Stability in totally nonlinear neutral differential equations with variable delay
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Abstract
In this paper, we use a fixed point technique and the concept of large contractions to prove asymptotic stability results of the zero solution of a class of the totally nonlinear neutral differential equation with functional delay. The study concerns the equation$$x'(t) =- a (t)h(x(t)) + c(t)x'(t - r(t)) + b(t)G(x(t),x(t - r(t))),$$which has proved very challenging in the theory of Liapunov's direct method. The stability results are obtained by means of Krasnoselskii-Burton's theorem and they improve and generalize the works of Burton [7], and Derrardjia, Ardjouni and Djoudi [16].
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Ardjouni, A., Derrardjia, I., & Djoudi, A.
(2014).
Stability in totally nonlinear neutral differential equations with variable delay.
Acta Mathematica Universitatis Comenianae, 83(1), 119-134.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/83/39
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References
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nonlinear delay dierential equations, Hacettepe Journal of Mathematics and Statistics, Volume 41 (1) (2012), 1-13.
[2] A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral dierential equations with variable delays.
Nonlinear Analysis 74 (2011) 2062-2070.
[3] T. A. Burton, Stability by Fixed Point Theory for Functional Dierential Equations, Dover Publications, New
York, 2006.
[4] T. A. Burton, Integral Equations, Implicit Functions, and Fixed Points, Proceedings of the A.M.S., 124 (1996),
2383-2390.
[5] T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett. 11(1998), 85-88.
[6] T. A. Burton, Krasnoselskii's inversion principle and fixed points, Nonlinear Analysis, 30(1997), 3975-3986.
[7] T. A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9
(2001) 181-190.
[8] T. A. Burton, Stability by fixed point theory or Liapunov's theory: A comparison, Fixed Point Theory 4 (2003) 15-32.
[9] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proceedings of the American Mathematical Society 132 (2004) 3679-3687.
[10] T. A. Burton, C. Kirk, A xed point theorem of Krasnoselskii-Schaefer type, Mathematische Nachrichten,
189(1998), 23-31.
[11] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Analysis 49(2002), 445-
454.
[12] T. A. Burton, T. Furumochi, A note on stability by Schauder's theorem, Funkcialaj Ekvacioj 44 (2001) 73-82.
[13] T. A. Burton, T. Furumochi, Fixed points and problems in stability theory, Dynamical Systems and Applications
10 (2001) 89-116.
[14] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point
theorems, Dynamic Systems and Applications 11 (2002) 499-519.
[15] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Analysis 49 (2002)
445-454.
[16] I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii's theorem in totally nonlinear neutral
differential equations, Opuscula Math. 33, no. 2 (2013), 255-272.
[17] A. Djoudi, R. Khemis, Fixed point techniques and stability for neutral nonlinear differential equations with un
bounded delays, Georgian Mathematical Journal, Vol. 13 (2006), Nu. 1, 25-34.
[18] J. K. Hale, Theory of functional differential equation, Springer, New York, 1077.
[19] C. H. Jin, J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear
Anal. 68 (2008) 3307-3315.
[20] C. H. Jin, J. W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proceedings of the American Mathematical Society, Vol. 136, Nu. 3 (2008) 909-918.
[21] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press,
London-New York, 1974.
[22] B. Zhang, Fixed points and stability in dierential equations with variable delays, Nonlinear Anal. 63 (2005)
233-242.
nonlinear delay dierential equations, Hacettepe Journal of Mathematics and Statistics, Volume 41 (1) (2012), 1-13.
[2] A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral dierential equations with variable delays.
Nonlinear Analysis 74 (2011) 2062-2070.
[3] T. A. Burton, Stability by Fixed Point Theory for Functional Dierential Equations, Dover Publications, New
York, 2006.
[4] T. A. Burton, Integral Equations, Implicit Functions, and Fixed Points, Proceedings of the A.M.S., 124 (1996),
2383-2390.
[5] T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett. 11(1998), 85-88.
[6] T. A. Burton, Krasnoselskii's inversion principle and fixed points, Nonlinear Analysis, 30(1997), 3975-3986.
[7] T. A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9
(2001) 181-190.
[8] T. A. Burton, Stability by fixed point theory or Liapunov's theory: A comparison, Fixed Point Theory 4 (2003) 15-32.
[9] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proceedings of the American Mathematical Society 132 (2004) 3679-3687.
[10] T. A. Burton, C. Kirk, A xed point theorem of Krasnoselskii-Schaefer type, Mathematische Nachrichten,
189(1998), 23-31.
[11] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Analysis 49(2002), 445-
454.
[12] T. A. Burton, T. Furumochi, A note on stability by Schauder's theorem, Funkcialaj Ekvacioj 44 (2001) 73-82.
[13] T. A. Burton, T. Furumochi, Fixed points and problems in stability theory, Dynamical Systems and Applications
10 (2001) 89-116.
[14] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point
theorems, Dynamic Systems and Applications 11 (2002) 499-519.
[15] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Analysis 49 (2002)
445-454.
[16] I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii's theorem in totally nonlinear neutral
differential equations, Opuscula Math. 33, no. 2 (2013), 255-272.
[17] A. Djoudi, R. Khemis, Fixed point techniques and stability for neutral nonlinear differential equations with un
bounded delays, Georgian Mathematical Journal, Vol. 13 (2006), Nu. 1, 25-34.
[18] J. K. Hale, Theory of functional differential equation, Springer, New York, 1077.
[19] C. H. Jin, J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear
Anal. 68 (2008) 3307-3315.
[20] C. H. Jin, J. W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proceedings of the American Mathematical Society, Vol. 136, Nu. 3 (2008) 909-918.
[21] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press,
London-New York, 1974.
[22] B. Zhang, Fixed points and stability in dierential equations with variable delays, Nonlinear Anal. 63 (2005)
233-242.