Full article: A note on existence results for noninstantaneous impulsive integrodifferential systems

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ABSTRACT

A class of noninstantaneous impulsive integrodifferential systems in Banach space will be the focus of the research presented in this paper. First, we formulate the existence and uniqueness of mild solutions for the considered equation with local initial conditions. The problem is then solved step by step using the resolvent operator in the sense of Grimmer combined with the Banach contraction theorem. In addition, considering the scenario in which we provide our system with nonlocal initial conditions, Krasnoselskii’s fixed point theorem is utilized to establish mild solutions. To illustrate the theory, we will look at two different examples.

KEYWORDS:

MATHEMATICS SUBJECT CLASSIFICATION:

1. Introduction

Differential equations with instantaneous impulses have been the subject of study for many scholars over the last several decades. These equations have been utilized to represent sudden changes, such as those that occur during harvests, natural catastrophes, and shocks. This theory of instantaneous impulsive equations, in particular, has a wide range of applications in control, mechanics, electrical engineering, biology, and medicine. For additional information regarding the differential equations that involve instantaneous impulses, one may have a look at (Bainov et al., Citation1989; Benchohra et al., Citation2006; Chen & Li, Citation2011; S. Liang & Mei, Citation2014; J. Liang et al., Citation2009).

Instantaneous impulsive dynamic systems cannot provide a complete description of many processes. When one considers, for instance, the hemodynamic balance of a person, introducing medications into the bloodstream and the subsequent absorption of those drugs by the body is a gradual and ongoing process. In (Hernández & O’Regan, Citation2013) Hernàdez and O’Regan started a study on the Cauchy problem for a new type first-order evolution equation with no-instantaneous impulses of the form:

\begin{array}{l} {\begin{cases} u^{'} (t) = A u (t) + f (t, u (t)), t \in (s_{i}, t_{i + 1}], i = 0, 1, \dots, m, \\ u (t) = g_{i} (t, u (t)), t \in (t_{i}, s_{i}], i = 1, 2, \dots, m, \\ u (0) = x_{0} \in X, \end{cases} \end{array}

where $A : D (A) \subset X \to X$ is the generator of C₀-semigroup of bounded linear operators ${(T (t))}_{t \geq 0}$ defined on a Banach space $(X, ‖ \cdot ‖), x_{0} \in X, 0 = t_{0} = s_{0} < t_{1} \leq s_{1} \leq t_{2} < \dots < t_{N} \leq s_{N} \leq t_{N} + 1 = a$ are pre-fixed numbers, $g_{i} \in C ((t_{i}, s_{i}] \times X, X)$ for all $i = 1, \dots, N$ and $f : [0, a] \times X \to X$ is a suitable function.

The reader can find recent results for evolution equations with non-instantaneous impulses in (Bai & Nieto, Citation2017; Chen et al., Citation2016; Wang & Li, Citation2014; Yu & Wang, Citation2015) and the references therein.

When modeling phenomena prone to sudden shifts, the nonlocal constraint significantly impacts applications more than the more conventional value problems that start with the starting point $v = v_{0}$ . The study of the abstract nonlocal conditions is due to Byszewski (Byszewski, Citation1991, Citation1993). The importance of problems involving nonlocal constraints is that they are more general and have better implications than the traditional initial constraint. Nonlocal constraints make it possible to collect more data on the phenomenon under study. K.Malar, A.anguraj (Malar & Anguraj, Citation2016), have studied the existence results of abstract impulsive integrodifferential systems with measure of noncompactness:

\begin{array}{l} {\begin{cases} u^{'} (t) = A u (t) + f (t, u (t), \int_{0}^{t} ρ (t, s) h (t, s, u (s)) ds), \\ t \in (s_{i}, t_{i + 1}], i = 0, 1, \dots, m, \\ u (t) = g_{i} (t, u (t)), t \in (s_{i}, t_{i}], i = 1, 2, \dots, m, \\ u (0) = u_{0} + k (u), \end{cases} \end{array}

where $A$ generates a C₀-semigroup of bounded linear operator ${T (t), t \geq 0}$ defined on a Banach space $(X, ‖ \cdot ‖) . u_{0} \in X, 0 = t_{0} = s_{0} < t_{1} \leq s_{1} \leq t_{2} < \dots < t_{n} \leq s_{n} \leq t_{n + 1} = b,$ are prefixed numbers, $k : X \to X,$ and $g_{i} \in C ((t_{i}, s_{i}] \times X, X)$ for all $i = 1, 2, \dots, n, f : [0, b] \times X \times X \to X$ is a given function, and $h \in C (D, R^{+}), D = {(t, s) ∖ t, s \in [0, b], t \geq 0} .$ As a result, most research in this area has focused on differential equations with nonlocal constraints. The reader interested in learning more about this topic can refer to the papers (Diallo et al., Citation2017; M. A. Diop et al., Citation2019; K. Ezzinbi & Ghnimi, Citation2019; G. Ezzinbi et al., Citation2015; Jeet, Citation2020; Yan, Citation2012) in which the authors examined impulsive equations with nonlocal constraints under various conditions.

In evolution system theory, integrodifferential equations are commonly used in modern applied mathematics and are essential in modeling. Recently, the number of research articles devoted to investigating integrodifferential equations has increased, which gives more effectiveness than classical differential equations and more opportunities in applications as well as in finance, biology, population dynamics, and other science and engineering domains. Solving integrodifferential equations with the theory of resolvent operators to establish their qualitative and quantitative properties, such as existence, uniqueness, controllability, and stability, is extensively used by many scholars. Employing the technique of the resolvent operator in the sense of Grimmer (Grimmer, Citation1982), a sufficient set of criteria is formulated for the existence of mild, strict, and classical solutions for many classes of integrodifferential equations. We refer the reader to (Dieye et al., Citation2017; M. A. Diop & Caraballo, Citation2015; A. Diop et al., Citation2022; Kumar et al., Citation2014; Bensatal et al., Citation2024).

Inspired by the above research, we consider the following non-instantaneous impulsive integrodifferential system:

(1.1)

{\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}

(1.1)

with local conditions $v (0) = v_{0}$ and nonlocal conditions $v (0) = v_{0} + q (v)$ over the interval $[0, b]$ in a Banach space $H$ . Here $A : D (A) \subset H \to H$ is the infinitesimal generator of a C₀-semigroup ${(S (t))}_{t \geq 0}$ on the Banach space $H, γ$ is a closed linear operator on $H$ with domain $D (γ) \supset D (A),$ which is independent of $t, t \geq 0$ . $K v = \int_{0}^{t} ζ (t, s, v (s)) ds$ is the nonlinear Volterra integral operator, $g : [0, b] \times H \times H \to H$ is a nonlinear function and $f_{k} : [0, b] \times H \to H$ are set of nonlinear functions applied in the interval $[t_{k}, s_{k})$ for all $k = 1, 2, \dots, m$ .

In this paper, we treat a more general situation than in some previous works (Hernández & O’Regan, Citation2013; Malar & Anguraj, Citation2016) and establish the existence of mild solutions. For example, when we chose γ = 0 and ζ = 0, problem(Equation1.1(1.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}$ (1.1) ) improves the results of the existence of mild solutions for the first-order evolution equations with noninstantaneous impulses discussed in (Hernández & O’Regan, Citation2013; Malar & Anguraj, Citation2016)

No work has been reported on the subject of noninstantaneous impulsive integrodifferential equations with nonlocal conditions involving the resolvent operator. We explore the existence of mild solutions to EquationEquation (1.1)(1.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}$ (1.1) in this paper. We assume that γ ≠ 0, indicating that the semigroup theory is insufficient to study this differential equations class. Our approach consists of applying the resolvent operator’s theory in the sense of Grimmer (see (Grimmer, Citation1982)) and transforming the problem (Equation1.1(1.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}$ (1.1) ) into a fixed-point problem of an appropriate operator and to apply Banach contraction theorem and Krasnoselskii’s fixed point theorem. For more details, see subsection 2.1.

The following is the overall structure of this paper. In Section 3, we provide a brief overview of some fundamental notions and preliminary results, valid for the proof of our results. The results in Section 4 are devoted to investigating the existence of mild solutions for the system (Equation1.1(1.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}$ (1.1) ) with local conditions. Section 6 discusses the case when our integrodifferential system has nonlocal conditions. Finally, we add two models for the description of the obtained theory.

2. Preliminaries

In this section, we briefly recall some fundamental results and definitions which are used in the sequel. Let $H$ be a Banach space with $| \cdot | .$ We denote $J = [0, b]$ for any constant b > 0. Let $C (J, H)$ be a Banach space of all continuous functions from J to $H$ endowed with supremum norm ${‖ v ‖}_{C} = sup_{t \in J} | v (t) | .$ Consider the space

\begin{array}{l} P C (J,) \equiv P C & = {v : J \to, v is continuous at t \neq t_{k}, v (t_{k}^{-}) \\ = v (t_{k}^{+}) and v (t_{k}^{+}) exists for all k = 1, 2, \dots, m}, \end{array}

in which $v (t_{k}^{-})$ and $v (t_{k}^{+})$ represent the left-hand side and the right-hand side limits of the function $v (t),$ respectively. Clearly, $PC (J, H)$ is a Banach space endowed with supremum norm ${‖ v ‖}_{PC} = sup_{t \in J} | v (t) | .$

2.1. Partial integrodifferential equations in Banach spaces

This section contains a brief exposition of some basic results needed throughout this work. Especially, we recall some fundamental facts for the existence of the resolvent operator. For more details on the theory of the resolvent operator we refer the reader to (Grimmer, Citation1982).

Let $A$ and $γ (t)$ be closed linear operators on $H$ . $\tilde{Y}$ represents the Banach space $D (A)$ equipped with the graph norm defined by

\begin{array}{l} | x |_{\tilde{Y}} := | A x | + | x | for x \in \tilde{Y} . \end{array}

The notations $C ([0, + \infty); \tilde{Y}), L (\tilde{Y}, H)$ and $L (H)$ stand for the space of all continuous functions from $[0, + \infty)$ into $\tilde{Y}$ , the space of all bounded linear operators from $\tilde{Y}$ into $H$ and the space of all bounded linear operators on $H,$ respectively.

We consider the following Cauchy problem

(2.1)

{\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}

(2.1)

Definition 2.1.

(Grimmer, Citation1982)

A resolvent operator for Equation (Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ) is a bounded linear operator valued function $R (t) \in L (H)$ for $t \geq 0$ , having the following properties:

$R (0) = I$ and ${| R (t) |}_{L (H} \leq M e^{αt}$ for some constants M and α.
For each $x \in H, R (t) x$ is strongly continuous for $t \geq 0$
$R (t) \in L (\tilde{Y})$ for $t \geq 0$ . For $x \in \tilde{Y}, R (\cdot) x \in C^{1} ([0, + \infty); H) \cap C ([0, + \infty); \tilde{Y})$ and

\begin{array}{l} R^{'} (t) x & = A R (t) x + \int_{0}^{t} γ (t - s) R (s) xds \\ = R (t) A x + \int_{0}^{t} R (t - s) γ (s) xds for t \geq 0. \end{array}

We have the following example of a resolvent operator for equation(Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ) in $R$ .

Example.

(Desch et al., Citation1984)

Let $H = R, A y = 2 y$ , and $γ (t) y = - 2 y$ in (Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ). Then in that case, we have

\begin{array}{l} R (t) y_{0} = e^{t} (\cos t + \sin t) y_{0} and S (t) y_{0} = e^{2 t} y_{0} . \end{array}

Remark 2.1.

Generally, the resolvent operator $(R (t))_{t \geq 0}$ for EquationEquation (2.1)(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) does not meet the semigroup law, as demonstrated by the example presented earlier. Namely,

\begin{array}{l} R (t + s) \neq R (t) R (s) for\ some t, s > 0. \end{array}

We impose the following assumptions:

$A$ is the infinitesimal generator of a strongly continuous semigroup ${S (t)}_{t \geq 0}$ on $H$ .
For all $t \geq 0$ , $γ (t)$ is a closed linear operator from $D (A)$ to $H$ , and $γ (t) \in L (\tilde{Y}, H)$ . For any $y \in \tilde{Y}$ , the map $t \to γ (t) y$ is bounded, differentiable and the derivative $t \to γ^{'} (t) y$ is bounded and uniformly continuous on $R^{+}$ .

The following theorem offers sufficient conditions to guarantee that the resolvent operator for the Cauchy problem (Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ) exists.

Theorem 2.2

(Grimmer, Citation1982)

Assume that $(H_{1}) - (H_{2})$ hold. Then, there exists a unique resolvent operator of the Cauchy problem (Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ).

Lemma 2.3.

(J. Liang et al., Citation2008)

Let $(H_{1}) - (H_{2})$ hold. Then, there exists a constant $Q = Q (b)$ such that

\begin{array}{l} {| R (t + ϵ) - R (ϵ) R (t) |}_{L (H)} \leq Qε, for 0 \leq ε \leq t \leq b . \end{array}

The next outcome provides a concise relation between the compactness of the C₀-semigroup generated by operator $A$ and the compactness of the resolvent operator.

Theorem 2.4

(Desch et al.,)

Let $S (t)$ be compact for t > 0. Then, the corresponding resolvent operator $R (t)$ of the Cauchy problem (Equation2.1(2.1) ${\begin{cases} y^{'} (t) = A y (t) + \int_{0}^{t} γ (t - s) y (s) ds, for t \geq 0, \\ y (0) = y_{0} \in H . \end{cases}$ (2.1) ) is compact for t > 0 and is continuous in the uniform operator topology for t > 0.

Now recall the two following fixed point theorems, which will be used in the proof of the main results.

Theorem

Banach fixed point theorem

(Borah & Nandan Bora, Citation2019) Let $H$ be a closed subset of a Banach space $(K, ‖ . ‖)$ and let $T : H \to H$ be a contraction mapping, then T has a unique fixed point in $H$ .

Theorem

Krasnoselskii’s fixed point theorem

(Borah & Nandan Bora, Citation2019) Let $H$ be a closed convex nonempty subset of a Banach space $(K, ‖ \cdot ‖)$ and $P$ and $G$ are two operators on $H$ satisfying:

$P u + G v \in H$ whenever $u, v \in H$ ,
$P$ is contraction,
$P$ is completely continuous,

then the equation $P u + G u = u$ has a unique solution.

Definition 2.7.

Completely continuous operators

Let $X$ and $Y$ be Banach spaces. Then the operator $T : E \subset X \to Y$ is called completely continuous if T is continuous and maps any bounded subset of $E$ to relatively compact subset of $Y$ .

3. Existence results for equation with local conditions

In this section, we will derive sufficient conditions to ensure the existence and uniqueness of mild solutions for the equation:

(3.1)

{\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0}, \end{cases}

(3.1)

over the interval $[0, b]$ in the Banach space $H$ .

Definition 3.1.

A function $v \in PC (J, H)$ is called mild solution of the system (Equation3.1(3.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0}, \end{cases}$ (3.1) ) over the interval $[0, b],$ if $v (t)$ satisfies the integral equation

(3.2)

\begin{array}{l} v (t) = {\begin{cases} R (t) v_{0} + \int_{0}^{t} R (t - s) g (t, v (s), K v (s)) ds, \\ t \in [0, t_{1}), \\ f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ R (t - s_{k}) f_{k} (s_{k}, v (s_{k})) \\ + \int_{s_{k}}^{t} R (t - s) g (t, v (s), K v (s)) ds, t \in [s_{k}, t_{k + 1}), \end{cases} \end{array}

(3.2)

where $K v (t) = \int_{0}^{t} ζ (t, s, v (s)) ds$ is the linear operator defined on $H$ .

Now, we will discuss the first primary result. In this order, the assumptions that we need to make are as follows:

There exists a constant M > 0 such that ${| R (t) |}_{L (H)} \leq M$ .
The function $g : [0, T] \times Y \times Y \to Y$ is continuous with respect to t, and there exist some positive constants $η_{g}$ and ${\hat{η}}_{g}$ such that $| g (t, v_{1}, w_{1}) - g (t, v_{2}, w_{2}) | \leq η_{g} ‖ v_{1} - v_{2} ‖ + {\hat{η}}_{g} ‖ w_{1} - w_{2} ‖ for v_{1}, w_{1}, v_{2}, w_{2} \in B_{τ_{0}} := {v \in PC (J, Y) : {‖ v ‖}_{PC} \leq τ_{0}} .$
The operator $K : [0, T] \times Y \to Y$ is continuous and there exists a constant $\hat{K}$ such that
$\begin{array}{l} | K v - K w | \leq \hat{K} ‖ v - w ‖ for v, w \in B_{τ_{0}} . \end{array}$
The functions $f_{k} : [t_{k}, s_{k}] \times H \to H$ are continuous and there exist positive constants $0 < η_{k} < 1, k = 1, 2, \dots, m$ such that
$\begin{array}{l} | f_{k} (t, v (t)) - f_{k} (t, w (t)) | \leq η_{k} ‖ v - w ‖, \\ ∀v, w \in PC (J, H) for\ all t \in J . \end{array}$

Theorem 3.2

Assume that conditions (H₁)-(H₂) and (C₁)-(C₄) are satisfied, then the integrodifferential system with noninstantaneous impulses (Equation3.1(3.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0}, \end{cases}$ (3.1) ) has a unique mild solution.

Proof.

Define the operator Φ on $PC (J, H)$ by

(3.3)

\begin{array}{l} Φ v (t) = {\begin{cases} Φ_{1} v (t), t \in [0, t_{1}), \\ Φ_{2} v (t), t \in [t_{k}, s_{k}), \\ Φ_{3} v (t), t \in [s_{k}, t_{k + 1}), \end{cases} \end{array}

(3.3)

where Φ₁, Φ₂ and Φ₃ are defined as follows

\begin{array}{l} Φ_{1} v (t) = & R (t) v_{0} + \int_{0}^{t} R (t - s) g (t, v (s), K v (s)) ds, t \in [0, t_{1}), \\ Φ_{2} v (t) = & f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ Φ_{3} v (t) = & R (t - s_{k}) f_{k} (s_{k}, v (s_{k})) + \int_{s_{k}}^{t} R (t - s) g (t, v (s), K v (s)) ds, \\ t \in [s_{k}, t_{k + 1}), \end{array}

for all $k = 1, 2, \dots, m$ . Then, it is evident that $Φ : PC \to PC .$

It is clear owing to the definition of the operator Φ, that the system (Equation3.1(3.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0}, \end{cases}$ (3.1) ) has a unique mild solution if only if the operator equation $v (t) = Φv (t)$ has a unique solution. This holds if only if each of equations $v (t) = Φ_{1} v (t)$ , $v (t) = Φ_{2} v (t)$ and $v (t) = Φ_{3} v (t)$ has a unique solution over the interval $[0, t_{1})$ , $[t_{k}, s_{k})$ and $[s_{k}, t_{k + 1})$ for all $k = 1, 2, \dots, m$ , respectively. Let $v_{1} (t), v_{2} (t)$ and $v_{3} (t)$ be the solutions of $v (t) = Φ_{1} v (t)$ , $v (t) = Φ_{2} v (t)$ and $v (t) = Φ_{3} v (t)$ respectively. Then, it is easily seen that v(t) is a unique solution of $v (t) = Φv (t)$ , where v(t) is defined as follows

\begin{array}{l} v (t) = {\begin{cases} v_{1} (t), t \in [0, t_{1}), \\ v_{2} (t), t \in [t_{k}, s_{k}), \\ v_{3} (t), t \in [s_{k}, t_{k + 1}) . \end{cases} \end{array}

For all $t \in [0, t_{1})$ and $v, w \in B_{τ_{0}}$ , by considering (C₁)-(C₃) we see that

(3.4)

\begin{array}{l} | Φ_{1} v (t) - Φ_{1} w (t) | \\ \leq \int_{0}^{t} {| R (t - s) |}_{L (H)} | g (t, v (s), K v (s)) \\ - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} | g (t, v (s), K v (s)) - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} | K v (s) - K w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} \hat{K} | v (s) - w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} ds {‖ v - w ‖}_{PC} \\ \leq t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K}) {‖ v - w ‖}_{PC} . \end{array}

(3.4)

In light of this, using the process of mathematical induction, we will demonstrate that for any positive integer n and $t \in [0, t_{1})$ ,

(3.5)

\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}

(3.5)

When n = 1, (Equation3.5(3.5) $\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}$ (3.5) ) holds by (Equation3.4(3.4) $\begin{array}{l} | Φ_{1} v (t) - Φ_{1} w (t) | \\ \leq \int_{0}^{t} {| R (t - s) |}_{L (H)} | g (t, v (s), K v (s)) \\ - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} | g (t, v (s), K v (s)) - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} | K v (s) - K w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} \hat{K} | v (s) - w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} ds {‖ v - w ‖}_{PC} \\ \leq t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K}) {‖ v - w ‖}_{PC} . \end{array}$ (3.4) ). For n = 2, by (Equation3.4(3.4) $\begin{array}{l} | Φ_{1} v (t) - Φ_{1} w (t) | \\ \leq \int_{0}^{t} {| R (t - s) |}_{L (H)} | g (t, v (s), K v (s)) \\ - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} | g (t, v (s), K v (s)) - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} | K v (s) - K w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} \hat{K} | v (s) - w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} ds {‖ v - w ‖}_{PC} \\ \leq t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K}) {‖ v - w ‖}_{PC} . \end{array}$ (3.4) ), we have

\begin{array}{l} | Φ_{1}^{2} v (t) - Φ_{1}^{2} w (t) | \\ = | Φ_{1} (Φ_{1} v (t)) - Φ_{1} (Φ_{1} w (t)) | \\ \leq \int_{0}^{t} | R (t - s) | | g (s, Φ_{1} v (s), K Φ_{1} v (s)) \\ - g (s, Φ_{1} w (s), K Φ_{1} w (s)) | ds \\ \leq M \int_{0}^{t} | g (s, Φ_{1} v (s), K Φ_{1} v (s)) \\ - g (s, Φ_{1} w (s), K Φ_{1} w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | Φ_{1} v (s) - Φ_{1} w (s) | \\ + {\hat{η}}_{g} | K Φ_{1} v (s) - K Φ_{1} w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | Φ_{1} v (s) - Φ_{1} w (s) | \\ + {\hat{η}}_{g} \hat{K} | Φ_{1} v (s) - Φ_{1} w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} | Φ_{1} v (s) - Φ_{1} w (s) | ds \\ \leq t_{1} M^{2} {(η_{g} + {\hat{η}}_{g} \hat{K})}^{2} \int_{0}^{t_{1}} ds {‖ v - w ‖}_{PC} \\ \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{2}}{2!} {‖ v - w ‖}_{PC} . \end{array}

Thus, (Equation3.5(3.5) $\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}$ (3.5) ) is true for n = 2. Now, suppose (Equation3.5(3.5) $\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}$ (3.5) ) holds for n = k, that is, for any $t \in [0, t_{1})$ ,

(3.6)

\begin{array}{l} | Φ_{1}^{k} v (t) - Φ_{1}^{k} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{k}}{k!} {‖ v - w ‖}_{PC} . \end{array}

(3.6)

Then, by (Equation3.4(3.4) $\begin{array}{l} | Φ_{1} v (t) - Φ_{1} w (t) | \\ \leq \int_{0}^{t} {| R (t - s) |}_{L (H)} | g (t, v (s), K v (s)) \\ - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} | g (t, v (s), K v (s)) - g (t, w (s), K w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} | K v (s) - K w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | v (s) - w (s) | + {\hat{η}}_{g} \hat{K} | v (s) - w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} ds {‖ v - w ‖}_{PC} \\ \leq t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K}) {‖ v - w ‖}_{PC} . \end{array}$ (3.4) ) and (Equation3.6(3.6) $\begin{array}{l} | Φ_{1}^{k} v (t) - Φ_{1}^{k} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{k}}{k!} {‖ v - w ‖}_{PC} . \end{array}$ (3.6) ), for any $t \in [0, t_{1})$ , we have

\begin{array}{l} | Φ_{1}^{k + 1} v (t) - Φ_{1}^{k + 1} w (t) | \\ = | Φ_{1} (Φ_{1}^{k} v (t)) - Φ_{1} (Φ_{1}^{k} w (t)) | \\ \leq \int_{0}^{t} | R (t - s) | | g (s, Φ_{1}^{k} v (s), K Φ_{1}^{k} v (s)) \\ - g (s, Φ_{1}^{k} w (s), K Φ_{1}^{k} w (s)) | ds \\ \leq M \int_{0}^{t} | g (s, Φ_{1}^{k} v (s), K Φ_{1}^{k} v (s)) \\ - g (s, Φ_{1}^{k} w (s), K Φ_{1}^{k} w (s)) | ds \\ \leq M \int_{0}^{t} [η_{g} | Φ_{1}^{k} v (s) - Φ_{1}^{k} w (s) | \\ + {\hat{η}}_{g} | K Φ_{1}^{k} v (s) - K Φ_{1}^{k} w (s) |] ds \\ \leq M \int_{0}^{t} [η_{g} | Φ_{1}^{k} v (s) - Φ_{1}^{k} w (s) | \\ + {\hat{η}}_{g} \hat{K} | Φ_{1}^{k} v (s) - Φ_{1}^{k} w (s) |] ds \\ \leq M (η_{g} + {\hat{η}}_{g} \hat{K}) \int_{0}^{t} | Φ_{1}^{k} v (s) - Φ_{1}^{k} w (s) | ds \\ \leq \frac{t_{1}^{k} M^{k + 1} {(η_{g} + {\hat{η}}_{g} \hat{K})}^{k + 1}}{k!} \int_{0}^{t_{1}} ds {‖ v - w ‖}_{PC} \\ \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{k + 1}}{(k + 1)!} {‖ v - w ‖}_{PC} . \end{array}

Hence, it follows by induction that for any positive integer n and $t \in [0, t_{1})$ , (Equation3.5(3.5) $\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}$ (3.5) ) is true.

From (Equation3.5(3.5) $\begin{array}{l} | Φ_{1}^{n} v (t) - Φ_{1}^{n} w (t) | \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}$ (3.5) ), considering supremum over interval $[0, t_{1})$ , for any integer n, we get

\begin{array}{l} {‖ Φ_{1}^{n} v - Φ_{1}^{n} w ‖}_{PC} \leq \frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} {‖ v - w ‖}_{PC} . \end{array}

Let n₀ be such that $\frac{[t_{1} M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n_{0}}}{n_{0}!} < 1$ . Then $Φ_{1}^{n_{0}}$ is a contraction on $B_{τ_{0}}$ . Then, accordingly to the general Banach contraction theorem, Φ₁ has a unique fixed point which is the unique solution of $v (t) = Φ_{1} v (t)$ over the interval $[0, t_{1})$ .

For all $k = 1, 2, \dots, m$ , $t \in [t_{k}, s_{k})$ and $v, w \in PC (J, H)$ , from (C₄), we have

\begin{array}{l} ‖ Φ_{2} v (t) - Φ_{2} w (t) ‖ \leq η_{k} {‖ v - w ‖}_{PC} . \end{array}

Since $η_{k} < 1$ , Φ₂ is a contraction. Therefore, by Banach’s fixed point theorem $v (t) = Φ_{2} v (t)$ has a unique solution over the interval $[t_{k}, s_{k})$ for all $k = 1, 2, \dots, m .$

For any integer n, $k = 1, 2, \dots, m$ , $t \in [s_{k}, t_{k + 1})$ and $v, w \in B_{τ_{0}}$ from (C₁)-(C₄) and applying the same procedure on the interval $[0, t_{1})$ , we get

\begin{array}{l} {‖ Φ_{3}^{n} v - Φ_{3}^{n} w ‖}_{PC} \\ \leq (\frac{[(sup {t_{k + 1} - s_{k}}) M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n}}{n!} + M^{n} η_{k}^{n}) {‖ v - w ‖}_{PC} . \end{array}

Let n₀ be such that $(\frac{[(sup {t_{k + 1} - s_{k}}) M (η_{g} + {\hat{η}}_{g} \hat{K})]^{n_{0}}}{n_{0}!} + M^{n_{0}} η_{k}^{n_{0}}) < 1$ . Then $Φ_{3}^{n_{0}}$ is a contraction on $B_{τ_{0}}$ . Thus, thanks to the general Banach contraction theorem, Φ₃ has a unique fixed point which is the unique solution of $v (t) = Φ_{3} v (t)$ over the interval $[s_{k}, t_{k + 1})$ for all $k = 1, 2, \dots, m .$ .

Hence, the operator equation $v (t) = Φv (t)$ has a unique solution over $[0, b]$ , which is the unique mild solution of the system (Equation3.1(3.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0}, \end{cases}$ (3.1) ).

4. Example

In this part, we give an illustration of the results proved in the previous section with the use of an example. Consider the following system:

(4.1)

\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}

(4.1)

where $Γ : R_{+} \to R$ is C¹-function such that and $Γ^{'} (\cdot)$ is bounded and uniformly continuous.

Let $b = t_{2} = 1, t_{0} = s_{0} = 0 t_{1} = \frac{1}{3}, s_{1} = \frac{2}{3}$ , $H = L^{2} ([0, 1], R))$ equipped with the norm $‖ \cdot ‖$ . Define $A : D (A) \subset H \to H$ by $Ay = y^{″},$ with domain

\begin{array}{l} D (A) & = {y \in H : y, y^{'} are\ absolutely\ continuous, \\ y^{′′} \in H, y (0) = y (1) = 0} . \end{array}

It is well known that $A$ generates a compact C₀-semigroup ${(T (t))}_{t \geq 0}$ given by

\begin{array}{l} T (t) y = - \sum_{n = 1}^{+ \infty} e^{- n^{2} t} ⟨ y, e_{n} ⟩ e_{n}, y \in H \end{array}

where $e_{n} (z) = \sqrt{\frac{2}{π}} \sin (nz), n = 1, 2, \dots$ denotes the orthonormal basis for the space $H .$ Thus, (H₁) holds.

Let $γ : D (A) \subset H \to H$ defined by $γ (t) z = Γ (t) A z$ for $z \in D (A) .$ In order to rewrite Eq.(Equation4.1(4.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}$ (4.1) ) in abstract form in $H$ , we introduce the following notation

(4.2)

\begin{array}{l} {\begin{cases} v (t) & = u (t, \cdot) for t \in [0, 1], \\ v_{0} & = u_{0} (t, \cdot) for t \in [0, 1] . \end{cases} \end{array}

(4.2)

Let the functions $g$ , $K$ and $f$ be defined as

\begin{array}{l} g (t, v (t), K v (t)) & = \frac{{(v (t))}^{'}}{2} + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} v (s)} ds, \\ K v (t) & = \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} v (s)} ds, \\ f (t, v (t)) & = \frac{v (t)}{2 (1 + v (t))} . \end{array}

Then (Equation4.1(4.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}$ (4.1) ) takes the following abstract form

(4.3)

\begin{array}{l} {\begin{cases} v^{'} (t) = A v (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [0, 1 / 3) \cup [2 / 3, 1], \\ v (t) = f (t, v (t)), t \in [0, 1 / 3) \cup [2 / 3, 1], \\ v (0) = v_{0} . \end{cases} \end{array}

(4.3)

for k = 1.

Moreover, we suppose that Γ is bounded and C¹-function such that $Γ^{'}$ is bounded and uniformly continuous. Then, the linear integrodifferential system corresponding to (Equation4.1(4.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}$ (4.1) ) has a resolvent operator ${(R (t))}_{t \geq 0}$ on $H .$

On the other hand, by the definitions of the functions $g$ , $K$ and $f$ above, observe that

The function $K : [0, 1] \times H \times H \to H$ is continuous with respect to t and differentiable with respect to v for all v. This means that there exists positive constant $\hat{K}$ such that
$\begin{array}{l} | K v - K w | \leq \hat{K} ‖ v - w ‖ for v, w \in B_{τ_{0}} . \end{array}$
The function $g : [0, 1] \times H \times H \to H$ is continuous with respect to t and is differentiable with respect to v and $K v$ . Therefore, there exist positive constants $η_{g}$ and ${\hat{η}}_{g}$ such that
$\begin{array}{l} | g (t, v_{1}, K v_{1}) - g (t, v_{2}, K v_{2}) | \\ \leq η_{g} ‖ v_{1} - v_{2} ‖ + {\hat{η}}_{g} ‖ K v_{1} - K v_{2} ‖, \\ v_{1}, v_{2} \in B_{τ_{0}} for\ some τ_{0}, \end{array}$
The impulse functions $f_{k}$ are continuous with respect to t and Lipschitz continuous with respect to v with Lipschitz constant $η_{k} = \frac{1}{2} < 1$ .

Therefore, all the assumptions of Theorem 3.2 are fulfilled. Hence, the existence of a unique mild solution of (Equation4.1(4.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}$ (4.1) ) follows on $[0, 1] .$

5. Existence results for equations with nonlocal conditions

In this section, we establish the existence of mild solutions for the following system:

(5.1)

{\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}

(5.1)

in the Banach space $H$ .

Definition 5.1.

A function $v \in PC (J, H)$ is called mild solution of system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) on $[0, b]$ if v(t) satisfies the integral equation

(5.2)

\begin{array}{l} v (t) = {\begin{cases} R (t) (v_{0} + q (v)) \\ + \int_{0}^{t} R (t - s) g (t, v (s), K v (s)) ds, t \in [0, t_{1}), \\ f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ R (t - s_{k}) f_{k} (s_{k}, v (s_{k})) \\ + \int_{s_{k}}^{t} R (t - s) g (t, v (s), K v (s)) ds, t \in [s_{k}, t_{k + 1}), \end{cases} \end{array}

(5.2)

where $K v (t) = \int_{0}^{t} ζ (t, s, v (s)) ds$ is the linear operator defined on $H$ .

To formulate our problem, we will make the following assumptions :

The resolvent operator ${(R (t))}_{t \geq 0}$ associated with the system (Equation1.1(1.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \end{cases}$ (1.1) ) is compact and there exists a constant M > 0 such that ${| R (t) |}_{L (H)} \leq M .$
The function $g (t, \cdot, \cdot)$ is continuous and $g (\cdot, v, w)$ is measurable on $[0, T]$ . In addition, there exists a function $χ_{g} \in L^{1} ([0, T], R)$ such that $| g (t, v, w) | \leq χ_{g} (t)$ for all $v, w \in Y$ .
The operator $K : [0, T] \times Y \to Y$ is continuous and there exists a constant $\hat{K}$ such that
$\begin{array}{l} ‖ K v - K w ‖ \leq \hat{K} ‖ v - w ‖ . \end{array}$
The functions $f_{k} : [t_{k}, s_{k}] \times Y \to Y$ are continuous and there exist positive constants $0 < η_{k} < 1$ such that
$\begin{array}{l} | f_{k} (t, v (t)) - f_{k} (t, w (t)) | \leq η_{k} ‖ v - w ‖ . \end{array}$
The operator $q : PC \to H$ is Lipschitz continuous with respect to v with Lipschitz constant $0 < η_{q} \leq 1$ .

Theorem 5.2

Assume that conditions (H₁)-(H₂) and (B₁)-(B₅) are satisfied, then the nonlocal integrodifferential system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) has a mild solution provided $M η_{k} < 1$ and $M η_{q} < 1$ .

Proof.

For any $v \in B_{τ_{0}}$ , we have by using (B₁) and (B₅)

(5.3)

\begin{array}{l} | R (t) (v_{0} + q (v)) | \leq M (| v_{0} | + η_{q} {‖ v ‖}_{PC} + | q (0) |) . \end{array}

(5.3)

Moreover for $t \in [0, t_{1})$ , by (B₁) and (B₂), we have the following estimate

(5.4)

\begin{array}{l} \int_{0}^{t} | R (t - s) g (t, v (s), K v (s)) | ds & \leq M \int_{0}^{t} χ_{g} (s) ds \\ \leq M N_{0}, \end{array}

(5.4)

where $N_{0} = {‖ χ_{g} ‖}_{L^{1} ([0, b])}$ .

For $t \in [0, t_{1})$ and for positive τ₀, we define Λ₁ and Λ₂ on $B_{τ_{0}}$ as

\begin{array}{l} {\begin{cases} Λ_{1} v (t) = R (t) (v_{0} + q (v)), \\ Λ_{2} v (t) = \int_{0}^{t} R (t - s) g (t, v (s), K v (s)) ds . \end{cases} \end{array}

It yields that $v (t)$ is the mild solution of the system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) if only if the operator equation $v = Λ_{1} v + Λ_{2} v$ has a solution for $v \in B_{τ_{0}}$ for some τ₀. Thus, for $t \in [0, t_{1})$ , we show there exists a positive constant τ₀ such that $Λ_{1} + Λ_{2}$ has a fixed point on $B_{τ_{0}}$ .

Step 1. ${‖ Λ_{1} v + Λ_{2} w ‖}_{PC} \leq τ_{0}$ for some positive τ₀.

Let $v, w \in B_{τ_{0}}$ , choose

\begin{array}{l} τ_{0} = \frac{M}{1 - M η_{q}} (| v_{0} | + | q (0) | + N_{0}), \end{array}

and consider

\begin{array}{l} | Λ_{1} v (t) + Λ_{2} w (t) | \\ \leq | R (t) (v_{0} + q (v)) | \\ + | \int_{0}^{t} R (t - s) g (t, w (s), K w (s)) ds | \\ \leq M (‖ v_{0} | + η_{q} {‖ v ‖}_{PC} + | q (0) |) + M N_{0} \\ \leq M (| v_{0} | + | q (0) | + N_{0}) + M η_{q} τ_{0} \\ \leq (1 - M η_{q}) τ_{0} + M η_{q} τ_{0} \\ \leq τ_{0} . \end{array}

Therefore, ${‖ Λ_{1} v + Λ_{2} w ‖}_{PC} \leq τ_{0}$ , for every pair $v, w \in B_{τ_{0}}$ .

Step 2. Λ₁ is a contraction on $B_{τ_{0}}$ .

For any $v, w \in B_{τ_{0}}$ and $t \in [0, t_{1})$ , using (B₅), we get:

\begin{array}{l} | Λ_{1} v (t) - Λ_{1} w (t) | & \leq | R (t) (q (v) - q (w)) | \\ \leq M η_{q} {‖ v - w ‖}_{PC} . \end{array}

Taking the supremum over $[0, t_{1})$ on the above inequality, we obtain

\begin{array}{l} {‖ Λ_{1} v - Λ_{1} w ‖}_{PC} \leq M η_{q} {‖ v - w ‖}_{PC} . \end{array}

Since $M η_{q} < 1,$ we conclude that Λ₁ is contraction.

Step 3. We show that Λ₂ is a completely continuous operator on $B_{τ_{0}}$ .

Claim 1. The operator Λ₂ is continuous.

Let ${v_{n}}$ be a sequence in $B_{τ_{0}}$ converging to $v \in B_{τ_{0}}$ . Thus,

\begin{array}{l} | Λ_{2} v_{n} (t) - Λ_{2} v (t) | \\ \leq \int_{0}^{t} | R (t - s) [g (t, v_{n} (s), K v_{n} (s)) \\ - g (t, v (s), K v (s))] | ds \\ \leq M \int_{0}^{t} sup_{s \in [0, t_{1})} | g (t, v_{n} (s), K v_{n} (s)) \\ - g (t, v (s), K v (s)) | ds \\ \leq M t_{1} sup_{s \in [0, t_{1})} | g (t, v_{n} (s), K v_{n} (s)) - g (t, v (s), K v (s)) |, \end{array}

which implies

\begin{array}{l} {‖ Λ_{2} v_{n} - Λ_{2} v ‖}_{PC} & \leq M t_{1} sup_{s \in [0, t_{1})} | g (t, v_{n} (s), K v_{n} (s)) \\ - g (t, v (s), K v (s)) | . \end{array}

By the continuity of $g$ and $K$ , the right-hand side of the above inequality tends to zero as $n \to \infty$ . From which yields that ${‖ Λ_{2} v_{n} - Λ_{2} v ‖}_{PC} \to 0$ as $n \to \infty .$ Hence, Λ₂ is continuous.

Claim 2. The set ${Λ_{2} v (t), v \in B_{τ_{0}}}$ is relatively compact.

In this situation, we will only need to show that the family of functions ${Λ_{2} v (t), v \in B_{τ_{0}}}$ is uniformly bounded and equicontinuous, and for any $t \in [0, t_{1})$ , ${Λ_{2} v (t), v \in B_{τ_{0}}}$ is relatively compact in $H$ .

It is easily seen that for any $v \in B_{τ_{0}},$ ${‖ Λ_{2} v ‖}_{PC} \leq τ_{0}$ . Thus, the family ${Λ_{2} v (t), v \in B_{τ_{0}}}$ is uniformly bounded. Next, we show that ${Λ_{2} v, v \in B_{τ_{0}}}$ is equicontinuous.

For any $0 \leq l_{1} < l_{2} < t_{1}$ and $v \in B_{τ_{0}}$ ,

\begin{array}{l} | Λ_{2} v (l_{2}) - Λ_{2} v (l_{1}) | \\ = | \int_{0}^{l_{2}} R (l_{2} - s) g (s, v (s), K v (s)) ds \\ - \int_{0}^{l_{1}} R (l_{1} - s) g (s, v (s), K v (s)) ds | \\ \leq \int_{0}^{l_{1}} | [R (l_{2} - s) - R (l_{1} - s)] g (s, v (s), K v (s)) | ds \\ + \int_{l_{1}}^{l_{2}} | R (l_{2} - s) g (s, v (s), K v (s)) | ds \\ \leq \int_{0}^{l_{1}} | [R (l_{2} - s) - R (l_{1} - s)] g (s, v (s), K v (s)) | ds \\ + M \int_{l_{1}}^{l_{2}} χ_{g} (s) ds . \end{array}

Using the strong continuity of the resolvent operator ${(R (t))}_{t \geq 0}$ and the fact that $g (s, \cdot, \cdot)$ is measurable, the Lebesgue dominated theorem implies that the right-hand side of the above inequality tends to zero as $l_{1} \to l_{2} .$ Thus, $| Λ_{2} v (l_{2}) - Λ_{2} v (l_{1}) |$ tends to zero as $l_{1} \to l_{2}$ independently of $v \in B_{τ_{0}} .$ Hence, ${Λ_{2} v, v \in B_{τ_{0}}}$ is equicontinuous.

Now, we show that the family $U (t) = {Λ_{2} v (t), v \in B_{τ_{0}}}$ for all $t \in [0, t_{1})$ is relatively compact. Of course, $U (0)$ is relatively compact.

Fix $t \in [0, t_{1})$ and take ϵ such that $0 < ϵ < t$ . For $v \in B_{τ_{0}}$ , we define the operators

\begin{array}{l} Λ_{2}^{* ϵ} v (t) = R (ϵ) \int_{0}^{t - ϵ} R (t - s - ϵ) g (t, v (s), K v (s)) ds, \end{array}

and

\begin{array}{l} {\tilde{Λ}}_{2}^{* ϵ} v (t) = \int_{0}^{t - ϵ} R (t - s) g (t, v (s), K v (s)) ds . \end{array}

By Lemma 2.3 and the compactness of the operator $R (ϵ),$ the set $U_{ϵ} (t) = {Λ_{2}^{* ϵ} v (t), v \in B_{τ_{0}}}$ is relatively compact in $H,$ for every $ϵ,$ $ϵ \in (0, t) .$ Furthermore, also using Lemma 2.3, for each $v \in B_{τ_{0}}$ , we have

\begin{array}{l} | Λ_{2}^{* ϵ} v (t) - {\tilde{Λ}}_{2}^{* ϵ} v (t) | \\ = | \int_{0}^{t - ϵ} R (ϵ) R (t - s - ϵ) g (t, v (s), K v (s)) ds \\ - \int_{0}^{t - ϵ} R (t - s) g (t, v (s), K v (s)) ds | \\ \leq \int_{0}^{t - ϵ} | R (ϵ) R (t - s - ϵ) - R (t - s) | | g (t, v (s), K v (s)) | ds \\ \leq Qϵ \int_{0}^{t - ϵ} χ_{g} (s) ds . \end{array}

Thus, the set ${\tilde{U}}_{ϵ} (t) = {{\tilde{Λ}}_{2}^{* ϵ} v (t), v \in B_{τ_{0}}}$ is precompact in $H$ by utilizing the total boundedness.

Applying this idea once more, we arrive at

\begin{array}{l} | Λ_{2} v (t) - {\tilde{Λ}}_{2}^{* ϵ} v (t) | \\ = | \int_{0}^{t} R (t - s) g (t, v (s), K v (s)) ds \\ - \int_{0}^{t - ϵ} R (t - s) g (t, v (s), K v (s)) ds | \\ \leq \int_{t - ϵ}^{t} | R (t - s) g (t, v (s), K v (s)) | ds \\ \leq M \int_{t - ϵ}^{t} χ_{g} (s) ds \to 0 as ϵ \to 0, \end{array}

and there are precompact sets arbitrarily close to the set $U (t)$ . Thus, ${Λ_{2} v (t), v \in B_{τ_{0}}}$ is precompact in $H .$

Hence, thanks to the Ascoli-Arzela theorem, the operator Λ₂ is completely continuous on $B_{τ_{0}}$ . Applying the Krasnoselskii’s fixed point theorem, $Λ_{1} + Λ_{2}$ has a fixed point on $B_{τ_{0}},$ corresponding to a mild solution of (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) on $[0, t_{1})$ .

Next, on the interval $[t_{k}, s_{k})$ , for $k = 1, 2, \dots, m$ , and for positive τ₀, define Λ₁ and Λ₂ on $B_{τ_{0}}$ as follows

\begin{array}{l} {\begin{cases} Λ_{1} v (t) = f_{k} (t, v (t)), \\ Λ_{2} v (t) = 0, \end{cases} \end{array}

therefore, $v (t)$ is a mild solution of the system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) if only if $v (t)$ is a fixed point of the operator equation $v = Λ_{1} v + Λ_{2} v$ on $B_{τ_{0}}$ for some τ₀. Consequently, the existence of mild solution of (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) over the interval $[t_{k}, s_{k})$ is equivalent to determining a positive constant τ₀ such that $Λ_{1} + Λ_{2}$ has a fixed point on $B_{τ_{0}}$ . This holds immediately by virtue of (B₄).

Now, on the interval $[s_{k}, t_{k + 1})$ , for all $k = 1, 2, \dots, m$ , and for positive τ₀, define Λ₁ and Λ₂ on $B_{τ_{0}}$ as follows

\begin{array}{l} {\begin{cases} Λ_{1} v (t) = R (t - s_{k}) f_{k} (s_{k}, v (s_{k})), \\ Λ_{2} v (t) = \int_{s_{k}}^{t} R (t - s) g (t, v (s), K v (s)) ds . \end{cases} \end{array}

Then, $v (t)$ is the mild solution of the system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) if only if there exist a solution for the operator equation $v = Λ_{1} v + Λ_{2} v$ for $v \in B_{τ_{0}}$ . Then, in order to show the existence of a mild solution for the system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) over the interval $[s_{k}, t_{k + 1})$ , we will only need to determine a positive constant τ₀ such that $Λ_{1} + Λ_{2}$ has a fixed point on $B_{τ_{0}}$ .

Choosing $τ_{0} = \frac{M}{1 - M η_{k}} (| v_{0} | + | f (t, 0) | + N_{0})$ and following the same procedure as for interval $[0, t_{1})$ , the Krasnoselskii’s fixed point theorem gives that $Λ_{1} + Λ_{2}$ has fixed point on $B_{τ_{0}}$ corresponding to a mild solution of system (Equation5.1(5.1) ${\begin{cases} v^{'} (t) = Av (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in [s_{k}, t_{k + 1}), k = 1, 2, \dots, m, \\ v (t) = f_{k} (t, v (t)), t \in [t_{k}, s_{k}), \\ v (0) = v_{0} + q (v), \end{cases}$ (5.1) ) on $[s_{k}, t_{k + 1})$ . This completes the proof.

6. Example

In this section, we used the results obtained in the previous area by analyzing the following noninstantaneous impulsive integro-differential equations with nonlocal conditions :

(6.1)

\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, x) = \frac{\partial^{2}}{\partial x^{2}} u (t, x) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial x^{2}} u (s, x) ds \\ + \frac{1}{25} \frac{e^{- t}}{1 + e^{t}} u (t, x) + \frac{1}{50} \int_{0}^{t} e^{- s} u (s, x) ds, \\ t \in (0, 1 / 3] \cup (2 / 3, 1], x \in [0, 1], \\ u (t, x) = \frac{u (t, x)}{10 (1 + u (t, x))}, t \in] 1 / 3, 2 / 3], x \in [0, 1], \\ u (0, x) = u_{0} (x) + \sum_{k = 1}^{2} \frac{1}{3^{k}} u (\frac{1}{k}, x), x \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \end{cases} \end{array}

(6.1)

where $b : [0, + \infty [\to R$ is continuously differentiable and b^ʹ is bounded and uniformly continuous.

Let $b = t_{2} = 1, t_{0} = s_{0} = 0 t_{1} = \frac{1}{3}, s_{1} = \frac{2}{3}$ , $H = L^{2} ([0, 1], R))$ equipped with the norm $‖ \cdot ‖$ and $A : D (A) \subset H \to H$ be defined by $A z = \partial^{2} / \partial z^{2}$ with domain

\begin{array}{l} D (A) & = {z \in H : z, z^{'} are\ absolutely\ continuous, \\ z^{′′} \in H, z (0) = z (1) = 0} . \end{array}

Then $A$ generates a compact C₀-semigroup ${(S (t))}_{t \geq 0}$ given by

\begin{array}{l} S (t) z = - \sum_{n = 1}^{+ \infty} e^{- n^{2} t} ⟨ z, e_{n} ⟩ e_{n}, z \in H \end{array}

where $e_{n} (y) = \sqrt{\frac{2}{π}} \sin (ny), n = 1, 2, \dots$ denotes the orthonormal basis for the space $H .$ Thus, (H₁) holds.

Take $γ : D (A) \subset H \to H$ defined by $γ (t) y = Γ (t) A y$ for $y \in D (A)$ . To rewrite Eq.(Equation4.1(4.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, ξ) = \frac{\partial^{2}}{\partial ξ^{2}} u (t, ξ) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial ξ^{2}} u (s, ξ) ds \\ + u (t, ξ) u_{ξ} (t, ξ) + \frac{1}{9} \int_{0}^{t} e^{- \frac{1}{4} u (s, ξ)} ds, \\ t \in [0, 1 / 3) \cup [2 / 3, 1], ξ \in [0, 1], \\ u (t, ξ) = \frac{u (t, ξ)}{2 (1 + u (t, ξ))}, t \in [1 / 3, 2 / 3), ξ \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \\ u (0, ξ) = u_{0} (ξ), ξ \in [0, 1], \end{cases} \end{array}$ (4.1) ) in abstract form in $H$ , we introduce the following notation

(6.2)

\begin{array}{l} {\begin{cases} v (t) = u (t, \cdot) for t \in [0, 1], \\ v_{0} = u_{0} (t, \cdot) for t \in [0, 1], \end{cases} \end{array}

(6.2)

Let the functions $g, f$ and q be defined as follows

\begin{array}{l} g (t, v (t), K v (t)) & = \frac{1}{25} \frac{e^{- t}}{(1 + e^{t})} u (t, x) + \frac{1}{50} \int_{0}^{t} e^{- s} v (s) ds, \\ K v (t) & = \frac{1}{50} \int_{0}^{t} e^{- s} v (s) ds, \\ f (t, v (t)) & = \frac{e^{- (t - \frac{1}{3})}}{4} \frac{| v (t) |}{(1 + | v (t)) |}, \\ q (v) & = \sum_{k = 1}^{2} \frac{1}{3^{k}} v (\frac{1}{k}) . \end{array}

Then (Equation6.1(6.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, x) = \frac{\partial^{2}}{\partial x^{2}} u (t, x) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial x^{2}} u (s, x) ds \\ + \frac{1}{25} \frac{e^{- t}}{1 + e^{t}} u (t, x) + \frac{1}{50} \int_{0}^{t} e^{- s} u (s, x) ds, \\ t \in (0, 1 / 3] \cup (2 / 3, 1], x \in [0, 1], \\ u (t, x) = \frac{u (t, x)}{10 (1 + u (t, x))}, t \in] 1 / 3, 2 / 3], x \in [0, 1], \\ u (0, x) = u_{0} (x) + \sum_{k = 1}^{2} \frac{1}{3^{k}} u (\frac{1}{k}, x), x \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \end{cases} \end{array}$ (6.1) ) takes the following abstract form

(6.3)

\begin{array}{l} {\begin{cases} v^{'} (t) = A v (t) + \int_{0}^{t} γ (t - s) v (s) ds \\ + g (t, v (t), \int_{0}^{t} ζ (t, s, v (s)) ds), \\ t \in) 0, 1 / 3] \cup] 2 / 3, 1], \\ v (t) = f_{k} (t, v (t)), t \in] 1 / 3, 2 / 3], \\ v (0) = v_{0} + q (v) . \end{cases} \end{array}

(6.3)

for k = 1. Moreover, we suppose b is bounded and C¹-function such that b^ʹ is bounded and uniformly continuous. Then, the linear part of the abstract formulation of (Equation6.1(6.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, x) = \frac{\partial^{2}}{\partial x^{2}} u (t, x) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial x^{2}} u (s, x) ds \\ + \frac{1}{25} \frac{e^{- t}}{1 + e^{t}} u (t, x) + \frac{1}{50} \int_{0}^{t} e^{- s} u (s, x) ds, \\ t \in (0, 1 / 3] \cup (2 / 3, 1], x \in [0, 1], \\ u (t, x) = \frac{u (t, x)}{10 (1 + u (t, x))}, t \in] 1 / 3, 2 / 3], x \in [0, 1], \\ u (0, x) = u_{0} (x) + \sum_{k = 1}^{2} \frac{1}{3^{k}} u (\frac{1}{k}, x), x \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \end{cases} \end{array}$ (6.1) ) has a resolvent operator ${(R (t))}_{t \geq 0}$ on $H .$ Accordingly, (H₂) holds. Since the semigroup generated by $A$ is compact, Theorem 2.4 implies that the corresponding resolvent operator is compact. In addition, from the expressions of the functions $g, f$ and q defined above, one can easily see that the assumptions (B₂)-(B₅) hold as well as with

\begin{array}{l} χ_{g} (t) := \frac{3}{50} \frac{e^{- t}}{(1 + e^{t})}, η_{k} = \frac{1}{4}, \hat{K} = \frac{1}{50}, η_{q} = \frac{4}{9} . \end{array}

We take M = 1, then $M η_{k} < 1 and M η_{q} < 1$ . Thus all the assumptions of Theorem 5.2 are satisfied. Hence, it follows the existence of mild solutions of (Equation6.1(6.1) $\begin{array}{l} {\begin{cases} \frac{\partial}{∂t} u (t, x) = \frac{\partial^{2}}{\partial x^{2}} u (t, x) + \int_{0}^{t} Γ (t - s) \frac{\partial^{2}}{\partial x^{2}} u (s, x) ds \\ + \frac{1}{25} \frac{e^{- t}}{1 + e^{t}} u (t, x) + \frac{1}{50} \int_{0}^{t} e^{- s} u (s, x) ds, \\ t \in (0, 1 / 3] \cup (2 / 3, 1], x \in [0, 1], \\ u (t, x) = \frac{u (t, x)}{10 (1 + u (t, x))}, t \in] 1 / 3, 2 / 3], x \in [0, 1], \\ u (0, x) = u_{0} (x) + \sum_{k = 1}^{2} \frac{1}{3^{k}} u (\frac{1}{k}, x), x \in [0, 1], \\ u (t, 0) = u (t, 1) = 0, t \in [0, 1], \end{cases} \end{array}$ (6.1) ) on $[0, 1] .$

7. Conclusion

In this paper, we have studied a class of non-instantaneous impulsive integro-differential system with local and nonlocal conditions. Using the resolvent operator as well as the general Banach contraction theorem, we proved the existence and uniqueness of a mild solution of the system with local conditions. On the other hand, the system with nonlocal conditions is investigated, and the existence of mild solutions is established via the Kranoselsk’iis fixed point theorem. In addition, two examples are worked out to illustrate the obtained theoretical results. One of the main goals of a future research study will be to look at the approximate controllability of non-instantaneous fractional impulsive stochastic integro-differential systems with local and nonlocal conditions that are driven by a Rosenblatt process.

Acknowledgements

The authors would like to thank the referees and the editor for their substantial comments and suggestions, which have significantly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Gaston Berger university [0045].

Notes on contributors

Mamadou Abdoul Diop

Mamadou Abdoul Diop received his MSc in applied mathematics from Gaston Berger University. His research interest is toward

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A note on existence results for noninstantaneous impulsive integrodifferential systems

ABSTRACT

1. Introduction

2. Preliminaries

2.1. Partial integrodifferential equations in Banach spaces

(Grimmer, Citation1982)

(Desch et al., Citation1984)

(Grimmer, Citation1982)

(J. Liang et al., Citation2008)

(Desch et al.,)

Banach fixed point theorem

Krasnoselskii’s fixed point theorem

Completely continuous operators

3. Existence results for equation with local conditions

4. Example

5. Existence results for equations with nonlocal conditions

6. Example

7. Conclusion

Acknowledgements

Disclosure statement

Notes on contributors

Mamadou Abdoul Diop

References

Information for

Open access

Opportunities

Help and information

A note on existence results for noninstantaneous impulsive integrodifferential systems

ABSTRACT

1. Introduction

2. Preliminaries

2.1. Partial integrodifferential equations in Banach spaces

(Grimmer, Citation1982)

(Desch et al., Citation1984)

(Grimmer, Citation1982)

(J. Liang et al., Citation2008)

(Desch et al.,)

Banach fixed point theorem

Krasnoselskii’s fixed point theorem

Completely continuous operators

3. Existence results for equation with local conditions

4. Example

5. Existence results for equations with nonlocal conditions

6. Example

7. Conclusion

Acknowledgements

Disclosure statement

Additional information

Funding

Notes on contributors

Mamadou Abdoul Diop

References

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