Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

In this paper, we discuss a class of fractional semilinear integrodi ﬀ erential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the ﬁ xed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.


Introduction
Fractional calculus can be used to describe some nonclassical phenomena in natural science and engineering applications. Fractional differential equations have been applied in different fields ranging from engineering, finance, and physics in the past few decades. Researchers have conducted extensive explorations on this subject and have achieved fruitful results for the fractional differential equations [1][2][3][4][5][6][7][8][9][10][11][12][13]. Zhu and Han [10] and Chadha and Pandey [11] studied the fractional integrodifferential equations and discussed the existence of mild solutions. Based on the theory of the resolvent family and fixed point theorems, Chen et al. [14][15][16][17] analyzed nonautonomous evolution equations in a Banach space. Moreover, some researchers considered sufficient conditions on the existence of mild solutions for fractional differential equations by the measure of noncompactness [4,18,19]. The initial boundary value problem for the fractional integrodifferential equations with delay has been investigated by using fixed point theorems [4,5,18,20]. In [3,[21][22][23][24], differential equations of mixed type have been studied and some results have been concluded.
Chen [22] studied the fractional nonautonomous evolution equations of mixed type: where the kernels K and H are linear functions. The operator T is an integral with a variable upper limit, and the operator S is an ordinary definite integral; accordingly, problem (1) is called fractional semilinear integrodifferential equations of mixed type. Li and Jia [25] investigated the existence of mild solutions for abstract delay fractional differential equations: is the Riemann-Liouville fractional integral, the linear operator A is independent on t, and the Lipschitz coefficient of f is constant.
To the best of our knowledge, there are no results on the fractional integrodifferential equations of mixed type with delay. Motivated by this idea, we consider the following problem: where K : D × Cð½−r, 0 ; EÞ ⟶ E and H : D 0 × Cð½−r, 0 ; EÞ ⟶ E are continuous and nonlinear functions, D = fðt, sÞ , f is to be specified later, and x t means the element of Cð½−r, 0 ; EÞ defined by x t ðθÞ = xðt + θÞ, −r ≤ θ ≤ 0, for x ∈ Cð½−r, T 0 ; EÞ, t ∈ J. We demonstrate the existence and uniqueness of global mild solutions for problem (4) under the conditions of the compact resolvent operator and noncompact resolvent operator, respectively. The kernels K and H of the operators K and H are nonlinear functions. In addition, the operator A ðtÞ is dependent on t: The rest of this paper is organized as follows. Basic definitions and auxiliary results are presented in Section 2. In Section 3, we prove the existence and uniqueness of mild solutions via various fixed point theorems, the measure of noncompactness, and the Banach contraction mapping principle. An example is provided to illustrate the main theorems in Section 4. Finally, Section 5 is the summary of our results.
Remark 2 [25]. D Definition 3 [26,27]. The Caputo fractional derivative of order β > 0 of a function f : ð0,∞Þ ⟶ R is given by Remark 4 [25]. For the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, the following conclusions are obtained: Definition 5 [28,29]. Let AðtÞ be a closed and linear operator with domain DðAÞ defined on a Banach space E and β > 0.
Let ρ½AðtÞ be the resolvent set of AðtÞ. AðtÞ is called the generator of a β-resolvent family if there exist ω ≥ 0 and a strongly continuous function U β : R 2 + ⟶ BðEÞ such that fλ β : Re λ > ωg ⊂ ρðAÞ and In this case, U β ðt, sÞ is called the β-resolvent family generated by AðtÞ.
where α denotes the measure of noncompactness.
Lemma 9 [21]. Let E be a Banach space and D ⊂ E be bounded; then, there exists a countable set D 0 ⊂ D such that αðDÞ ≤ 2αðD 0 Þ.
Lemma 10 [31]. Let E be a Banach space and D ⊂ E be a bounded closed and convex set. Assume that Q : D ⟶ D is a strict set contraction mapping; then, Q has at least one fixed point in D.

Definition 11.
A function x ∈ Cð½−r, T 0 ; EÞ is a mild solution of problem (4), if x satisfies the following equations:

Main Results
Let us introduce the operator Ψ : Theorem 12. Assume that the following conditions hold: (H 1 ). The resolvent operator U β ðt, sÞ is compact for all t, Then, problem (4) has at least one mild solution x ∈ C ð½−r, T 0 ; EÞ.
Proof. Let us set the notation R 1 > 0 such that where ϕ 0 = kϕð0Þk and ð Ð 0 First of all, we consider the set B R 1 = fx ∈ Cð½−r, T 0 ; EÞ : kxk Cð½−r,T 0 ;EÞ ≤ R 1 g and show that ΨB R 1 ⊂ B R 1 . By using conditions (H 2 ) and (H 3 So, we conclude that Ψ maps B R 1 into itself. Second, we prove that Ψ : for any t ∈ J uniformly. That is, for any ε > 0, there exists a natural number N 0 , for n > N 0 , t ∈ J, such that which implies that In consequence, Ψ : B R 1 ⟶ B R 1 is continuous. Furthermore, we prove that ΨðB R 1 Þ is equicontinuous.
Moreover, for any x ∈ B R 1 , one can find that Thus, YðtÞ = fðΨxÞðtÞ: x ∈ B R 1 g is totally bounded. Hence, YðtÞ is relatively compact in E, and so, based on the Arzelà-Ascoli theorem, Ψ : B R 1 ⟶ B R 1 is completely continuous. As all the assumptions of the Schauder fixed point theorem are satisfied, the conclusion implies that the operator Ψ has a fixed point x in Cð½−r ; T 0 , EÞ, which is a global mild solution of problem (4). This completes the proof.
Proof. By (H 4 ), there exists 0 < μ < 1/T 0 M ⋆ and R 0 > 0, for any R ≥ R 0 , such that Let R ⋆ = max fR 0 , M ⋆ ϕ 0 ð1 − M ⋆ T 0 μÞ −1 g; we first consider the set B R ⋆ = fx ∈ Cð½−r, T 0 ; EÞ: kxk Cð½−r,T 0 ;EÞ ≤ R ⋆ g and show that ΨB R ⋆ ⊂ B R ⋆ . From the above inequality, for all x ∈ B R ⋆ , we have Meanwhile, applying the arguments employed in the proof of Theorem 12, we conclude that Ψ is a continuous and bounded operator on B R ⋆ .
Finally, we prove that Ψ : CoΨðB R ⋆ Þ ⟶ CoΨðB R ⋆ Þ is a condensing operator. By Lemma 9, for any D ⊂ CoΨðB R ⋆ Þ, there exists a countable set D 0 = fx n g ⊂ D such that By using condition (H 5 ) and Lemma 8, we obtain In addition, using Lemma 8, we have Consequently, By (H 6 ), we obtain that Ψ is a condensing operator on CoΨðB R ⋆ Þ. By Lemma 10, there exists at least one fixed point x ∈ CoΨðB R ⋆ Þ ⊂ Cð½−r, T 0 ; EÞ for Ψ. In conclusion, problem (4) has at least one global mild solution. This completes the proof.

Journal of Function Spaces
These arguments enable us to conclude that the operator Ψ is a contraction mapping. Hence, the operator Ψ has a unique fixed point x ⋆ ∈ Cð½−r, T 0 ; EÞ, which implies that problem (4) has a unique global mild solution. This completes the proof.

An Application
In order to show the application of the main results, we consider the following problem: is the Riemann-Liouville fractional integral of order 1 − β, Ω ⊂ ℝ n is a bounded domain with regular boundary ∂Ω, and φ ∈ Cð½−r, 0 ; EÞ, E = Cð Ω ; ℝÞ, Ω = Ω S

Conclusion
In this paper, we study the existence and uniqueness of the global mild solutions for the fractional integrodifferential equations of mixed type with delay. Under the condition of the compact resolvent operator, we obtain Theorems 12 and 13, respectively, via various fixed point theorems and the measure of noncompactness. Theorem 15 is established by using the Banach contraction mapping principle under the condition of the noncompact resolvent operator. Furthermore, an example is provided to illustrate the main theorems.
The kernels K and H of the operators K and H are nonlinear functions; meanwhile, the operator AðtÞ is dependent on t. As a consequence, our main theorems improve and generalize many corresponding results by using different methods.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.