Stability in Functional Integro-differential Equations of Second Order with Variable Delay

Cemil Tunc, Melek Gozen


In this paper, we investigate the stability of the zero solution of an integro-differential equation of the second order with variable delay. By means of the fixed point theory and an exponential weighted metric, we find sufficient conditions under which the zero solution of the equation considered is stable.


fixed point; integro-differential equation; non-linear; second order; stability.

Full Text:



Burton, T.A., Fixed Points, Stability and Exact Linearization, Nonlinear Anal., 61(5), pp. 857-870, 2005.

Pi, D., Study the Stability of Solutions of Functional Differential Equations via Fixed Points, Nonlinear Anal., 74(2), pp. 639-651, 2011.

Pi, D., Fixed Points and Stability of A Class of Integro-differential Equations, Math. Probl. Eng., 2014, Art. ID 286214, 10 pp, 2014.

Pi, D., Stability Conditions of Second Order Integro-differential Equations with Variable Delay, Abstr. Appl. Anal., 2014(Art. ID 371639), 11 pp., 2014.

Abdollahpour, M.R., Aghayari, R., & Rassias, M.Th., Hyers-Ulam Stability of Associated Laguerre Differential Equations in a Subclass of Analytic Functions, J. Math. Anal. Appl. 437(1), pp. 605-612, 2016.

Ardjouni, A. & Djoudi, A., Fixed Points and Stability in Linear Neutral Differential Equations with Variable Delays, Nonlinear Anal., 74(6), pp. 2062-2070, 2011.

Burton, T.A., Stability and Periodic Solutions of Ordinary and Functional-differential Equations, Mathematics in Science and Engineering, 178, Academic Press, Inc., Orlando, FL, 1985.

Burton, T.A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., Mineola, New York, 2006.

Graef, J.R. & Tunç, C., 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), pp. 169-173, 2015.

Hale, J., Theory of Functional Differential Equations, 2nd Ed, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York - Heidelberg, 1977.

Jin, C. & Luo, J., Fixed Points and Stability in Neutral Differential Equations with Variable Delays, Proc. Amer. Math. Soc., 136(3), pp. 909-918, 2008.

Korkmaz, E. & Tunç, C., Convergence to Non-autonomous Differential Equations of Second Order, J. Egyptian Math. Soc., 23(1), pp. 27-30, 2015.

Levin, J.J. & Nohel, J.A., Global Asymptotic Stability for Nonlinear Systems of Differential Equations and Applications to Reactor Dynamics, Arch. Rational Mech. Anal., 5, pp. 194-211, 1960.

Levin, J.J. & Nohel, J.A., On A Nonlinear Delay Equation, J. Math. Anal. Appl., 8, pp. 31-44, 1964.

Pi, D., On the Stability of A Second Order Retarded Differential Equation, Appl. Math. Comput., 256, pp. 324-333, 2015.

Tunç, C. & Biçer, E., Stability to A Kind of Functional Differential Equations of Second Order with Multiple Delays by Fixed Points, Abstr. Appl. Anal., 2014(Art. ID 413037), 9 pp., 2014.

Tunç, C., Boundedness Results for Solutions of Certain Nonlinear Differential Equations of Second Order, J. Indones. Math. Soc., 16(2), pp. 115-126, 2010.

Tunç, C. & Tunç, O., A Note on Certain Qualitative Properties of A Second Order Linear Differential System, Appl. Math. Inf. Sci., 9(2), pp. 953-956, 2015.

Zhao, D., Yuan, S. & Zhang, T., Improved Stability Conditions for A Class of Stochastic Volterra-Levin Equations, Appl. Math. Comput., 231, pp. 39-47, 2014.

Zhao, D. & Yuan, S., 3/2-stability Conditions for A Class of Volterra-Levin Equations, Nonlinear Anal., 94, pp. 1-11, 2014.



  • There are currently no refbacks.

View my Stats

Creative Commons License
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.


ITB Journal Publisher, LPPM ITB, Center for Research and Community Services (CRCS) Building, 6th & 7th Floor, Jalan Ganesha 10, Bandung 40132, Indonesia, Phone: +62-22-86010080, Fax.: +62-22-86010051; E-mail: