Applications of Mathematics, Vol. 63, No. 4, pp. 423-437, 2018


Existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions and its applications

Sahbi Boussandel

Received August 26, 2017.   Published online May 20, 2018.

Abstract:  We establish the existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions. The energies associated to these evolution equations are quadratic forms. Our approach is based on application of the Schaefer fixed-point theorem combined with the continuity method.
Keywords:  existence; anti-periodic boundary condition; Schaefer fixed-point theorem; continuity method; diffusion equation
Classification MSC:  35K10, 35K55, 35K57, 35K59, 35K90, 47J35


References:
[1] S. Aizicovici, M. McKibben, S. Reich: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 43 (2001), 233-251. DOI 10.1016/S0362-546X(99)00192-3 | MR 1790104 | Zbl 0977.34061
[2] S. Boussandel: Global existence and maximal regularity of solutions of gradient systems. J. Differ. Equations 250 (2011), 929-948. DOI 10.1016/j.jde.2010.09.009 | MR 2737819 | Zbl 1209.47020
[3] Y. Chen: Anti-periodic solutions for semilinear evolution equations. J. Math. Anal. Appl. 315 (2006), 337-348. DOI 10.1016/j.jmaa.2005.08.001 | MR 2196551 | Zbl 1100.34046
[4] Y. Chen, Y. J. Cho, J. S. Jung: Antiperiodic solutions for semilinear evolution equations. Math. Comput. Modelling 40 (2004), 1123-1130. DOI 10.1016/j.mcm.2003.06.007 | MR 2113840 | Zbl 1074.34058
[5] Y. Chen, J. J. Nieto, D. O'Regan: Anti-periodic solutions for fully nonlinear first-order differential equations. Math. Comput. Modelling 46 (2007), 1183-1190. DOI 10.1016/j.mcm.2006.12.006 | MR 2376702 | Zbl 1142.34313
[6] Y. Chen, J. J. Nieto, D. O'Regan: Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl. Math. Lett. 24 (2011), 302-307. DOI 10.1016/j.aml.2010.10.010 | MR 2741034 | Zbl 1215.34069
[7] Y. Chen, D. O'Regan, R. P. Agarwal: Anti-periodic solutions for evolution equations associated with monotone type mappings. Appl. Math. Lett. 23 (2010), 1320-1325. DOI 10.1016/j.aml.2010.06.022 | MR 2718504 | Zbl 1208.34098
[8] Y. Chen, D. O'Regan, R. P. Agarwal: Anti-periodic solutions for semilinear evolution equations in Banach spaces. J. Appl. Math. Comput. 38 (2012), 63-70. DOI 10.1007/s12190-010-0463-y | MR 2886666 | Zbl 1302.34097
[9] Y. Chen, X. Wang, H. Xu: Anti-periodic solutions for semilinear evolution equations. J. Math. Anal. Appl. 273 (2002), 627-636. DOI 10.1016/S0022-247X(02)00288-3 | MR 1932511 | Zbl 1055.34113
[10] R. Chill, E. Fašangová: Gradient Systems. Lecture Notes of the 13th International Internet Seminar. Matfyzpress, Praha (2010).
[11] A. Haraux: Anti-periodic solutions of some nonlinear evolution equations. Manuscr. Math. 63 (1989), 479-505. DOI 10.1007/BF01171760 | MR 0991267 | Zbl 0684.35010
[12] H. Okochi: On the existence of periodic solutions to nonlinear abstract parabolic equations. J. Math. Soc. Japan 40 (1988), 541-553. DOI 10.2969/jmsj/04030541 | MR 0945351 | Zbl 0679.35046
[13] H. Okochi: On the existence of anti-periodic solutions to a nonlinear evolution equation associated with odd subdifferential operators. J. Funct. Anal. 91 (1990), 246-258. DOI 10.1016/0022-1236(90)90143-9 | MR 1058971 | Zbl 0735.35071
[14] H. Okochi: On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains. Nonlinear Anal., Theory Methods Appl. 14 (1990), 771-783. DOI 10.1016/0362-546X(90)90105-P | MR 1049120 | Zbl 0715.35091
[15] Ph. Souplet: An optimal uniqueness condition for the antiperiodic solutions of parabolic evolution equations. C. R. Acad. Sci., Paris, Sér. I 319 (1994), 1037-1041. (In French.) MR 1305673 | Zbl 0809.35036
[16] Ph. Souplet: Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Nonlinear Anal., Theory Methods Appl. 32 (1998), 279-286. DOI 10.1016/S0362-546X(97)00477-X | MR 1491628 | Zbl 0892.35078
[17] L. Zhenhai: Anti-periodic solutions to nonlinear evolution equations. J. Funct. Anal. 258 (2010), 2026-2033. DOI 10.1016/j.jfa.2009.11.018 | MR 2578462 | Zbl 1184.35184

Affiliations:   Sahbi Boussandel, Faculty of Sciences of Bizerte, Department of Mathematics, 7021 Jarzouna Bizerte, University of Carthage, Bizerte, Tunisia, Laboratore EDP et Applications LR03ES04, e-mail: sboussandels@gmail.com


 
PDF available at: