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New Henry–Gronwall Integral Inequalities and Their Applications to Fractional Differential Equations

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Abstract

Some new Henry–Gronwall integral inequalities are established, which generalize some former famous inequalities and can be used as powerful tools in the study of differential and integral equations.

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References

  • Agarwal, R.P., Meahan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  • Aghajani, A., Pourhadi, E., Trujillo, J.J.: Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 16, 962–977 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Baleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Barbu, V.: Differential Equations (in Romanian). Junimea, Iasi (1985)

    Google Scholar 

  • Daywish, M.A.: On integral equations of Urysohn–Volterra type. Appl. Math. Comput. 136, 93–98 (2003)

    MathSciNet  Google Scholar 

  • Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Denton, Z., Vatsala, A.S.: Fractional integral inequalities and applications. Comput. Math. Appl. 59, 1087–1094 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Granas, A., Dugudji, J.: Fixed Point Theory. Springer, Berlin (2003)

    Book  Google Scholar 

  • Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (eds.): Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elseiver, Amsterdam (2006)

    Google Scholar 

  • Laskhmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, Q., Pečarić, J.: Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential equations and integral equations. J. Math. Anal. Appl. 341, 894–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Martyniuk, A.A., Lakshmikanthan, V., Leela, S.: Motion Stability: The Method of Integral Inequalities. Naukova Dumka, Kiev (1977). [In Russian]

    Google Scholar 

  • Medveď, M.: A new approach to an analysis of Henry type integral inequalities and their Bihair type versions. J. Math. Anal. Appl. 214, 349–366 (1997)

    Article  MathSciNet  Google Scholar 

  • Medveď, M.: Integral inequalities and global solutions of semilinear evolution equations. J. Math. Anal. Appl. 267, 643–650 (2002)

    Article  MathSciNet  Google Scholar 

  • Millett, D.: Nonlinear vector integral equations as contraction mappings. Arch. Ration. Mech. Anal. 15, 79–86 (1964)

    Article  MathSciNet  Google Scholar 

  • Millett, D., Wong, J.S.W.: On discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69, 362–367 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  • O’Regan, D.: Existence results for nonlinear integral equations. J. Math. Anal. Appl. 192, 705–726 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Podlubny, I.: Fractional Differential Equations: An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  • Wang, J., Zhou, Y., Fečkan, M.: Abstract Cauchy problem for fractional differential equations. Nonlinear Dyn. 71, 685–700 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, C., Gao, G.: Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26–29 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, S.: The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 252, 804–812 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, Y.: Existence and uniqueness of solutions for a system of fractional differential equations. Fract. Calc. Appl. Anal. 12, 195–204 (2009)

    MathSciNet  MATH  Google Scholar 

  • Zhu, T., Zhong, C., Song, C.: Existence results for nonlinear fractional differential equations in \(C[0,T)\). J. Appl. Math. Comput. (2017). https://doi.org/10.1007/s12190-017-1094-3

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Acknowledgements

The research was supported by Scientific Research Foundation of Nanjing Institute of Technology (no: CKJB201508).

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Correspondence to Tao Zhu.

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Zhu, T. New Henry–Gronwall Integral Inequalities and Their Applications to Fractional Differential Equations. Bull Braz Math Soc, New Series 49, 647–657 (2018). https://doi.org/10.1007/s00574-018-0074-z

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  • DOI: https://doi.org/10.1007/s00574-018-0074-z

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