# American Institute of Mathematical Sciences

January  2018, 23(1): 219-238. doi: 10.3934/dcdsb.2018015

## Periodic solutions for nonlinear nonmonotone evolution inclusions

 1 Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland 2 National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

* Corresponding author: Leszek Gasiński

Received  August 2016 Revised  June 2017 Published  January 2018

Fund Project: The first author was supported by the National Science Center of Poland under Project no. 2015/19/B/ST1/01169.

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple $(X,H,X^*)$ of spaces. The operator $A(t,x)$ representing the spatial differential operator is not in general monotone. The reaction (source) term $F(t,x)$ is defined on $[0,b]× X$ with values in $2^{X^*}\setminus\{\emptyset\}$. Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Periodic solutions for nonlinear nonmonotone evolution inclusions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 219-238. doi: 10.3934/dcdsb.2018015
##### References:
 [1] H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear Analysis (Collection of Papers in Honor of Erich H. Rothe), (1978), 1-29. [2] 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. [3] R. I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl., 82 (1981), 33-48.  doi: 10.1016/0022-247X(81)90223-7. [4] F. E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1100-1103.  doi: 10.1073/pnas.53.5.1100. [5] R. Caşcaval and I. I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mat. Univ. Complut. Madrid, 7 (1994), 325-338. [6] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [7] F. H. Clarke, Optimization and Nonsmooth Analysis Second edition. Classics in Applied Mathematics, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. [8] A. Defranceschi, Asymptotic analysis of boundary value problems for quasi-linear monotone operators, Asymptotic Anal., 3 (1990), 221-247. [9] L. Gasiński, Existence result for hyperbolic hemivariational inequalities, Nonlinear Anal., 47 (2001), 681-686.  doi: 10.1016/S0362-546X(01)00211-5. [10] L. Gasiński, Existence of solutions for hyperbolic hemivariational inequalities, J. Math. Anal. Appl., 276 (2002), 723-746.  doi: 10.1016/S0022-247X(02)00431-6. [11] L. Gasiński, Evolution hemivariational inequalities with hysteresis, Nonlinear Anal., 57 (2004), 323-340.  doi: 10.1016/j.na.2004.02.016. [12] L. Gasiński, Evolution hemivariational inequality with hysteresis operator in higher order term, Acta Math. Sin. (Engl. Ser.), 24 (2008), 107-120.  doi: 10.1007/s10114-007-0997-6. [13] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006. [14] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 1 Problem Books in Mathematics. Springer, Cham, 2014. [15] L. Gasiński and M. Smolka, An existence theorem for wave-type hyperbolic hemivariational inequalities, Math. Nachr., 242 (2002), 79-90.  doi: 10.1002/1522-2616(200207)242:1<79::AID-MANA79>3.0.CO;2-S. [16] L. Gasiński and M. Smolka, Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping, J. Math. Anal. Appl., 270 (2002), 150-164.  doi: 10.1016/S0022-247X(02)00057-4. [17] 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. [18] S. Hu and N. S. Papageorgiou, On the existence of periodic solutions for a class of nonlinear evolution inclusions, Boll. Un. Mat. Ital. B (7), 7 (1993), 591-605. [19] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4. [20] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅱ. Applications Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17. [21] P. O. Kasyanov, V. S. Mel'nik and S. Toscano, Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{λ_0}$-pseudomonotone maps, J. Differential Equations, 249 (2010), 1258-1287.  doi: 10.1016/j.jde.2010.05.008. [22] V. Lakshmikantham and N. S. Papageorgiou, Periodic solutions of nonlinear evolution inclusions, J. Comput. Appl. Math., 52 (1994), 277-286.  doi: 10.1016/0377-0427(94)90361-1. [23] J. Leray and J.-L. Lions, Quelques résulatats de Višik sur les problémes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. [24] J. -L. Lions, Quelques Méthodes de Résolution Des Problémes Aux Limites Non Linéaires Dunod; Gauthier-Villars, Paris, 1969. [25] A. Paicu, Periodic solutions for a class of differential inclusions in general Banach spaces, J. Math. Anal. Appl., 337 (2008), 1238-1248.  doi: 10.1016/j.jmaa.2007.04.053. [26] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications to Mechanics and Engineering Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis Springer-Verlag, New York, 2009. doi: 10.1007/b120946. [28] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612.  doi: 10.1016/0362-546X(79)90089-0. [29] P. Sattayatham, S. Tangmanee and W. Wei, On periodic solutions of nonlinear evolution equations in Banach spaces, J. Math. Anal. Appl., 276 (2002), 98-108.  doi: 10.1016/S0022-247X(02)00378-5. [30] N. Shioji, Existence of periodic solutions for nonlinear evolution equations with pseudomonotone operators, Proc. Amer. Math. Soc., 125 (1997), 2921-2929.  doi: 10.1090/S0002-9939-97-03984-1. [31] I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc., 109 (1990), 653-661.  doi: 10.1090/S0002-9939-1990-1015686-4. [32] 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. [33] E. Zeidler, Nonlinear Functional Analysis and Its Applications Ⅱ/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

##### References:
 [1] H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear Analysis (Collection of Papers in Honor of Erich H. Rothe), (1978), 1-29. [2] 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. [3] R. I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl., 82 (1981), 33-48.  doi: 10.1016/0022-247X(81)90223-7. [4] F. E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1100-1103.  doi: 10.1073/pnas.53.5.1100. [5] R. Caşcaval and I. I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mat. Univ. Complut. Madrid, 7 (1994), 325-338. [6] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [7] F. H. Clarke, Optimization and Nonsmooth Analysis Second edition. Classics in Applied Mathematics, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. [8] A. Defranceschi, Asymptotic analysis of boundary value problems for quasi-linear monotone operators, Asymptotic Anal., 3 (1990), 221-247. [9] L. Gasiński, Existence result for hyperbolic hemivariational inequalities, Nonlinear Anal., 47 (2001), 681-686.  doi: 10.1016/S0362-546X(01)00211-5. [10] L. Gasiński, Existence of solutions for hyperbolic hemivariational inequalities, J. Math. Anal. Appl., 276 (2002), 723-746.  doi: 10.1016/S0022-247X(02)00431-6. [11] L. Gasiński, Evolution hemivariational inequalities with hysteresis, Nonlinear Anal., 57 (2004), 323-340.  doi: 10.1016/j.na.2004.02.016. [12] L. Gasiński, Evolution hemivariational inequality with hysteresis operator in higher order term, Acta Math. Sin. (Engl. Ser.), 24 (2008), 107-120.  doi: 10.1007/s10114-007-0997-6. [13] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006. [14] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 1 Problem Books in Mathematics. Springer, Cham, 2014. [15] L. Gasiński and M. Smolka, An existence theorem for wave-type hyperbolic hemivariational inequalities, Math. Nachr., 242 (2002), 79-90.  doi: 10.1002/1522-2616(200207)242:1<79::AID-MANA79>3.0.CO;2-S. [16] L. Gasiński and M. Smolka, Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping, J. Math. Anal. Appl., 270 (2002), 150-164.  doi: 10.1016/S0022-247X(02)00057-4. [17] 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. [18] S. Hu and N. S. Papageorgiou, On the existence of periodic solutions for a class of nonlinear evolution inclusions, Boll. Un. Mat. Ital. B (7), 7 (1993), 591-605. [19] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4. [20] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅱ. Applications Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17. [21] P. O. Kasyanov, V. S. Mel'nik and S. Toscano, Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{λ_0}$-pseudomonotone maps, J. Differential Equations, 249 (2010), 1258-1287.  doi: 10.1016/j.jde.2010.05.008. [22] V. Lakshmikantham and N. S. Papageorgiou, Periodic solutions of nonlinear evolution inclusions, J. Comput. Appl. Math., 52 (1994), 277-286.  doi: 10.1016/0377-0427(94)90361-1. [23] J. Leray and J.-L. Lions, Quelques résulatats de Višik sur les problémes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. [24] J. -L. Lions, Quelques Méthodes de Résolution Des Problémes Aux Limites Non Linéaires Dunod; Gauthier-Villars, Paris, 1969. [25] A. Paicu, Periodic solutions for a class of differential inclusions in general Banach spaces, J. Math. Anal. Appl., 337 (2008), 1238-1248.  doi: 10.1016/j.jmaa.2007.04.053. [26] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications to Mechanics and Engineering Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis Springer-Verlag, New York, 2009. doi: 10.1007/b120946. [28] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612.  doi: 10.1016/0362-546X(79)90089-0. [29] P. Sattayatham, S. Tangmanee and W. Wei, On periodic solutions of nonlinear evolution equations in Banach spaces, J. Math. Anal. Appl., 276 (2002), 98-108.  doi: 10.1016/S0022-247X(02)00378-5. [30] N. Shioji, Existence of periodic solutions for nonlinear evolution equations with pseudomonotone operators, Proc. Amer. Math. Soc., 125 (1997), 2921-2929.  doi: 10.1090/S0002-9939-97-03984-1. [31] I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc., 109 (1990), 653-661.  doi: 10.1090/S0002-9939-1990-1015686-4. [32] 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. [33] E. Zeidler, Nonlinear Functional Analysis and Its Applications Ⅱ/B. Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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