The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces
In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a <i>w</i>-weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"&g...
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MDPI AG
2024-02-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/3/144 |
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| author | Zainab Alsheekhhussain Ahmad Gamal Ibrahim Mohammed Mossa Al-Sawalha Yousef Jawarneh |
| author_facet | Zainab Alsheekhhussain Ahmad Gamal Ibrahim Mohammed Mossa Al-Sawalha Yousef Jawarneh |
| author_sort | Zainab Alsheekhhussain |
| collection | DOAJ |
| description | In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a <i>w</i>-weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup><mo>,</mo><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula>of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></semantics></math></inline-formula> in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup></semantics></math></inline-formula> of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∈</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the <i>w</i>-weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup></semantics></math></inline-formula> includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results. |
| first_indexed | 2024-04-24T18:15:52Z |
| format | Article |
| id | doaj.art-7d499b1874834dc9ad191596e8804ef7 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-04-24T18:15:52Z |
| publishDate | 2024-02-01 |
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| series | Fractal and Fractional |
| spelling | doaj.art-7d499b1874834dc9ad191596e8804ef72024-03-27T13:42:03ZengMDPI AGFractal and Fractional2504-31102024-02-018314410.3390/fractalfract8030144The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach SpacesZainab Alsheekhhussain0Ahmad Gamal Ibrahim1Mohammed Mossa Al-Sawalha2Yousef Jawarneh3Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics Al Ahsa, College of Science, King Fiasal University, Al-Ahsa 31982, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaIn this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a <i>w</i>-weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup><mo>,</mo><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula>of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>,</mo></mrow></semantics></math></inline-formula> in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup></semantics></math></inline-formula> of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∈</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the <i>w</i>-weighted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>D</mi><mrow><mn>0</mn><mo>,</mo><mi>t</mi></mrow><mrow><mi>σ</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>ψ</mi><mo>,</mo><mi>w</mi></mrow></msubsup></semantics></math></inline-formula> includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.https://www.mdpi.com/2504-3110/8/3/144differential inclusionsnon-instantaneous impulses<i>w</i>-weighted <i>ψ</i>-Hilfer fractional derivativemeasure of non-compactness |
| spellingShingle | Zainab Alsheekhhussain Ahmad Gamal Ibrahim Mohammed Mossa Al-Sawalha Yousef Jawarneh The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces Fractal and Fractional differential inclusions non-instantaneous impulses <i>w</i>-weighted <i>ψ</i>-Hilfer fractional derivative measure of non-compactness |
| title | The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces |
| title_full | The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces |
| title_fullStr | The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces |
| title_full_unstemmed | The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces |
| title_short | The Existence of Solutions for <i>w</i>-Weighted <i>ψ</i>-Hilfer Fractional Differential Inclusions of Order <i>μ</i> ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces |
| title_sort | existence of solutions for i w i weighted i ψ i hilfer fractional differential inclusions of order i μ i ∈ 1 2 with non instantaneous impulses in banach spaces |
| topic | differential inclusions non-instantaneous impulses <i>w</i>-weighted <i>ψ</i>-Hilfer fractional derivative measure of non-compactness |
| url | https://www.mdpi.com/2504-3110/8/3/144 |
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