Abstract 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∈W Δ,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.

中文翻译：

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时间尺度上Volterra方程解的存在唯一性

摘要 使用全局反演定理，我们研究了以下算子 V(x)(⋅) 的性质：=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 } 在时间刻度设置中。在对非线性项 v 的一些假设下，我们证明存在恰好一个解 xy∈W Δ,01,p([0,1]𝕋,𝕉N) {x_y} \in W_{\Delta ,0}^{ 1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right) 到相关的积分方程 { xΔ(t)+∫0tv (t,τ,x(τ))Δτ=y(t) 对于 Δ-ae 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 - ae\,\,\,t \in {{[0. 1]}_\mathbb{T}},} \cr {x(0) = 0,} \cr } } \right. 这是在合适的 Sobolev 空间上考虑的。