Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators
DOI:
https://doi.org/10.12775/TMNA.2021.010Keywords
Anti-periodic solution, Hille--Yosida operator, measure of noncompactness, fixed point theoryAbstract
In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.References
N.T.V. Anh and T.D. Ke, Asymptotic behavior of solutions to a class of differential variational inequalities, Ann. Polon. Math. 114 (2015), 147–164.
N.T.V. Anh, Periodic Solutions to Differential Variational Inequalities of Parabolicelliptic Type, Taiwanese J. Math. 24 (2020), no. 6, 1497–1527.
R.R. Akhmerov, M.I. Kamenskiı̆, A.S. Potapov, A.E. Rodkina and B.N. Sadovskiı̆, Measures of Noncompactness and Condensing Operators, Birkhäuser, Boston, Basel, Berlin, 1992.
W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc. (3) 54 (1987), no. 2, 321–349.
W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96, BirkhauserVerlag, Basel, 2001.
W. Arendt and P. Bénilan, Wiener regularity and heat semigroups on spaces of continuous functions, Topics in Nonlinear Analysis Progress in Nonlinear Differential Equations Application, vol. 35, Birkhauser, Basel, 1999, pp. 29–49.
J.P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.
M.T. Batchelor, R.J. Baxter, M.J. O’Rourke and C.M. Yung, Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions, J. Phys. A 28 (1995), 2759–2770.
L.L. Bonilla and F.J. Higuera, The onset and end of the Gunn effect in extrinsic semiconductors, SIAM J. Appl. Math. 55 (1995), 1625–1649.
D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math 108 (1998), 109–138.
C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.
Y. Chen, D. O’Regan and R.P. Agarwal, Anti-periodic solutions for semilinear evolution equations in Banach spaces, J. Appl. Math. Comput. 38 (2012), 63–70.
J.L. Dalecki and M.G. Krein, Stability of solutions of differential equations in Banach space, English transl. by S. Smith, Transl. Math. Monogr., Vol. 43, American Mathematical Society, Providence, R.I., 1974.
N.V. Dac and T.D. Ke, Asymptotic behavior for non-autonomous functional differential inclusions with measures of noncompactness, Topol. Methods Nonlinear Anal. 49 (2017), 383–400.
E.S. Polovinkin, On continuous variations of trajectories of differential inclusion with unbounded right-hand side, Optimization 69 (2020), 703–728.
K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, vol. 194, Springer–Verlag, New York, 2000.
A.F. Filippov, Differential equations with discontinuous righthand sides, English transl., Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988.
L. Górniewicz and M. Lassonde, Approximation and fixed points for compositions of Rδ -maps. Topology Appl. 55 (1994), 239–250.
A. Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989), 479–505.
A. Haraux and M.A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, Classical Methods and Recent Advances, Springer, Heidelberg, 2015.
D.S. Kulshreshtha, J.Q. Liang and H.J.W. Muller-Kirsten, Fluctuation equations about classical field configurations and supersymmetric quantum mechanics, Ann. Physics 225 (1993) 191–211.
M. Kamenskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin, New York, 2001.
T.D. Ke, On the dynamics generated by a class of functional evolution inclusions, J. Math. Anal. Appl. 402 (2013), 275–285.
T.D. Ke and D. Lan, Global attractor for a class of functional differential inclusions with Hille–Yosida operators, Nonlinear Anal. 103 (2014), 72–86.
H. Kellerman and M. Hieber, Integrated semigroup, J. Funct. Anal. 84 (1989), 160–180.
S. Kornev, V. Obukhovskiı̆ and J.C. Yao, Random integral guiding functions in the periodic problem for random differential inclusions with causal multioperators, J. Differential Equations 269 (2020), 5792–5810.
J.H. Liu, S.H. Cheng and L.T. Zhang, Anti-periodic mild solutions of semilinear fractional differential equations, J. Appl. Math. Comput. 48 (2015), no. 1–2, 381–393.
J.H. Liu, X. Song and L. Zhang, Existence of anti-periodic mild solutions to semilinear nonautonomous evolution equations, J. Math. Anal. Appl. 425 (2015), no. 1, 295–306.
Q. Liu, Existence of anti-periodic mild solution for semilinear evolution equations, J. Math. Anal. Appl. 377 (2011), 110–120.
Z.H. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal. 258 (2010), 2026–2033.
M. Meehan and D. O’Regan, Positive solutions of singular integral equations, J. Integral Equations Appl. 12 (2000), no 3, 271–280.
G. M. N’Guérékata and V. Valmorin, Anti-periodic solutions of semilinear integrodifferential equations in Banach spaces, Appl. Math. Comput. 218 (2012), 11118–11124.
V. Obukhovskiı̆ and J.-C. Yao, On impulsive functional differential inclusions with Hille–Yosida operators in Banach spaces, Nonlinear Anal. 73 (2010), 1715–1728.
H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 40 (1988), 541–553.
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.
H. Okochi, On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains, Nonlinear Anal. 14 (1990), 771–783.
H. Thieme, Integrated Semigroups and Integrated Solutions to Abstract Cauchy Problems, J. Math. Anal. Appl. 152 (1990), 416–447.
I.I. Vrabie, C0 -semigroups and applications, North-Holland Mathematics Studies, vol. 191, North-Holland Publishing Co., Amsterdam, 2003.
V. Vijayakumar, Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators, Internat. J. Control 92 (2019), 2210–2222.
R.N. Wang and D.H. Chen, Anti-periodic problem to semilinear partial neutral evolution equations, Electron. J. Qual. Theory Differ. Equ. 16 (2013), 1–16.
Y. Wang, Anti-periodic solutions for dissipative evolution equations, Math. Comput. Model. 51 (2010), 715–721.
Y. Zhou, R.N. Wang and L. Peng, Topological Structure of the Solution Set for Evolution Inclusions, Developments in Mathematics, vol. 51, Springer, Singapore, 2017.
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Topological Methods in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
