Glasnik Matematicki, Vol. 49, No. 2 (2014), 395-406.

INTEGRABLE SOLUTIONS OF A NONLINEAR INTEGRAL EQUATION RELATED TO SOME EPIDEMIC MODELS

Azzeddine Bellour, Mahmoud Bousselsal and Mohamed-Aziz Taoudi

Department of Mathematics, Ecole Normale Superieure de Constantine, Constantine, Algeria , 25000, Constantine-Algeria
e-mail: bellourazze123@yahoo.com

Department of Mathematics, Laboratoire d'EDP non linéaires, Ecole Normale Superieure, Vieux Kouba, 16050, Algiers-Algeria
e-mail: Bousselsal55@gmail.com

Université Cadi Ayyad, Centre Universitaire Kalaa des Sraghnas, Kalaa des Sraghnas, Morocco
e-mail: mataoudi@gmail.com


Abstract.   In this paper, we discuss the existence of integrable solutions for a nonlinear integral equation related to some epidemic models. The analysis uses the techniques of measures of noncompactness and relies on an improved version of the Krasnosel'skii fixed point theorem.

2010 Mathematics Subject Classification.   45D05, 45G10, 47H30.

Key words and phrases.   Integral equations, measure of weak noncompactness, fixed point theorem, integrable solutions.


Full text (PDF) (free access)

DOI: 10.3336/gm.49.2.12


References:

  1. A. Abdeldaim, On some new Gronwall-Bellman-Ou-Iang type integral inequalities to study certain epidemic models, J. Integral Equations Appl. 24 (2012), 149-166.
    MathSciNet     CrossRef

  2. R. P. Agarwal, D. O'Regan and M.-A. Taoudi, Fixed point theorems for ws-compact mappings in Banach spaces, Fixed Point Theory Appl. 2010, Art. ID 183596, 13 pp.
    MathSciNet    

  3. J. Appell and E. De Pascale, Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B (6) 3 (1984), 497-515.
    MathSciNet    

  4. J. Appell and P. P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, 1990.
    MathSciNet     CrossRef

  5. N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, Hafner Press, New York, 1975.
    MathSciNet    

  6. J. Banas and J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. (4) 151 (1988), 213-224.
    MathSciNet     CrossRef

  7. H. Brezis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983.
    MathSciNet    

  8. F. S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie 21 (1977), 259-262.

  9. O. Diekmann, Run for your life, A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations 33 (1979), 58-73.
    MathSciNet     CrossRef

  10. S. Djebali and Z. Sahnoun, Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in L1 spaces, J. Differential Equations 249 (2010), 2061-2075.
    MathSciNet     CrossRef

  11. J. Dieudonné, Sur les espaces de Köthe, J. Analyse Math. 1 (1951), 81-115.
    MathSciNet     CrossRef

  12. J. García-Falset, Existence of fixed points and measure of weak noncompactness, Nonlinear Anal. 71 (2009), 2625-2633.
    MathSciNet     CrossRef

  13. J. Garcia-Falset, K. Latrach, E. Moreno-Gálvez and M.-A. Taoudi, Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness, J. Differential Equations 252 (2012), 3436-3452.
    MathSciNet     CrossRef

  14. G. Gripenberg, Periodic solutions of an epidemic model, J. Math. Biol. 10 (1980), 271-280.
    MathSciNet     CrossRef

  15. G. Gripenberg, On some epidemic models, Quart. Appl. Math. 39 (1981/82), 317-327.
    MathSciNet    

  16. J. Jachymski, On Isac's fixed point theorem for selfmaps of a Galerkin cone, Ann. Sci. Math. Québec 18 (1994), 169-171.
    MathSciNet    

  17. K. Latrach and M.-A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1-spaces, Nonlinear Anal. 66 (2007), 2325-2333.
    MathSciNet     CrossRef

  18. K. Latrach, M.-A. Taoudi and A. Zeghal, Some fixed point theorems of the Schauder and Krasnosel'skii type and application to nonlinear transport equations, J. Differential Equations 221 (2006), 256-271.
    MathSciNet     CrossRef

  19. L. Li, F. Meng and P. Ju, Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay, J. Math. Anal. Appl. 377 (2011), 853-862.
    MathSciNet     CrossRef

  20. Y. C. Liu and Z. X. Li, Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl. 316 (2006), 237-255.
    MathSciNet     CrossRef

  21. M. A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlinear Anal. 71 (2009), 4131-4136.
    MathSciNet     CrossRef

  22. I. M. Olaru, Generalization of an integral equation related to some epidemic models, Carpathian J. Math. 26 (2010), 92-96.
    MathSciNet    

  23. B. G. Pachpatte, On a new inequality suggested by the study of certain epidemic models, J. Math. Anal. Appl. 195 (1995), 638-644.
    MathSciNet     CrossRef

  24. G. Scorza Dragoni, Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile, R Rend. Sem. Mat. Univ. Padova 17 (1948), 102-106.
    MathSciNet     CrossRef

  25. P. Waltman, Deterministic threshold models in the theory of epidemics, Springer-Verlag, New York, 1974.
    MathSciNet    

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