<abstract xmlns="http://www.w3.org/1999/xhtml">Using a global inversion theorem we investigate properties of the following operator<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0040_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>:</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mi>Δ</mi></msup><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>+</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mo>⋅</mo></msubsup><mrow><mi>v</mi><mrow><mo>(</mo><mrow><mo>⋅</mo><mo>,</mo><mi>τ</mi><mo>,</mo><mi>x</mi><mo>,</mo><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>Δ</mi><mi>τ</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math><tex-math>\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 }</tex-math></alternatives></disp-formula>in a time scale setting. Under some assumptions on the nonlinear term <italic>v</italic> we then show that there exists exactly one solution <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0040_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>x</mi></mrow><mi>y</mi></msub><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mrow><msub><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mi>𝕋</mi></msub><mo>,</mo><msup><mrow><mi>𝕉</mi></mrow><mi>N</mi></msup></mrow><mo>)</mo></mrow></mrow></math><tex-math>{x_y} \in W_{\Delta ,0}^{1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right)</tex-math></alternatives></inline-formula> to the associated integral equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2019-0040_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msup><mrow><mi>x</mi></mrow><mi>Δ</mi></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mrow><mi>v</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi>x</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>Δ</mi><mi>τ</mi><mo>=</mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi> </mi><mi> </mi><mi> </mi><mi>f</mi><mi>o</mi><mi>r</mi><mi> </mi><mi>Δ</mi><mo>-</mo><mi>a</mi><mo>.</mo><mi>e</mi><mo>.</mo><mi> </mi><mi> </mi><mi> </mi><mi>t</mi><mo>∈</mo><msub><mrow><mrow><mo>[</mo><mn>0.1</mn><mo>]</mo></mrow></mrow><mi>𝕋</mi></msub><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\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.</tex-math></alternatives></disp-formula>which is considered on a suitable Sobolev space.</abstract>

Using a global inversion theorem we investigate properties of the following operatorV(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.

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

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

Published Online: Dec 21, 2019

Page range: 177 - 194

Received: Dec 18, 2018

Accepted: Feb 25, 2019

DOI: https://doi.org/10.2478/auom-2019-0040

Keywords
Time scale, Diffeomorphism, Volterra, Operator, Proper operator, Sobolev’s space

© 2019 Bartłomiej Kluczyński, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Dec 21, 2019

Page range: 177 - 194

Received: Dec 18, 2018

Accepted: Feb 25, 2019

DOI: https://doi.org/10.2478/auom-2019-0040

KeywordsTime scale, Diffeomorphism, Volterra, Operator, Proper operator, Sobolev’s space

© 2019 Bartłomiej Kluczyński, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Time scale, Diffeomorphism, Volterra, Operator, Proper operator, Sobolev’s space