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February  2022, 27(2): 639-657. doi: 10.3934/dcdsb.2021059

Qualitative analysis of integro-differential equations with variable retardation

1. 

Missouri S & T, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA

2. 

Van Yuzuncu Yil University, Department of Mathematics, Faculty of Sciences, Van, 65080, Turkey

* Corresponding author: Martin Bohner

Received  June 2020 Revised  January 2021 Published  February 2022 Early access  February 2021

The paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs). By the Lyapunov–Krasovski$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $ functional method, two new results with weaker conditions related to uniform stability (US), uniform asymptotic stability (UAS), integrability, boundedness, and boundedness at infinity of solutions of the RIDEs are given. For illustrative purposes, two examples are provided. The study of the results of this paper shows that the given theorems are not only applicable to time-varying linear RIDEs, but also applicable to time-varying nonlinear RIDEs.

Citation: Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059
References:
[1]

R. P. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-3455-9.

[2]

R. P. AgarwalM. BohnerA. Domoshnitsky and Y. Goltser, Floquet theory and stability of nonlinear integro-differential equations, Acta Math. Hungar., 109 (2005), 305-330.  doi: 10.1007/s10474-005-0250-7.

[3]

S. Ahmad and M. R. Mohana Rao, Stability of Volterra diffusion equations with time delays, Appl. Math. Comput., 90 (1998), 143-154.  doi: 10.1016/S0096-3003(97)00395-0.

[4]

F. AlahmadiY. N. Raffoul and S. Alharbi, Boundedness and stability of solutions of nonlinear Volterra integro-differential equations, Adv. Dyn. Syst. Appl., 13 (2018), 19-31. 

[5]

J. A. D. Appleby and D. W. Reynolds, On necessary and sufficient conditions for exponential stability in linear Volterra integro-differential equations, J. Integral Equations Appl., 16 (2004), 221-240.  doi: 10.1216/jiea/1181075283.

[6]

N. V. Azbelev and P. M. Simonov, Stability of Differential Equations with Aftereffect, vol. 20 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003.

[7]

D. Ba$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $nov and A. Domoshnitsky, Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional-differential equations, Extracta Math., 8 (1993), 75-82. 

[8]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original. doi: 10.1137/1.9781611971262.

[9]

M. Bershadsky, M. V. Chirkov, A. Domoshnitsky, S. V. Rusakov and I. L. Volinsky, Distributed control and the Lyapunov characteristic exponents in the model of infectious diseases, Complexity, 2019 (2018), Art. ID 5234854, 12. doi: 10.1155/2019/5234854.

[10]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover Publications, Inc., Mineola, NY, 2005, Corrected version of the 1985 original.

[11]

T. A. Burton, Volterra Integral and Differential Equations, vol. 202 of Mathematics in Science and Engineering, 2nd edition, Elsevier B. V., Amsterdam, 2005.

[12]

C. Corduneanu and I. W. Sandberg (eds.), Volterra Equations and Applications, vol. 10 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Amsterdam, 2000, Papers from the Volterra Centennial Symposium held at the University of Texas, Arlington, TX, May 23–25, 1996.

[13]

A. Domoshnitsky and E. Fridman, A positivity-based approach to delay-dependent stability of systems with large time-varying delays, Systems Control Lett., 97 (2016), 139-148.  doi: 10.1016/j.sysconle.2016.09.011.

[14]

A. Domoshnitsky, M. Gitman and R. Shklyar, Stability and estimate of solution to uncertain neutral delay systems, Bound. Value Probl., 2014 (2014), 14pp. doi: 10.1186/1687-2770-2014-55.

[15]

A. Domoshnitsky and R. Shklyar, Positivity for non-Metzler systems and its applications to stability of time-varying delay systems, Systems Control Lett., 118 (2018), 44-51.  doi: 10.1016/j.sysconle.2018.05.009.

[16]

A. Domoshnitsky, I. L. Volinsky and M. Bershadsky, Around the model of infection disease: The Cauchy matrix and its properties,, Symmetry, 11 (2019), 1016. doi: 10.3390/sym11081016.

[17]

A. DomoshnitskyI. L. VolinskyA. Polonsky and A. Sitkin, Stabilization by delay distributed feedback control, Math. Model. Nat. Phenom., 12 (2017), 91-105.  doi: 10.1051/mmnp/2017067.

[18]

X. T. Du, Stability of Volterra integro-differential equations with respect to part of the variables, Hunan Ann. Math., 12 (1992), 110-115. 

[19]

X. T. Du, Some kinds of Liapunov functional in stability theory of RFDE, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 214-224.  doi: 10.1007/BF02013157.

[20]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.

[21]

M. FunakuboT. Hara and S. Sakata, On the uniform asymptotic stability for a linear integro-differential equation of Volterra type, J. Math. Anal. Appl., 324 (2006), 1036-1049.  doi: 10.1016/j.jmaa.2005.12.053.

[22]

T. Furumochi and S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 32 (1999), 25-40. 

[23]

K. Gopalsamy, A simple stability criterion for a linear system of neutral integro-differential equations, Math. Proc. Cambridge Philos. Soc., 102 (1987), 149-162.  doi: 10.1017/S0305004100067141.

[24]

W. M. Haddad and V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with time delay, Systems Control Lett., 51 (2004), 355-361.  doi: 10.1016/j.sysconle.2003.09.006.

[25]

C. Jin and J. Luo, Stability of an integro-differential equation, Comput. Math. Appl., 57 (2009), 1080-1088.  doi: 10.1016/j.camwa.2009.01.006.

[26]

I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations, J. Soviet Math., 43 (1988), 2259-2339. 

[27]

I. T. Kiguradze and B. Půža, Boundary Value Problems for Systems of Linear Functional Differential Equations, vol. 12 of Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica, Masaryk University, Brno, 2003.

[28]

M. A. Krasnosel$'$ski$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $, G. M. Va$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $nikko, P. P. Zabre$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $ko, Y. B. Rutitskii and V. Y. Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972, Translated from the Russian by D. Louvish.

[29]

V. Lakshmikantham and M. R. Mohana Rao, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, 1995.

[30]

W. E. Mahfoud, Boundedness properties in Volterra integro-differential systems, Proc. Amer. Math. Soc., 100 (1987), 37-45.  doi: 10.2307/2046116.

[31]

H. Matsunaga and M. Suzuki, Effect of off-diagonal delay on the asymptotic stability for an integro-differential system, Appl. Math. Lett., 25 (2012), 1744-1749.  doi: 10.1016/j.aml.2012.02.004.

[32]

M. R. Mohana Rao and V. Raghavendra, Asymptotic stability properties of Volterra integro-differential equations, Nonlinear Anal., 11 (1987), 475-480.  doi: 10.1016/0362-546X(87)90065-4.

[33]

M. R. Mohana Rao and P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc., 94 (1985), 55-60.  doi: 10.1090/S0002-9939-1985-0781056-5.

[34]

P. H. A. Ngoc, Novel criteria for exponential stability of functional differential equations, Proc. Amer. Math. Soc., 141 (2013), 3083-3091.  doi: 10.1090/S0002-9939-2013-11554-6.

[35]

P. H. A. Ngoc, On stability of a class of integro-differential equations, Taiwanese J. Math., 17 (2013), 407-425.  doi: 10.11650/tjm.17.2013.1699.

[36]

P. H. A. Ngoc, Stability of positive differential systems with delay, IEEE Trans. Automat. Control, 58 (2013), 203-209.  doi: 10.1109/TAC.2012.2203031.

[37]

Y. Raffoul, Exponential stability and instability in finite delay nonlinear Volterra integro-differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 95-106. 

[38]

Y. Raffoul and H. Rai, Uniform stability in nonlinear infinite delay Volterra integro-differential equations using Lyapunov functionals, Nonauton. Dyn. Syst., 3 (2016), 14-23.  doi: 10.1515/msds-2016-0002.

[39]

Y. Raffoul and M. Ünal, Stability in nonlinear delay Volterra integro-differential systems, J. Nonlinear Sci. Appl., 7 (2014), 422-428. 

[40]

M. Rahman, Integral Equations and Their Applications, WIT Press, Southampton, 2007.

[41]

H. L. Smith, Monotone Dynamical Systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995, An introduction to the theory of competitive and cooperative systems.

[42]

J. Tian, Z. Ren and S. Zhong, A new integral inequality and application to stability of time-delay systems,, Appl. Math. Lett., 101 (2020), 106058, 7pp. doi: 10.1016/j.aml.2019.106058.

[43]

C. Tunç, Properties of solutions of Volterra integro-differential equations with delay, Appl. Math. Inf. Sci., 10 (2016), 1775-1780. 

[44]

C. Tunç, Qualitative properties in nonlinear Volterra integro-differential equations with delay, J. Taibah Univ. Sci., 11 (2017), 309-314. 

[45]

C. Tunç, Stability and boundedness in Volterra integro-differential equations with delay, Dynam. Systems Appl., 26 (2017), 121-130. 

[46]

C. Tunç and O. Tunç, New qualitative criteria for solutions of Volterra integro-differential equations, Arab. J. Basic Appl. Sci., 25 (2018), 158-165. 

[47]

C. Tunç and O. Tunç, New results on the stability, integrability and boundedness in Volterra integro-differential equations, Bull. Comput. Appl. Math., 6 (2018), 41-58. 

[48]

C. Tunç and O. Tunç, On behaviours of functional Volterra integro-differential equations with multiple time-lags, J. Taibah Univ. Sci., 12 (2018), 173-179. 

[49]

C. Tunç and O. Tunç, On the exponential study of solutions of Volterra integro-differential equations with time lag, Electron. J. Math. Anal. Appl., 6 (2018), 253-265. 

[50]

C. Tunç and O. Tunç, A note on the qualitative analysis of Volterra integro-differential equations, J. Taibah Univ. Sci., 13 (2019), 490-496. 

[51]

O. Tunç, On the qualitative analyses of integro-differential equations with constant time lag, Appl. Math. Inf. Sci., 14 (2020), 57-63.  doi: 10.18576/amis/140107.

[52]

J. Vanualailai and S.-i. Nakagiri, Stability of a system of Volterra integro-differential equations, J. Math. Anal. Appl., 281 (2003), 602-619.  doi: 10.1016/S0022-247X(03)00171-9.

[53]

K. Wang, Uniform asymptotic stability in functional-differential equations with infinite delay, Ann. Differential Equations, 9 (1993), 325-335. 

[54]

L. Wang and X. T. Du, The stability and boundedness of solutions of Volterra integro-differential equations, Acta Math. Appl. Sinica, 15 (1992), 260-268. 

[55]

Q. Wang, Asymptotic stability of functional-differential equations with infinite time-lag, J. Huaqiao Univ. Nat. Sci. Ed., 19 (1998), 329-333. 

[56]

Q. Wang, The stability of a class of functional differential equations with infinite delays, Ann. Differential Equations, 16 (2000), 89-97. 

[57]

Z. C. WangZ. X. Li and J. H. Wu, Stability properties of solutions of linear Volterra integro-differential equations, Tohoku Math. J. (2), 37 (1985), 455-462.  doi: 10.2748/tmj/1178228588.

[58]

A.-M. Wazwaz, Linear and Nonlinear Integral Equations, Higher Education Press, Beijing; Springer, Heidelberg, 2011, Methods and applications. doi: 10.1007/978-3-642-21449-3.

[59]

P.-X. Weng, Asymptotic stability for a class of integro-differential equations with infinite delay, Math. Appl. (Wuhan), 14 (2001), 22-27. 

[60]

Z. D. Zhang, Asymptotic stability of Volterra integro-differential equations, J. Harbin Inst. Tech., 1990, 11–19.

show all references

References:
[1]

R. P. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-3455-9.

[2]

R. P. AgarwalM. BohnerA. Domoshnitsky and Y. Goltser, Floquet theory and stability of nonlinear integro-differential equations, Acta Math. Hungar., 109 (2005), 305-330.  doi: 10.1007/s10474-005-0250-7.

[3]

S. Ahmad and M. R. Mohana Rao, Stability of Volterra diffusion equations with time delays, Appl. Math. Comput., 90 (1998), 143-154.  doi: 10.1016/S0096-3003(97)00395-0.

[4]

F. AlahmadiY. N. Raffoul and S. Alharbi, Boundedness and stability of solutions of nonlinear Volterra integro-differential equations, Adv. Dyn. Syst. Appl., 13 (2018), 19-31. 

[5]

J. A. D. Appleby and D. W. Reynolds, On necessary and sufficient conditions for exponential stability in linear Volterra integro-differential equations, J. Integral Equations Appl., 16 (2004), 221-240.  doi: 10.1216/jiea/1181075283.

[6]

N. V. Azbelev and P. M. Simonov, Stability of Differential Equations with Aftereffect, vol. 20 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003.

[7]

D. Ba$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $nov and A. Domoshnitsky, Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional-differential equations, Extracta Math., 8 (1993), 75-82. 

[8]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original. doi: 10.1137/1.9781611971262.

[9]

M. Bershadsky, M. V. Chirkov, A. Domoshnitsky, S. V. Rusakov and I. L. Volinsky, Distributed control and the Lyapunov characteristic exponents in the model of infectious diseases, Complexity, 2019 (2018), Art. ID 5234854, 12. doi: 10.1155/2019/5234854.

[10]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover Publications, Inc., Mineola, NY, 2005, Corrected version of the 1985 original.

[11]

T. A. Burton, Volterra Integral and Differential Equations, vol. 202 of Mathematics in Science and Engineering, 2nd edition, Elsevier B. V., Amsterdam, 2005.

[12]

C. Corduneanu and I. W. Sandberg (eds.), Volterra Equations and Applications, vol. 10 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Amsterdam, 2000, Papers from the Volterra Centennial Symposium held at the University of Texas, Arlington, TX, May 23–25, 1996.

[13]

A. Domoshnitsky and E. Fridman, A positivity-based approach to delay-dependent stability of systems with large time-varying delays, Systems Control Lett., 97 (2016), 139-148.  doi: 10.1016/j.sysconle.2016.09.011.

[14]

A. Domoshnitsky, M. Gitman and R. Shklyar, Stability and estimate of solution to uncertain neutral delay systems, Bound. Value Probl., 2014 (2014), 14pp. doi: 10.1186/1687-2770-2014-55.

[15]

A. Domoshnitsky and R. Shklyar, Positivity for non-Metzler systems and its applications to stability of time-varying delay systems, Systems Control Lett., 118 (2018), 44-51.  doi: 10.1016/j.sysconle.2018.05.009.

[16]

A. Domoshnitsky, I. L. Volinsky and M. Bershadsky, Around the model of infection disease: The Cauchy matrix and its properties,, Symmetry, 11 (2019), 1016. doi: 10.3390/sym11081016.

[17]

A. DomoshnitskyI. L. VolinskyA. Polonsky and A. Sitkin, Stabilization by delay distributed feedback control, Math. Model. Nat. Phenom., 12 (2017), 91-105.  doi: 10.1051/mmnp/2017067.

[18]

X. T. Du, Stability of Volterra integro-differential equations with respect to part of the variables, Hunan Ann. Math., 12 (1992), 110-115. 

[19]

X. T. Du, Some kinds of Liapunov functional in stability theory of RFDE, Acta Math. Appl. Sinica (English Ser.), 11 (1995), 214-224.  doi: 10.1007/BF02013157.

[20]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.

[21]

M. FunakuboT. Hara and S. Sakata, On the uniform asymptotic stability for a linear integro-differential equation of Volterra type, J. Math. Anal. Appl., 324 (2006), 1036-1049.  doi: 10.1016/j.jmaa.2005.12.053.

[22]

T. Furumochi and S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 32 (1999), 25-40. 

[23]

K. Gopalsamy, A simple stability criterion for a linear system of neutral integro-differential equations, Math. Proc. Cambridge Philos. Soc., 102 (1987), 149-162.  doi: 10.1017/S0305004100067141.

[24]

W. M. Haddad and V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with time delay, Systems Control Lett., 51 (2004), 355-361.  doi: 10.1016/j.sysconle.2003.09.006.

[25]

C. Jin and J. Luo, Stability of an integro-differential equation, Comput. Math. Appl., 57 (2009), 1080-1088.  doi: 10.1016/j.camwa.2009.01.006.

[26]

I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations, J. Soviet Math., 43 (1988), 2259-2339. 

[27]

I. T. Kiguradze and B. Půža, Boundary Value Problems for Systems of Linear Functional Differential Equations, vol. 12 of Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica, Masaryk University, Brno, 2003.

[28]

M. A. Krasnosel$'$ski$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $, G. M. Va$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $nikko, P. P. Zabre$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $ko, Y. B. Rutitskii and V. Y. Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972, Translated from the Russian by D. Louvish.

[29]

V. Lakshmikantham and M. R. Mohana Rao, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, 1995.

[30]

W. E. Mahfoud, Boundedness properties in Volterra integro-differential systems, Proc. Amer. Math. Soc., 100 (1987), 37-45.  doi: 10.2307/2046116.

[31]

H. Matsunaga and M. Suzuki, Effect of off-diagonal delay on the asymptotic stability for an integro-differential system, Appl. Math. Lett., 25 (2012), 1744-1749.  doi: 10.1016/j.aml.2012.02.004.

[32]

M. R. Mohana Rao and V. Raghavendra, Asymptotic stability properties of Volterra integro-differential equations, Nonlinear Anal., 11 (1987), 475-480.  doi: 10.1016/0362-546X(87)90065-4.

[33]

M. R. Mohana Rao and P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc., 94 (1985), 55-60.  doi: 10.1090/S0002-9939-1985-0781056-5.

[34]

P. H. A. Ngoc, Novel criteria for exponential stability of functional differential equations, Proc. Amer. Math. Soc., 141 (2013), 3083-3091.  doi: 10.1090/S0002-9939-2013-11554-6.

[35]

P. H. A. Ngoc, On stability of a class of integro-differential equations, Taiwanese J. Math., 17 (2013), 407-425.  doi: 10.11650/tjm.17.2013.1699.

[36]

P. H. A. Ngoc, Stability of positive differential systems with delay, IEEE Trans. Automat. Control, 58 (2013), 203-209.  doi: 10.1109/TAC.2012.2203031.

[37]

Y. Raffoul, Exponential stability and instability in finite delay nonlinear Volterra integro-differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 95-106. 

[38]

Y. Raffoul and H. Rai, Uniform stability in nonlinear infinite delay Volterra integro-differential equations using Lyapunov functionals, Nonauton. Dyn. Syst., 3 (2016), 14-23.  doi: 10.1515/msds-2016-0002.

[39]

Y. Raffoul and M. Ünal, Stability in nonlinear delay Volterra integro-differential systems, J. Nonlinear Sci. Appl., 7 (2014), 422-428. 

[40]

M. Rahman, Integral Equations and Their Applications, WIT Press, Southampton, 2007.

[41]

H. L. Smith, Monotone Dynamical Systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995, An introduction to the theory of competitive and cooperative systems.

[42]

J. Tian, Z. Ren and S. Zhong, A new integral inequality and application to stability of time-delay systems,, Appl. Math. Lett., 101 (2020), 106058, 7pp. doi: 10.1016/j.aml.2019.106058.

[43]

C. Tunç, Properties of solutions of Volterra integro-differential equations with delay, Appl. Math. Inf. Sci., 10 (2016), 1775-1780. 

[44]

C. Tunç, Qualitative properties in nonlinear Volterra integro-differential equations with delay, J. Taibah Univ. Sci., 11 (2017), 309-314. 

[45]

C. Tunç, Stability and boundedness in Volterra integro-differential equations with delay, Dynam. Systems Appl., 26 (2017), 121-130. 

[46]

C. Tunç and O. Tunç, New qualitative criteria for solutions of Volterra integro-differential equations, Arab. J. Basic Appl. Sci., 25 (2018), 158-165. 

[47]

C. Tunç and O. Tunç, New results on the stability, integrability and boundedness in Volterra integro-differential equations, Bull. Comput. Appl. Math., 6 (2018), 41-58. 

[48]

C. Tunç and O. Tunç, On behaviours of functional Volterra integro-differential equations with multiple time-lags, J. Taibah Univ. Sci., 12 (2018), 173-179. 

[49]

C. Tunç and O. Tunç, On the exponential study of solutions of Volterra integro-differential equations with time lag, Electron. J. Math. Anal. Appl., 6 (2018), 253-265. 

[50]

C. Tunç and O. Tunç, A note on the qualitative analysis of Volterra integro-differential equations, J. Taibah Univ. Sci., 13 (2019), 490-496. 

[51]

O. Tunç, On the qualitative analyses of integro-differential equations with constant time lag, Appl. Math. Inf. Sci., 14 (2020), 57-63.  doi: 10.18576/amis/140107.

[52]

J. Vanualailai and S.-i. Nakagiri, Stability of a system of Volterra integro-differential equations, J. Math. Anal. Appl., 281 (2003), 602-619.  doi: 10.1016/S0022-247X(03)00171-9.

[53]

K. Wang, Uniform asymptotic stability in functional-differential equations with infinite delay, Ann. Differential Equations, 9 (1993), 325-335. 

[54]

L. Wang and X. T. Du, The stability and boundedness of solutions of Volterra integro-differential equations, Acta Math. Appl. Sinica, 15 (1992), 260-268. 

[55]

Q. Wang, Asymptotic stability of functional-differential equations with infinite time-lag, J. Huaqiao Univ. Nat. Sci. Ed., 19 (1998), 329-333. 

[56]

Q. Wang, The stability of a class of functional differential equations with infinite delays, Ann. Differential Equations, 16 (2000), 89-97. 

[57]

Z. C. WangZ. X. Li and J. H. Wu, Stability properties of solutions of linear Volterra integro-differential equations, Tohoku Math. J. (2), 37 (1985), 455-462.  doi: 10.2748/tmj/1178228588.

[58]

A.-M. Wazwaz, Linear and Nonlinear Integral Equations, Higher Education Press, Beijing; Springer, Heidelberg, 2011, Methods and applications. doi: 10.1007/978-3-642-21449-3.

[59]

P.-X. Weng, Asymptotic stability for a class of integro-differential equations with infinite delay, Math. Appl. (Wuhan), 14 (2001), 22-27. 

[60]

Z. D. Zhang, Asymptotic stability of Volterra integro-differential equations, J. Harbin Inst. Tech., 1990, 11–19.

Figure 1.  Solution $ x_1 $ of system of RIDEs (9) for different initial values
Figure 2.  Solution $ x_2 $ of system of RIDEs (9) for different initial values
Figure 3.  Bounded solution $ x_1 $ of system of RIDEs (11) for different initial values
Figure 4.  Bounded solution $ x_2 $ of system of RIDEs (11) for different initial values
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