Unbounded perturbation to evolution problems with time-dependent subdifferential operators

Yıl 2021, Sayı: 2, 101 - 112, 30.10.2021

Sarra Boudada Mustapha Fateh Yarou

Öz

In this paper, we consider a nonlinear evolution inclusion governed by the subdifferential of a proper convex lower semicontinuous function in a separable Hilbert space. The right-hand side contains a set-valued perturbation with nonempty closed convex and not necessary bounded values. The existence of absolutely continuous solution is stated under different assumptions on the perturbation.
The main purpose in this paper is to study, in the setting of infinite dimensional Hilbert space H, the perturbed problem (P), under various assumptions. Throughout the paper, H is a separable Hilbert space whose inner product is denoted by <.,.> and the associated norm by II. II and [0, T] is an interval of R. We will denote by B the closed unit ball of H; P_c(H) the family of all nonempty closed sets of H and P_cc(H) (resp. P_ck(H)) the set of nonempty closed (resp. compact) convex subsets of H.
We give some preliminaries and we recall some results which will be used in the paper. We establish the existence theorem for the considered problem (P) for a globally upper hemicontinuous perturbation, then we extend the result obtained in [0, T] to the whole interval R+. Finally, we weaken the result by taking the perturbation G measurable in the time t and upper semicontinuous in the state x.

Anahtar Kelimeler

Evolution equations, differential inclusion, subdifferential operator, unbounded perturbation

Destekleyen Kurum

General direction of scientific research and technological development (DGRSDT)

Proje Numarası

PRFU C00L03UN180120180001

Kaynakça

• D. Affane and M. F. Yarou, Perturbed first-order state dependent Moreau's sweeping process. Int. J. Nonlinear Anal. Appl. 12, Special Issue, (2021) 605-615.
• J. P. Aubin and A. Cellina, Differential Inclusions, Set-Valued maps and viability theory. Springer, Berlin, Heidelberg (1984).
• S. Boudada and M. F. Yarou, Sweeping process with right uniformly lower semicontinuousmappings. Positivity, 24 (2020) 207-228.
• H. Brezis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Es- paces de Hilbert. North-Holland, Amsterdam, (1973).
• C. Castaing, A. G. Ibrahim and M. F. Yarou, Existence problems in second order evolution inclusions: Discretization and variational approch, Taiwanese J. Math. 12 (6) (2008) 1435-1447.
• C.Castaing, A. G. Ibrahim and M. F. Yarou, Some contributions to nonconvex sweeping process, J. Nonlin. Convex Anal. 10 (2009) 1-20.
• C. Castaing and Manuel D. P. Monteiro Marques, Evolution problems associated with non convex closed moving sets with bounded variation, Portugal. Math. 53 Fasc.2 (1996) 73-78.
• C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Note in Math. 580. Springer, Berlin, (1997).
• F. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York, (1983).
• N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ. 30 (1981) 1-87.
• M. Kubo, Caracterisation of a class of evolution operators generated by time dependent subdifferential, Funkcial. Ekvac. 32 (1989) 301-321.
• J. J. Moreau, Evolution problems associated with a moving convex set in a Hilbert space, J. Diff. Equs. 26 (1977) 347-374.
• J. Noel and L. Thibault, Nonconvex sweeping process with a moving set depending on the state, Vietnam J. Math. 42 (2014) 595-612.
• M. Otani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Diff. Equs. 46 (1982) 268-299.
• N. S. Papageorgiou, F. Papalini, On the structure of the solution set of evolution inclusions with time-dependent subdierentials, Rend. Sem. Univ. Padova. 65 (1997) 163-187.
• J. C. Peralba, Un probleme d'evolution relatif a un operateur sous differentiel dependant du temps, Seminaire d'analyse convexe. Montpellier, expose No. 6 (1972).
• S. Saidi, L. Thibault and M. F. Yarou, Relaxation of optimal control problems involving time dependent subdierential operators, Numer. Funct. Anal. Optim. 34 (10) (2013) 1156-1186.
• S. Saidi and M. F. Yarou, Delay perturbed evolution problems involving time dependent subdierential operators, Discussiones Math. Diff. Inclus. Control Optim. 34 (2014) 61-87.
• S. Saidi and M. F. Yarou, Set-valued perturbation for time dependent subdifferential operator, Topol. Meth. Nonlin. Anal. 46 (2015) 447-470.
• A. A. Tolstonogov, Existence and relaxation of solutions for a subdifferntial inclusion with unbounded perturbation, J. Math. Anal. Appl. 447 (2017) 269-288.
• S. A. Timoshin, Existence and relaxation for subdifferential inclusions with unbounded perturbation, Math. Program. Ser. A. 166 (2017) 65-85.
• Y. Yamada, On evolution equations generated by subdierential operators, J. Math. Sci. Univ. Tokyo. 23 (1976) 491-515.
• N. Yamazaki, Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials, Elect. J. Diff. Equs. 107 (2004) 1-22.
• M. F. Yarou, Reduction approach to second order perturbed state-dependent sweeping process, Crea. Math. Infor., 28(02) (2019), 215-221.
• M. F. Yarou, Discretization methods for nonconvex differential inclusions, Elect. J. Qual. Theo. Diff. Equ. 12 (2009) 1-10.