On the two-point problem for implicit second-order ordinary differential equations

R E S E A R C H

Open Access

On the two-point problem for implicit

second-order ordinary differential equations

Paolo Cubiotti

_{and Jen-Chih Yao}

*_{Correspondence:}

yaojc@mail.cmu.edu.tw

2_{Center for General Education,}

China Medical University, Taichung, 40402, Taiwan, ROC

3_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

Given a nonempty setY⊆Rn_{and a functionf: [a,b]×R}n_×Rn_{×Y→R, we are}

interested in the problem of findingu∈W2,p_([a,b],Rn_{) such that}

f(t,u(t),u(t),u(t)) = 0 for a.e.t∈[a,b], u(a) =u(b) = 0Rn.

We prove an existence result where, for any fixed (t,y)∈[a,b]×Y, the function f(t,·,·,y) can be discontinuous even at all points (x,z)∈Rn_×Rn_{. The function}

f(t,x,z,·) is only assumed to be continuous and locally nonconstant.

We also show how the same approach can be applied to the implicit integral equationf(t,_abg(t,z)u(z)dz,u(t)) = 0. We prove an existence result (withf(t,x,y) discontinuous inxand continuous and locally nonconstant iny) which extends and improves in several directions some recent results in the field.

Keywords: implicit differential equations; two-point problem; discontinuity; discontinuous selections; implicit integral equations

1 Introduction

Let [a,b] be a compact interval, and letp∈[, +∞] andn∈N. As usual, we denote by

W,p_{([a,b], R}n_{) the set of allu∈C}_{([a,b], R}n_{) such thatuis absolutely continuous in [a,b]}

andu∈Lp_{([a,b], R}n_{). Given a nonempty setY⊆R}n_{and a functionf : [a,b]×R}n_×Rn_× Y→R, we are interested in the problem of findingu∈W,p_{([a,b], R}n_{) such that}

⎧ ⎨ ⎩

f(t,u(t),u(t),u(t)) =  for a.e.t∈[a,b],

u(a) =u(b) = Rn.

()

To the best of our knowledge, there are not many existence results in the literature as regards problem (). Some of them [–] deal with the special case wheren=  and

f(t,x,z,y) =y–g(t,x,z,y), wheregsatisfies some growth conditions with respect to (x,z), and very strong conditions (such as Lipschitzianity) are imposed ony. In general, there is little known about the existence of solutions for boundary value problems associated to the equation

ft,u(t),u(t), . . . ,u(k)(t)=  ()

(even whenf is continuous), unless it is possible to solve the equation with respect to the highest derivative. In this case, a variety of existence results based on the Leray-Schauder theory is available. As regards boundary value problems, a global continuous solvability inu(k)_{is needed, which is rarely available (we refer to [, ] and to the references therein}

for more details).

One approach, used in [, ], consists to reduce () (or, more generally, ()) to a differ-ential inclusion of the form

u(k)(t)∈t,u(t), . . . ,u(k–)(t),

with a ‘well-behaved’ multifunction , and then to apply existence results for differen-tial inclusions or selection arguments. Even in this case (see [, ]), the continuity off is assumed.

Our aim in this paper is to prove an existence result for problem (), in the general vector case, where we assume only that the functionf(t,x,z,y), with respect toy, is continuous and locally nonconstant. Moreover, our assumptions do not imply any continuity with respect to (x,z). In particular, a functionfsatisfying our assumptions can be discontinuous at each point (x,z)∈Rn×Rn, for all fixed (t,y)∈[a,b]×Y.

Our approach is based on a selection theorem (Theorem . below) for multifunctions defined on subsets of product spaces. Such result, which refines a former version originally proved in [], ensures the existence of a selection (for a given multifunction), whose set of discontinuities has a peculiar geometric property. That is, it is contained in the union of the set where the given multifunction fails to be lower semicontinuous, together with a finite family of sets, each of them with at least one projection of zero measure.

The selection theorem will be proved in Section , together with other preliminary re-sults. In Section , our main result will be stated and proved, together with examples of application. Finally, in Section , we show how the same approach can be usefully em-ployed to study an implicit integral equation of the form

f t, b

a

g(t,z)u(z)dz,u(t)

= .

We obtain an existence result where, as above, the functionf(t,z,y) is only assumed, with respect toy, to be continuous and locally nonconstant. As before, our assumptions allowf

(with respect to the second variable) to be discontinuous at all pointsx∈Rn. As showed in Section , our result extends and also improves in several directions some recent results.

2 Notations and preliminary results

In the following, givenn∈N, we denote byAnthe family of all subsetsU⊆Rnsuch that,

for everyi= , . . . ,n, the supremum and the infimum of the projection ofconv(U) on the

ith axis are both positive or both negative (‘conv’ standing for ‘closed convex hull’). Given any Lebesgue measurable set V ⊆Rn, we indicate by L(V) the family of all Lebesgue measurable subsets of V, and by mn the n-dimensional Lebesgue measure

in Rn. For alli∈ {, . . . ,n}, we shall denote byPn,i: Rn→Rthe projection over theith

axis. We also denote by Fn the family of all subsetsF⊆Rn such that there exist sets F,F, . . . ,Fn⊆Rn, withm(Pn,i(Fi)) =  for alli= , . . . ,n, such thatF=

n

i=Fi. Of course,

The space Rn_{will be considered with its Euclidean norm ·}

n. Ifp∈[, +∞], the space Lp_{([a,b], R}n_{) will be considered with the usual norm}

uLp_([a,b],Rn₎:=

b

a

u(t)p_ndt

 p

ifp< +∞,

uL∞([a,b],Rn₎:=esssup

t∈[a,b]

u(t)_n ifp= +∞.

As usual, we putLp([a,b]) :=Lp([a,b], R).

LetX be a topological space. We shall denote byB(X) the Borel family of X. In the following, ifTis a Polish space (that is, a separable complete metric space) endowed with a positive regular Borel measureμ, we shall denote byTμthe completion of theσ-algebra B(T) with respect to the measureμ.

For the basic definitions and facts about multifunctions, we refer to []. Here, we only recall (see also []) that if (S,D) is a measurable space andXis a topological space, then a multifunctionF:S→Xis said to beD-measurable(resp.,D-weakly measurable) if for any closed (resp., open) setA⊆Xthe set

F–(A) :=s∈S:F(s)∩A=∅

is measurable. Finally, ifx∈Rn_{andr> , we put}

Bn(x,r) :=v∈Rn:v–xn≤r

.

Our first goal in this section is to prove the following selection result, which is essential for our purposes (for definitions and basic properties as regards Souslin sets and spaces, the reader is referred to []).

Theorem . Let T and X,X, . . . ,Xk be Polish spaces,with k∈N,and let X:= k

j=Xj

(endowed with the product topology).Letμ,ψ, . . . ,ψkbe positive regular Borel measures over T,X,X, . . . ,Xk,respectively,withμfinite andψ, . . . ,ψkσ-finite.

Let S be a separable metric space,W ⊆X a Souslin set,and let F:T×W →S _{be a} multifunction with nonempty complete values.Let E⊆W be a given set.Finally,for all i∈ {, . . . ,k},let P∗,i:X→Xibe the projection over Xi.Assume that:

(i) the multifunctionFisTμ⊗B(W)-weakly measurable;

(ii) for a.e.t∈T,one has

x:= (x, . . . ,xk)∈W:F(t,·)is not lower semicontinuous atx

⊆E. ()

Then,there exist sets Q, . . . ,Qk,with

Qi∈B(Xi) and ψi(Qi) =  for all i= , . . . ,k,

and a functionφ:T×W→S such that:

(a) φ(t,x)∈F(t,x)for all(t,x)∈T×W; (b) for allx:= (x,x, . . . ,xk)∈W\[(

k

i=P∗–,i(Qi))∪E],the functionφ(·,x)is

(c) for a.e.t∈T,one has

x:= (x,x, . . . ,xk)∈W:φ(t,·)is discontinuous atx

⊆E∪

W∩

_k

i=

P–_∗,i(Qi)

.

As noted in Section , Theorem . is a refinement of Theorem . of []. Although the proof is quite similar to the one of [], for the reader’s convenience we shall give it explicitly. First of all, we prove the two following lemmas.

Lemma . Let T, X be two Polish spaces,and letμ be a finite positive regular Borel measure on T.Let W⊆X be a Souslin set,f :T×W→Ra given function,E⊆W another set.Assume that:

(i) f isTμ⊗B(W)-measurable;

(ii) infT×Wf > –∞; (iii) for allt∈T,one has

x∈W:f(t,·)is not lower semicontinuous atx⊆E.

Then,for each ε> there exists a compact set K ⊆T such that μ(T \K)≤εand the function f|K×Wis lower semicontinuous at each point(t,x)∈K×(W\E).

Proof Without loss of generality we can assume thatf ≥ inT×W. Letdbe the distance inX. Fixn∈N, and letfn:T×X→[, +∞[ be defined by putting, for all (t,x)∈T×X,

fn(t,x) :=_y∈Winf

nd(x,y) +f(t,y).

Let us observe the following facts.

(a) For allx∈X, the functionfn(·,x)isTμ-measurable overT. To see this, fixx∈X.

Our assumptions imply that the function

(t,y)∈T×W→nd(x,y) +f(t,y)

isTμ⊗B(W)-measurable. SinceW (endowed with the relative topology) is a

Souslin space, by Lemma III. of [] our claim follows.

(b) For each fixedt∈T, the functionfn(t,·)isn-Lipschitzian overX(the proof is straightforward).

(c) We have

fn(t,x)≤f(t,x) for alln∈Nand(t,x)∈T×W. ()

Now, letψ:T×W→[, +∞[ be defined by putting

ψ(t,x) :=sup n∈N

In particular, by () we have

ψ(t,x)≤f(t,x) for all (t,x)∈T×W. ()

We claim that

ψ(t,x) =f(t,x) for all (t,x)∈T×(W\E). ()

To see this, fix (t,x)∈T×(W\E) and assume thatψ(t,x) <f(t,x). Letδ:=f(t,x) –ψ(t,

x) > . Then, for eachn∈Nthere existsyn∈W such that

nd(x,yn) +f(t,yn) <f(t,x) –δ ,

hence

nd(x,yn) <f(t,x) –

δ

–f(t,yn)≤f(t,x) –

δ

.

This implies that{yn} →xinX(hence, in particular, inW). By assumption (iii), we get

f(t,x)≤lim inf

n→∞ f(t,yn)≤lim infn→∞ f(t,x) –

δ

–nd(x,yn)

≤f(t,x) – δ ,

a contradiction. Thus, () is proved.

Now, in order to prove the conclusion, fixε> . By Theorem  of [], for eachn∈N

there exists a compact setKn⊆T such thatμ(T\Kn) < (ε/n) and the functionfn|Kn×Xis continuous. LetK:=_n∈NKn. Of course,K is compact andμ(T\K) <ε. Moreover, for

alln∈Nthe functionfn|K×W is continuous.

Consequently, the functionψ|K×Wis l.s.c. Let us prove thatf|K×Wis l.s.c. at each point

(t,x)∈K×(W\E). To this aim, fix (t∗,x∗)∈K×(W\E) andγ > . LetUbe a neighbor-hood of (x∗,y∗) inK×W such that

ψ(t,x) >ψt∗,x∗–γ, ∀(t,x)∈U

(suchUexists sinceψ|K×W is lower semicontinuous). For all (t,x)∈U, by () and () we get

f(t,x)≥ψ(t,x) >ψt∗,x∗–γ =ft∗,x∗–γ,

hencef|K×W is l.s.c. at (t∗,x∗), as desired. This concludes the proof.

Lemma . Let T,X,μ,W and E be as in Lemma..Let S be a separable metric space,

F:T×W→S_{a multifunction with nonempty values.Assume that:}

(i) FisTμ⊗B(W)-weakly measurable;

(ii) for allt∈T,one has

Then,for each ε> there exists a compact set K ⊆T such that μ(T \K)≤εand the multifunction F|K×W is lower semicontinuous at each point(t,x)∈K×(W\E).

Proof LetdS be an equivalent distance overS such thatdS≤, and let{sn}be a dense sequence inS. Fixε> . For eachn∈N, lethn:T×W→[, +∞[ be defined by putting, for each (t,x)∈T×W,

hn(t,x) = –dSsn,F(t,x).

By assumption (i) and Theorem . of [], the functionhnisTμ⊗B(W)-measurable. More-over, by assumption (ii) and Lemma  of [], for allt∈Twe have

x∈W:hn(t,·) is not lower semicontinuous atx⊆E.

By Lemma ., there exists a compact setKn⊆Tsuch thatμ(T\Kn) <ε/n_andhn| Kn×W is lower semicontinuous at each point (t,x)∈Kn×(W\E). LetK:=_n∈NKn. Of course,

μ(T\K)≤εand, for alln∈N, the functionhn|K×Wis lower semicontinuous at each point

(t,x)∈K×(W\E). Again by Lemma  of [], this implies that the multifunctionF|K×W

is lower semicontinuous at each point (t,x)∈K×(W\E). This completes the proof.

The following results are proved in []. We state them for easy reading (in the following, the space NN_{of all infinite sequences of integers is considered with the product topology;}

we recall that NN_{is Polish and zero-dimensional).}

Lemma .(Lemma . of []) Let p∈N,with p≥,and let Y,Y, . . . ,Ypbe Polish spaces,

endowed withσ-finite positive regular Borel measuresμ,μ, . . . ,μp,respectively. Then,there exist sets Q,Q, . . . ,Qp,with

Qi∈B(Yi) and μi(Qi) =  for all i= , . . . ,p,

and two functionsπ: NN_→Y

×Y× · · · ×Ypandσ :Y×Y× · · · ×Yp→NN,such

that:

(a) the functionπis continuous and open; (b) the functionσis continuous at each point

x:= (x,x, . . . ,xp)∈(Y\Q)×(Y\Q)× · · · ×(Yp\Qp);

(c) one has

πσ(x)=x for allx∈Y×Y× · · · ×Yp.

Lemma .(Lemma . of []) Let p∈N,with p≥,and let Y,Y, . . . ,Ypbe Polish spaces,

endowed withσ-finite positive regular Borel measuresμ,μ, . . . ,μp,respectively.Let the sets Q,Q, . . . ,Qpand the functionsπandσ be as in the conclusion of Lemma..

Let B⊆Y× · · · ×Yp a nonempty set,V⊆B another given set,E a metric space and

F:B→E_{a multifunction with nonempty complete values.Assume that F is lower} semi-continuous at each point x= (x, . . . ,xp)∈B\V.

(a) g(x, . . . ,xp)∈F(x, . . . ,xp)for all(x, . . . ,xp)∈B; (b) gis continuous at each point

(x, . . . ,xp)∈

(Y\Q)×(Y\Q)× · · · ×(Yp\Qp)

∩B\V.

Proof of Theorem. It is matter of routine to check that, without loss of generality, we can assume that () holds for allt∈T.

Fixn∈N. By Lemma . (applied to the multifunctionF:T×W→S_{, withε= /n),}

there exists a compact setKˆn⊆Tsuch that

μ(T\ ˆKn)≤ 

n

and the multifunctionF|Kˆn×Wis lower semicontinuous at each point (t,x)∈ ˆKn×(W\E). Let us put

K:=Kˆ, Kn:=Kˆn

n–

j= ˆ

Kj, n≥.

The sets{Kn}are pairwise disjoint, the equality_n∈NKnˆ =

n∈NKnholds, and for alln∈N

the multifunctionF|Kn×Wis lower semicontinuous at each point (t,x)∈Kn×(W\E). Let

Y:=T n∈N

Kn.

HenceY∈B(T), and for allj∈None has

μ(Y) =μ T n∈N

Kn

=μ T

n∈N

ˆ

Kn

=μ

n∈N

(T\ ˆKn)

≤μ(T\ ˆKj)≤



j,

henceμ(Y) = . Now, let us apply Lemma . above to thek+  spacesT,X, . . . ,Xk. We

see that there exist sets

Q∈B(T), Q∈B(X), . . . , Qk∈B(Xk)

and functionsπandσsatisfying the conclusion of Lemma .. For eachn∈N, let us apply Lemma ., with

B=Kn×W, V:=Kn×E.

We see that there exists a functiongn:Kn×W→Ssuch that

gn(t,x)∈F(t,x), ∀(t,x)∈Kn×W

andgn:Kn×W→Sis continuous at each point

(t,x)∈(T\Q)×(X\Q)× · · · ×(Xk\Qk)

∩[Kn×W]\[Kn×E]

=(Kn\Q)×

(X\Q)× · · · ×(Xk\Qk)

= (Kn\Q)×

(X\Q)× · · · ×(Xk\Qk)

∩W\E

= (Kn\Q)×

(X\Q)× · · · ×(Xk\Qk)

∩(W\E).

For eacht∈Yandx∈W, choose anyh(t,x)∈F(t,x). Letφ:T×W→Sbe defined by

φ(t,x) = ⎧ ⎨ ⎩

gn(t,x) ift∈Kn,

h(t,x) ift∈Y.

()

It is immediate to check thatφ(t,x)∈F(t,x) for all (t,x)∈T×W.

Now, fixt∗∈T\(Q∪Y) (observe thatQ∪Y∈B(T) and has null measure) and let

n∈Nsuch thatt∗∈Kn. Hence, for allx∈W, we haveφ(t∗,x) =gn(t∗,x), that is,φ(t∗,·) =

gn(t∗,·). Consequently, we get

x∈W:φt∗,·is discontinuous atx⊆E∪

W k

i=

(Xi\Qi)

=E∪

W∩

_k

i=

P–_∗,i(Qi)

.

Finally, if we fix

x:= (x,x, . . . ,xk)∈W

k

i=

P–_∗,i(Qi)

∪E

=(X\Q)× · · · ×(Xk\Qk)

∩W\E,

by () we see that for alln∈Nthe functionφ(·,x)|Kn\Q isB(Kn\Q)-measurable (since

gn(·,x)|Kn\Qis continuous), andφ(·,x)|Kn∩QisTμ-measurable (sinceμ(Q) =  andTμis

complete). It follows that for alln∈Nthe functionφ(·,x)|KnisTμ-measurable. Hence, the functionφ(·,x) isTμ-measurable overT (sinceμ(Y) = ). The proof is complete.

Finally, we prove the following proposition.

Proposition . Letψ: [a,b]×Rn_→Rk_{be a given function,E⊆R}n_{be a Lebesgue} mea-surable set,with mn(E) = ,and let D be a countable dense subset ofRn_{,with D∩E=∅.} Assume that:

(i) for allt∈[a,b],the functionψ(t,·)is bounded;

(ii) for allx∈D,the functionψ(·,x)isL([a,b])-measurable.

Let G: [a,b]×Rn→Rk

be the multifunction defined by setting,for each(t,x)∈[a,b]×Rn,

G(t,x) =

m∈N

conv y∈D y–xn≤m

ψ(t,y).

Then one has:

(a) Ghas nonempty closed convex values;

(c) for allt∈[a,b],the multifunctionG(t,·)has closed graph;

(d) ift∈[a,b],andψ(t,·)|Rn_\Eis continuous atx∈Rn\E,then one has

G(t,x) =ψ(t,x).

Proof Conclusions (a), (b), and (c) can be proved by arguing exactly as in the proof of Proposition  of []. To prove conclusion (d), fixt∈[a,b], and letx∈Rn_{\Esuch that}

ψ(t,·)|Rn_\Eis continuous atx. Fixε> . By continuity, there existsδ>  such that ψ(t,y) –ψ(t,x)_k≤ε, ∀y∈Rn\E, withy–xn<δ.

Hence, ifm∈N, with 

m <δ, taking into account thatE∩D=∅, we get

conv y∈D y–xn≤_m

ψ(t,y)⊆Bkψ(t,x),ε.

Consequently, we getG(t,x)⊆Bk(ψ(t,x),ε). Sinceε>  was arbitrary andG(t,x)=∅, we

get the conclusion.

3 The main result

The following is our main result.

Theorem . Let[a,b]be a compact interval,and let Y∈Anbe a closed,connected and locally connected subset ofRn_{.Let f : [a,b]×R}n_×Rn_{×Y→Rbe a given function,⊆} Y×Y a countable set,dense in Y×Y,Dand Dtwo dense subset of Y.Assume that there existn sets

V,V, . . . ,Vn∈B(R),

with m(Vi) = for all i= , . . . , n,such that,if one puts

:=

n

i=

[R\Vi],

f∗: [a,b]××Y→R, f∗:=f|[a,b]××Y,

one has:

(i) for all(y,y)∈,one has

(t,x,z)∈[a,b]×:f∗(t,x,z,y) <  <f∗(t,x,z,y)

∈L[a,b]⊗B();

(ii) for a.e.t∈[a,b],and for ally∈D,the functionf∗(t,·,·,y)is l.s.c.over; (iii) for a.e.t∈[a,b],and for ally∈D,the functionf∗(t,·,·,y)is u.s.c.over; (iv) for a.e.t∈[a,b],and for all(x,z)∈,the functionf(t,x,z,·)is continuous overY,

∈intR

and

intY

y∈Y:f(t,x,z,y) = =∅;

(v) there existp∈[, +∞]and a positive functionβ∈Lp_{([a,b])such that,for a.e.} t∈[a,b],and for all(x,z)∈,one has

y∈Y:f(t,x,z,y) = ⊆BnRn,β(t).

Then,there exists u∈W,p_{([a,b], R}n_{)such that}

⎧ ⎨ ⎩

f(t,u(t),u(t),u(t)) =  for a.e. t∈[a,b],

u(a) =u(b) = Rn,

and

u(t),u(t)∈ for a.e. t∈[a,b].

Proof From now on, we assume that assumptions (ii)-(v) are satisfied for allt∈[a,b] (it is routine matter to check that it is not restrictive to do this). First of all, we assumep< +∞. Fixt∈[a,b]. Let

Vt:→Y, Et:→Y, Qt:→Y

be the multifunctions defined by setting, for all (x,z)∈,

Vt(x,z) :=y∈Y:f∗(t,x,z,y) = ,

Et(x,z) :=

y∈Y:yis a local extremum forf∗(t,x,z,·),

Qt(x,z) :=Vt(x,z)\Et(x,z).

By assumptions (ii), (iii), (iv), and Theorem . of [], the multifunctionQt:→Y

is lower semicontinuous inwith nonempty closed values (inY, hence in Rn_sinceYis

closed). Now, let

Q: [a,b]×→Y, Q(t,x,z) :=Qt(x,z).

By what precedes, the multifunctionQhas nonempty closed values and, for each fixed

t∈[a,b], the multifunctionQ(t,·,·) is lower semicontinuous in.

Now we prove that the multifunctionQisL([a,b])×B()-measurable. To this aim, let

A⊆Ybe a nonempty open connected set, such thatQ–_{(A)=∅. We claim that}

Q–(A) =

(y,y)∈(A×A)∩

(t,x,z)∈[a,b]×:f(t,x,z,y) <  <f(t,x,z,y)

. ()

To this aim, fix any (t,x,z)∈Q–(A). Therefore, (t,x,z)∈[a,b]×and there exists

the functionf(t,x,z,·). This implies that there exist two pointsy,y∈Asuch that

ft,x,z,y

<  and ft,x,z,y

> .

By the continuity of the functionf(t,x,z,·), there exist two open setsB,B⊆Y such

thaty∈B,y∈B, and

f(t,x,z,y) <  for ally∈B,

f(t,x,z,y) >  for ally∈B.

PutA:=B∩A,A:=B∩A. Of course, bothA andAare open inY and nonempty

(since they containyandy, respectively).

Let (y,y)∈(A×A)∩(this last set is nonempty sinceis dense inY×Y). We see

that

(t,x,z)∈

(t,x,z)∈[a,b]×:f(t,x,z,y) <  <f(t,x,z,y)

,

hence (t,x,z) belongs to the right-hand side of (). Conversely, let (t∗,x∗,z∗) belong to

the right-hand side of (). Thus, there exists (y,y)∈(A×A)∩such that

ft∗,x∗,z∗,y

<  <ft∗,x∗,z∗,y

. ()

SinceAis connected andf(t∗,x∗,z∗,·) is continuous inY, there existsy∈Asuch that

f(t∗,x∗,z∗,y) = . We now distinguish two cases.

() Ifyis not a local extremum for the functionf(t∗,x∗,z∗,·), then we get

y∈Q(t∗,x∗,z∗)∩A, hence(t∗,x∗,z∗)∈Q–(A), as desired.

() Ifyis a local extremum for the functionf(t∗,x∗,z∗,·)(not absolute by assumption (iv)), thenyis also a local extremum for the functionf(t∗,x∗,z∗,·)|A(not absolute

by ()). Moreover, sinceAis open inY, by assumption (v) we get

intA

y∈A:ft∗,x∗,z∗,y= =∅.

Consequently, by Lemma . of [], there exists a pointy∗∈Asuch that

f(t∗,x∗,z∗,y∗) = andy∗is not a local extremum for the functionf(t∗,x∗,z∗,·)|A.

This easily implies thaty∗is not a local extremum for the functionf(t∗,x∗,z∗,·)

(considered over the whole setY). Therefore, we havey∗∈Q(t∗,x∗,z∗)∩A, hence

(t∗,x∗,z∗)∈Q–_{(A), as desired.}

Thus, the equality () is proved, and therefore, by assumption (i), the set Q–_{(A) is}

L([a,b])⊗B()-measurable.

Our assumptions onYimply that it has a countable base of connected open sets. There-fore, it follows that the multifunctionQisL([a,b])×B()-weakly measurable. By Theo-rem . of [], the multifunctionQis alsoL([a,b])⊗B()-measurable.

By Corollary .. of [], the setis a Souslin set since it belongs toB(Rn_{). By}

The-orem . (where the spaces [a,b] and R are considered with the usual one-dimensional Lebesgue measure m over their Borel families), there exist Q, . . . ,Qn ∈B(R), with m(Q) =  for all i = , . . . , n, a set K∈L([a,b]), with m(K) = , and a function

(a) φ(t,x,z)∈Q(t,x,z)for all(t,x,z)∈[a,b]×(hence, in particular, the functionφ

takes its values inY); (b) for all(x,z)∈\[_i=nP–

n,i(Qi)], the functionφ(·,x,z)isL([a,b])-measurable;

(c) for allt∈[a,b]\K, one has

(x,z)∈:φ(t,·,·)is discontinuous at(x,z)⊆∩

_n

i=

P_–_n,i(Qi)

.

Now, letψ: [a,b]×Rn×Rn→Rnbe defined by

ψ(t,x,z) = ⎧ ⎨ ⎩

φ(t,x,z) if (x,z)∈, Rn if (x,z) /∈.

Observe that Rn\=_i_=nP_–_n,i(Vi), and also

n

i=

P–_n,i(Qi)

= Rn

n

i=

P–_n,i(Vi∪Qi) = n i=

R\(Vi∪Qi).

LetDbe a countable subset of\[_i=nP–

n,i(Qi)], dense in Rn. Of course, such a setD

exists since

mn

_n

i=

P_–_n,i(Vi∪Qi)

= .

Now we want to apply Proposition . to the functionψ, choosingE= Rn_{\. To this}

aim, observe that:

() for allt∈[a,b], and all(x,z)∈Rn_×Rn_{, we haveψ(t,x,z)∈Bn(}

Rn,β(t))(this

follows by the construction ofφandQand by assumption (v)), hence for all

t∈[a,b]the functionψ(t,·,·)is bounded;

() for all(x,z)∈D, the functionψ(·,x,z) =φ(·,x,z)isL([a,b])-measurable.

Consequently, ifG: [a,b]×Rn_×Rn_→Rn_{is the multifunction defined by setting, for}

all (t,x,z)∈[a,b]×Rn×Rn,

G(t,x,z) =

m∈N

conv

(v,w)∈D (v,w)–(x,z)n≤m

ψ(t,v,w)

=

m∈N

conv

(v,w)∈D

(v,w)–(x,z)n≤m

φ(t,v,w),

by Proposition . we see that:

(a) Ghas nonempty closed convex values;

(b) for all(x,z)∈Rn_×Rn_{, the multifunctionG(·,x,z)isL([a,b])-measurable;}

(d) ift∈[a,b], and the functionψ(t,·,·)|=φ(t,·,·)is continuous at(x,z)∈, then one has

G(t,x,z) =ψ(t,x,z)=φ(t,x,z).

Moreover, observe that by the above construction we see that

G(t,x,z)⊆BnRn,β(t)∩conv(Y) for all (t,x,z)∈[a,b]×Rn×Rn. ()

Consequently, by Theorem  of [], there existu∈W,p([a,b], Rn) and a setK∈L([a,b]),

withm(K) = , such that ⎧

⎨ ⎩

u(t)∈G(t,u(t),u(t)) for allt∈[a,b]\K,

u(a) =u(b) = Rn.

()

In particular, by () we get

u(t)∈conv(Y) for allt∈[a,b]\K. ()

Now, fixi∈ {, . . . ,n}. Of course, () implies that

u_i(t)∈infPn,i

conv(Y),supPn,i

conv(Y) for allt∈[a,b]\K

(of course, we are denoting byui(t) theith component ofu(t)). In particular, taking into account thatY∈An, this implies thatui(t) has constant sign for allt∈[a,b]\K. Assume

that

u_i(t) >  for allt∈[a,b]\K.

Sinceu_i(t) is absolutely continuous, we have

u_i(t) =u_i(a) + t

a

u_i(s)ds for allt∈[a,b],

henceu_i(t) is strictly increasing in [a,b]. Consequently, by Theorem  of [], the function

u_i–:u_i[a,b]→[a,b]

is absolutely continuous. Moreover, sinceui(a) =ui(b) = , there existsci∈]a,b[ such that

u_i(t) <  for allt∈[a,ci[

and

Again by Theorem  of [], the functions

(ui|[a,ci])

–_{:ui[a,ci]→[a,ci]}

and

(ui|[ci,b]) –_:u

i

[ci,b]

→[ci,b]

are absolutely continuous. Consequently, by Theorem . of [], for each Lebesgue measurableU⊆R, withm(U) = , the sets

u_i–(U) :=t∈[a,b] :u_i(t)∈U

and

(ui)–(U) :=t∈[a,b] :ui(t)∈U

=t∈[a,ci] :ui(t)∈U∪t∈[ci,b] :ui(t)∈U

have null Lebesgue measure. If, conversely, one has

u_i(t) <  for allt∈[a,b]\K,

by an analogous argument we can get the same conclusion. Now, for eachi= , . . . ,n, let us put

Si:=u–_i (Vi∪Qi), Wi:=u_i–(Vn+i∪Qn+i),

and

S:=K∪K∪ _n

i=

(Si∪Wi)

.

For what precedes, we havem(S) = . Now, fixt∈[a,b]\S. Sincet∈/ n

i=(Si∪Wi), for

alli= , . . . ,nwe get

ui(t) /∈Vi∪Qi, u_i(t) /∈Vn+i∪Qn+i,

hence

u(t),u(t)∈

n

i=

R\(Vi∪Qi)=

n

i=

P_–_n,i(Qi)

.

Sincet∈/K, this implies that the functionφ(t,·,·) :→Rnis continuous at (u(t),u(t)).

Hence, by the property (d)and the above construction, taking into account thatt∈/ K

and (), we get

Therefore, we getu(t)∈Yand

ft,u(t),u(t),u(t)= .

Thus, our conclusion is proved in the casep< +∞.

Now, assumep= +∞. Fix anyq∈[, +∞[. Sinceβ∈Lq_{([a,b]), by the first part of the}

proof there exist a functionu∈W,q_{([a,b], R}n_{) and a setU∈L([a,b]), withm}

(U) = ,

such that ⎧ ⎨ ⎩

f(t,u(t),u(t),u(t)) =  for allt∈[a,b]\U,

u(a) =u(b) = Rn,

and

u(t),u(t)∈ for allt∈[a,b]\U.

By (v) we get

y∈Y:ft,u(t),u(t),y= ⊆BnRn,β(t) for allt∈[a,b]\U,

hence

u(t)_n≤β(t) for allt∈[a,b]\U.

This impliesu∈L∞([a,b], Rn), henceu∈W,∞([a,b], Rn). The proof is now complete.

Remark Theorem . can be put in the following equivalent form.

Corollary . Let[a,b],Y,f,,Dand Dbe as in the statement of Theorem..Assume that there exists a set F∈Fnsuch that,if one puts:= Rn\F and f∗:=f|[a,b]××Y,then

assumptions(i)-(v)of Theorem.are satisfied.

Then the same conclusion of Theorem.holds.

Proof LetF,F, . . . ,Fn⊆Rn, withm(Pn,i(Fi)) =  for alli= , . . . , n, be such thatF=

n

i=Fi. For eachi= , . . . , n, letVi∈B(R) be such thatPn,i(Fi)⊆Viandm(Vi) = . If

one puts:=_i=n(R\Vi), then⊆, hence all the assumptions of Theorem . are

satisfied. The conclusion follows at once.

Now we give a simple example of application of Theorem ..

Example  Letn= ,Y= [, +∞[,p∈[, +∞], and letα∈Lp_{([, ]), withα(t) >  for all} t∈[, ]. Let

and letf: [, ]×R×R×Y→Rbe defined by putting

f(t,x,z,y) = ⎧ ⎨ ⎩

y–α(t) –  if (x,z)∈E,

cos(y+π

 – ) +y( +|cosx|) –|cos(x+z)|–α(t) –  if (x,z) /∈E.

It is immediate to check that for each fixed (t,y)∈[, ]×Yone has

f(t,x,z,y)≥y–α(t) –  for all (x,z)∈R\E.

Consequently, for each fixed (t,y)∈[, ]×Y, the functionf(t,·,·,y)is discontinuous at all points(x,z)∈R_.

Now, let us observe that such a functionf satisfies the assumptions of Theorem .. To this aim, chooseV=V= Q,D=D=Y, and letbe any countable dense subset of

Y×Y. In this case, we have

= (R\Q)×(R\Q),

hence in particular we get= R_\Eand

f∗(t,x,z,y) = cos y+π  – 

+y +|cosx|–cos(x+z)–α(t) – 

for all (t,x,z,y)∈[, ]××Y. At this point, observe what follows.

(a) For all fixedy∈Y, the functionf∗(·,·,·,y) isL([, ])⊗B()-measurable. Therefore, assumption (i) of Theorem . is satisfied.

(b) For all (t,y)∈[, ]×Y, the functionf∗(t,·,·,y) is continuous over.

(c) Lett∈[, ] and (x,z)∈be fixed. We see thatf(t,x,z,·) =f∗(t,x,z,·) is continuous inY, and also

f(t,x,z, )≤–α(t) < .

Since

lim

y→+∞f(t,x,z,y) = +∞,

we get ∈intR(f(t,x,z,Y)). Moreover, if we puts(·) :=f(t,x,z,·), we get

s(y) = –sin y+π  – 

+  +|cosx|, ∀y∈Y.

(d) Lett∈[, ] and (x,z)∈be fixed, and lety∈Y be such thatf(t,x,z,y) = . By the definition ofYandf we get

≤y≤ 



 +α(t).

Since the function

β(t) =  

 +α(t)

belongs toLp([, ]), assumption (v) of Theorem . is also satisfied.

Therefore, all the assumptions of Theorem . are fulfilled. Consequently, there exists

u∈W,p_{([, ]) such that}

⎧ ⎨ ⎩

f(t,u(t),u(t),u(t)) =  for a.e.t∈[, ],

u() =u() =  ()

and also

u(t),u(t)∈(R\Q)×(R\Q) for a.e.t∈[, ].

Finally, we observe that problem () does not admit the trivial solutionu(t)≡.

The following result is an immediate consequence of Theorem ..

Theorem . Let[a,b]be a compact interval,and let Y∈Anbe a closed,connected and locally connected subset ofRn.Letψ: [a,b]×Y→Rand g: [a,b]×Rn×Rn→Rbe two given functions,D⊆Y a countable set,dense in Y.Assume that for all y∈D,the function

ψ(·,y)isL([a,b])-measurable,and for a.e.t∈[a,b],the functionψ(t,·)is continuous in Y.

Moreover,assume that there existn sets

V,V, . . . ,Vn∈B(R),

with m(Vi) = for all i= , . . . , n,such that,if one puts

:=

n

i=

[R\Vi],

one has:

(i) for allt∈[a,b],the functiong(t,·,·)|is continuous over; (ii) for all(x,z)∈,the functiong(·,x,z)isL([a,b])-measurable; (iii) for a.e.t∈[a,b],one hasg(t,)⊆intR(ψ(t,Y));

(iv) for a.e.t∈[a,b]and for allv∈intR(ψ(t,Y)),one hasintY({y∈Y:ψ(t,y) =v}) =∅;

(v) there existp∈[, +∞]and a positive functionβ∈Lp_{([a,b])such that,for a.e.} t∈[a,b],and for all(x,z)∈,one has

Then,there exists u∈W,p_{([a,b], R}n_{)such that}

⎧ ⎨ ⎩

ψ(t,u(t)) =g(t,u(t),u(t)) for a.e. t∈[a,b],

u(a) =u(b) = Rn,

and

u(t),u(t)∈ for a.e. t∈[a,b].

Proof Let us apply Theorem ., with f(t,x,z,y) =ψ(t,y) –g(t,x,z), D=D=Y and

=D×D. Observe that by assumptions (i) and (ii) and by the lemma at p. of [], the functiong|[a,b]× isL([a,b])×B()-measurable. Consequently, assumption (i) of The-orem . is satisfied. It is a matter of routine to check that all the other assumptions of Theorem . are satisfied. Therefore, the conclusion follows at once.

Arguing exactly as for Theorem ., it can be easily seen that Theorem . can be put in the following equivalent form.

Corollary . Let[a,b],Y,ψ,g and D be as in the statement of Theorem..Assume that there exists a set F∈Fnsuch that,if one puts:= Rn\F,then assumptions(i)-(v)of Theorem.are satisfied.

Then,the same conclusion of Theorem.holds.

Remark Theorem . (or, equivalently, Corollary .) should be compared with Theo-rem . of [] (valid for the case whereψdoes not depend ontand the setYis bounded), where the functiongis assumed to be continuous in all variables. We now give an example of an application of Theorem . in the vector casen= .

Example  Letn= . We now denote vectors of R_by

x= (x,x), z= (z,z), y= (y,y).

Letp∈[, +∞] andα∈Lp_{([, ]) be fixed, withα(t)≥ for allt∈[, ]. Let}

E:=(x,z)∈R: at least one ofx,x,z,zis rational

,

and leth: [, ]×R_{→Randg: [, ]×R}_×R_{→Rbe defined by}

h(t,y,y) =sin

t(y– )

+y+y,

g(t,x,x,z,z) = ⎧ ⎨ ⎩

α(t) +cos(t xxzz) if (x,z) /∈E,

α(t) –  if (x,z)∈E.

Of course, for allt∈[, ] the functiong(t,·,·)is discontinuous at all points(x,z)∈R_.

However, Theorem . can be easily applied, by choosing

V=V=V=V= Q, Y:= [, +∞[×[, +∞[, ψ:=h|[,]×Y,

In particular, we observe that in this case we have= R_\Eand

g(t,x,x,z,z) =α(t) +cos(t xxzz) for all (x,z)∈.

Consequently, assumptions (i) and (ii) of Theorem . are satisfied. If we fixt∈[, ], then

ψ(t, , ) =  andψ(t,·,·) is upperly unbounded, hence

g(t,)⊆[, +∞[⊆], +∞[⊆ψ(t,Y).

Hence, assumption (iii) of Theorem . is also satisfied. We now prove that assumption (iv) is also satisfied (although this fact is quite intuitive, we provide a direct proof ).

To this aim, fix t ∈[, ] and v∗ ∈ int(ψ(t,Y)). Let y∗ := (y∗_,y∗_) ∈Y be such that

ψ(t,y∗_,y∗_) =v∗, and letbe any open set inY such that (y∗_,y∗_)∈. This implies that there existsδ>  such that

y∗_–δ,y∗_+δ×y∗_–δ,y∗_+δ∩Y⊆.

If we choose anyy∈]y∗,y∗+δ[, we get (y∗,y)∈and

ψt,y∗_,y

>ψt,y∗_,y∗_=v∗.

Hence, the set{y∈Y:ψ(t,y,y) =v∗}has empty interior inY, as desired.

Finally, lett∈[, ] and (x,z)∈. Takey∈Y such thatψ(t,y) =g(t,x,z). By the defini-tions ofY,ψ andgwe get

y≤α(t) + ,

hence assumption (v) of Theorem . is also satisfied. Consequently, there exists u∈ W,p_{([, ], R}_{) such that}

⎧ ⎨ ⎩

h(t,u(t)) =g(t,u(t),u(t)) for a.e.t∈[, ],

u(a) =u(b) = R,

()

and also

u(t)∈(R\Q)×(R\Q), u(t)∈(R\Q)×(R\Q), u(t)∈[, +∞[×[, +∞[

for a.e.t∈[, ]. We also observe that problem () does not admit the trivial solution

u(t)≡R.

4 Implicit integral equations

As announced in Section , our aim in this section it to apply the same approach as before to the study of implicit integral equations. From now on, letn∈Nbe fixed. In what follows, for simpler notation, we shall putPi:=Pn,i. The following theorem is the main result of

Theorem . Let[a,b]be a compact interval,and let Y∈Anbe a closed,connected and locally connected subset ofRn.Let f : [a,b]×Rn×Y→Rand g: [a,b]×[a,b]→[, +∞[

be two given functions,⊆Y×Y a countable set,dense in Y×Y,Dand Dtwo dense subset of Y.Moreover,let p∈], +∞],φ∈Lj([a,b]),with j> and j≥p (the conjugate

exponent of p),φ∈Lp

([a,b]).Assume that there exists a set F∈Fnsuch that:

(i) for all(y,y)∈,one has

(t,x)∈[a,b]×Rn\F:f(t,x,y) <  <f(t,x,y)

∈L[a,b]⊗BRn\F;

(ii) for a.e.t∈[a,b],and for ally∈D,the functionf(t,·,y)|Rn_\Fis l.s.c.;

(iii) for a.e.t∈[a,b],and for ally∈D,the functionf(t,·,y)|Rn_\Fis u.s.c.;

(iv) for a.e.t∈[a,b],and for allx∈Rn\F,the functionf(t,x,·)is continuous overY,

∈intR

f(t,x,Y)

and

intY

y∈Y:f(t,x,y) = =∅;

(v) there exists a positive functionβ∈Lp([a,b])such that,for a.e.t∈[a,b],and for all

x∈Rn_{\F,one has}

y∈Y:f(t,x,y) = ⊆BnRn,β(t).

Moreover,assume that:

(vi) for allt∈[a,b],the functiong(t,·)is measurable;

(vii) for a.e.z∈[a,b],the functiong(·,z)is continuous in[a,b],differentiable in]a,b[

and

g(t,z)≤φ(z),  <

∂g

∂t(t,z)≤φ(z) for allt∈]a,b[.

Then,there exists u∈Lp_{([a,b], R}n_{)such that}

f t, b

a

g(t,z)u(z)dz,u(t)

=  for a.e. t∈[a,b],

u(t)_n≤β(t) and

b

a

g(t,z)u(z)dz∈Rn\F for a.e. t∈[a,b].

Proof Without loss of generality we can assume that assumptions (ii)-(v) are satisfied for allt∈[a,b]. Moreover, taking into account thatp> , it is not restrictive to assumej< +∞.

LetF,F, . . . ,Fn⊆Rn, withm(Pi(Fi)) =  for alli= , . . . ,n, be such thatF= n

i=Fi. For

alli= , . . . ,n, letBi∈B(R) such thatPi(Fi)⊆Biandm(Bi) = . LetC:=

n

i=Pi–(Bi), and

:= Rn\C=

n

i=

Clearly, we haveF⊆Cand thus⊆Rn_{\F. Let}

V: [a,b]×→Y, E: [a,b]×→Y, Q: [a,b]×→Y

be the multifunctions defined by setting, for all (t,x)∈[a,b]×,

V(t,x) :=y∈Y:f(t,x,y) = ,

E(t,x) :=y∈Y:yis a local extremum forf(t,x,·),

Q(t,x) :=V(t.x)\E(t,x).

By assumptions (ii), (iii), (iv), and Theorem .. of [], it is immediately seen thatQhas nonempty closed values (inY, hence in Rn_{) and for allt∈[a,b] the multifunctionQ(t,·)}

is lower semicontinuous in.

Arguing exactly as in the proof of Theorem ., it can be checked that the multifunction

QisL([a,b])×B()-measurable. By Theorem . (applied withT= [a,b],Xi= R for all

i= , . . . ,n, where all the spaces are considered with the usual one-dimensional Lebesgue measuremover their Borel families), there existQ, . . . ,Qn∈B(R), withm(Qi) =  for all

i= , . . . ,n, a setK∈L([a,b]), withm(K) = , and a functionφ: [a,b]×→R n

, such that:

(a) φ(t,x)∈Q(t,x)for all(t,x)∈[a,b]×(hence the functionφtakes its values inY); (b) for allx∈\[n_i=P–_i (Qi)], the functionφ(·,x)isL([a,b])-measurable;

(c) for allt∈[a,b]\K, one has

x∈:φ(t,·)is discontinuous atx⊆∩

_n

i=

P_i–(Qi)

.

Letψ: [a,b]×Rn_→Rn_{be defined by}

ψ(t,x) = ⎧ ⎨ ⎩

φ(t,x) ifx∈, Rn ifx∈/.

Observe that

n

i=

P–_i (Qi)

= Rn

n

i=

P–_i (Bi∪Qi)

= n i=

R\(Bi∪Qi)

,

and letDbe any countable subset of\[n_i=P–_i (Qi)], dense in Rn. Of course, such a set

Dexists since

mn

_n

i=

P–_i (Bi∪Qi)

= .

LetG: [a,b]×Rn_→Rn_{be the multifunction defined by setting, for all (t,x)∈[a,b]×R}n_,

G(t,x) :=

m∈N

conv v∈D v–xn≤m

ψ(t,v)=

m∈N

conv v∈D v–xn≤m