Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Jaouad El Matloub; Khalil Ezzinbi

doi:10.1515/msds-2023-0102

Open Access Published by De Gruyter Open Access November 2, 2023

Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Jaouad El Matloub and Khalil Ezzinbi

From the journal Nonautonomous Dynamical Systems

https://doi.org/10.1515/msds-2023-0102

Abstract

We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.

Keywords: resolvent operator; measures of noncompactness and condensing mappings; integrodifferential inclusions; mild solution; asymptotically almost periodic solution

MSC 2010: 45K05; 47H10; 47D06

1 Introduction

Many studies have been published in recent years on the subject of differential inclusions in Banach spaces. Differential inclusions are a useful tool in the analysis of many dynamical processes that are represented by equations with a discontinuous or multivalued right-hand side. The progress of control theory of dynamical systems of the form:

(1) x ′ ( t ) = f ( t , x ( t ) , u ( t ) ) for t ≥ 0 , x ( 0 ) = x 0 ,

emerged as a fundamental motivator for the study of differential inclusions. In fact, introducing the set-valued map

F ( t , x ) = { f ( t , x , u ) : u ∈ U } ,

where U is the set of possible controls, we may infer that the solutions of the differential equation (1) are solutions to the following differential inclusion:

x ′ ( t ) ∈ F ( t , x ( t ) ) for t ≥ 0 , x ( 0 ) = x 0 ,

in which the controls are not explicitly visible.

Several real-world processes in physics, biology, and engineering may be described by partial differential equations. However, such equations can only describe the state of a system at a given time. They cannot reflect the accumulation of past effects and therefore may not provide a sufficiently accurate description of the system. Especially where heat conduction is involved, in porous elastic media, and in nuclear reactor dynamics, “memory” effects on the system often need to be taken into account (see [26]). One approach to dealing with this is to add an integral term to the conventional partial differential equation, to give a more general integro-differential equation as the following form:

(2) x ′ ( t ) = A x ( t ) + ∫ 0 t B ( t − s ) x ( s ) d s + f ( t ) for t ≥ 0 , x ( 0 ) = x 0 ∈ X ,

where A is an unbounded operator, ( B ( t ) ) t ≥ 0 is the kernel system, and f is the input function. Desch et al. [17] and Grimmer et al. [13] investigated the existence and regularity of mild solutions for (2) under various situations using the operator resolvent theory. Let us now go through the advantages of using the resolvent operator briefly. When B ( ⋅ ) = 0 , the problem can still be treated using the classical semigroup, and various articles dealing with this topic have been published, including those by Adimy et al. [1,2] when ( f ( t ) = f ( t , x ( t ) ) ). However, when B ( ⋅ ) ≠ 0 , additional constraints must be imposed, the simplest of which is that ( B ( t ) ) t ≥ 0 is a family of bounded operators on X , which refers to a particular situation. It is the resolvent operator that allows us to investigate the general situation where D ( B ( ⋅ ) ) is contained strictly in X . We recall that there is an extensive study of equation (2) in the literature. The reader may see the study of Grimmer [19] and references therein. As a result of this, numerous authors extend the quantitative and qualitative study of equation (2) to the inclusion type. For instance, Chang and Chalishajar [8] established a sufficient condition for the controllability of semilinear mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. Chang and Nieto [7] studied the existence of solutions for some impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. Vijayakumar [29] examined the approximate controllability via analytic resolvent for the following integro-differential inclusions:

d d t [ x ( t ) + g ( t , x ( t ) ) ] ∈ A x ( t ) + ∫ 0 t G ( t − s ) x ( s ) d s + B u ( t ) + F ( t , x ( t ) ) , t ∈ I = [ 0 , b ] x ( 0 ) = x 0 ,

where A is the infinitesimal generator of a C 0 -semigroup of operators ( R ( t ) ) t ≥ 0 in a Hilbert space X . The control function u ( ⋅ ) ∈ L 2 ( I , U ) , which is a Hilbert space of admissible control functions.

Due to its importance in the physical sciences, the existence of almost periodic (a.p.) and asymptotically almost periodic (a.a.p.) solutions is one of the most interesting topics in the qualitative theory of differential equations or inclusions. For that reason, many authors [4,11,30] have discussed the a.a.p. solutions of differential equations in practical problems. For instance, Hernández and Pelicer [23] studied the existence of a.a.p. solutions for some partial neutral functional differential equations with unbounded delay. Del campo et al. [12] established the existence of bounded a.a.p. solutions for functional difference equations with infinite delay on abstract phase space. Diagana et al. [14] investigated the existence of a.a.p. solutions to some classes of second-order partial functional differential equations with unbounded delay.

Gallegos and Henriquez [16] were concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Their main tool is a scale of Banach spaces and the theory of semigroups. In particular, they considered the following first-order differential inclusion:

x ′ ( t ) − A x ( t ) ∈ f ( t , x ( t ) ) for t ≥ 0 , x ( 0 ) = x 0 ∈ X ,

where A is the infinitesimal generator of a C 0 -semigroup and f is a set valued map defined on R + × X . Using the decomposition f = f 1 + f 2 , where f 1 is a single-valued function and f 2 is a multifunction defined on R + × X , they obtained the existence of mild solutions under different conditions on f 1 and f 2 .

Since many systems can be described in the form of an integrodifferential inclusion as we mentioned previously, it is natural to ask to discuss the existence of solutions of the following partial integrodifferential inclusion:

(3) z ′ ( t ) ∈ A z ( t ) + ∫ 0 t B ( t − s ) z ( s ) d s + F ( t , z ( t ) ) for t ≥ 0 , z ( 0 ) = z 0 ∈ X ,

where z ( ⋅ ) is the state variable taking values in a Banach space ( X , ‖ ⋅ ‖ ) . The operator A is the infinitesimal generator of a C 0 -semigroup ( T ( t ) ) t ≥ 0 on X and B ( t ) is a closed linear operator with domain D ( B ) ⊃ D ( A ) time-independent. Furthermore, F is a multifunction defined on R + × X , and its properties will be specified later.

Our discussion is based on the theory of resolvent operators, which can be seen as a generalization of the semigroup theory, even though it does not satisfy the semigroup law. This particular property holds significant importance, especially when considering issues related to compactness. In addition, we use the fixed point theory approach, which is related to a scale of Banach spaces, to obtain the existence of mild solutions and a.a.p. mild solutions for multivalued condensing maps under hypotheses given in terms of the measure of noncompactness.

The remainder of this work is as follows: in Section 2, we introduce the notations, definitions, and preliminary facts that will be used throughout this work. The existence results of mild solutions are proved in Section 3. In Section 4, we prove the existence of a.a.p. solutions for Problem (3). Finally, we provide an application for illustration.

2 Preliminaries

This section introduces several fundamental concepts, notations, and lemmas that will be needed throughout this work.

For a > 0 , C ( [ 0 , a ] , X ) is the Banach space of all continuous functions from [ 0 , a ] to X equipped with the following norm:

‖ x ‖ ∞ = sup t ∈ [ 0 , a ] ‖ x ( t ) ‖ .

C b ( J , X ) denotes the Banach space of all bounded continuous functions from J into X endowed with the sup-norm ‖ ⋅ ‖ ∞ . Here, J is either the real line R or the half real line R + .

For later use, we denote by C 0 ( [ 0 , + ∞ ) ; X ) the space formed by all the continuous functions that vanish at infinity, equipped with the sup-norm.

B ( X ) denotes the Banach space of all linear bounded operators L from X into X with norm:

‖ L ‖ = sup ‖ y ‖ = 1 ‖ L ( y ) ‖ .

And L p ( [ 0 , a ] , X ) , where p ≥ 1 , denotes the Banach space of functions f : [ 0 , a ] → X that are Bochner integrable, normed by:

‖ f ‖ p = ∫ 0 a ‖ f ( t ) ‖ p d t 1 p .

Let ( E , d ) be a metric space; for the rest of this work, we use the following notations:

P ( E ) denotes the collection of all nonempty subsets of E .
P c ( E ) = { C ∈ P ( E ) : C is closed } .
P b ( E ) = { C ∈ P ( E ) : C is bounded } .
P c b ( E ) = { C ∈ P ( E ) : C is closed and bounded } .
V ( E ) = { C ∈ P ( E ) : C is convex } .
K ( E ) = { C ∈ P ( E ) : C is compact } .
KV ( E ) = { C ∈ P ( E ) : C is convex and compact } .

2.1 Multivalued functions and measure of noncompactness

This subsection focuses on the fundamental concepts and properties of multivalued maps.

Definition 1

[24] A multivalued map (multi-map) F of a set X into a set C is a correspondence that associates with every x ∈ X a non-empty subset F ( x ) ⊂ C , called the value of x . We write this correspondence as F : X ⟶ P ( C ) .

For a multivalued map F : X → P ( C ) , we denote

F − 1 ( V ) = { x ∈ X : F ( x ) ⊆ V } , F + − 1 ( V ) = { x ∈ X : F ( x ) ∩ V ≠ ∅ } .

Definition 2

[24] Let X be a metric space. A multivalued map F : X → P ( C ) is said to be:

Upper semi-continuous (u.s.c.) if F − 1 ( V ) is an open subset of X for all open set V ⊆ C .
Closed if its graph G F = { ( x , y ) : y ∈ F ( x ) } is a closed subset of X × C .
Compact if its range F ( X ) is relatively compact in C .
Lower semi-continuous (l.s.c.) if F + − 1 ( V ) is an open subset of X for all open set V ⊆ C .

We also provide the definition of a Lipschitzian and locally Lipschitzian multimap as follows:

Definition 3

[16] Let X be a Banach space. A multivalued map f : [ 0 , a ] × X ⟶ P ( X ) is said to be locally Lipschitzian if:

for every ( t 0 , x 0 ) ∈ [ 0 , a ] × X , there exist ε > 0 and positive constants L 1 and L 2 such that

f ( t 2 , x 2 ) ⊂ f ( t 1 , x 1 ) + ( L 1 ∣ t 2 − t 1 ∣ + L 2 ‖ x 2 − x 1 ‖ ) B X ( 0 , 1 )

for all ∣ t i − t 0 ∣ ≤ ε , ‖ x i − x 0 ‖ ≤ ε for i = 1 , 2 and B X ( 0 , 1 ) is the unit ball on X .

Definition 4

[27] A multivalued map F : X → P ( C ) is said to be Lipschitzian if there exists a positive constant L F such that

F ( x 1 ) ⊂ F ( x 2 ) + L F d ( x 1 , x 2 ) B C ( 0 , 1 ) ,

where B C ( 0 , 1 ) = { y ∈ C : ‖ y ‖ ≤ 1 } .

Definition 5

[24] Let Y and Z be sets. A single-valued map f : Y → Z is said to be a selection of a multi-map F : Y → P ( Z ) if

f ( x ) ∈ F ( x )

for all x ∈ Y .

Let I ⊂ R be a compact interval, λ be the Lebesgue measure on I , and E be a Banach space.

Definition 6

[24] A function f : I ⟶ E is said to be a measurable selection of a multifunction F : I ⟶ K ( E ) if f is measurable and f ( t ) ∈ F ( t ) for μ a.e. t ∈ I .

The following theorem is the key to define the mild solutions of Problem (3).

Theorem 1

[24] Let E and E 0 be Banach spaces and F : I × E 0 → K ( E ) be such that

for every x ∈ E 0 , the multifunction F ( ⋅ , x ) : I → K ( E ) has a strongly measurable selection.
for λ -a.e. t ∈ I , the multi-map F ( t , ⋅ ) : E 0 → K ( E ) is u.s.c.

Then, for every strongly measurable function q : I → E 0 , there exists a strongly measurable selection z : I → E of the multifunction Φ : I → K ( E ) :

Φ ( t ) = F ( t , q ( t ) ) .

Definition 7

[24] Let ( Ω , A ) be a measurable space and let C be a Banach space. A multivalued map F : Ω → P ( C ) is said to be measurable if F + − 1 ( V ) ∈ A for all open set V ⊆ C .

The following Kuratowski-Ryll-Nardzewski selection theorem will be used subsequently.

Theorem 2

[18] Let C be a separable complete space. Then, every measurable F : Ω ⟶ P ( C ) has a (single-valued) selection.

Let ( M , d ) and ( N , δ ) be metric spaces. The Hausdorff metric d H on P c b ( M ) is defined by:

d H ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( A , b ) } ,

where d ( a , B ) = inf b ∈ B d ( a , b ) .

Definition 8

[16] Let F : M → P ( M ) be a multivalued map. A point x ∈ M is said to be a fixed point of F if x ∈ F ( x ) . We denote by Fix ( F ) the set consisting of fixed points of F .

Definition 9

[16] A multivalued map F : M → P c b ( N ) is said to be k -contraction, where 0 < k < 1 , if

δ H ( F x , F y ) ⩽ k d ( x , y ) , for all x , y ∈ M .

The Hausdorff measure of noncompactness plays a major role in our work. For this reason, we describe briefly this concept.

Definition 10

[24] Let ( E , d ) be a complete metric space. The Hausdorff measure of noncompactness of a nonempty and bounded subset Q of E , denoted by χ ( Q ) , is the infimum of all numbers ε > 0 such that Q can be covered by a finite number of balls with radii < ε , i.e.,

χ ( Q ) = inf ε > 0 : Q ⊂ ⋃ i = 1 n B ( x i , r i ) , x i ∈ E , r i < ε , i = 1 , 2 , … , n , n ∈ N ,

where B ( x i , r i ) denotes the open ball of center x i and radius r i . The function χ defined on the set of all nonempty and bounded subsets of ( E , d ) is called Hausdorff measure of noncompactness.

The following lemma contains several properties of χ that are useful for our purposes.

Lemma 1

Let B , C ⊆ E be bounded. Then,

χ ( B ) = 0 if and only if B is totally bounded.
χ ( B ) = χ ( B ¯ ) = χ ( c o ¯ B ) , where c o ¯ B is the closed convex hull of B.
χ ( B ) ≤ χ ( C ) when B ⊆ C .
χ ( B + C ) ≤ χ ( B ) + χ ( C ) , where B + C = { b + c : b ∈ B , c ∈ C } .
χ ( B ∪ C ) ≤ max { χ ( B ) , χ ( C ) } .
The function χ : P c b ( E ) ⟶ R + is d H -continuous.
For λ ∈ R , χ ( λ B ) = ∣ λ ∣ χ ( B ) .

For the proof of these properties, the reader may refer [3,5,9,20].

Let ( C , ‖ ⋅ ‖ ) be a Banach space. Our work is based on the following concept.

Definition 11

[24] We assume that η is any measure of noncompactness on E . A multivalued map F : E → P ( C ) is said to be a condensing map with respect to η (abbreviated, η -condensing) provided that if D ⊂ E with η ( F ( D ) ) ≥ η ( D ) , then D is relatively compact.

In the whole of this work, we denote by χ the Hausdorff measure of noncompactness in X and ϑ is the Hausdorff measure of noncompactness in the space of continuous functions with values in X . For later use, we recall the following results.

Lemma 2

[16] Let G : [ 0 , a ] → ℒ ( X ) be a strongly continuous operator-valued map. Let D ⊂ X be a bounded set. Then, ϑ ( { G ( ⋅ ) x : x ∈ D } ) ⩽ sup t ∈ [ 0 , a ] ‖ G ( t ) ‖ χ ( D ) .

Lemma 3

[5] Let ℋ ⊂ C ( [ 0 , a ] ; X ) be a bounded set. Then, χ ( ℋ ( t ) ) ≤ ϑ ( ℋ ) for any t ∈ [ 0 , a ] , where ℋ ( t ) = { u ( t ) : u ∈ ℋ } . Furthermore, if ℋ is equicontinuous on [ 0 , a ] , then t ⟼ χ ( ℋ ( t ) ) is continuous on [ 0 , a ] and

ϑ ( ℋ ) = sup { χ ( ℋ ( t ) ) : t ∈ [ 0 , a ] } .

Definition 12

[15] A set of functions ℋ ⊆ L 1 ( [ 0 , a ] ; X ) is said to be uniformly integrable if there exists a positive function κ ∈ L 1 ( [ 0 , a ] ; R + ) such that ‖ h ( t ) ‖ ≤ κ ( t ) a.e. for all h ∈ ℋ .

Lemma 4

[21] Let W ⊆ C ( [ 0 , a ] ; X ) be a bounded set. Then, there exists a countable set W 0 ⊆ W such that ϑ ( W 0 ) = ϑ ( W ) .

Lemma 5

[16] Let G : [ 0 , a ] → ℒ ( X ) be a strongly continuous operator-valued map such that G is continuous for the norm of operators on ( 0 , a ) , and G : L 1 ( [ 0 , a ] ; X ) → C ( [ 0 , a ] ; X ) be the map defined by:

G ( u ) ( t ) = ∫ 0 t G ( t − s ) u ( s ) d s .

Let W ⊂ L 1 ( [ 0 , a ] ; X ) be a uniformly integrable set. Assume that there is a positive function q ∈ L 1 ( [ 0 , a ] , R + ) such that χ ( W ( t ) ) ⩽ q ( t ) for a.e. t ∈ [ 0 , a ] . Then,

β ( G ( W ) ) ⩽ 2 sup 0 ⩽ t ⩽ a ‖ G ( t ) ‖ ∫ 0 a q ( t ) d t .

2.2 Resolvent operators

In this section, we recall the definition and basic results on theory of resolvent operators for the following linear integrodifferential problem:

(4) u ′ ( t ) = A u ( t ) + ∫ 0 t B ( t − s ) u ( s ) d s for t ≥ 0 , u ( 0 ) = u 0 ∈ X .

Definition 13

[19] A family of bounded linear operators ( ℛ ( t ) ) t ≥ 0 in X is called the resolvent operator of equation (4) if

ℛ ( 0 ) = I and ‖ ℛ ( t ) ‖ ⩽ N e β t for some N ⩾ 1 and β ∈ R .
For all x ∈ X , t ↦ ℛ ( t ) x is continuous for t ⩾ 0 .
ℛ ( t ) ∈ ℒ ( D ( A ) ) for t ⩾ 0 . For x ∈ D ( A ) , ℛ ( . ) x ∈ C 1 ( R + , X ) ∩ C ( R + , D ( A ) ) and for t ⩾ 0 , we have
ℛ ′ ( t ) x = A ℛ ( t ) x + ∫ 0 t B ( t − s ) ℛ ( s ) x d s = ℛ ( t ) A x + ∫ 0 t ℛ ( t − s ) B ( s ) x d s .

Throughout this work, we assume that:

A is the infinitesimal generator of a strongly continuous semigroup ( T ( t ) ) t ≥ 0 on X .
For all t ≥ 0 , B ( t ) is closed linear operator from D ( A ) to X and B ( t ) ∈ ℬ ( D ( A ) , X ) . For any y ∈ D ( A ) , the map t ↦ B ( t ) y is bounded, differentiable and the derivative t ↦ B ′ ( t ) y is bounded uniformly continuous on R + .

The following theorem establishes sufficient conditions for the existence of a resolvent operator for equation (4).

Theorem 3

[19] Assume that ( I ) and ( II ) hold. Then, there exists a unique resolvent operator of equation (4).

The following estimation will be needed in the sequel.

Theorem 4

[19] Assume that ( I ) – ( II ) hold. Then, for any a > 0 , there exists a positive constant M ¯ = M ¯ ( a ) such that for x ∈ X , we have

∥ ℛ ( t + h ) x − ℛ ( h ) ℛ ( t ) x ∥ ≤ M ¯ h ‖ x ‖ for 0 ≤ h < t ≤ a .

The next results are essential for our work.

Lemma 6

[15] Assume that ( I ) and ( II ) hold. Then, ( T ( t ) ) t ≥ 0 is norm continuous (or continuous in the uniform operator topology) for t > 0 if and only if the corresponding resolvent operator ( ℛ ( t ) ) t ≥ 0 is norm continuous for t > 0 .

Theorem 5

[7] Assume that ( I ) and ( II ) . Then, ( T ( t ) ) t ≥ 0 is compact for t > 0 if and only if the corresponding resolvent operator ( ℛ ( t ) ) t ≥ 0 is compact for t > 0 .

Now, we consider the operator G a : L 1 ( [ 0 , a ] , X ) ⟶ C ( [ 0 , a ] , X ) defined by:

G a ( u ) ( t ) = ∫ 0 t ℛ ( t − s ) u ( s ) d s .

Lemma 7

Let W ⊂ L 1 ( [ 0 , a ] , X ) be a uniformly integrable set. Assume that ( T ( t ) ) t ≥ 0 is compact for t > 0 . Then, G a ( W ) is relatively compact.

Proof

Without loss of generality, we assume that β < 0 . To show this result, we use the Ascoli-Arzela theorem. First, we shall prove that for all t ∈ J , the set

{ G a ( v ) ( t ) : v ∈ W } ¯

is compact in X .

For t = 0 , it is obvious. Let t > 0 and 0 < ε < t . We introduce the operators G a ε and G a ˜ ε by:

G a ε ( v ) ( t ) = ℛ ( ε ) ∫ 0 t − ε ℛ ( t − s − ε ) v ( s ) d s ;

G ˜ a ε ( v ) ( t ) = ∫ 0 t − ε ℛ ( t − s ) v ( s ) d s .

Since T ( t ) is compact for t > 0 , then by Theorem 5, the resolvent operator ℛ ( t ) is compact for t > 0 too. This implies that the set

{ G a ε ( v ) ( t ) : v ∈ W } ¯

is compact in X , for each 0 < ε < t .

Furthermore, using the fact that W is uniformly bounded, we can find a function μ ∈ L 1 ( [ 0 , a ] , R + ) such that

‖ v ( t ) ‖ ≤ μ ( t ) , for a.e. t ∈ [ 0 , a ] .

Then, we obtain, for all v ∈ W

∥ G a ε ( v ) ( t ) − G a ˜ ε ( v ) ( t ) ∥ ≤ ∫ 0 t − ε ( ℛ ( t − s ) − ℛ ( ε ) ℛ ( t − s − ε ) ) v ( s ) d s ≤ M ¯ ε ∫ 0 t − ε μ ( s ) d s ,

where we use Theorem 4.

This implies that the set

{ G a ˜ ε ( v ) ( t ) : v ∈ W } ¯

is compact in X , for each 0 < ε < t .

Moreover,

∥ G a ( v ) ( t ) − G a ˜ ε ( v ) ( t ) ∥ = ∫ t − ε t ℛ ( t − s ) v ( s ) d s ≤ M ∫ t − ε t μ ( s ) d s .

As we can easily observe, the right-hand side of the aforementioned inequality converges to 0 as ε → 0 + , independently of v ∈ W . Consequently, the set

{ G a ( v ) ( t ) : v ∈ W } ¯

is compact in X for every t ∈ [ 0 , a ] .

Now, we prove the equicontinuity of the sequence { G a ( v ) : v ∈ W } on J . For t = 0 , let t ′ > 0 . Then,

∥ G a ( v ) ( t ′ ) − G a ( v ) ( 0 ) ∥ ≤ ∫ 0 t ′ ∥ ℛ ( t ′ − s ) v ( s ) ∥ d s ≤ M ∫ 0 t ′ μ ( s ) d s .

We deduce that the right-hand side of the aforementioned inequality goes to 0 as t ′ → 0 independently of v ∈ W .

For 0 < t ≤ t ′ ≤ a , we have

‖ G a ( v ) ( t ′ ) − G a ( v ) ( t ) ‖ ≤ ∫ 0 t ∥ ( ℛ ( t ′ − s ) − ℛ ( t − s ) ) v ( s ) ∥ d s + ∫ t t ′ ‖ ℛ ( t ′ − s ) v ( s ) ‖ d s ≤ ∫ 0 t ‖ ℛ ( t ′ − s ) − ℛ ( t − s ) ‖ μ ( s ) d s + M ∫ t t ′ μ ( s ) d s .

Using again the compactness of T ( t ) for t > 0 and combining Lemma 6, we conclude that ℛ ( t ) is norm continuous for t > 0 . Thus, using the dominated convergence theorem, the right-hand side of the aforementioned expression goes to 0 as t ′ → t independently of v ∈ W , which ends the proof.□

Lemma 8

Let W ⊂ L 1 ( [ 0 , a ] , X ) be a uniformly integrable set. Assume that ( T ( t ) ) t ≥ 0 is norm continuous for t > 0 . Then, G a ( W ) is equicontinuous on [ 0 , a ] .

Proof

It is similar to the previous one of Lemma 7.□

Remark 1

Let v ∈ L 1 ( [ 0 , a ] , X ) be a function satisfying

∫ 0 t ℛ ( t − s ) v ( s ) d s = 0 for all t ∈ [ 0 , a ] .

Then, v ( t ) = 0 a.e t ∈ [ 0 , a ] .

Remark 2

The operator G a satisfies the following properties:

There exists d ≥ 0 such that
‖ G a f ( t ) − G a g ( t ) ‖ ≤ d ∫ 0 t ‖ f ( s ) − g ( s ) ‖ d s
for every f , g ∈ L 1 ( [ 0 , a ] ; X ) , 0 ≤ t ≤ a .
For any compact K ⊂ X and sequence { f p } p = 1 ∞ ⊂ L 1 ( [ 0 , a ] ; X ) such that { f p ( t ) } p = 1 ∞ ⊂ K for a.e. t ∈ [ 0 , a ] . We have
f p ⇀ f 0 ⇒ G a f p ⟶ G a f 0 as p → + ∞ .

For later use, we make the following assumption:

( ℛ ( t ) ) t ≥ 0 is uniformly stable; i.e., there exist M ≥ 1 and α > 0 such that for each t ≥ 0 ,
‖ ℛ ( t ) ‖ ≤ ℳ e − α t .

2.3 Scale of Banach spaces concept and fixed point theorem

In this section, we recall some results on the existence of fixed points via a scale of Banach spaces.

Definition 14

[6] A scale of Banach spaces ( X k ) k ∈ N is a family of Banach spaces ( X k , ‖ ⋅ ‖ k ) for k ∈ N such that X k ↪ X l densely and continuously for k ≥ l .

Let ( C , ‖ ⋅ ‖ ) be a Banach space and let F : C → P b ( C ) be a multi-map. Let Z be a closed vector subspace of C , which is invariant under F , that is to say, F : Z → P b ( Z ) . We assume that there exists a scale of Banach spaces

( C , ‖ ⋅ ‖ ) … ↪ … ( C n , ‖ ⋅ ‖ n ) ↪ R n − 1 , n ( C n − 1 , ‖ ⋅ ‖ n − 1 ) ↪ … ( C 1 , ‖ ⋅ ‖ 1 ) ,

where ( C n , ‖ ⋅ ‖ n ) are the Banach spaces for n ∈ N , R n − 1 , n : ( C n , ‖ ⋅ ‖ n ) → ( C n − 1 , ‖ ⋅ ‖ n − 1 ) are the bounded surjective linear maps, and there exist bounded surjective linear maps R n : ( C , ‖ ⋅ ‖ ) → ( C n , ‖ ⋅ ‖ n ) , and u.s.c. maps F n : C n → P c b ( C n ) for all n ∈ N . We assume that F , F n , R n − 1 , n , and R n are related as follows.

Let y , z ∈ C such that R n y ∈ F n ( R n z ) for all n ∈ N , then y ∈ F ( z ) .
For every n ∈ N , and for every y ∈ C n + 1 such that y n = R n , n + 1 y ∈ F n ( y n ) , there exists z ∈ F n + 1 ( y ) such that R n , n + 1 y = R n , n + 1 z .
If ( y n ) n ∈ N is a sequence such that y n ∈ C n , y n = R n , n + 1 y n + 1 , and ( ∥ y n ∥ n ) n ∈ N is a bounded set, then there exists y ∈ C such that y n = R n y for all n ∈ N .
For every n ∈ N ,
R n , n + 1 F n + 1 ⊆ F n R n , n + 1 .

Theorem 6

[16] Assume that F : C → P b ( C ) , F n : C n → KV ( C n ) , n ∈ N , satisfy conditions ( V 1 ) – ( V 4 ) , and { ‖ z ‖ n : z ∈ F n ( y ) , y ∈ C n , n ∈ N } is a bounded set. Assume further that F n is an u.s.c. ϑ -condensing multivalued map for all n ∈ N . Then, Fix ( F ) is a nonempty set.

Let x 0 be the fixed point of F whose existence was proved in Theorem 6. Then, we have the following consequence.

Corollary 1

[16] Assume that F : C → P b ( C ) and F n : C n → KV ( C n ) , n ∈ N , satisfy the conditions of Theorem 6. Let Z be a closed vector subspace of C such that F : Z → P c ( Z ) . Assume further that F satisfies the local Lipschitz condition:

d H ( F ( x 2 ) , F ( x 1 ) ) ⩽ L ( r , x ) ∥ x 2 − x 1 ∥ ,

for all x ∈ C , r > 0 , and x 1 , x 2 ∈ B C ( x , r ) . If there exists r 0 > 0 such that B C ( x 0 , r 0 ) ∩ Z ≠ ∅ and L ( r 0 , x 0 ) < 1 , then F has a fixed point in Z.

3 Existence of mild solutions of Problem (3)

In this section, we will show the existence of mild solutions under some compactness conditions. In particular, we focus on the case where F takes the form F = F 1 + F 2 , with F 1 : [ 0 , ∞ ) × X → X and F 2 being a multivalued map from [ 0 , ∞ ) × X into V ( X ) . First, we prove the existence of solutions for equation (3) under certain compactness conditions on F 2 . In the second part, we demonstrate the existence of solutions by avoiding the condition that F 2 has compact values. Instead, we replace it with measurability conditions, allowing us to apply the Kuratowski-Ryll-Nardzewski theorem [18].

We make the following assumptions:

( T ( t ) ) t ≥ 0 is norm continuous for t > 0 .
The function F 1 ( ⋅ , v ) : R + ⟶ X is strongly measurable for each v ∈ X , and the function F 1 ( ⋅ , 0 ) is bounded on R + .
For each t ⩾ 0 , the function F 1 ( t , ⋅ ) : X ⟶ X is continuous.
There is a function ξ ∈ L 1 ( R + ) such that
∥ F 1 ( t , v 1 ) − F 1 ( t , v 2 ) ∥ ⩽ ξ ( t ) ∥ v 1 − v 2 ∥ , a.e. t ⩾ 0 ,
for all v 1 , v 2 ∈ X .

3.1 Existence of mild solutions under compactness condition

In the whole of this part, we assume that the multi-map F 2 satisfies the following conditions:

The function F 2 ( ⋅ , v ) : [ 0 , + ∞ ) ⟶ KV ( X ) admits a strongly measurable selection for each v ∈ X .
For each t ⩾ 0 , the function F 2 ( t , ⋅ ) : X ⟶ KV ( X ) is u.s.c.
For each r > 0 , there is a function ω r ∈ L loc 1 ( R + ) such that
sup t ∈ R + ∫ 0 t e − α ( t − s ) ω r ( s ) d s < ∞
and
∥ F 2 ( t , v ) ∥ ≔ sup u ∈ F 2 ( t , v ) ‖ u ‖ ⩽ ω r ( t ) , a.e. t ⩾ 0 ,
for all v ∈ X with ‖ v ‖ ⩽ r .
There exists a positive function τ ∈ L 1 ( R + ) such that for all bounded set D ⊆ X , we have
χ ( F 2 ( t , D ) ) ⩽ τ ( t ) χ ( D ) , a.e. t ⩾ 0 ,
where
F 2 ( t , D ) = { F 2 ( t , x ) : x ∈ D } .

Remark 3

We point out that Deimling [10] showed that if F 2 is Lipschitzian, then Assumption ( H 9 ) is fulfilled.

For each x ∈ C b ( R + , X ) , let S F 2 ( x ) be the set of all locally Bochner integrable selectors of the multi-map F 2 , i.e.,

S F 2 ( x ) = { u ∈ L l o c 1 ( R + , X ) : u ( t ) ∈ F 2 ( t , x ( t ) ) a . e t ∈ R + } .

Now, we give the definition of the so-called mild solution of equation (3).

Definition 15

A function x ∈ C b ( R + ; X ) is said to be a mild solution of Problem (3) if the integral equation

x ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + ∫ 0 t ℛ ( t − s ) u ( s ) d s for t ≥ 0 ,

is satisfied for u ∈ S F 2 ( x ) .

Remark 4

Let x ∈ C b ( R + , X ) . Since ( H 6 ) – ( H 8 ) are satisfied, then by Theorem 1, the function F 2 ( ⋅ , x ( ⋅ ) ) : R + ⟶ KV ( X ) admits a locally integrable selection, which proves that S F 2 ( x ) ≠ ∅ .

Next, we present the necessary working tool to cite the main result of this section. From now on, let C = C b ( R + ; X ) equipped with the sup-norm ‖ ⋅ ‖ ∞ and ( L loc 1 ( [ 0 , ∞ ) ; X ) , N 1 ( ⋅ ) ) be the Banach space of all locally Bochner integrable functions v : [ 0 , ∞ ) → X such that

sup t ∈ R + ∫ 0 t e − α ( t − s ) ‖ v ( s ) ‖ d s < ∞ ,

equipped with the following norm:

N 1 ( v ) = sup t ∈ R + ∫ 0 t e − α ( t − s ) ‖ v ( s ) ‖ d s .

Now, we define the operator G : L loc 1 ( [ 0 , ∞ ) ; X ) ⟶ C given by:

(5) G u ( t ) = ∫ 0 t ℛ ( t − s ) u ( s ) d s , t ⩾ 0 .

It is clear that G is a bounded linear operator. Then, from Assumptions ( H 6 ) – ( H 8 ) , we deduce that the multi-operator P : C ⟶ V ( C ) defined by:

P = G o S F 2

is well defined.

We consider the solution map for problem (3) Θ : C ⟶ P ( C ) as:

Θ ( x ) = { ω ∈ C : ω ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G ( u ) ( t ) , u ∈ S F 2 ( x ) }

for x ∈ C .

Remark 5

It is clear that x is a mild solution of Problem (3) if and only if x is a fixed point of Θ .

The following property plays a crucial role in the proof of the result in this section.

Lemma 9

Assume that Conditions ( H 6 ) – ( H 9 ) hold. Then, P is an u.s.c. map with convex closed values.

Proof

Applying the same reasoning used in the proof of [16, Lemma 4.4], we obtain the result.□

Now, we are in a position to state and prove our first existence result.

Theorem 7

Assume that Conditions ( I ) – ( II ) and ( H 1 ) – ( H 9 ) hold. If the following conditions are satisfied:

(6) ℳ sup t ∈ R + ∫ 0 t e − α ( t − s ) ξ ( s ) d s + ℳ liminf r → + ∞ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ω r ( s ) r d s < 1 ,

(7) ℳ sup t ∈ R + ∫ 0 t e − α ( t − s ) ξ ( s ) d s + 2 ∫ 0 ∞ τ ( t ) d t < 1 .

Then, there exists a mild solution x of Problem (3) on R + .

Proof

Let n ∈ N . We define the following space:

C n = C ( [ 0 , n ] ; X )

equipped with the following norm:

‖ x ‖ n = sup 0 ≤ t ≤ n ‖ x ( t ) ‖ for x ∈ C n .

We define the maps R n : C ⟶ C n and R n , n + 1 : C n + 1 ⟶ C n as follows:

R n y = y ∣ [ 0 , n ] ,

R n , n + 1 y = y ∣ [ 0 , n ] .

For x ∈ C n , we consider the multioperator Θ n : C n ⟶ P ( C n ) as:

Θ n ( x ) = { ω ∈ C n : ω ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G n ( u ) ( t ) , t ∈ [ 0 , n ] and u ∈ S F 2 n ( x ) } ,

where

S F 2 n ( x ) = { u ∈ L 1 ( [ 0 , n ] ; X ) : u ( t ) ∈ F 2 ( t , x ( t ) ) , t ∈ [ 0 , n ] } .

We claim that conditions of Theorem 6 are fulfilled. The proof will be presented in several steps.

Step 1: The operators Θ , Θ n , R n , and R n , n + 1 satisfy conditions ( V 1 ) – ( V 4 ) .

First, we prove that ( V 1 ) holds. Let z 1 , z 2 ∈ C be such that for all n ∈ N R n z 1 ∈ Θ n ( R n z 2 ) , which gives

z 1 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , z 2 ( s ) ) d s + G n ( u n ) ( t ) , for t ∈ [ 0 , n ] ,

for u n ∈ S F 2 n ( z 2 ) .

Using Lemma 1, we obtain that

u n ( t ) = u n + 1 ( t )

for t ∈ [ 0 , n ] .

Thus, if we choose u = u n ∣ [ 0 , n ] , we infer that

z 1 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , z 2 ( s ) ) d s + G ( u ) ( t ) , for t ≥ 0 .

On the other hand, ( H 8 ) gives that u ∈ ( L loc 1 ( [ 0 , ∞ ) ; X ) , N 1 ( ⋅ ) ) .

Consequently, z 1 ∈ Θ ( z 2 ) .

Now, we prove that ( V 2 ) holds. Let z ∈ C n + 1 be such that z

z n = R n , n + 1 z ∈ Θ n ( z n ) .

This means that

z ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , z ( s ) ) d s + G n ( u n ) ( t ) for t ∈ [ 0 , n ] ,

for u n ∈ S F 2 n ( z n ) .

Since ( H 6 ) – ( H 8 ) are satisfied, it follows by Theorem 1 that there exists a locally integrable selection v such that v ∈ F 2 ( t , y ( t ) ) for t ≥ 0 .

If we put

w ( t ) = u n ( t ) for t ∈ [ 0 , n ] v ( t ) for t ∈ ( n , n + 1 ] .

Then, w ∈ S F 2 n + 1 ( z ) .

We define

Ψ ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , z ( s ) ) d s + G n + 1 ( w ) ( t ) , for t ∈ [ 0 , n + 1 ] .

Obviously, Ψ ∈ Θ n + 1 ( z ) and we have z = Ψ ∣ [ 0 , n ] , which implies that ( V 2 ) is satisfied.

We next shall prove that ( V 3 ) holds. For that, let ( z n ) n ≥ 0 be such that z n ∈ C n , z n = R n , n + 1 z n + 1 and the sequence ( ‖ z n ‖ n ) n ≥ 0 is bounded.

If we take z such that z = z n ∣ [ 0 , n ] , then the map t ↦ z ( t ) is bounded for t ≥ 0 , which assures that ( V 3 ) is satisfied.

Finally, by considering R n , n + 1 , one can observe that ( V 4 ) is satisfied.

Step 2: Θ n ( x ) is convex for each x ∈ C n .

Let ω 1 , ω 2 ∈ Θ n ( x ) . Then, there exist u 1 , u 2 ∈ S F 2 n ( x ) such that

ω 1 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G n ( u 1 ) ( t ) ,

ω 2 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G n ( u 2 ) ( t ) ,

for t ∈ [ 0 , n ] .

Let ϕ ∈ [ 0 , 1 ] . Then, for t ∈ [ 0 , n ] , we have that

( ϕ ω 1 + ( 1 − ϕ ) ω 2 ) ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G n ( ϕ u 1 + ( 1 − ϕ ) u 2 ) ( t ) .

Since F 2 has convex values, then it is easy to see that S F 2 n ( x ) is convex. This implies that ϕ u 1 + ( 1 − ϕ ) u 2 ∈ S F 2 n ( x ) .

Consequently, ϕ ω 1 + ( 1 − ϕ ) ω 2 ∈ Θ n ( x ) .

Step 3: Θ n is u.s.c. with compact values.

Based on Remark 2 and since ( H 6 ) – ( H 9 ) are satisfied, then from [24], the superposition multi-operator

P n : C n → V ( C n )

given by:

P n = G n o S F 2 n

is u.s.c. with compact values, which implies that Θ n is u.s.c. with compact values too.

Step 4: There exists a positive constant r such that

Θ n ( B C n ( 0 , r ) ) ⊂ B C n ( 0 , r ) ,

where B C n ( 0 , r ) = { v ∈ C n : ‖ v ‖ n ≤ r } .

We argue by contradiction. If it is false, then for each r > 0 , there exist x ∈ B C n ( 0 , r ) and t 1 ≥ 0 such that

r < ‖ ℛ ( t 1 ) x 0 + ∫ 0 t 1 ℛ ( t 1 − s ) F 1 ( s , x ( s ) ) d s + ∫ 0 t 1 ℛ ( t 1 − s ) u ( s ) d s ‖ < ℳ ‖ x 0 ‖ + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ∥ F 1 ( s , 0 ) ∥ d s + r sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ξ ( s ) d s + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ω r ( s ) d s < ℳ ‖ x 0 ‖ + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ∥ F 1 ( s , 0 ) ∥ d s + r sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ξ ( s ) d s + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ω r ( s ) d s ,

with u ∈ S F 2 n ( x ) .

Dividing by r , we obtain that

1 < ℳ ‖ x 0 ‖ r + 1 r sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ∥ F 1 ( s , 0 ) ∥ d s + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ξ ( s ) d s + sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ω r ( s ) r d s .

Taking liminf as r → ∞ , we obtain that

1 ≤ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ξ ( s ) d s + ℳ liminf r > 0 sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ω r ( s ) r d s ,

which contradicts Assumption (6).

Step 5: Θ n is ϑ -condensing on B C n ( 0 , r ) .

Let D ⊂ B C n ( 0 , r ) be such that

ϑ ( Θ n ( D ) ) ≥ ϑ ( D ) .

It follows from Lemma 4 that there exists a sequence ( w k ) k ∈ N in Θ n ( D ) such that

ϑ ( Θ n ( D ) ) = ϑ ( { w k : k ∈ N } ) .

We can write w k ∈ Θ n ( x k ) for some x k ∈ D . Then,

w k ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x k ) d s + G n ( u k ) ( t ) , 0 ⩽ t ⩽ n ,

for u k ∈ S F 2 n ( x k ) .

Using Lemma 2, we obtain

ϑ ( { w k : k ∈ N } ) ≤ ℳ sup t ∈ [ 0 , n ] ∫ 0 t e − α ( t − s ) ξ ( s ) d s ϑ ( { x k : k ∈ N } ) + ϑ ( { G n ( u k ) : k ∈ N } ) .

Since u k ∈ S F 2 n ( x k ) , then by Hypothesis ( H 8 ) , we infer that { u k : k ∈ N } is uniformly integrable. Using ( H 9 ) , we obtain that

χ ( { u k ( t ) : k ∈ N } ) ≤ τ ( t ) χ ( { x k ( t ) : k ∈ N } )

for a.e t ∈ [ 0 , n ] .

Using Assumption ( H 2 ) , then by Lemma 6, ( ℛ ( t ) ) t ≥ 0 is norm continuous for t > 0 . Therefore, applying Lemma 5, we obtain

ϑ ( { G n ( u k ) : k ∈ N } ) ≤ 2 ℳ ϑ ( D ) ∫ 0 n τ ( t ) d t .

We obtain

ϑ ( { w k : k ∈ N } ) ≤ ℳ sup t ∈ [ 0 , n ] ∫ 0 t e − α ( t − s ) ξ ( s ) d s + 2 ∫ 0 n τ ( t ) d t ϑ ( D ) .

Therefore,

ϑ ( D ) ≤ ℳ sup t ∈ [ 0 , n ] ∫ 0 t e − α ( t − s ) ξ ( s ) d s + 2 ∫ 0 n τ ( t ) d t ϑ ( D ) .

Using (7), we infer that ϑ ( D ) = 0 . Thus, Θ n is a ϑ -condensing mapping.

Consequently, by Theorem 6, Problem (3) has a mild solution.□

The following corollary gives the existence of mild solution to Problem (3) in a particular case. That is, the semigroup ( T ( t ) ) t ≥ 0 is compact for t > 0 .

Corollary 2

Let Assumptions ( H 1 ) – ( H 7 ) hold. Assume further that ( T ( t ) ) t ≥ 0 is compact and (6) holds. Then, Problem (3) has a mild solution on R + .

Proof

In view of Theorem 7, G n is compact. Using the compactness of G n , we infer that Θ n is u.s.c. with convex compact values. Moreover, using the compactness of ( ℛ ( t ) ) t ≥ 0 for all t > 0 , we can prove that Θ n ( B C n ( 0 , r ) ) ¯ is compact, where r is the one found in the proof of Theorem 7, which proves that Θ n is ϑ -condensing. We complete the proof by reasoning in the same manner as in the demonstration of Theorem 7.□

3.2 Existence of mild solutions without compact condition

As it cited earlier, the fact that F 2 has compact values serves in the existence of a Bochner locally integrable selection. However, we can avoid that condition by applying the Kuratowski-Ryll-Nardzewski Theorem 2. In the whole of this subsection, we assume that F 2 is defined on [ 0 , + ∞ ) × X with values in P c ( X ) .

In this part, instead of Hypotheses ( H 8 ) – ( H 9 ) , we assume the following assumptions:

For each r > 0 , there is a function ω r ∈ L loc 2 ( R + ) such that
∥ F 2 ( t , v ) ∥ ≔ sup u ∈ F 2 ( t , v ) ‖ u ‖ ⩽ ω r ( t ) , a.e. t ⩾ 0 ,
for all v ∈ X with ‖ v ‖ ⩽ r .
There exists a positive function τ ∈ L 2 ( R + ) such that for all bounded set D ⊆ X , we have
χ ( F 2 ( t , D ) ) ⩽ τ ( t ) χ ( D ) a.e. t ⩾ 0 .

Since the compactness condition on F 2 is avoided, we shall ensure the existence of a strongly measurable selection. To do so, we recall the following important result proved in [16], which is based on Theorem 2.

Proposition 1

Assume that X is a separable Banach space. Let F 2 : [ 0 , a ] × X ⟶ P ( X ) be a locally Lipschitzian map with closed values. Let x : [ 0 , a ] ⟶ X be a continuous function. Then, there exists a measurable function u : [ 0 , a ] ⟶ X with respect to the Lebesgue measure such that u ( t ) ∈ F 2 ( t , x ( t ) ) for all t ∈ [ 0 , a ] .

Remark 6

We point out that if ( H 8 ′ ) is fulfilled, one can see that u ∈ L 2 ( [ 0 , a ] , X ) .

Remark 7

Based on Proposition 1 and Remark 6, for a > 0 and x ∈ C a = C ( [ 0 , a ] , X ) , we can define

S F 2 a ( x ) = { u ∈ L 2 ( [ 0 , a ] ; X ) : u ( t ) ∈ F 2 ( t , x ( t ) ) , t ∈ [ 0 , a ] } ≠ ∅ .

Similarly, for x ∈ C , we define

S F 2 ( x ) = { u ∈ L loc 2 ( [ 0 , + ∞ ) ; X ) : u ( t ) ∈ F 2 ( t , x ( t ) ) , t ≥ 0 } ≠ ∅ .

In this case, let ( L loc 2 ( [ 0 , + ∞ ) ; X ) , N 2 ( ⋅ ) ) be the Banach space of all locally Bochner integrable functions v : [ 0 , ∞ ) → X such that:

sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ‖ v ( s ) ‖ 2 d s < ∞ ,

equipped with the following norm:

N 2 ( v ) = sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ‖ v ( s ) ‖ 2 d s 1 2 .

We define the operator G : L loc 2 ( [ 0 , + ∞ ) ; X ) ⟶ C given by (5).

Next, we give the definition of a weakly u.s.c map.

Definition 16

[16] Let Ω be a metric space and let Y be a Banach space. A multivalued map F : Ω ⟶ P ( Y ) is said to be weakly upper semi-continuous (w.u.s.c.) if F − 1 ( V ) is an open subset of Ω for all weakly open set V ⊂ Y .

Remark 8

From Definition 11, a multi-map F : Ω ⟶ P ( Y ) is w.u.s.c. if and only if F + − 1 ( V ) is a closed subset of Ω for all weakly closed set V ⊂ Y .

Remark 9

According to [16, Proposition 4.12], in a reflexive space X , a locally Lipschitzian multifunction f that satisfies ( H 8 ′ ) is necessarily w.u.s.c.

Lemma 10

[24] Assume that G a satisfies conditions in Remark 2 and the sequences

( x n ) n ≥ 1 ⊂ C ( [ 0 , a ] ; X ) , ( v n ) n ≥ 1 ⊂ L 1 ( [ 0 , a ] ; X ) ,

v n ∈ P a ( x n ) , n ≥ 1 are such that x n → x 0 , v n ⇀ v 0 . Then, v 0 ∈ P a ( x 0 ) .

Lemma 11

Assume that X is a separable reflexive Banach space. Let Assumptions ( H 2 ) and ( H 8 ′ ) hold and F 2 be a locally Lipschitzian map with closed values. Then, the multioperator P a : C a ⟶ KV ( C a ) is u.s.c.

Proof

The proof is similar to that of [16, Proposition 4.16].□

Now, we are in a position to state the main theorem of this part.

Theorem 8

Assume that X is reflexive. Let Assumptions ( H 1 ) – ( H 4 ) hold. Let F 2 be a locally Lipschitzian multimap that satisfies ( H 8 ′ ) – ( H 9 ′ ) . If (6) and (7) hold, then Problem (3) has a mild solution on R + .

Proof

Using the same main tool working that was used in the proof of Theorem 7, the only difficulty in the proof is to show that Θ n is an u.s.c. map with compact values and it is guaranteed by Lemma 11. The proof is then concluded by reasoning as in the proof of Theorem 7.□

4 Existence of a.a.p. solutions of Problem (3)

Our aim in this part is to study the existence of a.a.p. solutions for Problem (3). We start the study by giving some definitions and basic results.

Definition 17

[25] A set E of real numbers is called relatively dense if there exists a number l > 0 such that any interval ( α , α + l ) ⊂ R of length l contains at least one number from E .

Definition 18

[16] A continuous function f : R ⟶ X is called a.p. if for every ε > 0 , there exists a relatively dense subset P ε of R such that

‖ x ( t + τ ) − x ( t ) ‖ ≤ ε , t ∈ R , τ ∈ P ε .

Definition 19

[16] A function z ∈ C b ( [ 0 , + ∞ ) ; X ) is called a.a.p. if there exists w ∈ C 0 ( [ 0 , + ∞ ) ; X ) and an a.p. function x such that z ( t ) = x ( t ) + w ( t ) for all t ≥ 0 .

Theorem 9

[22] A function f ∈ C ( [ 0 , + ∞ ) ; X ) is a.a.p. if and only if for every ε > 0 , there exists t ε > 0 and a relatively dense subset P ε of [ 0 , + ∞ ) such that

‖ f ( t + ξ ) − f ( t ) ‖ ≤ ε , t ≥ t ε , ξ ∈ P ε .

Definition 20

[16] A function f ∈ C ( [ 0 , + ∞ ) × X ; X ) is called uniformly a.a.p. (u.a.a.p.) on compact sets if for every ε > 0 and every compact K ⊂ X , there exists a relatively dense subset P K , ε in [ 0 , + ∞ ) and t K , ε > 0 such that

‖ f ( t + τ , x ) − f ( t , x ) ‖ ≤ ε , t ≥ t K , ε , ( τ , x ) ∈ P K , ε × K .

In the sequel, A A P ( X ) denotes the space of consisting of a.a.p. functions equipped with the sup-norm.

The following lemma plays an important role in the next.

Lemma 12

[22] Assume that ( H 1 ) is satisfied. Let v ∈ A A P ( X ) and u : [ 0 , + ∞ ) ⟶ X be the function defined by:

u ( t ) = ∫ 0 t ℛ ( t − s ) v ( s ) d s , t ≥ 0 .

Then, u ∈ A A P ( X ) .

In order to state and prove the main result of this section, we need the following important results.

Lemma 13

Let F 1 ∈ C ( [ 0 , + ∞ ) × X ; X ) be a function satisfying ( H 5 ) and F 1 ( ⋅ , 0 ) be bounded on [ 0 , + ∞ ) . Let K ⊂ X be a compact set. Then,

lim r → + ∞ ∫ r t ℛ ( s ) F 1 ( t − s , v ) d s = 0 ,

uniformly for t ≥ r and v ∈ K .

Proof

Let v ∈ K and t ≥ r . Using Hypothesis ( H 1 ) , we obtain

∫ r t ℛ ( s ) F 1 ( t − s , v ) d s ≤ ℳ ∫ r t e − α s ξ ( t − s ) ‖ v ‖ d s + ∫ r t e − α s ‖ F 1 ( t − s , 0 ) ‖ d s ≤ ℳ ‖ v ‖ ∫ 0 + ∞ ξ ( s ) d s + 1 α sup t ≥ 0 ‖ F 1 ( t , 0 ) ‖ e − α r .

Since the right-hand side tends to 0 uniformly for t ≥ r and v ∈ K , we deduce the result.□

Lemma 14

[28] Let f : R + ⟶ X be an a.a.p. function. Then, the range of f is relatively compact.

Theorem 10

[22] Let F 1 ∈ C ( [ 0 , + ∞ ) × X ; X ) be an u.a.a.p. on compact sets function that satisfies Condition ( H 5 ) . Let x : R + ⟶ X be an a.a.p. function. Then, the function w : R + ⟶ X defined by:

w ( t ) = ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s for t ≥ 0

is a.a.p.

We make the following assumptions:

F 1 is u.a.a.p. on compact sets.
For every ρ > 0 , there exists a measurable function γ ρ : [ 0 , ∞ ) ⟶ [ 0 , ∞ ) such that the function ρ ↦ γ ρ is nondecreasing and satisfies the following property: for each v 1 , v 2 ∈ C with ∥ v 1 − v 2 ∥ ∞ ⩽ ρ , for every u 2 ∈ S F 2 ( v 2 ) , there exists u 1 ∈ S F 2 ( v 1 ) such that
∥ u 2 ( t ) − u 1 ( t ) ∥ ⩽ γ ρ ( t ) ∥ v 2 ( t ) − v 1 ( t ) ∥ for t ≥ 0 .
For each x ∈ A A P ( X ) , the set N F 2 ( x ) = S F 2 ( x ) ∩ A A P ( X ) ≠ ∅ .

Theorem 11

Assume that (4) has a resolvent operator ( ℛ ( t ) ) t ≥ 0 and ( H 1 ) – ( H 12 ) hold. If

(8) ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) [ ξ ( s ) + γ 2 ϱ ( s ) ] d s < 1 ,

where

ϱ = ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) [ ξ ( s ) r + ω r ( s ) ] d s

and r is the one obtained in the proof of Theorem 7, then, there exists an a.a.p. mild solution of Problem (3).

Proof

First, let y be the mild solution of Problem (3) obtained by Theorem 7, which is given by:

y ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , y ( s ) ) d s + G ( u ) ( t ) , t ≥ 0 ,

for u ∈ S F 2 ( y ) .

One can see that ‖ y ‖ ∞ ≤ r .

For x ∈ A A P ( X ) , we define Π ( x ) as the set formed by all functions Φ given by:

Φ ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , x ( s ) ) d s + G ( u ) ( t ) , t ⩾ 0

with u ∈ N F 2 ( x ) .

Then,

Π = Θ ∣ A A P ( X ) .

Then, as a consequence of Lemma 12 and Theorem 10, we have

Π ( A A P ( X ) ) ⊆ KV ( A A P ( X ) ) .

Thus,

Π ( A A P ( X ) ∩ B C ( 0 , r ) ) ⊆ KV ( A A P ( X ) ∩ B C ( 0 , r ) ) .

Consider the function Ψ given by:

Ψ ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , 0 ) d s , t ⩾ 0 .

By Lemma 12, Ψ is an a.a.p. function.

Then, we have the following estimation:

‖ y − Ψ ‖ ∞ ⩽ sup t ≥ 0 ∫ 0 t ℛ ( t − s ) ( F 1 ( s , y ( s ) ) − F 1 ( s , 0 ) ) d s + G ( u ) ( t ) ⩽ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ( ξ ( s ) ‖ y ( s ) ‖ + ω r ( s ) ) d s ⩽ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ( ξ ( s ) r + ω r ( s ) ) d s = ϱ ,

which implies that A A P ( X ) ∩ B C ( y , ϱ ) is a nonempty closed set.

We now claim that Π is a contraction. For this, let v , w ∈ A A P ( X ) ∩ B C ( y , ϱ ) . Let z 2 ∈ Π ( w ) and u 2 ∈ N F 2 ( w ) be such that

z 2 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , w ( s ) ) d s + G ( u 2 ) ( t ) , t ⩾ 0 .

Then, by ( H 11 ) , there exists u 1 ∈ S F 2 ( v ) such that

∥ u 2 ( t ) − u 1 ( t ) ∥ ⩽ γ 2 ϱ ( t ) ∥ w ( t ) − v ( t ) ∥ for t ⩾ 0 .

We now define z 1 by:

z 1 ( t ) = ℛ ( t ) x 0 + ∫ 0 t ℛ ( t − s ) F 1 ( s , v ( s ) ) d s + G ( u 1 ) ( t ) for t ⩾ 0 .

We obtain for t ≥ 0

∥ z 2 ( t ) − z 1 ( t ) ∥ ⩽ ℳ ∫ 0 t e − α ( t − s ) ξ ( s ) ∥ w ( s ) − v ( s ) ∥ d s + ℳ ∫ 0 t e − α ( t − s ) ∥ u 2 ( s ) − u 1 ( s ) ∥ d s ⩽ ℳ ∫ 0 t e − α ( t − s ) ξ ( s ) ∥ w ( s ) − v ( s ) ∥ d s + ℳ ∫ 0 t e − α ( t − s ) γ 2 ϱ ( s ) ∥ w ( s ) − v ( s ) ∥ d s ⩽ ℳ ∫ 0 t e − α ( t − s ) ( ξ ( s ) + γ 2 ϱ ( s ) ) d s sup s ∈ [ 0 , t ] ∥ w ( s ) − v ( s ) ∥ ≤ ℳ ∫ 0 t e − α ( t − s ) ( ξ ( s ) + γ 2 ϱ ( s ) ) d s ∥ w − v ∥ ∞ .

We obtain

d ( z 2 , Π ( v ) ) ⩽ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) [ ξ ( s ) + γ 2 ϱ ( s ) ] d s ∥ w − v ∥ ∞ .

Then, we deduce that

d H ( Π ( w ) , Π ( v ) ) ⩽ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ( ξ ( s ) + γ 2 ϱ ( s ) ) d s ∥ w − v ∥ ∞ .

Hence, Π is a contraction.

Next, we prove that

Π ( A A P ( X ) ∩ B C ( y , ϱ ) ) ⊆ A A P ( X ) ∩ B C ( y , ϱ ) .

To do so, let x ∈ A A P ( X ) ∩ B C ( y , ϱ ) and v ∈ Π ( x ) , then a similar reasoning as earlier, we infer that

‖ v − y ‖ ∞ ⩽ ℳ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ( ξ ( s ) + γ ϱ ( s ) ) d s ‖ x − y ‖ ∞ ⩽ ℳ ϱ sup t ⩾ 0 ∫ 0 t e − α ( t − s ) ( ξ ( s ) + γ 2 ϱ ( s ) ) d s ⩽ ϱ .

This implies that v ∈ B C ( y , ϱ ) . Thus, by applying Corollary 1, Π has a fixed point x , which is an a.a.p. mild solution for Problem (3).□

5 An example

In this part, we consider a reaction–diffusion system with memory of the following form:

(9) ∂ w ( t , x ) ∂ t ∈ ∂ 2 w ( t , x ) ∂ 2 x + ∫ 0 t e − ( t − s ) ∂ 2 w ( s , x ) ∂ 2 x d s + f 0 ( t , w ( t , x ) ) + [ f 1 ( t , w ( t , x ) ) ; f 2 ( t , w ( t , x ) ) ] for t ≥ 0 and x ∈ [ 0 , π ] w ( t , 0 ) = w ( t , π ) = 0 for t ≥ 0 w ( 0 , x ) = w 0 ( x ) for x ∈ [ 0 , π ] ,

where w 0 is a L 2 -function, σ : R + ⟶ R is a C 1 function, and the functions f 0 , f 1 , f 2 : R + × R → R ( f 1 ≤ f 2 ) satisfy the following conditions:

f 0 ( ⋅ , x ) : R + ⟶ R is measurable for each x ∈ R .
There exist l 0 , l 1 , l 2 ∈ L 1 ( R + ) such that
∣ f 0 ( t , x 1 ) − f 0 ( t , x 2 ) ∣ ≤ l 0 ( t ) ∣ x 1 − x 2 ∣ for x 1 , x 2 ∈ R and t ≥ 0 ,

∣ f 1 ( t , x 1 ) − f 1 ( t , x 2 ) ∣ ≤ l 1 ( t ) ∣ x 1 − x 2 ∣ for x 1 , x 2 ∈ R and t ≥ 0 ,

∣ f 2 ( t , x 1 ) − f 2 ( t , x 2 ) ∣ ≤ l 2 ( t ) ∣ x 1 − x 2 ∣ for x 1 , x 2 ∈ R and t ≥ 0 .
The function t ↦ f 0 ( t , 0 ) is bounded.

To treat System (9) in its abstract form, we take X = L 2 ( 0 , π ; R ) and

z ( t ) ( x ) = w ( t , x ) for t ≥ 0 and x ∈ [ 0 , π ] .

We define the operator A : D ( A ) ⊂ X ⟶ X by:

D ( A ) = H 2 ( 0 , π ) ∩ H 0 1 ( 0 , π ) , A z = z ″ for z ∈ D ( A ) .

It is well known that A is the generator of a compact C 0 -semigroup ( T ( t ) ) t ≥ 0 on X satisfying

‖ T ( t ) ‖ ≤ e − t .

B ( ⋅ ) is given by:

D ( B ) = D ( A ) , ( B ( t ) z ) ( x ) = e − t z ′ ′ ( x ) for t ∈ R + and x ∈ [ 0 , π ] .

It is clear that ( B ( t ) ) t ≥ 0 satisfies ( II ) .

We introduce the multifunction F by:

F ( t , z ) ( x ) = f 0 ( t , z ( x ) ) + [ f 1 ( t , z ( x ) ) ; f 2 ( t , z ( x ) ) ]

for ( t , z ) ∈ R + × X and x ∈ [ 0 , π ] .

Then, our problem takes the following abstract form:

(10) z ′ ( t ) ∈ A z ( t ) + ∫ 0 t B ( t − s ) z ( s ) d s + F ( t , z ( t ) ) for t ≥ 0 , z ( 0 ) = w 0 ∈ X .

We observe that F can be decomposed into the form F = F 1 + F 2 , where

F 1 ( t , z ) ( x ) = f 0 ( t , z ( x ) ) ,

F 2 ( t , z ) ( x ) = [ f 1 ( t , z ( x ) ) ; f 2 ( t , z ( x ) ) ] ,

for ( t , z ) ∈ R + × X and x ∈ [ 0 , π ] .

Moreover, by using [19, Theorem 4.1], we deduce that the following linear integrodifferential equation:

z ′ ( t ) = A z ( t ) + ∫ 0 t B ( t − s ) z ( s ) d s for t ≥ 0 , z ( 0 ) ∈ X ,

admits a uniformly stable resolvent operator ( ℛ ( t ) ) t ≥ 0 satisfying the following estimation:

‖ ℛ ( t ) ‖ ≤ e − 1 2 t .

Hence, Hypothesis ( H 1 ) is satisfied.

It is clear that F 1 satisfies ( H 3 ) – ( H 5 ) . Moreover, according to [10], Condition (ii) implies that F 2 ( t , ⋅ ) is u.s.c., which proves that ( H 7 ) is satisfied.

On the other hand, let z ∈ X , then, for u ∈ F 2 ( t , z ) and t ≥ 0 , we obtain

‖ u ‖ 2 ≤ ω r ( t ) ,

where ω r ( t ) = κ ( t ) + μ ( t ) r , where κ ( t ) = π ( ∣ f 1 ( t , 0 ) ∣ + ∣ f 2 ( t , 0 ) ∣ ) and μ ( t ) = max { l 1 ( t ) ; l 2 ( t ) } . This implies that F 2 satisfies ( H 8 ) .

In addition, by following the same reasoning used in [16, Example 4.13], we infer that F 2 is a Lipschitzian multimap. Hence, from [10], F 2 satisfies ( H 9 ) with τ ( t ) = μ ( t ) .

Proposition 2

(11) sup t ∈ R + ∫ 0 t e − 1 2 ( t − s ) l 0 ( s ) d s + 2 ∫ 0 ∞ μ ( t ) d t < 1 ,

then, (10) has a mild solution on R + .

Consequently, the function w defined by w ( t , x ) = z ( t ) ( x ) for t ≥ 0 and x ∈ [ 0 , π ] is a solution of Problem (9).

We assume that

There exists an a.a.p. function c : R + ⟶ R such that:
∣ f 0 ( t 1 , x ) − f 0 ( t 2 , x ) ∣ ≤ ∣ c ( t 1 ) − c ( t 2 ) ∣ ∣ x ∣ for t 1 , t 2 ∈ R + and x ∈ R .

Since c is a.a.p., then we can show easily that F 1 is uniformly a.a.p. on compact sets. Hence, ( H 10 ) is satisfied.

Let z 1 , z 2 ∈ X and u 2 ∈ S F 2 ( z 2 ) . Then, for every x ∈ [ 0 , π ] , we have

u 2 ( x ) ∈ [ f 1 ( t , z 2 ( x ) ) , f 2 ( t , z 2 ( x ) ) ] .

And

u 2 ( x ) ∈ [ f 1 ( t , z 1 ( x ) ) − μ ( t ) ∣ z 2 ( x ) − z 1 ( x ) ∣ , f 2 ( t , z 1 ( x ) ) + μ ( t ) ∣ z 2 ( x ) − z 1 ( x ) ∣ ] .

We now define the following sets:

I 1 = { x ∈ [ 0 , π ] : u 2 ( x ) < f 1 ( t , z 1 ( x ) ) } ,

I 2 = { x ∈ [ 0 , π ] : f 1 ( t , z 1 ( x ) ) ≤ u 2 ( x ) ≤ f 2 ( t , z 1 ( x ) ) } ,

I 3 = { x ∈ [ 0 , π ] : f 2 ( t , z 1 ( x ) ) < u 2 ( x ) } .

We choose the function u 1 ∈ X defined as follows:

u 1 ( x ) = u 2 ( x ) + μ ( t ) ∣ z 2 ( x ) − z 1 ( x ) ∣ for x ∈ I 1 u 2 ( x ) − μ ( t ) ∣ z 2 ( x ) − z 1 ( x ) ∣ for x ∈ I 3 .

One can see that u 1 ∈ F 2 ( t , z 1 ) . Furthermore, we have the following estimation:

‖ u 2 − u 1 ‖ 2 ≤ μ ( t ) ‖ z 2 − z 1 ‖ 2 for all t ≥ 0 .

We infer that ( H 11 ) is satisfied with γ ρ = μ .

Let z ∈ A A P ( X ) . Since f 1 is Lipschitzian with respect to the second argument, the function t ↦ f 1 ( t , z ( t ) ( x ) ) is a.a.p. [22], which proves that ( H 12 ) is satisfied.

Proposition 3

sup t ∈ R + ∫ 0 t e − 1 2 ( t − s ) ( l 0 ( s ) + μ ( s ) ) d s < 1 ,

then, the Problem (10) has an a.a.p. mild solution z.

Acknowledgement

The authors express their gratitude toward the referees for their valuable comments and suggestions.

Funding information: The authors state that there was no funding involved.
Conflict of interest: The authors state that there are no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article since no datasets were generated or analyzed during this study.

References

[1] M. Adimy, H. Bouzahir, and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal. 46 (2001), 91–112. 10.1016/S0362-546X(99)00447-2Search in Google Scholar

[2] M. Adimy, H. Bouzahir, and K. Ezzinbi, Local existence for a class of partial neutral functional differential equations with infinite delay, Differ. Equ. Dyn. Syst. 12 (2004), 353–370. 10.32917/hmj/1150998507Search in Google Scholar

[3] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of noncompactness and condensing operators, Birkhäuser, Basel, 1992. 10.1007/978-3-0348-5727-7Search in Google Scholar

[4] M. U. Akhmet, M. A. Tleubergenova, and A. Ğ. A. C. I. K. Zafer, Asymptotic equivalence of differential equations and asymptotically almost periodic solutions, Nonlinear Anal. 67 (2007), 1870–1877. 10.1016/j.na.2006.07.045Search in Google Scholar

[5] J. Banaś, On measures of noncompactness in Banach spaces, Commentationes Mathematicae Universitatis Carolinae, vol. 21, 1980, pp. 131–143. Search in Google Scholar

[6] O. Caps, Evolution Equations in Scales of Banach Spaces, Tuebner, Stuttgart-Leipzig-Wiesbaden, 2012. Search in Google Scholar

[7] Y. K. Chang and J. J. Nieto, Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators, Numer. Funct. Anal. Optim. 30 (2009), 227–244. 10.1080/01630560902841146Search in Google Scholar

[8] Y. K. Chang and D. N. Chalishajar, Controllability of mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces, J. Franklin Inst. 345 (2008), 499–507. 10.1016/j.jfranklin.2008.02.002Search in Google Scholar

[9] N. M. Chuong and T. D. Ke, Generalized Cauchy problems involving nonlocal and impulsive conditions, J. Evol. Equ. 12 (2012), 367–392. 10.1007/s00028-012-0136-4Search in Google Scholar

[10] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 2011. Search in Google Scholar

[11] B. De Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl. 382 (2011), 761–771. 10.1016/j.jmaa.2011.04.078Search in Google Scholar

[12] L. Del Campo, M. Pinto, and C. Vidal, Almost and asymptotically almost periodic solutions of abstract retarded functional difference equations in phase space, J. Difference Equ. Appl. 17 (2011), 915–934. 10.1080/10236190903460404Search in Google Scholar

[13] W. Desch, R. Grimmer, and W. Schappacher, Some considerations for linear integrodifferential equations, J. Math. Anal. Appl. 104 (1984), 219–234. 10.1016/0022-247X(84)90044-1Search in Google Scholar

[14] T. Diagana, H. Henriquez, and E. A. H. Morales, Asymptotically almost periodic solutions to some classes of second-order functional differential equations, Differential and Integral Equations 21 (2008), 575–600. 10.57262/die/1356038633Search in Google Scholar

[15] K. Ezzinbi, S. Ghnimi, and M. A. Taoudi, Existence results for some partial integrodifferential equations with nonlocal conditions, Glas. Mat. 51 (2016), 413–430. 10.3336/gm.51.2.09Search in Google Scholar

[16] C. A. Gallegos and H. R. Henríquez, Fixed points of multivalued maps under local Lipschitz conditions and applications, Proc. A. 150 (2020), 1467–1494. 10.1017/prm.2018.151Search in Google Scholar

[17] R. Grimmer and F. Kappel, Series expansions for resolvents of Volterra integrodifferential equations in Banach space, SIAM J. Math. Anal. 15 (1984), 595–604. 10.1137/0515045Search in Google Scholar

[18] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Springer, Dordrecht, 2006. Search in Google Scholar

[19] R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), 333–349. 10.1090/S0002-9947-1982-0664046-4Search in Google Scholar

[20] H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351–1371. 10.1016/0362-546X(83)90006-8Search in Google Scholar

[21] H. R. Henríquez, V. Poblete, and J. C. Pozo, Mild solutions of non-autonomous second order problems with nonlocal initial conditions, J. Math. Anal. Appl. 412 (2014), 1064–1083. 10.1016/j.jmaa.2013.10.086Search in Google Scholar

[22] H. R. Henríquez, E. Hernández, and J. C. DosSantos, Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations, Zeitschrift für Analysis und ihre Anwendungen 26 (2007), 363–375. 10.4171/ZAA/1329Search in Google Scholar

[23] E. Hernández and M. L. Pelicer, Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations, Appl. Math. Lett. 18 (2005), 1265–1272. 10.1016/j.aml.2005.02.015Search in Google Scholar

[24] K. Deimling, Multivalued differential equations, De Gruyter, Berlin, 1992. 10.1515/9783110874228Search in Google Scholar

[25] B. M. Levitan, and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, U.K., 1982. Search in Google Scholar

[26] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. 10.1090/qam/295683Search in Google Scholar

[27] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, Rhode Island, 2022. Search in Google Scholar

[28] S. Zaidman, Almost-periodic Functions in Abstract Spaces, Pitman Advanced Pub. Program, Boston, 1985. Search in Google Scholar

[29] V. Vijayakumar, Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces, Int. J. Control. 91 (2018), 204–214. 10.1080/00207179.2016.1276633Search in Google Scholar

[30] P. T. Xuan, N. T. Van, and B. Quoc, Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold, J. Math. Anal. Appl. 517 (2023), 126578. 10.1016/j.jmaa.2022.126578Search in Google Scholar

Received: 2023-04-09

Revised: 2023-07-14

Accepted: 2023-08-20

Published Online: 2023-11-02

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