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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 $C(\bbfR_+\times \bbfR^n,\bbfR^n)$, the set $D= \{(t,s)\in\bbfR^2; 0\le s\le t<+\infty\}$, the function $F\in C(D\times \bbfR^n,\bbfR^n)$, such that exists the partial derivative $$f_x\in C(\bbfR_+\times \bbfR^n, \bbfR^n).$$ Let $\Psi= \text{diag}[\Psi_1,\dots,\Psi_n]$, where $\Psi_i\in C(\bbfR_+,(0,+\infty))$ for $1\le i\le n$. The authors prove sufficient conditions for $\Psi$-uniform stability of trivial solutions of the nonlinear system $$y'= f(t,y)+ g(t,y)\tag1$$ and the nonlinear Volterra integro-differential system $$z'= f(t,x)+ \int^t_0 F(t,s,z(s))\,ds,\tag2$$ which can be seen as perturbed system of $$x'= f(t,x)\tag3$$ or the variational system $$u'= f_x(t,x(t, t_0,x_0))\,u\tag4$$ 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 $t_0\ge 0$. 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 $\bbfR_+$.

45J05Integro-ordinary differential equations
45G15Systems of nonlinear integral equations
45M10Stability theory of integral equations
45D05Volterra integral equations