June  2017, 6(2): 277-297. doi: 10.3934/eect.2017015

Periodic solutions for time-dependent subdifferential evolution inclusions

1. 

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

2. 

Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

3. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,014700 Bucharest, Romania

* Corresponding author:Vicenţiu D.Rădulescu

Received  May 2016 Revised  January 2017 Published  April 2017

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations and Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015
References:
[1]

G. Akagi and U. Stefanelli, Periodic solutions for double nonlinear evolution equations, J. Differential Equations, 251 (2011), 1790-1812.  doi: 10.1016/j.jde.2011.04.014.

[2]

R. Bader, A topological fixed point index theory for evolution inclusions, Z. Anal. Anwendungen, 20 (2001), 3-15.  doi: 10.4171/ZAA/1001.

[3]

R. Bader and N. S. Papageorgiou, On the problem of periodic evolution inclusions of the subdifferential type, Z. Anal. Anwendungen, 21 (2002), 963-984.  doi: 10.4171/ZAA/1120.

[4]

A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math., 90 (1988), 69-86. 

[5]

H. Brezis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert North Holland, Amsterdam, 1973.

[6]

K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure. Appl. Math., 33 (1980), 117-146.  doi: 10.1002/cpa.3160330203.

[7]

B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130-147.  doi: 10.1016/0022-247X(83)90032-X.

[8]

M. Frigon, Systems of first order differential inclusions with maximal monotone terms, Nonlinear Anal., 66 (2007), 2064-2077.  doi: 10.1016/j.na.2006.03.002.

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis Chapman & Hall/CRC, Boca Raton, FL, 2006.

[10]

L. Gasinski and N. S. Papageorgiou, Exercices in Analysis: Part 1 Springer, New York, 2014.

[11]

P. Hartman, On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509.  doi: 10.1090/S0002-9947-1960-0124553-5.

[12]

C. Henry, Differential equations with discontinuous right-hand side for planning procedures, J. Economic Theory, 4 (1972), 545-551.  doi: 10.1016/0022-0531(72)90138-X.

[13]

N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc., 120 (1994), 185-192.  doi: 10.1090/S0002-9939-1994-1174494-8.

[14]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅰ: Theory Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[15]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅱ: Applications Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[16]

M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61302-9.

[17]

S. Qin and X. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl., 424 (2015), 988-1005.  doi: 10.1016/j.jmaa.2014.11.057.

[18]

X. Xue and Y. Cheng, Existence of periodic solutions of nonlinear evolution inclusions in Banach spaces, Nonlinear Anal. Real World Appl., 11 (2010), 459-471.  doi: 10.1016/j.nonrwa.2008.11.020.

[19]

Y. Yamada, Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries, Nagoya Math. Jour., 70 (1978), 111-123.  doi: 10.1017/S0027763000021814.

[20]

N. Yamazaki, Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials, Electronic J. Differential Equations, 2004 (2004), 1-22. 

[21]

S. Yotsutani, Evolutions associated with subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.  doi: 10.2969/jmsj/03140623.

show all references

References:
[1]

G. Akagi and U. Stefanelli, Periodic solutions for double nonlinear evolution equations, J. Differential Equations, 251 (2011), 1790-1812.  doi: 10.1016/j.jde.2011.04.014.

[2]

R. Bader, A topological fixed point index theory for evolution inclusions, Z. Anal. Anwendungen, 20 (2001), 3-15.  doi: 10.4171/ZAA/1001.

[3]

R. Bader and N. S. Papageorgiou, On the problem of periodic evolution inclusions of the subdifferential type, Z. Anal. Anwendungen, 21 (2002), 963-984.  doi: 10.4171/ZAA/1120.

[4]

A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math., 90 (1988), 69-86. 

[5]

H. Brezis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert North Holland, Amsterdam, 1973.

[6]

K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure. Appl. Math., 33 (1980), 117-146.  doi: 10.1002/cpa.3160330203.

[7]

B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130-147.  doi: 10.1016/0022-247X(83)90032-X.

[8]

M. Frigon, Systems of first order differential inclusions with maximal monotone terms, Nonlinear Anal., 66 (2007), 2064-2077.  doi: 10.1016/j.na.2006.03.002.

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis Chapman & Hall/CRC, Boca Raton, FL, 2006.

[10]

L. Gasinski and N. S. Papageorgiou, Exercices in Analysis: Part 1 Springer, New York, 2014.

[11]

P. Hartman, On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509.  doi: 10.1090/S0002-9947-1960-0124553-5.

[12]

C. Henry, Differential equations with discontinuous right-hand side for planning procedures, J. Economic Theory, 4 (1972), 545-551.  doi: 10.1016/0022-0531(72)90138-X.

[13]

N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc., 120 (1994), 185-192.  doi: 10.1090/S0002-9939-1994-1174494-8.

[14]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅰ: Theory Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[15]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅱ: Applications Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[16]

M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61302-9.

[17]

S. Qin and X. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl., 424 (2015), 988-1005.  doi: 10.1016/j.jmaa.2014.11.057.

[18]

X. Xue and Y. Cheng, Existence of periodic solutions of nonlinear evolution inclusions in Banach spaces, Nonlinear Anal. Real World Appl., 11 (2010), 459-471.  doi: 10.1016/j.nonrwa.2008.11.020.

[19]

Y. Yamada, Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries, Nagoya Math. Jour., 70 (1978), 111-123.  doi: 10.1017/S0027763000021814.

[20]

N. Yamazaki, Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials, Electronic J. Differential Equations, 2004 (2004), 1-22. 

[21]

S. Yotsutani, Evolutions associated with subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.  doi: 10.2969/jmsj/03140623.

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