Meng, Fanwei; Li, Lianzhong; Bai, Yuzhen \(\Psi\)-stability of nonlinear Volterra integro-differential systems. (English) Zbl 1246.45009 Dyn. Syst. Appl. 20, No. 4, 563-574 (2011). Consider the functions \(f\), \(g\) belonging to the space \(C(\mathbb{R}_+\times \mathbb{R}^n,\mathbb{R}^n)\), the set \(D= \{(t,s)\in\mathbb{R}^2; 0\leq s\leq t<+\infty\}\), the function \(F\in C(D\times \mathbb{R}^n,\mathbb{R}^n)\), such that exists the partial derivative \[ f_x\in C(\mathbb{R}_+\times \mathbb{R}^n, \mathbb{R}^n). \] Let \(\Psi= \text{diag}[\Psi_1,\dots,\Psi_n]\), where \(\Psi_i\in C(\mathbb{R}_+,(0,+\infty))\) for \(1\leq i\leq n\). The authors prove sufficient conditions for \(\Psi\)-uniform stability of trivial solutions of the nonlinear system \[ y'= f(t,y)+ g(t,y)\tag{1} \] and the nonlinear Volterra integro-differential system \[ z'= f(t,x)+ \int^t_0 F(t,s,z(s))\,ds,\tag{2} \] which can be seen as perturbed system of \[ x'= f(t,x)\tag{3} \] or the variational system \[ u'= f_x(t,x(t, t_0,x_0))\,u\tag{4} \] associated with system (3), where \(x(t,t_0,x_0)\) is the solution of (3) with \(x(t_0, t_0, x_0)= x_0\) for \(t_0\geq 0\). The fundamental matrix solution \(\Phi(t,t_0,x_0)\) of (4) is given by \[ \Phi(t,t_0,x_0)= {\partial\over\partial x_0}\,(x(t,t_0, x_0)). \] The authors prove that the trivial solutions of (1), (2), (3), (4) are \(\Psi\)-uniformly stable and \(\Psi\)-stable on \(\mathbb{R}_+\). Reviewer: Dan-Mircea Borş (Iaşi) Cited in 5 Documents MSC: 45J05 Integro-ordinary differential equations 45G15 Systems of nonlinear integral equations 45M10 Stability theory for integral equations 45D05 Volterra integral equations Keywords:stability; nonlinear Volterra integro-differential system; fundamental matrix solution PDFBibTeX XMLCite \textit{F. Meng} et al., Dyn. Syst. Appl. 20, No. 4, 563--574 (2011; Zbl 1246.45009)