Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Nonlinear Analysis Research Group, Ton Duc #ang University, Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc #ang University, Ho Chi Minh City, Vietnam Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia


Introduction
In recent years, we see a great interest to consider fractional differential equations in various kinds of studies. Namely, these equations have striking applications in diverse science. Moreover, during the past decades, new areas for applications of fractional differential and integral equations with initial boundary conditions have been appeared. In the modeling of significant numbers of phenomena occurring in engineering and scientific subjects, we encountered a problem of such kinds. In multiple cases, modeling by fractional operators can also be done better than modeling by ordinary derivative operators. One can see fractional operators in applications of rheology, control, viscoelasticity, porus structures, chemical physics, electrochemistry, and significant number of other branches of science (see [1][2][3] for more details). For these reasons, there exist many articles about the existence and uniqueness of solutions of problems containing operators of fractional orders and using fixed point techniques . But, the number of research studies on the study of existence and uniqueness of solutions for FDEs with two or more terms of fractional operators is limited. Fujita [28] studied the following specific Cauchy problem: and established some existence and uniqueness results. In [29], the author considered an initial value problem of the following form: where n − 1 < β < α < n, and studied existence and uniqueness of related solutions. Based on the results of the above paper, the following exacting conditions should be satisfied on g for the existence of solutions: (C1) g is a continuous and differentiable function on [0, 1] × R ⟶ R. (C2) For t ∈ I, g(t, 0) ≠ 0 and g(0, 0) � 0, where I is any compact subinterval of [0, 1]. (C3) ere exists a continuous function q which satisfies C sup To detect a unique solution other than (C1)-(C3), the following restriction should hold. (C4) Let w 1 and w 2 be any real numbers and q ∈ C( [ 0, 1 ], [ 0, ∞ ) ) be the continuous function appeared in (4), then e authors in [16,30] have studied the same problem. In [16], we observe that the necessary conditions to exist at least one solution for (2) Also, the additional condition that g(t, x(t)) satisfies a Lipschitz condition is needed to prove the uniqueness of the solution: where M is the Lipschitz constant. In [31], the following general form is considered: where n − 1 < β < α < n, (n ∈ N) and D α and D β are the fractional derivative orders in the Caputo definition and g ∈ C([0, 1] × R). We have used an index fixed point theorem and obtained some new results about existence and also multiplicity of solutions. In this paper, we consider the same general form (8) and (9) and try to use new fixed point theorems proved for operators having mixed monotone property to establish new results with a wide range of applications for existence and uniqueness.

Preliminaries
For convenience of readers, we provide some useful definitions and previous results which we use throughout the paper.
Definition 1 (see [1,3]). Let α > 0. e operator defined by where and m ∈ N, is the fractional integral of y of order α in Riemann-Liouville definition.
After introducing the fractional integral operator, it is time to remind the fractional differentiation definitions.
ere are several ways to define fractional derivative. Here, we focus on the Caputo definition.
Definition 2 (see [1,3]). Let n � [α] + 1 where α ≥ 0. If y(t) ∈ AC n [0, 1], then the fractional derivative of y in Caputo sense exists on the interval [0, 1], almost everywhere, and is obtained by For the sake of convenience, here, we mention some properties of Caputo fractional operator. For the power function f(t) � t q , q ≥ 0, taking the Caputo derivative yields By virtue of this formula, one can deduce erefore, unlike Riemann-Liouville derivative, the Caputo derivative of a constant function is zero. In the following equality: the polynomial's degree is less than or equal to m − 1, i. e, r ≤ m − 1. From the following relation, it is clear that similar to the case of integer order, the α th fractional integral of the Caputo fractional derivative of order α requires to know the values of the function and its integer order derivatives: e results of the paper about the existence and uniqueness are expressed by the famous beta function given as follows.

Definition 3.
e following integral defines the beta function: Journal of Mathematics It is known that where Γ denotes the Gamma function. e following lemma transforms problems (8) and (9) to an integral equation [29,30], at which the nonlinear term g needs to satisfy the conditions (C1) and (C2).
Lemma 1 (see [29]). Let n ∈ N, β < α and α, β ∈ (n − 1, n). If (C1) and (C2) hold, then y ∈ C n [0, 1] is a solution of the following problem: if and only if where w is a continuous function and satisfies the following integral equation: where with Here and anywhere below, the function K(x, r) is the Green function. (17) and (18) if and only if where w ∈ C[0, 1] satisfies the following integral equation: Now, we present the following lemma which will be useful in proving an existence and uniqueness theorem of the solution of (8) and (9). G(t, s), the following relations hold:

Lemma 3. For the Green function
Proof. By making use of (14), it is easy and straightforward to prove the properties for K(x, r). Now, we provide some definitions and a fixed point theorem involving mixed monotone operators. We suggest references [19,20] for more details. If there exists a constant C > 0 such that, for any x, y ∈ P with θ ≤ x ≤ y, we have ‖x‖ ≤ C‖y‖, then P is called a normal cone.
en, we give the definition of a partially ordered Banach space. We say that E is partially ordered by P if x ≤ y iff y − x ∈ P, for any x, y ∈ P. For arbitrary x 1 , x 2 ∈ E, the ordered interval is defined by [x 1 , Let us briefly recall that the operator T: P × P ⟶ P is increasing in its first variable and decreasing in its second variable, if from x i , y i ∈ P (i � 1, 2), x 1 ≤ x 2 , y 1 ≥ y 2 , one can deduce T(x 1 , y 1 ) ≤ T(x 2 , y 2 ). If x ≥ θ implies that T(x) ≥ θ, then the operator T is called positive.
In the following definition, P is a cone on the Banach space (E, ‖.‖).
Definition 5 (see [19,20]). Suppose T: P × P ⟶ P is increasing in its first variable and decreasing in its second variable, then T is called a mixed monotone operator.
If for an element x ∈ P, T(x, x) � x, then x is called a fixed point of T.

Main Results
In this section, we prove new existence and uniqueness results for nonlinear fractional differential equations with two terms of fractional derivatives. Compared with the results of the previous studies, our results are simple and straightforward. ey are expressed with the values of beta and gamma functions. For these reasons, these results can be applied for a wide range of problems. To make effective our obtained result, some examples provided at the end of section cannot be considered by techniques of [16,30]. In order to apply the fixed point results concerning the mixed monotone operators for the study of (8) and (9), we first consider E � C([0, 1]) as a suitable Banach space, and let the norm on E be defined by Let us consider It is obvious that P is a normal cone in E. In the following theorem, g is the function appeared in the right side of equation (8).

Theorem 2. Assume that (C6) g(t, u, v) is a continuous function which is in-
creasing in its first variable u and decreasing in its second variable v. (C7) For any η ∈ (0, 1), let u and v be two arbitrary elements of P so that there exists λ(η, u, v) ∈ (1, ∞) such that (C8) Let u 0 and v 0 be two elements of P and μ ∈ (0, 1) such that .
Proof. First, we define the operator T as follows: where K(t, r) is the Green function (21). Taking the definition of the operator T into account and Lemma 3, it is clear that a certain function u(t) is a solution of (2) and (3) iff u(t) is a fixed point of T, that is, T(u, u) � u. Hence, it will be sufficient to show that T satisfies the conditions of eorem 1. By virtue of (C6), it immediately follows that T is increasing in its first variable u and decreasing in its second variable v. On the other hand, under the hypothesis of (C7), if η ∈ (0, 1) and u, v are two elements of P, then there exists λ(η, u, v) ∈ (1, ∞) such that Also, with the help of (C8), and using Lemma 3, we can choose u 0 and v 0 such that Consequently, it is obvious that all essentials of eorem 1 are provided by (C6)-(C8), and therefore, there is a unique 4 Journal of Mathematics positive solution (u * , u * ) so that T(u * , u * ) � u * . It is the unique solution of (8) and (9). e proof is complete. □ Example 1. Let us consider the following set of nonlinear differential equations of fractional order c D m+ (3/4) subject to where m is a positive integer, ψ(t) is a continuous function such that (HTML translation failed), and In connection with eorem 2, set α � m + (3/4) and β � m + (1/4). By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (37) and (38) is equivalent to a unique solution for the following integral equation: where with the help of Lemma 3, one can deduce erefore, using eorem 2, we conclude that problems (37) and (38) possess a unique positive solution such that 1 ≤ y(t) ≤ m.
Example 2. In this example, we find a unique positive solution for the nonlinear fractional initial value problems of the following form: where n ∈ Z + , χ(t) is a continuous function such that χ(0) ≠ 0, and τ 1 � 315/100 ≤ χ(t) ≤ 945/100 � τ 2 . In connection with eorem 2, set α � n + (2/3) and β � n + (1/3). By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (41) and (42) is equivalent to a unique solution for the following integral equation: where with the help of Lemma 3, one deduces 2 ≤ 1 0 K(t, r)dr ≤ 53/10. We note that if u, v ∈ P are defined by (30) increasing in u and decreasing in v.

Conclusion
In this research, we considered nonlinear fractional differential equations of arbitrary order with two terms of fractional orders. By virtue of recent fixed point theorems on normal cones for mixed monotone operators, we obtained some new results for existence and uniqueness of solutions of these equations involving initial conditions, which are more applicable compared with previous existing results.
e previous results can be proved based on very strict conditions on the function g. While, in our paper, the function g(t, u, D α u) ≡ g(t, u, v) needs to be increasing in its first variable u and decreasing in its second variable v. Also, establishing the conditions (C7) and (C8) is not difficult. On the other hand, our results provided a constructive approach, based on an iterative relation, to find the solution. Indeed, the interval (0, 1] can be generalized to (0, T]. e presented examples, at the end of the paper, cannot be considered by the previous works.