Coupled systems of subdifferential type with integral perturbation and fractional differential equations

Yıl 2023, , 253 - 271, 31.03.2023

Aya Bouabsa Soumia Saıdı

https://doi.org/10.31197/atnaa.1149751

Öz

This paper is mainly devoted to study a class of first-order differential inclusions governed by time-dependent subdifferential operators involving an integral perturbation. Employing then the constructive method used there, we also handle the associated second-order differential inclusion.
Our final topic, accomplished in infinite-dimensional Hilbert spaces, is to develop some variants related to coupled systems by such differential inclusions and fractional differential equations.

Anahtar Kelimeler

Differential inclusion, subdifferential operator, integral perturbation, fixed point, fractional derivative

Kaynakça

• [1] S. Adly, H. Attouch, Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping, SIAM J. Optim. 30(3) (2020) 2134-2162.
• [2] S. Adly, H. Attouch, A. Cabot, Finite time stabililization of nonlinear oscillators subject to dry friction, Nonsmooth Mechanics and Analysis 12 (2006) pp 289-304, Advances in Mechanics and Mathematics.
• [3] R.P. Agarwal, B. Ahmad, A. Alsaedi, N. Shahzad, Dimension of the solution set for fractional differential inclusion, J. Nonlinear Convex Anal. 14(2) (2013) 314-323.
• [4] R.P. Agarwal, S. Arshad, D. O'Regan, V. Lupulescu, Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal. 15(4) (2012) 572-590.
• [5] B. Ahmad, J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory 13(2) (2012) 329-336.
• [6] H. Attouch, D. Damlamian, Problèmes d'évolution dans les Hilberts et applications, J. Math. Pures Appl. 54 (1975) 53-74.
• [7] H. Attouch, P.E. Maingé, P. Redont, A second-order differential system with hessian driven damping; application to non-elastic shock laws, Di?er. Equ. Appl. 4 (1) (2012) 27-65.
• [8] M. Benchohra, J. Graef, F.Z. Mostefai, Weak solutions for boundary-value problems with nonlinear fractional differential inclusions, Nonlinear Dyn. Syst. Theory 11(3) (2011) 227-237.
• [9] A. Bouach, T. Haddad, B.S. Mordukhovich, Optimal control of nonconvex integro-differential sweeping processes, J. Dif- ferential Equations 329 (2022) 255-317.
• [10] A. Bouach, T. Haddad, L. Thibault, Nonconvex integro-differential sweeping process with applications, SIAM J. Control Optim 60 (5) (2022) 2971-2995.
• [11] A. Bouach, T. Haddad, L. Thibault, On the discretization of truncated integro-differential sweeping process and optimal control, J. Optim. Theory Appl. 193 (1) (2022) 785-830.
• [12] Y. Brenier, W. Gangbo, G. Savare, M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl. 99 (2013) 577-617.
• [13] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Lecture Notes in Math. North-Holland, 1973.
• [14] C. Castaing, Quelques résultats de compacité liés a l'intégration, C. R. Math. Acad. Sci. Paris 270 (1970) 1732-1735 and Bull. Soc. Math. France 31 (1972) 73-81.
• [15] C. Castaing, A. Faik, A. Salvadori, Evolution equations governed by m-accretive and subdi?erential operators with delay, Int. J. Appl. Math. 2(9) (2000) 1005-1026.
• [16] C. Castaing, C. Godet-Thobie, M.D.P. Monteiro Marques, A. Salvadori, Evolution problems with m-accretive operators and perturbations, Mathematics 2022.
• [17] C. Castaing, C. Godet-Thobie, F.Z. Mostefai, On a fractional differential inclusion with boundary conditions and application to subdi?erential operators, J. Nonlinear Convex Anal. 18 (9) (2017) 1717-1752.
• [18] C. Castaing, C. Godet-Thobie, P.D. Phung, L.T. Truong, On fractional differential inclusions with nonlocal boundary conditions, Fract. Calc. Appl. Anal. 22 (2019) 444-478.
• [19] C. Castaing, C. Godet-Thobie, P.D. Phung, L.T. Truong, Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator, Mathematics 8 (2020).
• [20] C. Castaing, C. Godet-Thobie, S. Saïdi, On fractional evolution inclusion coupled with a time and state dependent maximal monotone operator, Set-Valued Var. Anal. 30(2) (2022) 621-656.
• [21] C. Castaing, C. Godet-Thobie, L.T. Truong, F.Z. Mostefai, On a fractional differential inclusion in Banach space under weak compactness condition, Adv. Math. Econ. 20 (2016) 23-75.
• [22] C. Castaing, C. Godet-Thobie, L.T. Truong, B. Satco, Optimal control problems governed by a second order ordinary di?erential equation with m-point boundary condition, Adv. Math. Econ. 18 (2014) 1-59.
• [23] C. Castaing, M.D.P. Monteiro Marques, P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ. 22 (2018) 25-77.
• [24] C. Castaing, M.D.P. Monteiro Marques, S. Saïdi, Evolution problems with time-dependent subdifferential operators, Adv. Math. Econ. 23 (2019) 1-39.
• [25] C. Castaing, S. Saïdi, Lipschitz perturbation to evolution inclusion driven by time-dependent maximal monotone operators, Topol. Methods Nonlinear Anal. 58(2) (2021) 677-712.
• [26] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math, 580, Springer-Verlag Berlin Heidelberg, 1977.
• [27] G. Colombo, C. Kozaily, Existence and uniqueness of solutions for an integral perturbation of Moreau`s sweeping process, J. Convex Anal. 27 (2020) 227-236.
• [28] K. Deimling, Non Linear Functional Analysis, Springer, Berlin, 1985.
• [29] S. Hu, N.S. Papageorgiou, Handbook of multivalued analysis. Vol. II, volume 500 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000.
• [30] N. Kenmochi, Some nonlinear parabolic inequalities, Israel J. Math 22 (1975) 304-331.
• [31] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Math. Studies 204, North Holland: Amsterdam, The Netherlands, 2006.
• [32] J.C. Peralba, Équations d'évolution dans un espace de Hilbert, associées à des opérateurs sous-di?érentiels, Thèse de doctorat de spécialité, Montpellier, 1973.
• [33] P.H. Phung, L.X. Truong, On a fractional differential inclusion with integral boundary conditions in Banach space, Fract. Calc. Appl. Anal. 16(3) (2013) 538-558.
• [34] S. Saïdi, Some results associated to first-order set-valued evolution problems with subdifferentials, J. Nonlinear Var. Anal. 5(2) (2021) 227-250.
• [35] S. Saïdi, Set-valued perturbation to second-order evolution problems with time-dependent subdifferential operators, Asian- Eur. J. Math. 15(7) (2022) 1-20.
• [36] S. Saïdi, Second-order evolution problems by time and state-dependent maximal monotone operators and set-valued per- turbations, Int. J. Nonlinear Anal. Appl. 14(1) (2023) 699-715.
• [37] S. Saïdi, On a second-order functional evolution problem with time and state dependent maximal monotone operators, Evol. Equ. Control Theory 11 (4) (2022) 1001-1035.
• [38] S. Saïdi, F. Fennour, Second-order problems involving time-dependent subdi?erential operators and application to control, Math. Control Relat. Fields, doi:10.3934/mcrf.2022019
• [39] S. Saïdi, L. Thibault, M. Yarou, Relaxation of optimal control problems involving time dependent subdi?erential operators, Numer. Funct. Anal. Optim. 34(10) (2013) 1156-1186.
• [40] S. Saïdi, M.F. Yarou, Set-valued perturbation for time dependent subdi?erential operator, Topol. Methods Nonlinear Anal. 46 (1) (2015) 447-470.
• [41] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, New York, 1993.
• [42] A.A. Tolstonogov, Properties of attainable sets of evolution inclusions and control systems of subdi?erential type, Sib. Math. J. 45(4) 2004 763-784.
• [43] J. Watanabe, On certain nonlinear evolution equations, J. Math. Soc. Japan 25 (1973) 446-463.
• [44] Y. Yamada, On evolution equations generated by subdi?erential operators, J. Fac. Sci. Univ. Tokyo 23 (1976) 491-515.
• [45] S. Yotsutani, Evolution equations associated with the subdi?erentials, J. Math. Soc. Japan 31 (1978) 623-646.
• [46] Y. Zhou, Basic theory of fractional di?erential equations, World Scienti?c Publishing, 2014.