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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 |
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.
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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