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September  2019, 8(3): 621-631. doi: 10.3934/eect.2019029

Periodic solutions for implicit evolution inclusions

Nikolaos S. Papageorgiou 1,2, , Vicenţiu D. Rădulescu 2,3,4,, and  Dušan D. Repovš 2,5,

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

* Corresponding author: Vicenţiu D. Răadulescu

Received  August 2018 Revised  October 2018 Published  September 2019 Early access  May 2019

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.

Keywords: Evolution triple, compact embedding, pseudo-monotone map, coercive map, implicit inclusion, periodic solution.
Mathematics Subject Classification: Primary: 34A20; Secondary: 35A05, 35R70.
Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations and Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029
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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