On a New Class of Antiperiodic Fractional Boundary Value Problems

and Applied Analysis 3 where g(t, s) is g (t, s) = {{{{{{{{{{{{{{ {{{{{{{{{{{{{{ { (t − s) q−1 − (1/2) (T − s) q−1 Γ (q) + (T − 2t) (T − s) q−2 4Γ (q − 1) + t (T − t) (T − s) q−3 4Γ (q − 2) , s ≤ t, − (T − s) q−1 2Γ (q) + (T − 2t) (T − s) q−2 4Γ (q − 1) + t (T − t) (T − s) q−3 4Γ (q − 2) , t < s. (14) If we let p → 1− in (7), we obtain G T (t, s) 󵄨󵄨󵄨󵄨p→1− = {{{{{{{{{{{{{{{{{{{ {{{{{{{{{{{{{{{{{{{ { (t − s) q−1 − (1/2) (T − s) q−1 Γ (q) + (T − 2t) (T − s) q−2 2Γ (q − 1) + (−2t 2 − T 2 + 4tT) (T − s) q−3 4Γ (q − 2) , s ≤ t, − (T − s) q−1 2Γ (q) + (T − 2t) (T − s) q−2 2Γ (q − 1) + (−2t 2 − T 2 + 4tT) (T − s) q−3 4Γ (q − 2) , t < s. (15) We note that the solutions given by (14) and (15) are different. As amatter of fact, (15) contains an additional term: (−t2−T2+ 3tT)(T − s) q−3 /4Γ(q − 2). Therefore the fractional boundary conditions introduced in (2) give rise to a new class of problems. Remark 5. When the phenomenon of antiperiodicity occurs at an intermediate point η ∈ (0, T), the parametric-type antiperiodic fractional boundary value problem takes the form c D q x (t) = f (t, x (t)) , t ∈ [0, T] , 2 < q ≤ 3, x (0) = −x (η) , c D p (0) = − c D p (η) , c D p+1 (0) = − c D p+1 (η) , (16) whose solution is x (t) = ∫ T 0 G η (t, s) f (s, x (s)) ds, (17) where G η (t, s) is given by (7). Notice that G η (t, s) → G T (t, s) when η → T. 3. Existence Results LetC = C([0, T], R) denotes a Banach space of all continuous functions defined on [0, T] into R endowed with the usual supremum norm. In relation to (2), we define an operatorF : C → C as