Skip to main content
SpringerLink
Log in

Vector-valued measures of noncompactness and the Cauchy problem with delay in a scale of Banach spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, by developing a new method we advance to prove the existence results for the Cauchy problem with delay \(d_tu=f(t,u(t),u(h(t))),\;t\in (0,T),\;u(0)=u_0\), in a scale of Banach spaces. The arguments are based upon Darbo–Sadovskii fixed point theorem for condensing operators on Fréchet spaces and the technique of vector-valued measure of noncompactness.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Sadovskii, B.N.: Measure of Noncompactness and Condensing Operators. Birkhäuser, Basel (1992)

    Book  Google Scholar 

  2. Augustynowicz, A., Leszczyński, H., Walter, W.: Cauchy–Kovalevskaya theory for nonlinear equations with deviating variables. Nonlinear Anal. 45, 743–753 (2001)

    Article  MathSciNet  Google Scholar 

  3. Ayerbe Toledano, J.M., Dominguez Benavides, T., Lopez Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997)

    Book  Google Scholar 

  4. Barkova, E.A., Zabreiko, P.P.: An analog of the Peano theorem for fractional-order quasilinear equations in compactly embedded scale of Banach spaces. Differ. Equ. 40(4), 565–570 (2004)

    Article  MathSciNet  Google Scholar 

  5. Barkova, E.A., Zabreiko, P.P.: Fractional differential equations with worsening right-hand sides. Differ. Equ. 46(2), 208–213 (2010)

    Article  MathSciNet  Google Scholar 

  6. Barostichi, R.F., Himonas, A.A., Petronilho, G.: Autonomous Ovsyanniikov theorem and applications to nonlocal evolution equations and systems. J. Funct. Anal. 270, 330–358 (2016)

    Article  MathSciNet  Google Scholar 

  7. Barostichi, R.F., Himonas, A.A., Petronilho, G.: The power series method for nonlocal and nonlinear evolution equations. J. Math. Anal. Appl. 443(2), 834–847 (2016)

    Article  MathSciNet  Google Scholar 

  8. Finkelshtein, D.: Around Ovcyannikov’s method. Methods Funct. Anal Topol. 21, 131–150 (2015)

    Google Scholar 

  9. Finkelshtein, D., Kondratiev, Y., Kutoviy, O., Zhizhina, E.: On an aggregation in birth-and-death stochastic dynamics. Nonlinearity 27, 1105–1133 (2014)

    Article  MathSciNet  Google Scholar 

  10. Ghisi, M.: The Cauchy–Kowalevsky theorem and noncompactness measure. J. Math. Sci. Univ. Tokyo 4, 627–647 (1994)

    MATH  Google Scholar 

  11. Gourdin, D., Gramchev, T.: Global in time solutions of evolution equations in scales of Banach function spaces in \({\mathbb{R}}^n\). Bull. Sci. Math. 131, 761–786 (2007)

    Article  MathSciNet  Google Scholar 

  12. Huy, N.B., Sum, N.A., Tuan, N.A.: A second-order Cauchy problem in a scale of Banach spaces and application to Kirchhoff equations. J. Differ. Equ. 206, 253–264 (2004)

    Article  MathSciNet  Google Scholar 

  13. Kawagishi, M., Yamanaka, T.: On the Cauchy problem for PDEs in the Gevrey class with shrinking. J. Math. Soc. Jpn. 54, 649–677 (2002)

    Article  MathSciNet  Google Scholar 

  14. Maekava, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane. Commun. Pure Appl. Math. 67, 1045–1128 (2014)

    Article  MathSciNet  Google Scholar 

  15. Nirenberg, L.: An abstract form of the nonlinear Cauchy–Kowalewski theorem. J. Differ. Geom. 6, 561–576 (1972)

    Article  MathSciNet  Google Scholar 

  16. Nishida, T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12, 629–633 (1977)

    Article  MathSciNet  Google Scholar 

  17. Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61, 119–148 (1978)

    Article  MathSciNet  Google Scholar 

  18. Ovcyannikov, L.V.: A nonlinear Cauchy problem in a scale of Banach spaces. Sov. Math. Dokl. 12, 1497–1502 (1971)

    Google Scholar 

  19. Safonov, M.V.: The abstract Cauchy–Kowalevskaya theorem in a weighted Banach space. Commun. Pure Appl. Math. VXLVII I, 629–637 (1995)

    Article  Google Scholar 

  20. Sammartino, M., Caflish, R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on half-space I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192, 433–461 (1998)

    Article  MathSciNet  Google Scholar 

  21. Yamanaka, T.: Note on Kowalewskaya system of partial differential equations. Commun. Math. Univ. St Paul 9, 7–10 (1960)

    Google Scholar 

  22. Yamanaka, T., Kawagishi, M.: A Cauchy–Kowalevskaya type theory in the Gevrey class for PDEs with shrinking. Nonlinear Anal. 64, 1860–1884 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the referees for their careful reading the work that improved the paper. This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.02-2019.327.

Author information

Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Huy Nguyen Bich

  2. Faculty of Applied Sciences, HCMC University of Technology and Education, Ho Chi Minh City, Vietnam

    Hien Pham Van

Authors
  1. Huy Nguyen Bich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hien Pham Van
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huy Nguyen Bich.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bich, H.N., Van, H.P. Vector-valued measures of noncompactness and the Cauchy problem with delay in a scale of Banach spaces. J. Fixed Point Theory Appl. 22, 36 (2020). https://doi.org/10.1007/s11784-020-0771-2

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-020-0771-2

Keywords

Mathematics Subject Classification

Search

Navigation