Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space
In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order <inline-formula>&l...
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MDPI AG
2022-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/8/1332 |
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| author | Ahmed Salem Kholoud N. Alharbi Hashim M. Alshehri |
| author_facet | Ahmed Salem Kholoud N. Alharbi Hashim M. Alshehri |
| author_sort | Ahmed Salem |
| collection | DOAJ |
| description | In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> and the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>ρ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> are used to establish our model. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation. |
| first_indexed | 2024-03-09T10:32:08Z |
| format | Article |
| id | doaj.art-8424a1c82bd34ff7986530dc3acef2f9 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T10:32:08Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
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| spelling | doaj.art-8424a1c82bd34ff7986530dc3acef2f92023-12-01T21:12:21ZengMDPI AGMathematics2227-73902022-04-01108133210.3390/math10081332Fractional Evolution Equations with Infinite Time Delay in Abstract Phase SpaceAhmed Salem0Kholoud N. Alharbi1Hashim M. Alshehri2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaIn the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> and the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>ρ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> are used to establish our model. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation.https://www.mdpi.com/2227-7390/10/8/1332generalized Liouville–Caputo fractional derivative<i>ρ</i>-Laplace transformationinfinite time delaymild solutionLeray–Schauder alternative |
| spellingShingle | Ahmed Salem Kholoud N. Alharbi Hashim M. Alshehri Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space Mathematics generalized Liouville–Caputo fractional derivative <i>ρ</i>-Laplace transformation infinite time delay mild solution Leray–Schauder alternative |
| title | Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space |
| title_full | Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space |
| title_fullStr | Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space |
| title_full_unstemmed | Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space |
| title_short | Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space |
| title_sort | fractional evolution equations with infinite time delay in abstract phase space |
| topic | generalized Liouville–Caputo fractional derivative <i>ρ</i>-Laplace transformation infinite time delay mild solution Leray–Schauder alternative |
| url | https://www.mdpi.com/2227-7390/10/8/1332 |
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