Stability of solutions of almost periodic functional differential systems of neutral type (Q2386982)

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 \(s>0\). \smallskip 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 \textit{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)].