## Abstract

##### Publisher Name

## Authors

Calin-Ioan **Gheorghiu**

Tiberiu Popoviciu Institute of Numerical Analysis

Damian **Trif**

Babes-Bolyai University, Faculty of Mathematics and Computer Science

## Keywords

*Dual variational methods in critical point theory and applications*, J. Funct. Anal., 14 , pp. 349–381, 1973.

*Nonlinear diffusion in population genetics, combustion and nerve pulse propagation*, Lecture Notes in Math., 446, Springer-Verlag, 1975.

*Nonlinear Sturm-Liouvil le eigenvalue problems and topological degree*, J. Math. Mech., 19 , pp. 1083–1102, 1970.

*Bifurcation, perturbation of simple eigenvalues, and linearized stability*, Arch. Rat. Mech. Anal., 52, pp. 161–180, 1973.

*Differential Equations and the Calculus of Variations*, Mir Publishers, Moscow, 1980.

*Solution to problem 97-8 by Ph. Korman,*SIAM Review, 39, 1997, SIAM Review, 40 , no. 2, pp. 382–385, 1998.

*On the bifurcation and variational approximation of the positive solution of a nonlinear*

*reaction-diffusion problem*, Studia UBB, Mathemat- ica, XLX, pp. 29–37, 2000.

*The Theory and Applications of Reaction-Diffusion Equations*; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.

*Some positone problems suggested by nonlinear heat generation*, J. Math. Mech., 16, pp. 1361–1376, 1967.

*Average temperature in a reaction-diffusion process,*Problem 97-8, SIAM Review, 39 , p. 318, 1997.

*The number of solutions of a nonlinear two point boundary value problem*, Indiana Univ. Math. J., 20, pp. 1–13, 1970.

*A simple nonlinear dynamic stability problem*, Bull. Amer. Math.Soc., 76, pp. 620–625, 1970.

*Nonoscil lation theorems for a class of nonlinear differential equations*, Trans. Amer. Math. Soc., 93 , pp. 30–52, 1959.

*Topics in Stability and Bifurcation Theory*, Springer-Verlag, 1973.

*Stability of bifurcating solutions by Leray-Schauder degree*, Arch. Rat. Mech. Anal., 43, pp. 155–165, 1970.

*Monotone methods in nonlinear el liptic and parabolic boundary value problems*, Indiana Univ. Math. J., 21, pp. 979–1000, 1972.

*A physically motivated further note on the mean value theorem for integrals,*Amer. Math. Monthly, 126 , pp. 559–564, 1999.

*Positive solutions of nonlinear el liptic eigenvalue problems,*J. Math. Mech., 19, pp. 895–910, 1970.

*Nonlinear eigenvalue problems with nonlocal operators*, Comm. Pure Appl. Math., 23, pp. 963–972, 1970.

*Weakly nonlinear stability analysis of prototype reaction-diffusion model equations*, SIAM Review, 36 , no. 2, pp. 176– 214, 1994.

## Cite this paper as

C.I. Gheorghiu, D. Trif, *Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. **Part I. Direct (variational) approximation*, Rev. Anal. Numér. Théor. Approx. **31** (2002) 61-70.

## About this paper

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1222-9024

##### Online ISSN

2457-8126

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[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, pp. 349-381, 1973.

[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.

[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19, pp. 1083-1102, 1970.

[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161-180, 1973.

[5] Elsgolts, L., Differential Equations and the Calculus of Variations, Mir Publishers, Moscow, 1980.

[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40, no. 2, pp. 382-385, 1998.

[7] Gheorghiu, C. I. and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem, Studia UBB, Mathematica, XLV, pp. 29-37, 2000.

[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.

[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361-1376, 1967.

[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39, pp.318, 1997.

[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1-13, 1970.

[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., 76, pp. 620-625, 1970.

[13] Moore, R. A. and Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93, pp. 30-52, 1959.

[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.

[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155-165, 1970.

[16] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979-1000, 1972.

[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126, pp. 559-564, 1999.

[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, pp. 895-910, 1970.

[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963-972, 1970.

[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations, SIAM Review, 36, no. 2, pp. 176-214, 1994.