### Abstract

In this study, we investigate the existence of integro-differential solutions for a class of abstract partial impulsive differential equations.

##### Keywords:

integro-differential equations; neutral differential equations; analytic semigroup of compact operators; non-autonomous operators; family of evolution operators; mild solutions### 1. Introduction

In this study, we established two existence results of solutions for a class of impulsive functional differential equations which can be described in the following form

where
*t *∈ [0, *b*], *A*(*t*) is the infinitesimal generator of analytic semigroup of linear bounded operators
(*S _{t}*(

*s*))

_{s ≥ 0 }on a Banach space

*u*(

_{t}*θ*) =

*u*(

*t*+

*θ*),

*θ*≤ 0;

*φ*: (-∞, 0] →

*X*endowed with appropriate seminorm; the operator

*D*(

*t*,

*ϕ*) is defined as

*D*(

*t*,

*ϕ*) =

*ϕ*(0) +

*g*(

*t*,

*ϕ*), where the functions

*i*∈ ℤ are appropriate functions for all

*i*∈ {1, ...,

*m*}; 0

*< t*

_{1 }

*<*⋯

*< t*is a sequence of fixed real numbers and the symbol Δ

_{m }< b*ξ*(

*t*) represents the jump of the function

*ξ*at the moment

*t*, this means that Δ

*ξ*(

*t*) =

*ξ*(

*t*

^{+}) -

*ξ*(

*t*

^{-}), where the notation

*ξ*(

*t*

^{+}) and

*ξ*(

*t*

^{-}) represent, respectively, the right and the left-hand side limits of the function

*ξ*at

*t*.

There are many physical phenomena that are described by means of impulsive differential equations, for instance, biological systems, electrical engineering, chemical reactions, among others can be modeled by impulsive differential equations, a good survey on impulsive differential equations can be found in [1] see also [2,3]. However, impulsive actions can influence the behavior of solutions making the analysis more difficult. Motivated by this facts, the studies of such systems have drawn the attention of many researchers during last years.

Recently, Park et al. [4] have investigated the problem

In this model, the operator
*T*(*t*))_{t ≥ 0 }on a Banach space
*ρ*(*A*), where *ρ*(*A*) is the resolvent set of the operator
*
*,

*i*= 1, 2,...,

*m*are given functions that satisfy suitable conditions. Using the theory of fractional powers and priori estimates for compact operators, the authors established some existence result for the problem (1.4). Lately, Balachandran and Annapoorani [5] investigated the following class of abstract problem (1.5)

In the system (1.5), it was assumed that for each *t *∈ [0, *b*] the operator *A*(*t*) is the infinitesimal generator of compact analytic semigroup of bounded linear operators
on a Banach space
*D*(*A*(*t*)), of the operators *A*(*t*) is assumed to be independent of *t *∈ [0, *b*] and dense in

Then, using fractional powers and operators theory the authors get some existence
result based on *a priori *bounded estimates for compact operators.

Other authors have studied problems involving impulsive act, for retarded and neutral functional differential equations we cite [7-17], for applications of impulsive differential equations on biology and neural networks we cite [18-21]. On the other hand impulsive fractional differential equations is a topic treated in [22,23].

In this article, the study of a class of neutral impulsive integro-differential equations
is proposed. To get our results, we used the technique involving the fixed point theory
of compact and condensing operators. We pointed out that the problem studied in this
article has not been considered in the literature, once that the approach used in
this study is totally different from those studies mentioned above. Actually, the
main difference is that in our study we need to use an assumptions of compactness
on the nonlinear equation, and in applications, these assumptions make all differences,
because, even in infinite dimensional Hilbert space it is not straightforward handedly
with compact sets. However, in our applications, we overcome this difficult using
a well-known criterion of compactness in *L ^{p}*(Ω) space [[24], Kolmogorov-Riesz-Weil theorem]. This is the principal motivation of this study.

We now turn to a summary of this study. The second section provides tools which are necessary to establish the main results that are the Theorems 2.3 and 2.4. In third section, we apply our abstract results in concrete examples.

### 2. Preliminaries

In this study, the symbols
*A*(*t*), *t *∈ [0. *b*].

**Definition 2.1**. A family of operators *U*(*t*, *s*), *t *≥ *s*, *t*, *s *∈ *I *is said to be an evolution family associated to the problem (2.1) if the following
conditions hold:

(a) *U*(*t*, *s*)*U*(*s*, *r*) = *U*(*t*, *r*) for all *r *≤ *s *≤ *t*.

(b) For each
*t*, *s*) → *U*(*t*, *s*)*x *is continuous from {(*t*, *s*), *t *≥ *s*, *t*, *s *∈ *I*} into

(c) For each *t > s*, the function *t *→ *U*(*t*, *s*) is continuous differentiable with respect to *t *and

The family of evolution system *U*(*t*, *s*) is called exponential stable if there are positive constants
*α *such that
*t*, *s *∈ [0, *b*].

Throughout this study,
*D*, which is independent of *t *and dense in

has an associated evolution family of operators *U*(*t*, *s*), *t *≥ *s*, *t*, *s *∈ *I*. For additional details and more properties about the family *U*(*t*, *s*), we refer the reader to [6,25,26].

To study the problem (2.1), we consider the space of normalized piecewise continuous
functions
*u *is continuous at *t *≠ *t _{i}*,

*i*= 1,...,

*m*. It is well known that if it is equipped with the norm

The technique used in this study is based on the compactness criterion. For this reason, we will make the following assumptions.

Put *t*_{0 }= 0, *t*_{n+1 }= *τ *and for

In particular,

**Lemma 2.1**. ( [12]) A set
*for every i *= 0, 1,..., *n*.

The next step is to define the phase space. This will be done in the following way.
The space
*b *> 0 be a function such that *x*_{0 }= *φ*,

(i) *x _{t }*is in

(ii)

(iii)
*H > *0 is a constant; *K*, *M *: [0, ∞) → [1, ∞), *K*(·) is continuous, *M*(·) is locally bounded and *H*, *K*, *M *are independent of *x*(*·*).

**Remark 2.1**. To treat retarded impulsive differential equation we suitable modified the axioms
of the abstract phase space
*t *→ *x _{t}*, since

Following the ideas of [15], we used the notations

In what following we give some examples of phase spaces whose the above axioms are satisfied.

**Example 2.1**. Consider the function *g*(*θ*) = *e ^{γθ}*,

*θ*≤ 0,

*γ*≥ 0, and let

*L*

^{2}([0,

*π*], ℝ) be the space of square integrable Lebesgue measure functions endowed with the norm

If
*i*)-(*iii*) are fulfilled. In these particular example it is possible to show that *H *= 1, *K*(*t*) = 1 and *M*(*t*) = *e*^{-γt}, for all *t *≥ 0.

Motivated by Pazy [16,25] we adopt the following concept of mild solution to problem (1.1)-(1.3).

**Definition 2.2**. A function

(i) *u*_{0 }= *ϕ*,

(ii) the function
*i *= 1,..., *m*;

(iii) the integral equation below is satisfied,

is satisfied.

The tools used in this study are based on point fixed theory. For this reason, the next two theorems play important role in the development of our results.

**Theorem 2.1**. [[27], *Leray-Schauder Alternative*]* Let C be a convex subset of a Banach space *
*and assume *0 ∈ *C*. *Let F *: *C *→ *C be a completely continuous operator, and let*

*Then either
*.

**Theorem 2.2**. [[28]*, Corollary 4.3.2*]* Suppose that D is a closed bounded convex subset of the Banach space *
* and that B and C are continuous function from D to *
* with*

(*ma*1) *Bx *+ *Cx *∈ *D*, *for all x *∈ *D*.

(*ma*2)
*is compact set; and*

(*ma*3) *there is a number *0 ≤ *γ < *1 *such that *|| *Bx *- *By *|| ≤ γ || *x *- *y *||, *for all x*, *y *∈ *D*.

*Then there is z *∈ *D such that Bz *+ *Cz *= *z*.

Next, we stated some important conditions used in the proof of our results.

(*H*_{1}) The function

(*H*_{1.1}) Let
*ϕ *which is given by

Then, each bounded set *B *of
*g*(*t*, *y _{t }*+

*u*),

_{t}*t*∈ [0,

*b*],

*u*∈

*B*} is equi-continuous.

(*H*_{1.2}) There are constants *c*_{1 }and *c*_{2 }such that
*t *≥ 0 and

(*H*_{2}) The function

(*H*_{2.1}) The function (*x*, *ϕ*) → *f*(*t*, *ϕ*, *x*) is continuous for almost everywhere *t *∈ [0, *b*].

(*H*_{2.2}) The function *t *→ *f*(*t*, *ϕ*, *x*) is strong measurable for each

(*H*_{2.3}) There is a positive continuous function *m *: [0, *b*] → [0, ∞) and a nondecreasing positive continuous function *ψ*: ℝ → [0, ∞) such that

for every

(*H*_{3}) The function

(*H*_{3.1}) The function *ϕ *→ *e*(*t*, *s*, *ϕ*) is continuous almost everywhere for all *t*, *s *∈ [0, *b*].

(*H*_{3.2}) The function (*t*, *s*) → *e*(*t*, *s*, *ϕ*) is strong measurable for each

(*H*_{3.3}) There is a positive continuous function *p *: [0, *b*] → [0, ∞) and a nondecreasing integrable positive function Ω: ℝ → [0, ∞) such that

for all

(*H*_{4}) For each function
*τ *→ *e*(*s*, *τ*, *u _{τ}*),

*τ*∈ [0,

*b*] and

*t*→

*f*(

*t*,

*u*,

_{t}*x*),

*t*∈ [0,

*b*] are measurable functions for almost everywhere

*s*∈ [0,

*b*] and

Now we are already to state and prove the main result of this article.

**Theorem 2.3**. *Assume that the conditions *(*H*_{1}) - (*H*_{4}) *are satisfied. In addition, suppose that the following assumptions hold*.

(*t*1) *The function *
* is completely continuous*.

(*t*2) *The operators I _{i }are completely continuous and there are positive constants, L_{i }such that*

*for all *
*and i *= 1,..., *m*.

(t3) *For each bounded subsets *
* and *
* the set*

*is relatively compact for each t *≥ *s*, *t*, *s *∈ [0, *b*], *where *
* is an extension of x in such manner that *
*t *≤ 0 *and *
*t *∈ [0, *b*].

*If *
* and*

*where*

*and*

*then, the problems (1.1)-(1.3) have a mild solution*.

*Proof*. Suppose that
*ϕ *given in (*H*_{1}). If we written the solution *u*(·) of the problem (1.1)-(1.3) as *u*(*t*) = *x*(*t*) + *y*(*t*), *t *∈ (-∞, *b*], then we can see that *x*(*t*) = 0, *t *≤ 0 and for *t *∈ [0, *b*] the following integral equation hold true

Motivated by this remark we consider the space

endowed with the norm
* _{i }*: Λ → Λ,

*i*= 1, 2, 3 given by

for all *t *∈ [0, *b*]. Using the fact that (*U*(*t*, *s*))_{t ≥ s }is a evolution family of operators and assuming the conditions on *f*, *g *and the family of operator *I _{i}*,

*i*= 1,...,

*m*, it is not difficult see that

*t*→ Γ

*(*

_{i}*t*),

*t*∈ [0,

*b*] is a normalized piecewise continuous function for all

*i*= 1,...,

*m*. This shows that Γ is well defined. In the next, we prove that the operator Γ = Γ

_{1 }+ Γ

_{2 }+ Γ

_{3 }satisfies all conditions of Theorem 2.1. As the proof is very long we split it into various steps.

#### Step 1. The operator 1 is completely continuous

Let *x _{n }*∈ Λ,

*n*∈ ℕ be a sequence of elements of Λ such that

*x*→

_{n }*x*as

*n*→ ∞ for some

*x*∈ Λ. From the boundedness of operators

*U*(

*t*,

*s*) and the axioms of the phase space

*H*

_{1}) the uniformity convergence of

on [0, *b*]. Thus, we have the continuity of Γ_{1}. From condition (*H*1), and the axiom (*iii*) follows immediately the Γ_{1 }applies bounded sets of Λ into equi-continuous sets of Λ. On the other hand, again
by (*iii*), and using the fact that *f *is a completely continuous function, soon as infers that, for each *t *∈ [0, *b*], the set {*g*(*t*, *x _{t }*+

*y*),

_{t}*x*∈

*B*} is compact in

_{1 }is a completely continuous operator is complete.

#### Step 2. The operator Γ_{2 }is complete continuous

The condition (*H*_{3.1}) permit us conclude that
*n *→ ∞ almost everywhere for *t*, *s *∈ [0, *b*]. By (*H*_{3.3}), and the Lebesgue's dominated convergence theorem we conclude that

uniformly for *t *∈ [0, *b*]. From the strong continuity of the operators (*U*(*t*, *s*))_{t ≥ s}, we can conclude that

as *n *→ ∞, uniformly for *t *∈ [0, *b*]. This fact and the properties of the evolution family *U*(*t*, *s*) lead us to the continuity of the operator Γ_{2}. Next, we show that Γ_{2 }takes bounded sets into equi-continuous sets. First, we observe from conditions (*H*_{2.3}), (*H*_{3.3}) and the axioms of phase space that

and

are bounded sets in

Let *ε > *0 be the arbitrary positive real number and *t*_{1}, *t*_{2 }∈ [0, *b*], *t*_{1 }*> t*_{2}. Thus, take into account the previous notes and using the assumption (*iii*) we see that the set

is relatively compact in

thus, from the continuity of *U*(*t*, *s*) and the assumptions of compactness contained on the condition (*t*3) we can infer the existence of 0 *< δ < ε *such that if |*t*_{1 }- *t*_{2}| *< δ *then

This shows the equi-continuity of Γ_{2}. In what follows, we show that for each *t *∈ [0, *b*] the set

where *B *∈ Λ, is pre-compact in Λ. To do that, we observe from (2.3) that for each *s *∈ [0, *t*] the set

is a bounded set. Then,

which implies by [[28], Lemma 1.3]

with *diam*(*C _{ε}*)

*< ε*, where

*diam*(·) denotes the diameter of the set

*C*and

_{ε }*co*{·} the convex hull. Taking all this into account we see that for each fixed

*t*∈ [0,

*b*], the set Θ(

*t*) in (2.4) is relatively compact set in

_{2 }is completely continuous.

#### Step 3. The operator Γ_{3 }is completely continuous

To show that is Γ_{3 }is a completely continuous, consider a bounded subset *B *of Λ and for each *i *= 1,..., *m*, define the set

To prove that the sets
*i *= 1,..., *m*, are precompacts in
*t*_{1}, *t*_{2 }∈ (*t _{i}*,

*t*

_{i+1}],

*t*

_{1 }

*> t*

_{2}. Using the continuity of (

*t*,

*s*) →

*U*(

*t*,

*s*)

*x*, and the compactness of sets

*I*(

_{j}*B*),

*j*= 1,...,

*m*, given

*ε >*0 there is 0

*< δ < ε*such that if |

*t*

_{1 }-

*t*

_{2}|

*< δ*we have

uniformly for *x *∈ *B*. On the other hand for *t *∈ (*t _{i}*,

*t*

_{i+1}) fixed, from our hypothesis it is not difficult see that the set

is relatively compact in

On the other hand, if *t *= *t _{i}*, the set

and proceeding as in the early case we infer that the set

is an equi-continuous set of functions is done in the same manner as at the beginning

of the section. The proof that Γ_{3 }is completely continuous is finished.

In the next, we obtain *a priori *estimative of the solutions for the equation *λ*Γ*x _{λ }*=

*x*, for

_{λ}*λ*∈ (0, 1) and = Γ = Γ

_{1 }+ Γ

_{2 }+ Γ

_{3}. Let

*x*be a solution of the equation

*λ*Γ(

*x*) =

_{λ}*x*, in addition we use the notation

_{λ}

this implies that

If we take the right-hand side of the previous inequalities and call it of *v*(*t*) we have that *m _{λ}*(

*t*) ≤

*v*(

*t*), for all

*t*∈ [0,

*b*]. This leads us to the following inequality:

this yields

Next, we considered the function
*v*(0) = *ϖ*(0) and *v*(*t*) ≤ *ϖ*(*t*), for all *t *∈ [0, *b*], using this and the non-decreasingly properties of the function *ψ*(·), we get

Observe that if we define the function

which implies that

for all *t *∈ [0, *b*]. Integrating the early inequality from 0 to *t *we have

The early inequalities enable us to conclude that the set {*x _{λ}*,

*x*= Γ

_{λ }*x*,

_{λ}*λ*∈ (0, 1)} is bounded. From Theorem 2.1 the problem (1.1)-(1.3) has a mild solution. The proof of theorem is completed. □

In the next result, the following conditions are used.

(**G _{1}**) There is a positive constant

*L*such that

_{f }

for every *t *∈ [0, *b*] and

(**G _{2}**) There are positive constants

*d*,

_{i}*i*= 1,...,

*m*, such that

for every *x*,

**Theorem 2.4**. *Assume that the condition *(*H*2)-(*H*3) *and *(**G1**)-(**G2**) *are satisfied. In addition, suppose that the assumption *(*iii*) *of Theorem (2.3) is satisfied. Then if*

*and*

*then the problem (1.1)-(1.3) has a mild solution*.

*Proof*. Let us consider the operator Γ: Λ → Λ defined as in Theorem 2.3. We claim that there
is *r > *0 such that Γ(*B _{r}*) ⊂

*B*. Suppose by contradiction that this assumption is false. Then for each

_{r}*r >*0 there are

*t*∈ [0,

_{r }*b*] and

*u*(·) ∈

_{r}*B*such that

_{r }

take the lim inf in the previous inequality, we have

which is contradictory with our assumptions. So let *r > *0 be such a number and consider the restriction
*B _{r}*, that is,

_{1 }+ Γ

_{2}, where

As shown in the proof of Theorem 2.3, it is not difficult to see that Γ_{2 }is completely continuous and for *u*_{1}, *u*_{2 }∈ Λ we have that

The previous inequality shows that Γ_{1 }is contractive. Now, by Theorem 2.2, we can conclude that the problem (1.1)-(1.3)
has a mild solution. □

### 3. Applications

The main aim of this section is to apply our abstract results in concrete examples.
To this end, we handle with a very special kind of operators. To be more specific,
on the Banach space
*Ax*(*ξ*) = *x*"(*ξ*), *ξ *∈ [0, *π*] with domain

It is well known that in this case *A *has a discrete spectrum which is given by -*n*^{2}, *n *∈ ℕ. Moreover,
*A *associated with the eigenvalues -*n*^{2}, which is given
*n *∈ ℕ. This implies that the following conditions are satisfied.

(i) For each

(ii) For each *f *∈ *D*(*A*), we have

where 〈·,·〉 represents the inner product in
*A *is the infinitesimal generator of a compact semigroup of bounded linear operators
(*T*(*t*))_{t ≥ 0}, which is given by

To guarantee the existence of an evolution family associated with the problem

the following assumptions on the function *a*_{0 }: [0, *b*] × [0, *π*] → ℝ are made

*c > *0 and *α *∈ (0, 1) such that

for all *t*, *s *∈ [0, *b*] and almost everywhere *ξ *∈ [0, *π*].

*c*_{0 }such that

for all *τ *∈ [0, ∞) and *ξ *∈ [0, *π*].

Letting *D*(*A*(*t*)) = *D*(*A*) for all *t *≥ 0 and *A*(*t*)*x*(*ξ*) = *a*_{0}(*t*, *ξ*)*x*"(*ξ*), *ξ *∈ [0, *π*], we have that the system (3.1) has an associated evolution family of operators (*U*(*t*, *s*))_{t ≥ s }which is given explicitly by the following formula:

Using the properties of semigroup (*T*(*t*))_{t ≥ 0 }it is straightforward to show that *U*(*t*, *s*) satisfies the condition

Next, we consider the following partial differential equations

To model the problem (3.2) we choose as the phase space the set formed by all piecewise
continuous functions
_{s ≤ 0 }*h*(*θ*) || *φ*(*s*) || *< *∞, where *h*(*θ*) = *e ^{βθ}*,

*θ*≤ 0, and we denote this space by

*H*

_{1})-(

*H*

_{4}) are satisfied we needed to consider the following assumptions.

*P*_{2 }: (-∞, 0] × ℝ → [0, ∞) satisfies the following conditions:

*η *∈ ℝ, *s *→ *P*_{2}(*s*, *η *) is a measurable and bounded function,

for all *s *≤ 0, and *η _{i }*∈ ℝ,

*i*= 1, 2.

*b*, 0] × [0, *π*],
*b*, 0].

*k*_{2}(·) ∈ *L*((-∞, *π*]) and *s *→ *P*_{2}(*s*, *η *) is measurable and bounded function for each *η *∈ ℝ. In addition we assume the existence of positive constant

for almost everywhere *s *∈ (-∞, 0] and *η _{i }*∈ [0,

*π*],

*i*= 1, 2.

*P*_{3 }: [-*π*, ∞) × ℝ → ℝ satisfies the following conditions.

*η *∈ ℝ, *s *→ *P*_{3}(*s*, *η *), *s *∈ [-∞, *b*), is a measurable and bounded function,

for all *s *∈ [-*π*, ∞) and *η _{i }*∈ ℝ,

*i*= 1, 2.

*k*_{3}(·) ∈ *L*([-*π*, ∞)).

To transform the problem (3.2) into the abstract system (1.1), we define the functions
*i *= 1, 2,..., *n*, respectively, given by,

We shall show that the condition (*H*_{1}) hold true. In fact, let *x *: (-∞, *π*] → *L*^{1}(0, *π*) be a bounded function such that

The previous inequalities jointly with the assumption
*t *→ *g*(*t*, *x _{t}*) is uniformly continuous on bounded subsets of

*L*

^{2}(0,

*π*)) which implies that the condition (

*H*

_{1.1}) hold true. To prove that the condition (

*H*

_{1.2}) is satisfied, we observe that

which implies the condition (*H*_{1.2}).

The next step is a proof that the function (*x*, *ϕ*) → *f*(*t*, *ϕ*, *x*) is continuous. However, with the help of condition

for all
*i *= 1, 2. Thus we have shown that the condition (*H*_{2.1}) is fulfilled. In particular, as *P*_{2 }is continuous in the second variable we have that for each