PDF《domain》

2 <

q <

3, t ∈ J := [0, ∞), u(0) = u (0) = 0, Dp?1u(∞) = m?2 i=1 βiu(ξi), v(0) = v (0) = 0, Dq?1v(∞) = m?2 i=1 γiv(ξi), (5) where f, g ∈ C(J * R, R),

0 <

ξ1 <

ξ2 ξm?2 <

∞, Dp and Dq denote RiemannCLiouville fractional derivatives of order p and q, respectively, as well as βi >

0, γi >

0 are such that

0 <

m?2 i=1 βiξp?1 i <

Γ(p) and

0 <

m?2 i=1 γiξq?1 i <

Γ(q). Our aim in this paper is to generalize the above works on an in?nite interval and more general boundary conditions, so we discuss the existence of the solutions of a coupled system of nonlinear fractional di?erential equations on an unbounded domain ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Dαu(t) = f(t, v(t), Dβ?1v(t)), t ∈ J := [0, ∞), Dβv(t) = g(t, u(t), Dα?1u(t)), t ∈ J := [0, ∞), u(0) = 0, v(0) = 0, Dα?1u(∞) = u0 + m?2 i=1 aiu(ξi) + m?2 i=1 biDα?1u(ξi), Dβ?1v(∞) = v0 + n?2 i=1 civ(ηi) + n?2 i=1 diDβ?1v(ηi), (6) EJQTDE,

2013 No. 73, p.

2 where

1 <

α, β ≤ 2,

0 <

ξ1 <

ξ2 ξm?2 <

∞,

0 <

η1 <

η2 ηn?2 <

∞, ai, bi, ci, di ≥ 0, u0, v0 ≥

0 are real numbers and f, g ∈ C(J * R * R, R) and D is the RiemannCLiouville fractional derivative. This paper is organized as follows: in Section 2, some facts and results about fractional calculus are given, while inspired by [19] we prove the main result and some corollaries in Section 3, and we conclude this paper by considering an example in Section 4.

2 Preliminaries In this section, we present some de?nitions and results which will be needed later. De?nition 2.1. [11] The RiemannCLiouville fractional integral of order α >

0 of a function f : (0, ∞) → R is de?ned by Iα f(t) =

1 Γ(α) t

0 (t ? s)α?1 f(s)ds, t >

0, provided that the right-hand side is pointwise de?ned. De?nition 2.2. [11] The RiemannCLiouville fractional derivative of order α >

0 of a continuous function f : (0, ∞) → R is de?ned by Dα f(t) =

1 Γ(n ? α) d dt n t

0 (t ? s)n?α?1 f(s)ds t >

0, where n = [α] + 1, provided that the right-hand side is pointwise de?ned. In particular, for α = n, Dnf(t) = f(n)(t). Remark 1. The following properties are well known: DαIαf(t) = f(t), α >

0, f(t) ∈ L1(0, ∞), DβIαf(t) = Iα?βf(t), α >

β >

0, f(t) ∈ L1(0, ∞). The following two lemmas can be found in [5, 11]. Lemma 2.1. Let α >

0 and u ∈ C(0, 1)∩L1(0, 1). Then the fractional di?erential equation Dαu(t) =

0 has a unique solution u(t) = c1tα?1 + c2tα?2 cntα?n , ci ∈ R, i = 1,n, where n = [α] +

1 if α / ∈ N and n = α if α ∈ N. Lemma 2.2. Assume that u ∈ C(0, 1) ∩ L1(0, 1) with a fractional derivative of order α >

0 that belongs to C(0, 1) ∩ L1(0, 1). Then Iα Dα u(t) = u(t) + c1tα?1 + c2tα?2 cntα?n , for some ci ∈ R, i = 1,n and n = [α] +

1 if α / ∈ N and n = α if α ∈ N. For the forthcoming analysis, we de?ne the spaces X = u(t) u(t), Dα?1 u(t) ∈ C(J, R), sup t∈J |u(t)|

1 + tα?1 <

∞, sup t∈J |Dα?1 u(t)| <

∞ , EJQTDE,

2013 No. 73, p.

3 with the norm ||u||X = max sup t∈J |u(t)|

1 + tα?1 , sup t∈J |Dα?1 u(t)| , and Y = v(t) v(t), Dβ?1 v(t) ∈ C(J, R), sup t∈J |v(t)|

1 + tβ?1 <

∞, sup t∈J |Dβ?1 v(t)| <

∞ , with the norm ||v||Y = max sup t∈J |v(t)|

1 + tβ?1 , sup t∈J |Dβ?1 v(t)| . By Lemma 2.2 of [19], (X, ||・||X), (Y, ||・||Y ) are Banach spaces. For (u, v) ∈ X *Y , let ||(u, v)||X*Y = max{||u||X, ||v||Y }, then (X * Y,X*Y ) is a Banach space. The Arzel` aCAscoli theorem fails to work in the Banach space X, Y due to the fact that the in?nite interval [0, ∞) is noncompact. The following compactness criterion will help us to resolve this problem. Lemma 2.3. [19] Let Z ? X(Y ) be a bounded set. Then Z is relatively compact in X(Y ) if the following conditions hold. (i) For any u(t) ∈ Z, u(t) 1+tα?1 and Dα?1u(t) are equicontinuous on any compact interval of J. (ii) Given >