POLYNOMIAL DECAY SOLUTIONS TO FRACTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAYS

Abstract. We prove the existence of decay global solutions to a class of fractional differential inclusions with infinite delays and estimate their decay rate. For this purpose, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of differential variational inequalities is also given

DIFFERENTIAL INCLUSIONS WITH INFINITE DELAYS

CUNG THE ANH, TRAN DINH KE \

Abstract We prove the existence of decay global solutions to a class of

frac-tional differential inclusions with infinite delays and estimate their decay rate.

For this purpose, we have to construct a suitable regular measure of

noncom-pactness on the space of solutions and then deploy the fixed point theory for

condensing multivalued maps An application to a class of differential

varia-tional inequalities is also given.

1 IntroductionLet X be a Banach space We consider the Cauchy problem for fractional dif-ferential inclusions with infinite delays of the following form

Dα0x(t) ∈ Ax(t) + F (t, x(t), xt), t > 0, (1.1)

where D0α, 0 < α ≤ 1, is the Caputo derivative of order α, A is the infinitesimalgenerator of a C0-semigroup on X, ϕ ∈ B with B being an admissible phase spacethat will be specified later, F : [0, ∞) × X × B → P(X) is a multi-valued function,and xt: (−∞, 0] → X is the history of the state function defined by xt(θ) = x(t+θ)for θ ∈ (−∞, 0]

Differential inclusions (DIs) appeared as in (1.1) arise, for instance, from controltheory in which control factor is taken in the form of feedbacks In such controlproblems, the presence of delay terms is an inherent feature Recently, the theory

of differential variational inequalities (DVIs) has been an increasingly interestedsubject since DVIs come from various realistic problems (see [23]) In dealing withDVIs, an effective method is converting them to DIs These brief mentions tell usthat the study of DIs is able to range over many applications

Problem (1.1)-(1.2) in case α = 1 (with/without retarded terms) has been ied extensively For a complete reference to DIs in infinite dimensional spaces, werefer the reader to monograph [15] In addition, there are many contributions forsemilinear DIs published in the last few years (see e.g [1, 8, 9, 13, 20]) Concern-ing fractional DIs in infinite dimensional sapces, one can find a number of worksdevoted to the questions of solvability and controllability Let us quote some inves-tigations in [16, 21, 25, 28, 29] that are close to the problem under consideration

stud-An important question raised for problem (1.1)-(1.2) is to study the existence ofdecay global solutions and estimate their decay rate However, up to the best of

2010 Mathematics Subject Classification 34A08, 35B35, 37C75, 47H08, 47H10.

Key words and phrases Decay solution; Fractional differential inclusion; Infinite delay; densing map; Fixed point; Measure of noncompactness; Differential variational inequality.

Con-\ Corresponding author: ketd@hnue.edu.vn.

our knowledge, no such a result has been known This is the motivation of thepresent paper.

To study the stability of solutions to differential equations and functional ferential equations, Burton and Furumochi [6, 7] introduced a new approach thatdeploys the fixed point theory to search for solutions lying in a stable subset of statespaces We will exploit this idea to prove the existence of decay global solutions

dif-to problem (1.1)-(1.2) and determine the decay rate of solutions To do this, wehave to find a satisfactory space of solutions and construct on this space a regularmeasure of noncompactness (MNC) We also have to use some new asymptotic es-timates on the family of operators {Sα(t), Pα(t), t ≥ 0} established in our previouswork [3] The designed solution space and MNC enable us to utilize the fixed pointtheory for condensing multivalued maps Consequently, we obtain a compact set ofdecay solutions x with ||x(t)|| = O(t−γ) as t → ∞ for some γ < α Since the case

α ∈ (0, 1) is more involved than the case α = 1, in this paper we only focus on theformer one and make a note that our technique can be applied to the latter case

by the same manner

The paper is organized as follows In the next section, we recall some notions andfacts related to fractional calculus, including some properties of fractional resolventoperators We also recall concept of measure of noncompactness and the fixed pointtheory for condensing multivalued maps For the sake of completeness, in Section 3

we prove the existence result of problem (1.1)-(1.2) on the interval (−∞, T ] Section

4 is devoted to proving the existence of decay global solutions with a polynomialdecay rate In the last section, we apply the abstract results to a class of DVIsconsisting of a functional partial differential equations in unbounded domain and afinite-dimensional variational inequality

2 Preliminaries2.1 Fractional calculus Let L1(0, T ; X) be the space of integrable functions on[0, T ] in the sense of Bochner

Definition 2.1 The fractional integral of order α > 0 of a function f ∈ L1(0, T ; X)

is defined by

I0αf (t) = 1

Γ(α)

Z t 0

(t − s)α−1f (s)ds,where Γ is the Gamma function, provided the integral converges

Definition 2.2 For a function f ∈ CN([0, T ]; X), the Caputo fractional derivative

Z t 0

(t − s)N −α−1f(N )(s)ds, if α ∈ (N − 1, N ),

D0αf (t) = f(N )(t), if α = N

It should be noted that there are some notions of fractional derivatives, in whichthe Riemann-Liouville and Caputo definitions have been used widely Many ap-plication problems, expressed by differential equations of fractional order, requireinitial conditions related to u(0), u0(0), etc., and the Caputo fractional derivative

satisfies these demands For u ∈ CN([0, T ]; X), we have the following formulas

k.Consider the linear problem

(λα− A)−1=

Z t 0

e−λttα−1Pα(t)dt (2.4)Then we have the following representation of solution for the linear problem (2.1)-(2.2):

x(t) = Sα(t)x0+

Z t 0

(t − s)α−1Pα(t − s)f (s)ds (2.5)Let {S(t)} be the C0-semigroup generated by A We have the formulas for Sαand

Pαas follows (see [30])

Sα(t)z =

Z ∞ 0

Pα(t)z = α

Z ∞ 0

θφα(θ)S(tαθ)zdθ, z ∈ X, (2.7)where φαis a probability density function defined on (0, ∞), that is, φα(θ) ≥ 0 and

Proposition 2.1 [27] We have the following properties

(1) If the semigroup S(·) is norm continuous, that is t 7→ S(t) is continuousfor t > 0, then Sα(·) and Pα(·) are norm continuous as well;

(2) If S(·) is a compact semigroup then Sα(t) and Pα(t) are compact for t > 0.Let p > α1 We define the operator Qα: Lp(0, T ; X) → C([0, T ]; X) as follows:

Qα(f )(t) =

Z t 0

(t − s)α−1Pα(t − s)f (s)ds (2.8)Using Proposition 2.1, we prove the following result

Proposition 2.2 If S(·) is a norm continuous semigroup, then the operator Qα

defined by (2.8) maps any bounded set in Lp(0, T ; X) into an equicontinuous one

in C([0, T ]; X)

Proof Since Pα is norm continuous, the operator

Φ(t, s) = (t − s)α−1Pα(t − s)satisfies the assumption of Lemma 1 in [22] It follows that Qα(Ω) is equicontinuousfor each bounded set Ω ⊂ Lp(0, T ; X) The proof is complete 

To study the decay rate of solutions to problem (1.1)-(1.2), we need the followingresult

Proposition 2.3 [3] If the semigroup S(·) is exponentially stable, i.e

||S(t)|| ≤ M e−at for some a, M > 0,then there exist two positive numbers CS and CP such that

||Sα(t)|| ≤ CSt−α, ||Pα(t)|| ≤ CPt−α, ∀t > 0

2.2 Phase space As is known, when we consider the differential equation withinfinite delays, the way of choosing phase spaces plays an essential role In whatfollows, we recall the axioms of phase spaces of Hale and Kato [17]

Let (B, | · |B) be a semi-normed linear space, consisting of functions mapping(−∞, 0] into a Banach space X The definition of a phase space B, introduced in[17], is based on the following axioms stating that if a function v : (−∞, T +σ] → X

is such that v|[σ,T +σ]∈ C([σ, T + σ]; X) and vσ ∈ B, then

(B1) vt∈ B for t ∈ [σ, T + σ];

(B2) the function t 7→ vtis continuous on [σ, T + σ];

(B3) |vt|B≤ K(t − σ) sup{kv(s)kX : σ ≤ s ≤ t} + M (t − σ)|vσ|B, where K, M :[0, ∞) → [0, ∞), K is continuous, M is locally bounded, and they areindependent of v

Let us give some examples of phase spaces The first one is given by

Cγ = {φ ∈ C((−∞, 0]; X) : lim

θ→−∞eγθ||φ(θ)|| exists in X},where γ is a positive number This phase space satisfies (B1)-(B3) with

K(t) = 1, M (t) = e−γt,and it is a Banach space with the norm

|φ|γ= sup

θ≤0

eγθ||φ(θ)||

Regarding another typical example, suppose that 1 ≤ p < +∞, 0 ≤ r < +∞ and

g : (−∞, −r] → R is nonnegative, Borel measurable function on (−∞, −r) Let

CLp is a class of functions ϕ : (−∞, 0] → X such that ϕ is continuous on [−r, 0]and g(θ)kϕ(θ)kpX∈ L1(−∞, −r) A seminorm in CLp is given by

Assume further that

Z −r s

g(θ)dθ < +∞, for every s ∈ (−∞, −r), and (2.10)g(s + θ) ≤ G(s)g(θ) for s ≤ 0, θ ∈ (−∞, −r), (2.11)

where G : (−∞, 0] → R+ is locally bounded From [18], we know that if (2.11) hold, then CLp satisfies (B1)-(B3) Moreover, one can take

1/p

, G(−t)1/p} for t > r

(2.13)

For more examples of phase spaces, see [18]

2.3 Measure of noncompactness and condensing multivalued maps Let

E be a Banach space Denote

P(E) = {B ⊂ E : B 6= ∅},

Pb(E) = {B ∈ P(E) : B is bounded},K(E) = {B ∈ P(E) : B is compact},Kv(E) = {B ∈ K(E) : B is convex}

We will use the following definition of measure of noncompactness ([15])

Definition 2.3 A function β : Pb(E) → R+is called a measure of noncompactness(MNC) on E if

β(co Ω) = β(Ω) for every Ω ∈ Pb(E),where co Ω is the closure of the convex hull of Ω An MNC β is called

i) monotone if Ω0, Ω1∈ Pb(E), Ω0⊂ Ω1 implies β(Ω0) ≤ β(Ω1);

ii) nonsingular if β({a} ∪ Ω) = β(Ω) for any a ∈ E, Ω ∈ Pb(E);

iii) invariant with respect to union with compact set if β(K ∪ Ω) = β(Ω) forevery relatively compact set K ⊂ E and Ω ∈ Pb(E);

iv) algebraically semi-additive if β(Ω0+ Ω1) ≤ β(Ω0) + β(Ω1) for any Ω0, Ω1∈

Pb(E);

v) regular if β(Ω) = 0 is equivalent to the relative compactness of Ω

An important example of MNC is the Hausdorff MNC χ(·), which is defined asfollows: for Ω ∈ Pb(E) put

χ(Ω) = inf{ε > 0 : Ω has a finite ε-net}

This MNC satisfies all properties given in Definition 2.3

We now give some basic estimates based on MNCs We first recall the sequentialMNC χ0defined by

χ0(Ω) = sup{χ(D) : D ∈ ∆(Ω)}, (2.14)where ∆(Ω) is the collection of all at-most-countable subsets of Ω (see [2]) Weknow that

1

for all bounded set Ω ⊂ E Then the following property is evident

Proposition 2.4 Let χ be the Hausdorff MNC in E and Ω ⊂ E be a bounded set.Then for every  > 0, there exists a sequence {xn} ⊂ Ω such that

χ(Ω) ≤ 2χ({x }) + 

We need the following assertion, whose proof can be found in [15].

Proposition 2.5 If {wn} ⊂ L1(0, T ; X) such that

||wn(t)||X ≤ ν(t), for a.e t ∈ [0, T ],for some ν ∈ L1(0, T ), then we have

χ({

Z t 0

wn(s)ds}) ≤ 2

Z t 0

χ({wn(s)})dsfor t ∈ [0, T ]

Using Propositions 2.4 and 2.5, we get

Proposition 2.6 Let D ⊂ L1(0, T ; X) such that

(1) ||ξ(t)|| ≤ ν(t), for all ξ ∈ D and for a.e t ∈ [0, T ],

(2) χ(D(t)) ≤ q(t) for a.e t ∈ [0, T ],

where ν, q ∈ L1(0, T ) Then

χ

Z t 0

D(s)ds≤ 4

Z t 0

ξ(s)ds : ξ ∈ D}

Proof For  > 0, there exists a sequence ξn∈ D such that

χ

Z t 0

D(s)ds≤ 2χ{

Z t 0

ξn(s)ds}+ ,thanks to Proposition 2.4 Applying Proposition 2.5 for the last expression, wehave

χ({ξn(s)})ds +  ≤ 4

Z t 0

q(s)ds + 

Since  is arbitrary, we get the desired conclusion 

We make use of some notions and facts of set-valued analysis Let Y be a metricspace

Definition 2.4 A multivalued map (multimap) F : Y → P(E) is said to be:i) upper semicontinuous (u.s.c) if F−1(V ) = {y ∈ Y : F (y) ∩ V 6= ∅} is aclosed subset of Y for every closed set V ⊂ E;

ii) weakly upper semicontinuous (weakly u.s.c) if F−1(V ) is closed subset of Yfor all weakly closed set V ⊂ E;

iii) closed if its graph ΓF = {(y, z) : z ∈ F (y)} is a closed subset of Y × E;iv) compact if F (Y ) is relatively compact in E;

v) quasicompact if its restriction to any compact subset A ⊂ Y is compact.The following lemmas give criteria for checking a given multimap is (weakly)u.s.c

Lemma 2.7 ([15, Theorem 1.1.12]) Let G : Y → P(E) be a closed quasicompactmultimap with compact values Then G is u.s.c

Lemma 2.8 ([5, Proposition 2]) Let X be a Banach space and Ω be a nonemptysubset of another Banach space Assume that G : Ω → P(X) is a multimap withweakly compact, convex values Then G is weakly u.s.c if and only if {xn} ⊂ Ω with

xn→ x0∈ Ω and yn∈ G(xn) implies yn * y0∈ G(x0), up to a subsequence

We now introduce the concept of condensing multimaps

Definition 2.5 A multimap F : Z ⊆ E → P(E) is said to be condensing withrespect to an MNC β (β-condensing) if for any bounded set Ω ⊂ Z, the relation

β(Ω) ≤ β(F (Ω))implies the relative compactness of Ω

Let β be a monotone nonsingular MNC in E The application of the topologicaldegree theory for condensing maps (see, e.g., [2, 15]) yields the following fixed pointprinciple

Theorem 2.9 [15, Corollary 3.3.1] Let M be a bounded convex closed subset

of E and let F : M → Kv(M) be a u.s.c and β-condensing multimap ThenFix(F ) := {x ∈ E : x ∈ F (x)} is nonempty and compact

Consequently, we prove the following result which will be used later for ourexistence theorem in Sect 3

Theorem 2.10 Let M be a compact convex subset of E and let F : M → P(M)

be a closed multimap with convex values Then Fix(F ) 6= ∅

Proof Since F : M → P(M) is quasicompact and has closed, convex and compactvalues, Lemma 2.7 ensures that F is u.s.c Obviously, F is χE-condensing with χE

being the Hausdorff MNC on E Then Fix(F ) 6= ∅ thanks to Theorem 2.9 

3 Existence result

To prove existence results for problem (1.1)-(1.2), we assume that

(A) A is the infinitesimal generator of a C0-semigroup {S(t)}t≥0which is normcontinuous

(B) The phase space B verifies (B1)-(B3)

(F) F : [0, T ] × X × B → Kv(X) is a multimap satisfying that

(1) t 7→ F (t, v, w) admits a strongly measurable selection for each (v, w) ∈

X × B and (v, w) 7→ F (t, v, w) is u.s.c for a.e t ∈ (0, T );

(2) there exists a function m ∈ Lp(0, T ), p > α1, such that

||F (t, v, w)|| ≤ m(t)(||v|| + |w|B), ∀v ∈ X, w ∈ B,and for a.e t ∈ (0, T ), here ||F (t, v, w)|| = sup{||ξ|| : ξ ∈ F (t, v, w)};(3) if the semigroup S(·) is non-compact, then for any bounded sets B ⊂

X, C ⊂ B, we have

χ(F (t, B, C)) ≤ k(t)

χ(B) + sup

θ≤0

χ(C(θ))

,for a.e t ∈ (0, T ), where k ∈ Lp(0, T ) is a nonnegative function

For x ∈ Cϕ, we denote

PFp(x) = {f ∈ Lp(0, T ; X) : f (t) ∈ F (t, x(t), x[ϕ]t)} (3.1)Motivated by formula (2.5), we introduce the following definition

Definition 3.1 A function x : (−∞, T ] → X is said to be an integral solution

of problem (1.1)-(1.2) if and only if x(t) = ϕ(t), t ≤ 0, and there exists a function

f ∈ PFp(x) such that

x(t) = Sα(t)ϕ(0) +

Z t 0

(t − s)α−1Pα(t − s)f (s)ds, t > 0

For ϕ ∈ B given, we define the solution operator Σ : Cϕ→ P(Cϕ) as followsΣ(x)(t) = Sα(t)ϕ(0) +

Z t 0

(t − s)α−1Pα(t − s)f (s)ds : f ∈ PFp(x)

, (3.2)

or equivalently,

Σ(x) = Sα(·)ϕ(0) + Qα◦ PFp(x),where Qαis defined by (2.8) Since F has convex values, so does PFp This impliesthat Σ has convex values as well

It is obvious that if x is a fixed point of Σ, then x[ϕ] is an integral solution of(1.1)-(1.2) on (−∞, T ]

To establish the existence result, we need some properties of PFp Arguing as in[14], PFp is well-defined Moreover, we have the following lemma

Lemma 3.1 Under assumption (F), the multimap PFp is weakly u.s.c

Proof We use Lemma 2.8 Let {xk} ⊂ Cϕ such that xk → x∗, fk ∈ PFp(xk) Wesee that {fk(t)} ⊂ C(t) := F (t,{xk(t), xk[ϕ]t}), and C(t) is a compact set for a.e

t ∈ (0, T ) Furthermore, by (F)(2), {fk} is integrably bounded (bounded by an Lpintegrable function) Therefore {fk} is weakly compact in Lp(0, T ; X) (see [10]).Let fk* f∗ Then by Mazur’s lemma (see, e.g [11]), there are ˜fk ∈ co{fi: i ≥ k}such that ˜fk → f∗ in Lp(0, T ; X) and then ˜fk(t) → f∗(t) for a.e t ∈ (0, T ), up to

-a subsequence Since F h-as comp-act v-alues, the upper semicontinuity of F (t, ·, ·)means that

F (t, xk(t), xk[ϕ]t) ⊂ F (t, x∗(t), x∗[ϕ]t) + B,for all large k, here  > 0 is given and B is the ball in X centered at origin withradius  So

f (t) ∈ F (t, x∗(t), x∗[ϕ] ) + B , for a.e t ∈ (0, T ),

and the same inclusion holds for ˜fk(t) thanks to the convexity of F (t, x∗(t), x∗[ϕ]t)+

B Accordingly, f∗(t) ∈ F (t, x∗(t), x∗[ϕ]t) + B for a.e t ∈ (0, T ) Since  isarbitrary, one gets f∗∈ PFp(x∗) The lemma is proved Using the last lemma, we prove the following property for the solution operator.Lemma 3.2 Under the assumptions (A), (B) and (F), the solution operator Σ isclosed

Proof Let {xn} ⊂ Cϕ, xn → x∗, zn ∈ Σ(xn) and zn → z∗ We show that z∗ ∈Σ(x∗) Take fn∈ PFp(xn) such that

zn(t) = Sα(t)ϕ(0) + Qα(fn)(t), (3.3)where Qα is defined in (2.8) By Lemma 3.1, we get that fn * f∗ ∈ Lp(0, T ; X)and f∗ ∈ PFp(x∗) In addition, C(t) = {fn(t) : n ≥ 1} is relatively compact andthen

χ({Qα(fn)(t)}) ≤ χ

Z t 0

(t − s)α−1Pα(t − s)fn(s)ds



≤ 2

Z t 0

(t − s)α−1||Pα(t − s)||χ({fn(s)})ds

= 0,according to Proposition 2.5 Due to Proposition 2.2, {Qα(fn)} is equicontinuous.Then by the Arzela-Ascoli theorem, we have the relative compactness of {Qα(fn)}.Since fn(t) → f∗(t) for a.e t ∈ (0, T ), one has Qα(fn) → Qα(f∗) Therefore, itfollows from (3.3) that

z∗(t) = Sα(t)ϕ(0) + Qα(f∗)(t), ∀t ∈ [0, T ],where f∗∈ PFp(x∗) Thus z∗∈ Σ(x∗) The proof is complete Now we prove the main result of this section

Theorem 3.3 Let (A), (B) and (F) hold Then problem (1.1)-(1.2) has at leastone integral solution on (−∞, T ]

Proof We first look for a compact convex set M such that Σ(M) ⊂ M For x ∈ Cϕand z ∈ Σ(x), we have

||z(t)|| ≤ ||Sα(t)|| · ||ϕ(0)|| +

Z t 0

(t − s)α−1||Pα(t − s)||m(s)[||x(s)|| + |x[ϕ]s|B]ds

≤ SαT||ϕ(0)|| + PαT

Z t 0

(m(s))p[||x(s)|| + |x[ϕ]s|B]pds

1p

,(3.4)thanks to the H¨older inequality, here

SαT = sup ||S(t)||, PT

α = sup ||Pα(t)||

Taking into account that

(m(s))p[(1 + KT)p( sup

r∈[0,s]

||x(r)||)p+ MTp|ϕ|pB]ds

!p1,(3.5)where

( sup

r∈[0,t]

||z(r)||)p≤ C3+ C4

Z t 0

ψ(t) = C3+ C4

Z t 0

(m(s))pψ(s)ds, t ≥ 0,

and M0 = {x ∈ Cϕ : supr∈[0,t]||x(r)|| ≤ (ψ(t))1p} It is easy to see that M0 is

a closed convex subset of Cϕ If x ∈ M0, then z ∈ M0 according to (3.6) ThusΣ(M0) ⊂ M0

Now for k ≥ 1, let Mk = co Σ(Mk−1) Then we see that Mk is closed convexand

as k → ∞, where µk(t) = χ(Mk(t)) Observing that

µk(t) = χ(Σ(Mk−1)(t))

≤ χ

Z t 0

(t − s)α−1Pα(t − s)PFp(Mk−1)(s)ds



≤ 4

Z t 0

(t − s)α−1χ (Pα(t − s)PFp(Mk−1)(s)) ds,thanks to Proposition 2.6, here for Λ ⊂ L1(0, T ; X),

Z t 0

Λ(s)ds := {

Z t 0

(k(s))p( sup

r∈[0,s]

µk−1(r))pds

!1p,

where we have used the H¨older inequality Since the last expression is nondecreasing

(k(s))pνk−1(s)ds

Since {νk(t)} is decreasing, one can take the limit of the last inequality to get

ν∞(t) ≤ C5

Z t 0

(k(s))pν∞(s)ds,where ν∞(t) = lim

k→∞νk(t) This implies that ν∞(t) = 0 by using the Gronwallinequality Taking into account that 0 ≤ µk(t) ≤ (νk(t))p1, we have lim

k→∞µk(t) =

0, ∀t ∈ [0, T ], as desired

We have proved that M ⊂ Cϕ is a compact convex set Consider Σ : M →P(M) Applying Theorem 2.10, we have Fix(Σ) 6= ∅ The proof is complete 

4 Existence of decay global solutions

In this section, we prove the existence of decay global solutions to problem (1.2) To do this, we will consider the solution operator Σ on the following space:

(1.1)-BCϕγ= {y ∈ C([0, +∞); X) : y(0) = ϕ(0) and sup

t≥0

tγ||y(t)|| < ∞},where γ is a positive number chosen later This space is endowed with the supremumnorm

||y||BC = sup

t≥0

||y(t)||,and it becomes a closed subspace of the Banach space

BC = {y ∈ C([0, +∞); X) : ||y||BC < ∞}

To apply fixed point theorems for condensing multimaps, one of our main task is

to construct a regular MNC on the space BC([0, +∞); X)

We first recall some usual MNCs on C([0, T ]; X) For given L > 0 and D ⊂C([0, T ]; X), put

• ωT(D) = 0 iff D(t) is relatively compact for all t ∈ [0, T ];

• modT(D) = 0 iff D is equicontinuous

Let

χT(D) = ωT(D) + modT(D)

Then one can check that χT is a regular MNC on C([0, T ]; X)

Consider the space BC([0, ∞); X) of bounded continuous functions on [0, ∞)taking values on X Denote by πT the restriction operator on this space, that is,

πT(x) is the restriction of x on [0, T ] Then the function

ωT(πT({fk})) = 0 and modT(πT({fk})) = 0 for any T > 0, and therefore χ∞({fk}) =

0, but {f } is non-compact Thus, the MNC χ is not regular