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

Yıl 2023, Cilt: 7 Sayı: 1, 253 - 271, 31.03.2023
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.

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.
Yıl 2023, Cilt: 7 Sayı: 1, 253 - 271, 31.03.2023
https://doi.org/10.31197/atnaa.1149751

Öz

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.
Toplam 46 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Aya Bouabsa Bu kişi benim 0000-0003-4923-0109

Soumia Saıdı 0000-0002-4242-8940

Yayımlanma Tarihi 31 Mart 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 7 Sayı: 1

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