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0, there exists a constant T = T( ) >

0 such that u(t1)

1 + tα?1

1 ? u(t2)

1 + tα?1

2 <

, and Dα?1 u(t1) ? Dα?1 u(t2) <

, for any t1, t2 ≥ T and u(t) ∈ Z.

3 Main result In this section, we investigate su?cient conditions for the existence and uniqueness solutions for the boundary value problem (6). Before we state our main result, for the convenience, we introduce the following notations: ?α = Γ(α) ? m?2 i=1 aiξα?1 i ? Γ(α) m?2 i=1 bi, χ1 β = ∞

0 (1 + sβ?1 )a(s) + b(s) ds, χ2 φ = ∞

0 φ(s)ds, χ3 α,β = ξi

0 (ξi ? s)α?1 (1 + sβ?1 )a(s) + b(s) ds, χ4 α = ξi

0 (ξi ? s)α?1 φ(s)ds. By replacing ai, bi, ξi, φ, a(s), b(s) with ci, di, ηi, ψ, c(s), d(s) respectively, and α with β we can introduce ?β, χ1 α, χ2 ψ, χ3 β,α and χ4 β. Now, we state su?cient conditions which allow us to establish the existence results for the system (6). EJQTDE,

2013 No. 73, p.

4 (H1) There exist nonnegative functions a(t), b(t), φ(t) ∈ C[0, ∞) such that |f(t, x, y)| ≤ a(t)|x| + b(t)|y| + φ(t), ?α >

0, χ2 ........