Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations

• Autorzy: Derrardjia I. , Ardjouni A. , Djoudi A.
• Języki publikacji: EN

Abstrakty

EN

In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns x'(t)= -a(t)x3(t) + c(t)x'(t-r(t)) + b(t)x3(t-r(t)). The equation 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 on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9 (2001), 181–190]) in which he takes c=0 in the above equation

Słowa kluczowe

EN

fixed point; stability; nonlinear neutral equation; Krasnoselskii-Burton theorem

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2013
• Tom: Vol. 33, no. 2
• Strony: 255--272
• Opis fizyczny: Bibliogr 23 poz.

Twórcy

autor

Derrardjia I.

• University of Annaba Faculty of Sciences Department of Mathematics P.O. Box 12, 23000 Annaba, Algeria

autor

Ardjouni A.

• University of Annaba Faculty of Sciences Department of Mathematics P.O. Box 12, 23000 Annaba, Algeria

autor

Djoudi A.

• University of Annaba Faculty of Sciences Department of Mathematics P.O. Box 12, 23000 Annaba, Algeria

Bibliografia

• [1] A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal. 74 (2011), 2062-2070.
• [2] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
• [3] T.A. Burton, Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc. 124 (1996), 2383-2390.
• [4] T.A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998), 85-88.
• [5] T.A. Burton, Krasnoselskii's inversion principle and fixed points, Nonlinear Anal. 30 (1997), 3975-3986.
• [6] T.A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2001), 181-190.
• [7] T.A. Burton, Stability by fixed point theory or Liapunov's theory: A comparison, Fixed Point Theory 4 (2003), 15-32.
• [8] T.A. Burton, Fixed points and stability of a nonconvolution equation, Proc. Amer. Math. Soc. 132 (2004), 3679-3687.
• [9] T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23-31.
• [10] T.A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445-454.
• [11] T.A. Burton, T. Furumochi, A note on stability by Schauder's theorem, Funkcial. Ekvac. 44 (2001), 73-82.
• [12] T.A. Burton, T. Furumochi, Fixed points and problems in stability theory, Dynam. Systems Appl. 10 (2001), 89-116.
• [13] T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems Appl. 11 (2002), 499-519.
• [14] T.A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445-454.
• [15] H. Deham, A. Djoudi, Periodic solutions for nonlinear differential equation with functional delay, Georgian Math. J. 15 (2008) 4, 635-642.
• [16] H. Deham, A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electron. J. Differential Equations 2010 (2010) 127, 1-8.
• [17] A. Djoudi, R. Khemis, Fixed point techniques and stability for neutral nonlinear differential equations with unbounded delays, Georgian Math. J. 13 (2006) 1, 25-34.
• [18] J.K. Hale, Theory of Functional Differential Equation, Springer, New York, 1077.
• [19] L. Hatvani, Annulus arguments in the stability theory for functional differential equation, Differential Integral Equations 10 (1997), 975-1002.
• [20] C.H. Jin, J.W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68 (2008), 3307-3315.
• [21] C.H. Jin, J.W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proc. Amer. Math. Soc. 136 (2008) 3, 909-918.
• [22] D.R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics, No. 66, Cambridge University Press, London-New York, 1974.
• [23] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63 (2005), 233-242.