A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), 1001–1022.Google Scholar
5.Atici, F. and Peterson, A. C., Bounds for positive solutions for a focal boundary value problem, in Advances in Difference Equations II, Comput. Math. Appl., to appear.Google Scholar
6.Bandle, C., Coffman, C. V. and Marcus, M., Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), 322–345.Google Scholar
7.Bandle, C. and Kwong, M. K., Semilinear elliptic problems in annular domains, J. Appl. Math. Phys. 40 (1989), 245–257.Google Scholar
8.Choi, Y. S. and Ludford, G. S., An unexpected stability result of the near-extinction diffusion flame for non-unity lewis numbers, Quart. J. Mech. Appl. Math. 42 (1989), 143–158.Google Scholar
9.Chyan, C. J. and Henderson, J., Positive solutions for singular higher order nonlinear equations, Diff. Eqs. Dyn. Sys. 2 (1994), 153–160.Google Scholar
10.Cohen, D. S., Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory, SIAM J. Appl. Math. 20 (1971), 1–13.CrossRefGoogle Scholar
11.Dancer, E. N., On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37 (1980), 404–437.CrossRefGoogle Scholar
12.Eloe, P. W. and Henderson, J., Positive solutions for (n – 1, 1) boundary value problems, Nonlinear Anal. 28(1997), 1669–1680.Google Scholar
13.Eloe, P. W. and Henderson, J., Positive solutions and nonlinear (k, n – k) conjugate eigenvalue problems.Google Scholar
14.Erbe, L. H., Hu, S. and Wang, H., Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), 640–648.Google Scholar
15.Erbe, L. H. and Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743–748.Google Scholar
16.Fink, A. M., The radial Laplacian Gel'fand problem, in Delay and Differential Equations (Ames, IA, 1991, World Sci. Publishing, River Edge, NJ, 1992), 93–98.Google Scholar
17.Fink, A. M., Gatica, J. A. and Hernandez, G. E., Eigenvalues of generalized Gel'fand models, Nonlinear Anal. 20 (1993), 1453–1468.CrossRefGoogle Scholar
18.Fujita, H., On the nonlinear equations δu + eu = 0 and , Bull. Amer. Math. Soc. 75(1969), 132–135.CrossRefGoogle Scholar
19.Garaizar, X., Existence of positive radial solutions for semilinear elliptic problems in the annulus, J. Differential Equations 70 (1987), 69–72.CrossRefGoogle Scholar
20.Gatica, J. A., Oliker, V. and Waltman, P., Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations 79 (1989), 62–78.CrossRefGoogle Scholar
21.Gel'fand, I. M., Some problems in the theory of quasilinear equations, Uspehi Mat. Nauka 14 (1959), 87–158. English translation, Trans. Amer. Math. Soc. 29 (1963), 295–381.Google Scholar
22.Guo, D. and Lakshmikantham, V., Nonlinear Problems in Abstract Cones (Academic Press, San Diego, 1988).Google Scholar
23.Gustafson, G. B., A Green's function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math. 6 (1976), 457–492.Google Scholar
24.Hankerson, D. and Peterson, A. C., Comparison of eigenvalues for focal point problems for n-th order difference equations, Differential Integral Equations 3 (1990), 363–380.CrossRefGoogle Scholar
25.Henderson, J., Singular boundary value problems for difference equations, Dynamic Systems Appl. 1 (1992), 271–282.Google Scholar
26.Henderson, J., Singular boundary value problems for higher order difference equations, Lakshmikantham, V., ed., in Proceedings of the First World Congress on Nonlinear Analysts (Walter de Gruyter and Co., 1996), 1139–1150.CrossRefGoogle Scholar
27.Krasnosel'skii, M. A., Positive Solutions of Operator Equations (Noordhoff, Groningen, 1964).Google Scholar
28.Merdivenci, F., Two positive solutions of a boundary value problem for difference equations. J Differ. Equations Appl. 1 (1995), 262–270.Google Scholar
29.Merdivenci, F., Green's matrices and positive solutions of a discrete boundary value problem, Panamer. Math. J. 5 (1995), 25–42.Google Scholar
30.Merdivenci, F., Positive solutions for focal point problems for 2n-th order difference equations, Panamer. Math. J. 5 (1995), 71–82.Google Scholar
31.O'regan, D., Theory of Singular Boundary Value Problems (World Scientific, Singapore, 1994).Google Scholar
32.Parter, S., Solutions of differential equations arising in chemical reactor processes, SIAM J. Appl. Math. 26 (1974), 687–716.CrossRefGoogle Scholar
33.Peterson, A. C., Boundary value problems for an n-th order difference equation, SIAM J. Math. Anal. 15 (1984), 124–132.Google Scholar
34.Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J Differential Equations 109 (1994), 1–7.Google Scholar
35.Wong, P. J. Y. and Agarwal, R. P., Eigenvalues of boundary value problems for higher order differential equations, Mathematical Problems in Engineering 2 (1996), 401–434.CrossRefGoogle Scholar
36.Wong, P. J. Y. and Agarwal, R. P., Double positive solutions of (n, p) boundary value problems for higher order difference equations, Comput. Math. Appl. 32 (8) (1996), 1–21.CrossRefGoogle Scholar
37.Wong, P. J. Y. and Agarwal, R. P., On the existence of solutions of singular boundary value problems for higher order difference equations, Nonlinear Anal. 28 (1997), 277–287.CrossRefGoogle Scholar
38.Wong, P. J. Y. and Agarwal, R. P., On the existence of positive solutions of higher order difference equations, Topological Methods in Nonlinear Analysis 10 (1997), 339–351.Google Scholar
39.Wong, P. J. Y. and Agarwal, R. P., On the eigenvalues of boundary value problems for higher order difference equations, Rocky Mountain J. Math., to appear.Google Scholar
40.Wong, P. J. Y. and Agarwal, R. P., Eigenvalue characterization for (n, p) boundary value problems, J. Austral. Math. Soc. Ser. B. 39 (1998), 386–407.CrossRefGoogle Scholar
41.Wong, P. J. Y. and Agarwal, R. P., Eigenvalues of an nth order difference equation with (n, p) type conditions, Dynam. Contin. Discrete Impuls. Systems 4 (1998), 149–172.Google Scholar
42.Wong, P. J. Y. and Agarwal, R. P., On eigenvalues and twin positive solutions of (n, p) boundary value problems, Functional Differential Equations 4 (1997), 443–476.Google Scholar
43.Wong, P. J. Y. and Agarwal, R. P., Extension of continuous and discrete inequalities due to Eloe and Henderson, Nonlinear Anal. 34 (1998), 479–487.CrossRefGoogle Scholar