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

Abstract

Abstract Using a global inversion theorem we investigate properties of the following operator V ( x ) ( ⋅ ) : = x Δ ( ⋅ ) + ∫ 0 ⋅ v ( ⋅ , τ , x , ( τ ) ) Δ τ ,                                                       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 x y ∈ W Δ , 0 1 , 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 { x Δ ( t ) + ∫ 0 t v ( t , τ , x ( τ ) ) Δ τ = y ( t )       f o r   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.