Uniform stability of integro-differential inequalities with nonlinear control inputs and delay

Document Type : Research Paper


1 Department of Mathematics, College of Education, Mustansiriyah University, Baghdad, Iraq

2 Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq


In this paper, the uniform stability for the solution of integro-differential inequalities, with nonlinear control inputs and delay functions, is investigated by using some inequality estimator conditions. Moreover, we apply the obtained results on the solutions of some proposed classes of integro-differential inequalities with nonlinear control input functions as problem formulations examples. The results show that the stability technique used in this work is efficient and robust and it can be applied to a general class and various types of integro-differential inequalities.


[1] A.T. Ademola and P.O. Arawomo, Uniform Stability and Boundedness of solutions of nonlinear delay differential
equations of the third order , Math. J. Okayama Univ. 55 (2013) 157–166.
[2] M. Adivar and Y. N Raffoul, Inequalities and exponential stability and instability in ?nite delay Volterra integrodifferential equations, Rend. Circ. Mat. Palermo 61 (2012) 321–330.
[3] B.C. Dhage and S.B. Dhage, Differential inequalities and comparison theorems for nonlinear first order volterra
Integro-differential equations, Commun. Appl. Anal. 19 (2015) 287–306.
[4] T. Hara, T. Yoneyama and R. Miyazaki, Volterra Integro-Differential Inequality Asymptotic Criteria, Diff. Integral
Equ. 5(1) (1992) 201–212.
[5] S.Q. Hasan , Estimators of some inequality dynamical system, J. Iraqi Al-Khwarizmi Soc. 3 (Special issue) (2019)
[6] D. He and L. Xu, Integro-differential inequality for stability od singularly perturbed impulsive delay integrodifferential equations , J. Inequal. Appl. 2009 (2009) 1–11.
[7] P. C. Jackreece1 and S. Aniaku, Stability Results of Nonlinear Integro-differential Equations, Math. Theory Model.
8(1) (2018) 27–33.
[8] M. Kazemia, V. Torkashvandb and R. Ezzatic, A new method for solving three-dimensional nonlinear Fredholm
integral equations by Haar wavelet, Int. J. Nonlinear Anal. Appl. 12(2) (2021) 115–133.
[9] Z.A. Khan, Integro-differential inequalities arising in the theory of differential equations, Abst. Appl. Anal. 2016
(2016) 1–6.
[10] G. Ladas and I.P. Stavroulakis, On delay differential inequalities of first Order, Funkcialaj Ekvacioj 25 (1982)
[11] L. Li, F. Meng and P. Ju, Some new integral inequalities and their applications in studying the stability of nonlinear
integro-differential equations with time delay, Math. Anal. Appl. 377 (2011) 853–862.
[12] Y. Raffoul and H. Rai, Uniform Stability In Nonlinear Infinite Delay Volterra Integro-differential Equations Using
Lyapunov Functionals, University of Dayton, 2016.
[13] J. Szarski, Differential Inequalities, Polish Scientific Publishers: Warsaw, 1965.
[14] Z. Taheria, Sh. Javadia and E. Babolian, About solving stochastic It\∧{o}-Volterra integral equations using the
spectral collocation method, Int. J. Nonlinear Anal. Appl. 12(2) (2021) 11–24.
[15] C. Tunc and S.A. Mohammed, On the stability and uniform stability of retarded integro-di?erential equations,
Alexandria Engin. J. 57 (2018) 3501–3507.
[16] D. Xu and X. Wang, A new nonlinear integro-differential inequality and its application, Appl. Math. Lett. 22
(2009) 1721–1726.
[17] J. Zhao and F. Meng, Stability analysis of solutions for a kind of Integro-differential equations with a Delay,
Math. Prob. Engin. 2018 (2018) 1–6.
Volume 13, Issue 1
March 2022
Pages 421-430
  • Receive Date: 03 May 2021
  • Revise Date: 08 June 2021
  • Accept Date: 28 June 2021