On Antiperiodic Boundary Value Problems for Higher-Order Fractional Differential Equations

and Applied Analysis 3 Lemma 2.3. For any y ∈ C 0, T , the unique solution of the boundary value problem: Dx t y t , t ∈ 0, T , 4 < q ≤ 5, x 0 − x T , x′ 0 −x′ T , x′′ 0 −x′′ T , x′′′ 0 − x′′′ T , x iv 0 −x iv T 2.3


Introduction
In the preceding years, there has been a great advancement in the study of fractional calculus.A variety of results on initial and boundary value problems of fractional order, ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, have appeared in the literature.It is mainly due to the extensive application of fractional differential equations in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, and fitting of experimental data 1-5 .For an updated account of mathematical tools for fractional models and methods of solutions for fractional differential equations, we refer the reader to a recent text 6 by Baleanu et al.Fractional derivatives are also regarded as an excellent tool for the description of memory and hereditary properties of various materials and processes 7 .These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models.For more details and examples, see 8-20 .Antiperiodic boundary value problems occur in the mathematical modeling of a variety of physical processes and have received a considerable attention.Examples include where c D q denotes the Caputo fractional derivative of order q and f is a given continuous function.
The main objective of the present work is to develop the existence theory for problem 1.1 and relate problem 1.1 with lower-order fractional antiperiodic boundary value problems.Our results are new and give further insight into the characteristics of fractionalorder antiperiodic boundary value problems.

Preliminaries
Definition 2.1 see 4 .The Riemann-Liouville fractional integral of order q for a continuous function g is defined as provided the integral exists.
Definition 2.2 see 4 .For at least n-times continuously differentiable function g : 0, ∞ → R, the Caputo derivative of fractional order q is defined as where q denotes the integer part of the real number q.
To study the nonlinear problem 1.1 , we need the following lemma, which deals with a linear variant of problem 1.1 .

Lemma 2.3. For any y ∈ C 0, T , the unique solution of the boundary value problem:
c D q x t y t , t ∈ 0, T , 4 < q ≤ 5, where G t, s is the Green's function given by 48Γ q − 4 , 0 < t < s < T.

2.5
Proof.It is well known 4 that the solution of c D q x t y t can be written as where G t, s is given by 2.5 .This completes the proof.

Relationship with Lower-Order Problems
We observe that the first term in expressions for G t, s given by 2.5 corresponds to the Green's function for the problem: the first two terms in 2.5 form Green's function for the problem 21 : the first three terms in 2.5 give the Green's function for the problem 22 : while the first four terms in 2.5 yield the Green's function for the antiperiodic problem 23 : From the above deductions, it can easily be concluded that Green's function 2.5 for an antiperiodic boundary value problem of fractional order q ∈ 4, 5 contains Green's function or solution for lower-order fractional antiperiodic problems.We can further interpret that the last term in expressions for Green's function 2.5 arises due to consideration of the order q ∈ 4, 5 , whereas the remaining terms correspond to the lower-order problems.This observation gives a useful insight into the study of antiperiodic fractional boundary value problems that a unit-increase in the fractional order of the problem gives rise to a new term in expressions for Green's function, preserving the terms corresponding to lower-order antiperiodic problems.In other words, one can say that Green's function or solution for a higher-order antiperiodic fractional boundary value problem inherits all the characteristics of lower-order fractional antiperiodic problems.Hence, our results generalize the existing results on antiperiodic fractional boundary value problems 21-23 .

Existence Results
Let E : C 0, T , R denotes the Banach space of all continuous functions defined on 0, T ×R endowed with a topology of uniform convergence with the norm x sup t∈ 0,T |x t |.To prove the existence results for problem 1.1 , we need the following known results 28 .
Theorem 3.1.Let X be a Banach space.Assume that T : X → X is completely continuous operator and the set Theorem 3.2.Let X be a Banach space.Assume that Ω is an open-bounded subset of X with θ ∈ Ω and let T : Ω → X be a completely continuous operator such that Tu ≤ u , ∀u ∈ ∂Ω.

3.2
Then T has a fixed point in Ω.

3.3
Observe that the problem 1.1 has a solution if and only if the operator equation Ux x has a fixed point.

Theorem 3.3. Assume that there exists a positive constant
Then the problem 1.1 has at least one solution.
Proof.First of all, we show that the operator U is completely continuous.Note that the operator U is continuous in view of the continuity of f.Let B ⊂ E be a bounded set.By the assumption that |f t, x | ≤ L 1 , for x ∈ B, we have Abstract and Applied Analysis 7 which implies that Ux ≤ L 2 .Further, we find that

3.6
This implies that U is equicontinuous on 0, T .Thus, by the Arzela-Ascoli theorem, the operator U : E → E is completely continuous.Next, we consider the set and show that the set V is bounded.Let x ∈ V , then x μUx, 0 < μ < 1.For any t ∈ 0, T , we have 3.9 Thus, x ≤ M 1 for any t ∈ 0, T .So, the set V is bounded.Thus, by the conclusion of Theorem 3.1, the operator U has at least one fixed point, which implies that 1.1 has at least one solution.

3.10
Then the problem 1.1 has at least one solution.
Proof.Let us define B τ {x ∈ E | x < τ} and take x ∈ E such that x τ, that is, x ∈ ∂B τ .As before, it can be shown that U is completely continuous and which in view of 3.10 yields Ux ≤ x , x ∈ ∂B τ .Therefore, by Theorem 3.2, the operator U has at least one fixed point, which in turn implies that the problem 1.1 has at least one solution.
Our next existence result is based on Krasnoselskii's fixed point theorem 29 .
Theorem 3.5.Let M be a closed convex and nonempty subset of a Banach space X.Let A and B be the operators such that (i) Ax By ∈ M whenever x, y ∈ M; (ii) A is compact and continuous; (iii) B is a contraction mapping.Then there exists z ∈ M such that z Az Bz.
Theorem 3.6.Let f : 0, T × R → R be a jointly continuous function.Further, we assume that Then the problem 1.1 has at least one solution on 0, T if

3.15
Thus, U 1 x U 2 y ∈ B r .It follows from the assumption A 1 that U 2 is a contraction mapping for

3.16
Continuity of f implies that the operator U 1 is continuous.Also, U 1 is uniformly bounded on B r as 3.17 Now we prove the compactness of the operator U 1 .In view of A 1 , we define and consequently, for t 1 , t 2 ∈ 0, T with t 1 < t 2 , we have which is independent of x and tends to zero as t 2 − t 1 → 0. So U 1 is relatively compact on B r .Hence, By the Arzela-Ascoli theorem, U 1 is compact on B r .Thus all the assumptions Theorem 3.5 are satisfied.Therefore, the conclusion of Theorem 3.5 applies and the antiperiodic fractional boundary value problem 1.1 has at least one solution on 0, T .This completes the proof.Theorem 3.7.Assume that f : 0, T × R → R is a jointly continuous function satisfying the condition q q − 1 5q 2 − 9q 46 384 .

3.22
Then the antiperiodic boundary value problem 1.1 has a unique solution.
Then we show that UB r κ ⊂ B r κ , where B r κ {x ∈ E : x ≤ r κ }.For x ∈ B r κ , we have where 3.22 is used.Now, for x, y ∈ E, we obtain

y s ds
Abstract and Applied Analysis 13 x 0 −x 1 , x iv 0 −x iv 1 ,
Clearly, |f t, x | ≤ L 1 e 1 2 ln 22 , and the hypothesis of Theorem 3.3 holds.Therefore, the conclusion of Theorem 3.3 applies to problem 3.25 .
For sufficiently small x ignoring x 2 and higher powers of x , we have

3.31
Thus, all the assumptions of by Theorem 3.7 are satisfied.Hence, the fractional boundary value problem 3.30 has a unique solution on 0, π .

2 . 7
Substituting the values of b o , b 1 , b 2 , b 3 , and b 4 in 2.6 , we obtain