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On a certain nonlinear retarded Volterra integrodifferential equation. (English) Zbl 1211.45008
From the text: The author studies the nonlinear retarded Volterra integrodifferential equation of the form $$x'(t)=A(t)x(t)+F\left(t,x(t-h(t)), \int\sb 0\sp t k(t,s,x(s-h(s)))\,ds\right)\tag1$$ with initial condition $$x(0)=x\sb 0,\tag 2 $$ where $t\in I=[0,\beta],\ x\in {\Bbb R}\sp n$, $k$ and $F$ are ${\Bbb R}\sp n$-valued functions continuous respectively on $I\sp 2\times {\Bbb R}\sp n$ and $I \times {\Bbb R}\sp n \times {\Bbb R}\sp n$, and $h$ is a real-valued, differentiable, positive and nonincreasing function on $I$ with $t-k(t)\ge 0$ on $I$. He gives conditions for the boundedness, uniqueness, and continuous dependence of solutions on initial conditions and parameters appearing in $F$.

45J05Integro-ordinary differential equations
45D05Volterra integral equations