Existence Results for Nonlinear Multiorder Fractional Differential Equations with Integral and Antiperiodic Boundary Conditions

In this paper, we study the solvability of a class of nonlinear multiorder Caputo fractional differential equations with integral and antiperiodic boundary conditions. By using some fixed point theorems including the Banach contraction mapping principle and Schaefer’s fixed point theorem, we obtain new existence and uniqueness results for our given problem. Also, we give some examples to illustrate our main results.


Introduction
Fractional calculus has a history of several hundred years, and many valuable results that have contributed to the development of mathematical theories and their application to practice have been created during its historical process (see [1]). Also, fractional differential equations are one of the powerful means to model and solve scientific and technological problems that have been arisen in physics, chemistry, biology, mechanics, and many other fields, and it has developed more and more in-depth (see [2]). In particular, boundary value problems of fractional differential equations are often used as mathematical models for many phenomena in a variety of physical, biological, mechanical, and chemical studies such as analysis of turbulent flow, simulation of chemical reaction, and image processing technique (see [3][4][5][6]).
In recent years, antiperiodic boundary value problems have been put forward in various practical phenomena and have attracted the attention of a large number of researchers because of the specific properties of their solutions (see [7]). Based on many works about the solvability of antiperiodic boundary value problems for integer-order differential equations (see [8][9][10]), a lot of attempts have been made to extend the existence results for them to the case of fractional differential equations (see [11][12][13][14][15][16][17][18][19][20]). For instance, Agarwal and Ahmad [11] established the existence of solutions of the following single-term Caputo fractional differential equations with antiperiodic boundary conditions by using the nonlinear alternative of Leray-Schauder type and Leray-Schauder degree theory: In [17], Choudhary and Daftardar-Gejji considered the antiperiodic boundary value problem of the nonlinear multiorder Caputo fractional differential equation where λ i ∈ R, i = 0, 1, ⋯, n, λ n ≠ 0, 0 ≤ α 0 < α 1 <⋯<α n < 1. They proved the existence and uniqueness of solutions to their problem in terms of the two-parametric functions of Mittag-Leffler type. Their equation in problem (2) is a generalization of the classical relaxation equation and governs some fractional relaxation processes. Analyzing the higher-order fractional differential equations like that in problem (1), some new research papers considered not only antiperiodic boundary conditions but also mixed-type boundary conditions which are composed of both integral and antiperiodic boundary conditions (see [21][22][23][24][25]). Xu [24] obtained new existence and uniqueness results for the following single-term fractional differential equations with integral and antiperiodic boundary conditions by means of the Krasnosel'skii fixed point theorem, contraction mapping principle, and Leray-Schauder degree theory: Taking the previous results together, we can know that very little has been done on the multiterm fractional differential equations with integral and antiperiodic boundary conditions. In particular, as far as we know, the research on the mixed-type boundary value problems of nonlinear multiorder fractional differential equations like that in problem (2) which is of great significance in practice has not been carried out at all.

Derivation of the Integral Equation
The Riemann-Liouville fractional integral and the Caputo fractional derivative of order α > 0 of a function f : ð0,∞Þ ⟶ R are given by where n = ½α + 1, provided that the right-hand sides are pointwise defined on ð0, ∞Þ (see [1]).

Lemma 2. If a function u is a solution of problem (4)-(5), then
in C½0, 1, and conversely, if y ∈ C½0, 1 is a solution of the integral equation (7), then a function u which is given by is a solution of problem (4)-(5), where

Journal of Applied Mathematics
Proof. Let a function u be a solution of problem (4)- (5).
Applying I α 0+ on both sides of the expression yðtÞ = c D α 0+ uðtÞ, it is obvious that (see [24]). We can rewrite (10) as From (11), we have some equalities as follows: Substituting the above c D (4), it can be easily seen that This yields the integral equation (7). Conversely, let a function y ∈ C½0, 1 be a solution of the integral equation (7). Substituting the expression into (7), we can get that On the other hand, using the expression of Green's function Gðt, sÞ, we can see that So, we know that c D α 0+ uðtÞ, c D α i 0+ uðtÞ, and c D β i 0+ uðtÞ exist for any t ∈ ½0, 1 and c D α 0+ u ∈ C½0, 1.

Considering the relations
we can rewrite (15) as That is, u satisfies equation (4). Now, we prove that u satisfies the boundary conditions (5). By simple calculation, we have

Journal of Applied Mathematics
This means that uð1Þ = μ Ð 1 0 uðsÞds. Also, since we can see that u ′ ð0Þ + u ′ ð1Þ = 0. Therefore, it is proved that u satisfies the boundary conditions (5). The proof is completed.

Remark 4.
As we can see from the expression the function Ð 1 0 Gðt, sÞyðsÞds is continuous in ½0, 1 for any y ∈ C½0, 1. Also, for any y ∈ C½0, 1, the following inequality holds: where w α,μ is denoted as

Main Results
Define the operators P and Q on C½0, 1 as follows: Then, the integral equation (7) can be regarded as the operator equation Lemma 5. The following hold: (i) For any α ∈ R + , the fractional integral operator I α
(i) It is sufficient to prove that for any bounded set Ω ≔ fu ∈ C½0, 1jkuk ≤ rg, I α 0+ Ω is relatively compact. Obviously, we can see So, we can know that I α 0+ Ω is uniformly bounded. Also, we have that for any t 1 , t 2 ∈ ½0, 1 ðt 1 < t 2 Þ, This yields that I α 0+ Ω is equicontinuous. Therefore, using the Ascoli-Arzelà theorem, it can be easily seen that I α 0+ Ω is relatively compact.
(ii) It is obvious that for any d ∈ C½0, 1, This completes the proof of (ii).
Proof. As in the proof of Lemma 5, put Ω ≔ fu ∈ C½0, 1jku k ≤ rg. Then, we have that for any u ∈ Ω, This implies that PΩ is uniformly bounded.
On the other hand, we can get that for any t 1 , t 2 ∈ ½0, 1 ðt 1 < t 2 Þ, Since the following inequality holds: This yields that PΩ is equicontinuous. The conclusion then follows from the Ascoli-Arzelà theorem. Lemma 7. The operator Q : C½0, 1 ⟶ C½0, 1 is compact.
Proof. From Lemma 5 and the fact that the composition of the bounded linear operator and compact operator is also compact, every term of the operator Q is compact. Since the sum of two compact operators is also compact, the proof is completed.
Lemma 8 (see [26]). 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 compact operator such that Then, T has a fixed point in Ω.
Denote ω * as follows: Here, we list more hypotheses to be used throughout this paper.
Theorem 9. Assume that hypotheses H1 and H2 hold. If the nonlinear function f satisfies that then problem (4)-(5) has at least one solution.
Proof. Since lim Lemma 10 (Schaefer's fixed point theorem) (see [27]). Let X be a Banach space and A : X ⟶ X a compact operator. Then, either (i) the equation x = λAx has a solution for λ = 1 or (ii) the set of all such solutions x, for 0 < λ < 1, is unbounded.
If k Ð 1 0 Gð⋅ ,sÞuðsÞdsk ≤ r, then because of condition (ii), we have f t, Thus, it follows that And we obtain Therefore, it holds that These inequalities (49) and (52) imply the boundedness of the set S. By using Lemma 10., the operator P + Q has a fixed point. This completes the proof.