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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 \(y\).

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