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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[45]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[45]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

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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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