Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

and Applied Analysis 3 Definition 2.2. For f, g ∈ E the order interval 〈f, g〉 is defined as 17 〈 f, g 〉 { h ∈ E : f ≤ h ≤ g. 2.1 Definition 2.3. A subset E ⊂ Π is called order bounded if E is contained in some order interval. Definition 2.4. A coneK is called normal if there exists a positive constant μ such that f, g ∈ V and θ ≺ f ≺ g implies that ‖f‖ ≤ μ‖g‖. We state the two fixed point results which will be needed in this paper. Our first result is a nonlinear alternative of Leray-Schauder type in a cone whereas our second is Krasnoselskii‘s fixed point theorem. Theorem 2.5 Leray-Schauder Theorem . Let E be a Banach space with C ⊆ E closed and convex. Assume that U is relatively open subset of C with 0 ∈ U and W : U → C is a continuous, compact map. Then either i W has fixed point inU or ii there exist u ∈ ∂U and γ ∈ 0, 1 with u γWu. Theorem 2.6 Krasnoselskii‘s fixed point theorem 16 . Let E E, ‖ · ‖ be a Banach space and letK ⊂ E be a cone in E. Assume thatΩ1 andΩ2 are open subsets of E with 0 ∈ Ω1 andΩ1 ⊂ Ω2 and letW : K ∩ Ω2 \Ω1 → K be continuous and completely continuous. In addition suppose that either i ‖Wu‖ ≤ ‖u‖ for u ∈ K ∩ ∂Ω1 and ‖Wu‖ ≥ ‖u‖ for u ∈ K ∩Ω2 or ii ‖Wu‖ ≤ ‖u‖ for u ∈ K ∩ ∂Ω1 and ‖Wu‖ ≤ ‖u‖ for u ∈ K ∩Ω2. Then, W has a fixed point in K ∩ Ω2 \Ω1 . In this paper the Beta function B α, β is used also. B α, β is closely related to the Gamma function 1 . If α, β > 0, then it has the integral representation