## Asymptotic stability for functional differential equations.(English)Zbl 0805.34068

The authors consider the system of functional differential equations with finite delay (1) $$x'(t) = f(t,x_ t)$$, where $$f : [0, \infty) \times C_ H \to \mathbb{R}^ m$$ is continuous and takes bounded sets into bounded sets and $$f(t,0) = 0$$. Here $$(C, \| \cdot \|)$$ is the Banach space of continuous functions $$\varphi : [ - h,0] \to \mathbb{R}^ m$$ with the supremum norm, $$h$$ is a non-negative constant, $$C_ H$$ is the open $$H$$- ball in $$C$$, and $$x_ t(s) = x(t+s)$$ for $$-h \leq s \leq 0$$. Criteria for various types of stability (asymptotic stability, partial stability, uniform asymptotic stability) are obtained via Lyapunov’s direct method. The following corollary is representative of their results: “Suppose there is a Lyapunov functional $$V$$, wedges $$W_ i$$, positive constants $$K$$ and $$J$$, a sequence $$\{t_ n\} \uparrow \infty$$ with $$t_ n - t_{n-1} \leq K$$ such that (i) $$V(t_ n, \varphi) \leq W_ 2 (\| \varphi \|)$$, (ii) $$V_{(1)}' (t,x_ t) \leq -W_ 3 (| x(t) |)$$ if $$t_ n - h \leq t \leq t_ n$$, (iii) $$| f(t, \varphi) | \leq J(t + 1) \ln (t + 2)$$ for $$t \geq 0$$ and $$\| \varphi \|<H$$. Then $$x=0$$ is asymptotically stable.” An example is given that illustrates the main results. The authors note that the conclusion that the zero solution of the equation considered is asymptotically stable can not be obtained by the known stability criteria.
Reviewer: V.Petrov (Plovdiv)

### MSC:

 34K20 Stability theory of functional-differential equations
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### References:

 [1] R. J. Ballieu and K. Pfeiffer, Attractivity of the origin for the equationx”+f(t, x, x’)|x’| ax’+g(x)=0,J. Math. Anal. Appl.,65 (1978), 321–332. · Zbl 0387.34038 · doi:10.1016/0022-247X(78)90183-X [2] T. A. Burton, Uniform asymptotic stability in functional differential equations,Proc. Amer. Math. Soc.,68 (1978), 195–199. · Zbl 0378.34058 · doi:10.1090/S0002-9939-1978-0481371-5 [3] L. Becker, T. A. Burton and S. Zhang, Functional differential equations and Jensen’s inequality,J. Math. Anal. Appl.,138 (1989), 137–156. · Zbl 0669.34076 · doi:10.1016/0022-247X(89)90325-9 [4] T. A. Burton, A. Casal and A. Somolinos, Upper and lower bounds for Liapunov functionals,Funkcial. Ekvac.,32 (1989), 23–55. · Zbl 0687.34067 [5] T. A. Burton and L. Hatvani, Stability theorems for nonautonomous functional differential equations by Liapunov functionals,Tohoku Math. J.,41 (1989), 65–104. · Zbl 0677.34060 · doi:10.2748/tmj/1178227868 [6] S. N. Busenberg and K. L. Cooke, Stability conditions for linear non-autonomous delay differential equations,Quart. Appl. Math.,42 (1984), 295–306. · Zbl 0558.34059 [7] N. N. Krasovskii,Stability of Motion, Stanford University Press (1963). · Zbl 0109.06001 [8] G. Makay, On the asymptotic stability in terms of two measures for functional differential equations,J. Nonlinear Anal.,16 (1991), 721–727. · Zbl 0719.34130 · doi:10.1016/0362-546X(91)90178-4 [9] M. Marachkov, On a theorem on stability,Bull. Soc. Phy. Math., Kazan,12 (1940), 171–174. [10] R. A. Smith, Asymptotic stability ofx”+a(t)x’+x=0,Quart. J. Math. Oxford Ser. (2),12 (1961), 123–126. · Zbl 0103.05604 · doi:10.1093/qmath/12.1.123 [11] L. H. Thurston and J. S. W. Wong, On global stability of certain second order differential equations with integrable forcing terms,SIAM J. Appl. Math.,24 (1973), 50–61. · Zbl 0279.34041 · doi:10.1137/0124007 [12] T. Wang, Asymptotic stability and the derivatives of solutions of functional differential equations,Rocky Mountain J., to appear.
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