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Existence of solutions for a class ofnonconvex differential inclusionsChems Eddine Arrouda & Tahar Haddadb

a Faculté des Sciences Exactes, Département de Mathématiques,Université de Jijel, Jijel, Algérie.b Laboratoire LMPA, Département de Mathématiques, Universitéde Jijel, Jijel, Algérie.Published online: 17 Dec 2013.

To cite this article: Chems Eddine Arroud & Tahar Haddad (2014) Existence of solutions for a classof nonconvex differential inclusions, Applicable Analysis: An International Journal, 93:9, 1979-1988,DOI: 10.1080/00036811.2013.866228

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Applicable Analysis, 2014Vol. 93, No. 9, 1979–1988, http://dx.doi.org/10.1080/00036811.2013.866228

Existence of solutions for a class of nonconvex differential inclusionsChems Eddine Arrouda and Tahar Haddadb∗

aFaculté des Sciences Exactes, Département de Mathématiques, Université de Jijel, Jijel, Algérie;bLaboratoire LMPA, Département de Mathématiques, Université de Jijel, Jijel, Algérie

Communicated by Robert P. Gilbert

(Received 31 May 2013; accepted 31 October 2013)

We prove the existence of solutions to the functional differential inclusion of theform x(t) ∈ F(T (t)x) + f (t,T (t)x), where f is a Carathéodory function andF an upper semicontinuous multifunction with compact values in a Hilbert spacesuch that F(T (t)x) ⊂ γ (∂g(x(t))), with g a regular locally Lipschitz functionand γ a linear operator, and [T (t)x](s) = x(t + s).

Keywords: functional differential inclusion; uniformly regular function; delay

AMS Subject Classifications: 34A60; 434K05; 49J52

1. Introduction

Functional differential inclusions with delay, express the fact that the velocity of the systemdepends not only on the state of the system at a given instant but depends upon the history ofthe trajectory until this instant. The class of differential inclusions with delay encompassesa large variety of differential inclusions and control systems. In particular, this class coversthe differential inclusions, the differential inclusions with delay and the Volterra inclusions.A detailed discussion on this topic may be found in [1–5]. Let H be a real separable Hilbertspace with the norm ‖·‖ and scalar product 〈·, ·〉, I an interval of R and a positive scalar.Denote by CH (I ) the Banach space of continuous functions from I into H . By C0 we meanthe Banach space CH ([−τ, 0]) with the norm ‖ϕ‖∞ := sup ‖ϕ(s)‖s∈[−τ,0]. For T > 0,x ∈ CH ([−τ, T ]), and for any t ∈ [0, T ] we define a map T (t) from CH ([−τ, T ]) into C0as follows

T (t)x(s) = x(t + s), s ∈ [−τ, 0].T (t)x represents the history of the state from the time t − τ to the present time t .Let� be a nonempty subset in C0. For a given multifunction F : � → 2H we consider thefollowing functional differential inclusion:{

x(t) ∈ F(T (t)x))+ f (t,T (t)x) a.e. on [0, T ];T (0)x(s) = ϕ0, for all s ∈ [−τ, 0]. (1)

∗Corresponding author. Email: thaddadunivjijel@gmail.com

© 2013 Taylor & Francis

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Let ϕ0 ∈ �. By a solution of the functional differential inclusion (1) we mean anycontinuous function x : [−τ, T ] → H , which is absolutely continuous on [0.T ], (T (t)x) ∈� for all t ∈ [0, T ] and satisfying (1).

The existence of solutions for functional differential inclusion (1) was proved whenF is upper semicontinuous and with convex compact values by Haddad [6] for the firsttime with f ≡ 0. Recently in [7] and [8], the situation when the multifunction F is notconvex valued is considered, the existence of solution for the problem (1) was obtainedin the finite dimensional case by assuming F(·) upper semicontinuous, compact valuedmultifunction such that F(ψ) ⊂ ∂g(ψ(0)) for everyψ ∈ � and g is a convex proper lowersemicontinuous function. In this paper, we extend this result in two ways: we consider theinfinite dimensional case and we relax the convexity assumption on the function g, namelywe replace the convexity by the existence of a locally Lipschitz and uniformly regularfunction g and a linear operator γ such that F(ψ) ⊂ γ (∂g(ψ(0))) for every ψ ∈ � andso the usual subdifferentials will be replaced by the Clarke subdifferentials. This conditionenssures the L2-norm convergence of the derivatives of approximate solutions. The classof proper convex lower semicontinuous functions and the class of lower-C2 functions (seeExamples 2.2 and 2.3) are strictly contained within the class of uniformly regular functions.The paper is organized as follows: in Section 2 we recall some preliminary facts that weneed in the sequel and in Section 3 we prove our main result.

2. Notation and preliminaries

Let H be a real separable Hilbert space with the norm ‖ · ‖ and scalar product 〈·, ·〉. Wedenote by B(x, r) (resp. Br (x)) the closed unit ball of H (resp. of C0) centered in x withradius r > 0.We denote by δ∗(., A) the support function of A, by d(x, A) the distance fromx ∈ H to A. For any two subsets A, B of H , dH (A, B) stands to the Hausdorff distancebetween A and B.

Let σ the weak topology in H . Let us (en)n≥1 be a dense sequence in B(0, 1) and weconsider the linear application γ : H → H defined by

∀x ∈ H, γ (x) =∞∑

n=1

2−n〈x, en〉en .

Note that this series is absolutely convergent. According to the specialists of the theory oflinear operators the application γ belongs to the class of the nuclear operators of H . Further,γ satisfies the two following properties:

(a) The restriction of γ to B(0, 1) is continuous from (B(0, 1), σ ) into H .(b) For all x ∈ H\{0}, 〈x, γ (x)〉 > 0.

Indeed (b) is obvious. This condition is equivalent to

x ∈ H → 〈x, γ (x)〉is a strictly convex function (see [9]).

In the sequel we note by (H) the set of linear applications γ : H → H verifying theconditions (a) and (b). (H) ⊂ K (H) the space of compact operators of H . If H = R

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then (H) coincides with the set of the automorphism of Rm associated to positive definite

matrices.

Definition 2.1 Let g : H → R ∪ {+∞} be a lower semicontinuous function and let� ⊂ domg be a nonempty open subset. We will say that g is uniformly regular over � ifthere exists a positive number β ≥ 0, such that for all x ∈ � and for all ξ ∈ ∂ P g(x) onehas

〈ξ, x ′ − x〉 ≤ g(x ′)− g(x)+ β‖x ′ − x‖2 for all x ′ ∈ �.

Here ∂ P g(x) denotes the proximal subdifferential of g at x (for its definition the readeris refereed for instance to [10,11]). We will say that g is uniformly regular over closed set Sif there exists an open set O containing S such that g is uniformly regular over O . The classof functions that are uniformly regular over sets is so large. We state here some examples.

Example 2.2 Any lower semicontinuous proper convex function g is uniformly regularover any nonempty subset of its domain with β = 0.

Example 2.3 Any lower-C2 function g is uniformly regular over any nonempty convexcompact subset of its domain. Indeed, let g be a lower-C2 function over a nonempty convexcompact set S ⊂ dom f . By Rockafellar’s result (see for instance [12, Theorem 10.33])there exists a positive real number β such that f := g + β

2 ‖.‖2 is a convex function on S.Using the definition of the subdifferential of convex functions and the fact that the Clarkesubdifferential of g is ∂C g(x) = ∂ f (x) − βx for any x ∈ S, we get the inequality inDefinition 2.1 and so g is uniformly regular over S.

The following proposition summarizes some important properties for uniformly regularlocally Lipschitz functions over sets needed in the sequel. For the proof of these results werefer the reader to [10,13].

Proposition 2.4 Let g : H → R be a locally Lipschitz function and� a nonempty openset. If g is uniformly regular over �, then the following hold:

(i) The proximal subdifferential of g is closed over �, that is, for every xn → x ∈ �with xn ∈ � and every ξn → ξ with ξn ∈ ∂ P g(xn) one has ξ ∈ ∂ P g(x)

(ii) The proximal subdifferential of g coincides with ∂C g(x) the Clarke subdifferentialfor any point x (see for instance [11] for the definition of ∂C g)

(iii) The proximal subdifferential of g is upper hemicontinuous over S, that is, the supportfunction x → 〈v, ∂ P g(x)〉 is u.s.c. over S for every v ∈ H

(iv) For any absolutely continuous map x : [0, T ] → � one has

d

dt(g ◦ x)(t) = 〈∂C g(x(t)); x(t)〉.

3. Main result

The following theorem establishes our main result in this paper.

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Theorem 3.1 Let O be an open subset in H, � ⊂ C0 be an open subset, and ϕ0 ∈ �

with ϕ0(0) ∈ O. Consider F : � → 2H , f : R+ × � → H, g : H → R and γ ∈ (H)

satisfying the hypothesis:

(H1) F : � → 2H is upper semicontinuous (i.e. for all ε > 0 there exists δ > 0 suchthat ‖z − z′‖ ≤ δ implies F(z′) ⊂ F(z)+ εB) with compact values.

(H2) There exist γ ∈ (H) and a locally Lipschitz β-uniformly regular function g :H → R over O such that

F(ϕ) ⊂ γ (∂C g(ϕ(0))) for all ϕ ∈ �.(H3) f : R

+ ×� → H is a Carathéodory function, (i.e. for every ϕ ∈ C0, t −→ f (t, ϕ)is measurable, and for t ∈ R

+, ϕ −→ f (t, ϕ) is continuous) and for any boundedsubset B of C0, there is a compact set K such that f (t, ϕ) ∈ K for all (t, ϕ) ∈R

+ × B.

Then, there exist T > 0 and x(·) : [−τ, T ] → H solution to the problem{x(t) ∈ F(T (t)x))+ f (t,T (t)x) a.e. on [0, T ];T (0)x(s) = ϕ0, for all s ∈ [−τ, 0].

Proof Let ϕ0 ∈ �with ϕ0(0) ∈ O.There exist r > 0 and L > 0 such that g is L-Lipschitzon B(ϕ0(0), r). Therefore, ∂cg(x) ⊂ LB for all x ∈ B(ϕ0(0), r). Since� is an open set wecan choose r such that Br (ϕ0) ⊂ �. By our assumption (H3), there is a positive constantm such that f (t, ϕ) ∈ K ⊂ mB for all (t, ϕ) ∈ R

+ × Br (ϕ0).According to the choice of γ the set K1 := γ (LB) is convex compact in H , therefore

contained in a certain ball m1 B (m1 > 0 ) of H . Since ϕ0 is continuous on [−τ, 0], we cantake η > 0 such that

∀t, s ∈ [−τ, 0], |t − s| < η ⇒ |ϕ0(t)− ϕ0(s)| < r

4. (2)

Choose T > 0 such that

0 < T < min

{η,

r

2(m + m1)

}.

and put I := [0, T ]. For each integer n ≥ 1 and for 1 ≤ i ≤ n − 1 we set tni := iT

n ,I ni := [tn

i−1, tni [ and tn

n = T , I nn = {T }. First we set

xn(t) = ϕ0(t) on [−τ, 0]. (3)

Let define the following approximate sequence

xn(t) = xn(tni )+

∫ t

tni

[uni + f (s,T (tn

i )xn)]ds

whenever t ∈ I ni+1, 0 ≤ i ≤ n − 1, where xn(0) = ϕ0(0) := ϕ0(0), and un

i ∈ F(T (tni )xn).

For every 0 ≤ i ≤ n − 1, take zni ∈ ∂C g(xn(tn

i )) such that uni = γ (zn

i ). Now let usdefine the step functions from [0, T ] to [0, T ] by

θn(t) = tni , un(t) = un

i , zn(t) = zni t ∈ I n

i+1.

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Now let us define the step functions from [0, T ] to [0, T ] by

θn(t) = tni , un(t) = un

i , zn(t) = zni t ∈ I n

i+1.

Then, for all n ∈ N∗ and all t ∈ [0, T ], we have the following properties:

0 ≤ t − θn(t) ≤ T

n(4)

xn(t) = ϕ0(0)+∫ t

0[un(s)+ f (s,T (θn(s))xn)]ds (5)

un(t) ∈ F(T (θn(t))xn) (6)

zn(t) ∈ ∂C g(xn(θn(t))) (7)

un(t) = γ (zn(t)). (8)

Observe that

‖xn(t)− xn(t′)‖ =

∥∥∥∥∥∫ t

′

t[un(s)+ f (s,T (θn(s))xn)] ds

∥∥∥∥∥ ≤ (m1 + m)|t − t′ |

whenever 0 ≤ t ≤ t′ ≤ T and n ∈ N

∗. Hence,

xn(t) ∈ B

(ϕ0(0),

r

2

)for all n ∈ N

∗and all t ∈ [0, T ] (9)

and (xn)n∈N∗ is equi-Lipschitz subset of CH ([0, T ]).Now, we must show that T (θn(t))xn ∈ Br (ϕ0) for every t ∈ [0, T ].If −θn(t) ≤ s ≤ 0, then θn(t) + s ≥ 0 and so there exists j ∈ {0, 1, . . . , n − 1} such thatθn(t) + s ∈ I n

j+1. Thus, by (2), (9) and by the fact that |θn(t) − t | ≤ T and |s| ≤ T , wehave

‖(T (θn(t))xn)(s)− ϕ0(s)‖ = ‖xn(θn(t)+ s)− ϕ0(s)‖≤ ‖xn(θn(t)+ s)− ϕ0(0)‖ + ‖ϕ0(s)− ϕ0(0)‖<

3r

4< r.

If −τ ≤ s ≤ −θn(t) then s + θn(t) ≤ 0 and by (2) we have

‖(T (θn(t))xn)(s)− ϕ0(s)‖ = ‖ϕ0(θn(t)+ s)− ϕ0(s)‖ ≤ r

4< r.

Hence,T (θn(t))xn ∈ Br (ϕ0), for every t ∈ [0, T ]. (10)

Therefore, the set {xn(t) : n ∈ N∗} is relatively compact in H for every t ∈ [0, T ]. Indeed,

we have for all n ∈ N∗ and all t ∈ [0, T ]

xn(t) ∈ ϕ0(0)+ [0, T ](K + K1) := K2

which is compact. Then by Ascoli’s theorem, (xn)n∈N∗ is relatively compact in the Banachspace CH([0, T ]). Further, the sequences (un)n∈N and (zn)n∈N are relativelyσ(L1([0, T ],H);L∞([0, T ],H))-compact andσ(L∞([0, T ],H); L1([0, T ],H))-compact, respectively, sincewe have a.e.

∀n ∈ N∗ un(t) ∈ K1 and zn(t) ∈ LB .

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Therefore, by extracting subsequences if necessary, we can assume that there exist x inCH ([0, T ]), u in L1([0, T ],H), and z in L∞([0, T ],H) such that xn → x in C([0, T ],H),un → u for σ(L1([0, T ], H); L∞([0, T ],H))-topology and zn → z for σ(L∞([0, T ],H);L1([0, T ],H))-topology.

Also, since |θn(t) − t | ≤ Tn for every t ∈ [0, T ], then θn(t) → t uniformly on [0, T ].

Moreover, by the uniformly convergence of (xn) and (θn),we deduce that xn(θn(t)) → x(t)uniformly on [0, T ].

Now, we have to estimate ‖T (θn(t))xn −T (t)xn‖∞ for each s ∈ [−τ, 0]. Let us denotethe modulus continuity of a function ψ defined on interval I of R by

ω(ψ, I, ε) := sup{‖ψ(t)− ψ(s)‖; s, t ∈ I, |s − t | < ε}, ε > 0.

Then we have:

‖T (θn(t)) xn − T (t)xn‖∞ = sup−τ≤s≤0

‖xn (θn(t)+ s)− xn(t + s)‖

≤ ω

(xn, [−τ, T ], T

n

)

≤ ω

(ϕ0, [−τ, 0], T

n

)+ ω

(xn, [0, T ], T

n

)

≤ ω

(ϕ0, [−τ, 0], T

n

)+ T

n(m1 + m) ;

hence‖T (θn(t))xn − T (t)xn‖∞ ≤ δn for every t ∈ [0, T ], (11)

where δn := ω(ϕ0, [−τ, 0], Tn )+ T

n (m1 + m). Thus, by continuity of ϕ0, we have δn → 0as n → ∞ and hence

‖T (θn(t))xn − T (t)xn‖∞ → 0 as n → ∞.

Therefore, the uniform convergence of xn to x on [−τ, T ] implies

T (t)xn → T (t)x uniformly on [−τ, 0], (12)

we deduce thatT (θn(t))xn → T (t)x in C0 . (13)

Moreover, by (9), (10), and (13), we have that T (t)x ∈ Br (ϕ0) ⊂ �. Consequently, for allt ∈ [0, T ],

ϕ0(0)+∫ t

0x(s)ds = x(t) = lim

n→∞ xn(t)

= ϕ0(0)+ limn→∞

∫ t

0[un(s)+ f (s,T (θn(s))xn)]ds

= ϕ0(0)+∫ t

0[u(s)+ f (s,T (s)x)]ds.

which gives the equality

x(t) = u(t)+ f (t,T (t)x) for almost t ∈ [0, T ]. (14)

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Now we assert that u = γ (z) a.e. Indeed, for any w ∈ H and any measurable set A in[0, T ], one has

⟨w,

∫A

u(η)dη

⟩=

∫A

〈w, u(η)〉 dη

= limn→∞

∫A

〈w, un(η)〉 dη

= limn→∞

⟨w,

∫Aγ (zn(η))dη

⟩

= limn→∞

⟨w, γ

(∫A

zn(η)dη

)⟩

=⟨w, γ

(∫A

z(η)dη

)⟩

=⟨w,

∫Aγ (z(η)dη)

⟩.

Hence, u(t) = γ (z(t)) for almost every t ∈ [0, T ]. By construction, we have for a.et ∈ [0, T ],

xn(t)− f (t,T (θn(t))xn) = un(t) ∈ F(T (θn(t))xn)

⊂ γ (∂C g(xn(θn(t))))

= γ (∂ P g(xn(θn(t)))). (15)

The last equality follows from the uniform regularity of g over O and the part (ii) inProposition 2.4. The convergence of zn to z for σ(L∞([0, T ],H); L1([0, T ],H))-topologyand Mazur’s Lemma entails

z ∈⋂

n

coσ {zm : m ≥ n}, for a.e. t ∈ [0, T ]

(here σ = σ(L∞([0, T ], H); L1([0, T ], H)). Fix any such t and consider any ξ ∈ H .Then, the last relation above yields

〈ξ, z(t)〉 ≤ infn

supm≥n

〈ξ, zn(t)〉

and by Proposition 2.4 part (iii) and (7) yield

〈ξ, z(t)〉 ≤ lim supn

δ∗(ξ, ∂ P g(xn(θn(t))))

≤ δ∗(ξ, ∂ pg(x(t))) for any ξ ∈ H,

So, by [14, Theorem VI.4], the convexity and the closeness of the set ∂ pg(x(t)) ensures

z(t) ∈ ∂ pg(x(t))

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Now, since g is uniformly regular over O and x(t) ∈ B(ϕ0(0), r) ⊂ O for all t ∈ [0, T ],we have by Proposition 2.4 part (iv)

d

dt(g ◦ x)(t) = 〈∂ pg(x(t)), x(t)〉 = 〈z(t), x(t)〉

= 〈z(t), f (t,T (t)x)+ u(t)〉.Consequently,

g(x(T ))− g(ϕ0(0)) =∫ T

0〈z(t), f (t,T (t)x)〉dt +

∫ T

0〈z(t), u(t)〉dt (16)

On the other hand, since xn(θn(t)) ∈ B(ϕ0(0), r2 ) ⊂ O and by (7) and Definition 2.1, we

have for all i ∈ {0, . . . , n − 1}g(xn(t

ni+1)))− g(xn(t

ni )))

≥ 〈zni , xn(t

ni+1)− xn(t

ni )〉 − β‖xn(t

ni+1)− xn(t

ni )‖2

=⟨

zni ,

∫ tni+1

tni

xn(s)ds

⟩− β‖xn(t

ni+1)− xn(t

ni )‖2

=⟨

zn(t),∫ tn

i+1

tni

[ f (s,T (θn(s))xn)+ un(s)]ds

⟩− β‖xn(t

ni+1)− xn(t

ni )‖2

≥∫ tn

i+1

tni

〈zn(s), f (s,T (θn(s))xn)〉ds +∫ tn

i+1

tni

〈zn(s), un(s)〉ds

− β(m1 + m)2(tni+1 − tn

i )2

By adding, we obtain

g(xn(T ))− g(ϕ0(0))

≥∫ T

0〈zn(s), f (s,T (θn(s))xn)〉ds +

∫ T

0〈zn(s), un(s)〉ds − εn (17)

with

εn = β(m1 + m)2T 2

n→ 0

as n → ∞. We have also have

limn→∞

∫ T

0〈zn(s), f (s,T (θn(s))xn)〉ds =

∫ T

0〈z(s), f (s,T (s)x)〉ds.

Indeed, for all t ∈ [0, T ] and all n ∈ N∗,

〈zn(t), f (t,T (θn(t))xn)〉 − 〈z(t), f (t,T (t)x)〉 = αn(t)+ βn(t)

where

αn(t) = 〈zn(t), f (t,T (θn(t))xn)− f (t,T (t)x)〉,βn(t) = 〈zn(t)− z(t), f (t,T (t)x)〉

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Since zn(t)− z(t) → 0 for σ(L∞([0, T ],H); L1([0, T ],H)),∫ T

0βn(s)ds → 0

and f (t,T (θn(t))xn) → f (t,T (t)x) strongly in L1([0, T ],H) which implies∫ T

0αn(s)ds → 0.

Taking the limit superior in (17) when n → ∞ and using the continuity of g, we obtain

g(x(T ))− g(ϕ0(0)) ≥∫ T

0〈z(s), f (s,T (s)x)〉ds + lim

nsup

∫ T

0〈zn(s), un(s)〉ds

This inequality compared with (16) yields

lim supn

∫ T

0〈zn(s), un(s)〉ds ≤

∫ T

0〈z(s), u(s)〉ds (18)

The values of the function zn are in the convex weakly compact C := LB, further theapplication � : (H, σ ) → [0.+ ∞] defined by

�(α) ={ 〈α, γ (α)〉 if α ∈ C

+∞ otherwise

is lower semicontinuous and strictly convex on C (According to (a) and (b) ). The condition(18) is equivalent to

lim supn

∫ T

0�(zn(s))ds ≤

∫ T

0�(z(s))ds.

Then [5, Proposition 3.2] yields

z(t) ∈⋂

n

coσ {zm(t) : m ≥ n}, for a.e t ∈ [0, T ].

Hence, there is a negligible N such that for t /∈ N , we have

u(t) = γ (z(t))

z(t) ∈⋂

n

coσ {zm(t) : m ≥ n}.

Now let t /∈ N be fixed. Then we can extract from (zn(t))n∈N a subsequence (znk (t))k∈N,such that znk (t) ⇀ z(t) weakly in H so that γ (znk (t)) → γ (z(t)) for the norm topologysince γ ∈ (H). By (6) and (8), recalling that

un(t) = γ (zn(t)) ∈ F(T (θn(t))xn)

for every t ∈ [0, T ] and every n ∈ N∗, that limn→∞ T (θn(t))xn = T (t)x , for all t ∈ [0, T ]

and that the graph of F is closed, we obtain

u(t) = γ (z(t)) ∈ F(T (t)x) a.e. (19)

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Since x(t) = f (t,T (t)x)+ u(t) for almost t ∈ [0, T ] it follows from (19) that

x(t) ∈ F(T (t)x)+ f (t,T (t)x) a.e. on [0, T ].This completes the proof of the theorem. �

Remark 1 If we take the operator T := I d in the above theorem, we obtain a new proofof the existence of solutions in infinite dimensional setting for the functional differentialinclusion {

x(t) ∈ F(x(t)))+ f (t, x(t)) a.e. on [0, T ];x(0) = x0.

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