Regularity results for some class of nonautonomous partial neutral functional differential equations with finite delay

Regularity results for some class of nonautonomous partialneutral functional differential equations with finite delay Bila Adolphe Kyelem 1,2 Received: 30 October 2020 / Accepted: 12 Feb

Regularity results for some class of nonautonomous partial

neutral functional differential equations with finite delay

Bila Adolphe Kyelem 1,2

Received: 30 October 2020 / Accepted: 12 February 2021

© The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada 2021

Keywords Nonautonomous operator· C0semigroup· Neutral partial differential

equations· Mild solution · Strict solution · Evolution system

Mathematics Subject Classification Primary 34G20· 34K30 · 34K40; Secondary 47N20

1 Introduction

In the population dynamics theory and applications, the ordinary neutral differential equationsunderstandably received most attention in the vast literature In this vast domain, one can citethe works done in [4,12,13] Also, it is well known that the partial differential neutral systemsappear in transmission line theory For example, Wu and Xia in [17] proposed the following

system of partial neutral functional differential difference equations on the unit circle S1

∂

∂t [x(., t) − qx(., t − r)] = K

∂2

∂ξ2[x(., t) − qx(., t − r)] + f (x t ) t ≥ 0

whereξ ∈ S1, K is a positive constant and 0 ≤ q < 1 which models a circular array of

identical resistively coupled lossless transmission lines

B Bila Adolphe Kyelem

kyeleadoc@yahoo.fr

1 Unité de Formation et de Recherche en Sciences et Technologies, Département de Mathématiques

et Informatique, Université de Ouahigouya, 01 B.P 346, Ouahigouya 01, Burkina Faso

2 LAboratoire de Mathématiques et d’Informatique (LAMI), Université Joseph-Ki-Zerbo, 03 B.P 7021, Ouagadougou 03, Burkina Faso

In this work which focus on the non-autonomous partial neutral differential equationswith finite delay, we denote by(X, .) the Banach space X endowed with the norm .

andL (X, Z) the space of linear operators from X to Z Also, we refer to D(A(t)), the

domain of the operator A (t) : X → X for every t ∈ [0, T ] For the convenience, we assume

that there exists a Banach space Y densely and continuously embedded in X The space

C = C([−r, 0], X) endowed with the uniform norm topology

C0-semigroup with T be some fixed positive real number.

We denote by u t for t ∈ [0, T ], the historic function defined on [−r, 0] by

ut (θ) = u(t + θ) for θ ∈ [−r, 0],

where u is a function from [0, T ] into X.

Dis a bounded linear operator fromC = C([−r, 0], X) into X defined by

andη : [−r, 0] → L (X) is of bounded variation and non atomic at zero; that is, there exists

a continuous nondecreasing functionγ : [0, r] → [0, +∞) such that γ (0) = 0 and

In the same track, Friendman in [5] imposed optimal conditions to the family{A(t)} t ∈[0,T ]

and obtained the regularity results

It is also important to note that the study of non-autonomous evolution equations staysactively the subject of many theoretical and applied branches of mathematics Among theclassical relative works in the subject, we refer explicitly to [2,3,6,8,10,14] Note also that

in the autonomous case where A (t) = A, the problem (1.1) has been the subject of variousquantitative and qualitative studies (see [9,16])

Our paper is organized as follows: in Sect.2, we made some preliminary results andassumptions which play an important role in this paper The Sect.3did essentially the study

of existence and uniqueness of mild solution of Eq (1.1) via some fixed point theory Wedealed in Sect.4with the existence and uniqueness of strict solution of (1.1) The last sectionfocused on an application to our studied theoretical results

norm. Y)

Definition 2 Let X be a Banach space A family {A(t)} t∈[0,T ]of infinitesimal generators of

C0semigroups on X is said stable if there are constants M ≥ 1 and ω (called the stability

constants) such that

and any every sequence 0≤ t1 ≤ t2≤ · · · ≤ t k ≤ T , k = 1, 2,

Here,ρ(A(t)) is the resolvent set of the operator A(t) and R(λ; A(t)) defines the resolvent

operator associated to A (t) at the point λ.

Remark 1 The stability of a family {A(t)} t∈[0,T ] of infinitesimal generators of C0semigroups

on X is preserved when we replace the norm in X by an equivalent norm.

For t ∈ [0, T ], let A(t) be the infinitesimal generator of a C0semigroup{T t (s)}s≥0 on X

Pazy did the following assumptions to obtain some useful results for the study of classicalsolutions in the non-autonomous hyperbolic problem:

(H1 ) {A(t)}t ∈[0,T ] is a stable family with the stability constants M , ω.

(H2 ) Y ⊂ X is A(t)-admissible for t ∈ [0, T ] and the family { ˜A(t)}t∈[0,T ]of parts ˜A(t) of

A (t) in Y is a stable family in Y with the stability constants ˜ M , ˜ω.

(H3

A(t) is continuous in the space L (Y , X) equipped with the uniform norm topology

. L(Y ,X)

Proposition 1 [14] Let {A(t)} t∈[0,T ] be the infinitesimal generator of a C0 semigroup

{T t(s)}s≥0 on X If the family {A(t)} t∈[0,T ] satisfies the conditions (H1 ), (H2) and (H3),

then there exists a unique evolution system

{U(t, s) : 0 ≤ s ≤ t ≤ T } in X verifying

U(t, s) L(X) ≤ Me ω(t−s) for 0 ≤ s ≤ t ≤ T

∂

∂t U (t, s)v = A(s)v for v ∈ Y , 0 ≤ s ≤ t ≤ T

∂s U (t, s)v = −U(t, s)A(s)v for v ∈ Y , 0 ≤ s ≤ t ≤ T

In order to obtain an evolution system that satisfies the two last conditions of the sition1, Pazy formulated this additional assumption:

Propo-(H+2) There exists a family {Q(t)}t∈[0,T ] of isomorphisms of Y onto X such that, for every

v ∈ Y , Q(t)v is continuously differentiable in X on [0, T ] and

where{B(t)} t∈[0,T ] is a strongly continuous family of bounded operators on X

Using the above additional condition, the following two results were obtained by Pazy

Lemma 1 [14, Lemma 4.4] The conditions (H1) and (H+2) imply the condition (H2).

Theorem 1 [14] Let {A(t)} t ∈[0,T ] be the infinitesimal generator of a C0 semigroup {T t (s)}s≥0

on X If the family A(t) t ∈ [0, T ] satisfies the conditions (H1), (H+2) and (H3) then there

exists a unique evolution system {U(t, s) : 0 ≤ s ≤ t ≤ T } in X satisfying the following:

(a1) U(t, s)L(X) ≤ Me ω(t−s) for 0 ≤ s ≤ t ≤ T

(a2) ∂t ∂ U (t, s)v = A(s)v for v ∈ Y , 0 ≤ s ≤ t ≤ T

(a3) ∂s ∂ U (t, s)v = −U(t, s)A(s)v for v ∈ Y , 0 ≤ s ≤ t ≤ T

(a4) U(t, s)Y ⊂ Y for 0 ≤ s ≤ t ≤ T

(a5) For every v ∈ Y , U(t, s)v is continuous in Y for 0 ≤ s ≤ t ≤ T

Let us give the notion of solutions which will be studied in this paper

Definition 3 Letφ ∈ C A function u : [−r, T ] → X is called a mild solution of Eq (1.1)associated toφ if:

Definition 4 Letφ ∈ C A continuous function u : [−r, T ] → X is called a strict solution

of Eq (1.1) associated toφ if:

t D (ut ) is continuously differentiable on [0, T ]

D (ut ) ∈ D(A(t)) for t ∈ [0, T ]

u (t) satisfies the system (1.1) for t ∈ [0, T ].

Now, we will make the following assumptions which give us some sufficient conditions

to obtain the regularity results:

(C1 ) The domain D(A(t)) = D is independent of t ∈ [0, T ].

In this case, we define on D a norm . by

Using the closedness of A (0), then Y = (D, .Y ) is a Banach space.

(C3 ):  D0L(C,X) < 1.

Remark 2 It is well known that the operator A(0) ∈ L (Y , X) Using the fact that the family

{A(t)} t ∈[0,T ] has the common closed domain, the closed graph theorem gives that A (t) ∈

L

condition lead to supt∈[0,T ] A(t) L(Y ,X) < +∞ via the principle of uniform boundedness.

The similar argument gives supt ∈[0,T ] A(t)L(Y ,X) < +∞.

Remark 3 [14] Letλ0 ∈ ρ(A(t)) Using (C1), {Q(t)}t∈[0,T ]given by

Q(t) = λ0 I − A(t) is a family of isomorphism operators from Y to X and satisfies

where B (t) = 0 for all t ∈ [0, T ] is a strongly continuous and bounded operator on X.

Remark 4 [14] When(C1

entiable on[0, T ] then, the conditions (H+2) and (H3) are verified.

Now, we are able to make our first result which is the existence and uniqueness of mildsolution to the problem (1.1)

3 Existence and uniqueness of mild solution

The main result of this section is the following

Theorem 2 Let {A(t)} t ∈[0,T ] be a stable family of infinitesimal generators of C0-semigroups

on X and assume (C1), (C2 ) and (C3) hold Furthermore, suppose that the continuous

f : R+×C → X is lipschitzian with respect to its second argument i.e., there exists positive

constant L > 0 such that for ϕ, ψ ∈ C



 f (t, ϕ) − f (t, ψ) ≤ Lϕ − ψ

C for all t ≥ 0.

Then, for all φ ∈ C , there exists a unique mild solution associated to (1.1) on [−r, +∞).

Proof The parameter set {A(t)} t∈[0,T ]is assumed to be a stable family of infinitesimal

gener-ators of C0-semigroups on X Consequently, the hypothesis (H1 ) hods Also, the conditions

(C1) and (C2) imply the hypothesis (H+2) and (H3) via the Remarks3and4 Hence, using orem1, one obtains the existence of the unique evolution system{U(t, s) : 0 ≤ s ≤ t ≤ T }

the-associated to the family of linear operators{A(t)} t∈[0,T ]and satisfying(a1)–(a5).

Now, let a > 0 and M a = C([0, a], X) be the space of continuous functions from [0, a]

to X provided with the uniform norm topology Let us set for φ ∈ C

K0(φ) = {z ∈M a : z(0) = φ(0)} For z ∈ K0(φ), we introduce the extension ˜z of z on [−r, a] by

Moreover, consider the operatorT defined on K0(φ) by

U (t, s) f (s, ˜zs )ds

 ≤t t1

HenceT (z) ∈ K0(φ) for all z ∈ K0(φ).

Now, let us show thatT (z) is a strict contraction on K0(φ) For that, let z, u ∈ K0(φ) and

wherez − u

C denotes the supremum norm in C ([0, a], X) Using (C3), one can choose a

small enough such that

Then,T is a strict contraction on K0(φ) Therefore,T has a unique fixed point u which

is the unique mild solution of Eq (1.1) on[0, a] Moreover, one can extend the solution u to [a, 2a] To prove this extension of the obtained solution, we consider the following equation

⎧

⎪

⎪

d

dt D (zt ) = A(t) D (zt ) + f (t, zt) for t ∈ [a, 2a]

z (t) = u(t) for t ∈ [−r, a].

U (t, s) f (s, ˜zs )ds for t ∈ [a, 2a],

where the function˜z is defined by

Using the similar argument, one obtains thatT ais a strict contraction on[a, 2a] that gives

a unique mild solution of (3.2) on[a, 2a] which is an extension of u Proceeding inductively, the solution u is uniquely and continuously extended to [na, (n + 1)a] for all n ≥ 1 Finally,

we obtain that Eq (1.1) has a unique mild solution on[−r, +∞).

Now, we make the following lemma which will play an important role for the study ofexistence of strict solution

Lemma 2 Let {A(t)} t∈[0,T ] be a stable family of infinitesimal generators of C0-semigroups

on X and assume (C1), (C2) and (C3 ) hold Consider φ ∈ C and h ∈ C(R+, X) such that

D (φ) = h(0) Then, there exists a unique continuous function x on R+ which solves the

Moreover, there exist two functions α and β in L∞

loc (R+, R+) such that

x tC ≤ α(t)φ C + β(t) sup

0≤s≤th(s) for t ≥ 0. (3.4)

Proof We define for p > 0 the space

W = {x ∈ C([0, p], X) : x(0) = φ(0)}

endowed with the uniform norm topology For x ∈ W, we define its extension ˜x on [−r, 0]

We have prove that the applicationK has a unique fixed point on W

Firstly, let us showK (W) ⊂ W.

It is known that h ∈ C(R+, X) Consequently, h ∈ C([0, p], X) Also, it is clear to note

that h (0) = D (φ) = φ(0) − D0(φ) It follows that (K (x))(0) = φ(0) Thus,

K (W) ⊂ W.

Moreover, we have to prove that the applicationKis a strict contraction

To do this, let x , y ∈ W with their respective extensions ˜x and ˜y associated to φ Then,

where. W is the supremum norm defined on the functional space W Taking account to the

assumption(C3), one obtains that Kis a strict contraction Consequently, the problem (3.3)

has a unique solution x defined on [−r, p].

Now, let us show the existence of the two functionsα and β in L∞

Using(C3), on obtains that 1 −D

loc ([0, p], R+) such that

We define the applicationK1on W1by

( K1(u))(t) =D0( ˜ut) + h(t), for t ∈ [p, 2p].

Using the same arguments as above, we show thatK1is a strict contraction on W1 That

leads to the existence of a unique solution u of problem (3.3) defined on[−r, 2p] and u is the extension of x on [−r, 2p].

Moreover, for t ∈ [p, 2p] and x ∈ W1one has,

Using the fact thatD0L(C,X) < 1, it follows

Hence, there exist two functionsα1andβ1 in L∞

loc ([0, 2p], R+) such that

x tC ≤ α1(t)φC + β1(t) sup

0≤s≤th(s) for t ∈ [p, 2p]

whereα1 is some extension ofα on [0, 2p] and β1, some extension ofβ on [0, 2p] It is

equivalently to say there exist two functionsα and β in L+∞loc ([0, 2p], R+) such that

x tC ≤ α(t)φ C + β(t) sup

0≤s≤th(s) for t ∈ [p, 2p].

Inductively, one can show the existence of an extension u of x on [np, (n + 1)p] with

n ∈ N and n ≥ 2, the extension a np of a, b np of b on [0, (n + 1)p] Finally, the solution

x is unique and is continuously defined onR+ Also, the existence of functions a and b in

L∞

loc (R+, R+) is inductively proved and the proof is complete.

In the next section, we will focus on the existence of strict solution of Eq (1.1) Thisregularity result can be estabilished under some smooth condition of the nonlinearity term of

Eq (1.1) When the operator A (t) = A is not dependent of the parameter t, some regularity

results equations of the form Eq (1.1) were made in the Banach space and also in theα-norm.

For more details, we refer reader to the work of Adimy and Ezzinbi in [1] and the referencestherein To our knowledge, the study of the general class of problem (1.1) in the case where

the operator A (t) depending on the parameter remains always open.

4 Existence and uniqueness of strict solution

In this section, we have to study the existence of strict solution of (1.1) Thereby, we will

assume that the function f : [0, T ] × C → X is continuous and is locally Lipschitz with

respect to the second argument and is continuously differentiable with partial derivatives

D1f and D2f both locally Lipschitz continuous with respect to the second variable It is

worth to do some necessary techniques to exhibit the strict solution of (1.1) So, let us givethe main result of this work

Theorem 3 Let {A(t)} t∈[0,T ] be a stable family of infinitesimal generators of C0-semigroups

on X and assume (C1 ), (C2 ) and (C3) hold Furthermore, suppose that f : [0, T ] × C →

X is continuously differentiable with partial derivatives D1f and D2f locally Lipschitz continuous with respect to the second variable i e., there exist some positive constants L, L1 and L2such that for all ϕ, ψ ∈ C ,

Then, the mild solution u of the problem (1.1) is a strict solution of the problem (1.1).

Proof Let λ0 ∈ ρ(A(t)) Then, using remark 3, we have the existence of the family

{Q(t)} t∈[0,T ] of isomorphism operators from Y to X defined as follows

Q(t) = λ0 I − A(t).

Moreover, let p ∈ (0, T ] and u be the mild solution of the problem (1.1) associated toφ.

Furthermore, we consider the following Cauchy problem of unknown function y given by

dt D (yt ) = A(t) D (yt ) + Q−1(t)[D1 f (t, ut ) + D2 f (t, ut )Φu(y)t]

+Q−1(t)[A(t)Q−1(t) f (t, ut) − λ0 f (t, ut)] for t ∈ [0, T ] y0(θ) = Q−1(0)(φ(θ) − λ0φ(θ)) for θ ∈ [−r, 0]

(4.1)

whereΦu : C([−r, T ], X) → C([−r, T ], X) is some continuous function defined as

fol-lows: for allw ∈ C([−r, T ], X)

Using the fact that f , D1f , D2f are lipschitz functions with respect to their second

argument and also the uniform boundedness of the operators A (t), Q−1(t) for t ∈ [0, T ]

then, the Eq (4.1) has a unique mild and continuous solution y on [−r, T ] given by

Now, we consider the function z ∈ C([−r, T ], X) associated to the above mild solution

D (φ) = A(0) D (φ) + f (0, φ). (4.7)Using (4.7) in (4.6), it follows,