New Sequential Fractional Differential Equations with Mixed-Type Boundary Conditions

where q ∈ ð1, 2 is a real number and k, r > 0, αi, βi, γi, εi ∈R, i = 1, 2, CDq is the Caputo fractional derivative of order q. The nonlinearity term f contains the unknown function and its lower order fractional derivatives. The new boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases which can be regarded as the linear combination of the values of the unknown function and its first derivatives at the end points of interval, and the Riemann-Liouville fractional integral value of the unknown function and its first derivatives at an interior point of interval. Fractional differential equations have attracted significant attention for their wide application in many fields of engineering and applied sciences (see [1–10]). Sequential fractional differential equations as an importance branch have also received wide attention; for instance, see [11–16]. Motivated by the HIV infection model and its application background in [12], the existence and uniqueness of solutions for the following sequential fractional differential system are obtained by means of Leray-Schauder’s alternative and Banach’s contraction principle


Introduction
In this paper, we focus on sequential fractional differential equations with mixed-type boundary conditions.
where q ∈ ð1, 2 is a real number and k, r > 0, α i , β i , γ i , ε i ∈ ℝ, i = 1, 2, C D q is the Caputo fractional derivative of order q.
The nonlinearity term f contains the unknown function and its lower order fractional derivatives. The new boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases which can be regarded as the linear combination of the values of the unknown function and its first derivatives at the end points of interval, and the Riemann-Liouville fractional integral value of the unknown function and its first derivatives at an interior point of interval. Fractional differential equations have attracted significant attention for their wide application in many fields of engineering and applied sciences (see [1][2][3][4][5][6][7][8][9][10]). Sequential fractional differential equations as an importance branch have also received wide attention; for instance, see [11][12][13][14][15][16]. Motivated by the HIV infection model and its application background in [12], the existence and uniqueness of solutions for the following sequential fractional differential system are obtained by means of Leray-Schauder's alternative and Banach's contraction principle where λ i > 0 (i = 1, 2) is a parameter; 2 < p, q ≤ 3, C D p , and C D q are the Caputo fractional derivatives; and the nonlinearity terms f 1 , f 2 : ½0, 1 × ℝ × ℝ ⟶ ℝ are the given continuous function. Antiperiodic boundary conditions arise in the mathematical problems of certain physical phenomena and pro-cesses. Recently, many scholars paid attention to solvability for fractional differential equations involving antiperiodic boundary conditions (see [17][18][19][20][21]). For example, in [21], the authors considered the nonlinear antiperiodic boundary value problems where C D q is the Caputo fractional derivatives of order q, α i , Integral boundary conditions are believed to be more reasonable than the local boundary conditions, which can describe modeling of blood flow, cellular systems, population dynamics, heat transmission, etc. There are a number of results about fractional differential equations and partial differential equations with integral boundary condition; we refer the reader to see [17,20, and the references cited therein. In [20], the authors discussed the following fractional differential equation with integral boundary conditions given by where C D α and C D β are the Caputo fractional derivatives; 0 < α < 1, 1 < β ≤ 2, k > 0, and μ > 0 are real numbers; and f is a given continuous function.
Observing the results of the above literature, an interesting and important question is whether antiperiodic and integral boundary conditions can be unified in a system. If we have unified the conditions, how can we obtain the existence of the solutions? Through a literature search, the sequential fractional differential equation (1) has not been given up to now. Now in this paper, we shall discuss the problem (1) by using the standard tools of fixed point theory and some new inequalities of norm from.

Preliminary and Lemmas
In this paper, we provide some necessary definitions and lemmas of the Caputo fractional calculus; for more information, see the books [1][2][3]. Definition 1. The Caputo derivative of fractional order α > 0 for a n-times continuously differentiable function f : ½0, ∞Þ ⟶ ℝ is defined as where n = ½α + 1 and ½α denotes the integer part of number α.
Definition 2. The Riemann-Liouville fractional integral of order α for a function f is defined as where c i ∈ ℝ, i = 1, 2, ⋯, n, and n is given as in Definition 1.

Lemma 4.
Let hðtÞ ∈ C ½0, 1 and u ∈ C 2 ½0, 1. Then, the following sequential fractional differential equations 2 Journal of Function Spaces have a unique solution where Proof. Using Lemma 3, the general solution of the fractional differential equations ð C D q + k C D q−1 ÞuðtÞ = hðtÞ can be written as where A 0 , A 1 ∈ ℝ. Differentiating (11) with respect to t, we get Applying the boundary condition (8) in (11) and (12), we obtain A simultaneous solution of equation (13) leads to Substituting A 0 and A 1 to (11), we obtain the desired solution in (9). The converse of the lemma follows by direct computation. The proof is completed.
Remark 5. Caputo fractional differentiating (11) with respect to t, we obtain where A 0 is defined as (14). Set C½0, 1 is all the continuous functions on ½0, 1, C q−1 ½ 0, 1 = fu ∈ C ½0, 1: C D q−1 u ∈ C ½0, 1g. Let E = ðC q−1 ½0, 1, k·k q−1 Þ denotes the Banach space endowed with the norm defined by kuk q−1 = kuk + k C D q−1 uk = sup 0≤t≤1 juðtÞj + sup 0≤t≤1 j C D q−1 uðtÞj. For the convenience of the proofs in the next main results, we first give the bounds for integrals arising from the sequel, which are very important for us to establish the existence of solutions for problem (1).
(i) Obviously, we have Journal of Function Spaces Furthermore, Hence, When the proof of (ii) is similar to (i), we omit it. In view of Lemma 4 and Remark 5, replacing hðtÞ by f ðt, uðtÞ, C D q−1 uðtÞÞ in (9) and (15), we transform the solution of problem (1) into the fixed point of operator equation Lemma 7 (see [44]). Let T : E → E be completely continuous (i.e., a map restricted to any bounded set in E is compact). Let Then, either set F is unbounded or F has at least one fixed point.
Lemma 8 (see [45]). Let E be a Banach space, D ⊂ E be closed and F : D ⟶ D a strict contraction, i.e., |Fx − Fy| ≤ k|x − y| for some k ∈ ð0, 1Þ and all x, y ∈ D. Then, F has a unique fixed point.

Main Results
Before starting and introducing the main results, we list our assumption for f : (iii) H2: there exist positive constants a 21 , a 22 , a 23 ∈ ℝ + such that (iv) H3: there exist positive constants a 31 , a 32 , a 33 ∈ ℝ + such that (v) H4: there exist positive constants a 41 , a 42 , a 43 ∈ ℝ + such that (vi) H5: there exist positive constants a 51 , a 52 ∈ ℝ + such that

Journal of Function Spaces
For convenience, we introduce the following symbols: Theorem 9. Suppose that (H0) and (H1) hold. Then, problem (1) has at least one solution.
Proof. We first define a ball in E as Then, we show that T : B R ⟶ B R . For ∀u ∈ B R , using Lemma 6 and the condition (H1), we have From (29) and (30), we obtain This means F : From the formula (19), it is easy to know that operators FðuÞðtÞ, C D q−1 FðuÞðtÞ are continuous on ½0, 1. Now, we show that operator F is equicontinuous. Set Let t 1 , t 2 ∈ ½0, 1ðt 1 < t 2 Þ, we have

Journal of Function Spaces
In (33) and (34), letting t 1 ⟶ t 2 , then, That is, as t 1 ⟶ t 2 , Therefore, FðB R Þ is an equicontinuous set. Furthermore, it is uniformly bounded because of FðB R Þ ⊂ B R . Applying the Arzelà-Ascoli theorem, we can infer that F is a completely continuous operator.
Proof. The proof is similar to Theorem 9, we omit it.
Proof. The proof is similar to Theorem 10, we omit it.