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Wang Wu-Sheng. Estimation of unknown function of weakly singular iterated integral inequality[J]. Journal of South China Normal University (Natural Science Edition), 2017, 49(4): 111-114.
Citation: Wang Wu-Sheng. Estimation of unknown function of weakly singular iterated integral inequality[J]. Journal of South China Normal University (Natural Science Edition), 2017, 49(4): 111-114.

Estimation of unknown function of weakly singular iterated integral inequality

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  • Corresponding author:

    Wang Wu-Sheng

    Wang Wu-Sheng

  • Received Date: December 23, 2015
  • Revised Date: March 04, 2016
  • A class of nonlinear weakly singular iterated integral inequalities is given. There is a nonconstant term outside the integrals in the integral inequality. The upper bound of the embedded unknown function of the inequalities is estimated explicitly by adopting novel analysis techniques, such as:discrete Jensen inequality, H\"older integral inequality, the techniques of change of variable, and the method of amplification. The derived results can be applied in the study of qualitative properties of solutions of fractional integral equations.

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    parameter of the solutions of a system of differential equations[J].
    Ann Math. 1919, {\bf 20}: 292-296.



    Medve\v d M. A new approach to an analysis of Henry type integral inequalities and their Bihari type versions[J]. J Math
    Anal Appl. 1997, {\bf 214}: 349-366.





    Agarwal R.P. Deng S. and Zhang W., Generalization of a retarded Gronwall-like inequality and its
    applications[J]. Appl Math Comput. 2005, {\bf 165}: 599-612.



    Ma Q.H., Pe\v cari\'c J. Some new explicit bounds for weakly singular integral
    inequalities with applications to fractional differential and integral equations[J]. J.
    Math. Anal. Appl. 2008, \textbf{341}(2), 894-905.


    Abdeldaim A. and Yakout M. On some new integral inequalities of Gronwall-Bellman-Pachpatte
    type[J]. Appl Math Comput. 2011, {\bf 217}: 7887-7899.



    Zheng B. Explicit bounds derived by some new inequalities and applications in fractional integral equations[J]. Journal of Inequalities and Applications, 2014, {\bf 2014}(4), 1-12.



    Kuczma M. {\it An introduction to the theory of functional equations and inequalities:
    Cauchy’s equation and Jensen’s inequality}[M], University of Katowice, Katowice, 1985.

    Medve\v d M. Nonlinear singular integral inequalities for functions in two and n independent
    variables[J]. J. Inequal. Appl. 2000, {\bf 5}(3): 287-308.

    马庆华, 杨恩浩. 弱奇性Volterra积分不等式解的估计[J]. 应用数学学报, 2002, {\bf 25}: 505-515.

    吴宇.关于一类弱奇性Volterra 积分不等式的注记[J].四川师范大学学报(自然科学版), 2008, {\bf 31}(5): 534-537.

    王五生, 李自尊. 含多个非线性项的时滞积分不等式及其应用[J]. 数学进展, 2012, {\bf 41}(5): 597-604.




    梁英. 一类时滞弱奇异Wendroff型积分不等式[J]. 四川师范大学学报(自然科学版), 2014, {\bf 37}(4): 493-496.






    卢钰松, 王五生. 一类含有p次幂的Volterra-Fredholm型非线性迭代积分不等式[J]. 西南大学学报(自然科学版), 2015, {\bf 27}(8): 76-80.




    Yong Y. On some new weakly singular Volterra integral inequalities with maxima and their applications, Journal of Inequalities and Applications, 2015, {\bf 2015}:369.






    Gronwall T.H. Note on the derivatives with respect to a
    parameter of the solutions of a system of differential equations[J].
    Ann Math. 1919, {\bf 20}: 292-296.



    Medve\v d M. A new approach to an analysis of Henry type integral inequalities and their Bihari type versions[J]. J Math
    Anal Appl. 1997, {\bf 214}: 349-366.





    Agarwal R.P. Deng S. and Zhang W., Generalization of a retarded Gronwall-like inequality and its
    applications[J]. Appl Math Comput. 2005, {\bf 165}: 599-612.



    Ma Q.H., Pe\v cari\'c J. Some new explicit bounds for weakly singular integral
    inequalities with applications to fractional differential and integral equations[J]. J.
    Math. Anal. Appl. 2008, \textbf{341}(2), 894-905.


    Abdeldaim A. and Yakout M. On some new integral inequalities of Gronwall-Bellman-Pachpatte
    type[J]. Appl Math Comput. 2011, {\bf 217}: 7887-7899.



    Zheng B. Explicit bounds derived by some new inequalities and applications in fractional integral equations[J]. Journal of Inequalities and Applications, 2014, {\bf 2014}(4), 1-12.



    Kuczma M. {\it An introduction to the theory of functional equations and inequalities:
    Cauchy’s equation and Jensen’s inequality}[M], University of Katowice, Katowice, 1985.

    Medve\v d M. Nonlinear singular integral inequalities for functions in two and n independent
    variables[J]. J. Inequal. Appl. 2000, {\bf 5}(3): 287-308.

    马庆华, 杨恩浩. 弱奇性Volterra积分不等式解的估计[J]. 应用数学学报, 2002, {\bf 25}: 505-515.

    吴宇.关于一类弱奇性Volterra 积分不等式的注记[J].四川师范大学学报(自然科学版), 2008, {\bf 31}(5): 534-537.

    王五生, 李自尊. 含多个非线性项的时滞积分不等式及其应用[J]. 数学进展, 2012, {\bf 41}(5): 597-604.




    梁英. 一类时滞弱奇异Wendroff型积分不等式[J]. 四川师范大学学报(自然科学版), 2014, {\bf 37}(4): 493-496.






    卢钰松, 王五生. 一类含有p次幂的Volterra-Fredholm型非线性迭代积分不等式[J]. 西南大学学报(自然科学版), 2015, {\bf 27}(8): 76-80.




    Yong Y. On some new weakly singular Volterra integral inequalities with maxima and their applications, Journal of Inequalities and Applications, 2015, {\bf 2015}:369.





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