A coupled system of fractional differential equations on the half-line

• Published in: Boundary value problems Vol. 2019; no. 1; pp. 1 - 22
• Main Authors: Zhai, Chengbo, Ren, Jing
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 05.07.2019; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Banach spaces; Boundary value problem; Boundary value problems; Difference and Functional Equations; Differential equations; Existence and uniqueness; Existence theorems; Fixed points (mathematics); Fractional differential equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Standard fixed point theorems; Unbounded domain; Standard fixed point theorems; 34B15; Unbounded domain; Boundary value problem; Fractional differential equations; 34B40; Existence and uniqueness; 34A08
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-019-1230-0

In this paper, we consider a new fractional differential system on an unbounded domain D α u ( t ) + φ ( t , v ( t ) , D γ 1 v ( t ) ) = 0 , t ∈ [ 0 , + ∞ ) , α ∈ ( 2 , 3 ] , D β v ( t ) + ψ ( t , u ( t ) , D γ 2 u ( t ) ) = 0 , t ∈ [ 0 , + ∞ ) , β ∈ ( 2 , 3 ] , subject to the conditions I 3 − α u ( t ) | t = 0 = 0 , D α − 2 u ( t ) | t = 0 = ∫ 0 h g 1 ( s ) u ( s ) d s , D α − 1 u ( + ∞ ) = M u ( ξ ) + a , I 3 − β v ( t ) | t = 0 = 0 , D β − 2 v ( t ) | t = 0 = ∫ 0 h g 2 ( s ) v ( s ) d s , D β − 1 v ( + ∞ ) = N v ( η ) + b . The nonlinear terms φ and ψ are dependent on the fractional derivative of lower order γ i ∈ ( 0 , 1 ) , i = 1 , 2 , which creates additional complexity to verify the existence of solutions. Moreover, a proper choice of Banach space allows the solutions to be defined on the half-line. From some standard fixed point theorems, sufficient conditions for the existence and uniqueness of solutions to boundary value problems are developed. Finally, the main result is applied to an illustrative example.