Qualitative analysis of integro-differential equations with variable retardation
The paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs), which reads as \[ \frac{\mathrm{d} x}{\mathrm{~d} t}=A(t) x+B F(x(t-\tau(t)))+\int_{t-\tau(t)}^{t} \Omega(t, s) F(x(s)) \mathrm{d} s+G(t, x), \] where \(x \in \mathbb{R}^{n},\ t \in \mathbb{R}^{+}:=[0, \infty), \ \tau \in \mathrm{C}^{1}\left(\mathbb{R}^{+}, \ \mathbb{R}^{+}\right), \ A=\left(a_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+},\ \mathbb{R}^{n \times n}\right)\), \(\Omega=\left(\Omega_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \ \mathbb{R}^{n \times n}\right)\), \(B=\left(b_{i j}\right) \in \mathbb{R}^{n \times n}, \ F \in \mathrm{C}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right), \ F(0)=0\), and \(G\in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \mathbb{R}^{n}\right)\). The authors focus on the uniform stability and uniform asymptotic stability of the zero solution and integrability and boundedness of solutions in the case \(G(x,t)=0\). The boundedness of solutions at infinity is discussed for \(G(x,t)\neq 0\) too. Also, the authors provide two illustrative examples. Remarkably the given theorems are not only applicable to time-varying linear RIDEs, but also to nonlinear RIDEs depending on time.
scientific article; zbMATH DE number 7461151
scientific article