Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable
The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of...
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2024-11-01
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| author | Cheng Li Limin Guo |
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| description | The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular <i>p</i>-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>π</mi><msup><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfrac></mstyle></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>π</mi><msup><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></msup></mrow></mfrac></mstyle></semantics></math></inline-formula> in Example 1, and by the integrable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>,</mo><mover><mi>θ</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>ψ</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. |
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| publishDate | 2024-11-01 |
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| series | Fractal and Fractional |
| spelling | doaj-art-e4efb0fceacc452baffc105e52a8a43a2025-08-20T02:57:07ZengMDPI AGFractal and Fractional2504-31102024-11-0181268210.3390/fractalfract8120682Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time VariableCheng Li0Limin Guo1School of Automotive Engineering, Changzhou Institute of Technology, Changzhou 213002, ChinaSchool of Science, Changzhou Institute of Technology, Changzhou 213002, ChinaThe mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular <i>p</i>-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>π</mi><msup><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfrac></mstyle></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>π</mi><msup><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mo form="prefix">ln</mo><mi>t</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></msup></mrow></mfrac></mstyle></semantics></math></inline-formula> in Example 1, and by the integrable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>,</mo><mover><mi>θ</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>ψ</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus.https://www.mdpi.com/2504-3110/8/12/682Hadamard fractionalsingular nonlinear termcoupled differential systempositive solutionintegral and infinite-point boundary condition |
| spellingShingle | Cheng Li Limin Guo Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable Fractal and Fractional Hadamard fractional singular nonlinear term coupled differential system positive solution integral and infinite-point boundary condition |
| title | Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable |
| title_full | Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable |
| title_fullStr | Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable |
| title_full_unstemmed | Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable |
| title_short | Positive Solution Pairs for Coupled <i>p</i>-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable |
| title_sort | positive solution pairs for coupled i p i laplacian hadamard fractional differential model with singular source item on time variable |
| topic | Hadamard fractional singular nonlinear term coupled differential system positive solution integral and infinite-point boundary condition |
| url | https://www.mdpi.com/2504-3110/8/12/682 |
| work_keys_str_mv | AT chengli positivesolutionpairsforcoupledipilaplacianhadamardfractionaldifferentialmodelwithsingularsourceitemontimevariable AT liminguo positivesolutionpairsforcoupledipilaplacianhadamardfractionaldifferentialmodelwithsingularsourceitemontimevariable |