We study the existence of solutions of impulsive semilinear differential equation in a Banach space X in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.

1. Introduction

In a few decades, impulsive differential equations have received much attention of researchers mainly due to its demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, medicine and so on. The real world processes and phenomena which are subjected during their development to short-term external inuences can be modeled as impulsive differential equation. Their duration is negligible compared to the total duration of the entire process or phenomena. Impulsive differential equations are an appropriate model to hereditary phenomena for which a delay argument arises in the modelling equations. To further study on impulsive differential equations, we refer to books [1, 2] and papers [3–11].

In this paper, our purpose is to establish the existence of a solution to the following differential equations with non instantaneous impulses

()

where −A : D(A) ⊂ X → X is a closed and bounded linear operator with dense domain D(A) ⊂ X. We assume that −A is the infinitesimal generator of a strongly continuous semigroup

in a Banach space (X, ∥·∥). Here, x₀ ∈ X, 0 = t₀ = s₀ < t₁ ≤ s₁ ≤ t₂ < ⋯<t_δ ≤ s_δ < t_δ+1 = T₀ and F : [0, T₀] × X → X, G_i : (t_i, s_i] × X → X for all i = 1, …, δ are suitable functions to be specified later.

In [4], authors have introduced a new class of abstract impulsive differential equations in which impulses are not instantaneous and established the existence of solutions to the problem (1)–(3) with the assumption that operator A generates a C₀-semigroup of bounded linear operators. In this system of (1)–(3), the impulses begin all of a sudden at the points t_i and their proceeding continues on a finite time interval [t_i, s_i] [5]. To concern the hemodynamical harmony of an individual we think about the following simplified situation. One can recommend a few intravenous sedates (insulin) on account of a decompensation (e.g., high or low level of glucose). Since the presentation of the medications in the bloodstream and the ensuing retention for the form are progressive and continuous processes, we can depict this circumstance as an impulsive activity which begins abruptly and stays animated on a finite time interval.

In [12], the generalization of the condensing operator as convex-power condensing operator has been introduced by Sun and Zhang and a new fixed point theorem for convex-power condensing operator has been established. The new fixed point theorem for convex-power condensing operator, defined by Sun and Zhang, is the generalization of the famous Schauder’s fixed point theorem and Sadovskii’s fixed point theorem. Sun and Zhang [12] have considered the following problem:

()

in a Banach space and established existence theorems for positive mild solutions and global mild solutions to the problem (4) with noncompact semigroup. The nonlinear function f fulfills the suitable conditions on the measure of noncompactness as

for any bounded set B ⊂ X, where α is the Kuratowski measure of noncompactness and

is a positive constant. For more details about measure of noncompactness, we refer to [12–20].

In the present work, our aim is to obtain results concerning the existence of mild solutions to problem (1)–(3) by using convex-power condensing operator and fixed point theorem for convex-power condensing operator Sun and Zhang [12].

The organization of the paper is as follows. We provide some basic definitions, Lemmas and theorems in Section 2 as “preliminaries.” We prove the existence of a mild solution for system (1) in Section 3 as “existence of mild solution.” In the last section, we present an example to illustrate the application of the abstract results.

2. Preliminaries

In this section, we give some definitions, notations, theorems, and lemmas which will be used in later sections.

Let (X, ∥·∥) be a real Banach space. The symbol C([0, T₀]; X) stands for the Banach space of all the continuous functions from [0, T₀] into X equipped with the norm ∥z∥ = sup⁡{∥z(s)∥, s ∈ [0, T₀]} and L¹([0, T₀], X) stands for the space of X-valued Bochner integrable functions on [0, T₀] endowed with the norm , f ∈ L¹([0, T₀], X).

The operator −A : D(A) ⊂ X → X is the infinitesimal generator of a uniformly continuous semigroup

and D(A) denotes the domain of A, which is densely defined, endowed with the graph norm. A semigroup

is said to be equicontinuous if

is equicontinuous at arbitrary t, 0 < t < ∞, for any bounded subset K ⊂ X. Throughout the paper we assume that

(H1)
the operator −A generates the equicontinuous semigroup and there exists a positive number M such that .

For the study of impulsive differential equation, we define the space

which contains all the functions y : [0, T₀] → X such that y(·) is continuous at t ≠ t_j,

and

exists for all j = 1,2, …, δ. Clearly,

is a Banach space endowed with norm

. For a function

and i ∈ {0,1, …, δ}, we define the function

such that

()

For

, we have

and we have following Accoli-Arzelà type criteria.

Lemma 1 (see [4].)A set is relatively compact in if and only if each set (j = 0,1, …, δ) is relatively compact in C([t_j, t_j+1], X) (j = 0,1, …, δ).

Now we present the following definition of mild solution.

Definition 2. A piecewise continuous function is said to be a mild solution of the (1)–(3) if u(0) = x₀, u(t) = G_i(t, u(t)), for all t ∈ (t_i, s_i], i = 1, …, δ, and

()

for all t ∈ [0, t₁], and

()

for all t ∈ [s_i, t_i+1] and every i = 1, …, δ.

Next, we give the definition of the Hausdorff measure of noncompactness (MNC).

Definition 3 (see [14].)The Hausdorff measure of noncompactness β of the set E in Banach space X is the greatest lower bound of those ε > 0 for which the set E has in the space X a finite ε-net; that is,

()

for every bounded subset E in a Banach space X.

Definition 4 (see [14].)The Kuratowski measure of noncompactness α defined on each bounded subset E of X as

()

The relation between Kuratowski measure of noncompactness α and the Hausdorff measure of noncompactness β is given by

()

To set the structure for our primary existence results, we review some essential properties about the Kuratowski and Hausdorff measure of noncompactness.

Lemma 5. For any bounded set U, V ⊂ Y, where Y is a Banach space. Then, we have following results:

(i)
β(U) = 0 if and only if U is precompact;
(ii)
, where conv U and denote the convex hull and closure of U, respectively;
(iii)
β(U) ⊂ β(V), when U ⊂ V;
(iv)
β(U + V) ≤ β(U) + β(V), where U + V = {u + v : u ∈ U, v ∈ V};
(v)
β(U ∪ V) ≤ max⁡{β(U), β(V)};
(vi)
β(λU) = λ · β(U), for any ;
(vii)
If the map is continuous and satisfies the Lipschitsz condition with constant κ, then, we have that for any bounded subset , where Y and are Banach space.

Definition 6. A continuous and bounded map is called β-condensing if, for any noncompact bounded subset E ⊂ D,

()

where X is a Banach space.

To avoid confusion, we denote by β(·) the Hausdorff measure of noncompactness on set X, C([0, T₀]; X) and .

Lemma 7 (see [14], Darbo-Sadovskii.)Let D ⊂ X be bounded, closed, and convex. If the continuous map is a β-contraction, then the map has a fixed point in D.

In [12], authors have presented the generalization of the condensing operator and suggested a new fixed point theorem for such operators. Firstly, we present some recognition. Let D ⊂ X be a bounded, closed, and convex set and let

be a continuous map from D into itself with u₀ ∈ D. For every E ⊂ D, we set

()

Definition 8 (see [12].)Let D ⊂ X be bounded, closed, and convex. A bounded and continuous operator is called a convex-power condensing operator if for any bounded nonprecompact subset E ⊂ D, there exist x₀ ∈ D and an integer n₀ > 0 such that

()

A β-convex-power condensing operator is β-condensing for n₀ = 1. Therefore, it is clear that convex-power condensing operator is a generalization of the condensing operator.

Lemma 9 (see [12].)Let D ⊂ X be bounded, closed, and convex set. If the continuous map is β-convex-power condensing, then there exists a fixed point of map in D.

Lemma 10 (see [11], [14].)If E⊆C([0, T₀]; X) is bounded, then β(E(t)) ≤ β(E), ∀t ∈ [0, T₀], where E(t) = {u(t); u ∈ E}⊆X. In addition, if E is equicontinuous on [0, T₀], then β(E(t)) is continuous on [0, T₀] and

()

Lemma 11 (see [11].)If is bounded, then β(E(t)) ≤ β(E), ∀t ∈ [0, T₀]. Besides, suppose the following conditions are satisfied:

(1)
E is equicontinuous on [0, t₁] and each (t_i, s_i], [s_i, t_i+1], i = 1, …, δ,
(2)
E is equicontinuous at , i = 1, …, δ.

Then, we have

Lemma 12 (see [14].)If E ⊂ C([0, T₀]; X) is bounded and equicontinuous, then β(E(t)) is continuous and

()

where

Lemma 13. Let be a sequence of functions in . Suppose that there exist satisfying ∥u_n(t)∥ ≤ γ(t) for almost all t ∈ [0, T₀] and every n ≥ 1. Then, we have

()

Lemma 14. We assume that (H1) holds. Then the set , ∥u(ζ)∥ ≤ η(ζ) for a.e. ζ ∈ [0, T₀]} is equicontinuous for all t ∈ [0, T₀].

Proof. Let h be a positive constant such that 0 ≤ t < t + h ≤ T₀. For , we have

()

It is obvious for t = 0. Let ϵ > 0 be arbitary integer with 0 < ϵ < t. For t > 0, we have

()

Since

, t > 0 is equicontinuous; therefore,

()

as h → 0, uniformly for u. Second and third terms of (18) tend to zero when ϵ → 0 since ϵ is arbitrary small.

Then from (17), (18), and (19) and the absolute continuity of integrals, we obtain that , ∥u(ζ)∥ ≤ η(ζ) for a.e. ζ ∈ [0, T₀]} is equicontinuous for all t ∈ [0, T₀].

3. Main Results

In this section, the existence of the mild solution of (1)–(3) under some specified conditions on G_i is established by using the measure of noncompactness and fixed point theorem for convex-power condensing operator.

Now, we made the following hypothesis which will be useful for proving our results.

(HG)
The functions G_i (i = 1, …, δ) are compact and continuous.
(HF)
F : [0, T] × X → X is a nonlinear function such that F satisfies the Carathèodary condition; that is, we have the following.
- (i)
  F(τ, ·) : X → X is continuous for a.e. τ ∈ [0, T₀].
- (ii)
  For each x ∈ X, F(·, x):[0, T₀] → X is strongly measurable. For any k > 0, there exists a function such that for u ∈ X
  ()
- (iii)
  There exists a constant such that for any bounded set B ⊂ X,
  ()
(HG1)
The functions G_i are continuous and there exist positive constants such that
()
for all x, y ∈ X, τ ∈ (t_i, s_i], (i = 1, …, δ).
(Hk)

()
(H)
.

Theorem 15. Suppose that (H1), (HG), (HF), and (Hk) are satisfied. Then, there exists at least one mild solution on [0, T₀] for the problem (1)–(3).

Proof. We define the operator as , , for all t ∈ (t_i, s_i] and

()

for i = 0,1, …, δ and for t ∈ [s_i, t_i+1], where i = 1, …, δ,

()

To prove the result, we show that the operator

has a fixed point. Firstly, we show that the

is continuous on

. Let

be a sequence in

such that lim⁡_n→∞u_n = u in

. For t ∈ [s_i, t_i+1], we have

()

By the continuity of F and G_i (i = 1 … , δ), we have

()

Therefore from (26), (27), (28), and Lebesgue dominated convergence theorem, we get

()

which implies that

is continuous on [s_i, t_i+1]. For t ∈ [0, t₁], we get

()

From the (27), we get

()

thus,

is continuous on [0, t₁]. Hence, we conclude that

is continuous on [0, T₀].

Secondly, we claim that , where . For each and t ∈ [0, t₁], we get

()

For t ∈ (s_i, t_i+1], (i = 1, …, δ), we have

()

which implies that

, for all i = 1, …, δ. On the other hand, by the property of G_i(·), we get

()

for t ∈ (t_i, s_i]. By the assumption (Hk), we have

Therefore, we conclude that ; that is, has values in B_k.

Now we show the equicontinuity of on [0, T₀]. Since G_i(·) is compact, therefore it is obvious that is equicontinuous on (t_i, s_i]. Assume t ∈ [0, t₁]. Let h > 0 be a constant such that 0 < t < t + h ≤ t₁ < T₀. For u ∈ B_k, we get

()

Using the semigroup property, we have

()

By the strong continuity of

and Lemma 14, we conclude that

is equicontinuous on [0, t₁].

For t ∈ [s_i, t_i+1] we have

()

Since we have

()

G_i(·) is compact and

is strongly continuous, which implies that

is equicontinuous on [s_i, t_i+1]. Hence,

is equicontinuous on each [0, T₀].

Set , where conv and denote the convex hull and closure of the convex hull, respectively. It can be shown easily that maps B into itself and B is equicontinuous on each [0, t₁], (t_i, s_i], [s_i, t_i+1], i = 1,2, …, δ. Next we prove that is a convex-power condensing operator. We take u₀ ∈ B and show that there exists a positive integer n₀ such that

()

for every nonprecompact bounded subset H ⊂ B. From (6), (15), and compactness of G_i, for t ∈ [s_i, t_i+1], where i = 1, …, δ, we have

()

and similarly for t ∈ [0, t₁], we have

()

For t ∈ [t_i, s_i], we get

by the fact that G_i(·) are compact.

Further for t ∈ [s_i, t_i+1], we have

()

Proceeding with this iterative method, we get

()

for t ∈ [s_i, t_i+1] and similarly for t ∈ [0, t₁], we get

. Thus, we obtain

()

for all t ∈ [0, T₀].

We have that is equicontinuous on [0, T₀] by Lemma 14. Therefore, from Lemma 11, we get

()

since we have that

as n → ∞, which infers that there exists a substantial enough positive integer n₀ such that

()

which means that

is a convex-power condensing operator. Therefore, from Lemma 10, we get that

has at least one fixed point in B which is just a mild solution to the problem (1)–(3). This completes the proof of the theorem.

In the next result, the existence of the solution for problem (1)–(3) under Lipschitz conditions of G_i is established by using Darbo-Sadovskii’s fixed point theorem.

Theorem 16. Suppose that assumptions (H1), (HF), (HG1), and (H) hold. Then, the impulsive problem (1)–(3) has at least one mild solution on [0, T₀].

Proof. Firstly, we decompose the map such that , where , i = 0,1, 2, …, δ, j = 1,2, are defined as

()

For t ∈ (t_i, s_i] and u, v ∈ B_k, we have

()

and for t ∈ (s_i, t_i+1]

()

therefore, for all t ∈ [0, T₀] we conclude that

()

Since

. By Lemma 5 (vii), we have that for any bounded set

()

For operator

, we have

()

From (51)–(53), we have

()

From the assumption (H), we get that

, which implies that map

is β-condensing in B_k. Therefore, by Darbo-Sadovskii’s fixed point theorem, the solution map Q has a fixed point in B_k which is a mild solution of the nonlocal problem (1)–(3). This completes the proof of the theorem.

4. Example

Let us consider the following impulsive problem:

()

where 0 = t₀ = s₀ < t₁ ≤ s₁ < ⋯<t_N ≤ s_N < t_N+1 = T₀ are fixed numbers, z ∈ X,

, and

, for all i = 1, …, N.

To convert problem (55) into the abstract form (1)–(3), where u(t) = w(t, ·), that is, u(t)x = w(t, x), we introduced the functions F : [0, T₀] × X → X and G_i : (t_i, s_i] × X → X such that F(t, u(t))(x) = f(t, w(t, x)) and G_i(t, u(t))(x) = g_i(t, w(t, x)). We consider X = L²[0, π] and the operator A defined as −Au = u^′′ with the the domain

()

It is well known (Pazy [21]) that −A is the infinitesimal generator of an analytic semigroup

, t ≥ 0 and every analytic semigroup is equicontinuous. This means that A satisfies the assumption (H1).

Next we have that is a mild solution of the problem (55).

Case 1. We take

()

and here

are continuous functions and

. Now, we show that F satisfies the assumption (HF). Now, we have that for u ∈ X

()

and for u₁, u₂ ∈ X

()

Therefore, for any bounded sets D₁ ⊂ X, we obtain

()

where

. For u ∈ X, we have

()

Clearly, G_i, i = 1, …, δ are compact and satisfy the assumption (HG). Then by Theorem 15, problem (55) has at least a mild solution.

Case 2. Take f(t, w(t, x)) = f₁(t)w(t, x) and g_i(t, w(t, x)) = (1/(5 + α_it³))[|w(t, x)|/(1+|w(t, x)|)], ; here, is a bounded and continuous function and α_i > 0.

For t ∈ (s_i, t_i+1], i = 1, …, δ and , we have

()

where m_k(t) = |f₁(t)|. For any bounded set D₁ ⊂ X, we have

()

where

. Thus, (HF) holds. For

, we have

()

Thus, (HG1) holds. Then by Theorem 16, problem (55) has at least a mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi.

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Existence of the Mild Solution for Impulsive Semilinear Differential Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

Conflict of Interests

Acknowledgments

References

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Existence of the Mild Solution for Impulsive Semilinear Differential Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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