Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point

MESMOULI, Mouataz Billah   ; Ardjouni, Abdelouaheb   ; Djoudi, Ahcene

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 2, pp. 129-142

Abstract (VO)

In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay \[ x^{\prime }\left( t\right) =-a\left( t\right) h\left( x\left( t\right) \right) +\frac{d}{dt}Q\left( t,x\left( t-\tau \left( t\right) \right) \right) +G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right) \right) . \] The stability of the zero solution of this eqution provided that $h\left(0\right) =Q\left( t,0\right) =G\left( t,0,0\right) =0$. The Caratheodory condition is used for the functions $Q$ and $G$.

Détail

Classification : 34K20, 34K30, 34K40, 47H10
Keywords: Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation

Bibliographie

[1] Adıvar, M., Islam, M. N., Raffoul, Y. N.: Separate contraction and existence of periodic solution in totally nonlinear delay differential equations. Hacettepe Journal of Mathematics and Statistics, 41, 1 (2012), 1–13. | MR

[2] Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Nonlin. Anal., T.M.A., 74, 6 (2011), 2062–2070. | DOI | MR | Zbl

[3] Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Opuscula Mathematica, 32, 1 (2012), 5–19. | DOI | MR

[4] Ardjouni, A., Derrardjia, I., Djoudi, A.: Stability in totally nonlinear neutral differential equations with variable delay. Acta Math. Univ. Comenianae, 83, 1 (2014), 119–134. | MR | Zbl

[5] Burton, T. A.: Integral equations, implicit functions, and fixed points. Proc. Amer. Math. Soc., 124, 8 (1996), 2383–2390. | DOI | MR | Zbl

[6] Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190. | MR

[7] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, New York, 2006. | MR | Zbl

[8] Burton, T. A.: Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem. Nonlinear Stud., 9 (2001), 181–190. | MR

[9] Burton, T. A.: Stability by fixed point theory or Liapunov’s theory: A comparison. Fixed Point Theory, 4 (2003), 15–32. | MR

[10] Burton, T. A.: Fixed points and stability of a nonconvolution equation. Proc. Amer. Math. Soc., 132 (2004), 3679–3687. | DOI | MR | Zbl

[11] Burton, T. A., Kirk, C.: A point theorem of Krasnoselskii–Schaefe type. Math. Nachr., 189 (1998), 23–31. | DOI | MR

[12] Burton, T. A., Furumochi, T.: Krasnoselskii’s fixed point theorem and stability. Nonlinear Anal., 49 (2002), 445–454. | DOI | MR | Zbl

[13] Burton, T. A., Furumochi, T.: A note on stability by Schauder’s theorem. Funkcial. Ekvac., 44 (2001), 73–82. | MR | Zbl

[14] Burton, T. A., Furumochi, T.: Fixed points and problems in stability theory. Dynam. Systems Appl., 10 (2001), 89–116. | MR | Zbl

[15] Burton, T. A., Furumochi, T.: Krasnoselskii’s fixed point theorem and stability. Nonlinear Anal., 49 (2002), 445–454. | DOI | MR | Zbl

[16] Burton, T. A., Furumochi, T.: Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dynam. Systems Appl., 11 (2002), 499–519. | MR | Zbl

[17] Deham, H., Djoudi, A.: Periodic solutions for nonlinear differential equation with functional delay. Georgian Math. J., 15, 4 (2008), 635–642. | MR | Zbl

[18] Deham, H., Djoudi, A.: Existence of periodic solutions for neutral nonlinear differential equations with variable delay. Electron. J. Differential Equations, 2010, 127 (2010), 1–8. | Zbl

[19] Djoudi, A., Khemis, R.: Fixed point techniques and stability for neutral nonlinear differential equations with unbounded delays. Georgian Math. J., 13, 1 (2006), 25–34. | MR | Zbl

[20] Derrardjia, I., Ardjouni, A., Djoudi, A.: Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equation. Opuscula Math., 33, 2 (2013), 255–272. | DOI | MR

[21] Dib, Y. M., Maroun, M. R., Raffoul, Y. N.: Periodicity and stability in neutral nonlinear differential equations with functional delay. Electronic Journal of Differential Equations, 2005, 142 (2005), 1–11. | MR | Zbl

[22] Hatvani, L.: Annulus arguments in the stability theory for functional differential equations. Differential and Integral Equations, 10 (1997), 975–1002. | MR | Zbl

[23] Hale, J. K.: Theory of Functional Differential Equation. Springer, New York, 1977. | MR

[24] Raffoul, Y. R.: Stability in neutral nonlinear differential equations with functional delays using fixed-point theory. Mathematical and Computer Modelling, 40 (2004), 691–700. | DOI | MR | Zbl

[25] Smart, D. S.: Fixed point theorems; Cambridge Tracts in Mathematics. 66, Cambridge University Press, London–New York, 1974. | MR

[26] Tunc, C.: Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. Appl. Comput. Math., 10, 3 (2011), 449–462. | MR | Zbl

[27] Tunc, C.: On the stability and boundedness of solutions of a class of non-autonomous differential equations of second order with multiple deviating arguments. Afr. Mat., 23, 2 (2012), 249–259. | DOI | MR

[28] Tunc, C.: A note on the stability and boundedness of non-autonomous differential equations of second order with a variable deviating argument. Afr. Mat., 25, 2 (2014), 417–425. | DOI | MR | Zbl

[29] Zhang, B.: Fixed points and stability in differential equations with variable delays. Nonlinear Anal., 63 (2005), 233–242. | DOI | Zbl