Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems
In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with <i>m</i>-product of fractional operators on a half-line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics>&...
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MDPI AG
2022-04-01
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| author | Ateq Alsaadi Mieczysław Cichoń Mohamed M. A. Metwali |
| author_facet | Ateq Alsaadi Mieczysław Cichoń Mohamed M. A. Metwali |
| author_sort | Ateq Alsaadi |
| collection | DOAJ |
| description | In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with <i>m</i>-product of fractional operators on a half-line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mo>+</mo></msup><mo>=</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>L</mi><mn>1</mn><mi>N</mi></msubsup></semantics></math></inline-formula>-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems. |
| first_indexed | 2024-03-09T11:38:09Z |
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| id | doaj.art-7514f82c98ee4b34a6d12321ef54646e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:38:09Z |
| publishDate | 2022-04-01 |
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| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7514f82c98ee4b34a6d12321ef54646e2023-11-30T23:38:12ZengMDPI AGMathematics2227-73902022-04-01107117210.3390/math10071172Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value ProblemsAteq Alsaadi0Mieczysław Cichoń1Mohamed M. A. Metwali2Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi ArabiaFaculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, PolandDepartment of Mathematics, Faculty of Sciences, Damanhour University, Damanhour 22514, EgyptIn this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with <i>m</i>-product of fractional operators on a half-line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mo>+</mo></msup><mo>=</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>L</mi><mn>1</mn><mi>N</mi></msubsup></semantics></math></inline-formula>-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.https://www.mdpi.com/2227-7390/10/7/1172weighted Lebesgue spacesmeasure of noncompactnessfractional calculusGripenberg-type equationsinitial value problemgeneralized Hölder inequality |
| spellingShingle | Ateq Alsaadi Mieczysław Cichoń Mohamed M. A. Metwali Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems Mathematics weighted Lebesgue spaces measure of noncompactness fractional calculus Gripenberg-type equations initial value problem generalized Hölder inequality |
| title | Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems |
| title_full | Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems |
| title_fullStr | Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems |
| title_full_unstemmed | Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems |
| title_short | Integrable Solutions for Gripenberg-Type Equations with <i>m</i>-Product of Fractional Operators and Applications to Initial Value Problems |
| title_sort | integrable solutions for gripenberg type equations with i m i product of fractional operators and applications to initial value problems |
| topic | weighted Lebesgue spaces measure of noncompactness fractional calculus Gripenberg-type equations initial value problem generalized Hölder inequality |
| url | https://www.mdpi.com/2227-7390/10/7/1172 |
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