The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative
Given an injective closed linear operator <i>A</i> defined in a Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo></mrow></semant...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-09-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/19/3540 |
| _version_ | 1797478230697967616 |
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| author | Jennifer Bravo Carlos Lizama |
| author_facet | Jennifer Bravo Carlos Lizama |
| author_sort | Jennifer Bravo |
| collection | DOAJ |
| description | Given an injective closed linear operator <i>A</i> defined in a Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo></mrow></semantics></math></inline-formula> and writing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>C</mi><mi>F</mi></mrow></msub><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup></mrow></semantics></math></inline-formula> the Caputo–Fabrizio fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> we show that the unique solution of the abstract Cauchy problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msub><mspace width="0.166667em"></mspace><mrow><mi>C</mi><mi>F</mi></mrow></msub><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <i>f</i> is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>B</mi><mi>α</mi></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>F</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mo>;</mo><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where the family of bounded linear operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>α</mi></msub></semantics></math></inline-formula> constitutes a Yosida approximation of <i>A</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>→</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Moreover, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mfrac><mo>∈</mo><mi>ρ</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and the spectrum of <i>A</i> is contained outside the closed disk of center and radius equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>1</mn><mrow><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></mfrac></semantics></math></inline-formula> then the solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow></semantics></math></inline-formula> converges to zero as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> in the norm of <i>X</i>, provided <i>f</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mo>′</mo></msup></semantics></math></inline-formula> have exponential decay. Finally, assuming a Lipchitz-type condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> (and its time-derivative) that depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula> we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>:</mo><mo>=</mo><mo>{</mo><mi>x</mi><mo>∈</mo><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>:</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>=</mo><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>}</mo><mo>.</mo></mrow></semantics></math></inline-formula> |
| first_indexed | 2024-03-09T21:29:00Z |
| format | Article |
| id | doaj.art-e0715847e3044023b3077dd02fe387ca |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:29:00Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e0715847e3044023b3077dd02fe387ca2023-11-23T21:03:16ZengMDPI AGMathematics2227-73902022-09-011019354010.3390/math10193540The Abstract Cauchy Problem with Caputo–Fabrizio Fractional DerivativeJennifer Bravo0Carlos Lizama1Departamento de Matemática y Estadística, Facultad de Ciencias de la Educación, Universidad San Sebastian, Bellavista 7, Santiago 8420524, ChileDepartamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago 9170124, ChileGiven an injective closed linear operator <i>A</i> defined in a Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo></mrow></semantics></math></inline-formula> and writing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>C</mi><mi>F</mi></mrow></msub><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup></mrow></semantics></math></inline-formula> the Caputo–Fabrizio fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> we show that the unique solution of the abstract Cauchy problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msub><mspace width="0.166667em"></mspace><mrow><mi>C</mi><mi>F</mi></mrow></msub><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <i>f</i> is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>B</mi><mi>α</mi></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>F</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><mn>0</mn><mo>;</mo><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where the family of bounded linear operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>α</mi></msub></semantics></math></inline-formula> constitutes a Yosida approximation of <i>A</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>→</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Moreover, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mfrac><mo>∈</mo><mi>ρ</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and the spectrum of <i>A</i> is contained outside the closed disk of center and radius equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>1</mn><mrow><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></mfrac></semantics></math></inline-formula> then the solution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow></semantics></math></inline-formula> converges to zero as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>→</mo><mo>∞</mo><mo>,</mo></mrow></semantics></math></inline-formula> in the norm of <i>X</i>, provided <i>f</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mo>′</mo></msup></semantics></math></inline-formula> have exponential decay. Finally, assuming a Lipchitz-type condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> (and its time-derivative) that depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula> we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>:</mo><mo>=</mo><mo>{</mo><mi>x</mi><mo>∈</mo><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>:</mo><mspace width="0.166667em"></mspace><mi>x</mi><mo>=</mo><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>}</mo><mo>.</mo></mrow></semantics></math></inline-formula>https://www.mdpi.com/2227-7390/10/19/3540Caputo–Fabrizio fractional derivativeYosida approximationstabilitylinear and semilinear abstract Cauchy problemone-parameter semigroups of operators |
| spellingShingle | Jennifer Bravo Carlos Lizama The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative Mathematics Caputo–Fabrizio fractional derivative Yosida approximation stability linear and semilinear abstract Cauchy problem one-parameter semigroups of operators |
| title | The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative |
| title_full | The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative |
| title_fullStr | The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative |
| title_full_unstemmed | The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative |
| title_short | The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative |
| title_sort | abstract cauchy problem with caputo fabrizio fractional derivative |
| topic | Caputo–Fabrizio fractional derivative Yosida approximation stability linear and semilinear abstract Cauchy problem one-parameter semigroups of operators |
| url | https://www.mdpi.com/2227-7390/10/19/3540 |
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