Lipschitz perturbation to evolution inclusion driven by time-dependent maximal monotone operators
DOI:
https://doi.org/10.12775/TMNA.2021.012Keywords
Evolution inclusion, fractional, maximal monotone operator, relaxation, subdifferential, viscosity, Young measureAbstract
An evolution inclusion driven by a time-dependent maximal monotone operator and a Lipschitz closed valued perturbation, in a separable Hilbert space is considered. The inclusion with a convexified perturbation term is also studied. Then, the existence of solutions and the relaxation property between these evolution inclusions are proved. Applications to dynamical systems governed by a couple of a fractional equation and an evolution inclusion involving time-dependent maximal monotone operators with a Lipschitz perturbation are presented.References
D. Azzam-Laouir, W. Belhoula, C. Castaing and M.D.P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Theory Appl. 21 (2019).
D. Azzam-Laouir and I. Boutana-Harid, Mixed semicontinuous perturbation to an evolution problem with time-dependent maximal monotone operator, J. Nonlinear Convex Anal. 20 (1) (2019), 39–52.
D. Azzam-Laouir, A. Makhlouf and L. Thibault, Existence and relaxation theorem for a second order differential inclusion, Numer. Funct. Anal. Optim. 31 (2010), 1103–1119.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ, Leyden, 1976.
H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Lecture Notes in Math., North-Holland, 1973.
C. Castaing, Quelques résultats de compacité liés à l’intégration, C.R. Acad. Sci. Paris 270 (1970), 1732–1735; Bulletin Soc. Math. France 31 (1972), 73–81.
C. Castaing, C. Godet-Thobie, P.D. Phung, L.X. Truong, On fractional differential inclusions with nonlocal boundary conditions, Fract. Calc. Appl. Anal. 22 (2) (2019), 444–478.
C. Castaing, M.D.P. Monteiro Marques and S. Saı̈di, Evolution problems with timedependent subdifferential operators, Adv. Math. Econ. 23 (2019), 1–39.
C. Castaing, P. Raynaud de Fitte and A. Salvadori, Some variational convergence results with applications to evolution inclusions, Adv. Math. Econ. 8 (2006), 33–73.
C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004.
C. Castaing and L.T. Truong, Some topological properties of solution sets in a second order differential inclusion with m-point boundary conditions, Set-Valued Var. Anal. 20 (2012), 249–277.
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., vol. 580, Springer–Verlag, Berlin, Heidelberg, 1977.
J.R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions – A Fixed Point Approach, de Gruyter GmbH Company, KG, 2013.
M. Guessous, An elementary proof of Komlós–Révész Theorem in Hilbert spaces, J. Convex Anal. 4 (1997), no. 2, 321–332.
J. Henderson and A. Ouahab, A Filippov’s theorem, some existence results and the compactness of solution sets of impulsive fractional order differential inclusions, Mediterr. J. Math. 9 (2012), 453–485.
F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149–182.
A. Idzik, Almost fixed points theorems, Proc. Amer. Math. Soc. 104 (1988), 779–784.
A. Ioffe, Existence and relaxation theorems for unbounded differential inclusions, J. Convex Anal. 13 (2006), no. 2, 353–362.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Math. Studies, vol. 204, North Holland, 2006.
M. Kunze and M.D.P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal. 5 (1997), 57–72.
P.D. Loewen and R.T. Rockafellar, Optimal control of unbounded differential inclusions, SIAM J. Control Optim. 32 (1994), no. 2, 442–470.
A. Makhlouf, D. Azzam-Laouir and C. Castaing, Existence and relaxation of solutions for evolution differential inclusions with maximal monotone operators, J. Fixed Point Theory Appl. 23 (2021), no. 2.
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, NewYork, 1993.
A. Ouahab, Some results for fractional boundary value problem of differential inclusions, Nonlinear Anal. 69 (2008), 3877–3896.
S. Park, Fixed points of approximable or Kakutani maps, J. Nonlinear Convex Anal. 7 (2006), no. 1, 1–17.
I. Podlubny, Fractional Differential Equation, Academic Press, New York, 1999.
S. Saı̈di, L. Thibault and M. Yarou, Relaxation of optimal control problems involving time dependent subdifferential operators, Numer. Funct. Anal. Optim. 34 (2013), no. 10, 1156–1186.
A.A. Tolstonogov, Relaxation in non-convex control problems described by first-order evolution equations, Sb. Math. 190 (1999), no. 11, 1689–1714.
A.A. Tolstonogov, Properties of attainable sets of evolution inclusions and control systems of subdifferential type, Sib. Math. J. 45 (2004), no. 4, 763–784.
A.A. Tolstonogov, Relaxation in control systems of subdifferential type, Izv. Math. 70 (2006), no. 1, 121–152.
A.A. Tolstonogov, Properties of the set of ”trajectory-control” pairs of a control systems with subdifferential operators, J. Math. Sci. 162 (2009), no. 3, 407–442.
A.A. Tolstonogov, Variational stability of optimal control problems involving subdifferential operators, Sb. Math. 202 (2011), no. 4, 583–619.
A.A. Tolstonogov, Differential inclusions with unbounded right-hand side. Existence and relaxation theorems, Proc. Steklov Inst. Math. 291 (2015), 190–207.
A.A. Tolstonogov, Relaxation in nonconvex optimal control problems containing the difference of two subdifferentials, SIAM J. Control Optim. 54 (2016), no. 1, 175–197.
A.A. Tolstonogov, Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation, J. Math. Anal. Appl. 447 (2017), 269–288.
A.A. Tolstonogov, Filippov-Ważewski theorem for subdifferential inclusions with an unbounded perturbation, SIAM J. Control Optim. 56 (2018), no. 4, 2878–2900.
A.A. Tolstonogov and D. Tolstonogov, Lp Continuous Extreme Selectors of Multifunctions with Decomposable Values: Relaxation Theorems, Set-Valued Anal. 4 (1996), no. 3, 237–269.
A.A. Vladimirov, Nonstationary dissipative evolution equations in Hilbert space, Nonlinear Anal. 17 (1991), 499–518.
I.I. Vrabie, Compactness Methods for Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific and Technical, John Wiley and Sons, Inc., New York, 1987.
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Charles Castaing, Soumia Saïdi
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0