Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both φ − ðh, eÞ-concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.

In [23], Song and Cui concerned the existence of solutions of nonlinear mixed fractional differential equation with the integral boundary value problem under resonance: where C D α 1 is the left Caputo fractional derivative of order α ∈ ð1, 2, and D β 0+ is the right Riemann-Liouville fractional derivate of order β ∈ ð0, 1. The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems. In recent years, there have been some studies on the existence of solutions for mixed fractional differential equations (see [24,25]).
By using the fixed-point theorem for mixed monotone operators, Jong et al. [26] dealt with the existence of positive solutions of the following multipoint boundary value problems for nonlinear fractional differential equations where D α 0+ , D Based on above works, this paper investigates the existence of solutions for the fractional differential equations where C D α 1− is the right-sided Caputo fractional derivative with 0 < α ≤ 1, D β 0+ is the left-sided Riemann-Liouville fractional derivative with 1 < β ≤ 2 and α + β > 2. Here, f : ½0, 1 × ð−∞,+∞Þ ⟶ ð−∞,+∞Þ is continuous, and b > 0 is a constant real number.
In Section 2, there are some definitions, properties, and lemmas related to this article. Then, we obtain the Green's function and prove some lemmas of Green's function. In Section 3, two important theorems are obtained. In Theorem 13, set P h,e is defined. According to the fixed-point theorem of increasing concave operator, the existence of the unique solution of boundary value problem (3) is obtained. In Theorem 14, set P h is defined, the existence of solutions is also obtained by the fixed-point theorem of mixed monotone operators. In the last section, some examples are given to illustrate the validity of the theorems.

Preliminaries
In this part, we present some basic definitions, properties, and lemmas.
Definition 1 (see [20]). The left-sided and right-sided Riemann-Liouville fractional integrals of order αðα > 0Þ of a function g : ð0,∞Þ ⟶ R are given by where ΓðαÞ is the Gamma function.

Lemma 4.
Let α ∈ (0,1], β ∈ (1,2]. For y ∈ C[0,1], then the unique solution of the fractional differential equation Proof. Applying the right-sided fractional integral I α 1− to both sides of the Equation (7) and by Property 3, we can obtain that where C 0 ∈ R is arbitrary constant. Applying the left-sided fractional integral I β 0+ to both sides Equation (9) above and by Property 3, we can obtain that Journal of Function Spaces where Ci ∈ Rði = 0, 1, 2Þ. By xð0Þ = 0 and D β 0+ xð1Þ = 0, we get C 2 = C 0 = 0, then Finding the derivative of (11), we have Since x′ð1Þ = 0, it follows Exchanging the order of the above double integral, we have Substitute C 1 into (11), we know that Exchange the order of the first double integral in the above formula, we get that Calculated that the Green's function of the fractional differential Equation (7) is The proof is completed.
Consequently, the boundary value problem (3) has a unique solution if and only if xðtÞ satisfies the integral equation: Proof. First, we prove that the function Gðt, sÞ is nonnegative.
For any t, s ∈ ½0, 1, The proof is completed.
Next, we summarize two fixed-point lemmas and some basic concepts in ordered Banach space.
Let ðE, k·kÞ be a real Banach space which is partially ordered by a cone P ⊂ E, i.e., x ≤ y if and only if y − x ∈ P. If x ≤ y and x ≠ y, then we denote x < y or y > x. θ denotes the zero element of E. P is called normal if there exists M > 0 such that, for all x, y ∈ E, θ ≤ x ≤ y implies jjxjj ≤ Mjjyjj; in this case, M is called the normality constant of P. We say that an operator A : E ⟶ E is increasing if x ≥ y implies Ax ≥ Ay [29].
Given h ∈ E and h > θ, we define the set Let e ∈ P and θ ≤ e ≤ h, we define the set Definition 7 (see [29,31]). Let A : P ⟶ P be a given operator. For any x ∈ P and r ∈ ð0, 1Þ, there exists φðrÞ ∈ ðr, 1Þ such that AðrxÞ ≤ φðrÞAx. Then, A is called a generalized concave operator.
Definition 8 (see [29]). Let A : P h,e ⟶ E be a given operator.
Definition 11 (see [28]). Let A : P × P ⟶ E a mixed monotone operator. Assume that for all 0 < t < 1, there exists 0 < σ = σðtÞ < 1 such that holds for all x, y ∈ P; then, A is called a t − σðtÞ mixed monotone model operator.
Lemma 12 (see [28]). Let h > θ. A : P h × P h ⟶ P h is a t − σðtÞ mixed monotone operator. Then, A has exactly one fixed-point x * in P h . Moreover, constructing successively the sequences for any initial point x 0 , y 0 ∈ P h , we have kx n − x * k ⟶ 0 and ky n − x * k ⟶ 0 as n ⟶ ∞.
Then, the fractional differential equations (3) have a unique nontrivial solution x * in P h,e . Moreover, for any given initial value x 0 ∈ P h,e , making the sequence x n = Ax n−1 , n = 1, 2, ⋯, then we obtain kx n − x * k ⟶ 0 as n ⟶ ∞.
Proof. For any ∈½0, 1, it is easy to see eðtÞ ≥ 0, that is, e ∈ P. Further, Hence, 0 ≤ eðtÞ ≤ hðtÞ. Let Þds − e t ð Þ, t ∈ 0, 1 ½ : ð29Þ The boundary (3) has an integral formulation given by So, xðtÞ is the solution of the problem (3) if and only if xðtÞ is the fixed point of the operator of A.
Firstly, it is apparent from the definition of A that A is P h,e ⟶ E.
From (H2), we know that That is to say, l 2 ≥ l 1 > 0. Then, we have l 1 h ≤ Ah + e ≤ l 2 h. So, Ah ∈ P h,e .