### Evolution inclusions in non separable Banach spaces

F.S. de Blasi, G. Pianigiani

Abstract. We study a Cauchy problem for non-convex valued evolution inclusions in non separable Banach spaces under Filippov type assumptions. We establish existence and relaxation theorems.

Keywords: evolution inclusions, mild solutions, Lusin measurable multifunctions, Ba- nach spaces, relaxation

Classification: 34A60, 34G20

1. Introduction

Let E be a real Banach space with norm k · k, and let C(E) be the space of all closed bounded nonempty subsets ofEendowed with the Pompeiu-Hausdorff distanceh. LetI= [0,1].

In this paper we consider the Cauchy problem for evolution inclusions of the form

(Ca,F)

x^{′}(t)∈Ax(t) +F(t, x(t))
x(0) =a.

Here A is the infinitesimal generator of a strongly continuous semigroup S(t), t ≥ 0, of bounded linear operators on E, F is a multifunction from I×E to C(E), anda∈E.

WhenEis finite dimensional, Filippov [4] (see also Hermes [6]) proved that the
Cauchy problem (C_{a,F}), withA= 0, has solutions provided thatF is continuous
in (t, x) and Lipschitzian inx, i.e.

h F(t, x), F(t, y)

≤k(t)kx−yk (t, x), (t, y)∈I×E,

for somek∈L^{1}(I). The more general case in whichF is Carath´eodory-Lipschitz,
i.e.Fis measurable intand Lipschitzian inx, was studied by Himmelberg and Van
Vleck [9]. It is worth while to observe that a crucial step in the proof of Filippov
theorem is the construction, for a C(E) valued multifunction, of a measurable
selector, which is usually obtained by virtue of a selection theorem of Kuratowski
and Ryll-Nardzewski [12]. More recently Frankowska [5], Tolstonogov [16] and
Papageorgiou [13] have shown that if E is infinite dimensional, Filippov’s ideas
can be suitably adapted in order to prove the existence of mild solutions to the

Cauchy problem (C_{a,F}), provided thatEis separable. This restriction is actually
unavoidable if one has to apply in an infinite dimensional setting either selection
theorem, of Kuratowski and Ryll-Nardzewski [12] or of Bressan and Colombo [1].

In the present paper we will establish the existence of mild solutions for the
Cauchy problem (C_{a,F}) in an arbitrary, not necessarily separable, Banach space
E, under assumptions onF of Filippov type. Our approach follows essentially the
pattern introduced by Filippov [4] and developed by Frankowska [5], Tolstonogov
[16], and Papageorgiou [13], however with the basic difference that measurable
selectors of multifunctions, when needed, will be constructed without relying on
either of the above mentioned selection theorems. Actually our existence result
(see Theorem 3.1) covers also the case ofF Carath´eodory-Lipschitz, where mea-
surability in tis understood in the sense of Lusin. Furthermore, for the Cauchy
problem (Ca,F) we shall prove a corresponding relaxation result (see Theorem 4.1)
without assuming the Banach spaceEto be separable. This is made possible by
an argument which, unlike the ones of [5], [16], [13], again does not depend on
the above mentioned selection theorems.

Our existence and relaxation results for the Cauchy problem (C_{a,F}) are only a
partial generalization of analogous results proved by Frankowska [5], Tolstonogov
[16] and Papageorgiou [13] under slightly different assumptions on A and F, in
separable Banach spaces. So far it is not clear if an analogous existence and
relaxation theory, in absence of separability assumptions, might hold also for
more general classes of systems, of the type considered by Papageorgiou [14] and
by Hu, Lakhsmikantham and Papageorgiou [10].

This paper consists of four sections. Notation and some properties of Lusin
measurable multifunctions are contained in Section 2. The existence and relax-
ation theorems for the Cauchy problem (C_{a,F}) are discussed in Section 3 and
Section 4, respectively.

2. Lusin measurable multifunctions

Throughout this paper E denotes an arbitrary real Banach space with norm k k, andC(E) the space of all closed bounded nonempty subsets ofE. Forx∈E and A ⊂ E, A 6= ∅, set d(x, A) = infa∈Akx−ak. C(E) is endowed with the Pompeiu-Hausdorff metric

h(A, B) = max{e(A, B), e(B, A)} A, B∈ C(E).

Here e(A, B) is the metric excess ofA over B ande(B, A) the metric excess of
B overA, that ise(A, B) = sup_{a∈A}d(a, B) ande(B, A) = sup_{b∈B}d(b, A)

If A⊂E, A6= ∅, and r ≥0 we setN(A, r) ={x∈ E|d(x, A) ≤r}. Clearly N(A, r) is closed inE.

We recall below some properties of the metric excess functions, that we shall use later.

Let A, B, C ∈ C(E). We have: (a_{1}) e(A, B) = 0 if and only if A ⊂ B;

(a_{2})e(A, B)≤e(A, C) +e(C, B) (a_{3})e(A, C)≤e(B, C) ande(C, A)≥e(C, B),

if A ⊂B; (a_{4})e N(A, r), C

≤ e(A, C) +r and e C, N(A, r)

≥ e(C, A)−r;

(a_{5})e(A, B)≤rif and only ifA⊂N(B, r),r≥0.

ForA ⊂E, by coA and coA, we mean respectively the convex hull and the closed convex hull ofA.

Let X be a metric space. An open (resp. closed) ball in X with center x
and radiusris denoted byU_{X}(x, r) (resp. ˜U_{X}(x, r)). For any set A⊂X, intA
and Astand, respectively, for the interior ofA, and the closure ofA in X. For
convenience we setU =UE(0,1) andI= [0,1].

A multifunction F : X → C(E) is said to be h-upper semicontinuous (resp.

h-lower semicontinuous, h-continuous) atx0 ∈X if for everyε >0 there exists δ > 0 such that for every x ∈ UX(x0, δ) we have e F(x), F(x0)

≤ ε (resp.

e F(x_{0}),F(x)

≤ε, h F(x), F(x_{0})

≤ε). For brevity we writeh-u.s.c. and h- l.s.c. to mean, respectively,h-upper semicontinuous andh-lower semicontinuous.

F is calledh-u.s.c. (resp.h-l.s.c.,h-continuous) if it is so at each pointx0∈X. Let L be the σ-algebra of the (Lebesgue) measurable subsets of R and, for A∈ L, letµ(A) be the Lebesgue measure ofA.

For any setA⊂X, we denote byχ_{A} the characteristic function ofA.

Let A ∈ L, with µ(A)< +∞. A multifunction F : A → C(E) is said to be Lusin measurable if for every ε > 0 there exists a compact set Kε ⊂ A, with µ(ArKε)< ε, such thatF restricted toKεish-continuous.

It is clear that ifF, G :A→ C(E) andf :A→E are Lusin measurable, then so are F restricted to B (B ⊂A measurable), F +G, and t → d f(t), F(t)

. Moreover, the uniform limit F : A → C(E) of a sequence of Lusin measurable multifunctionsFn:A→ C(E) is also Lusin measurable.

Further details about other notions of measurability for multifunctions and their relations can be found in Castaing and Valadier [2], Himmelberg [8], Klein and Thompson [11], and in [3].

The above definitions ofh-upper orh-lower semicontinuity,h-continuity, Lusin measurability are unchanged if the spaceC(E) is replaced by P(E), the space of all bounded nonempty subsets of Eendowed with the Pompeiu-Hausdorff pseu- dometrich.

The following propositions show that h-u.s.c. and h-l.s.c. multifunctions are Lusin measurable.

Proposition 2.1. If F:I→ C(E)ish-u.s.c., thenF is Lusin measurable.

Proof: Forn∈Nset I_{i}^{n} =

(i−1)/2^{n}, i/2^{n}

, i= 1, . . . ,2^{n}−1,I_{2}^{n} =
2^{n}−
1

/2^{n},1

. The family

I_{i}^{n} ^{2}_{i=1}^{n} is a partition ofI. Now forn∈NdefineGn:I→
C(E) by

Gn(t) =

2^{n}

X

i=1

[

s∈I_{i}^{n}

F(s)
χ_{I}^{n}

i(t).

It is clear thatGnis piecewise constant, and thatGn(t)∈ C(E), forF is bounded

onI. Moreover, we have:

(i) G1(t)⊃G2(t)⊃ · · · ⊃Gn(t)⊃ · · · ⊃F(t) for everyt∈I;

(ii) for eachn∈N,t→h Gn(t),F(t)

is measurable;

(iii) h Gn(t),F(t)

→0 asn→+∞, for everyt∈I.

Property (i) follows immediately from the definition ofGn. To prove (ii), fix
n∈ Nand let t_{0} ∈I, t_{0} 6=i/2^{n}, i = 0,1, . . . ,2^{n}. Clearly t_{0} ∈ intI_{i}^{n}, for some
1≤i≤2^{n}. SinceF ish-u.s.c., givenε >0 there isδ >0, withU_{I}(t_{0}, δ)⊂ intI_{i}^{n},
such thatt ∈U_{I}(t_{0}, δ) implies e F(t),F(t_{0})

≤ε. Hence for everyt ∈U_{I}(t_{0}, δ)
we have

e Gn(t_{0}), F(t_{0})

≤e Gn(t_{0}), F(t)

+e F(t), F(t_{0})

≤e Gn(t), F(t)
+ε,
as Gn is constant on I_{i}^{n}. On the other hand, by (i), e F(t), Gn(t)

= 0 for each t ∈ I. Consequently, h Gn(t), F(t)

≥ h Gn(t0), F(t0)

−ε, for every t∈UI(t0, δ), and (ii) follows, as a lower semicontinuous function is measurable.

It remains to prove (iii). Lett_{0}∈Iandε >0 be arbitrary. SinceF ish-u.s.c.,
there is a δ >0 such that t ∈ U_{I}(t_{0}, δ) implies F(t) ⊂N F(t_{0}), ε

. For every
nlarge enough, say n≥n_{0}, there is 1 ≤in ≤2^{n} such thatt_{0} ∈I_{i}^{n}_{n} ⊂U_{I}(t_{0}, δ).

Thus ifn≥n0 we have Gn(t) = [

s∈I^{n}

in

F(s)⊂N F(t_{0}), ε

for every t∈I_{i}^{n}_{n},

and hence,e Gn(t_{0}),F(t_{0})

≤ε. On the other hand, from (i),e F(t_{0}),Gn(t_{0})

=
0 for everyn∈N, and soh Gn(t_{0}),F(t_{0})

≤εfor everyn≥n_{0}, and also (iii) is
proved.

We are ready to show thatF is Lusin measurable. Letσ > 0. Since eachGn

is piecewise constant, there is a compact set Hσ ⊂ I, with µ(IrHσ) < σ/2, such that each Gn restricted to Hσ is h-continuous. In view of (ii) and (iii), using Egoroff-Severini theorem, one can construct a compact setKσ ⊂Hσ, with µ(Hσ rKσ)< σ/2, such thath Gn(t), F(t)

→ 0 as n → +∞, uniformly on Kσ. ThereforeF restricted toKσ is h-continuous, as each Gn restricted to Kσ

is so, and the convergence is uniform. Clearlyµ(IrKσ)< σ. HenceF is Lusin

measurable, completing the proof.

Proposition 2.2. If F:I→ C(E)ish-l.s.c., thenF is Lusin measurable.

Proof: Forn∈N, let

I_{i}^{n} ^{2}_{i=1}^{n} be as in the proof of Proposition 2.1.

We claim that for everyε >0 there is ak∈Nsuch that ifn≥kwe have

(2.1) \

t∈I_{i}^{n}

F(t) +εU

6=∅ for each i= 1, . . . ,2^{n}.

Indeed, in the contrary case, there is an ε > 0 such that for everyk ∈N there
existn_{k}∈Nand 1≤ink ≤2^{n}^{k} such that

(2.2) \

t∈I^{nk}

ink

F(t) +εU

=∅.

Passing to a subsequence, without change of notation, we can suppose that
I_{i}^{n}^{k}

nk

converges to some point t ∈ I. Since F is h-l.s.c., there is δ > 0 such that
t ∈ UI(t, δ) implies F(t) ⊂ F(t) +εU. But for k large enough, say k ≥ k0,
I_{i}^{n}^{k}

nk ⊂UI(t, δ), and soF(t) +εU ⊃F(t) for every t∈I_{i}^{n}^{k}

nk. As this contradicts (2.2), the claim is proved.

Letε >0. Letkcorrespond to εaccording to the claim, thus (2.1) holds with
n=k. Ifn > k, each intervalI_{i}^{n}, 1≤i≤2^{n}, is contained exactly in one interval
I_{j}^{k}, for some 1≤j≤2^{k}, and hence

\

t∈I_{j}^{k}

F(t) +εU

⊂ \

t∈I_{i}^{n}

F(t) +εU .

Now for eachn≥kdefine G^{ε}_{n}:I→ C(E) by
G^{ε}_{n}(t) =

2^{n}

X

i=1

\

s∈I_{i}^{n}

F(s) +εU

χ_{I}^{n}

i(t).

By definition eachG^{ε}_{n} is piecewise constant. Moreover the sequence

G^{ε}_{n} _{n≥k}
has the following properties:

(i) G^{ε}_{k}(t)⊂G^{ε}_{k+1}(t)⊂. . .⊂G^{ε}_{n}(t)⊂. . .⊂F(t) +εU for everyt∈I;

(ii) for eachn≥k,t→h G^{ε}_{n}(t),F(t) +εU

is measurable onI;

(iii) h G^{ε}_{n}(t),F(t) +εU

→0 asn→+∞, for everyt∈I.

Property (i) follows at once from the definition ofG^{ε}_{n}. To prove (ii), fixn≥k
and lett_{0}∈I,t_{0} 6=i/2^{n},i= 0,1, . . . ,2^{n}. Clearlyt_{0}∈ intI_{i}^{n}, for some 1≤i≤2^{n}.
SinceF ish-l.s.c., givenσ >0 there isδ >0, with U_{I}(t_{0}, δ)⊂ intI_{i}^{n}, such that
t∈U_{I}(t_{0}, δ) impliese F(t_{0}),F(t)

≤σ. Hence for everyt∈U_{I}(t_{0}, δ) we have:

e F(t_{0}) +εU , G^{ε}_{n}(t_{0})

≤e F(t_{0}) +εU , F(t) +εU

+e F(t) +εU , G^{ε}_{n}(t_{0})

≤e F(t_{0}), F(t)

+e F(t) +εU , G^{ε}_{n}(t_{0})

≤σ+e F(t) +εU , G^{ε}_{n}(t)
,

forG^{ε}_{n}is constant onI_{i}^{n}. On the other hand, by (i),e G^{ε}_{n}(t),F(t) +εU

= 0 for
eacht∈I. Consequently,h G^{ε}_{n}(t), F(t) +εU

≥h G^{ε}_{n}(t_{0}),F(t_{0}) +εU

−σfor

everyt ∈U_{I}(t_{0}, δ), and hence (ii) follows, as a lower semicontinuous function is
measurable.

It remains to prove (iii). Let t0 ∈ I and 0 < σ < ε be arbitrary. Since F
is h-l.s.c., there is δ > 0 such that t ∈ U_{I}(t_{0}, δ) implies F(t_{0}) ⊂ F(t) +σU.
For every n large enough, say n ≥ n_{0} ≥ k, there is 1 ≤ in ≤ 2^{n} such that
t_{0} ∈I_{i}^{n}_{n} ⊂U_{I}(t_{0}, δ). Thus for everyn≥n_{0}ands∈I_{i}^{n}_{n}we haveF(t_{0})+(ε−σ)U ⊂
F(s) +σU + (ε−σ)U =F(s) +εU, which implies

F(t_{0}) + (ε−σ)U ⊂ \

s∈I^{n}

in

(F(s) +εU) =G^{ε}_{n}(t_{0}).

Hence for everyn≥n_{0},F(t_{0}) +εU ⊂G^{ε}_{n}(t_{0}) +σU. This and (i) implyh G^{ε}_{n}(t_{0}),
F(t_{0}) +εU

≤σfor everyn≥n_{0}, and thus (iii) is proved.

We are ready to show thatF is Lusin measurable. For eachj ∈ N consider the sequence

G^{ε}n^{j} n≥kj, whereεj = 1/j and kj corresponds to εj. Each G^{ε}n^{j} is
piecewise constant, thus there is a compact setHσ ⊂I independent ofj andn,
withµ(IrHσ)< σ/2, such that everyG^{ε}n^{j} restricted toHσ ish-continuous. In
view of (ii) and (iii), withε=εj, using Egoroff-Severini theorem, one can find a
compact setKσ ⊂Hσ independent of j, withµ(HσrKσ)< σ/2, such that for
each fixedj∈Nwe have

h G^{ε}n^{j}(t), F(t) +εjU)→0 as n→+∞,

uniformly onKσ. Since eachG^{ε}n^{j} restricted to Kσ is h-continuous and the con-
vergence is uniform, one has that the multifunction t → F(t) +ε_{j}U restricted
to Kσ is h-continuous. But the sequence of these multifunctions, as j → +∞,
converges toF uniformly onKσ, hence also F restricted to Kσ is h-continuous.

Clearlyµ(IrKσ)< σ. ThereforeF is Lusin measurable, completing the proof.

3. A Filippov type existence theorem

In this section we prove a theorem on the existence of mild solutions for the
Cauchy problem (C_{a,F}) in an arbitrary (not necessarily separable) Banach space,
under assumptions onF of Filippov type ([4]).

About the operatorA and the multifunctionF :I×E→ C(E),I= [0,1], we shall use the following assumptions.

(H_{1}) A is the infinitesimal generator of a strongly continuous semigroup S(t),
t≥0, of bounded linear operators fromEinto itself.

(H_{2}) For eachx∈E, t→F(t, x) is Lusin measurable onI.

(H3) There exists a summable functionk:I→[0,+∞[ such that h F(t, x), F(t, y)

≤k(t)kx−yk for every (t, x), (t, y)∈I×E.

(H_{4}) There exists a summable function q : I → [0,+∞[ such that F(t,0) ⊂
U˜E 0, q(t)

, for allt∈I.

As is well known (see Pazy [15, p. 4]), under the assumption (H_{1}) there is a
constantM ≥1 such that

kS(t)k ≤M for every t∈I.

Furthermore, if (H_{3}) is satisfied, we denote bym:I→[0,+∞[ the function given
by

m(t) = Z t

0

k(s)ds.

Given a multifunctionGdefined onI×E with nonempty valuesG(t, x)⊂E, consider the Cauchy problem (Ca,G). By mild solution of the Cauchy problem (Ca,G) we mean a functionx:I→Esatisfying the following conditions: (i)xis continuous onIwithx(0) =a, (ii) there is a Lusin measurable functionv:I→E integrable in the sense of Bochner such that:

v(t)∈G t, x(t)

for each t∈I x(t) =S(t)a+

Z t 0

S(t−s)v(s)ds for each t∈I.

Remark 3.1. In the above definition the requirement that “v : I → E is Lusin measurable” is equivalent to “v : I → E is strongly measurable” (in the sense of Hille and Phillips [7, p. 72]). In fact ifv is Lusin measurable then, by a stan- dard iterative procedure one can easily construct a sequence of countably-valued functions converging tov a.e. in I, thusv is strongly measurable. Conversely, if v is strongly measurable then, by Hille and Phillips [7, Corollary 1, p. 73], v is the uniform limit a.e. of a sequence of countably valued functions, from which it follows thatv is Lusin measurable.

Lemma 3.1. LetFi : I → P(E), i = 1,2, be two Lusin measurable multifunc- tions and letε1, ε2>0be such that

(3.1) G(t) = F_{1}(t) +ε_{1}U

∩ F_{2}(t) +ε_{2}U

6=∅ for every t∈I.

Then the multifunctionG : I → P(E)defined by (3.1) has a Lusin measurable selectorv:I→E

Proof: Since F1 and F2 are Lusin measurable, one can construct a sequence {Jn} of pairwise disjoint compact sets Jn ⊂ I satisfying, for each n ∈ N, the following properties:

(i) F_{1} andF_{2} restricted toJnareh-continuous;

(ii) J_{n+1}⊂IrSn
i=1Ji;

(iii) µ(IrSn

i=1J_{i})<1/2^{n}.
Set J_{0} = I rS

nJn and observe that, by (iii), µ(J_{0}) = 0. It is evident that
{Jn}_{n≥}_{0} is a partition ofI.

We claim that for each n = 0,1, . . . there is a Lusin measurable function
vn:Jn→Ewhich is a selector of the multifunction Grestricted toJn. To show
this, fix an arbitrary n ∈ N (the case n = 0 is trivial). For each t ∈ Jn pick
out a point ut ∈ G(t). Since G(t) is open and F_{1} and F_{2} restricted to Jn are
h-continuous, there is aδt>0 such that

(3.2) ut∈ F_{1}(s) +ε_{1}U

∩ F_{2}(s) +ε_{2}U

for every s∈U_{J}_{n}(t, δt).

The family

UJn(t, δt) _{t∈J}

nis an open covering ofJn. AsJnis compact, it admits a finite subcovering, say

UJn(t_{k}, δtk) ^{q}_{k=1}. Now, consider the partition{I_{k}}^{q}_{k=1}
ofJn given by

I1=UJn(t1, δt_{1}) Ik=UJn(tk, δtk)r

k−1

[

i=1

Ii 2≤k≤q, and definevn:Jn→Eby

vn(t) =

q

X

k=1

utkχ_{I}_{k}(t).

It is evident that vn is Lusin measurable. Further,vn is a selector of the multi-
functionGrestricted to Jn. In fact let s∈Jnbe arbitrary, thuss∈I_{k} for some
1≤k≤q. Sinces∈I_{k}⊂U_{J}_{n}(t_{k}, δtk), in view of (3.2) (witht=t_{k}) we have

utk∈ F1(s) +ε1U

∩ F2(s) +ε2U ,

thusvn(s)∈G(s), forvn(s) =utk. Hence vnis a Lusin measurable selector ofG restricted toJn. Then the functionv:I→Egiven by

v(t) =X

n≥0

vn(t)χ_{J}_{n}(t)

is a Lusin measurable selector ofG, completing the proof.

Lemma 3.2. LetF :I×E→ C(E)satisfy the hypotheses(H_{2})and(H_{3}). Then
for arbitraryx:I → Econtinuous, u: I →E Lusin measurable, and ε >0 we
have:

(a_{1}) the multifunctiont→F(t, x(t))is Lusin measurable onI;

(a2) the multifunctionG:I→ P(E)defined by G(t) = F(t, x(t)) +εU

∩UE u(t), d(u(t), F(t, x(t))) +ε has a Lusin measurable selector v:I→E.

Proof: (a_{1}) Let{xn}be a sequence of piecewise constant functionsxn:I→E
converging to x uniformly on I. Given ε > 0, let Kε ⊂ I be a compact set,
with µ(IrKε) < ε, such that k restricted to Kε is continuous and, for each
n ∈N, the multifunction t → F t, xn(t)

restricted to Kε is h-continuous. Set
Mε= sup_{t∈K}_{ε}k(t).

Lett_{0},t∈Kεbe arbitrary. We have:

h F(t, x(t)), F(t_{0}, x(t_{0}))

≤h F(t, x(t)), F(t, xn(t))

+h F(t, xn(t)), F(t_{0}, xn(t_{0}))
+h F(t_{0}, xn(t_{0})), F(t_{0}, x(t_{0}))

≤Mεkx_{n}(t)−x(t)k+h F(t, xn(t)), F(t_{0}, xn(t_{0}))

+Mεkx_{n}(t_{0})−x(t_{0})k

≤2Mεσn+h F(t, xn(t)), F(t_{0}, xn(t_{0}))
,

whereσn= sup_{t∈I}kx_{n}(t)−x(t)k. Sinceσn→0 asn→+∞, andt→F t, xn(t)
restricted to Kε is h-continuous, the multifunction t → F t, x(t)

restricted to
Kεish-continuous, and (a_{1}) is proved.

(a2) For t ∈I set G1(t) = F t, x(t)

, G2(t) = ˜UE u(t), d(u(t), G1(t)) , and observe thatG1 andG2 are Lusin measurable onI. Furthermore, for eacht∈I we haveG(t) = G1(t) +εU

∩ G2(t) +εU

andG(t)6=∅. Hence, by Lemma 3.1,
Ghas a Lusin measurable selectorv :I→E, thus also (a_{2}) holds, and the proof

is complete.

Theorem 3.1. If (H_{1})–(H_{4}) are satisfied, then for every a ∈ E the Cauchy
problem(C_{a,F})has a mild solutionx:I→E.

Proof: We will adapt a construction due to Filippov [4]. First we observe that if z : I → E is continuous, then every Lusin measurable selector u : I → E of the multifunctiont→F t, z(t)

+U is Bochner integrable onI. In fact, for each t∈Iwe have

ku(t)k ≤h F(t, z(t)) +U, 0

≤h F(t, z(t)), F(t,0)

+h F(t,0), 0
+ 1
and hence, in view of (H_{3}) and (H_{4}),

(3.3) ku(t)k ≤k(t)kz(t)k+q(t) + 1, t∈I.

By Hille and Phillips [7, Theorem 3.7.4, p. 80], in view of Remark 3.1 and the above inequality (3.3), if follows thatuis Bochner integrable onI.

Let 0< ε <1 and, forn≥0, setεn=ε/2^{n+2}. Definex_{0}:I→Eby

(3.4) x_{0}(t) =S(t)a+

Z t 0

S(t−s)v_{0}(s)ds,

where v_{0} : I → E is an arbitrary Lusin measurable function, integrable in the
sense of Bochner. Since x_{0} is continuous, by Lemma 3.2 there exists a Lusin
measurable function, sayv1:I→E, satisfying

v_{1}(t)∈ F(t, x_{0}(t)) +ε_{1}U

∩UE v_{0}(t), d(v_{0}(t), F(t, x_{0}(t))) +ε_{1}

t∈I.

Clearly, by (3.3),v_{1} is also Bochner integrable onI. Definex_{1}:I→Eby
x_{1}(t) =S(t)a+

Z t 0

S(t−s)v_{1}(s)ds.

Now by recurrence one can construct a sequence {xn} of continuous functions xn:I→E,n= 1,2, . . ., given by

(3.5)n xn(t) =S(t)a+ Z t

0 S(t−s)vn(s)ds, wherevn:I→Eis a Lusin measurable function satisfying (3.6)n

vn(t)∈ F(t, x_{n−1}(t)) +εnU

∩UE v_{n−1}(t), d(v_{n−1}(t), F(t, x_{n−1}(t))
+εn

t∈I.

Furthermorevn is also Bochner integrable onI because, by (3.6)n and (3.3), we have

(3.7) kvn(t)k ≤k(t)kx_{n−1}(t)k+q(t) + 1, t∈I.

Now from (3.6)n, forn= 2,3, . . . andt∈I we have
kvn(t)−v_{n−1}(t)k ≤d v_{n−1}(t), F(t, x_{n−1}(t))

+εn

≤d vn−1(t), F(t, xn−2(t))

+h(F(t, xn−2(t)), F(t, xn−1(t)) +εn

≤εn−1+k(t)kxn−1(t)−xn−2(t)k+εn. Hence, for eachn= 2,3, . . . andt∈I,

(3.8)n kvn(t)−v_{n−1}(t)k ≤ε_{n−2}+k(t)kx_{n−1}(t)−x_{n−2}(t)k,
asεn−1+εn< εn−2. Setp_{0}(t) =d v_{0}(t),F(t, x_{0}(t))

,t∈I.

We claim that for eachn= 2,3, . . . andt∈I we have:

(3.9)n kxn(t)−xn−1(t)k ≤

n−2

X

k=0

Z t 0 εn−2−k

M^{k+1} m(t)−m(u)k

k! du

+ε_{0}
Z t

0

M^{n} m(t)−m(u)n−1

(n−1)! du+ Z t

0

M^{n} m(t)−m(u)n−1

(n−1)! p_{0}(u)du.

First we verify the above inequality when n = 2. In view of (3.5)n, (3.8)n, (3.4) and (3.6)n, for eacht∈I we have:

kx_{2}(t)−x_{1}(t)k ≤
Z t

0

kS(t−s)kkv_{2}(s)−v_{1}(s)kds

≤ Z t

0

M

ε_{0}+k(s)kx_{1}(s)−x_{0}(s)k
ds

≤ε_{0}M t+
Z t

0

M k(s)

Z s 0

kS(s−u)kkv_{1}(u)−v_{0}(u)kdu

ds

≤ε_{0}M t+
Z t

0

M^{2}k(s)

Z s 0

p_{0}(u) +ε_{1}
du

ds

≤ε0M t+ Z t

0

M^{2} p0(u) +ε0
Z t

u

k(s)ds

du

=ε_{0}M t+ε_{0}
Z t

0

M^{2} m(t)−m(u)
du

+ Z t

0

M^{2} m(t)−m(u)

p_{0}(u)du,
and so (3.9)_{2} is verified.

Now, assuming (3.9)n, we shall show that (3.9)n+1 holds. In view of (3.8)n

and (3.9)n, for eacht∈I we have:

kx_{n+1}(t)−xn(t)k ≤
Z t

0

kS(t−s)kkv_{n+1}(s)−vn(s)kds

≤ Z t

0

M

ε_{n−1}+k(s)kxn(s)−x_{n−1}(s)k
ds

≤ε_{n−1}M t+
Z t

0

k(s)

"_{n−2}
X

k=0

Z s 0

ε_{n−}_{2}_{−k}M^{k+2} m(s)−m(u)k

k! du

+ε0

Z s 0

M^{n+1} m(s)−m(u)n−1

(n−1)! du

+ Z s

0

M^{n+1} m(s)−m(u)n−1

(n−1)! p_{0}(u)du

# ds

=εn−1M t+

n−2

X

k=0

Z t 0

"

Z s 0

ε_{n−2−k}M^{k+2} m(s)−m(u)k

k! k(s)du

# ds

+ε_{0}
Z t

0

"

Z s 0

M^{n+1} m(s)−m(u)n−1

(n−1)! k(s)du

# ds

+ Z t

0

"

Z s 0

M^{n+1} m(s)−m(u)n−1

(n−1)! k(s)p_{0}(u)du

# ds

=εn−1M t+

n−2

X

k=0

Z t 0

"

Z t u

ε_{n−2−k}M^{k+2} m(s)−m(u)k

k! k(s)

# du

+ε0

Z t 0

"

Z t u

M^{n+1} m(s)−m(u)n−1

(n−1)! k(s)ds

# du

+ Z t

0

"

Z t u

M^{n+1} m(s)−m(u)n−1

(n−1)! k(s)ds

#

p0(u)du

=εn−1M t+

n−2

X

k=0

Z t

0 ε_{n−2−k}M^{k+2} m(t)−m(u)k+1

(k+ 1)! du

+ε_{0}
Z t

0

M^{n+1} m(t)−m(u)n

n! du

+ Z t

0

M^{n+1} m(t)−m(u)n

n! p_{0}(u)du

=

n−1

X

k=0

Z t 0

ε_{n−1−k}M^{k+1} m(t)−m(u)k

k! du

+ε0

Z t 0

M^{n+1} m(t)−m(u)n

n! du

+ Z t

0

M^{n+1} m(t)−m(u)n

n! p_{0}(u)du.

Thus (3.9)_{n+1} holds true, and the claim is proved.

Now from (3.9)n, forn= 2,3, . . . and everyt∈I, we have
(3.10) kx_{n}(t)−xn−1(t)k ≤an,

where (3.11) an=

n−2

X

k=0

ε_{n−2−k}M^{k+1}L^{k}

k! +ε_{0}M^{n}L^{n−}^{1}

(n−1)! +M^{n}L^{n−}^{1}
(n−1)!

Z _{1}

0

p_{0}(u)du
and L=m(1).

Clearly the series whosenth term is the first quantity on the right side of (3.11) is convergent, as Cauchy product of absolutely convergent series. Thus the series

whosenth term is anconverges as well. From this and (3.10) it follows that the
sequence{x_{n}} converges uniformly onI to a continuous function, sayx:I→E.
On the other hand, in view of (3.8)n, forn= 3,4, . . . and everyt∈I

kvn(t)−v_{n−1}(t)k ≤ε_{n−2}+k(t)a_{n−1},

which implies that{vn}converges onIto a Lusin measurable function, sayv:I→ E. As{xn}is bounded by a constant, sayH, (3.7) yieldskvn(t)k ≤k(t)H+q(t)+1 forn= 1,2, . . . and eacht∈I, and hencevis also Bochner integrable onI. Then from (3.5)n, lettingn→+∞and using Lebesgue dominated convergence theorem, we obtain

x(t) =S(t)a+ Z t

0 S(t−s)v(s)ds for each t∈I.

On the other hand, by (3.6)n,vn(t)∈F t, xn−1(t)

+εnU for n= 1,2, . . . and t∈I, whence lettingn→+∞we have

v(t)∈F t, x(t)

for each t∈I.

Thereforexis a mild solution of the Cauchy problem (C_{a,F}). This completes the

proof.

WhenA= 0 the Cauchy problem (C_{a,F}) takes the form
(D_{a,F})

x^{′}(t)∈F t, x(t)
x(0) =a.

By solution of the Cauchy problem (D_{a,F}) we mean a continuous function x :
I →Esuch that there exists a Lusin measurable function v :I →E, integrable
in the sense of Bochner, satisfying:

v(t)∈F t, x(t)

for each t∈I x(t) =a+

Z t

0 v(s)ds for each t∈I.

WhenA= 0, Theorem 3.1 yields the following:

Corollary 3.1. If (H_{2})–(H_{4}) are satisfied, then for every a ∈ E the Cauchy
problem(D_{a,F})has a solutionx:I→E.

4. A relaxation theorem

In this section we prove a relaxation theorem for the Cauchy problem (C_{a,F}).

More precisely, we associate to (C_{a,F}) the convexified Cauchy problem
(C_{a,}_{co}_{F})

x^{′}(t)∈Ax(t) + coF t, x(t)
x(0) =a,

and we show that, if (H_{1})–(H_{4}) are satisfied, then the set of the mild solutions of
(C_{a,F}) is dense in the set of the mild solutions of (C_{a,}_{co}_{F}).

Lemma 4.1. Let G : A → C(E) be a Lusin measurable multifunction defined on a measurable setA⊂R, withµ(A)<+∞. ThenGhas a Lusin measurable selectorg:A→E.

Proof: By virtue of [3], Propositions 6 and 4, the statement holds true if A is
compact. IfAis measurable, it suffices to consider a countable partition{Kn}_{n≥0}
ofA, where allKn,n≥1, are compact and K0 is of measure zero.

The following lemma plays a crucial role in the proof of the relaxation theorem.

Lemma 4.2. Let(H1)–(H4)be satisfied. Leta∈E, and lety:I→Ebe a mild
solution of the convexified Cauchy problem (C_{a,}_{co}_{F}). Then given 0 < ε < 1,
there is a mild solutionx_{0}:I→Eof the Cauchy problem

(C_{a,F}_{+ϕ}_{ε}_{U})

x^{′}(t)∈Ax(t) +F t, x(t)

+ϕε(t)U x(0) =a,

whereϕε(t) =ε[k(t)/(L+ 1) + 1]andL=R_{1}

0 k(s)ds, such thatkx_{0}(t)−y(t)k ≤
ε/(L+ 1)≤εfor everyt∈I.

Proof: The proof, rather long, will be divided into four steps.

By hypothesisy:I→Eis a mild solution of (C_{a,}_{co}_{F}). Thusy is continuous,
and there is a Lusin measurable function u: I → E, integrable in the sense of
Bochner, satisfying

u(t)∈ coF t, y(t)

t∈I (4.1)

y(t) =S(t)a+ Z t

0

S(t−s)u(s)ds t∈I.

(4.2)

Let ε > 0. Our aim is to construct a Lusin measurable function v_{0} : I → E
integrable in the sense of Bochner, and a continuous functionx0 :I→Esatisfying

v_{0}(t)∈F t, x_{0}(t)

+ϕε(t)U t∈I

x0(t) =S(t)a+ Z t

0

S(t−s)v0(s)ds t∈I,
such thatkx_{0}(t)−y(t)k ≤ε/(L+ 1) for everyt∈I.

Step 1. Construction of v_{0} andx_{0}.

Let 0< ε <1 be arbitrary. Fixδsuch that

(4.3) 0< δ < ε

4(M+ 1)^{2}(L+ 1),

where M ≥ 1 is a constant satisfying kS(t)k ≤ M for every t ∈ I. Clearly δ < ε <1. Likewise in the proof of Theorem 3.1, one can show that each Lusin measurable selectorw:I→Eof the multifunctiont→ coF t, y(t)

+δU satisfies

(4.4) kw(t)k ≤ψ(t) t∈I,

where ψ(t) =k(t)ky(t)k+q(t) + 1. As ψ is summable,w is Bochner integrable onI.

Takeα >0 such that for each measurable setA⊂I,

(4.5) µ(A)< α implies

Z

A

ψ(t)dt < δ.

The mappings t →u(t) andt → F t, y(t)

are Lusin measurable, the latter by Lemma 3.2, thus there is a compact setK⊂I, with

(4.6) µ(IrK)< α,

such that, when restricted to K, u is continuous and t → F t, y(t)

is h-conti- nuous.

ForN∈N, denote by

I_{i} ^{N}_{i=1} the partition ofIgiven by

I_{i}= [t_{i−}_{1}, t_{i}[ i= 1, . . . , N−1 I_{N} = [t_{N−}_{1}, t_{N}] where t_{i}= i
N .
Now fix N ∈ Nlarge enough so that for each i = 1, . . . , N we have: µ(I_{i})< α
and, furthermore,

(4.7) ku(t^{′})−u(t^{′′})k< δ andh F(t^{′}, y(t^{′})

, F(t^{′′}, y(t^{′′}))

< δ,

for every t^{′}, t^{′′}∈Ii∩K.

Setℑ^{′} ={1 ≤i ≤N|Ii∩K 6=∅}and ℑ^{′′} ={1≤i ≤N|Ii∩K =∅}. In each
intervalIi, withi∈ ℑ^{′}, choose a pointτi∈Ii∩K. Sinceu(τi)∈ coF τi, y(τi)

, there exists a finite set

e^{i}_{n} ^{p}_{n=1}^{i} of points
(4.8) e^{i}_{n}∈F τ_{i}, y(τ_{i})

n= 1, . . . , p_{i},

and there existp_{i} numbersλ^{i}_{n}≥0, withλ^{i}_{1}+· · ·+λ^{i}_{p}_{i} = 1, such that

(4.9) ku(τi)−

pi

X

n=1

λ^{i}_{n}e^{i}_{n}k< δ.

By Pazy [15, Corollary 2.3, p. 4], for eachi∈ ℑ^{′} the functionst→S(ti−t)u(τi)
and t → S(ti−t)e^{i}_{n}, n = 1, . . . , pi, are continuous on the compact interval Ii.
Consequently, for eachi∈ ℑ^{′} we can construct a partition

J_{j}^{i} ^{r}_{j=1}^{i} ofI_{i}, where
J_{j}^{i}= [s^{i}_{j−1}, s^{i}_{j}[ j = 1, . . . , ri, and s^{i}_{j} =ti−1+ j

riN ,

(ifi=N,J_{r}^{N}_{N} is closed) so that the following inequalities are satisfied:

(4.10) kS(t_{i}−t)u(τ_{i})−

ri

X

j=1

S(t_{i}−s^{i}_{j})u(τ_{i})χ_{J}i

j(t)k ≤δ for each t∈I_{i}, i∈ ℑ^{′}
(4.11) kS(ti−t)e^{i}_{n}−

ri

X

j=1

S(ti−s^{i}_{j})e^{i}_{n}χ_{J}i

j(t)k ≤δ for each t∈Ii, i∈ ℑ^{′}
n= 1, . . . , pi.
Furthermore, fori∈ ℑ^{′} and 1≤j≤ri consider a partition

Kn^{ij} pi

n=1 ofJ_{j}^{i}∩K
by measurable setsKn^{ij} such that

(4.12) µ(K_{n}^{ij}) =λ^{i}_{n}µ(J_{j}^{i}∩K) n= 1, . . . , p_{i}.
By Lemma 4.1, the multifunction t → F t, y(t)

restricted to IrK admits a
Lusin measurable selector, sayw0 :IrK→E. Moreover, for eachi∈ ℑ^{′}, denote
byv_{i}:I_{i}∩K→Ethe function given by

v_{i}(t) =

ri

X

j=1 pi

X

n=1

e^{i}_{n}χ_{K}ij
n(t).

Now definev_{0} :I→Eandx_{0} :I→Eby
v_{0}(t) = X

i∈ℑ^{′}

v_{i}(t)χ_{I}_{i}_{∩K}(t) +w_{0}(t)χ_{I}rK(t) t∈I
(4.13)

x_{0}(t) =S(t)a+
Z t

0

S(t−s)v_{0}(s)ds t∈I.

(4.14)

Clearlyv0 is Lusin measurable, and also Bochner integrable, because
(4.15) v_{0}(t)∈F t, y(t)

+δU t∈I.

To show (4.15) let t ∈ I be arbitrary, thus t ∈ I_{i}, for some 1 ≤ i ≤ N. If
t ∈IrK, we have v_{0}(t) = w_{0}(t)∈ F t, y(t)

. If t ∈ Ii∩K, then t ∈ Kn^{ij} for
some 1≤j≤ri and 1≤n≤pi, hencev_{0}(t) =e^{i}_{n}χ_{K}ij

n(t) =e^{i}_{n}∈F τi, y(τi)
, by
(4.8). Sincet, τi ∈Ii∩K, (4.7) impliesF τi, y(τi)

⊂F t, y(t)

+δU. Whence
ift∈Ii∩K, we havev_{0}(t)∈F t, y(t)

+δU and (4.15) is proved.

Step 2. For eachi∈ ℑ^{′} we have:

(4.16) Z

Ii∩K

S(t_{i}−s)u(s)ds−
Z

Ii∩K

S(t_{i}−s)v_{0}(s)ds

≤2(M+ 1)δµ(I_{i}).

Denoting by Λ_{i}the quantity on the left side of (4.16), we have
Λi≤

Z

Ii∩K

S(ti−s)u(s)ds− Z

Ii∩K

S(ti−s)u(τi)ds

+ Z

Ii∩K

S(ti−s)u(τi)ds−

ri

X

j=1

Z

J_{j}^{i}∩K

S(ti−s^{i}_{j})u(τi)ds
(4.17)

+

ri

X

j=1

Z

J_{j}^{i}∩K

S(ti−s^{i}_{j})u(τi)ds−

ri

X

j=1 pi

X

n=1

Z

Kn^{ij}

S(ti−s^{i}_{j})vi(s)ds

+

ri

X

j=1 pi

X

n=1

Z

K_{n}^{ij}

S(t_{i}−s^{i}_{j})v_{i}(s)ds−
Z

Ii∩K

S(t_{i}−s)v_{0}(s)ds
.

Let Λ^{I}_{i,...},Λ^{IV}_{i} be the first, . . ., fourth term on the right side of (4.17). Clearly,
by virtue of (4.7), we have

(4.18) Λ^{I}_{i} ≤
Z

Ii∩K

kS(ti−s)kku(s)−u(τi)kds≤M δµ(Ii).

Further,
Λ^{II}_{i} =

Z

Ii∩K

S(ti−s)u(τi)ds− Z

Ii∩K

ri

X

j=1

S(ti−s^{i}_{j})u(τi)χ_{J}i
j(s)

ds

and so, by (4.10), we have

(4.19) Λ^{II}_{i} ≤
Z

Ii∩K

S(ti−s)u(τi)−

ri

X

j=1

S(ti−s^{i}_{j})u(τi)χ_{J}i
j(s)

ds≤δµ(Ii).

As far as Λ^{III}_{i} is concerned we have:

Λ^{III}_{i} ≤

ri

X

j=1

Z

J_{j}^{i}∩K

S(ti−s^{i}_{j})u(τi)ds−

ri

X

j=1

S(ti−s^{i}_{j})

pi

X

n=1

λ^{i}_{n}e^{i}_{n}µ(J_{j}^{i}∩K)

+

r_{i}

X

j=1

S(ti−s^{i}_{j})

pi

X

n=1

λ^{i}_{n}e^{i}_{n}µ(J_{j}^{i}∩K)−

r_{i}

X

j=1 pi

X

n=1

Z

K^{ij}n

S(ti−s^{i}_{j})vi(s)ds

≤

ri

X

j=1

S(ti−s^{i}_{j}) u(τi)−

pi

X

n=1

λ^{i}_{n}e^{i}_{n}

!

µ(J_{j}^{i}∩K)

+

ri

X

j=1 pi

X

n=1

S(t_{i}−s^{i}_{j})λ^{i}_{n}e^{i}_{n}µ(J_{j}^{i}∩K)−

ri

X

j=1 pi

X

n=1

S(t_{i}−s^{i}_{j})e^{i}_{n}µ(K_{n}^{ij})
.

The last term on the right side of the above inequality is zero because, by (4.12),
µ(Kn^{ij}) =λ^{i}_{n}µ(J_{j}^{i}∩K) for everyj= 1, . . . , ri andn= 1, . . . , pi. Thus, in view of
(4.9), it follows:

(4.20)

Λ^{III}_{i} ≤

ri

X

j=1

kS(t_{i}−s^{i}_{j})kku(τ_{i})−

pi

X

n=1

λ^{i}_{n}e^{i}_{n}kµ(J_{j}^{i}∩K)

≤M δ

ri

X

j=1

µ(J_{j}^{i}∩K)≤M δµ(Ii).

It remains to evaluate Λ^{IV}_{i} . Taking into account the definition ofv_{0}, we have
Z

Ii∩K

S(ti−s)v_{0}(s)ds−

ri

X

j=1 pi

X

n=1

Z

Kn^{ij}

S(ti−s^{i}_{j})vi(s)ds

=

pi

X

n=1
r_{i}

X

j=1

Z

K_{n}^{ij}

S(ti−s)e^{i}_{n}ds−

pi

X

n=1
r_{i}

X

j=1

Z

K_{n}^{ij}

S(ti−s^{i}_{j})e^{i}_{n}ds

=

pi

X

n=1

Z

K_{n}^{i}

S(t_{i}−s)e^{i}_{n}ds−

pi

X

n=1

Z

K_{n}^{i}

^{r}i

X

j=1

S(t_{i}−s^{i}_{j})e^{i}_{n}χ_{K}ij
n(s)

ds,

whereK_{n}^{i} =Sri

j=1Kn^{ij}. Thus
Λ^{IV}_{i} ≤

pi

X

n=1

Z

K_{n}^{i}

S(ti−s)e^{i}_{n}−

ri

X

j=1

S(ti−s^{i}_{j})e^{i}_{n}χ_{K}ij
n(s)

ds.

Now eachs∈K_{n}^{i} is in one set, say Kn^{ij}, for some 1≤j ≤ri, and thus s∈J_{j}^{i}.
Hence by (4.11)

(4.21) Λ^{IV}_{i} ≤δ

pi

X

n=1

µ(K_{n}^{i})≤δµ(Ii).

From (4.17), by virtue of (4.18)–(4.21), it follows

Λ_{i} ≤M δµ(I_{i}) +δµ(I_{i}) +M δµ(I_{i}) +δµ(I_{i}) = 2(M+ 1)δµ(I_{i}),
and Step 2 is proved.

Step 3. We havekx_{0}(t)−y(t)k ≤ε/(L+ 1)for every t∈I.

Lett ∈I be arbitrary, thus t ∈I_{h} for some 1 ≤h≤N. By virtue of (4.14)
and (4.2) we have

kx_{0}(t)−y(t)k ≤

Z th−1

0

S(t−s) v_{0}(s)−u(s)
ds

+

Z t
t_{h−1}

S(t−s) v_{0}(s)−u(s)
ds

≤

h−1

X

i=1

S(t−ti) Z ti

t_{i−1}

S(ti−s) v_{0}(s)−u(s)
ds

+
Z t_{h}

t_{h−1}

kS(t−s)k kv_{0}(s)k+ku(s)k
ds,
and hence

(4.22)

kx_{0}(t)−y(t)k ≤

h−1

X

i=1

kS(t−t_{i})k

Z ti

t_{i−1}

S(t_{i}−s) v_{0}(s)−u(s)
ds

+M Z th

t_{h−1}

kv_{0}(s)k+ku(s)k
ds.

The last term on the right side of (4.22) is not greater than 2M δ. In fact v_{0}
and u are selectors of the multifunction t → coF t, y(t)

+δU, by (4.15) and
(4.1), therefore satisfy (4.4) i.e.kv_{0}(s)k ≤ψ(s) andku(s)k ≤ψ(s),s∈I. Further
µ(I_{h})< α, thus by virtue of (4.5) the statement holds true. From (4.22), in view
of Step 2, we have

kx_{0}(t)−y(t)k ≤M

h−1

X

i=1

Z

Ii

S(ti−s) v_{0}(s)−u(s)
ds

+ 2M δ

≤M X

i∈ℑ^{′}
i≤h−1

Z

Ii∩K

S(t_{i}−s) v_{0}(s)−u(s)
ds

+M X

i∈ℑ^{′′}

i≤h−1

Z

IirK

S(ti−s) v0(s)−u(s) ds

+ 2M δ

≤M X

i∈ℑ^{′}
i≤h−1

2(M+ 1)δµ(I_{i})

+M Z

IrK

kS(t_{i}−s)k kv_{0}(s)k+ku(s)k

ds+ 2M δ

≤2M(M+ 1)δ+ 2M^{2}
Z

IrK

ψ(s)ds+ 2M δ.

By (4.6)µ(IrK)< α, hence (4.5) implies that the latter integral is less than δ.

Consequently,

kx_{0}(t)−y(t)k ≤2M(M + 1)δ+ 2M^{2}δ+ 2M δ <4(M+ 1)^{2}δ < ε
L+ 1
for, by (4.3),δ < ε/[4(M+ 1)^{2}(L+ 1)]. Sincet∈Iis arbitrary, Step 3 is proved.

Step 4. x_{0} is a mild solution of the Cauchy problem(C_{a,F}_{+ϕ}_{ε}_{U}).

In view of the definition ofx_{0} andv_{0} (see (4.14) and (4.13)),x_{0} is continuous
on I, with x(0) = a, and v_{0} is Lusin measurable and integrable in the sense of
Bochner onI. To prove the statement we have only to show that

(4.23) v_{0}(t)∈F t, x_{0}(t)

+ϕε(t)U t∈I.

Lett∈IrK. From (4.13),v0(t) =w0(t)∈F t, y(t)
, thus
d v_{0}(t), F t, x_{0}(t))

≤h F(t, y(t)), F(t, x_{0}(t))

≤k(t)ky(t)−x_{0}(t)k.

Since, by Step 3,ky(t)−x_{0}(t)k ≤ε/(L+ 1), we have
d v_{0}(t), F(t, x_{0}(t))

< ε[k(t)/(L+ 1) + 1] =ϕε(t), and hence (4.23) is satisfied, for eacht∈IrK.

Let t ∈ K. Then for some i ∈ ℑ^{′}, 1 ≤ j ≤ r_{i}, and 1 ≤ n ≤ p_{i} we have
t ∈Kn^{ij}. By virtue of (4.13) and (4.8), v_{0}(t) = e^{i}_{n}∈F τ_{i}, y(τ_{i})

. On the other hand, by (4.7),F τi, y(τi)

⊂F t, y(t)

+δU asτi,t∈Ii∩K and, consequently,
v_{0}(t)∈F t, y(t)

+δU. By virtue of Step 3 we have:

d v0(t), F t, x0(t)

≤h F(t, y(t)) +δU, F(t, x0(t))

≤h F(t, y(t)), F(t, x_{0}(t))
+δ

≤k(t)ky(t)−x_{0}(t)k+δ≤ε k(t)

L+ 1 +δ < ϕε(t) asδ < ε, by (4.3). It follows that (4.23) is satisfied also fort∈K, and Step 4 is

proved. This completes the proof.

Theorem 4.1. Let(H_{1})–(H_{4})be satisfied. Leta∈E, and let y:I →E be an
arbitrary mild solution of the convexified Cauchy problem(C_{a,}_{co}_{F}). Then, for
everyσ >0, there exists a mild solutionx:I→Eof the Cauchy problem(C_{a,F})
such thatkx(t)−y(t)k ≤σfor everyt∈I.

Proof: Let y : I → E be an arbitrary mild solution of the Cauchy problem
(C_{a,}_{co}_{F}), and let 0< σ <1. Fixεso that

0< ε < σ
7M e^{LM} ,

where M ≥ 1 is a constant such that M ≥ kS(t)k for each t ∈ I, and L = R1

0 k(t)dt.

By Lemma 4.2, with the above choice of ε, there exists a mild solutionx_{0} :
I → E of the Cauchy problem (Ca,F+ϕεU), where ϕε(t) = ε[k(t)/(L+ 1) + 1],
such that

(4.24) kx_{0}(t)−y(t)k ≤ ε

L+ 1 ≤ε t∈I.

By definition of mild solution, x_{0} is continuous, with x_{0}(0) = a, and there is
a Lusin measurable function v_{0} : I → E, integrable in the sense of Bochner,
satisfying

(4.25)

v_{0}(t)∈F t, x_{0}(t)

+ϕε(t)U t∈I

x_{0}(t) =S(t)a+
Z t

0

S(t−s)v_{0}(s)ds t∈I.

With this choice ofx_{0} andv_{0}, following the argument and retaining the notation
of Theorem 3.1, we can construct a sequence{xn} of continuous functions xn :
I→E,n= 1,2, . . . given by

xn(t) =S(t)a+ Z t

0

S(t−s)vn(s)ds t∈I,

where vn : I → E is a Lusin measurable function, integrable in the sense of Bochner, such that

(4.26)n

vn(t)∈ F(t, x_{n−1}(t)) +εnU

∩UE v_{n−1}(t), d(v_{n−1}(t), F(t, x_{n−1}(t))
+εn

t∈I,
andεn=ε/2^{n+2}. Letp_{0}:I→Randm:I→Rbe, respectively, given by

p_{0}(t) =d v_{0}(t), F(t, x_{0}(t))

m(t) = Z t

0

k(s)ds.