Abstract
We consider a system of functional difierential equations with infinite delay and derive conditions on Liapunov functionals to ensure that solutions are uniformly bounded and uniformly ultimately bounded. The analysis is based on the method of finding a bound on the average values of unknown solutions and Jensen's inequality. Comparisons between our theorems and those existing in the literature are also given.
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References
O. A. Arino, T. A. Burton and J. R. Haddock, Periodic solutions of functional differential equations, Roy. Soc. Edinburgh Proc. A., 101A (1985), 253-271.
F. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J., 26 (1959), 291-303.
T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press (Orlando, FL, 1985).
T. A. Burton and R. H. Hering, Boundedness in infinite delay systems, J. Math. Anal. Appl., 144 (1989), 486-502.
T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Ann. Mat. Pura Appl., 145 (1986), 129-158.
J. K. Hale, Theory of Functional Differential Equations, Springer (New York, 1977).
R. H. Hering, Boundedness and periodic solutions in infinite delay systems, J. Math. Anal. Appl., 163 (1992), 521-535.
W. A. Horn, Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc., 149 (1970), 391-404.
G. S. Jones, Stability and asymptotic fixed point theory, Contrib. Differential Equations, 3 (1964), 75-106.
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. I, Academic Press (New York, 1969).
N. Levinson, Transformation theory of nonlinear differential equations of the second order, Annals of Math., 45 (1944), 724-737.
G. Makay, Periodic solutions of dissipative functional differential equations, Tohoku Math. J., 46 (1994), 417-426.
G. Makay, Uniform boundedness and uniform ultimate boundedness for functional differential equations, Funkcialaj Ekvacioj, 38 (1995), 283-296.
G. Sansone and R. Conti, Non-linear Differential Equations, MacMillan (New York, 1964).
T. Yoshizawa, Stability Theory by Liapunov's Second Method, Math. Soc. Japan (Tokyo, 1966).
B. Zhang, Space Cg and uniform behavior of solutions of functional differential equations with infinite delay, Acta Mathematica Sinica, New Series, 4 (1988), 177-188.
B. Zhang, Boundedness in functional differential equations, Nonlinear Analysis, 22 (1994), 1511-1527.
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Burton, T., Zhang, B. Uniform Boundedness by Averaging. Acta Mathematica Hungarica 92, 263–283 (2001). https://doi.org/10.1023/A:1013881809859
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DOI: https://doi.org/10.1023/A:1013881809859