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Stability of solutions of almost periodic functional differential systems of neutral type. (English. Russian original) Zbl 1081.34526

Russ. Math. 47, No. 6, 73-77 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No. 6, 77-81 (2003).
The paper deals with the stability of the zero solution of the delayed system of neutral type \[ {d\over dt} [x(t)-g(x_t,t)]=f(x_t,t), \] where \(f,g:C[J]\times \mathbb{R}\to \mathbb{R}\) (here, \(J=[-a,0]\), \(a>0\), and \(C(J)\) is the Banach space of continuous functions \(\varphi:[-a,0]\to \mathbb{R}^N\)) satisfy the conditions: 1) The mappings \((\varphi,t)\to f(\varphi,t)\) and \((\varphi,t)\to g(\varphi,t)\) are continuous and Lipschitzian with respect \(\varphi\). 2) The mappings \(f\) and \(g\) are almost-periodic in \(t\). 3) \(f(0,t)=g(0,t)=0\).
The solution of this problem is understood as a continuous function \(x:[-a,\infty)\to \mathbb{R}^N\), which satisfies for \(t\geq 0\) the integral equation \[ [x(\tau)-g(x_\tau,\tau)]_0^t=\int_0^t f(x_\tau,\tau)\,d\tau. \] A Lyapunov-type theorem for the asymptotic stability of the zero solution is proved in which the Lyapunov functions \(v_0(y,t)\), \(v_1(z,t,\theta)\) defined on the sets \(\{| y| \leq 2r\}\times \mathbb{R}\) and \(\{| z| \leq r\}\times \mathbb{R}\times J\) with \(r>0\), respectively, are supposed to satisfy the conditions: 1) \(v_0\) is \(C^1\)-smooth, \(v_1\) is continuous and \(C^1\) smooth with respect to \(t\), \(\theta\). 2) \(v_0\), \(v_1\) and their first derivatives are almost-periodic in \(t\) uniformly with respect to the other variables. 3) \(v_0(0,t)=v_1(0,t,\theta)=0\). 4) \[ \alpha_1(| y| )\leq v_0(y,t)\leq\alpha_2(| y| ),\quad 0\leq v_1(z,t,\theta)\leq\alpha_3(| z| ), \] where \(\alpha_k(s)\) are continuous nondecreasing functions \([0,\infty)\to \mathbb{R}\), \(\alpha_k(0)=0\), and \(\alpha_k(s)>0\) for \(s>0\).
The main result of the paper is the following Theorem: Suppose that for the considered system there exist functions \(v_0\), \(v_1\) with properties 1)-4) such that \(1^0.\) \(\dot V(\phi,t)\leq 0\) for \((\varphi,t)\in B_r\times \mathbb{R}^+\). \(2^0.\) \(\dot V\) is not identically zero at any essential nontrivial solution. Then the zero solution is asymptotically stable.
The paper extends from delay to neutral equations the result of N. V. Aleksenko [Russ. Math. 44, No. 2, 1–4 (2000); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2000, No. 2, 3–6 (2000; Zbl 0964.34062)].

MSC:

34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations

Citations:

Zbl 0964.34062
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