## Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional differential equations.(English)Zbl 0989.34063

Summary: The authors consider the system $x'_i(t)+ \sum^n_{j=1} q_{ij}(t)x' _j\bigl(g_{ij}(t)\bigr) +\sum^n_{j=1} p_{ij} (t)x_j\bigl(h _{ij}(t)\bigr) =f_i(t),\;t\in[0,+\infty),x_i(s)=x'_i(s)=0,\tag{1}$ if $$s<0$$, $$i=1,\dots,n$$. $$f_i,p_{ij},q_{ij}, h_{ij}, g_{ij}: [0,+\infty) \to\mathbb{R}$$ are measurable essentially bounded functions, $$h_{ij}(t)\leq t$$, $$g_{ij} (t)\leq t$$, $$A \subseteq\mathbb{R}$$, $$\text{mes} A=0$$ implies $$\text{mes} g^{-1} _{ij}(A)=0$$, where $$\text{mes}$$ is the Lebesgue measure, $$i,j=1,\dots,n$$, $$\text{vrai} \sup _{t\in [0,+\infty)} \sum^n_{j=1}|q_{ij}(t)|<1$$, $$1,\dots, n$$. Let $$X(t)$$ be the fundamental matrix of (1) satisfying the condition $$X(0)= E$$ and $$C(t,s)$$ the Cauchy matrix.
The authors obtain sufficient conditions for the validity of exponential estimates of the form $|C_{ij} (t,s)|\leq N\exp \{-\alpha(t-s)\},0\leq s\leq t<+\infty,i,j=1,\dots,n, |X_{ij}(t)|\leq N\exp\{-\alpha t\},\tag{2}$ with positive numbers $$N$$ and $$\alpha$$. The estimates (2) are based on results on the nonnegativity of the entries of the Cauchy matrix [see A. I. Domoshnitsky and M. V. Sheina, Differ. Equations 25, No. 2, 145-150 (1989); translation from Differ. Uravn. 25, No. 2, 201-208 (1989; Zbl 0694.34060)]. Furthermore, the authors’ method allows one to estimate $$\lim_{t\to+\infty}\int^t_0|C_{ij}(t,s)|ds$$.

### MSC:

 34K20 Stability theory of functional-differential equations

Zbl 0694.34060
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