Existence of solutions for a nonlinear fractional order differential equation, Electron

Let D denote the Riemann-Liouville fractional differential operator of order α. Let 1 < α < 2 and 0 < β < α. Define the operator L by L = D − aD where a ∈ R. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem Lu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) = 0.


Introduction
For u ∈ L p [0, T ], 1 ≤ p < ∞, the Riemann-Liouville fractional integral of order α > 0 is defined as For n − 1 ≤ α < n, the Riemann-Liouville fractional derivative of order α is defined Also, when α < 0, we will sometimes use the notation I α = D −α . Define the operator L by L = D α − aD β where a ∈ R. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem Lu(t) + f (t, u(t)) = 0, 0 < t < 1, While much attention has focused on the Cauchy problem for fractional differential equations for both the Reimann-Liouville and Caputo differential operators, see [3,6,8,9,10,11,12,13,14] and references therein, there are few papers devoted to the study of fractional order boundary value problems, see for example [1,2,4,5,15].
In the remainder of this section we present some fundamental results from fractional calculus that will be used later in the paper. For more information on fractional calculus we refer the reader to the manuscripts [9,11,12,13]. In Section 2 we use the properties given below to find an equivalent integral operator to (1), (2). We also state the fixed point theorems that we employ to find solutions. In Section 3 we present our main results.
From (5) and (6) we have that Furthermore, by (4) we see that At this point we need to consider three cases. If β < 1, then In any case, equation (9) simplifies to for some constant c.
We seek a fixed point of an operator associated with (1), (2), using a Nonlinear Alternative of Leray-Schauder type and the Krasnosel'skiȋ-Zabeiko fixed point theorem [7]. For completeness we state these theorems below.
Then T has a fixed point in B.

Main Results
Define Note that fixed points of (13) are solutions of (1), (2) and vice versa.
Assume that the function f satisfies the following conditions. Proof. It follows trivially that T : B → B.
Let {v i } ⊂ B be such that v i → v as i → ∞. By (H 2 ) and the continuity of f we have, Hence T is continuous.   Let v ∈ V and suppose that t 1 , t 2 ∈ [0, 1] are such that t 1 ≤ t 2 . Then, where ε 1 > 0 and ε 2 > 0 are such that |G(t 1 , s) − G(t 2 , s)| ≤ ε 1 |t 1 − t 2 | and |G * (t 1 , s) − G * (t 2 , s)| ≤ ε 2 |t 1 − t 2 | respectively. As such, the operative T is equicontinuous. By the Arzela-Ascoli Theorem, the operator T is compact and the proof is complete. EJQTDE, 2009 No. 71, p. 5 These quantities will be used in our first main theorem. .
Suppose there exists a u ∈ ∂U and a λ ∈ (0, 1) such that u = λT u, then for this u and λ we have which is a contradiction.
By Theorem 2.2, there exists a fixed point u ∈ U of T . This fixed point is a solution of (1), (2) and the proof is complete.
In our next theorem we replace condition (H 2 ) with the following condition.
We use Theorem 2.3 to establish a fixed point for the operator T .
Then there exists a solution of (1), (2). EJQTDE, 2009 No. 71, p. 6 Proof. We use the same operator defined in (13) and note that under condition (H 3 ) standard arguments can be used to show that T is compact.