Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory
DOI:
https://doi.org/10.4067/S0716-09172015000100003Keywords:
Fixed points, Stability, Neutral differential equations, Variable delays.Abstract
The totally nonlinear neutral differential equation
(d/ dt) (x(t))=−a(t)g(x(t−τ (t))) + (d/ dt)( G(t,x(t−τ (t)))),
with variable delay τ(t) ≥ 0 is investigated. We find suitable conditions for t, a, g and G so that for a given continuous initial function 0 a mapping P for the above equation can be defined on a carefully chosen complete metric space S0ψ ; and in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Becker and Burton [6]. An example is given to illustrate our main result.
References
[1] A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Analysis 74, pp. 2062-2070, (2011).
[2] A. Ardjouni, A. Djoudi, Stability in nonlinear neutral differential with variable delays using fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 43, pp. 1-11, (2011).
[3] A. Ardjouni, A. Djoudi, Fixed point and stability in neutral nonlinear differential equations with variable delays, Opuscula Mathematica, Vol. 32, No. 1, pp. 5-19, (2012).
[4] A. Ardjouni, A. Djoudi, I. Soualhia, Stability for linear neutral integrodifferential equations with variable delays, Electronic journal of Differential Equations, No. 172, pp. 1-14, (2012).
[5] A. Ardjouni, A. Djoudi, Fixed points and stability in nonlinear neutral Volterra integro-differential equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 28, pp. 1-13, (2013)
[6] L. C. Becker, T. A. Burton, Stability, fixed points and inverse of delays, Proc. Roy. Soc. Edinburgh 136A, pp. 245-275, (2006).
[7] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proceedings of the American Mathematical Society 132, pp. 3679— 3687, (2004).
[8] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, (2006).
[9] T. A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9, pp. 181—190, (2001).
[10] T. A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4, pp. 15—32, (2003).
[11] T. A. Burton, T. Furumochi, A note on stability by Schauder’s theorem, Funkcialaj Ekvacioj 44, pp. 73—82, (2001).
[12] T. A. Burton, T. Furumochi, Fixed points and problems in stability theory, Dynamical Systems and Applications 10, pp. 89—116, (2001).
[13] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11, pp. 499—519, (2002).
[14] T. A. Burton, T. Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Analysis 49, pp. 445—454, (2002).
[15] Y. M. Dib, M. R. Maroun, Y. N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. No. 142, pp. 1-11, (2005).
[16] A. Djoudi, R. Khemis, Fixed point techniques and stability for neutral nonlinear differential equations with unbounded delays, Georgian Mathematical Journal, Vol. 13, No. 1, pp. 25-34, (2006).
[17] C. H. Jin, J. W. Luo, Stability of an integro-differential equation, Computers and Mathematics with Applications 57, pp. 1080-1088, (2009).
[18] C. H. Jin, J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68, pp. 3307- 3315, (2008).
[19] 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, pp. 909-918, (2008).
[20] J. Luo, Fixed points and exponential stability for stochastic Volterra— Levin equations, Journal of Computational and Applied Mathematics, Vol. 234, Issue 3, 1 June 2010, pp. 934—940, (2010).
[21] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. Comput. Modelling 40, pp. 691—700, (2004).
[22] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974).
[23] J. A. Yorke, Asymptotic stability for one dimensional differential delay equations, J. Diff. Eqns 7, pp. 189-202, (1970).
[24] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63, pp. e233—e242, (2005).
[25] B. Zhang, Contraction mapping and stability in a delay differential equation, Dynamical systems and appl. 4, pp. 183—190, (2004).
[2] A. Ardjouni, A. Djoudi, Stability in nonlinear neutral differential with variable delays using fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 43, pp. 1-11, (2011).
[3] A. Ardjouni, A. Djoudi, Fixed point and stability in neutral nonlinear differential equations with variable delays, Opuscula Mathematica, Vol. 32, No. 1, pp. 5-19, (2012).
[4] A. Ardjouni, A. Djoudi, I. Soualhia, Stability for linear neutral integrodifferential equations with variable delays, Electronic journal of Differential Equations, No. 172, pp. 1-14, (2012).
[5] A. Ardjouni, A. Djoudi, Fixed points and stability in nonlinear neutral Volterra integro-differential equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 28, pp. 1-13, (2013)
[6] L. C. Becker, T. A. Burton, Stability, fixed points and inverse of delays, Proc. Roy. Soc. Edinburgh 136A, pp. 245-275, (2006).
[7] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proceedings of the American Mathematical Society 132, pp. 3679— 3687, (2004).
[8] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, (2006).
[9] T. A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9, pp. 181—190, (2001).
[10] T. A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4, pp. 15—32, (2003).
[11] T. A. Burton, T. Furumochi, A note on stability by Schauder’s theorem, Funkcialaj Ekvacioj 44, pp. 73—82, (2001).
[12] T. A. Burton, T. Furumochi, Fixed points and problems in stability theory, Dynamical Systems and Applications 10, pp. 89—116, (2001).
[13] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11, pp. 499—519, (2002).
[14] T. A. Burton, T. Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Analysis 49, pp. 445—454, (2002).
[15] Y. M. Dib, M. R. Maroun, Y. N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electronic Journal of Differential Equations, Vol. No. 142, pp. 1-11, (2005).
[16] A. Djoudi, R. Khemis, Fixed point techniques and stability for neutral nonlinear differential equations with unbounded delays, Georgian Mathematical Journal, Vol. 13, No. 1, pp. 25-34, (2006).
[17] C. H. Jin, J. W. Luo, Stability of an integro-differential equation, Computers and Mathematics with Applications 57, pp. 1080-1088, (2009).
[18] C. H. Jin, J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68, pp. 3307- 3315, (2008).
[19] 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, pp. 909-918, (2008).
[20] J. Luo, Fixed points and exponential stability for stochastic Volterra— Levin equations, Journal of Computational and Applied Mathematics, Vol. 234, Issue 3, 1 June 2010, pp. 934—940, (2010).
[21] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. Comput. Modelling 40, pp. 691—700, (2004).
[22] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974).
[23] J. A. Yorke, Asymptotic stability for one dimensional differential delay equations, J. Diff. Eqns 7, pp. 189-202, (1970).
[24] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63, pp. e233—e242, (2005).
[25] B. Zhang, Contraction mapping and stability in a delay differential equation, Dynamical systems and appl. 4, pp. 183—190, (2004).
How to Cite
[1]
A. Ardjouni and A. Djoudi, “Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory”, Proyecciones (Antofagasta, On line), vol. 34, no. 1, pp. 25-44, 1.
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