Periodic boundary value problem involving sequential fractional derivatives in Banach space

: In this paper, by the method of upper and lower solutions coupled with the monotone iterative technique, we investigate the existence and uniqueness results of solutions for a periodic boundary value problem of nonlinear fractional di ﬀ erential equation involving conformable sequential fractional derivatives in Banach space. An example is given to illustrate our main result.


Introduction
In this work, we consider the following periodic boundary value problem (PBVP for short) in a Banach space x(t), D α x(t)), t ∈ (0, 1], 0 < α ≤ 1, where f (t, x, y) is a continuous E-value function on [0, 1] × E × E, D α is the conformable fractional derivative of order α, D 2α = D α D α is the conformable sequential fractional derivative. Sequential fractional derivative for a sufficiently smooth function g(t) due to Miller and Ross [1] is defined as D δ g(t) = D δ 1 D δ 2 . . . D δ k g(t), here δ = (δ 1 , δ 2 , . . . , δ k ) is a multi-index. In general, the operator D δ can be Riemann-Liouville or Caputo or any other kind of differential operators. There is a close connection between the sequential fractional derivatives and the non-sequential derivatives [2]. Many research papers have appeared recently concerning the existence of solutions for fractional differential equations involving Riemann-Liouville or Caputo sequential fractional derivatives by techniques of nonlinear analysis such as fixed point theorems, coincidence degree continuation theorems and nonlinear alternatives, see, for example, the papers [3][4][5][6][7][8][9][10][11] and the references therein.
In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives due to their wide range of applications in varied fields of science and engineering. Generally, for most of the fractional differential equations, it is difficult to find exact solutions in closed forms. In most cases, only approximate solutions or numerical solutions can be expected. Therefore, many iterative methods have been designed to be one of the suitable and successful classes of numerical techniques for obtaining the solutions of numerous types of fractional differential equations, see, for instance, [12][13][14][15][16] and the references therein. The monotone iterative technique, combined with the method of upper and lower solutions, is an effective technique for proving the existence of solutions for initial and boundary value problems of nonlinear differential equations. The basic idea of this method is that by choosing upper and lower solutions as two initial iterations, one can construct the monotone sequences for a corresponding linear equation and that converge monotonically to the extremal solutions of the nonlinear equation. Not only does this method give constructive proof for existence theorems but also the monotone behavior of iterative sequences is useful in the treatment of numerical solutions of various initial and boundary value problems. So many authors developed the upper and lower solutions method to investigate fractional differential equations, see, for example, [17][18][19][20][21][22][23] and the references therein.
In [2], using the method of upper and lower solutions and its associated monotone iterative technique, the authors considered the existence of minimal and maximal solutions and uniqueness of solution of the following initial value problem for fractional differential equation involving Riemann-Liouville sequential fractional derivative The nonlinear impulsive fractional differential equation with periodic boundary conditions is studied in [24], where D α is the standard Riemann-Liouville fractional derivative and D 2α = D α D α is the Riemann-Liouville sequential fractional derivative. I j , I j ∈ C(R, R), j = 1, 2, . . . , m. f is continuous at every point (t, u, v) ∈ [0, 1] × R × R. The existence and uniqueness results of solutions are obtained by the method of upper and lower solutions and its associated monotone iterative technique.
A new simple well-behaved definition of the fractional derivative called conformable fractional derivative has been presented very recently in [25]. This new definition is a natural extension of the usual derivative, and it satisfies some similar properties to the integer order calculus such as derivative of the product of two functions, derivative of the quotient of two functions and the chain rule. In [26] the author developed further the definitions and properties of conformable fractional derivative and integral.
The existence of solutions for periodic boundary value problem of impulsive conformable fractional integro-differential equation . . , m, is studied in [33], where t k D α denotes the conformable fractional derivative of order α starting from . By the method of upper and lower solutions in reversed order coupled with the monotone iterative technique, some sufficient conditions for the existence of solutions are established.
In [42], applying the upper and lower solutions method and the monotone iterative technique, the authors investigated the existence of solutions to antiperiodic boundary value problem for impulsive conformable fractional functional differential equation where f ∈ C(J × R 2 , R), t k D α denotes the conformable fractional derivative of order α starting from t k , ω ∈ C(J, J + ), J + = [−r, T ], r > 0, t − r ≤ ω(t), t ∈ J and t k < ω(t) ≤ t, t ∈ (t k , t k+1 ], I k ∈ C(R, R).
However, to the best of our knowledge, the existence of minimal and maximal solutions and uniqueness of solution for fractional PBVP (1.1) involving conformable sequential fractional derivatives in ordered Banach spaces have not been considered up to now. Inspired by above works, we apply the theory of noncompactness measure and the method of upper and lower solutions coupled with the monotone iterative technique to construct two monotone iterative sequences, and then prove that the sequences converge to the extremal periodic solutions of PBVP (1.1), respectively, under some monotonicity conditions and noncompactness measure conditions of f . Also, we prove minimal and maximal solutions are equal and thus we obtain the uniqueness of solution. Moreover, we give the existence and uniqueness results of periodic solutions of the non-sequential fractional differential equation.

Preliminaries
In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper. Let J = [0, 1] and E be an ordered Banach space with the norm · and the partial order "≤", whose positive cone P = {x ∈ E, x ≥ θ} is normal with normal constant L, where θ denotes the zero element of E. Generally, C(J, E) denotes the ordered Banach space of all continuous E-value functions on the interval J with the norm x c = max t∈J x(t) and the partial order "≤" deduced by the positive cone Property 2.2. The Kuratowski measure of noncompactness satisfies some properties (for more details see [45]).
Denote the Kuratovski noncompactness measures of bounded sets in C(J, E) and C α (J, E) by µ c and µ c α , respectively. For any H ⊂ C(J, E) and t ∈ J, set We can deduce the following useful result by Lemma 2.3.

Lemma 2.4. Let H ⊂ C α (J, E) be bounded and equicontinuous. Then
Proof. Firstly, we prove that µ c α (H) ≤ d =: max{max t∈J µ(H(t)), max t∈J µ(D α H(t))}. Noting that H ⊂ C(J, E) and D α H ⊂ C(J, E) are bounded and equicontinuous, by Lemma 2.3, we know where diam c (·) denotes the diameter of the bounded subset of C(J, E). At the same time, for any Similarly, for y 1 , y 2 ∈ W j , we have On the other hand, for any ε > 0, there exist Hence, for any t ∈ J and any x 1 , x 2 ∈ H i , i = 1, 2, . . . , k, we have The following lemma also will be used in the proof of our main results.
Conformable calculus satisfies the following properties.
Property 2.6. ( [25]) Let α ∈ (0, 1] and f, g be conformable differentiable of order α. Then Property 2.7. By the definition of conformable fractional integral, it is easy to know I α f : Lemma 2.9. ( [35,39]) For 0 < α ≤ 1, the general solution of the fractional nonhomogeneous equation is expressed by where C is a constant. Further, the unique solution of the linear initial value problem has the following form From Lemma 2.9 we can obtain the solution of linear boundary value problem.
Lemma 2.10. For σ(t) ∈ C(J) and 0 < α ≤ 1, the unique solution x ∈ C(J) of the linear boundary value problem has the following form The continuity of the solution x(t) in Lemma 2.10 is guaranteed by Property 2.7.
Remark 2.11. It is easy to know from the expression of G λ (t, s) that for λ < 0, . Furthermore, from Lemma 2.10 we can deduce the solution of linear PBVP with sequential derivative. where and By Lemma 2.10, we obtain that the problems (2.8) and (2.9) have the following representation of solutions and respectively. Substituting (2.11) into (2.10), we get (2.7).
The following comparison result plays an important role in the proofs of our main results.
Then x(t) ≥ 0 on J.

Main results
and w is called an upper solution of PBVP (1.1) if it satisfies In the following, we assume that v(t) ≤ w(t), t ∈ J. Define the ordered interval in space C(J, E) . We work with the following conditions on the function f in (1.1).
(H1) There exist constants M, N > 0 with M 2 ≥ 4N such that where v, w ∈ C α (J, E) are lower and upper solutions of (1.1).
This completes the proof of Theorem 3.3. By (3.15) and the normality of cone P, we can derive that (3.16) From (3.16) and the definition of Kuratowski measure of noncompactness, it follows that If {x n } ⊂ [v, w] is a decreasing sequence and {y n } ⊂ [D 1 (t), D 2 (t)], the above inequality is also valid. Hence (H3) is satisfied. Therefore, Theorem 3.3 asserts that the PBVP (1.1) has minimal and maximal solutions p and q between v and w. In the following we show that p = q.
Thus, p = q by (H4), which means that there exists a unique solution of PBVP (1.1) in [v, w]. This completes the proof of Theorem 3.4.
Remark 3.5. Using the methods of our main results Theorem 3.3 and Theorem 3.4, we can easy to obtain the existence of solutions of the following PBVP for fractional differential equation x(t)), t ∈ (0, 1], 0 < α ≤ 1, Let v, w ∈ C(J, E). We say that the function v is a lower solution of problem (3.23) if Analogously, w is an upper solution for problem (3.23) if it verifies similar conditions for the inequalities reversed.
(G1) There exists a constant N > 0 such that f (t,   .23) and v ≤ w. The conditions (G1) and (G2) are valid. Then there exist p(t), q(t) ∈ C(J, E) such that p(t), q(t) are minimal and maximal solutions on the ordered interval [v, w] for BPVP (3.23), respectively, that is, for any solution x(t) of BPVP (3.23) such that x ∈ [v, w], we have v(t) ≤ p(t) ≤ x(t) ≤ q(t) ≤ w(t), t ∈ J.