×

Asymptotic stability of neutral set-valued functional differential equation by fixed point method. (English) Zbl 1459.34165

Summary: This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.

MSC:

34K20 Stability theory of functional-differential equations
34A06 Generalized ordinary differential equations (measure-differential equations, set-valued differential equations, etc.)
47H04 Set-valued operators
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Burton, T. A., Liapunov functionals fixed points and stability by Krasnoselskii’s theorem, Nonlinear Studies, 9, 181-190 (2001) · Zbl 1084.47522
[2] Burton, T. A., Stability by fixed point theory or Liapunov’s theory: a Comparison, Fixed Point Theory, 4, 15-32 (2003) · Zbl 1061.47065
[3] Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations (2006), New York, NY, USA: Dover Publications, New York, NY, USA · Zbl 1160.34001
[4] Abdelouaheb, A.; Ahcene, D., Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis, 74, 2062-2070 (2011) · Zbl 1216.34069
[5] Burton, T. A.; Furumochi, T., Asymptotic behavior of solutions of functional differential equations by fixed points theorem, Dynamic Systems and Applications, 11, 499-519 (2002) · Zbl 1044.34033
[6] Chen, G.; Li, D.; van Gaans, O.; Verduyn Lunel, S., Stability results for nonlinear functional differential equations using fixed point methods, Indagationes Mathematicae, 29, 2, 671-686 (2018) · Zbl 1388.34064 · doi:10.1016/j.indag.2017.11.004
[7] Ge, F.; Kou, C., Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations, Applied Mathematics and Computation, 257, 308-316 (2015) · Zbl 1338.34103 · doi:10.1016/j.amc.2014.11.109
[8] Jin, C. H.; Luo, J. W., Fixed points and stability in neutral differential equations with variable delays, Proceedings of the American Mathematical Society, 136, 909-918 (2008) · Zbl 1136.34059
[9] Manuel, P.; Daniel, S., h-Asymptotic stability by fixed point in neutral nonlinear differential equations with delay, Nonlinear Analysis, 74, 3926-3933 (2011) · Zbl 1237.34129
[10] Mesmouli, M. B.; Ardjouni, A.; Djoudi, A., Study of the stability in nonlinear neutral differential equations with functional delay using Krasnoselskii-Burton’s fixed-point, Applied Mathematics and Computation, 243, 492-502 (2014) · Zbl 1335.34115 · doi:10.1016/j.amc.2014.05.135
[11] Rao, R.; Zhong, S., Stability analysis of impulsive stochastic reaction-diffusion cellular neural network with distributed delay via fixed point theory, Complexity, 2017, 1-9 (2017) · Zbl 1377.93169 · doi:10.1155/2017/6292597
[12] Raffoul, Y. N., Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Mathematical and Computer Modelling, 40, 7-8, 691-700 (2004) · Zbl 1083.34536 · doi:10.1016/j.mcm.2004.10.001
[13] Wang, C. S.; Li, Y. M., Krasnoselskii fixed point and exponential p-stability of neutral stochastic dynamic systems with time-varying delays, Chinese Journal of Applied Mechanics, 36, 4, 901-905 (2019)
[14] Zhao, D.; Han, D., Stability of linear neutral differential equations with delays and impulses established by the fixed points method, Nonlinear Analysis: Theory, Methods & Applications, 74, 18, 7240-7251 (2011) · Zbl 1232.34103 · doi:10.1016/j.na.2011.07.041
[15] Lakshmikantham, V.; Bhaskar, T. G.; Devi, J. V., Theory of Set Differential Equations in Metric Spaces (2005), Cambridge, UK: Cambridge Scientific Publisher, Cambridge, UK · Zbl 1141.01003
[16] Azzam-Laouir, D.; Boukrouk, W., A delay second-order set-valued differential equation with Hukuhara derivatives, Numerical Functional Analysis and Optimization, 36, 6, 704-729 (2015) · Zbl 1326.34105 · doi:10.1080/01630563.2015.1017646
[17] Azzam-Laouir, D.; Boukrouk, W., Second-order set-valued differential equations with boundary conditions, Journal of Fixed Point Theory and Applications, 17, 1, 99-121 (2015) · Zbl 1410.34077 · doi:10.1007/s11784-015-0236-1
[18] Bashir, A.; Sivasundaram, S., Setvalued perturbed Hybrid integro-differential equations and stability in terms of two measures, Dynamic Systems and Applications, 16, 299-310 (2007) · Zbl 1151.45006
[19] Bashir, A.; Sivasundaram, S., Basic results and stability criteria for set valued differential equations on time scales, Communications in Applied Analysis, 11, 419-428 (2007) · Zbl 1148.34002
[20] Drice, Z.; Mcrae, F. A.; Devi, J. V., Set differential equations with causal operators, Mathematical Problems in Engineering, 2, 185-194 (2005) · Zbl 1108.34011
[21] Nguyen, D. P.; Le, T. Q.; Tran, T. T., Stability criteria for set control differential equations, Nonlinear Analysis, 69, 3715-3721 (2008) · Zbl 1167.93025
[22] Bhaskar, T. G.; Devi, J. V., Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84, 2, 131-143 (2005) · Zbl 1074.34010 · doi:10.1080/00036810410001724346
[23] Lakshmikantham, V.; Leela, S.; Devi, J. V., Stability theory for set differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 11, 181-190 (2004) · Zbl 1068.34047
[24] Nguyen, N. T.; Tran, T. T., Stability of set differential euqations and applications, Nonlinear Analysis, 71, 1526-1533 (2009) · Zbl 1188.34068
[25] Abbas, U.; Lupulescu, V.; O’Regan, D.; Younus, A., Neutral set differential equations, Czechoslovak Mathematical Journal, 65, 140, 593-615 (2015) · Zbl 1363.34205 · doi:10.1007/s10587-015-0199-9
[26] Abbas, U.; Lupulescu, V., Set functional differential equations, Communications on Applied Nonlinear Analysis, 18, 97-110 (2011) · Zbl 1243.34108
[27] Bashir, A.; Sivasundaram, S., Stability in terms of two measures for setvalued perturbed impulsive delay differential equations, Communications in Applied Analysis, 12, 1, 57-68 (2008) · Zbl 1161.34043
[28] Devi, J. V.; Vatsala, A. S., A study of set differential equations with delay, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 11, 287-300 (2004) · Zbl 1069.34112
[29] Malinowski, M. T., Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Applied Mathematics and Computation, 218, 18, 9427-9437 (2012) · Zbl 1252.34071 · doi:10.1016/j.amc.2012.03.027
[30] Bhaskar, T. G.; Devi, J. V., Stability criteria for set differential equations, Mathematical and Computer Modelling, 41, 11-12, 1371-1378 (2005) · Zbl 1093.34542 · doi:10.1016/j.mcm.2004.01.012
[31] Bhaskar, T. G.; Devi, J. V., Set differential systems and vector Lyapunov functions, Applied Mathematics and Computation, 165, 539-548 (2005) · Zbl 1080.34040
[32] Liu, G.; Yan, J., Global asymptotic stability of nonlinear neutral differential equation, Communications in Nonlinear Science and Numerical Simulation, 19, 4, 1035-1041 (2014) · Zbl 1457.34110 · doi:10.1016/j.cnsns.2013.08.035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.