Abstract
In this paper, we consider a new class of random differential variational inequalities (RDVIs) with nonlocal boundary conditions in Hilbert spaces. We apply the projection operator, Gronwall’s lemma and a result on the existence of a random differential inclusion to establish uniqueness and Hyers–Ulam stability results of the abstract inequality. As an illustrative application, the linear random differential complementarity systems and a price control model are investigated.
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Andres, J., Górniewicz, L.: Random topological degree and random differential inclusions. Topol. Method Nonl. Ana. 40, 337–358 (2012)
Anh, N.T.V., Ke, T.D.: On the differential variational inequalities of parabolic-elliptic type. Math. Methods Appl. Sci. 40, 4683–4695 (2017)
Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Marcel Dekker Inc., New York (1980)
Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10(4), 643–647 (1943)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer-Verlag, Berlin, New York (1977)
Chen, X.J., Wang, Z.Y.: Differential variational inequality approach to dynamic games with shared constraints. Math. Program. Ser. A 146, 379–408 (2014)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Springer-Verlag, New York (2003)
Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers Inc, New York (2003)
Gwinner, J.: On a new class of differential variational inequalities and a stability result. Math. Program. Ser. B 139, 205–221 (2013)
Kandilakis, D.A., Papageorgiou, N.S.: On the existence of solutions for random differential inclusions in a Banach space. J. Math. Anal. Appl. 126, 11–23 (1987)
Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivaluedmaps and Semilinear Differential Inclusions in Banach Spaces. Walter de Gruyter, Berlin, New York (2001)
Ke, T.D., Loi, N.V., Obukhovskii, V.: Decay solutions for a class of fractional differential variational inequalities. Fract. Calc. Appl. Anal. 18, 531–553 (2015)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1984)
Li, X.S., Huang, N.J., O’Regan, D.: Differential mixed variational inqualities in finite dimensional spaces. Nonlinear Anal. 72, 3875–3886 (2010)
Li, X.S., Huang, N.J., O’Regan, D.: A class of impulsive differential variational inequalities in finite dimensional spaces. J. Franklin Inst. 353(13), 3151–3175 (2016)
Liu, Z.H., Loi, N.V., Obukhovskii, V.: Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Int. J. Bifurcat. Chaos 23, ID 1350125 (2013)
Liu, Z.H., Migorski, S., Zeng, S.D.: Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces. J. Differ. Equ. 263, 3989–4006 (2017)
Liu, Z.H., Motreanu, D., Zeng, S.D.: Nonlinear evolutionary systems driven by mixed variational inequalities and its applications. Nonlinear Anal. Real World Appl. 42, 409–421 (2018)
Loi, N.V., Ke, T.D., Vu, M.Q., Obukhovskii, V.: Random integral guiding functions with application to random differential complementarity systems. Discussiones Mathematicae Differential Inclusions, Control and Optimization 38, 113–132 (2018)
Loi, N.V., Vu, M.Q.: Uniqueness and Hyers–Ulam stability results for differential variational inequalities with nonlocal conditions, Differ. Equ. Dynam. Syst. https://doi.org/10.1007/s12591-018-0429-3, (2018)
Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic Publisher, Boston, The Netherlands (1999)
Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program. Ser. A 113, 345–424 (2008)
Raghunathan, A.U., Pérez-Correa, J.R., Agosin, E., Biegler, L.T.: Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities. Ann. Oper. Res. 148, 251–270 (2006)
Stewart, D.E.: Uniqueness for index-one differential variational inequalities. Nonlinear Anal. Hybrid Syst. 2, 812–818 (2008)
Tasora, A., Anitescu, M., Negrini, S., Negrut, D.: A compliant visco-plastic particle contact model based on differential variational inequalities. Int. J. Nonlin. Mech. 53, 2–12 (2013)
Wang, X., Huang, N.J.: Differential vector variational inequalities in finite dimensional spaces. J. Optim. Theory Appl. 158, 109–129 (2013)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11661030, 11961014), Natural Science Foundation of Guangxi Province (2018GXNSFAA281021) and Technology Base Foundation of of Guangxi Province (AD20159017), and the Foundation of Guilin University of Technology (GUTQDJJ2017062). The authors are grateful to the editor and the referees for their valuable comments and suggestions.
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Communicated by Lorenz Biegler.
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Jiang, Y., Song, Q. & Zhang, Q. Uniqueness and Hyers-Ulam Stability of Random Differential Variational Inequalities with Nonlocal Boundary Conditions. J Optim Theory Appl 189, 646–665 (2021). https://doi.org/10.1007/s10957-021-01850-x
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DOI: https://doi.org/10.1007/s10957-021-01850-x
Keywords
- Random differential variational inequality
- Hyers–Ulam stability
- Nonlocal boundary condition
- Linear complementarity system
- Price control