Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition

In this paper, we apply Krasnosel'skii's cone expansion and compression fixed point theorem to show the existence of at least one positive solution to the nonlinear fractional boundary value problem $D^\alpha_{0^+} u + a(t)f(u)=0$, $0 < t < 1$, $1 < \alpha \le 2$, satisfying boundary conditions $u(0)=D^\beta_{0^+} u(1)=0$, $0\le\beta\le1$.


Preliminary Definitions and Theorems.
We start with the definition of the Riemann-Liouville fractional integral and fractional derivative.
Definition 2.1. Let ν > 0. The Riemann-Liouville fractional integral of a function u of order ν, denoted I ν 0 + u, is defined as provided the right-hand side exists. Moreover, let n denote a positive integer and assume n − 1 < α ≤ n. The Riemann-Liouville fractional derivative of order α of the function u : [0, 1] → R, denoted D α 0 + u, is defined as provided the right-hand side exists.
In the analysis, we will make use of a cone in a Banach space.
Definition 2.2. Let B be a Banach space over R. A closed nonempty subset K of B is said to be a cone provided (i) αu + βv ∈ K, for all u, v ∈ K and all α, β ≥ 0, and (ii) u ∈ K and −u ∈ K implies u = 0.
The following is the well-known cone expansion and compression fixed point theorem. These results will be utilized in this paper. [10]). Let B be a Banach space and let K ⊂ B be a cone in 3. The Green's Function. The Green's function for −D α 0 + u = 0 satisfying the boundary conditions (2) is given by (see [6]) Therefore, u is a solution of (1), (2) if and only if The following Lemma gives properties of the Green's function that are integral to our work. These properties can be utilized when applying many different fixed point theorems, such as the Leggett-Williams fixed point theorem [11], to (1), (2), making this result the major contribution of the paper. Lemma 3.1. Let γ ∈ (0, 1) be fixed. G(t, s) satisfies the following properties: and min Proof. Notice that (4) holds trivially. Next, we show (5) holds. Define which has a maximum value at t = α − 1 α − β . Therefore, (5) holds.
If there exist positive constants r and R, where r < R and Br < AR, such that f satisfies then (1),(2) has at least one positive solution u with r < u ≤ R.
The final result uses the expansive result of Theorem 2.3 to give at least one solution to (1), (2). Theorem 4.2. Suppose that (H1)-(H2) are satisfied and suppose γ ∈ (0, 1) is such that (H3) is satisfied. Let M = |a| ∞ and let A, B ∈ R with If there exist positive constants r and R, where r < R and Ar > BR, such that f satisfies then (1),(2) has at least one positive solution u with r < u ≤ R.
Proof. Again, we define the cone and note that T : K → K and T is completely continuous. Define the open set Ω 1 = {u ∈ B : u < r}. Let u ∈ K ∩ ∂Ω 1 . Then assumption (i) and (5) give   Therefore, T u ≥ u for all u ∈ K ∩ ∂Ω 2 . Therefore, since 0 ∈ Ω 1 ⊂ Ω 2 , part (i) of Theorem 2.3 gives the existence of at least one fixed point of T in K ∩ (Ω 2 \ Ω 1 ). So there exists at least one solution u of (1), (2) with r < u ≤ R.