Results for retarded nonlinear integral inequalities with mixed powers and their applications to delay integro-differential equations

А.  Shakoor; M.  Samar; T.  Athar; M.  Saddique

doi:10.37863/umzh.v75i3.6903

А. Shakoor Department of Mathematics, the Islamia University of Bahawalpur, Pakistan
M. Samar College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, China
T. Athar Department of Computer Science, Shaheed Zul kar Ali Bhutto Institute of Science and Technology, Islamabad, Pakistan
M. Saddique Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan

DOI: https://doi.org/10.37863/umzh.v75i3.6903

Keywords: Retarded integral inequality, Boundedness, Delay integro-differential equation

Abstract

UDC 517.9

We present new retarded nonlinear integral inequalities with mixеd powers. The obtained inequalities can be used to study the boundedness and global existence of the solutions of integro-differential equation with delay and Volterra-type integral equation with delay. These inequalities extend some results available in the literature. Finally, we present two examples to demonstrate the usefulness of our main results.

References

A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary value problems, Second ed., Walter de Gruyter, Berlin, Boston (2016). DOI: https://doi.org/10.1515/9783110378443

A. M. Samoilenko, A. A. Boichuk, S. A. Krivosheya, Boundary-value problems for systems of integro-differential equations with degenerate kernel, Ukr. Math. J., 48, № 12, 1785–1789 (1996). DOI: https://doi.org/10.1007/BF02529500

A. A. Boichuk, L. M. Shegda, Bifurcation of solutions of singular Fredholm boundary value problems, Different. Equat., 47, 453–461 (2011). DOI: https://doi.org/10.1134/S001226611104001X

G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, 2nd ed., Cambridge Univ. Press, London, New York (1952).

B. G. Pachpatte, Integral and finite difference inequalities and applications, Elsevier Sci. (2006). DOI: https://doi.org/10.1016/S0304-0208(06)80018-6

B. G. Pachpatte, Inequalities for differential and integral equations, Acad. Press, London (1998).

D. S. Mitrinovic, Analytic inequalities, Springer-Verlag, Berlin, New York (1970).

D. D. Bainov, P. Simeonov, Integral inequalities and applications, Kluwer Acad. Publ., Dordrecht (1992). DOI: https://doi.org/10.1007/978-94-015-8034-2

T. H. Hilderbrandt, Introduction to the theory of integration, Acad. Press, New York (1963).

T. H. Gronwall, Note on the derivative with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20, 292–296 (1919). DOI: https://doi.org/10.2307/1967124

R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10, 643–647 (1943). DOI: https://doi.org/10.1215/S0012-7094-43-01059-2

A. Shakoor, I. Ali, M. Azam, A. Rehman, M. Z. Iqbal, Further nonlinear retarded integral inequalities for Gronwall–Bellman type and their applications, Iran. J. Sci. and Technol. Trans. A Sci., 43, 2559–2568 (2019). DOI: https://doi.org/10.1007/s40995-019-00745-x

O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. and Appl., 252, 389–401 (2000). DOI: https://doi.org/10.1006/jmaa.2000.7085

A. Abdeldaim, A. A. El-Deeb, On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations, Appl. Math. and Comput., 256, 375–380 (2015). DOI: https://doi.org/10.1016/j.amc.2015.01.047

Z. Z. Li, W. S. Wang, Some nonlinear Gronwall–Bellman type retarded integral inequalities with power and their applications, Appl. Math. and Comput., 347, 839–852 (2019). DOI: https://doi.org/10.1016/j.amc.2018.10.019

Y. Z. Tian, M. Fan, Nonlinear integral inequality with power and its application in delay integro-differential equations, Adv. Different. Equat., 142 (2020). DOI: https://doi.org/10.1186/s13662-020-02596-y

Y. Z. Tian, Y. L. Cai, Y. G. Sun, Asymptotic behavior of switched delay systems with nonlinear disturbances, Appl. Math. and Comput., 268, 522–533 (2015). DOI: https://doi.org/10.1016/j.amc.2015.06.073

I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7, 71–94 (1956). DOI: https://doi.org/10.1007/BF02022967

A. Shakoor, I. Ali, S. Wali, A. Rehman, Some generalizations of retarded nonlinear integral inequalities and its applications, J. Math. Inequal., 14, 1223–1235 (2020). DOI: https://doi.org/10.7153/jmi-2020-14-79

H. El-Owaidy, A. Ragab, W. Abuelela, A. A. El-Deeb, On some new nonlinear integral inequalities of Gronwall–Bellman type, Kyungpook Math. J., 54, 555–575 (2014). DOI: https://doi.org/10.5666/KMJ.2014.54.4.555