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An application of a diffeomorphism theorem to Volterra integral operator. (English) Zbl 1463.45056
This paper studies Volterra operators $V(x)(t)=x(t)+\int_0^t v(t,\tau,x(\tau))d\tau,$ on the space $$\tilde{W}_0^{1,p}([0,1],\mathbb{R}^n)$$ of absolutely continuous functions $$x$$ with $$x'\in L^p$$ and $$x(0)=0$$. Under appropriate assumptions on the function $$v$$, it is proved, using a global diffeomorphism theorem, that the operator $$V$$ is a diffeomorphism. This provides existence and uniqueness of the solution of the equation $$V(x)=y$$, as well as its smooth dependence on $$y$$.

##### MSC:
 45P05 Integral operators 45D05 Volterra integral equations 26B10 Implicit function theorems, Jacobians, transformations with several variables 47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
##### Keywords:
Volterra equation; global diffeomorphism theorem