Feng, Hanying; Zhang, Xiaofeng Existence of solutions for a coupled system of nonlinear fractional differential equations at resonance. (English) Zbl 1493.34023 Topol. Methods Nonlinear Anal. 58, No. 2, 389-401 (2021). In this paper, the authors study the following three-point boundary value problem at resonance for the coupled system of nonlinear fractional differential equations \[ \begin{cases} D^{\alpha}_{0^+}u(t)=f(t,v(t), D^{\gamma}_{0^+}v(t)),\quad 0<t<1,\\ D^{\beta}_{0^+}v(t)=g(t,u(t), D^{\delta}_{0^+}u(t)),\quad 0<t<1,\\ u(0)=u'(0)=0, u'(1)=\mu_1 u'(\xi_1),\\ v(0)=v'(0)=0, v'(1)=\mu_2 v'(\xi_2), \end{cases} \] where \(2<\alpha,\beta\leq 3,\) \(1<\gamma,\delta\leq 2\) and \(\alpha-\delta\geq 1,\) \(\beta-\gamma\geq 1,\) \(\mu_1, \mu_2>0,\) \(0<\xi_1, \xi_2<1,\) \(\mu_1\xi_1^{\alpha-2}=\mu_2\xi_2^{\beta-2}=1.\) \( D^{\phi}_{0^+},\) \(\phi\in\{\alpha,\beta,\gamma,\delta\}\) are the standard Riemann-Liouville fractional derivatives and \(f, g: [0,1]\times \mathbb{R}^2\to \mathbb{R}\) are given continuous functions. Existence results are obtained via coincidence degree theory. An example illustrating the main results is also presented. Reviewer: Sotiris K. Ntouyas (Ioannina) Cited in 3 Documents MSC: 34A08 Fractional ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations Keywords:fractional differential equation; existence; at resonance; coincidence degree theory × Cite Format Result Cite Review PDF Full Text: DOI References: [1] B. Ahmad and J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009), 1838-1843. · Zbl 1205.34003 [2] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505. · Zbl 1079.34048 [3] Y. Chen and X. Tang, Solvability of sequential fractional order multi-point boundary value problems at resonance, Appl. Math. Comput. 218 (2012), 7638-7648. · Zbl 1252.34004 [4] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, New York, 2008. · Zbl 1154.26007 [5] W. Ge, Boundary Value Problems for Ordinary Nonlinear Differential Equations, Science Press, Beijing, 2007 (in Chinese). [6] Z. Hu and W. Liu, Solvability for fractional order boundary value problems at resonance, Bound. Value Probl. 2011 (2011), no. 20. · Zbl 1273.34009 [7] Z. Hu, W. Liu and T. Chen, Existence of solutions for a coupled system of fractional differential equations at resonance, Bound. Value Probl. 2012 (2012), no. 98. · Zbl 1281.34009 [8] L. Hu and S. Zhang, Existence and uniqueness of solutions for a higher-order coupled fractional differential equations at resonance, Adv. Differ. Equ. 2015 (2015), no. 202. · Zbl 1422.34036 [9] W. Jiang, Solvability for a coupled system of fractional differential equations at resonance, Nonlinear Anal. 13 (2012), 2285-2292. · Zbl 1257.34005 [10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 [11] W. Rui, Existence of solutions of nonlinear fractional differential equations at resonance, Electron. J. Qual. Theory Differ. Equ. 66 (2011), 1-12. · Zbl 1340.34032 [12] K. Shah, A. Ali and R. Ali Khan, Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Bound. Value Probl. 2016 (2016), no. 43. · Zbl 1339.34013 [13] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009), 64-69. · Zbl 1163.34321 [14] Y. Wu and W. Liu, Positive solutions for a class of fractional differential equations at resonance, Adv. Differ. Equ. 2015 (2015), no. 241. · Zbl 1422.34068 [15] S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ. 2006 (2006), no. 36, 1-12. · Zbl 1096.34016 [16] Y. Zhang, Z. Bai and T. Feng, Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance, Comput. Math. Appl. 61 (2011), 1032-1047. · Zbl 1217.34031 [17] X. Zhang and H. Feng, Eigenvalue for a system of Caputo fractional differential equations, J. Comput. Anal. Appl. 3 (2018), no. 25, 544-551. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.