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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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/4/261 |
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| Summary: | 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. |
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| ISSN: | 2504-3110 |