Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations
The equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/1/64 |
| _version_ | 1827384687706767360 |
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| author | Evgeny Zhukovskiy Elena Panasenko |
| author_facet | Evgeny Zhukovskiy Elena Panasenko |
| author_sort | Evgeny Zhukovskiy |
| collection | DOAJ |
| description | The equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mover accent="true"><mi>y</mi><mo>˜</mo></mover><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mo>+</mo></msub><mo>,</mo><msub><mi>M</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> of a Banach space <i>E</i> and a linear space <i>M</i>, respectively), is considered. This operator equation is compared with a “model” equation, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>E</mi><mo>+</mo></msub><mo>×</mo><msub><mi>E</mi><mo>+</mo></msub><mo>→</mo><msub><mi>M</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. |
| first_indexed | 2024-03-08T15:02:56Z |
| format | Article |
| id | doaj.art-f586b2e217c34cf7a0eabec92f79b93e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T15:02:56Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f586b2e217c34cf7a0eabec92f79b93e2024-01-10T15:03:28ZengMDPI AGMathematics2227-73902023-12-011216410.3390/math12010064Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential EquationsEvgeny Zhukovskiy0Elena Panasenko1Department of Functional Analysis, Derzhavin Tambov State University, Tambov 392000, RussiaDepartment of Functional Analysis, Derzhavin Tambov State University, Tambov 392000, RussiaThe equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mover accent="true"><mi>y</mi><mo>˜</mo></mover><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mo>+</mo></msub><mo>,</mo><msub><mi>M</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> of a Banach space <i>E</i> and a linear space <i>M</i>, respectively), is considered. This operator equation is compared with a “model” equation, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>E</mi><mo>+</mo></msub><mo>×</mo><msub><mi>E</mi><mo>+</mo></msub><mo>→</mo><msub><mi>M</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation.https://www.mdpi.com/2227-7390/12/1/64operator equation in vector metric spaceexistence and estimates of solutionsfunctional differential equation |
| spellingShingle | Evgeny Zhukovskiy Elena Panasenko Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations Mathematics operator equation in vector metric space existence and estimates of solutions functional differential equation |
| title | Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations |
| title_full | Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations |
| title_fullStr | Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations |
| title_full_unstemmed | Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations |
| title_short | Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations |
| title_sort | extension of the kantorovich theorem to equations in vector metric spaces applications to functional differential equations |
| topic | operator equation in vector metric space existence and estimates of solutions functional differential equation |
| url | https://www.mdpi.com/2227-7390/12/1/64 |
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