Integro-differential equations and delay integral inequalities. (English) Zbl 0760.34059
Sufficient conditions for the boundedness, asymptotic properties, and exponential decay are first obtained for solutions of linear system of integral inequalities with infinite delay. Then nonlinear integro- differential equations are reduced to delay integral inequalities by the variation of parameter formula, and some new criteria are given for asymptotic stability, uniformly asymptotic stability and exponential asymptotic stability. The results obtained here are illustrated by examples which have been particularly difficult to treat by means of the Lyapunov theory.
Reviewer: D.Xu (Sichuan)
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 45J05 | Integro-ordinary differential equations |
Keywords:
boundedness; asymptotic properties; exponential decay; linear system of integral inequalities with infinite delay; nonlinear integro-differential equations; exponential asymptotic stabilityReferences:
| [1] | T. A. BURTON, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, New York, 1985. · Zbl 0635.34001 |
| [2] | T. A. BURTON, A. CASAL AND A. SOMOLINOS, Upper and lower bounds for Liapunov functionals, Funkcial. Ekvac. 32 (1989), 23-55. · Zbl 0687.34067 |
| [3] | T. A. BURTON AND L. HATVANI, On nonuniform asymptotic stability for nonautonomous functiona differential equations, Differential and Integral Equations 3 (1990) 285-293. · Zbl 0736.34068 |
| [4] | S. BUSENBERG AND K. L. COOKE, Stability conditions for linear nonautonomous delay differentia equations, Q. Appl. Math. 42 (1984), 295-306. · Zbl 0558.34059 |
| [5] | K. GOPALSAMY, Stability and decay rates in a class of linear integrodiferential systems, Funkcial 378D. -Y. xu Ekvac. 26(1983), 251-261. · Zbl 0545.45006 |
| [6] | T. KARA, T. YONEYAMA AND T. ITOH, Asymptotic stability criteria for nonlinear Volterr integro-differential equations, Funkcial. Ekvac. 33 (1990), 39-57. · Zbl 0702.45009 |
| [7] | C. J. HARRIS AND J. F. MILES, Stability of Linear Systems: Some Aspects of Kinematic Similarity, Academic Press, London, 1980. · Zbl 0453.34005 |
| [8] | J. KATO, Liapunov functional vs. Liapunov function, in ”Functional Differential Equations” (T. Yoshizawa and J. Kato, eds.), World Scientific, Singapore, 1991, 161-170. · Zbl 0713.54035 · doi:10.1016/0166-8641(90)90078-G |
| [9] | J. P. LASALLE, The Stability of Dynamical Systems, Regional Conference Series in Appl. Math., SIAM, Philadelphia, 1976. · Zbl 0364.93002 |
| [10] | S. MURAKAMI, Exponential asymptotic stability for scalar linear Volterra equations, Differential an Integral Equations, 4 (1991), 519-525. · Zbl 0726.45013 |
| [11] | D. -Y. Xu, Robust stability analysis of uncertain neutral delay-differential systems via differenc inequality, Control-Theory & Advanced Technology 5 (1989), 301-313. · doi:10.1007/s11768-005-5190-9 |
| [12] | D. -Y. Xu, Stability and convergence of solutions of functional differential equations, in ”Functiona Differential Equations” (T. Yoshizawa and J. Kato, eds.), World Scientific, Singapore, 1991, 362-367. · Zbl 0796.34073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
