Existence theory of fractional order three-dimensional differential system at resonance
<p>This paper deals with three-dimensional differential system of nonlinear fractional order problem</p> <p class="disp_formula"> $ \begin{align*} D^{\alpha}_{0^{+}}\upsilon(\varrho) = f(\varrho,\omega(\varrho),\omega^{\prime}(\varrho),\omega^{\prime\prime}(\varrho),...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-06-01
|
| Series: | Mathematical Modelling and Control |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mmc.2023012?viewType=HTML |
| _version_ | 1797690909593174016 |
|---|---|
| author | M. Sathish Kumar M. Deepa J Kavitha V. Sadhasivam |
| author_facet | M. Sathish Kumar M. Deepa J Kavitha V. Sadhasivam |
| author_sort | M. Sathish Kumar |
| collection | DOAJ |
| description | <p>This paper deals with three-dimensional differential system of nonlinear fractional order problem</p>
<p class="disp_formula"> $ \begin{align*} D^{\alpha}_{0^{+}}\upsilon(\varrho) = f(\varrho,\omega(\varrho),\omega^{\prime}(\varrho),\omega^{\prime\prime}(\varrho),...,\omega^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\beta}_{0^{+}}\nu(\varrho) = g(\varrho, \upsilon(\varrho),\upsilon^{\prime}(\varrho),\upsilon^{\prime\prime}(\varrho),...,\upsilon^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\gamma}_{0^{+}}\omega(\varrho) = h(\varrho,\nu(\varrho),\nu^{\prime}(\varrho),\nu^{\prime\prime}(\varrho),...,\nu^{(n-1)}(\varrho)), \; \varrho \in (0,1), \end{align*} $ </p>
<p>with the boundary conditions,</p>
<p class="disp_formula">$ \begin{align*} \upsilon(0) = \upsilon^{\prime}(0) = ... = \upsilon^{(n-2)}(0) = 0,\; \upsilon^{(n-1)}(0) = \upsilon^{(n-1)}(1),\\ \nu(0) = \nu^{\prime}(0) = ... = \nu^{(n-2)}(0) = 0,\; \nu^{(n-1)}(0) = \nu^{(n-1)}(1),\\ \omega(0) = \omega^{\prime}(0) = ... = \omega^{(n-2)}(0) = 0,\; \omega^{(n-1)}(0) = \omega^{(n-1)}(1), \end{align*} $</p>
<p>where $ D^{\alpha}_{0^{+}}, D^{\beta}_{0^{+}}, D^{\gamma}_{0^{+}} $ are the standard Caputo fractional derivative, $ n-1 < \alpha, \beta, \gamma \leq n, \; n \geq 2 $ and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.</p> |
| first_indexed | 2024-03-12T02:06:54Z |
| format | Article |
| id | doaj.art-2abac944952848e4b115370e11a06658 |
| institution | Directory Open Access Journal |
| issn | 2767-8946 |
| language | English |
| last_indexed | 2024-03-12T02:06:54Z |
| publishDate | 2023-06-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Modelling and Control |
| spelling | doaj.art-2abac944952848e4b115370e11a066582023-09-07T03:47:27ZengAIMS PressMathematical Modelling and Control2767-89462023-06-013212713810.3934/mmc.2023012Existence theory of fractional order three-dimensional differential system at resonanceM. Sathish Kumar0M. Deepa1J Kavitha2V. Sadhasivam 31. Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal - 637 018, Tamil Nadu, India2. Department of Mathematics, Pavai Arts and Science College for Women, Namakkal - 637 401, Tamil Nadu, India3. Department of Mathematics, Sona College of Technology (Autonomous), Salem - 636 005, Tamil Nadu, India4. Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram - 637 401, Namakkal, Tamil Nadu, India<p>This paper deals with three-dimensional differential system of nonlinear fractional order problem</p> <p class="disp_formula"> $ \begin{align*} D^{\alpha}_{0^{+}}\upsilon(\varrho) = f(\varrho,\omega(\varrho),\omega^{\prime}(\varrho),\omega^{\prime\prime}(\varrho),...,\omega^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\beta}_{0^{+}}\nu(\varrho) = g(\varrho, \upsilon(\varrho),\upsilon^{\prime}(\varrho),\upsilon^{\prime\prime}(\varrho),...,\upsilon^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\gamma}_{0^{+}}\omega(\varrho) = h(\varrho,\nu(\varrho),\nu^{\prime}(\varrho),\nu^{\prime\prime}(\varrho),...,\nu^{(n-1)}(\varrho)), \; \varrho \in (0,1), \end{align*} $ </p> <p>with the boundary conditions,</p> <p class="disp_formula">$ \begin{align*} \upsilon(0) = \upsilon^{\prime}(0) = ... = \upsilon^{(n-2)}(0) = 0,\; \upsilon^{(n-1)}(0) = \upsilon^{(n-1)}(1),\\ \nu(0) = \nu^{\prime}(0) = ... = \nu^{(n-2)}(0) = 0,\; \nu^{(n-1)}(0) = \nu^{(n-1)}(1),\\ \omega(0) = \omega^{\prime}(0) = ... = \omega^{(n-2)}(0) = 0,\; \omega^{(n-1)}(0) = \omega^{(n-1)}(1), \end{align*} $</p> <p>where $ D^{\alpha}_{0^{+}}, D^{\beta}_{0^{+}}, D^{\gamma}_{0^{+}} $ are the standard Caputo fractional derivative, $ n-1 < \alpha, \beta, \gamma \leq n, \; n \geq 2 $ and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.</p>https://www.aimspress.com/article/doi/10.3934/mmc.2023012?viewType=HTMLfractional differential equationcoincidence degree theoryresonance |
| spellingShingle | M. Sathish Kumar M. Deepa J Kavitha V. Sadhasivam Existence theory of fractional order three-dimensional differential system at resonance Mathematical Modelling and Control fractional differential equation coincidence degree theory resonance |
| title | Existence theory of fractional order three-dimensional differential system at resonance |
| title_full | Existence theory of fractional order three-dimensional differential system at resonance |
| title_fullStr | Existence theory of fractional order three-dimensional differential system at resonance |
| title_full_unstemmed | Existence theory of fractional order three-dimensional differential system at resonance |
| title_short | Existence theory of fractional order three-dimensional differential system at resonance |
| title_sort | existence theory of fractional order three dimensional differential system at resonance |
| topic | fractional differential equation coincidence degree theory resonance |
| url | https://www.aimspress.com/article/doi/10.3934/mmc.2023012?viewType=HTML |
| work_keys_str_mv | AT msathishkumar existencetheoryoffractionalorderthreedimensionaldifferentialsystematresonance AT mdeepa existencetheoryoffractionalorderthreedimensionaldifferentialsystematresonance AT jkavitha existencetheoryoffractionalorderthreedimensionaldifferentialsystematresonance AT vsadhasivam existencetheoryoffractionalorderthreedimensionaldifferentialsystematresonance |