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Zbl 0639.34071
Becker, L.C.; Burton, T.A.; Zhang, S.
Functional differential equations and Jensen's inequality. (English)
[A] Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, 31-38 (1987).

[For the entire collection see Zbl 0623.00009.] \par This paper describes how Jensen's inequality for convex functions can aid in obtaining stability results when Lyapunov's direct method is applied to a system of functional differential equations (1) $x'=F(t,x\sb t)$ having a finite delay $h>0$. It is assumed that F: [0,$\infty)\times C\sb H\to R\sp n$ is continuous and that it takes bounded sets into bounded sets, where $C\sb H$ is an open H-ball in the Banach space of continuous functions $\psi$ : [-h,0]$\to R\sp n$ with the supremum norm. The Jensen inequality is used to separate certain integrals associated with a Lyapunov functional into two parts, integrals heretofore regarded as intractable if Lyapunov type conditions are sought that would establish the asymptotic stability of the zero solution of (1). These new Lyapunov conditions are used to complement other previously known conditions to obtain stability results. One of the results is applied to the scalar equation $(2)\quad x'=-a(t)x(t)+b(t)x(t-h)$ in which a and b are continuous functions and each allowed to change sign but with the restriction that $-a(t)+b(t+h)\le -\beta <0.$Conditions are then given to ensure that solutions tend to zero.
[L.C.Becker]

MSC 2000:: *34K20 Stability theory of functional-differential equations

Keywords: first order differential equation; Jensen's inequality; convex functions; Lyapunov's direct method; functional differential equations; Lyapunov functional

Citations: Zbl 0623.00009

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