On Impulsive Boundary Value Problems of Fractional Differential Equations with Irregular Boundary Conditions

and Applied Analysis 3 To prove the existence of solutions of problem 1.1 , we need the following fixed-point theorems. Theorem 2.2 see 51 . Let E be a Banach space. Assume that Ω is an open bounded subset of E with θ ∈ Ω and let T : Ω → E be a completely continuous operator such that ‖Tu‖ ≤ ‖u‖, ∀u ∈ ∂Ω. 2.3 Then T has a fixed point in Ω. Lemma 2.3 see 1 . For α > 0, the general solution of fractional differential equation Du t 0 is u t C0 C1t C2t · · · Cn−1tn−1, 2.4 where Ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1 ( α denotes integer part of α). Lemma 2.4 see 1 . Let α > 0. Then I Du t u t C0 C1t C2t · · · Cn−1tn−1 2.5 for some Ci ∈ R, i 1, 2, . . . , n − 1, n α 1. Lemma 2.5. For a given y ∈ C 0, T , a function u is a solution of the following impulsive irregular boundary value problem Du t y t , 1 < α ≤ 2, t ∈ J ′, Δu tk Ik u tk , Δu′ tk I∗ k u tk , k 1, 2, . . . , p, u′ 0 −1 θu′ T bu T 0, u 0 −1 θ u T 0, θ 1, 2, b / 0, 2.6 4 Abstract and Applied Analysis if and only if u is a solution of the impulsive fractional integral equation u t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∫ t 0 t − s α−1 Γ α y s ds 1 − −1 θ 1 bT ∫T tp T − s α−1 Γ α y s ds [ 1 −1 θ 1 ] t bT ∫T tp T − s α−2 Γ α − 1 y s ds − b ∫T tp T − s α−2 Γ α − 1 y s ds − t T ∫T tp T − s α−1 Γ α y s ds A, t ∈ J0; ∫ t tk t − s α−1 Γ α y s ds 1 − −1 θ 1 bT ∫T tp T − s α−1 Γ α y s ds [ 1 −1 θ 1 ] t bT ∫T tp T − s α−2 Γ α − 1 y s ds − b ∫T tp T − s α−2 Γ α − 1 y s ds − t T ∫T tp T − s α−1 Γ α y s ds