sequential fractional differential equations

In this paper, we shall discuss the properties of the well-known Mittag–Leffler function, and consider the existence of solution of the periodic boundary value problem for a fractional differential equation involving a Riemann–Liouville sequential fractional derivative by means of the method of upper and lower solutions and Schauder fixed point theorem.


.). (1.2)
There is a close connection between the sequential fractional derivatives and the non sequential Riemann-Liouville derivatives.For example, in the case k = 2, 0 < α < 1/2 and the Riemann-Liouville derivatives, the relationship between D kα a+ y and D kα a+ y is given by (D 2α a+ y)(x) = D 2α a+ y(t) − (I 1−α a+ y)(a+) (t − a) α−1 Γ(α) (x).(1.3)We shall consider the existence of solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville sequential fractional derivative, by using the method of upper and lower solutions and Schauder fixed point theorem.
(1.5) Differential equations of fractional order occur more frequently in different research areas and engineering, such as physics, chemistry, control of dynamical systems etc.Recently, many researchers paid attention to existence result of solution of the initial value problem for fractional differential equations, such as [4][5][6][7].Some recent contributions to the theory of fractional differential equations can be seen in [8][9][10][11][12].
In [4], the existence and uniqueness of solution of the following initial value problem for a fractional differential equation was discussed by using the method of upper and lower solutions and its associated monotone iterative.
In [5], the global existence results for an initial value problem associated to a large class of fractional differential equations was presented by means of a comparison result and the fixed point theory.In [7], the authors considered the existence of minimal and maximal solutions and uniqueness of solution of the initial value problem for a fractional differential equation involving a Riemann-Liouville sequential fractional derivative, by using the method of upper and lower solutions and its associated monotone iterative method.
where 0 < T < +∞, and While for the existence of solution of the periodic boundary value problem (1.4) for a fractional differential equation a involving Riemann-Liouville sequential fractional derivative EJQTDE, 2011 No. 87, p. 2 has not been given up to now, the research proceeds slowly and appears some new difficulties in obtaining comparison results.Now, in this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence of solution of the periodic boundary value problem (1.4) for a fractional differential equation involving Riemann-Liouville sequential fractional derivative by using the method of upper and lower solutions and Schauder fixed point theorem. Let Definition 1.1.We call a function y(x) a classical solution of problem (1.4), if: (i) y(x) ∈ C α 1−α ([0, T ]) and its fractional integral (I 1−α y(t))(x), (I 1−α D α 0+ y(t))(x) are continuously differentiable for (0, T ]; (ii) y(x) satisfies problem (1.4).
For problem (1.4), we have the following definitions of upper and lower solutions.
Analogously, a function q ∈ C α 1−α ([0, T ]) is called an upper solution of problem (1.4), if it satisfies 0+ q)(x) ≥ f (x, q, D α 0+ q), x ∈ (0, T ], In what follows, we assume that p(x) ≤ q(x), x ∈ (0, T ] : and define that the ordered interval in space (1.9) The following is an existence result of the solution for the linear periodic boundary value problem for a fractional differential equation and a property of Riemann-Liouville fractional EJQTDE, 2011 No. 87, p. 3 calculus, which are important for us to obtain existence of solutions for problem (1.4).
and the solution given by (1.11) is valid (it is the classical solution using the variation of constants formula).
Lemma 1.2 .Suppose that u ∈ C 1−α ([0, T ]), then the linear periodic boundary value problem where M ∈ R is a constant and σ ∈ C 1−α [0, T ], has the following integral representation of solution (1.15) Proof By Lemma 1.1, we have that the linear initial value problem (1.14) has the integral representation of solution (1.11).By the condition of periodic boundary value problem (1.14), we have Substituting (1.16) into (1.11),we obtain (1.15).The proof of Lemma 1.2 is completed.where Then the problem (1.17) is equivalent to and (1.24) By the Lemma 1.2, we have that the linear periodic boundary value problems (1.23) and (1.24) have the following representation of solutions where ȳ0 , u 0 are given by (1.21) and (1.22).Substituting (1.25) into (1.26),we obtain (1.18).The proof of Lemma 1.3 is completed.
[e α (λ 2 , t) * e α (λ This paper is organized as follows.In Section 2 we give some preliminaries, including a property of Mittag-Leffler function which will be used in our main result, a comparison result.The main results are established in Section 3.
Hence, lim Therefore, 0 < F (x) < α −x for x < 0, and lim The following results will play a very important role in this paper. where . By the formula (1.11) of Lemma 1.1, we obtain that w(t) ≥ 0, t ∈ (0, T ]. Remark 2.1 In this result, we delete the condition M > − Γ(1 + α) T α of the Lemma 2.1 of paper [4], so this result is an essential improvement of the paper [4].

Main results
On the basis of Lemmas 1.2-1.4 and 2.3-2.6, using the method of upper and lower solutions and Schauder fixed point theorem, we shall show the existence theorem of solutions for PBVP (1.4).For convenience, we list the following conditions: p, q ∈ C α 1−α ([0, T ]) are lower and upper solutions of problem (1.4); (H2): there exist constants N > 0, M ∈ R, N 2 > 4M such that (H1) holds, and for where In view of (3.2), the function Lemma 3.1.Let (H1) be satisfied.Then Hence, where λ 2 < 0 is given by (3.4).