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Existence and uniqueness of solutions for a coupled system of sequential fractional differential equations with initial conditions Hamid Baghani1 · Jehad Alzabut2 · Javad Farokhi-Ostad3 · Juan J. Nieto4 Received: 22 April 2020 / Revised: 20 July 2020 / Accepted: 23 July 2020 © Springer Nature Switzerland AG 2020

Abstract In this paper, we consider a coupled system of sequential fractional differential equations associated with initial conditions. The main theorems provide new existence and uniqueness conditions for solutions of the proposed coupled system. We conclude an immediate consequence that establishes weaker conditions to ensure the existence and uniqueness of solutions for the corresponding sequential fractional differential equation. Meanwhile, an iterative sequence is constructed in terms of solution operator that converges to the unique fixed point which corresponds to the unique solution. The consistency of the main results is verified by presenting two numerical examples. For the sake of completeness, we end the paper with a concluding remark. Keywords Coupled fractional system · Sequential fractional differential equations · Banach fixed point theorem Mathematics Subject Classification 26A33 · 34A08 · 34A12

B

Hamid Baghani [email protected] Jehad Alzabut [email protected] Javad Farokhi-Ostad [email protected] Juan J. Nieto [email protected]

1

Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

2

Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

3

Department of Basic Sciences, Birjand University of Technology, Birjand, Iran

4

Department of Statistics, Mathematical Analysis and Optimization, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

H. Baghani et al.

1 Introduction In [1], the existence and uniqueness of solutions for the following initial value problem of the nonlinear sequential fractional differential equation were considered 

D2α x(t) = h(t, x(t), Dα x(t)), t ∈ (0, T ], 0 < α ≤ 1, 0 < T < ∞, limt→0 t 1−α x(t) = x0 , limt→0 t 1−α Dα x(t) = x1 ,

(1)

where Dν is the sequential fractional derivative of order ν ∈ {α, 2α} and h : [0, T ] × R2 → R is a given function. The sequential fractional derivative Dμ has the properties Dμ x = D μ x and Dkμ x = Dμ D(k−1)μ x, k = 2, 3, · · · , where D μ is the classical Riemann–Liouville fractional derivative of order μ. For their purposes, the authors introduced the weighted space of continues functions C1−α [0, T ] := {x ∈ C(0, T ] : t 1−α x ∈ C[0, T ]} with the norm x1−α := max0≤t≤T |t 1−α x(t)| and the space α [0, T ] := {x ∈ C α C1−α 1−α [0, T ] : D x ∈ C 1−α [0, T ]} equipped with the norm α α [0, T ] is a Banach α x1−α := x1−α + D x1−α . They verified that the space C1−α space and then proved existence and uniqueness results via a fixed point theorem for mixed monotone mappings in partially ordered metric spaces. Equation (1) could be utilized as a typical example that describes several mathematical model