On a Certain Generalized Functional Equation for Set-Valued Functions
The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function <inline-formula><math display="inline">...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-12-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/12/2243 |
| _version_ | 1797544180235370496 |
|---|---|
| author | Yaroslav Bazaykin Dušan Bednařík Veronika Borůvková Tomáš Zuščák |
| author_facet | Yaroslav Bazaykin Dušan Bednařík Veronika Borůvková Tomáš Zuščák |
| author_sort | Yaroslav Bazaykin |
| collection | DOAJ |
| description | The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>c</mi><mi>c</mi><mo>(</mo><mi>Y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>X</i> is a real vector space and <i>Y</i> is a locally convex real linear metric space with an invariant metric. Most general results are described in the case of a compact topological group <i>G</i> equipped with the right-invariant Haar measure acting on <i>X</i>. Further results are found if the group <i>G</i> is finite or <i>Y</i> is Asplund space. The main results are applied to an example where <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, and <i>G</i> is the unitary group <inline-formula><math display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T13:55:47Z |
| format | Article |
| id | doaj.art-fd99bf3a3a74400996bd5d97ed6dcaeb |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T13:55:47Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-fd99bf3a3a74400996bd5d97ed6dcaeb2023-11-21T01:38:28ZengMDPI AGMathematics2227-73902020-12-01812224310.3390/math8122243On a Certain Generalized Functional Equation for Set-Valued FunctionsYaroslav Bazaykin0Dušan Bednařík1Veronika Borůvková2Tomáš Zuščák3Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicThe aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>c</mi><mi>c</mi><mo>(</mo><mi>Y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>X</i> is a real vector space and <i>Y</i> is a locally convex real linear metric space with an invariant metric. Most general results are described in the case of a compact topological group <i>G</i> equipped with the right-invariant Haar measure acting on <i>X</i>. Further results are found if the group <i>G</i> is finite or <i>Y</i> is Asplund space. The main results are applied to an example where <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, and <i>G</i> is the unitary group <inline-formula><math display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/8/12/2243integral-functional equationset-valued functiontopological vector spacecompact topological group |
| spellingShingle | Yaroslav Bazaykin Dušan Bednařík Veronika Borůvková Tomáš Zuščák On a Certain Generalized Functional Equation for Set-Valued Functions Mathematics integral-functional equation set-valued function topological vector space compact topological group |
| title | On a Certain Generalized Functional Equation for Set-Valued Functions |
| title_full | On a Certain Generalized Functional Equation for Set-Valued Functions |
| title_fullStr | On a Certain Generalized Functional Equation for Set-Valued Functions |
| title_full_unstemmed | On a Certain Generalized Functional Equation for Set-Valued Functions |
| title_short | On a Certain Generalized Functional Equation for Set-Valued Functions |
| title_sort | on a certain generalized functional equation for set valued functions |
| topic | integral-functional equation set-valued function topological vector space compact topological group |
| url | https://www.mdpi.com/2227-7390/8/12/2243 |
| work_keys_str_mv | AT yaroslavbazaykin onacertaingeneralizedfunctionalequationforsetvaluedfunctions AT dusanbednarik onacertaingeneralizedfunctionalequationforsetvaluedfunctions AT veronikaboruvkova onacertaingeneralizedfunctionalequationforsetvaluedfunctions AT tomaszuscak onacertaingeneralizedfunctionalequationforsetvaluedfunctions |