Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Method of quasilinearization and positivity of solutions in abstract cones

  • 47 Accesses

  • 2 Citations


In this paper, the method of quasilinearization is extended to differential equations in Banach spaces using abstract cones. Frequently, it is useful to apply this technique to a comparison system. Several applications are given indicating the flexibility obtained by appropriate choice of the comparison system and cone.

This is a preview of subscription content, log in to check access.


  1. 1.

    Bellman, R.,Functional Equations in the Theory of Dynamic Programming—V: Positivity and Quasilinearity, Proceedings of the National Academy of Sciences, USA, Vol. 41, pp. 743–746, 1955.

  2. 2.

    Kalaba, R.,On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence, Journal of Mathematics and Mechanics, Vol. 8, pp. 519–574, 1959.

  3. 3.

    Bellman, R., andKalaba, R.,Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier Publishing Company, New York, New York, 1965.

  4. 4.

    Lakshmikantham, V., andLeela, S.,Differential and Integral Inequalities, Vols. 1 and 2, Academic Press, New York, New York, 1969.

  5. 5.

    Cronin, J.,Periodic Solutions in N-Dimensions and Volterra Equations, Journal of Differential Equations, Vol. 19, pp. 21–35, 1975.

  6. 6.

    Ladde, G. S.,Competitive Processes and Comparison Differential Systems, Transactions of the American Mathematical Society (to appear).

  7. 7.

    Lakshmikantham, V., Mitchell, A. R., andMitchell, R. W.,Maximal and Minimal Solutions and Comparison Results for Differential Equations in Abstract Cones, Annales Polonici Mathematici (to appear).

  8. 8.

    Volkmann, P., Gewohnliche Differentialungleichungen mit Quasimonoton Wachsenden Functionen in Topologischen Vektorraumen, Mathematische Zeitschrift, Vol. 127, pp. 157–164, 1972.

  9. 9.

    Ladas, G. E., andLakshmikantham, V.,Differential Equations in Abstract Spaces, Academic Press, New York, New York, 1972.

  10. 10.

    Lakshmikantham, V.,Existence and Comparison Results for Differential Equations, Proceedings of the International Conference on Differential Equations, Academic Press, New York, New York, pp. 459–473, 1975.

  11. 11.

    Lakshmikantham, V.,Stability and Asymptotic Behavior of Solutions of Differential Equations in a Banach Space, Lecture Notes, CIME, Edizioni Cremonese, Rome, Italy, 1974.

  12. 12.

    Bronson, E., Mitchell, A. R., andTennison, R. L.,On the Existence of Solutions of Differential Equations and Zeros of Operators in K-Banach Spaces (to appear).

  13. 13.

    Bernfeld, S. R., andLakshmikantham, V.,An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, New York, 1974.

  14. 14.

    Perov, A., andKibenko, A.,On a Certain General Method for Investigation of Boundary Value Problems, Izvestija Akademii Nauk SSSR, Vol. 30, pp. 249–264, 1966.

Download references

Author information

Additional information

Communicated by R. E. Kalaba

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lakshmikantham, V., Mitchell, A.R. & Sety, D.D. Method of quasilinearization and positivity of solutions in abstract cones. J Optim Theory Appl 22, 353–372 (1977). https://doi.org/10.1007/BF00932860

Download citation

Key Words

  • Quasilinearization methods
  • abstract spaces
  • differential equations
  • inequality theory