Ailing Shi

The method of lower and upper solutions for fractional differential equation Du(t)+ g(t,u(t)) = 0,t 2 (0,1),1 < � � 2, with Dirichlet boundary condition u(0) = a,u(1) = b is used to give sufficient conditions for the existence of at least onesolution.


Introduction
In this paper, we consider the two-point boundary value problem      D δ u(t) + g(t, u) = 0, t ∈ (0, 1), 1 < δ ≤ 2 u(0) = a, u(1) = b, (1.1) where g : [0, 1] × R → R, a, b ∈ R, and D δ is Caputo fractional derivative of order 1 < δ ≤ 2 defined by (see [1]) is the Riemann-Liouville fractional integral of order 2 − δ, see [1].Differential equations of fractional order occur more frequently in different research areas and engineering, such as physics, chemistry, etc.Recently, many people pay attention to the existence of solution to boundary value problem for fractional differential equations, such as [2] − [7], by means of some fixed point theorems.However, as far as we know, there are no papers dealing with the existence of solution to boundary value problem for fractional differential equations, by means of the lower and upper solutions method.The lower and upper solutions method plays very important role in investigating the existence of solutions to ordinary differential equation problems of integer orders, for example, [8] − [11].
In this paper, by generalizing the concept of lower and upper solutions to boundary value problem for fractional differential equation (1.1), we shall present sufficient conditions for the existence of at least one solution satisfying (1.1)

Extremum principle for the Caputo derivative
In order to apply the upper and lower solutions method to fractional differential equation two-point boundary value problem (1.1), we need the following results about Caputo derivative. (2.1) For ε > obtained in (2.1), let us consider the following two cases: Case (i): For case (i), we consider an auxiliary function Because the function f attains its maximum over the interval [0, 1] at the point t 0 , t 0 ∈ (0, 1], the Caputo derivative is a linear operator and D α c ≡ 0(c being a constant), hence, function h possesses the following properties: where c 1 > 0, c 2 > 0 are positive constants, and t ∈ (t, t 0 ).Hence, we have which leads to the relation I 2 ≥ 0 that together with (2.1) complete the proof of the theorem.
We consider case (ii) in the remaining part of the proof.Here, we consider the following auxiliary function where ϕ(t) is infinitely differentiable function on R, defined by and A is a positive constant satisfies By calculation(applying the Law of L'Hospital), we easily obtain that ϕ(t 0 ) = 0, ϕ (t 0 ) = 0, ϕ (t 0 ) = 0 and that, Hence, the Riemann-Liouville fractional integral I 2−α ϕ (t 0 ) ≤ 0. And that, it follows from where c 1 > 0, c 2 > 0 are positive constants, and t ∈ (t, t 0 ).By the same arguments as case (i), we can obtain that which leads to the relation I 2 ≥ 0 that together with (2.1) produce D α h(t 0 ) ≥ 0. Hence, we have which implies that Thus, we complete this proof.
, attain its minimum over the interval [0, 1] at the point t 0 , t 0 ∈ (0, 1].Then the Caputo derivative of the function f is nonnegative at the point t 0 for any α, (2.3) For ε > obtained in (2.3), let us consider the following two cases: Case (i): For case (i), we consider an auxiliary function Because the function f attains its minimum over the interval [0, 1] at the point t 0 , t 0 ∈ (0, 1], the Caputo derivative is a linear operator and D α c ≡ 0(c being a constant), hence, function h possesses the following properties: (2.4) where c 1 > 0, c 2 > 0 are positive constants, and t ∈ (t, t 0 ).And that 3); On the other hand, we have which leads to the relation that together with (2.3) complete the proof of the theorem.
We consider case (ii) in the remaining part of the proof.Here, we consider the following auxiliary function where ϕ(t) is infinitely differentiable function on R, defined by and A is a positive constant satisfies By calculation(applying the Law of L'Hospital), we easily obtain that ϕ(t 0 ) = 0, ϕ (t 0 ) = 0, ϕ (t 0 ) = 0, and that, , so, for t ∈ [0, t 0 ), there is Hence the Riemann-Liouville fractional integral I 2−α ϕ (t 0 ) ≤ 0. Since h ∈ C 2 (0, 1), h(t 0 ) = h (t 0 ) = 0, there are where c 1 > 0, c 2 > 0 are positive constants, and t ∈ (t, t 0 ).Thus, by the same arguments as case (i), we can obtain that which leads to the relation I 2 ≥ 0 that together with (2.3) produce D α h(t 0 ) ≥ 0. Hence, we have Thus, we complete this proof.

Existence result
In this section, we shall apply the lower and upper solutions method to consider the existence of solution to problem (1.1).Definition 3.1 we call a function α(t) a lower solution for problem (1.1) The following theorem is our main result.
Thus, we have Then, the claim is proved and now it is sufficient to prove that problem (3.3) has at least one solution.
From the standard argument, we can know that the solution of (3.3) has the form where EJQTDE, 2009 No. 30, p. 10 In fact, we may consider the solution of the linear problem of (3.3) where ρ(t) ∈ C[0, 1].Applying the fractional integral I δ on both sides of equation in (3.7) and Using the following relationship (Lemma 2.22 [1]): 1).EJQTDE, 2009 No. 30, p. 3