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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
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