EXISTENCE AND STABILITY ANALYSIS OF EVOLUTIONARY DIFFERENTIAL HEMIVARIATIONAL INEQUALITIES IN BANACH SPACES

doi:10.1007/s10473-026-0203-5

Abstract

Abstract: In this paper, we introduce and study a new class of evolutionary differential hemivariational inequalities in Banach spaces. Using a fixed-point theorem, we first establish the existence and uniqueness of mild solutions for the evolutionary differential hemivariational inequality. Building on this result, we then investigate the stability of the evolutionary differential hemivariational inequality problem under perturbations of the mappings, constraint set, and infinitesimal generator of the $C_0$-semigroup ${\rm e}^{tB}$.

Key words: evolutionary differential hemivariational inequality, mild solution, stability, $C_0$-semigroup

CLC Number:

49J40

Wei LI, Ke XU, Rong HU, Zhiyang MA. EXISTENCE AND STABILITY ANALYSIS OF EVOLUTIONARY DIFFERENTIAL HEMIVARIATIONAL INEQUALITIES IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2026, 46(2): 549-567.

Trendmd

References

[1] Pang J S, Stewart D. Differential variational inequaliities. Mathematical Programming Series A, 2008, 113: 345-424
[2] Stewart D E. Uniqueness for index-one differential variational inequalities. Nonlinear Analysis Hybrid Systems, 2008, 2(3): 812-818
[3] Li X S, Huang N J, O'Regan D. Differential mixed variational inequalities in finite dimensional spaces. Nonlinear Analysis, 2010, 72(9): 3875-3886
[4] Li X, Huang N J. A class of impulsive differential variational inequalities in finite demensional space. Journal of the Franklin Institute, 2016, 353: 3151-3175
[5] Li W, Wang X, Huang N J. Differential inverse variational inequalities in finite dimensional spaces. Acta Mathematica Scientia, 2015, 35B(2): 407-422
[6] Wang X, Qi Y W, Tao C Q, Xiao Y B. A class of delay differential variational inequalities. Journal of Optimization Theory and Applications, 2017, 172(1): 56-69
[7] Liu Z H, Zeng S D, Motreanu D. Evolutionary problems driven by variational inequalities. Journal of Differential Equations, 2016, 260(9): 6787-6799
[8] Liu Z H, Zeng S D. Differential variational inequalities in infinite Banach spaces. Acta Mathematica Scientia, 2017, 37B(1): 26-32
[9] Migórski S, Zeng S D. A class of generalized evolutionary problems driven by variational inequalities and fractional operators. Set-Valued and Variational Analysis,2019, 27(4): 949-970
[10] Migórski S, Zeng S D. Mixed variational inequalities driven by fractional evolutionary equations. Acta Mathematica Scientia,2019, 39(2): 461-468
[11] Liu Z H, Zeng S D, Motreanu D. Partial differential hemivariational inequalities. Advances in Nonlinear Analysis, 2018, 7(4): 571-586
[12] Zeng S D, Liu Z H, Migórski S. A class of fractional differential hemivariational inequalities with application to contact problem. Zeitschrift für angewandte Mathematik und Physik,2018, 69: 1-23
[13] Jiang Y R, Huang N J, Wei Z C. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems-Series B, 2020, 25(4): 1193-1212
[14] Liu Z H, Nguyen V T, Yao J C, Zeng S D. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations and Control Theory, 2020, 9(4): 1073-1087
[15] Migórski S, Zeng S D. A class of differential hemivariational inequalities in Banach spaces. Journal of Global Optimization,2018, 72(4): 761-779
[16] Chen T, Huang N J, Li X S, Zhou Y Z. A new class of differential ninliner system involving parabolic variational and history-dependent hemi-variational inequalities arising in contact mechanics. Commun Nonlinear Science Numerical Simulation, 2021, 101: 105886
[17] Zeng S D, Migórski S, Han W M. A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation. Izvestiya: Mathematics,2023, 87(2): 326-361
[18] Cen J X, Sousa J V D, Wu W. Fractional partial differential variational inequality. Communications in Nonlinear Science and Numerical Simulation, 2024, 128: 107600
[19] Chang S S, Salahuddin, Ahmadini A A H, et al. The existence problems of solutions for a class of differential variational-hemivariational inequality problems. Mathematics, 2023, 11(9): 2066
[20] Han W M, Nashed M Z. On variational-hemivariational inequalities in Banach spaces. Communications in Nonlinear Science and Numerical Simulation, 2023, 124: 107309
[21] Wang X, Chen S H, Qi H Y. A class of delay differential variational inequalities with boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 2024, 129: 107684
[22] Li W, Wang X, Li X S, Huang N J. A class of differential inverse mixed variational inequalities in finite dimensional spaces. Pacific Journal of Optimization, 2017, 13(4): 571-591
[23] Xiao Y B, Liu M T, Chen T, Huang N J. Stability analysis for evolutionary variational-hemivariational inequalities with constraint sets. Science China Mathematics, 2022, 65(7): 1469-1484
[24] Wang X, Li W, Li X S, Huang N J. Stability for differential mixed variational inequalities. Optimization Letters, 2014, 8(6): 1873-1887
[25] Gwinner J. On a new class of differential variational inequalities and a stability result. Mathematical Programming Series B, 2013, 139: 205-221
[26] Wang X, Huang N J. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial and Management Optimization, 2013, 9(1): 57-74
[27] Li W, Xiao Y B, Wang X, Feng J. Existence and stability for a generalized differential mixed quasi-variational inequality. Carpathian Journal of Mathematics, 2018, 34(3): 347-354
[28] Li W, Wang X F. Stability analysis of partial differential set-valued variational inequalities on Banach spaces. Journal of Convex Analysis, 2020, 27(2): 423-442
[29] Han Y, Huang N J, Lu J, Xiao Y B. Existence and stability of solutions to inverse variational inequality problems. Applied Mathematics and Mechanics, 2017, 38(5): 749-764
[30] Guo F, Li W, Xiao Y B, Migórski S. Stability analysis of partial differential variational inequalities in Banach spaces. Nonlinear Analysis: Modelling and Control,2020, 25(1): 69-83
[31] Jiang Y R, Wei Z C. Weakly asymptotic stability for fractional delay differential mixed variational inequalities. Applied Mathematics and Optimization, 2021, 84(1): 273-297
[32] Tang G J, Cen J X, Nguyen V T, Zeng S D. Differential variational-hemivariational inequalities: existence, uniqueness, stability,convergence. Journal of Fixed Point Theory and Applications, 2020, 22(4): 1-30
[33] Tam V M, Wu W. Caputo fractional differential variational-emivariational inequalities involving history-dependent operators: Global error bounds and convergence. Communications in Nonlinear Science and Numerical Simulation, 2024, 128: 107654
[34] Sofonea M, Migórski S. Variational-Hemivariational Inequalities with Applications. London: Chapman and Hall/CRC, 2018
[35] Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. New York: Springer Science and Business Media, 2012
[36] Li X J, Yong J M.Optimal Control Theory for Infinite Dimensional Systems. Boston: Springer Science and Business Media, 2012
[37] Liu Z H, Migorski S, Zeng S D. Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces. Journal of Differential Equations, 2017, 263(7): 3989-4006
[38] Lang S.Real and Functional Analysis. New York: Springer Science and Business Media, 2012
[39] Pazy A.Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer Science and Business Media, 2012
[40] Mosco U. Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics, 1969, 3(4): 510-585