Finite-time and predefined-time neurodynamic approach with time-varying coefficients for solving inverse quasi-variational inequality problems

Yan-Yu Xie; Vajahat Karim Khan; Md. Kalimuddin Ahmad; Qing-Bo Cai; Yan-Yu Xie; Vajahat Karim Khan; Md. Kalimuddin Ahmad; Qing-Bo Cai

doi:10.3934/math.2026915

AIMS Mathematics

2026, Volume 11, Issue 7: 22658-22687. doi: 10.3934/math.2026915

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Research article

Finite-time and predefined-time neurodynamic approach with time-varying coefficients for solving inverse quasi-variational inequality problems

1.
College of General Education, Minnan Science and Technology College, Quanzhou 362332, China
2.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, U.P., India
3.
Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China

Received: 06 May 2026 Revised: 13 July 2026 Accepted: 15 July 2026 Published: 27 July 2026
MSC : 47J30, 49J53, 49M37

This paper proposes finite-time (FT) and predefined-time (PdT) neurodynamic models with time-varying coefficients for solving inverse quasi-variational inequality (IQVI) problems. Two projection-based neurodynamic models are developed, in which time-varying gains are designed to regulate the convergence speed and improve transient performance. A nominal projected model is introduced to establish the equivalence between its equilibrium points and the solutions of IQVI problems. Under Lipschitz continuity and strong monotonicity assumptions, the existence, uniqueness, and global convergence of the proposed models are established. By employing Lyapunov theory, FT and PdT convergence are rigorously proved together with explicit settling-time estimates. In particular, the PdT model guarantees convergence within a user-prescribed time bound independent of the initial conditions. Furthermore, a forward Euler discretization is developed to obtain implementable numerical algorithms, and robustness of the PdT model under bounded disturbances is established. Numerical experiments, including an additional higher-dimensional IQVI problem example and a sparse signal recovery application, validate the theoretical results and demonstrate the effectiveness, robustness, and computational efficiency of the proposed framework.
- finite-time stability,
- predefined-time stability,
- neurodynamic approach,
- time-varying coefficients,
- inverse quasi-variational inequality
Citation: Yan-Yu Xie, Vajahat Karim Khan, Md. Kalimuddin Ahmad, Qing-Bo Cai. Finite-time and predefined-time neurodynamic approach with time-varying coefficients for solving inverse quasi-variational inequality problems[J]. AIMS Mathematics, 2026, 11(7): 22658-22687. doi: 10.3934/math.2026915

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Abstract

This paper proposes finite-time (FT) and predefined-time (PdT) neurodynamic models with time-varying coefficients for solving inverse quasi-variational inequality (IQVI) problems. Two projection-based neurodynamic models are developed, in which time-varying gains are designed to regulate the convergence speed and improve transient performance. A nominal projected model is introduced to establish the equivalence between its equilibrium points and the solutions of IQVI problems. Under Lipschitz continuity and strong monotonicity assumptions, the existence, uniqueness, and global convergence of the proposed models are established. By employing Lyapunov theory, FT and PdT convergence are rigorously proved together with explicit settling-time estimates. In particular, the PdT model guarantees convergence within a user-prescribed time bound independent of the initial conditions. Furthermore, a forward Euler discretization is developed to obtain implementable numerical algorithms, and robustness of the PdT model under bounded disturbances is established. Numerical experiments, including an additional higher-dimensional IQVI problem example and a sparse signal recovery application, validate the theoretical results and demonstrate the effectiveness, robustness, and computational efficiency of the proposed framework.

References

[1]	J. Zhao, T. Lu, Y. Zhang, Chaoticity of time-varying discrete dynamic systems via equicontinuity, AIMS Math., 10 (2025), 29406–29423. https://doi.org/10.3934/math.20251291 doi: 10.3934/math.20251291
[2]	T. Gan, V. K. Khan, M. K. Ahmad, Q. B. Cai, A second-order dynamical system for solving inverse quasi-variational inequalities and its application, PlOS One, 21 (2026), e0344815. https://doi.org/10.1371/journal.pone.0344815 doi: 10.1371/journal.pone.0344815
[3]	D. Filali, M. Dilshad, M. Akram, M. F. Khan, S. S. Irfan, Hybrid inertial viscosity-type forward-backward splitting algorithms for variational inclusion problems, AIMS Math., 10 (2025), 28829–28860. https://doi.org/10.3934/math.20251269 doi: 10.3934/math.20251269
[4]	M. Akram, M. Dilshad, A. K. Rajpoot, F. Babu, R. Ahmad, J. C. Yao, Modified iterative schemes for a fixed point problem and a split variational inclusion problem, Mathematics, 10 (2022), 2098. https://doi.org/10.3390/math10122098 doi: 10.3390/math10122098
[5]	F. Facchinei, J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer-Verlag, 2003. https://doi.org/10.1007/b97543
[6]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, 2011. https://doi.org/10.1007/978-3-319-48311-5
[7]	Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119–1132. https://doi.org/10.1080/02331934.2010.539689 doi: 10.1080/02331934.2010.539689
[8]	Y. V. Malitsky, V. V. Semenov, An extragradient for monotone variational inequalities, Cybern. Syst. Ananl., 50 (2014), 271–277. https://doi.org/10.1007/s10559-014-9614-8 doi: 10.1007/s10559-014-9614-8
[9]	M. V. Solodov, B. F. Svaiter, A new projectional algorithm for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765–776. https://doi.org/10.1137/S0363012997317475 doi: 10.1137/S0363012997317475
[10]	F. O. Nwawuru, O. K. Narain, M. Dilshad, J. N. Ezeora, Splitting method involving two-step inertial for solving inclusion and fixed point problems with applications, Fixed Point Theory Algorithms Sci. Eng., 2025 (2025), 8. https://doi.org/10.1186/s13663-025-00781-w doi: 10.1186/s13663-025-00781-w
[11]	Y. Han, N. Huang, J. Lu, Y. Xiao, Existence and stability of solutions to inverse variational inequality problems, Appl. Math. Mech., 38 (2017), 749–764. https://doi.org/10.1007/s10483-017-2191-9 doi: 10.1007/s10483-017-2191-9
[12]	P. T. Vuong, X. He, D. V. Thong, Global exponential stability of a neural network for inverse variational inequalities, J. Optim. Theory Appl., 190 (2021), 915–930. https://doi.org/10.1007/s10957-021-01915-x doi: 10.1007/s10957-021-01915-x
[13]	X. Zou, D. Gong, L. Wang, Z. Chen, A novel method to solve inverse variational inequality problems based on neural networks, Neurocomputing, 173 (2016), 1163–1168. https://doi.org/10.1016/j.neucom.2015.08.073 doi: 10.1016/j.neucom.2015.08.073
[14]	B. He, X. He, H. X. Liu, Solving a class of constrained 'black-box' inverse variational inequalities, Eur. J. Oper. Res., 204 (2010), 391–401. https://doi.org/10.1016/j.ejor.2009.07.006 doi: 10.1016/j.ejor.2009.07.006
[15]	X. He, H. X. Liu, Inverse variational inequalities with projection-based solution methods, Eur. J. Oper. Res., 208 (2011), 12–18. https://doi.org/10.1016/j.ejor.2010.08.022 doi: 10.1016/j.ejor.2010.08.022
[16]	D. Aussel, R. Gupta, A. Mehra, Gap functions and error bounds for inverse quasi-variational inequality problems, J. Math. Anal. Appl., 407 (2013), 270–280. https://doi.org/10.1016/j.jmaa.2013.03.049 doi: 10.1016/j.jmaa.2013.03.049
[17]	S. Dey, S. Reich, A dynamical system for solving inverse quasi-variational inequalities, Optimization, 73 (2024), 1681–1701. https://doi.org/10.1080/02331934.2023.2173525 doi: 10.1080/02331934.2023.2173525
[18]	X. Hu, J. Wang, Global stability of a recurrent neural network for solving pseudomonotone variational inequalities, 2006 IEEE International Symposium on Circuits and Systems, 2006. https://doi.org/10.1109/ISCAS.2006.1692695
[19]	C. Chen, L. Li, H. Peng, Y. Yang, L. Mi, H. Zhao, A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks, Neural Networks, 123 (2020), 412–419. https://doi.org/10.1016/j.neunet.2019.12.028 doi: 10.1016/j.neunet.2019.12.028
[20]	P. T. Vuong, The global exponential stability of a dynamical system for solving variational inequalities, Netw. Spat. Econ., 22 (2022), 395–407. https://doi.org/10.1007/s11067-019-09457-6 doi: 10.1007/s11067-019-09457-6
[21]	P. K. Anh, T. N. Hai, Regularized dynamics for monotone inverse variational inequalities in Hilbert spaces, Optim. Eng., 25 (2024), 2295–2313. https://doi.org/10.1007/s11081-024-09882-8 doi: 10.1007/s11081-024-09882-8
[22]	S. P. Bhat, D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751–766. https://doi.org/10.1137/S0363012997321358 doi: 10.1137/S0363012997321358
[23]	X. Ju, C. Li, X. He, G. Feng, A proximal dynamic approach to equilibrium problems with finite-time convergence, IEEE Trans. Autom. Control, 69 (2023), 1773–1780. https://doi.org/10.1109/TAC.2023.3326713 doi: 10.1109/TAC.2023.3326713
[24]	H. Wang, P. X. Liu, X. Zhao, X. Liu, Adaptive fuzzy finite-time control of nonlinear systems with actuator faults, IEEE Trans. Cybern., 50 (2019), 1786–1797. https://doi.org/10.1109/TCYB.2019.2902868 doi: 10.1109/TCYB.2019.2902868
[25]	J. Cortés, Finite-time convergent gradient flows with applications to network consensus, Automatica, 42 (2006), 1993–2000. https://doi.org/10.1016/j.automatica.2006.06.015 doi: 10.1016/j.automatica.2006.06.015
[26]	H. Wang, W. Bai, X. Zhao, P. X. Liu, Finite-time-prescribed performance-based adaptive fuzzy control for strict-feedback nonlinear systems with dynamic uncertainty and actuator faults, IEEE Trans. Cybern., 52 (2021), 6959–6971. https://doi.org/10.1109/TCYB.2020.3046316 doi: 10.1109/TCYB.2020.3046316
[27]	S. Lushate, S. Liu, J. You, R. Tohti, H. Jiang, A. Abdurahman, Predefined-time convergence of proximal dynamics methods for equilibrium problems, 2025 37th Chinese Control and Decision Conference (CCDC), 2025,545–550. https://doi.org/10.1109/CCDC65474.2025.11091126
[28]	E. Jimenez-Rodriguez, J. D. Sanchez-Torres, A. G. Loukianov, On optimal predefined-time stabilization, Int. J. Robust Nonlinear Control, 27 (2017), 3620–3642. https://doi.org/10.1002/rnc.3757 doi: 10.1002/rnc.3757
[29]	J. Zheng, X. Ju, N. Zhang, D. Xu, A novel predefined-time neurodynamic approach for mixed variational inequality problems and applications, Neural Netw., 174 (2024), 106247. https://doi.org/10.1016/j.neunet.2024.106247 doi: 10.1016/j.neunet.2024.106247
[30]	J. Xu, C. Li, X. He, X. Zhang, Projection neural networks with finite-time and fixed-time convergence for sparse signal reconstruction, Neural Comput. Appl., 36 (2024), 425–443. https://doi.org/10.1007/s00521-023-09015-9 doi: 10.1007/s00521-023-09015-9
[31]	L. Yu, G. Zheng, J. P. Barbot, Dynamical sparse recovery with finite-time convergence, IEEE Trans. Signal Process., 65 (2017), 6146–6157. https://doi.org/10.1109/TSP.2017.2745468 doi: 10.1109/TSP.2017.2745468
[32]	L. T. Nguyen, A. Eberhard, X. Yu, A. Y. Kruger, C. Li, Finite-time nonconvex optimization using time-varying dynamical systems, J. Optim. Theory Appl., 203 (2024), 844–879. https://doi.org/10.1007/s10957-024-02536-w doi: 10.1007/s10957-024-02536-w
[33]	D. Yu, S. Lin, G. Zhang, H. Yin, Predefined-time with time-varying coefficients neurodynamic for composite optimization problems, Chaos Solitons Fract., 199 (2025), 116792. https://doi.org/10.1016/j.chaos.2025.116792 doi: 10.1016/j.chaos.2025.116792
[34]	S. Xu, S. Li, A strongly convergent alternated inertial algorithm for solving equilibrium problems, J. Optim. Theory Appl., 206 (2025), 35. https://doi.org/10.1007/s10957-025-02720-6 doi: 10.1007/s10957-025-02720-6
[35]	S. Xu, S. Li, X. Qin, An improved alternating inertial algorithm with adaptive step-sizes for equilibrium problems with data classification experiments, J. Comput. Appl. Math., 483 (2026), 117393. https://doi.org/10.1016/j.cam.2026.117393 doi: 10.1016/j.cam.2026.117393
[36]	Y. Cao, V. K. Khan, M. Sarfaraz, A. Arbi, M. K. Ahmad, Second-order dynamical system to solve the sum of monotone operators via the KM algorithm, Alex. Eng. J., 123 (2025), 341–345. https://doi.org/10.1016/j.aej.2025.03.040 doi: 10.1016/j.aej.2025.03.040
[37]	B. Tan, X. Qin, Two relaxed inertial forward-backward-forward algorithms for solving monotone inclusions and an application to compressed sensing, Can. J. Math., 78 (2026), 572–593. https://doi.org/10.4153/S0008414X24000889 doi: 10.4153/S0008414X24000889
[38]	S. Karamardian, S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl., 66 (1990), 37–46. https://doi.org/10.1007/BF00940531
[39]	D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Society for Industrial and Applied Mathematics, 2000. https://doi.org/10.1137/1.9780898719451
[40]	M. Pappalardo, M. Passacantando, Stability for equilibrium problems: from variational inequalities to dynamical systems, J. Optim. Theory Appl., 113 (2002), 567–582. https://doi.org/10.1023/A:1015312921888 doi: 10.1023/A:1015312921888
[41]	A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Autom. Control, 57 (2011), 2106–2110. https://doi.org/10.1109/TAC.2011.2179869 doi: 10.1109/TAC.2011.2179869
[42]	N. V. Tran, T. T. H. Le, Finite-time and fixed-time stable dynamical systems for solving inverse quasi-variational inequality problems, Optimization, 2026. https://doi.org/10.1080/02331934.2026.2615249
[43]	N. Buong, A first order dynamical system and its discretization for a class of variational inequalities, J. Comput. Appl. Math., 458 (2025), 116341. https://doi.org/10.1016/j.cam.2024.116341 doi: 10.1016/j.cam.2024.116341

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