Approximate solvability method for nonlocal impulsive evolution equation

Weifeng Ma; Yongxiang Li

doi:10.1515/math-2023-0155

Article Open Access

Approximate solvability method for nonlocal impulsive evolution equation

Weifeng Ma and Yongxiang Li

Published/Copyright: December 9, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C₀-semigroup.

Keywords: impulsive evolution equation; C 0-semigroup; Yosida approximations; approximate solvability method; Hausdorff measure of noncompactness

MSC 2010: 34G20; 34K45; 47D03

1 Introduction

This article discusses the following nonlocal problem for the first-order semilinear impulsive evolution equation in a reflexive Banach space E :

(1.1) u ′ ( t ) = A u ( t ) + f ( t , u ( t ) ) , t ∈ J , t ≠ t k , Δ u ( t k ) = I k ( u ( t k ) ) , k = 1 , 2 , … , p , u ( 0 ) = g ( u ) ,

where A : D ( A ) ⊂ E → E is a linear operator and A generates a nonexpansive C 0 -semigroup T ( t ) ( t ≥ 0 ) . J = [ 0 , b ] , b > 0 is a constant, 0 = t 0 < t 1 < t 2 < ⋯ < t p < t p + 1 = b , Δ u ∣ t = t k denotes the jump of u ( t ) at t = t k , i.e., Δ u ∣ t = t k = u ( t k + ) − u ( t k − ) , where u ( t k + ) and u ( t k − ) represent the right and left limits of u ( t ) at t = t k , respectively. g : P C ( J , E ) → E is a linear operator, which is a nonlocal function.

Many dynamic evolving processes are subject to abrupt changes (called “impulsive”) that involve short-term perturbations from continuous and smooth dynamics. The duration of such systems is negligible in comparison with the duration of an entire evolution. One may consult the monographs [1,2] for background on this topic. The nonlocal initial problem has better effects in applications compared with the classical initial value problems, and nonlocal problems have been extensively investigated in the literature [3–8]. Therefore, introducing a classical method of analysis to this growing field of study is meaningful. By applying the operator semigroup theory, Byszewski et al. [8] obtained the uniqueness of a mild solution for semilinear parabolic evolution equation with nonlocal condition

u ( 0 ) + g ( t 1 , t 2 , … , t m , u ( ⋅ ) ) = u 0 ,

which is a pioneering work about the study of the nonlocal initial value problem. Under the conditions that impulsive terms are continuous and nonlocal function satisfies continuous or Lipschitz continuous, Fan and Li [9] obtained the existence of mild solutions for impulsive evolution equation (1.1) by using the Darbo-Sadovskii fixed-point theorem and the technique of approximate solutions, which solves the difficulty that the compactness problem of solution operator at zero.

In recent years, Benedetti et al. [10] developed a new approximation solvability method, which is a generalization of the well-known strong approximation used by Brower and Petryshyn [11] for solving semilinear nonlocal differential equations. By applying the weak topology [12] and Yosida approximations of the infinitesimal generator of C 0 -semigroup, they obtained the existence of mild solutions for problem

u ′ ( t ) = A u ( t ) + f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( 0 ) = M u

in a reflexive Banach space. Then, they proved that the set of mild solutions is strongly compact when the C 0 -semigroup is compact. In addition, without assuming compactness on the evolution system, Xu et al. [13] investigated the existence of mild solutions for a nonlocal, non-autonomous evolution equation

u ′ ( t ) = A ( t ) u ( t ) + f ( t , u ( t ) ) , t > 0 , u ( 0 ) = M u

on an unbounded interval by applying an approximation solvability method that is based on the reduction to a finite dimensional problem by virtue of the projections.

Inspired by the above results and overcoming the influence of discontinuous points on the systems, we deal with the existence of mild solutions for nonlocal problem (1.1) in reflexive Banach space with noncompact C 0 -semigroup. The transversality condition of nonlinear term f guarantees the absence of fixed points of the solution operator in the boundary of a given candidate set of solutions, which is a key ingredient when attaching solvability by means of fixed-point theorems. Compared with the existing literature on the argument, we construct the weak convergence of impulsive functions in continuous space, which is an innovation point of this article, and then, by exploiting the Yosida approximations of the infinitesimal generator of C 0 -semigroup and the approximation solvability method, the existence theorem of mild solutions for nonlocal problem (1.1) is proved. Furthermore, by applying the Hausdorff measure of noncompactness, we obtain the uniqueness result of mild solution.

In the next section, we recall and introduce some basic concepts and Lemmas. In Section 3, we prove main results of this article. An example is given to demonstrate how to utilize our results in the last section.

2 Preliminaries

Let E σ and ⟨ ⋅ , ⋅ ⟩ be the space E endowed with the weak topology and the dual product between E and E * (the dual of E ), respectively. For every x ∈ E , if there exists a unique scalars sequence { α m ( x ) } ⊂ R such that

x = lim m → ∞ ∑ i = 1 m α i ( x ) e i ,

we say the sequence { e m } ⊂ E is a Schauder basis of E . Given a basis { e i } i = 1 m generating the m -dimensional subspace of E , for every x ∈ E , define the projection P m : E → E m as P m x = ∑ i = 1 m α i ( x ) e i with α i ∈ E * , i ∈ N . For every m ∈ N , if ‖ P m ‖ = 1 , we said that the Schauder basis { e m } satisfies ( π 1 ) (see [14,15]).

Lemma 2.1

The projection P m : E → E m has the following properties:

P m : E σ → E m is continuous.
If x m ⇀ x , then P m x m ⇀ x as m → ∞ in E .
I − P m → 0 , as m → ∞ , where I is an identity map.

Let C ( J , E ) be the Banach space of all continuous E -value functions on interval J with the maximum norm ‖ x ‖ C = max t ∈ J ‖ x ( t ) ‖ and L 1 ( J , E ) be the Banach space of all integrable functions with the norm ‖ x ‖ 1 = ∫ 0 b ∣ x ( t ) ∣ d t . Let P C ( J , E ) = { u : J → E , u ( t ) is continuous at t ≠ t k , left continuous at t = t k , and u ( t k + ) exists, k = 1 , 2 , … , p } . Evidently, P C ( J , E ) is also a Banach space with the norm ‖ x ‖ P C = max t ∈ J ‖ x ( t ) ‖ . Let E be a reflexive Banach space, and we define the duality map

J ( x ) = { j ( x ) ∈ E * : ‖ j ( x ) ‖ = x and ⟨ x , j ( x ) ⟩ = ‖ x ‖ 2 } .

Then, for every m ∈ N , introduce the adjoint projection P m * : E * → E m * as

⟨ x , P m * j ( x ) ⟩ = ⟨ P m x , j ( x ) ⟩ .

Obviously, for every m ∈ N , we have P m * j ( x ) → j ( x ) (see [7]). B E ( 0 , R ) denotes the closed ball B E ( 0 , R ) = { x ∈ E : ‖ x ‖ ≤ R } .

Lemma 2.2

[16] If f m ⇀ f in L 1 ( J , E ) , then P m f m ⇀ f in L 1 ( J , E ) .

Lemma 2.3

If I k m ⇀ I k in C ( J , E ) , then P m I k m ⇀ I k in C ( J , E ) .

Proof

Since I k m ⇀ I k , for any Ψ ∈ C * ( J , E ) , there exists a bounded variation function v , such that

Ψ ( I k ) = ∫ 0 b I k ( t ) d v ( t ) .

By the Lebesgue-Stieltjes dominated convergence theorem, we conclude that Ψ ( I k m ) → Ψ ( I k ) . From the linearity of Ψ and P m , it follows that

Ψ ( P m I k ) = ∫ 0 b P m I k ( t ) d v ( t ) = ∫ 0 b ( P m I k ( t ) − I k ( t ) ) d v ( t ) + ∫ 0 b I k ( t ) d v ( t ) = ∫ 0 b ( P m − I ) I k ( t ) d v ( t ) + Ψ ( I k ) .

Again by the Lebesgue-Stieltjes dominated convergence theorem, we have Ψ ( P m I k m ) → Ψ ( I k ) . Then, P m I k m ⇀ C ( J , E ) I k .□

Lemma 2.4

[17] A sequence of continuous functions { x n } n ⇀ x ∈ C ( J , E ) if and only if there exists N > 0 such that for every n ∈ N , t ∈ J , ‖ x n ( t ) ‖ ≤ N , and x n ( t ) ⇀ x ( t ) in E .

From { x n } n ⇀ x ∈ C ( J , E ) , we can deduce that { x n } n ⇀ x ∈ L 1 ( J , E ) . Therefore, we obtain that P m I k m ⇀ I k ∈ L 1 ( J , E ) . To prove our main results, we need the following continuation principle:

Lemma 2.5

[16] Let ℱ be a closed convex subset of a Banach space E with nonempty interior ℱ and let Q : [ 0 , 1 ] × ℱ → E be a compact map having a closed graph such that Q ( 0 , ℱ ) ⊂ ℱ and Q ( ξ , ⋅ ) is fixed point free on ∂ ℱ for all ξ ∈ [ 0 , 1 ] . Then, there exists y ∈ ℱ such that y = Q ( 1 , y ) .

3 Main results

In this section, we give our main results. First, we introduce the following assumptions:

For every x ∈ E , the function f ( t , x ) : J × E → E satisfies conditions:
1. the function f ( ⋅ , x ) : J → E is measurable for every x ∈ E ;
2. the function f ( t , ⋅ ) : E σ → E σ is continuous for a.e. t ∈ J ;
3. for a.e. t ∈ J and every x ∈ B E ( 0 , R ) ,
  ⟨ f ( t , x ) , j ( x ) ⟩ ≤ 0 ;
4. for every bounded subset ℬ ⊂ E , there exists a function ν ℬ ∈ L 1 ( J , R ) , such that
  ‖ f ( t , x ) ‖ ≤ ν ℬ ( t )
  for a.e. t ∈ J and all x ∈ ℬ .
the maps I k : E σ → E σ is continuous.
g : P C ( J , E ) → E is a linear and bounded operator with ‖ g ‖ ≤ 1 .

Let B = B E ( 0 , R ) and ℱ = C ( J , B ) . For every m ∈ N , set ℱ m = ℱ ∩ C ( J , E m ) . Define the continuous operator Q : J × ℱ → P C ( J , E ) by

Q ( x ) ( t ) = T ( t ) g ( x ) + ∫ 0 t T ( t − τ ) f ( τ , x ( τ ) ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( x ( t k ) ) .

The mild solution for problem (1.1) on ℱ implies that

x = Q ( x ) .

Theorem 3.1

Assume conditions (H1)–(H3) hold. Then, the set of mild solutions for problem (1.1) on ℱ is nonempty and weakly compact in P C ( J , E ) .

Proof

For each λ ∈ N , let A λ = λ 2 R ( λ , A ) − λ I denote the Yosida approximation of linear operator A , where R ( λ , A ) = ( λ I − A ) − 1 and I denotes the identity operator. It is well known (see [6,18]) that A λ ∈ ℒ ( E ) and { e t A λ } t ≥ 0 is a semigroup of contractions such that

lim λ → ∞ e t A λ x = T ( t ) x

for every t ∈ J and x ∈ E . For every λ ∈ N , we consider the problem

(3.1) u ′ ( t ) = A λ u ( t ) + f ( t , u ( t ) ) , t ∈ J , t ≠ t k , Δ u ( t k ) = I k ( u ( t k ) ) , k = 1 , 2 , … , p , u ( 0 ) = g ( u ) .

Applying the bounding function technique and approximation solvability method, we prove that problem (3.1) has a solution u ∈ ℱ . From this point on, we construct the auxiliary problem in P C ( J , E m ) :

(3.2) u ′ ( t ) = P m A λ u ( t ) + P m f ( t , u ( t ) ) , t ∈ J , t ≠ t k , Δ u ( t k ) = P m I k ( q ( t k ) ) , k = 1 , 2 , … , p , u ( 0 ) = P m g ( u ) ,

I. Now, we prove that problem (3.2) has a solution u m ∈ ℱ for each m ∈ N .

Fix m ∈ N and q ∈ ℱ m . For the linear Cauchy problem

(3.3) u ′ ( t ) = ζ P m A λ q ( t ) + ζ P m f ( t , q ( t ) ) , t ∈ J , t ≠ t k , Δ u ( t k ) = ζ P m I k ( q ( t k ) ) , k = 1 , 2 , … , p , u ( 0 ) = ζ P m g ( q ) ,

where ζ ∈ [ 0 , 1 ] , we can determine that problem (3.3) has a unique solution, see [19]. Thus, we can define the solution operator Σ : [ 0 , 1 ] × ℱ m → P C ( J , E m ) as follows:

Σ ( ζ , q ) ( t ) = ζ P m g ( q ) + ζ ∫ 0 t ( P m A λ q ( τ ) + P m f ( τ , q ( τ ) ) ) d τ + ζ ∑ 0 < t k < t P m I k ( q ( t k ) ) .

Obviously, Σ is well defined and the fixed point of Σ is equivalent to the mild solution of problem (3.3).

In the sequel, we conclude that operator Σ has closed and compact graph. In fact, let { q n } ∈ N ⊂ ℱ m and { ζ n } ∈ N ⊂ [ 0 , 1 ] be two strong convergent sequences with q n → q 0 ∈ ℱ m and ζ n → ζ 0 ∈ [ 0 , 1 ] . Assume the sequence Σ ( q n , ζ n ) strongly converges to x 0 ∈ P C ( J , E m ) , and we prove that x 0 = Σ ( q 0 , ζ 0 ) . In virtue of the linearity and continuity of the operators g , P m , and A λ , we can deduce that ζ n P m g ( q n ) → E ζ 0 P m g ( q 0 ) and P m A λ q n ( t ) → E P m A λ q 0 ( t ) for every t ∈ J . Combining the continuity of f and I k with the finiteness of the dimensional of space E m , we have that P m f ( t , q n ( t ) ) → E P m f ( t , q 0 ( t ) ) for a.e. t ∈ J and ζ n P m I k ( q n ( t k ) ) → E ζ 0 P m I k ( q 0 ( t k ) ) . By the assumption (iv) of (H1), we can obtain that

‖ P m A λ q n ( t ) + P m f ( t , q n ( t ) ) ‖ ≤ R ‖ A λ ‖ ℒ + ν B ( t )

for a.e. t ∈ J . Applying the Lebesgue dominated convergence theorem, we have

ζ n ∫ 0 t ( P m A λ q n ( τ ) + P m f ( τ , q n ( τ ) ) ) d τ → E ζ 0 ∫ 0 t ( P m A λ q 0 ( τ ) + P m f ( τ , q 0 ( τ ) ) ) d τ

for a.e. t ∈ J . By the uniqueness of the limit, for every t ∈ J ,

x 0 ( t ) = ζ 0 P m g ( q 0 ) + ζ 0 ∫ 0 t ( P m A λ q 0 ( τ ) + P m f ( τ , q 0 ( τ ) ) ) d τ + ζ 0 ∑ 0 < t k < t P m I k ( q 0 ( t k ) ) .

Denote x n = Σ ( q n , ζ n ) and let { x n } be a sequence of solutions for problem (3.3), for a.e. t ∈ J , and we can conclude that

‖ x n ′ ( t ) ‖ ≤ R ‖ A λ ‖ ℒ + ν B ( t ) .

Then, { x n ′ } is uniformly integrable. It follows the equicontinuity of the sequence { x n } . By the condition (H2), there exists a constant a 1 , such that ‖ I k ( x ) ‖ ≤ a 1 ‖ x ‖ . Furthermore, from the condition (H3), it follows that

‖ x n ( t ) ‖ ≤ R + ∫ 0 t ‖ x n ′ ( τ ) ‖ d τ + p a 1 R ,

which implies the equiboundedness of { x n } . By means of the Arzela-Ascoli theorem, we can obtain the precompact of { x n } .

Since Σ ( ⋅ , 0 ) = 0 ⊂ int ℱ m , assume that there exists ( ζ , q ) ∈ ( 0 , 1 ) × ∂ ℱ m such that q = Σ ( ζ , q ) , or equivalently,

q ′ ( t ) = ζ P m A λ q ( t ) + ζ P m f ( t , q ( t ) ) , t ∈ J , t ≠ t k , Δ q ( t k ) = ζ P m I k ( q ( t k ) ) , k = 1 , 2 , … , p , q ( 0 ) = ζ P m g ( q ) .

For q ∈ ∂ ℱ m , there exists t ˜ ∈ [ 0 , b ] such that ‖ q ( t ˜ ) ‖ = R . If t ˜ = 0 , then

R = ‖ q ( 0 ) ‖ = ‖ ζ P m g ( q ) ‖ ≤ ζ ‖ q ‖ C < R .

Thus, t ˜ ∈ ( 0 , b ] . Taking a sufficiently small ε > 0 such that ‖ q ( t ˜ − ε ) ‖ < R , and for all t ∈ ( t ˜ − ε , t ˜ ) , ‖ q ( t ) k ‖ ≤ R . Since A λ is dissipative, i.e.,

⟨ A λ q ( t ) , j ( q ( t ) ) ⟩ ≤ 0 ,

and combining (iii) of (H1) with the continuity of f and A λ , we can conclude that

∫ t ˜ − ε t ˜ ⟨ P m A λ q ( t ) + P m f ( t , q ( t ) ) , j ( q ( t ) ) ⟩ d t ≤ 0 .

However,

∫ t ˜ − ε t ˜ ⟨ P m A λ q ( t ) + P m f ( t , q ( t ) ) , j ( q ( t ) ) ⟩ d t = ∫ t ˜ − ε t ˜ ⟨ q ′ ( t ) , j ( q ( t ) ) ⟩ d t = ∫ t ˜ − ε t ˜ d d t ( ‖ q ( t ) ‖ 2 ) = ‖ q ( t ˜ ) ‖ 2 − ‖ q ( t ˜ − ε ) ‖ 2 > 0 ,

which gives a contradiction. Thus, for each m ∈ N , from Lemma 2.5, there exists a solution u m ∈ ℱ m for problem (3.2).

II. From now on, we show that for every λ ∈ N , problem (3.1) has a solution.

Let f m ( t ) = A λ u m ( t ) + f ( t , u m ( t ) ) and I k m ( t k ) = I k ( u m ( t k ) ) , { u m } ∈ ℱ . According to conditions (iv) of (H1) and (H2), there exist a function ν * ∈ L 1 ( J , E ) and a constant a 1 ∈ R such that ‖ f m ( t ) ‖ ≤ ν * ( t ) , ‖ I k m ( t k ) ‖ ≤ a 1 R ( k = 1 , 2 , … , p ) for a.e. t ∈ J and every m ∈ N . Therefore, { f m } is a bounded and uniformly integrable sequence in L 1 ( J , E ) , and { I k m } are equibounded in C ( J , E ) for m big. Then, from Lemma 2.4, we have that { f m } and { I k m } are weak relatively compact in L 1 ( J , E ) . Without loss of generality, let

f m ⇀ L 1 ( J , E ) f 0 , I k m ⇀ L 1 ( J , E ) I k .

By Lemmas 2.2 and 2.3, we have that u m ′ = P m f m ⇀ L 1 ( J , E ) f 0 and P m I k m ⇀ L 1 ( J , E ) I k . Furthermore, the set { u m ( 0 ) : m ∈ N } is bounded in the reflexive Banach space E . Assuming

(3.4) u m ( 0 ) ⇀ E γ 0 ,

consider the continuous function

u 0 ( t ) ≔ γ 0 + ∫ 0 t f 0 ( τ ) d τ + ∑ 0 < t k < t I k 0 ( t k ) , t ∈ J .

It is easy to see that u 0 is continuous and u 0 ′ ( t ) = f 0 ( t ) for a.e. t ∈ J . Clearly, for all t ∈ J ,

u m ( t ) = u m ( 0 ) + ∫ 0 t u m ′ ( τ ) d τ + ∑ 0 < t k < t I k m ( t k ) ⇀ E u 0 ( t ) .

From Lemma 2.1 and the continuity of operator A λ , for every weak neighborhood U of A λ u 0 ( t ) + f ( t , u 0 ( t ) ) and a.e. t ∈ J , there exists m 0 = m 0 ( t , U ) such that for m > m 0 , f m ( t ) ∈ U . By Mazur lemma (see [15]), there exists a sequence of convex combinations { f ¯ ( m ) } ,

f ¯ ( m ) = ∑ k = m ∞ σ m k f k , σ m k ≥ 0 , ∑ k = m ∞ σ m k = 1

which converges to u ′ in L 1 ( J , X ) . In virtue of [20], without loss of generality, we assume

f ¯ ( m ) ( t ) → X u ′ ( t ) , a.e. t ∈ J .

Thus, from the convexity of the set V , it follows that

f ¯ ( m ) ( t ) ∈ V , m ≥ m 0 .

Therefore, by the uniqueness of weak limit and Lemma 2.4, we have

u m ⇀ P C ( J , E ) u 0 .

Thus, g ( u m ) ⇀ E g ( u 0 ) . By equation (3.4), we obtain g ( u 0 ) = γ 0 = u 0 ( 0 ) , and it follows that u 0 is a solution of problem (3.1).

III. Then, we show that problem (1.1) has a mild solution on ℱ .

Let u n ∈ ℱ be a solution of problem (3.1), namely,

u n ( t ) = e t A λ g ( u n ) + ∫ 0 t e ( t − τ ) A λ f ( τ , u n ( τ ) ) d τ + ∑ 0 < t k < t e ( t − t k ) A λ I k ( u n ( t k ) )

for each λ ∈ N . For { u n } ∈ ℱ and by condition (H3) and the reflexivity of space E , there exists u ¯ ∈ E such that, up to subsequence, g ( u n ) ⇀ E u ¯ . Denote f n ( t ) = f ( t , u n ( t ) ) and I k n ( t k ) = I k ( u n ( t k ) ) , the condition (iv) of (H1) implies that there exists a function ν * ∈ L 1 ( J , R ) such that

‖ f n ( t ) ‖ ≤ ν * ( t ) , a.e. t ∈ J .

Then, from condition (H2), there exists a constant a 1 , such that ‖ I k n ( t k ) ‖ ≤ a 1 R ( k = 1 , 2 , … , p ) . Hence, { f n } are bounded and uniformly integrable in L 1 ( J , E ) for n big and { I k m } are equibounded in C ( J , E ) for m big. Lemma 2.4 implies that { f m } and { I k m } are weak relatively compact in L 1 ( J , E ) . Without loss of generality, assume that

f n ⇀ L 1 ( J , E ) f , I k n ⇀ L 1 ( J , E ) I k ,

For every t ∈ J , one obtains

∫ 0 t ‖ e ( t − τ ) A λ f ( τ , u n ( τ ) ) ‖ d τ ≤ ∫ 0 t ‖ f n ( τ ) ‖ d τ ≤ ‖ ν * ‖ 1 , ‖ e ( t − t k ) A λ I k n ( t k ) ‖ ≤ a 1 R ,

similarly,

∫ 0 t ‖ T ( t − τ ) f ( τ ) ‖ d τ ≤ ‖ ν * ‖ 1 , ‖ T ( t − t k ) I k ( t k ) ‖ ≤ a 1 R .

Therefore, the maps e ( t − ⋅ ) A λ f n ∈ L 1 ( J , E ) and T ( t − ⋅ ) f ∈ L 1 ( J , E ) for a.e. t ∈ J . In addition, e ( t − t k ) A λ I k n ∈ L 1 ( J , E ) and T ( t − t k ) I k ∈ L 1 ( J , E ) for every t ∈ J . Combining the Lebesgue dominated convergence theorem with the definition of weakly convergence, we conclude that

e ( t − ⋅ ) A λ f n ⇀ L 1 ( J , E ) T ( t − ⋅ ) f ,

and

e ( t − t k ) A λ I k n ⇀ L 1 ( J , E ) T ( t − t k ) I k .

Therefore, for every t ∈ J , we have

u n ( t ) ⇀ T ( t ) u ¯ + ∫ 0 t T ( t − τ ) f ( τ ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u ( t k ) ) ≔ u ( t ) .

Then, u n ⇀ P C ( J , E ) u ∈ ℱ . Furthermore, if x n ⊂ E , x n ⇀ E x , then, for each t ∈ J , e t A λ x n ⇀ E T ( t ) x . Thus, by (H1) and (H3), we have

e t A λ g ( u n ) ⇀ E T ( t ) g ( u ) ,

∫ 0 t e ( t − τ ) A λ f ( τ , u n ( τ ) ) d τ ⇀ E ∫ 0 t T ( t − τ ) f ( τ , u ( τ ) ) d τ ,

and

e ( t − t k ) A λ I k ( u n ( t k ) ) ⇀ E T ( t − t k ) I k ( u ( t k ) ) .

So, for t ∈ J , again by the uniqueness of weak limit, we can obtain that

u ( t ) = T ( t ) g ( u ) + ∫ 0 t T ( t − τ ) f ( τ , u ( τ ) ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u ( t k ) )

is a solution of problem (1.1).

IV. In sequel, we show that the set of solutions for problem (1.1) is weakly compact in P C ( J , E ) .

Let { u n } ⊂ ℱ be a sequence of solutions to problem (1.1), i.e.,

u n ( t ) = T ( t ) g ( u ) + ∫ 0 t T ( t − τ ) f ( τ , u n ( τ ) ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u n ( t k ) ) .

Denoting f n ( t ) = f ( t , u n ( t ) ) and I k n ( t k ) = I k ( u n ( t k ) ) , n ∈ N , by means of conditions (iv) of (H1) and (H2), there exist a function ν * ∈ L 1 ( J , R ) and a constant a 1 ∈ R such that

‖ f n ( t ) ‖ ≤ ν * ( t ) , a.e. t ∈ J , ‖ I k n ( t k ) ‖ ≤ a 1 R .

So, we have that { f m } and { I k m } are weak relatively compact in L 1 ( J , E ) . In general, we assume that

f n ⇀ L 1 ( J , E ) f , I k n ⇀ L 1 ( J , E ) I k .

For every t ∈ J , we have

∫ 0 t T ( t − τ ) f n ( τ ) d τ ⇀ E ∫ 0 t T ( t − τ ) f ( τ ) d τ ,

and

T ( t − t k ) I k n ⇀ E T ( t − t k ) I k .

For { u n } ⊂ ℱ , by condition (H1) and the reflexivity of space E , we obtain that there exists u ¯ ∈ E such that, up to subsequence, g ( u n ) ⇀ E u ¯ . Then, for every t ∈ J , we have

u n ( t ) ⇀ T ( t ) u ¯ + ∫ 0 t T ( t − τ ) f ( τ ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u ( t k ) ) ≔ u ( t ) .

Therefore, u n ⇀ P C ( J , E ) u ∈ ℱ . Again, by the linearity and continuity of the semigroup { T ( t ) } t ≥ 0 and conditions (ii) of (H1), (H2), and (H3), we can obtain that

T ( t ) g ( u n ) ⇀ E T ( t ) g ( u ) ,

∫ 0 t T ( t − τ ) f ( τ , u n ( τ ) ) d τ ⇀ E ∫ 0 t T ( t − τ ) f ( τ , u ( τ ) ) d τ ,

and

T ( t − t k ) I k ( u n ( t k ) ) ⇀ E T ( t − t k ) I k ( u ( t k ) ) .

Thus, from the uniqueness of the weak limit, for every t ∈ J , we obtain that

u ( t ) = T ( t ) g ( u ) + ∫ 0 t T ( t − τ ) f ( τ , u ( τ ) ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u ( t k ) ) .

Then, we can claim that the set of mild solutions of problem (1.1) is weakly compact.□

Let Ω ⊂ C ( J , E ) be a bounded set. Consider the modulus of fiber of noncompactness as the measure of noncompactness in the space P C ( J , E ) , namely,

γ ( Ω ) = sup t ∈ J χ ( Ω ( t ) ) ,

with χ as the Hausdorff measure of noncompactness in E (more details, see [21–23]).

In the sequel, we discuss the uniqueness of mild solution for problem (1.1). Now, we assume the following Lipschitz conditions:

There exists a constant a 2 , such that
‖ I k ( x ) − I k ( y ) ‖ ≤ a 2 ‖ x − y ‖ , k = 1 , 2 , … , p
for all x , y ∈ B E ( 0 , R ) ;
There exists a function L ( t ) ∈ L 1 ( J , E ) such that
‖ f ( t , x ) − f ( t , y ) ‖ ≤ L ( t ) ‖ x − y ‖
for a.e. t ∈ J and all x , y ∈ B E ( 0 , R ) ,

with the following condition:

(3.5) ‖ g ‖ + ‖ L ‖ 1 + a 2 p < 1 .

Theorem 3.2

Let conditions (i)–(iii) of (H1), (H2) ′ , (H3), (H4), and (3.5) hold. Then, problem (1.1) has a unique mild solution on ℱ .

Proof

Fix λ ∈ N . From the proof of Theorem 3.1, we can obtain that problem (3.2) has a solution u m ∈ ℱ m and m ∈ N with

u m ( t ) = e t P m A λ P m g ( u m ) + ∫ 0 t e ( t − τ ) P m A λ P m f ( τ , u m ( τ ) ) d τ + ∑ 0 < t k < t e ( t − t k ) P m A λ P m I k ( u m ( t k ) ) .

For any t ∈ J 1 = [ 0 , t 1 ] , since g is a bounded linear operator and { u m } is a bounded sequence, we obtain

χ ( { u m ( t ) } ) ≤ χ ( { e t P m A λ P m g ( u m ) ( t ) } ) + χ ∫ 0 t e ( t − τ ) P m A λ P m f ( τ , u m ( τ ) ) d τ ≤ ‖ e t P m A λ ‖ χ ( g { u m ( t ) } ) + χ ∫ 0 t e ( t − τ ) P m A λ P m f ( τ , u m ( τ ) ) d τ ≤ ‖ g ‖ γ ( { u m } ) + χ ∫ 0 t e ( t − τ ) P m A λ P m f ( τ , u m ( τ ) ) d τ .

For every t ∈ J 1 , λ ∈ N , since A λ generates a contraction semigroup, we can obtain that ‖ e t P m A λ ‖ ≤ 1 . By ( ii ) ′ , for t ∈ J , one has

χ ( { e ( t − τ ) P m A λ P m f ( τ , u m ( τ ) ) d τ } ) ≤ ‖ e ( t − τ ) P m A λ ‖ ⋅ χ ( { P m f ( τ , u m ( τ ) ) d τ } ) ≤ L ( τ ) χ ( { u m ( t ) } ) ≤ L ( τ ) γ ( { u m } ) .

Hence,

χ ( { u m ( t ) } ) ≤ ‖ g ‖ γ ( { u m } ) + γ ( { u m } ) ∫ 0 t L ( τ ) d τ .

Thus,

γ ( { u m } ) ≤ γ ( { u m } ) ( ‖ g ‖ + ‖ L ‖ 1 ) .

From equation (3.5), it follows that γ ( { u m } ) = 0 , namely, for every t ∈ J 1 , χ ( { u m ( t ) } ) = 0 , therefore, χ ( { u m ( t 1 ) } ) = 0 , then χ ( { I 1 ( u m ( t 1 ) ) } ) = 0 .

For every t ∈ J 2 = ( t 1 , t 2 ] , λ ∈ N , we have

χ ( { e ( t − τ ) P m A λ P m f ( τ , u n ( τ ) ) d τ } ) ≤ ‖ e ( t − τ ) P m A λ ‖ ⋅ χ ( { P m f ( τ , u n ( τ ) ) d τ } ) ≤ L ( τ ) χ ( { u m ( τ ) } ) ≤ L ( τ ) γ ( { u m } ) .

Hence, since χ ( { I 1 ( u m ( t 1 ) ) } ) = 0 , we have

χ ( { u m } ( t ) ) ≤ ‖ g ‖ γ ( { u m } ) + γ ( { u m } ) ∫ t 1 t L ( τ ) d τ

Therefore,

γ ( { u m } ) ≤ γ ( { u m } ) ( ‖ g ‖ + ‖ L ‖ 1 ) .

Then, we have γ ( { u m } ) = 0 . Obviously, for every t ∈ J 2 , χ ( { u m ( t ) } ) = 0 ; furthermore, we have that χ ( { u m ( t 2 ) } ) = 0 , so, χ ( { I 2 ( u m ( t 2 ) ) } ) = 0 .

Continuing such process interval by interval, for every t ∈ J , the set { u m ( t ) } is relatively compact. For u m ∈ ℱ m , from (H4), it follows that { u m } is relatively compact in P C ( J , E ) and { u m ′ } is bounded and uniformly integrable in L 1 ( J , E ) , so u m → u in ℱ and u m ′ ⇀ u ′ in L 1 ( J , E ) . By (H3), we have that

u ( 0 ) = lim m → ∞ u m ( 0 ) = lim m → ∞ P m g ( u m ) = g ( u ) .

Moreover, from the continuity of A λ , for every t ∈ J , we can obtain that f ( t , u m ( t ) ) → f ( t , u ( t ) ) , A λ u m ( t ) → A λ u ( t ) , and I k ( u m ( t k ) ) → I k ( u ( t k ) ) . Then,

u m ′ ( t ) = P m A λ u m ( t ) + P m f ( t , u m ( t ) ) → A λ u ( t ) + f ( t , u ( t ) ) .

From the convergence of dominated and the uniqueness of weak limit, it follows that

u ′ ( t ) = A λ u ( t ) + f ( t , u ( t ) ) .

Thus, for every λ ∈ N , we conclude that u n ∈ ℱ and

(3.6) u n ( t ) = e t A λ g ( u n ) + ∫ 0 t e ( t − τ ) A λ f ( τ , u n ( τ ) ) d τ + ∑ 0 < t k < t e ( t − t k ) A λ I k ( u n ( t k ) )

is a solution to problem (3.1). Reasoning as above, we can prove that u n → u 0 in ℱ . From assumptions (H2) ′ and (H3), for every t ∈ J and a.e. s ≤ t , we can obtain that g ( u n ) → g ( u 0 ) , e ( t − s ) A λ f ( s , u n ( s ) ) → T ( t − s ) f ( s , u 0 ( s ) ) , and e ( t − s ) A λ I k ( u n ( t k ) ) → T ( t − s ) I k ( u 0 ( t k ) ) and the convergence is dominated. Taking the limit of equation (3.6), we obtain that

u 0 ( t ) = T ( t ) g ( u 0 ) + ∫ 0 t T ( t − τ ) f ( τ , u 0 ( τ ) ) d τ + ∑ 0 < t k < t T ( t − t k ) I k ( u 0 ( t k ) ) .

From the proof of Theorem 3.1, we have that u 0 is a solution to problem (1.1). Assume that v 0 is also a solution to problem (1.1). For every t ∈ J , we can conclude that

‖ u 0 ( t ) − v 0 ( t ) ‖ ≤ ( ‖ g ‖ + ‖ L ‖ 1 + p a 2 ) ‖ u 0 − v 0 ‖ C ≤ ‖ u 0 − v 0 ‖ C .

Therefore, u 0 ≡ v 0 , namely, problem (1.1) has a unique mild solution on ℱ .□

4 Example

In this section, we give an example to demonstrate the applicability of the abstract results. Let Ω ⊂ R N be a bounded domain with a sufficiently smooth boundary. J = [ 0 , T ] , 0 < t 1 < t 2 < ⋯ < t p < T . R is a nonnegative constant. Consider the following semilinear diffusion model:

(4.1) ∂ u ( t , x ) ∂ t + i ∇ 2 u ( t , x ) = h ( t , x , u ( t , x ) ) , ( t , x ) ∈ J × Ω , t ≠ t k , u ( t k + , x ) = u ( t k − , x ) + φ k ( u ( t k , x ) ) , k = 1 , 2 , … , p , ∂ u ( t , x ) ∂ n = 0 , u ( 0 , x ) = ∑ i = 1 n α i u ( t i , x ) , x ∈ Ω ,

where i is virtual unit.

Assume that nonlinear map h : [ 0 , T ] × Ω × R → R satisfies the following conditions:

for every η ∈ R , h ( ⋅ , ⋅ , η ) : [ 0 , T ] × Ω → R is measurable;
there exists a function ℓ : [ 0 , T ] → R + , such that for every η , μ ∈ R and a.e. ( t , ξ ) ∈ [ 0 , T ] × Ω ,
∣ h ( t , ξ , η ) − h ( t , ξ , μ ) ∣ ≤ ℓ ( t ) ∣ η − μ ∣ ;
for every η ∈ R and a.e. ( t , ξ ) ∈ [ 0 , T ] × Ω , η h ( t , ξ , η ) ≤ 0 ;

and impulsive functions φ k satisfy Lipschitz condition:

for every η , μ ∈ R , there exist constants M k such that
‖ φ k ( η ) − φ k ( μ ) ‖ ≤ M k ∣ η − μ ∣ ;

and nonlocal diffusion phenomenon is described by multipoint initial condition:

0 < t 1 < ⋯ < t n ≤ T and α i ∈ R such that ∑ i = 1 n ∣ α i ∣ = 1 .

Let E = L 2 ( Ω ; R ) with the norm ‖ ⋅ ‖ . Let operator A : D ( A ) ⊆ E → E be defined as follows:

D ( A ) = { u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) : ∂ u ∂ n = 0 on J × ∂ Ω } , A u = − i ∇ 2 u .

From the literature [6], we can obtain that the linear operator A generates a unitary operator semigroup { e − i ∇ 2 t } t ≥ 0 , which is a nonexpansive C 0 -semigroup. Then, from the Lumer-Philips theorem, it follows that operator A is dissipative. So, problem (4.1) can be transformed into problem (1.1). Then, we have the following uniqueness result of solution.

Theorem 4.1

Let conditions (h1)–(h3), ( φ ), and (g) hold if

∑ i = 1 n ∣ α i ∣ + T L + ∑ k = 1 p M k < 1 .

Then, problem (4.1) has a unique solution in B C ( 0 , R ) .

Proof

By the Vainberg theorem, with a sequence { f n } ∈ L p ( Ω ) such that f n → f ∈ L p ( Ω ) , there exist a subsequence { f n j } ⊂ { f n } and g ∈ L p ( Ω ) such that

∣ f n j ( x ) ∣ ≤ g ( x ) , and f n j ( x ) → f ( x ) , a.e. , x ∈ Ω .

Define the Nemytskii operator f : [ 0 , T ] × L 2 ( Ω ; R ) → L 2 ( Ω ; R ) by f ( t , u ) ( x ) = h ( t , x , u ( x ) ) . For any n ∈ N , by assumption (h2), for every t ∈ [ 0 , T ] and ξ ∈ Ω , we can obtain that h ( t , ξ , η n ) → h ( t , ξ , η 0 ) in R , i.e., f ( t , u n ) → f ( t , u 0 ) . Let ψ ∈ L 2 ( Ω ; R ) and f n = f ( t , u n ) , f = f ( t , u 0 ) . Then, we can deduce that ⟨ ψ , f n ⟩ → ⟨ ψ , f ⟩ . So, the function f : L 2 ( Ω ; R ) → L 2 ( Ω ; R ) is weakly convergent for a.e. t ∈ [ 0 , T ] . Assumption (h1) implies that f ( ⋅ , u ) : [ 0 , T ] → L 2 ( Ω ; R ) is measurable for every u ∈ L 2 ( Ω ; R ) . Then, the conditions (i) and (ii) of (H1) hold. It is obvious that f satisfies assumption (iii) of (H1) for a.e. t ∈ J and every u ∈ L 2 ( Ω ; R ) because of condition (h3). Furthermore, I k satisfy the condition (H2) ′ with a 1 = max k ∈ N M k . The multi-point condition (g) can be replaced by condition (H3). Next, we prove the assumption (H4). For every u , v ∈ L 2 ( Ω ; R ) and ( t , x ) ∈ [ 0 , T ] × Ω , we have

‖ f ( t , u ) ( x ) − f ( t , v ) ( x ) ‖ = ∣ h ( t , x , u ( x ) ) − h ( t , x , v ( x ) ) ∣ ≤ ℓ ( t ) ∣ u ( x ) − v ( x ) ∣ ≤ L ‖ u − v ‖ ,

where L = max t ∈ [ 0 , T ] ℓ ( t ) . Therefore, from the proof of Theorem 3.2, we can obtain that problem (4.1) has a unique solution.□

Acknowledgements

The authors are most grateful to the editor Professor and anonymous referees for the careful reading of the manuscript and valuable suggestions that helped in significantly improving an earlier version of this article.

Funding information: This work was supported by National Natural Science Foundation of China (Nos. 11661071 and 12061062) and the “Innovation Star” Project of Gansu Province (No. 2022CXZX-240).
Author contributions: W. Ma completed the proof of the main results and the writing of the first draft. Y. Li revised the first draft and put forward some suggestions for revision. All authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Not applicable.

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Received: 2023-04-17

Revised: 2023-10-01

Accepted: 2023-11-02

Published Online: 2023-12-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0155

Keywords for this article

impulsive evolution equation; C 0-semigroup; Yosida approximations; approximate solvability method; Hausdorff measure of noncompactness

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