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\(\Psi\)-stability of nonlinear Volterra integro-differential systems. (English) Zbl 1246.45009

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 \(\mathbb{R}_+\).

MSC:

45J05 Integro-ordinary differential equations
45G15 Systems of nonlinear integral equations
45M10 Stability theory for integral equations
45D05 Volterra integral equations
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