Equilibrium of multivalued permanent systems with nonconvex right-hand sides
Authors:
Yong Li, Huai Zhong Wang and Xian Rui Lü
Journal:
Quart. Appl. Math. 53 (1995), 673-678
MSC:
Primary 34C25; Secondary 34A60
DOI:
https://doi.org/10.1090/qam/1359502
MathSciNet review:
MR1359502
Full-text PDF Free Access
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Abstract: The existence of periodic solutions for differential inclusions with nonconvex right-hand sides is proved. As an application, an affirmative answer to the existence of an equilibrium for permanent multivalued systems with nonconvex righthand sides is given.
T. F. Bridgland, Jr., Contributions to the theory of generalized differential equations I, Math. Systems Theory 3, 17–50 (1969)
F. E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26, 291–303 (1959)
T. A. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137, 240–263 (1989)
T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Ann. Mat. Pura Appl. CXLV, 129–258 (1986)
G. Butler, H. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96, 425–430 (1986)
R. M. Colombo, A. Fryszkowski, T. Rzežuchowski, and V. Staicu, Continuous selections of solutions sets of Lipschitzean differential inclusions, Funkcial Ekvac. 34, 321–330 (1991)
H. Freedman and J. So, Persistence in discrete semi-dynamical systems, SIAM J. Math. Anal. 20, 930–938 (1989)
R. E. Gaines, R. B. Guenther, and J. W. Lee, Some existence results for differential inclusions, C. R. Acad. Sci. Paris 307, 391–396 (1988)
R. E. Graines and J. K. Peterson, Periodic solutions to differential inclusions, Nonlinear Analysis 5, 1109–1131 (1981)
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)
J. K. Hale and O. Lopes, Fixed point theorem and dissipative processes, J. Differential Equations 13, 391–402 (1973)
J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mount. J. Math. 20, 1017–1031 (1990)
J. Hofbauer, V. Hutson, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol. 25, 553–570 (1987)
J. Hofbauer and K. Sigmund, Dynamical Systems and the Theory of Evolution, Cambridge University Press, 1988
W. A. Horn, Some fixed point theorem for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc. 149, 391–404 (1970)
V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98, 267–275 (1984)
V. Hutson, The existence of an equilibrium for permanent systems, Rocky Mountain J. Math. 20, 1033–1040 (1990)
V. Hutson and W. Moran, Persistence of species obeying difference equations, Math. Biosci. 15, 203–213 (1982)
V. Hutson and W. Moran, Repellers in reaction-diffusion systems, Rocky Mountain J. Math. 17, 301–314 (1987)
W. G. Kelley, Periodic solutions of generalized differential equations, SIAM J. Appl. Math. 30, 70–74 (1976)
Li Yong, Wang Huaizhong, and Lü Xianrui, Equilibrium of permanent multivalued systems, Quart. Appl. Math. 51, 791–795 (1993)
L. E. Miller, Generalized boundary-value problems, J. Math. Anal. Appl. 74, 233–246 (1980)
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)
R. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 177, 349–375 (1972)
S. Plaskacz, Periodic solutions of nonlinear functional differential inclusions on compact subsets of $R^{p}$ , J. Math. Anal. Appl. 148, 202–212 (1990)
V. Staicu, Continuous selections of solutions sets to evolution equations, Proc. Amer. Math. Soc. 113, 403–413 (1991)
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo, 1966
S. Zaremba, Sur les équations au paratingent, Bull. Sci. Math. 60, 139–160 (1963)
T. F. Bridgland, Jr., Contributions to the theory of generalized differential equations I, Math. Systems Theory 3, 17–50 (1969)
F. E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26, 291–303 (1959)
T. A. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137, 240–263 (1989)
T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Ann. Mat. Pura Appl. CXLV, 129–258 (1986)
G. Butler, H. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96, 425–430 (1986)
R. M. Colombo, A. Fryszkowski, T. Rzežuchowski, and V. Staicu, Continuous selections of solutions sets of Lipschitzean differential inclusions, Funkcial Ekvac. 34, 321–330 (1991)
H. Freedman and J. So, Persistence in discrete semi-dynamical systems, SIAM J. Math. Anal. 20, 930–938 (1989)
R. E. Gaines, R. B. Guenther, and J. W. Lee, Some existence results for differential inclusions, C. R. Acad. Sci. Paris 307, 391–396 (1988)
R. E. Graines and J. K. Peterson, Periodic solutions to differential inclusions, Nonlinear Analysis 5, 1109–1131 (1981)
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)
J. K. Hale and O. Lopes, Fixed point theorem and dissipative processes, J. Differential Equations 13, 391–402 (1973)
J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mount. J. Math. 20, 1017–1031 (1990)
J. Hofbauer, V. Hutson, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol. 25, 553–570 (1987)
J. Hofbauer and K. Sigmund, Dynamical Systems and the Theory of Evolution, Cambridge University Press, 1988
W. A. Horn, Some fixed point theorem for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc. 149, 391–404 (1970)
V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98, 267–275 (1984)
V. Hutson, The existence of an equilibrium for permanent systems, Rocky Mountain J. Math. 20, 1033–1040 (1990)
V. Hutson and W. Moran, Persistence of species obeying difference equations, Math. Biosci. 15, 203–213 (1982)
V. Hutson and W. Moran, Repellers in reaction-diffusion systems, Rocky Mountain J. Math. 17, 301–314 (1987)
W. G. Kelley, Periodic solutions of generalized differential equations, SIAM J. Appl. Math. 30, 70–74 (1976)
Li Yong, Wang Huaizhong, and Lü Xianrui, Equilibrium of permanent multivalued systems, Quart. Appl. Math. 51, 791–795 (1993)
L. E. Miller, Generalized boundary-value problems, J. Math. Anal. Appl. 74, 233–246 (1980)
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)
R. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 177, 349–375 (1972)
S. Plaskacz, Periodic solutions of nonlinear functional differential inclusions on compact subsets of $R^{p}$ , J. Math. Anal. Appl. 148, 202–212 (1990)
V. Staicu, Continuous selections of solutions sets to evolution equations, Proc. Amer. Math. Soc. 113, 403–413 (1991)
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo, 1966
S. Zaremba, Sur les équations au paratingent, Bull. Sci. Math. 60, 139–160 (1963)
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© Copyright 1995
American Mathematical Society
