Abstract
In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.
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(Communicated by Jozef Džurina )
References
[1] Ambrossetti, A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349–360.Search in Google Scholar
[2] Ardjouni, A.—Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal. 74 (2011), 2062–2070.10.1016/j.na.2010.10.050Search in Google Scholar
[3] Ardjouni, A.—Djoudi, A.: Stability in nonlinear neutral differential with variable delays using fixed point theory, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Art. No. 43.10.14232/ejqtde.2011.1.43Search in Google Scholar
[4] Ardjouni, A.—Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays, Opuscula Math. 32(1) (2012), 5–9.Search in Google Scholar
[5] Ardjouni, A.—Djoudi, A.—Soualhia, I.: Stability for linear neutral integro-differential equations with variable delays, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), Art. No. 172.10.7494/OpMath.2012.32.1.5Search in Google Scholar
[6] Boe, E.—Chang, H. C.: Dynamics of delayed systems under feedback control, Chem. Engng. Sci. 44 (1989), 1281–1294.10.1016/0009-2509(89)85002-XSearch in Google Scholar
[7] Brayton, R. K.—Willoughby, R. A.: On the numerical integration of a symmetric system of difference-differential equations of Neutral type, J. Math. Anal. Appl. 18 (1976), 182–189.10.1016/0022-247X(67)90191-6Search in Google Scholar
[8] Banaś, J.—Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure Appl. Math. 60, Mercel Dekker, New York and Basel, 1980.Search in Google Scholar
[9] Becker, L. C.—Burton, T. A.: Stability, fixed points and inverse of delay, Proc. Roy. Soc. Edinb. A 136 (2006), 245–275.10.1017/S0308210500004546Search in Google Scholar
[10] Burton, T. A.: Fixed points and stability of nonconvolution equation, Proc. Amer. Math. Soc. 132 (2004), 3679–3687.10.1090/S0002-9939-04-07497-0Search in Google Scholar
[11] Burton, T. A.: Stability for Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.Search in Google Scholar
[12] Burton, T. A.: Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2001), 181–190.Search in Google Scholar
[13] Burton, T. A.: Stability by fixed point theory or Lyapunov theory: A comparison, Fixed Point Theory 4 (2003), 15–32.Search in Google Scholar
[14] Burton, T. A.—Furumochi, T.: A note on stability by Schauder's theorem, Funkcial. Ekvac. 44 (2001), 73–82.Search in Google Scholar
[15] Burton, T. A.—Furumochi, T.: Fixed points and problems in stability theory, Dyn. Sys. Appl. 10 (2001), 89–116.Search in Google Scholar
[16] Burton, T. A.—Furumochi, T.: Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dyn. Sys. Appl. 11 (2002), 499–519.Search in Google Scholar
[17] Burton, T. A.—Furumochi, T.: Krasnosielskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445–454.10.1016/S0362-546X(01)00111-0Search in Google Scholar
[18] Dib, Y. M.—Maroun, M. R.—Raffoul, Y. N.: Periodicity and stability in neutral nonlinear differential equations with functional delay, Electron. J. Qual. Theory Differ. Equ. 2005 (2005), Art. No. 142.Search in Google Scholar
[19] Djoudi, A.—Khemis, R.: Fixed point techniques and stability for neutral nonlinear differential equations with unbounded delays, Georgian Math. J. 13 (2006), 25–34.10.1515/GMJ.2006.25Search in Google Scholar
[20] Driver, D. R.: A mixed neutral systems, Nonlinear Anal. 8 (1984), 155-158.10.1016/0362-546X(84)90066-XSearch in Google Scholar
[21] Jin, C. H.—Luo, J. W.: Stability of an integro-differential equation, Comp. Math. Appl. 57 (2009), 1080–1088.10.1016/j.camwa.2009.01.006Search in Google Scholar
[22] Jin, C. H.—Luo, J. W.: Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68 (2008), 3307–3315.10.1016/j.na.2007.03.017Search in Google Scholar
[23] Jin, C. H.—Luo, J. W.: Fixed points and stability in neutral differential equations with variable delays, Proc. Amer. Math. Soc. 136 (2008), 909–918.10.1090/S0002-9939-07-09089-2Search in Google Scholar
[24] Hale, J. K.: Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.10.1007/978-1-4612-9892-2Search in Google Scholar
[25] Popov, E. P.: Automatic Regulation and Control, Nauka, Moscow, 1966 (in Russian).Search in Google Scholar
[26] Raffoul, Y. N.: Stability in neutral nonlinear differential equations with functional delays using fixed point theory, Math. Comput. Modelling 40 (2004), 691–700.10.1016/j.mcm.2004.10.001Search in Google Scholar
[27] Sadovskii, B. N.: Limit-compact and condensing operators, Russian Math. Surveys 27 (1972), 86–144.10.1070/RM1972v027n01ABEH001364Search in Google Scholar
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