Existence of Positive Solutions for a Class of Delay Fractional Differential Equations with Generalization to N-Term

and Applied Analysis 3 Let E be a real Banach space with a cone K. K introduces a partial order ≤ in E as x ≤ y if y − x ∈ K. Definition 2.1. For x, y ∈ E, the order interval 〈x, y〉 is defined as 〈 x, y 〉 { z ∈ E : x ≤ z ≤ y. 2.1 Theorem 2.2 Leray-Schauder Theorem 17 . Let E be a Banach space with C ⊆ E closed and convex. Assume U is relatively open subset of C with 0 ∈ U and F : U → C is a continuous, compact map. Then, either i F has fixed point in U or ii there exists u ∈ ∂U and γ ∈ 0, 1 with u γFu. Theorem 2.3 Banach fixed point theorem 17 . Let K be a closed subspace of a Banach space Ω. Let F be a contraction mapping with Lipschitz constant l < 1 from K to itself. Then, F has a unique fixed point x∗ in K. Moreover, if x0 is an arbitrary point in K and {xn} is defined by xn 1 Fxn, n 0, 1, 2, . . . , then limn→∞xn x∗ ∈ K and d xn, x∗ ≤ k/ 1 − k d x1, x0 . The complete Gamma function Γ α plays an important role in the theory of fractional integral and derivatives. A comprehensive definition of Γ α is provided by the Euler limit as Γ α lim n→∞ n!n α α 1 · · · α n . 2.2 If α > 0, then Γ α has the following familiar integral representation: