Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.

Meanwhile, differential equations with impulsive effects have been used widely as mathematical models for the study of many phenomena in physical, biology, optimal control model of economics, etc. Much attention has been paid to the existence of solutions for the differential equations with impulses in abstract space. For details, see [7,[20][21][22][23][24][25][26][27][28].
In [20], the author studied the following integer order integrodifferential equations with instantaneous impulses in a Banach space E: where for any t ≥ 0, the linear operator A is the infinitesimal generator of a compact, analytic semigroup, and the nonlinear term is Lipschitz continuous. The existence of mild solutions has been proved.
In this paper, we investigate the following fractional semilinear integrodifferential equations with noninstantaneous impulses and delay: where β ∈ ð0, 1, c D β t is the Caputo's fractional derivative of order β, AðtÞ is a closed and linear operator with domain D ðAÞ defined on a Banach space E, and the fixed points s i and t i satisfying 0 = s 0 < t 1 ≤ s 1 < t 2 ≤ ⋯ < t N ≤ s N < t N+1 = T 0 are prefixed numbers. f , l k , h, g ðk = 1, 2, ⋯, NÞ and τ i ði = 1, 2, ⋯, n + 1Þ are to be specified later.
Inspired by the results mentioned above, by the semigroup theory and fixed point theorems, we consider the existence of mild solutions for the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay (3). In [7], the authors discussed the existence of solutions for the fractional ordinary differential equation with a generalized impulsive term. In [20][21][22][23], the authors discussed the integer or fractional differential equations with instantaneous impulses and the linear operator A is independent of t. In [26,29,30], the authors discussed the integer-order differential equations with noninstantaneous impulses and the linear operator A is independent of t. In [31][32][33], the authors discussed the fractional differential equations with noninstantaneous impulses and the linear operator A is also independent of t. In this paper, we consider the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay, and the linear operator AðtÞ is assumed to be dependent on t. Therefore, the mentioned results above are special cases of the problem investigated in this paper. Our results improve and generalize the results in References [7, 20-23, 26, 29-33].
The rest of this paper is organized as follows. In Section 2, we present the basic notation and preliminary results. In Section 3, we prove the existence of mild solutions. In Section 4, we give an illustrative example, followed by the conclusion of this paper in Section 5.
Definition 1 [34]. The Riemann-Liouville fractional integral of order α > 0 of a function f : ð0,∞Þ → R is given by provided that the right-hand side is pointwise defined on ð0, ∞Þ.
Definition 2 [34]. The Caputo fractional derivative of order α > 0 of a function f : ð0,∞Þ ⟶ R is given by where Γð·Þ denotes the Gamma function, α is a fractional number, n = ½α + 1, provided that the right-hand side is pointwise defined on R + .
Definition 3 [35,36]. 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, we call AðtÞ the generator of an 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.

Main Results
For convenience in presentation, we give here the basic assumptions to be used later throughout the paper.

Journal of Function Spaces
Let B r = fu ∈ PCðJ, EÞ: ∥u∥ PC ≤ rg. We will prove QðB r Þ ⊂ B r . Let u ∈ B r , for t ∈ ½0, t 1 , we have For all t ∈ ðs k , t k+1 ðk = 1, 2, ⋯, NÞ, we have For all t ∈ ðt k , s k ðk = 1, 2, ⋯, NÞ, we have From the above inequalities, we have that Qu ∈ B r : Next, we prove that Q is a contraction map on B r . For all u, v ∈ PCðJ, EÞ and t ∈ ½0, t 1 , we have For all t ∈ ðs k , t k+1 ðk = 1, 2, ⋯, NÞ, we have For all t ∈ ðt k , s k ðk = 1, 2, ⋯, NÞ, we have From the above results, for all u, v ∈ PCðJ, EÞ, we have Therefore, Q : B r → B r is a contraction map and there exists a unique mild solution u * of the problem (3) in B r and ∥u * ∥ PC ≤ r.
Then, the problem (3) has a mild solution u * ∈ PCðJ, EÞ and ∥u * ∥ PC ≤ r, where Proof. We introduce the decomposition Q = Q 1 + Q 2 , where Q j : PCðJ, EÞ → PCðJ, EÞ ðj = 1, 2Þ are defined by Let B r = fu ∈ PCðJ, EÞ: ∥u∥ PC ≤ rg, we then prove that the operator Q = Q 1 + Q 2 is a condensing map on B r . It is easy to see that B r is a closed, bounded, and convex subset of PCðJ, EÞ.
Step 2. We prove that Q 1 is a contraction on B r . For any u, v ∈ B r and t ∈ ½0, t 1 , we have For all t ∈ ðt k , s k ðk = 1, 2, ⋯, NÞ, we have For all t ∈ ðs k , t k+1 ðk = 1, 2, ⋯, NÞ, we have Thus, since ϱ < 1, we get that Q 1 is a contraction on B r .
Step 3. We prove that Q 2 is completely continuous on B r . Firstly, we prove that the operator Q 2 is continuous. Let fu m g ∞ 0 ⊂ B r , and u m → u ∈ B r . For any t ∈ ½0, t 1 , by ðH 3 Þ, we have By the Lebesgue dominated convergence theorem, we have