Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side
Authors:
Yong Li, Qinde Zhou and Xianrui Lu
Journal:
Quart. Appl. Math. 55 (1997), 57-68
MSC:
Primary 34K15; Secondary 34A60, 34C25
DOI:
https://doi.org/10.1090/qam/1433752
MathSciNet review:
MR1433752
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Abstract: In this paper, we present a general existence theorem for periodic solutions for functional differential inclusions with nonconvex right-hand side. As an application, we prove that for a multivalued system with nonconvex right-hand side, permanence implies the existence of a pre-equilibrium state, which answers an open problem proposed by Hutson.
- Josef Hofbauer and Karl Sigmund, The theory of evolution and dynamical systems, London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge, 1988. Mathematical aspects of selection; Translated from the German. MR 1071180
- V. Hutson and W. Moran, Persistence of species obeying difference equations, J. Math. Biol. 15 (1982), no. 2, 203–213. MR 684934, DOI https://doi.org/10.1007/BF00275073
- V. Hutson, The existence of an equilibrium for permanent systems, Rocky Mountain J. Math. 20 (1990), no. 4, 1033–1040. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1988). MR 1096569, DOI https://doi.org/10.1216/rmjm/1181073060
- Yong Li, Huai Zhong Wang, and Xian Rui Lü, Equilibrium of permanent multivalued systems, Quart. Appl. Math. 51 (1993), no. 4, 791–795. MR 1247442, DOI https://doi.org/10.1090/qam/1247442
- Georges Haddad and Jean-Michel Lasry, Periodic solutions of functional-differential inclusions and fixed points of $\sigma $-selectionable correspondences, J. Math. Anal. Appl. 96 (1983), no. 2, 295–312. MR 719317, DOI https://doi.org/10.1016/0022-247X%2883%2990042-2
- Jack W. Macki, Paolo Nistri, and Pietro Zecca, The existence of periodic solutions to nonautonomous differential inclusions, Proc. Amer. Math. Soc. 104 (1988), no. 3, 840–844. MR 931741, DOI https://doi.org/10.1090/S0002-9939-1988-0931741-X
- Sławomir Plaskacz, Periodic solutions of differential inclusions on compact subsets of ${\bf R}^n$, J. Math. Anal. Appl. 148 (1990), no. 1, 202–212. MR 1052055, DOI https://doi.org/10.1016/0022-247X%2890%2990038-H
- Marlène Frigon and Andrzej Granas, Théorèmes d’existence pour des inclusions différentielles sans convexité, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 12, 819–822 (French, with English summary). MR 1058503
- Alberto Bressan and Giovanni Colombo, Selections and representations of multifunctions in paracompact spaces, Studia Math. 102 (1992), no. 3, 209–216. MR 1170551, DOI https://doi.org/10.4064/sm-102-3-209-216
- W. A. Horn, Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc. 149 (1970), 391–404. MR 267432, DOI https://doi.org/10.1090/S0002-9947-1970-0267432-1
- Alberto Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), no. 3, 459–470. MR 987798
- Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR 0208086
- T. A. Burton and Shu Nian Zhang, Unified boundedness, periodicity, and stability in ordinary and functional-differential equations, Ann. Mat. Pura Appl. (4) 145 (1986), 129–158. MR 886710, DOI https://doi.org/10.1007/BF01790540
- Alberto Bressan, On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 77 (1989), no. 2, 379–391. MR 983301, DOI https://doi.org/10.1016/0022-0396%2889%2990150-2
J. Hofbauer and K. Sigmund, Dynamical Systems and the Theory of Evolution, Cambridge University Press, 1988
V. Hutson and W. Moran, Persistence of species obeying difference equations, Math. Biosci. 15, 203–213 (1982)
V. Hutson, The existence of an equilibrium for permanent systems, Rocky Mountain J. Math. 20, 1033–1040 (1990)
Li Yong, Wang Huaizhong, and Lu Xianrui, Equilibrium of permanent multivalued systems, Quart. Appl. Math. 51, 791–795 (1993)
G. Haddad and M. Lasry, Periodic solutions of functional differential inclusions and fixed points of $\sigma$-selectionable correspondences, J. Math. Anal. Appl. 96, 295–312 (1983)
Jack W. Macki, P. Nistri, and P. Zecca, The existence of periodic solutions to nonautonomous differential inclusions, Proc. Amer. Math. Soc. 104, 840–844 (1988)
S. Plaskacz, Periodic solutions of nonlinear functional differential inclusions on compact subsets of $R^{n}$, J. Math. Anal. Appl. 148, 202–212 (1990)
M. Frigon and A. Granas, Existence theorems for differential inclusions without convexity, C. R. Acad. Sci. Paris Ser. I. Math. 310, 819–822 (1990) (in French)
A. Bressan and G. Colombo, Selections and representations of multifunctions in paracompact spaces, Studia Math. 102, 20–216 (1992)
W. A. Horn, Some fixed point theorem for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc. 149, 391–404 (1970)
A. Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31, 459–470 (1988)
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo, 1966
T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Ann. Math. Pure Appl. CXLV, 129–258 (1986)
A. Bressan, On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 77, 379–391 (1989)
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© Copyright 1997
American Mathematical Society