| Title:
|
New sufficient conditions for global asymptotic stability of a kind of nonlinear neutral differential equations (English) |
| Author:
|
Benhadri, Mimia |
| Author:
|
Caraballo, Tomás |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
385-405 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper addresses the stability study for nonlinear neutral differential equations. Thanks to a new technique based on the fixed point theory, we find some new sufficient conditions ensuring the global asymptotic stability of the solution. In this work we extend and improve some related results presented in recent works of literature. Two examples are exhibited to show the effectiveness and advantage of the results proved. (English) |
| Keyword:
|
contraction mapping principle |
| Keyword:
|
asymptotic stability |
| Keyword:
|
neutral differential equation\looseness 1 |
| MSC:
|
34K13 |
| MSC:
|
34K20 |
| MSC:
|
92B20 |
| idZBL:
|
Zbl 07584132 |
| idMR:
|
MR4482313 |
| DOI:
|
10.21136/MB.2021.0079-20 |
| . |
| Date available:
|
2022-09-05T09:40:15Z |
| Last updated:
|
2022-12-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151015 |
| . |
| Reference:
|
[1] Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays.Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 2062-2070. Zbl 1216.34069, MR 2781737, 10.1016/j.na.2010.10.050 |
| Reference:
|
[2] Ardjouni, A., Djoudi, A.: Fixed points and stability in neutral nonlinear differential equations with variable delays.Opusc. Math. 32 (2012), 5-19. Zbl 1254.34110, MR 2852465, 10.7494/OpMath.2012.32.1.5 |
| Reference:
|
[3] Ardjouni, A., Djoudi, A.: Global asymptotic stability of nonlinear neutral differential equations with variable delays.Nonlinear Stud. 23 (2016), 157-166. Zbl 1348.34126, MR 3524420 |
| Reference:
|
[4] Ardjouni, A., Djoudi, A.: Global asymptotic stability of nonlinear neutral differential equations with infinite delay.Transylv. J. Math. Mech. 9 (2017), 125-133. MR 3524420 |
| Reference:
|
[5] Brayton, R. K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type.Q. Appl. Math. 24 (1966), 215-224. Zbl 0143.30701, MR 0204800, 10.1090/qam/204800 |
| Reference:
|
[6] Burton, T. A.: Integral equations, implicit functions, and fixed points.Proc. Am. Math. Soc. 124 (1996), 2383-2390. Zbl 0873.45003, MR 1346965, 10.1090/S0002-9939-96-03533-2 |
| Reference:
|
[7] Burton, T. A.: Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem.Nonlinear Stud. 9 (2002), 181-190. Zbl 1084.47522, MR 1898587 |
| Reference:
|
[8] Burton, T. A.: Stability by fixed point theory or Liapunov theory: A comparison.Fixed Point Theory 4 (2003), 15-32. Zbl 1061.47065, MR 2031819 |
| Reference:
|
[9] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations.Dover Publications, New York (2006). Zbl 1160.34001, MR 2281958 |
| Reference:
|
[10] Dib, Y. M., Maroun, M. R., Raffoul, Y. N.: Periodicity and stability in neutral nonlinear differential equations with functional delay.Electron. J. Differ. Equ. 2005 (2005), Article ID 142, 11 pages. Zbl 1097.34049, MR 2181286 |
| Reference:
|
[11] Djoudi, A., Khemis, R.: Fixed point techniques and stability for neutral differential equations with unbounded delays.Georgian Math. J. 13 (2006), 25-34. Zbl 1104.34052, MR 2242326, 10.1515/GMJ.2006.25 |
| Reference:
|
[12] Fan, M., Xia, Z., Zhu, H.: Asymptotic stability of delay differential equations via fixed point theory and applications.Can. Appl. Math. Q. 18 (2010), 361-380. Zbl 1237.34125, MR 2858144 |
| Reference:
|
[13] Guo, Y.: A generalization of Banach's contraction principle for some non-obviously contractive operators in a cone metric space.Turk. J. Math. 36 (2012), 297-304. Zbl 1252.54033, MR 2912045, 10.3906/mat-1005-267 |
| Reference:
|
[14] Guo, Y., Xu, C., Wu, J.: Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle.Int. J. Control 90 (2017), 1555-1560. Zbl 1367.93697, MR 3658462, 10.1080/00207179.2016.1213524 |
| Reference:
|
[15] Hale, J. K., Meyer, K. R.: A class of functional equations of neutral type.Mem. Am. Math. Soc. 76 (1967), 65 pages. Zbl 0179.20501, MR 0223842, 10.1090/memo/0076 |
| Reference:
|
[16] Hale, J. K., Lunel, S. M. Verduyn: Introduction to Functional Differential Equations.Applied Mathematical Sciences 99. Springer, New York (1993). Zbl 0787.34002, MR 1243878, 10.1007/978-1-4612-4342-7 |
| Reference:
|
[17] Jin, C., Luo, J.: Fixed points and stability in neutral differential equations with variable delays.Proc. Am. Math. Soc. 136 (2008), 909-918. Zbl 1136.34059, MR 2361863, 10.1090/S0002-9939-07-09089-2 |
| Reference:
|
[18] Kolmanovskii, V. B., Myshkis, A. D.: Applied Theory of Functional Differential Equations.Mathematics and Its Applications. Soviet Series 85. Kluwer Academic, Dordrecht (1992). Zbl 0785.34005, MR 1256486, 10.1007/978-94-015-8084-7 |
| Reference:
|
[19] Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics.Mathematics in Science and Engineering 191. Academic Press, Boston (1993). Zbl 0777.34002, MR 1218880, 10.1016/s0076-5392(08)x6164-8 |
| Reference:
|
[20] Lisena, B.: Global attractivity in nonautonomous logistic equations with delay.Nonlinear Anal., Real World Appl. 9 (2008), 53-63. Zbl 1139.34052, MR 2370162, 10.1016/j.nonrwa.2006.09.002 |
| Reference:
|
[21] Liu, G., Yan, J.: Global asymptotic stability of nonlinear neutral differential equation.Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1035-1041. Zbl 1457.34110, MR 3119279, 10.1016/j.cnsns.2013.08.035 |
| Reference:
|
[22] Luo, J.: Fixed points and stability of neutral stochastic delay differential equations.J. Math. Anal. Appl. 334 (2007), 431-440. Zbl 1160.60020, MR 2332567, 10.1016/j.jmaa.2006.12.058 |
| Reference:
|
[23] Monje, A. A. Z. Mahdi, Ahmed, B. A. A.: Using Banach fixed point theorem to study the stability of first-order delay differential equations.Al-Nahrain J. Sci. 23 (2020), 69-72. 10.22401/ANJS.23.1.10 |
| Reference:
|
[24] Pinto, M., Sepúlveda, D.: $h$-asymptotic stability by fixed point in neutral nonlinear differential equations with delay.Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 3926-3933. Zbl 1237.34129, MR 2802978, 10.1016/j.na.2011.02.029 |
| Reference:
|
[25] Raffoul, Y. N.: Stability in neutral nonlinear differential equations with functional delays using fixed-point theory.Math. Comput. Modelling 40 (2004), 691-700. Zbl 1083.34536, MR 2106161, 10.1016/j.mcm.2004.10.001 |
| Reference:
|
[26] Seifert, G.: Liapunov-Razumikhin conditions for stability and boundedness of functional differential equations of Volterra type.J. Differ. Equations 14 (1973), 424-430. Zbl 0248.34078, MR 0492745, 10.1016/0022-0396(73)90058-2 |
| Reference:
|
[27] Smart, D. R.: Fixed Points Theorems.Cambridge Tracts in Mathematics 66. Cambridge University Press, Cambridge (1980). Zbl 0427.47036, MR 0467717 |
| Reference:
|
[28] Tunç, C.: Stability and boundedness of solutions of non-autonomous differential equations of second order.J. Comput. Anal. Appl. 13 (2011), 1067-1074. Zbl 1227.34054, MR 2789545 |
| Reference:
|
[29] Tunç, C.: Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments.Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 57 (2014), 121-130. Zbl 1340.34273, MR 3204786 |
| Reference:
|
[30] Tunç, C.: Convergence of solutions of nonlinear neutral differential equations with multiple delays.Bol. Soc. Mat. Mex., III. Ser. 21 (2015), 219-231. Zbl 1327.34126, MR 3377988, 10.1007/s40590-014-0050-6 |
| Reference:
|
[31] Tunç, C., Sirma, A.: Stability analysis of a class of generalized neutral equations.J. Comput. Anal. Appl. 12 (2010), 754-759. Zbl 1197.34145, MR 2649294 |
| Reference:
|
[32] Tunç, C., Tunç, O.: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order.J. Adv. Research 7 (2016), 165-168. 10.1016/j.jare.2015.04.005 |
| Reference:
|
[33] Yazgan, R., Tunç, C., Atan, Ö.: On the global asymptotic stability of solutions to neutral equations of first order.Palest. J. Math. 6 (2017), 542-550. Zbl 1369.34096, MR 3646266 |
| Reference:
|
[34] Zhang, B.: Contraction mapping and stability in a delay-differential equation.Dynamic Systems and Applications. Volume 4 Dynamic Publishers, Atlanta (2004), 183-190. Zbl 1079.34543, MR 2117781 |
| Reference:
|
[35] Zhang, B.: Fixed points and stability in differential equations with variable delays.Nonlinear Anal., Theory Methods Appl., Ser. A 63 (2005), e233--e242. Zbl 1159.34348, 10.1016/j.na.2005.02.081 |
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