Instantaneous and Noninstantaneous Impulsive Integrodifferential Equations in Banach Spaces

The existence of mild solutions is developed in [1, 2] for some semilinear functional differential equations. There has been a significant development in functional evolution equations in recent years (see the monographs [3–5], the papers [6–12], and the references therein). The study of an abstract nonlocal Cauchy problem was initiated by Byszewski [13] in 1991. Evolution equations with nonlocal initial conditions were motivated by physical problems. As a matter of fact, it is demonstrated that the evolution equations with nonlocal initial conditions have better effects in applications than the classical Cauchy problems. For example, it was pointed in [14, 15] that the nonlocal problems are used to represent mathematical models for evolution of various phenomena, such as nonlocal neural networks, nonlocal pharmacokinetics, nonlocal pollution, and nonlocal combustion. Due to nonlocal problems having a wide range of applications in real-world applications, evolution equations with nonlocal initial conditions were studied by many authors. Xue [16] studied the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in separable Banach spaces. Xue discussed the semilinear nonlocal differential equations when the semigroup TðtÞ generated by the coefficient operator is compact and the nonlocal term g is not compact. Fan and Li [4] discussed the existence for impulsive semilinear differential equations with nonlocal conditions by using Sadovskii’s fixed point theorem and Schauder’s fixed point theorem. Recently, several researchers obtained other results by application of the technique of measure of noncompactness (see [17–19] and the references therein). Impulsive differential equations have become more important in recent years in some mathematical models of real phenomena, especially in biological or medical domains, and in control theory (see, for example, the monographs [20–22] and the papers [23–25]). In this paper, we first discuss the existence of mild solutions for the following nonlocal problem of impulsive integrodifferential equations

The study of an abstract nonlocal Cauchy problem was initiated by Byszewski [13] in 1991. Evolution equations with nonlocal initial conditions were motivated by physical problems. As a matter of fact, it is demonstrated that the evolution equations with nonlocal initial conditions have better effects in applications than the classical Cauchy problems. For example, it was pointed in [14,15] that the nonlocal problems are used to represent mathematical models for evolution of various phenomena, such as nonlocal neural networks, nonlocal pharmacokinetics, nonlocal pollution, and nonlocal combustion. Due to nonlocal problems having a wide range of applications in real-world applications, evolution equations with nonlocal initial conditions were studied by many authors. Xue [16] studied the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in separable Banach spaces. Xue discussed the semilinear nonlocal differential equations when the semigroup TðtÞ generated by the coefficient operator is compact and the nonlocal term g is not compact. Fan and Li [4] dis-cussed the existence for impulsive semilinear differential equations with nonlocal conditions by using Sadovskii's fixed point theorem and Schauder's fixed point theorem.
Recently, several researchers obtained other results by application of the technique of measure of noncompactness (see [17][18][19] and the references therein).
Impulsive differential equations have become more important in recent years in some mathematical models of real phenomena, especially in biological or medical domains, and in control theory (see, for example, the monographs [20][21][22] and the papers [23][24][25]). In this paper, we first discuss the existence of mild solutions for the following nonlocal problem of impulsive integrodifferential equations where In [26][27][28][29] the authors initially offered to study some classes of impulsive differential equations with noninstantaneous impulses. Motivated by the above papers, we next discuss the existence of mild solutions for the following nonlocal problem of noninstantaneous impulsive integrodifferential equations: where I 0 ≔ ½0, t 1 , J k ≔ ðt k , s k , I k ≔ ðs k , t k+1 ; k = 1, ⋯, m, f : I k × E ⟶ E, g k : J k × E ⟶ E are given functions such that g k ðt, uðt − k ÞÞj t=s k = u k ∈ E ; k = 1, ⋯, m, g : P C ⟶ E is a given function, the set P C is given later, and 0 = s 0 < t 1 ≤ s 1 < t 2 ≤ s 2 <⋯≤s m−1 < t m ≤ s m < t m+1 = T.

Preliminaries
By BðEÞ, we denote the space of the bounded linear operator from E into itself. Let CðIÞ ≔ CðI, EÞ be the Banach space of continuous functions from I ≔ ½0, T into E. Let L ∞ ðIÞ be the Banach space of measurable functions v : I ⟶ ℝ that are essentially bounded and equipped with the norm Consider the Banach space with the norm A semigroup of bounded linear operators TðtÞ is uniformly continuous if Here, I denotes the identity operator in E. We note that if a semigroup TðtÞ is of class ðC 0 Þ, then it satisfies the growth condition ∥TðtÞ∥ BðEÞ ≤ Me βt , for 0 ≤ t<∞ with some constants M > 0 and β ≥ 0.
Let M X denote the class of all bounded subsets of a metric space X: Definition 2 [32]. LetXbe a Banach space and letΩ X be the family of bounded subsets ofX. The Kuratowski measure of noncompactness is the mapμ : where M ∈ Ω X .
For our purpose, we will need the following fixed point theorem: [33]). Let Dbe a bounded, closed, and convex subset of a Banach space such that0 ∈ Dand letNbe a continuous mapping ofDinto itself. If the implication holds for every subset V of D, then N has a fixed point.

Mild Solutions with Instantaneous Impulses
In this section, we are concerned with the existence results of the problem (1).
Definition 4 [10]. A resolvent operator for the Cauchy problem 2 Abstract and Applied Analysis is a bounded linear operator-valued function RðtÞ ∈ BðEÞ ; t ≥ 0, verifying the following conditions: (i) Rð0Þ = I (the identity map of E) and ∥RðtÞ∥≤Ne νt for some constants N > 0, and ν ∈ ℝ , ∞Þ Let us introduce the following hypotheses: ðR 1 Þ The operator A is the infinitesimal generator of a uniformly continuous semigroup ðSðtÞÞ t≥0 ðR 2 Þ For all t ≥ 0, YðtÞ is a closed linear operator from DðAÞ to E and YðtÞ ∈ BðEÞ: For any u ∈ E, the map t ↦ YðtÞ u is bounded differentiable and the derivative t ↦ Y ′ ðtÞu is bounded uniformly continuous on ℝ + . Theorem 5 [10,34]. Assume that ðR 1 Þ and ðR 2 Þ hold. Then, there exists a unique uniformly continuous resolvent operator for the Cauchy problem (9). Definition 6 [34]. By a mild solution of the problem (1), we mean a function u ∈ PC that satisfies The following hypotheses will be used in the sequel.
The function t ↦ f ðt, uÞ is measurable on I for each u ∈ E, and the function u ↦ f ðt, uÞ is continuous on E for a.e. t ∈ I k , H 2 There exists a function p ∈ L ∞ ðIÞ, such that and and for each bounded set B 1 ⊂ PC, we have where B 1 ðtÞ = fuðtÞ: u ∈ B 1 g ; t ∈ I. Set Theorem 7. Assume that the hypotheses ðR 1 Þ, ðR 2 Þ, ðH 1 Þ -ðH 4 Þ hold. If then problem (1) has at least one mild solution defined on I.
Proof. Transform problem (1) into a fixed point problem.
Consider the operator N : PC ⟶ PC defined by Let ρ > 0, such that and consider the ball B ρ ≔ Bð0, ρÞ = fw ∈ PC : ∥w∥ PC ≤ ρg. For any u ∈ B ρ and each t ∈ I, we have Thus, This proves that N transforms the ball B ρ into itself. We shall show that the operator N : B ρ ⟶ B ρ satisfies all the assumptions of Theorem 3. The proof will be given in three steps.

Abstract and Applied Analysis
Step 1. N : B ρ ⟶ B ρ is continuous.
Let fu n g n∈ℕ be a sequence such that u n ⟶ u as n ⟶ ∞ in B ρ . Then, for each t ∈ I, we have Since u n ⟶ u as n ⟶ ∞ and f , g, L i are continuous, the Lebesgue-dominated convergence theorem implies that Step 2. NðB ρ Þis bounded and equicontinuous.
Next, let t, τ ∈ I, τ < t and let u ∈ B ρ . Thus, we have Hence, we get As the resolvent operator Rð·Þ is uniformly continuous, the right-hand side of the above inequality tends to zero as τ ⟶ t.
Step 3. The implication (8) holds. Now let V be a subset of B ρ such that V ⊂ NðVÞ ∪ f0g. V is bounded and equicontinuous, and therefore, the function t ⟶ vðtÞ = μðVðtÞÞ is continuous on I. By ðH 3 Þ and the properties of the measure μ, for each t ∈ I, we have Hence, From (17), we get ∥v∥ ∞ = 0, that is, vðtÞ = μðVðtÞÞ = 0, for each t ∈ I, and then VðtÞ is relatively compact in E. In view of the Ascoli-Arzelà theorem, V is relatively compact in B ρ . Applying now Theorem 3, we conclude that N has a fixed point which is a mild solution of our problem (1).
Definition 8 [34]. By a mild solution of problem (2), we mean a function u ∈ P C that satisfies The following hypotheses will be used in the sequel: The functions t ↦ f ðt, uÞ and t ↦ g k ðt, uÞ are measurable on I k and J k , respectively, for each u ∈ E, and the functions u ↦ f ðt, uÞ and u ↦ g k ðt, uÞ are continuous on E for a.e. t in I k and J k , respectively.
Proof. Transform problem (2) into a fixed point problem.
Consider the operator N : PC ⟶ PC defined by Let L > 0, such that For any u ∈ PC and each t ∈ I k , we have Thus, Next, for each t ∈ J k ; k = 1, ⋯, m, it is clear that Hence, This proves that N transforms the ball B ρ ≔ fw ∈ P C : ∥w∥ P C ≤ ρg into itself.
We shall show that the operator N : B ρ ⟶ B ρ satisfies all the assumptions of Theorem 3. The proof will be given in three steps.
Let fu n g n∈ℕ be a sequence such that u n ⟶ u as n ⟶ ∞ in B ρ : Then, for each t ∈ J k ; k = 1, ⋯, m, we have and for each t ∈ I k ; k = 0, ⋯, m, we have Since u n ⟶ u as n ⟶ ∞ and f , g, g k are continuous, the Lebesgue-dominated convergence theorem implies that Step 2. NðB ρ Þis bounded and equicontinuous.
Next, let t, τ ∈ I k , τ < t and let u ∈ B ρ . Thus, we have Hence, we get

Abstract and Applied Analysis
As τ ⟶ t, the right-hand side of the above inequality tends to zero.
Step 3. The implication (8) holds. Now let V be a subset of B R such that V ⊂ NðVÞ ∪ f0g. V is bounded and equicontinuous, and therefore, the function t ⟶ vðtÞ = μðVðtÞÞ is continuous on I. By ðH 03 Þ and the properties of the measure μ, Next, for each t ∈ I k , we have Thus, for each t ∈ I, we get v t ð Þ ≤ ℓ∥v∥ ∞ : Hence, From (36), we get ∥v∥ ∞ = 0; that is, vðtÞ = μðVðtÞÞ = 0, for each t ∈ I, and then VðtÞ is relatively compact in E. In view of the Ascoli-Arzelà theorem, V is relatively compact in B ρ . Applying now Theorem 3, we conclude that N has a fixed point which is a mild solution of problem (2).
We define the strongly elliptic operator A : DðAÞ ⊂ H ⟶ H by where a μ ∈ C 2m ð½0, πÞ and DðAÞ = H 2m ð½0, πÞ ∩ H m 0 ð½0, πÞ. It is well known (see [31]) that A generates a uniformly continuous semigroup TðtÞ ; t ≥ 0 in the Hilbert space H.