Stability of Nonlinear Neutral Mixed Type Liven-Nohel Integro-Differential Equations
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Authors: K. BESSIOUD, A. ARDJOUNI AND A. DJOUDI
DOI: 10.46793/KgJMat2205.721B
Abstract:
In this paper, we use the contraction mapping theorem to obtain asymptotic stability results about the zero solution for a nonlinear neutral mixed type Levin-Nohel integro-differential equation. An asymptotic stability theorem with a necessary and sufficient condition is proved. An example is also given to illustrate our main results.
Keywords:
Asymptotic stability, contraction mapping theorem, neutral integro-differential equations, mixed type.
References:
[1] A. Ardjouni and A. Djoudi, Stability in nonlinear neutral integro-differential equations with variable delay using fixed point theory, J. Appl. Math. Comput. 44 (2014), 317–336.
[2] A. Ardjouni and A. Djoudi, Fixed point and stability in neutral nonlinear differential equations with variable delays, Opuscula Math. 32 (2012), 5–19.
[3] A. Ardjouni, A. Djoudi and I. Soualhia, Stability for linear neutral integro-differential equations with variable delays, Electron. J. Diff. Equ. 2012(172) (2012), 1–14.
[4] L. C. Becker and T. A. Burton, Stability, fixed points and inverse of delays, Proc. Roy. Soc. Edinburgh 136A (2006), 245–275.
[5] K. Bessioud, A. Ardjouni and A. Djoudi, Asymptotic stability in nonlinear neutral Levin-Nohel integro-differential equations, Journal of Nonlinear Functional Analysis 2017 (2017), 1–12.
[6] E. Bicer, On the asymptotic behavior of solutions of neutral mixed type differential equations, Results Math. 73(144) (2018), 1–12.
[7] T. A. Burton, Fixed points and stability of a nonconvolution equation, Proc. Amer. Math. Soc. 132 (2004), 3679–3687.
[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 Stud. 9 (2001), 181–190.
[10] T. A. Burton, Stability by fixed point theory or Liapunov’s theory, a comparison, Fixed Point Theory 4 (2003), 15–32.
[11] T. A. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems Appl. 11 (2002), 499–519.
[12] T. A. Burton and T. Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445–454.
[13] I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equation, Opuscula Math. 33 (2013), 255–272.
[14] L. Ćirić, Some Recent Results in Metrical Fixed Point Theory, University of Belgrade, Beograd, Serbia, 2003.
[15] N. T. Dung, Asymptotic behavior of linear advanved differential equations, Acta Math. Sci. 35 (2015), 610–618.
[16] N. T. Dung, New stability conditions for mixed linear Levin-Nohel integro-differential equations, J. Math. Phys. 54 (2013), 1–11.
[17] C. H. Jin and J. W. Luo, Stability of an integro-differential equation, Comput. Math. Appl. 57 (2009), 1080–1088.
[18] C. H. Jin and J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68 (2008), 3307–3315.
[19] C. H. Jin and J. W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proc. Amer. Math. Soc. 136 (2008), 909–918.
[20] R. Klen, V. Manojlović, S. Simić and M. Vuorinen, Bernoulli inequality and hypergeometric functions, Proc. Amer. Math. Soc. 142 (2014), 559–573.
[21] E. Malkowski and V. Rakocević, Advanced Functional Analysis, CRS Press, Taylor and Francis Group, Boca Raton, FL, 2019.
[22] M. B. Mesmouli, A. Ardjouni and A. Djoudi, Study of the stability in nonlinear neutral differential equations with functional delay using Krasnoselskii-Burton’s fixed-point, Appl. Math. Comput. 243 (2014), 492–502.
[23] M. B. Mesmouli, A. Ardjouni and A. Djoudi, Stability in neutral nonlinear differential equations with functional delay using Krasnoselskii-Burton’s fixed-point, Nonlinear Stud. 21 (2014), 601–617.
[24] S. Pinelas, Asymptotic behavior of solutions to mixed type differential equations, Electron. J. Differ. Equ. 2014(210) (2014), 1–9.
[25] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63 (2005), e233–e242.
[26] D. R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics 66, Cambridge University Press, London, New York, 1974.
[27] S. Thenmozhi, M. Marudai and S. Radenović, Existence of positive solution for the eighth-order boundary value problem by Leray-Schauder alternative fixed point theorem, Axioms 8(129) (2019), 1–13.
[28] V. Todorčević, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer Nature, Cham, Switzerland, 2019.