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

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.

Keywords

• fractional differential equation
• coincidence degree theory
• resonance