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 \(\lim_{t\to+\infty}\int^t_0|C_{ij}(t,s)|ds\).


34K20 Stability theory of functional-differential equations


Zbl 0694.34060
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