Several Integral Inequalities

  • Shuli Guo
  • Lina Han


In this chapter, several new integral inequalities are presented, which are effective in dealing with the integrodifferential inequalities whose variable exponents are greater than 1. Compared with existed integral inequalities, those proposed here can be applied to more complicated differential equations.



This work is Supported by National Key Research and Development Program of China (2017YFF0207400).


  1. 1.
    Guo S, Irene M, Si L, Han L. Several integral inequalities and their applications in nonlinear differential systems. Appl Math Comput. 2013;219:4266–77.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Hong J. A result on stability of time-varying delay differential equation. Acta Math Sin. 1983;26(3):257–61.Google Scholar
  3. 3.
    Si L. Boundness, stability of the solution of time-varying delay neutral differential equation. Acta Math Sin. 1974;17(3):197–204.Google Scholar
  4. 4.
    Si L. Stability of delay neutral differential equations. Huhhot: Inner Mongolia Educational Press; 1994. p. 106–41.Google Scholar
  5. 5.
    Alekseev VM. An estimate for the perturbation of the solution of ordinary differential equation. Vestnik Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika. 1961;2:28–36.Google Scholar
  6. 6.
    Li Y. Boundness, stability and error estimate of the solution of nonlinear different equation. Acta Math Sin. 1962;12(1):28–36 In Chinese.Google Scholar
  7. 7.
    Brauer F. Perturbations of nonlinear systems of differential equations. J Math Anal Appl. 1966;14:198–206.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Elaydi S, Rao M, Rama M. Lipschitz stability for nonlinear Volterra integro differential systems. Appl Math Comput. 1988;27(3):191–9.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Giovanni A, Sergio V. Lipschitz stability for the inverse conductivity problem. Adv Appl Math. 2005;35(2):207–41.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hale JK. Ordinary differential equations. Interscience, New York: Wiley; 1969.zbMATHGoogle Scholar
  11. 11.
    Jiang F, Meng F. Explicit bounds on some new nonlinear integral inequalities with delay. J Comput Appl Math. 2007;205(1):479–86.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Soliman AA. Lipschitz stability with perturbing Liapunov functionals. Appl Math Lett. 2004;17(8):939–44.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Soliman AA. On Lipschitz stability for comparison systems of differential equations via limiting equation. Appl Math Comput. 2005;163(3):1061–7.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Huang L. Stability theory. Beijing: Peking University Press; 1992. p. 235–83.Google Scholar
  15. 15.
    Horn RA, Johnson CR. Topics in matrix analysis. Cambridge University press; 1991.Google Scholar
  16. 16.
    Kumpati SN, Jeyendran B. A common lyapunov function for stable LTI systems with commuting A-matrixes. IEEE Trans Autom Control. 1994;39(12):2469–71.CrossRefGoogle Scholar
  17. 17.
    Lee SH, Kim TH, Lim JT. A new stability analysis of switched systems. Automatica. 2000;36:917–22.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liberzon D, Hespanta JP, Morse AS. Stability of switched systems: a Lie-algebraic condition. Syst Control Lett N-Holl. 1999;37:117–22.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mareada KS, Balakrishan J. A common Lyapunov function for stable LTI systems with commuting \(A\)-martices. IEEE Trans Autom Control. 1994;39(12):2469–71.CrossRefGoogle Scholar
  20. 20.
    Molchanov AP, Pyatnitskiy YS. Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst Control Lett N-Holl. 1989;13:59–64.MathSciNetCrossRefGoogle Scholar
  21. 21.
    La Salle J, Lefschetz S. Stability by Lyapunov’s direct method. New York, N.Y.: Academic Press; 1961.zbMATHGoogle Scholar
  22. 22.
    Polanski K. On absolute stability analysis by polyhydric Lypunov functions. Automatica. 2000;36:573–8.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schmitendorf WE, Barmish BR. Null controllability of linear systems with constrained controls. SIAM J Control Optim. 1980;18:327–45.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shevitz D, Paden B. Lyapunov stability theory of non-smooth systems. In: Proceeding of the 32nd conference on decision and control, San Antonio, Texas; 1993. p. 416–421.Google Scholar
  25. 25.
    Tatsushi O, Yasuyuki F. Two conditions concerning common quadratic lyapunov functions for linear systems. IEEE Trans Autom Control. 1997;42(5):719–21.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Complex System Intelligent Control and Decision, School of AutomationBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Cardiovascular Internal Medicine of Nanlou Branch, National Clinical Research Center for Geriatric DiseasesChinese PLA General HospitalBeijingChina

Personalised recommendations