Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications
In this article, we study a fractional lower-order differential equation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mi>D</mi><mrow><mn>0&...
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2025-04-01
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| description | In this article, we study a fractional lower-order differential equation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mi>D</mi><mrow><mn>0</mn><mo>+</mo></mrow><mi>α</mi></msubsup><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>+</mo><mi mathvariant="normal">a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><mi mathvariant="normal">y</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>ξ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> with a Dirichlet-type boundary condition, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> permits singularity. When the coefficient of perturbation term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">a</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is continuous on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, Graef et al. derived the associated Green’s function under certain conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">a</mi></semantics></math></inline-formula>, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mi>D</mi><mrow><mn>0</mn><mo>+</mo></mrow><mi>α</mi></msubsup><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>ξ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> is also considered. The existence of positive solutions is determined under a more general condition, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>(</mo><mi>ξ</mi><mo>,</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>≥</mo><mo>−</mo><mi mathvariant="normal">b</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mi mathvariant="normal">x</mi><mo>−</mo><mi mathvariant="normal">e</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">b</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>,</mo><mi mathvariant="normal">e</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> are non-negative functions. Relevant examples are listed to manifest the theoretical results. |
| format | Article |
| id | doaj-art-9906823b174747be80ca2deeff55b6fe |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-9906823b174747be80ca2deeff55b6fe2025-08-20T02:28:14ZengMDPI AGFractal and Fractional2504-31102025-04-019426110.3390/fractalfract9040261Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its ApplicationsYongqing Wang0School of Mathematical Sciences, Qufu Normal University, Qufu 273165, ChinaIn this article, we study a fractional lower-order differential equation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mi>D</mi><mrow><mn>0</mn><mo>+</mo></mrow><mi>α</mi></msubsup><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>+</mo><mi mathvariant="normal">a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><mi mathvariant="normal">y</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>ξ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> with a Dirichlet-type boundary condition, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> permits singularity. When the coefficient of perturbation term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">a</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is continuous on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, Graef et al. derived the associated Green’s function under certain conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">a</mi></semantics></math></inline-formula>, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mi>D</mi><mrow><mn>0</mn><mo>+</mo></mrow><mi>α</mi></msubsup><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mo>Υ</mo><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>ξ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> is also considered. The existence of positive solutions is determined under a more general condition, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>(</mo><mi>ξ</mi><mo>,</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>≥</mo><mo>−</mo><mi mathvariant="normal">b</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mi mathvariant="normal">x</mi><mo>−</mo><mi mathvariant="normal">e</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">b</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>,</mo><mi mathvariant="normal">e</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> are non-negative functions. Relevant examples are listed to manifest the theoretical results.https://www.mdpi.com/2504-3110/9/4/261Green’s functionpositivityDirichlet boundary conditionfractional differential equation |
| spellingShingle | Yongqing Wang Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications Fractal and Fractional Green’s function positivity Dirichlet boundary condition fractional differential equation |
| title | Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications |
| title_full | Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications |
| title_fullStr | Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications |
| title_full_unstemmed | Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications |
| title_short | Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications |
| title_sort | positive properties of green s function for fractional dirichlet boundary value problem with a perturbation term and its applications |
| topic | Green’s function positivity Dirichlet boundary condition fractional differential equation |
| url | https://www.mdpi.com/2504-3110/9/4/261 |
| work_keys_str_mv | AT yongqingwang positivepropertiesofgreensfunctionforfractionaldirichletboundaryvalueproblemwithaperturbationtermanditsapplications |