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Electronic Journal of Qualitative Theory of Di?erential Equations 2013, No.

73, 1C15;

http://www.math.u-szeged.hu/ejqtde/ Existence results for a coupled system of nonlinear fractional di?erential equations with boundary value problems on an unbounded domain Neda Khodabakhshi, S. Mansour Vaezpour? Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran. [email protected], [email protected] Abstract. This paper deals with the existence results for solutions of coupled system of non- linear fractional di?erential equations with boundary value problems on an unbounded domain. Also, we give an illustrative example in order to indicate the validity of our assumptions. Keywords: fractional di?erential equations;

boundary value problem;

?xed point theorem.

2010 AMS Subject Classi?cation: 34A08, 34B10, 47H10.

1 Introduction Recently, fractional di?erential calculus has attracted a lot of attention by many researchers of di?erent ?elds, such as: physics, chemistry, biology, economics, control theory and biophysics, etc. [11, 15, 16]. In particular, study of coupled systems involving fractional di?erential equations is also important in several problems. Many authors have investigated su?cient conditions for the existence of solutions for the following coupled systems of nonlinear fractional di?erential equations with di?erent boundary conditions on ?nite domain. Dαu(t) = f(t, v(t)), Dβv(t) = g(t, u(t)), and more generally, Dαu(t) = f(t, v(t), D?v(t)), Dβv(t) = g(t, u(t), Dνu(t)), see for example [1, 2, 4, 7, 8, 9, 10, 17, 21, 22, 23]. However, to the best of our knowledge few papers consider the existence of solutions of fractional di?erential equations on the half-line. Arara et al. [3] studied the existence of bounded solutions for di?erential equations involving the Caputo fractional derivative on the unbounded domain given by ? ? ? ? ? cDαu(t) = f(t, u(t)), t ∈ [0, ∞), u(0) = u0, u is bounded on [0, ∞), (1) ? Corresponding author. Fax: +98-21-66497930. EJQTDE,

2013 No. 73, p.

1 where α ∈ (1, 2], cDα is the Caputo fractional derivative of order α, u0 ∈ R, and f : [0, ∞) * R → R is continuous. Zhao and Ge [24] considered the following boundary value problem for fractional di?erential equations ? ? ? ? ? Dαu(t) + f(t, u(t)) = 0, t ∈ (0, ∞), u(0) = 0, limt→∞ Dα?1u(t) = βu(ξ), (2) where α ∈ (1, 2),

0 <

ξ <

∞, β ≥ 0, f is a given function and Dα is the RiemannCLiouville fractional derivative. Su, Zhang [19] considered the following boundary value problem ? ? ? ? ? Dαu(t) = f(t, u(t), Dα?1u(t)), t ∈ [0, ∞), u(0) = 0, Dα?1u(∞) = u0, u0 ∈ R, (3) where α ∈ (1, 2], f ∈ C([0, ∞) * R * R, R) and Dα, Dα?1 are the RiemannCLiouville fractional derivatives. In [13], Liang and Zhang investigated the existence of three positive solutions for the following m-point fractional boundary value problem ? ? ? ? ? Dαu(t) + a(t)f(u(t)) = 0, t ∈ (0, ∞), u(0) = u (0) = 0, Dα?1u(∞) = m?2 i=1 βiu(ξi), (4) where α ∈ (2, 3),

0 <

ξ1 <

ξ2 ξm?2 <

∞, βi ≥

0 such that

0 <

m?2 i=1 βiξα?1 i <

Γ(α) and Dα is the RiemannCLiouville fractional derivative. Wang et al. [20] by using Schauder'

s ?xed point theorem investigated the existence and unique- ness of solutions for the following coupled system of nonlinear fractional di?erential equations on an unbounded domain ? ? ? ? ? ? ? ? ? ? ? Dpu(t) + f(t, v(t)) = 0,

2 <

p <

3, t ∈ J := [0, ∞), Dqv(t) + g(t, u(t)) = 0,