Existence theory of fractional order three-dimensional differential system at resonance

M. Sathish Kumar; M. Deepa; J Kavitha; V. Sadhasivam; M. Sathish Kumar; M. Deepa; J Kavitha; V. Sadhasivam

doi:10.3934/mmc.2023012

Mathematical Modelling and Control

2023, Volume 3, Issue 2: 127-138. doi: 10.3934/mmc.2023012

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Existence theory of fractional order three-dimensional differential system at resonance

1.
Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal - 637 018, Tamil Nadu, India
2.
Department of Mathematics, Pavai Arts and Science College for Women, Namakkal - 637 401, Tamil Nadu, India
3.
Department of Mathematics, Sona College of Technology (Autonomous), Salem - 636 005, Tamil Nadu, India
4.
Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram - 637 401, Namakkal, Tamil Nadu, India

Received: 18 November 2022 Revised: 14 March 2023 Accepted: 19 March 2023 Published: 08 June 2023

This paper deals with three-dimensional differential system of nonlinear fractional order problem

$ \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*} $

with the boundary conditions,

$ \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*} $

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.
- fractional differential equation,
- coincidence degree theory,
- resonance
Citation: M. Sathish Kumar, M. Deepa, J Kavitha, V. Sadhasivam. Existence theory of fractional order three-dimensional differential system at resonance[J]. Mathematical Modelling and Control, 2023, 3(2): 127-138. doi: 10.3934/mmc.2023012

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Abstract

This paper deals with three-dimensional differential system of nonlinear fractional order problem

$ \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*} $

with the boundary conditions,

$ \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*} $

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.

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