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Optimal control of evolution differential inclusions with polynomial linear differential operators
Periodic solutions for implicit evolution inclusions
1. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece |
2. | Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia |
3. | Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland |
4. | Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania |
5. | Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia |
We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.
References:
[1] |
K. Andrews, K. Kuttler and M. Schillor,
Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.
doi: 10.1006/jmaa.1996.0053. |
[2] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1976.
doi: 10.1007/978-1-4615-4665-8_17. |
[3] |
V. Barbu and A. Favini,
Existence for implicit differential equations in Banach spaces, Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Natur. Rend. Mat. Appl., 3 (1992), 203-215.
|
[4] |
V. Barbu and A. Favini,
Existence for an implicit differential equation, Nonlinear Anal., 32 (1998), 33-40.
doi: 10.1016/S0362-546X(97)00450-1. |
[5] |
E. DiBenedetto and R. Showalter,
A pseudo-parabolic variational inequality and Stefan problem, Nonlinear Anal., 6 (1982), 279-291.
doi: 10.1016/0362-546X(82)90095-5. |
[6] |
A. Favini and A. Yagi,
Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., 163 (1993), 353-384.
doi: 10.1007/BF01759029. |
[7] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[8] |
L. Gasinski and N. S. Papageorgiou,
Anti-periodic solutions for nonlinear evolution inclusions, J. E Equ., 18 (2018), 1025-1047.
doi: 10.1007/s00028-018-0431-9. |
[9] |
S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-6359-4. |
[10] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[11] |
Z. Liu,
Existence for implicit differential equations with nonmonotone perturbations, Israel J. Math., 129 (2002), 363-372.
doi: 10.1007/BF02773170. |
[12] |
N.S. Papageorgiou, F. Papalini and F. Renzacci,
Existence of solutions and periodic solutions for nonlinear evolution inclusions, Rend. Circ. Mat. Palermo, 48 (1999), 341-364.
doi: 10.1007/BF02857308. |
[13] |
N. S. Papageorgiou and V. D. Rădulescu,
Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.
doi: 10.3934/eect.2017015. |
[14] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions, Adv. Nonlinear Anal., 6 (2017), 199-235.
doi: 10.1515/anona-2016-0096. |
[15] |
R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys and Monographs, 49, American Math. Soc., Providence, RI, 1997. |
[16] |
E. Zeidler, Nonlinear Functional Analysis and its Applications Ⅱ/B, Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
show all references
References:
[1] |
K. Andrews, K. Kuttler and M. Schillor,
Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.
doi: 10.1006/jmaa.1996.0053. |
[2] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1976.
doi: 10.1007/978-1-4615-4665-8_17. |
[3] |
V. Barbu and A. Favini,
Existence for implicit differential equations in Banach spaces, Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Natur. Rend. Mat. Appl., 3 (1992), 203-215.
|
[4] |
V. Barbu and A. Favini,
Existence for an implicit differential equation, Nonlinear Anal., 32 (1998), 33-40.
doi: 10.1016/S0362-546X(97)00450-1. |
[5] |
E. DiBenedetto and R. Showalter,
A pseudo-parabolic variational inequality and Stefan problem, Nonlinear Anal., 6 (1982), 279-291.
doi: 10.1016/0362-546X(82)90095-5. |
[6] |
A. Favini and A. Yagi,
Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., 163 (1993), 353-384.
doi: 10.1007/BF01759029. |
[7] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006. |
[8] |
L. Gasinski and N. S. Papageorgiou,
Anti-periodic solutions for nonlinear evolution inclusions, J. E Equ., 18 (2018), 1025-1047.
doi: 10.1007/s00028-018-0431-9. |
[9] |
S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
doi: 10.1007/978-1-4615-6359-4. |
[10] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[11] |
Z. Liu,
Existence for implicit differential equations with nonmonotone perturbations, Israel J. Math., 129 (2002), 363-372.
doi: 10.1007/BF02773170. |
[12] |
N.S. Papageorgiou, F. Papalini and F. Renzacci,
Existence of solutions and periodic solutions for nonlinear evolution inclusions, Rend. Circ. Mat. Palermo, 48 (1999), 341-364.
doi: 10.1007/BF02857308. |
[13] |
N. S. Papageorgiou and V. D. Rădulescu,
Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.
doi: 10.3934/eect.2017015. |
[14] |
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš,
Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions, Adv. Nonlinear Anal., 6 (2017), 199-235.
doi: 10.1515/anona-2016-0096. |
[15] |
R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys and Monographs, 49, American Math. Soc., Providence, RI, 1997. |
[16] |
E. Zeidler, Nonlinear Functional Analysis and its Applications Ⅱ/B, Springer, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
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