Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Uniformly Boundedness in Nonlinear Volterra Integro-Differential Equations with Delay

Journal of Environmental Accounting and Management 8(2) (2019) 279--290 | DOI:10.5890/JAND.2019.06.010

Cemil Tunç$^{1}$, Sizar Abid Mohammed$^{2}$

$^{1}$ Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yıl University, 65080, Van - Turkey

$^{2}$ Department of Mathematics, College of Basic Education, University of Duhok, Zakho Street 38, 1006 AJ, Duhok- Iraq

Download Full Text PDF



We pay our attention to a number of nonlinear Volterra integrodifferential equations of first order, VIDEs, with constant retardation. The uniform boundedness, UB, of the solutions, to that VIDEs is investigated via the Lyapunov functionals, LFs, method. The results obtained improve and complement a number of works found in the literature. The novelty and originality of this article is that it improves and extends the results found in the literature from the cases of the without delay to the more general cases with delay by constructing some new LFs.


The authors of this paper would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper.


  1. [1]  Driver, R.D. (1977), Ordinary and delay differential equations, Applied Mathematical Sciences, 20, Springer- Verlag, New York-Heidelberg.
  2. [2]  Gripenberg, G., Londen, S.Q., and Staffans, O. (1990), Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge.
  3. [3]  Burton, T.A. (2005), Volterra integral and differential equations, Second edition. Mathematics in Science and Engineering, 202, Elsevier B. V., Amsterdam.
  4. [4]  Wazwaz, A.M. (2011), Linear and nonlinear integral equations, Methods and applications, Higher Education Press, Beijing; Springer, Heidelberg.
  5. [5]  Rama Mohana Rao, M. and Srinivas, P. (1985), Asymptotic behavior of solutions of Volterra integrodifferential equations, Proc. Amer. Math. Soc., 94(1), 55-60.
  6. [6]  Mahfoud, W.E. (1987), Boundedness properties in Volterra integro-differential systems, Proc. Amer. Math. Soc., 100(1), 37-45.
  7. [7]  Furumochi, T. and Matsuoka, S. (1999), Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 32, 25-40.
  8. [8]  Wang, Q. (2000), The stability of a class of functional differential equations with infinite delays, Ann. Differential Equations, 16(1), 89-97.
  9. [9]  Vanualailai, J. and Nakagiri, S. (2003), Stability of a system of Volterra integro-differential equations, J. Math. Anal. Appl., 281(2), 602-619.
  10. [10]  Becker, L.C. (2009), Uniformly continuous L1 solutions of Volterra equations and global asymptotic stability, Cubo, 11(3), 1-24.
  11. [11]  Rama Mohana Rao, M. and Raghavendra, V. (1987), Asymptotic stability properties of Volterra integrodifferential equations, Nonlinear Anal., 11(4), 475-480.
  12. [12]  Adıvar, M. and Raffoul, Y.N. (2012), Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations, Rend. Circ. Mat. Palermo, (2) 61(3), 321-330.
  13. [13]  Tunç, C. (2016), A note on the qualitative behaviors of non-linear Volterra integro-differential equation, J. Egyptian Math. Soc., 24(2), 187-192.
  14. [14]  Tunç, C. (2016), New stability and boundedness results to Volterra integro-differential equations with delay, J. Egyptian Math. Soc. 24(2), 210-213.
  15. [15]  Tunç, C. (2016), Properties of solutions to Volterra integro-differential equations with delay, Appl. Math. Inf. Sci,. 10 (5), 1775-1780.
  16. [16]  Tunç, C. (2017), On qualitative properties in Volterra integro-differential equations, AIP Conference Proceedings, Vol.1798, Article Number: UNSP 020164, 1-9.
  17. [17]  Graef, J.R. and Tunç, C. (2015), Continuability and boundedness of multi-delay functional integro-differential equations of the second order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM , 109 (1), 169-173.
  18. [18]  Graef, J.R., Tunç, C., and Sevgin, S. (2016), Behavior of solutions of non-linear functional Voltera integro -differential equations with multiple delays, Dynam. Systems Appl., 25(1-2), 39-46.
  19. [19]  Tunç, C. and Mohammed, S.A. (2017), On the stability and instability of functional Volterra integrodifferential equations of first order, Bull. Math. Anal. Appl., 9(1), 151-160.
  20. [20]  Tunç, C. and Mohammed, S.A. (2017), New results on exponential stability of nonlinear Volterra integrodifferential equations with constant time-lag, Proyecciones, 36(4), 615-639.
  21. [21]  Tunç, C. and Mohammed, S.A. (2017), A remark on the stability and boundedness criteria in retarded Volterra integro-differential equations, J. Egyptian Math. Soc., 25(4), 363-368.
  22. [22]  Tunç, C. and Tunç, O. (2018), On the exponential study of solutions of Volterra integro-differential equations with time lag, Electron. J. Math. Anal. Appl., 6(1), 253-265.
  23. [23]  Tunç, C. and Tunç, O. (2018), New results on the stability, integrability and boundedness in Volterra integrodifferential equations, Bull. Comput. Appl. Math., 6(1), 41-58.
  24. [24]  Balachandran, K. and Divya, S. (2017), Controllability of nonlinear neutral fractional integro-differential systems with infinite delay, J. Appl. Nonlinear Dyn., 6(3), 333-344.
  25. [25]  Mondal, A. and Islam, N. (2016), Study on dynamical system with time-delay, J. Appl. Nonlinear Dyn., 5(4), 441-456.
  26. [26]  Raffoul, Y. (2004), Boundedness in nonlinear functional differential equations with applications to Volterra integro-differential equations, J. Integral Equations Appl. 16(4), 375-388. differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20(1), 95-106.
  27. [27]  Rahman, M. (2007), Integral equations and their applications, WIT Press, Southampton.
  28. [28]  Lakshmikantham, V. and Rama Mohan Rao, M. (1987), Stability in variation for nonlinear integro-differential equations, Applicable Analysis, 24(3), 165-173.
  29. [29]  Chang, X. and Wang, R. (2011), Stability of perturbed n-dimensional Volterra differential equations, Nonlinear Anal., 74(5), 1672-1675.
  30. [30]  Wang, Q. (2000), The stability of a class of functional differential equations with infinite delays, Ann. Differential Equations, 16(1), 89-97.
  31. [31]  Tunç, C. and Mohammed, S.A. (2018), On the stability and uniform stability of retarded integro-differential equations, Alexandria Engineering Journal, 57(4), 3501-3507.
  32. [32]  Tuncç, C. and Tuncç, O. (2018), New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences, 25(3), 158-165.
  33. [33]  Tuncç, C. and Tuncç, O. (2019), A note on the qualitative analysis of Volterra integro-differential equations, Journal of Taibah University for Science, 13(1), 490-496.
  34. [34]  Hristova, S. and Tunç, C. (2019), Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays, Electron. J. Differential Equations, 2019(30), 11 pp.