Skip to main content
Log in

On the Solvability of a Nonlinear Integro-Differential Equation on the Half-Axis

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this work, we combine the iterative techniques with fixed point theory to investigate the existence of absolutely continuous solutions to a class of nonlinear integro-differential equations. Existence results are obtained under fairly simple conditions of Carathéodory type.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Appell J., De Pascale E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili. Boll. Unicone Math. Ital. B 3(6), 497–515 (1984)

    MathSciNet  Google Scholar 

  2. Appell J., Kalitvin A.S., Zabrejko P.P.: Partial Integral Operators and Integro-Differential Equations. Macel Dekker, Inc., New York (2000)

    MATH  Google Scholar 

  3. Banas J., Rivero J.: On measures of weak noncompactness. Ann. Math. Pure. Appl. 151, 213–224 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berenguer, M.I., Garralda-Guillem, A. I., Ruiz Galán, M.: Biorthogonal systems approximating the solution of the nonlinear Volterra integro-differential equation. Fixed Point Theory Appl. 470149, 1–9. doi:10.1155/2010/470149

  5. Berenguer M.I., Garralda-Guillem A.I., Ruiz Galán M.: An approximation method for solving systems of Volterra integro-differential equations. Appl. Numer. Math. 67, 126–135 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cascales, B., Orihuela, J., Ruiz Galn, M.: Compactness, optimality, and risk. In: Computational and Analytical Mathematics. Springer Proc. Math. Stat., vol. 50, pp. 161–218. Springer, New York (2013)

  7. Cascales B., Pérez A., Raja M.: Radon–Nikodm indexes and measures of weak noncompactness. J. Funct. Anal. 267(10), 3830–3858 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cushing, J.M.: Integrodifferential equations and delay models in population dynamics. In: Lecture Notes in Biomathematics. Springer, New York (1977)

  9. Cascales, B.; Orihuela, J.; Ruiz Galán, M.: Compactness, optimality and risk. In: Computational and Analytical Mathematics. Springer Proceedings in Mathematics and Statistics, vol. 50, pp. 161–218. Springer, New York (2013)

  10. Cascales B., Pérez A., Raja M.: Radon–Nikodým indexes and measures of weak noncompactness. J. Funct. Anal. 267, 3830–3858 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. De Blasi F.S.: On a property of the unit sphere in Banach spaces. Bull. Math. Soc. Sci. Math. Roum. 21, 259–262 (1977)

    MathSciNet  MATH  Google Scholar 

  12. Dieudonné J.: Sur les espaces de Kothe. J. Anal. Math. 1, 81–115 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jachymski J.: On Isac’s fixed point theorem for selfmaps of a Galerkin cone. Ann. Sci. Math. Qubec 18(2), 169–171 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Jain P.K., Gupta V.P.: Lebesgue Measure and Integration. Wiley Eastern Limited, New Delhi (1989)

    MATH  Google Scholar 

  15. Jerri, A.J.: Introduction to Integral Equations with Applications. Wiley, New York (1999)

  16. Kostin, J.V., Saurin, V.V.: Integrodifferential Relations in Linear Elasticity. De Gruyter, Berlin (2012)

  17. Latrach K., Taoudi M.A., Zeghal A.: Some fixed point theorems of the Schauder and the Krasnoselskii type and application to nonlinear transport equations. J. Differ. Equ. 221, 256–271 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton (2008)

  19. Royden K.L.: Real Analysis. Macmillan Publishing Co., Inc., New York (1968)

    MATH  Google Scholar 

  20. Shakeri F., Dehghan M.: Solution of a model describing biological species living together using the variational iteration method. Math. Comput. Model. 48, 685–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Azzeddine Bellour.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bellour, A., Bousselsal, M. & Taoudi, MA. On the Solvability of a Nonlinear Integro-Differential Equation on the Half-Axis. Mediterr. J. Math. 13, 2887–2896 (2016). https://doi.org/10.1007/s00009-015-0662-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-015-0662-8

Mathematics Subject Classification

Keywords

Navigation