On the existence and uniqueness of solution to Volterra equation on a time scale

Using a global inversion theorem we investigate properties of the following operator

V(x)(⋅):=xΔ(⋅)+∫0⋅v(⋅,τ,x,(τ))Δτ, x(0)=0,\matrix{\matrix{ V(x)( \cdot ): = {x^\Delta }( \cdot ) + \int_0^ \cdot {v\left( { \cdot ,\tau ,x,\left( \tau \right)} \right)} \Delta \tau , \hfill \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x(0) = 0, \hfill \cr}\cr {} \cr }

in a time scale setting. Under some assumptions on the nonlinear term v we then show that there exists exactly one solution xy∈WΔ,01,p([0,1]𝕋,𝕉N){x_y} \in W_{\Delta ,0}^{1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right) to the associated integral equation

{xΔ(t)+∫0tv(t,τ,x(τ))Δτ=y(t) for Δ-a.e. t∈[0.1]𝕋,x(0)=0,\left\{ {\matrix{{{x^\Delta }(t) + \int_0^t {v\left( {t,\tau ,x\left( \tau \right)} \right)} \Delta \tau = y(t)\,\,\,for\,\Delta - a.e.\,\,\,t \in {{[0.1]}_\mathbb{T}},} \cr {x(0) = 0,} \cr } } \right.

which is considered on a suitable Sobolev space.