A boundary value problem for implicit vector differential inclusions without assumptions of lower semicontinuity

R E S E A R C H

Open Access

A boundary value problem for implicit

vector differential inclusions without

assumptions of lower semicontinuity

Paolo Cubiotti

_{and Jen-Chih Yao}

*_{Correspondence:} [email protected] 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 multifunctionF: [a,b]×Rn_×Rn_→2R_{and a functionh:X→R(with} X⊆Rn), we consider the following implicit two-point problem: find

u∈W2,p([a,b],Rn) such thatuh((ua) =(t))u(∈b) = 0F(t,u_R(tn),.u(t)) a.e. in[a,b],We prove an existence theorem where, for eacht∈[a,b], the multifunctionF(t,·,·) can fail to be lower semicontinuous even at all points (x,y)∈Rn_×Rn_{. The functionhis assumed to be continuous and}

locally nonconstant.

MSC: 34A60; 54C65; 28B20

Keywords: differential inclusions; inductively open functions; discontinuous selections; lower semicontinuity; null-measure projections

1 Introduction

Let [a,b] be a compact interval. Given a multifunctionF: [a,b]×Rn_×Rn_→R_{and a}

functionh:X→R(withX⊆Rn_{), we are interested in the following implicit two-point}

problem: findu∈W,p_{([a,b], R}n_{) such that} ⎧

⎨ ⎩

h(u(t))∈F(t,u(t),u(t)) a.e. in [a,b],

u(a) =u(b) = Rn.

()

As usual,W,p_{([a,b], R}n_{) denotes the space of all functionsu∈C}_{([a,b], R}n_{) such thatu}

is absolutely continuous in [a,b] andu∈Lp_{([a,b], R}n_).

Very recently, in [], the explicit form of problem () has been considered, and some new results have been proved for both the multivalued problem

⎧ ⎨ ⎩

u(t)∈F(t,u(t),u(t)) a.e. in [a,b],

u(a) =u(b) = Rn

()

and the single-valued problem

⎧ ⎨ ⎩

u(t) =f(t,u(t),u(t)) a.e. in [a,b],

u(a) =u(b) = Rn,

()

whereF: [a,b]×Rn_×Rn_→Rn_{is a multifunction andf : [a,b]×R}n_×Rn_→Rn_{is a given}

single-valued map.

The main peculiarity of the results proved in [] resides in the light regularity which is required forF andf. In particular, if the multifunctionFsatisfies the assumptions of Theorem . of [], then for allt∈[a,b] the multifunctionF(t,·,·) can fail to be lower semicontinuous even at all points (x,y)∈Rn×Rn. Similarly, if a functionf satisfies the assumptions of Theorem . of [], then for allt∈[a,b] the functionf(t,·,·) can be dis-continuous even at all points (x,y)∈Rn_×Rn_{. Moreover, the single-valued case allows to}

impose less stringent requirements of measurability onf. For a more detailed discussion and examples, as well as for comparison with the existing literature, we refer to [] and to the references therein.

The aim of this short note is simply to show how the results of [], together with a deep result of B Ricceri on inductively open functions, can be put together in order to solve the implicit vector problem (). More specifically, our aim is to prove the following existence theorem, which is in the same spirit of the results of [] (in the sequel, ifn∈Nandi∈

{, . . . , n},Pi: Rn_{→Rwill denote the projection from R}n_{over theith axis; moreover,}

we shall denote byB(Rn),L([a,b]) andmthe Borel family of Rn, the family of all Lebesgue

measurable subsets of [a,b], and the -dimensional Lebesgue measure on the real line, respectively).

Theorem . Let X⊆Rnbe a nonempty,closed,connected,and locally connected subset ofRn_{,and let h:X→Rbe a continuous function.}

Let F: [a,b]×Rn_×Rn_→R_{be a given multifunction,p∈[, +∞[,α: [a,b]→R}n_and

β∈Lp([a,b], Rn_{)two given functions.Assume that there exist a multifunction G: [a,b]×} Rn_×Rn_→R_{and sets E}

, . . . ,En⊆Rn,with m(Pi(Ei)) = for all i= , . . . , n,such that:

(i) GisL([a,b])⊗B(Rn_)⊗B(Rn_{)-measurable with nonempty closed values;}

(ii) for a.e.t∈[a,b],one has

(x,y)∈Rn×Rn:G(t,·,·)is not lower semicontinuous at(x,y)

∪(x,y)∈Rn×Rn:G(t,x,y)F(t,x,y)⊆

n

i=

Ei;

(iii) intX(h–(t)) =∅,for allt∈intR(h(X));

(iv) for a.e.t∈[a,b]and for all(x,y)∈Rn×Rn,one has

G(t,x,y)⊆h(X),  <αi(t)≤βi(t) for alli= , . . . ,n

and

h–G(t,x,y) ⊆

n

i=

αi(t),βi(t)

(whereαi(t)andβi(t)denote theith components of the vectorsα(t)andβ(t),

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

⎨ ⎩

h(u(t))∈F(t,u(t),u(t)) for a.e. t∈[a,b],

u(a) =u(b) = Rn.

It is immediate to check, by very simple examples, that if a multifunctionFsatisfies the assumptions of Theorem ., then for allt∈[a,b] the multifunctionF(t,·,·) can fail to be lower semicontinuous even at all points (x,y)∈Rn_×Rn_{. In particular, whenFis single}

valued, the functionF(t,·,·) can be discontinuous at all points (x,y)∈Rn×Rn. In the next section, two simple examples are provided in order to illustrate these circumstances. In this connection, Theorem . can be compared with Theorem . of [] (and with the references therein), where the same problem is studied (in the single-valued case, and under the same assumptions onh) by assuming the continuity ofF.

For the definitions and basic facts as regards multifunctions, the reader is referred to [].

2 The proof of Theorem 1.1

Without loss of generality we can assume that assumptions (ii) and (iv) are satisfied for all

t∈[a,b]. Moreover, let

E:=

n

i=

Ei.

Now, observe that by assumption (iii) and Theorem . of [] the functionhis induc-tively open. That is, there exists a setY⊆Xsuch that the function

h|Y:Y→h(X)

is open and h(Y) =h(X). It follows that the multifunction T :h(X)→Y defined by putting, for eachs∈h(X),

T(s) =h–(s)∩Y

is lower semicontinuous inh(X) with nonempty values. To see this, fix any set⊆Y,

withopen in the relative topology ofY. We get

T–₍ ) :=

s∈h(X) :T(s)∩=∅

=s∈h(X) :h–(s)∩Y∩=∅

=s∈h(X) :h–(s)∩=∅

=h().

Since the functionh|Y:Y→h(X) is open, the seth() is open inh(X). It follows that the

setT–₍

) is open inh(X), henceTis lower semicontinuous inh(X), as claimed.

Let: [a,b]×Rn_×Rn_→Y _{be defined by}

(note that is well defined sinceG(t,x,y)⊆h(X) for all (t,x,y)∈[a,b]×Rn_×Rn_{). We}

observe the following facts:

(a) the multifunctionisL([a,b])⊗B(Rn_)⊗B(Rn_{)-weakly measurable; that is, for}

each set⊆Y, withopen in the relative topology ofY, the set

–() :=(t,x,y)∈[a,b]×Rn×Rn:(t,x)∩=∅

belongs toL([a,b])⊗B(Rn_)⊗B(Rn_{)(this follows from Proposition .. of [],}

sinceGisL([a,b])⊗B(Rn_)⊗B(Rn_{)-measurable andTis lower semicontinuous);}

(b) has nonempty values and for allt∈[a,b]one has

(x,y)∈Rn×Rn:(t,·,·)is not lower semicontinuous at(x,y)⊆E.

Indeed, ift∈[a,b] and (x,y)∈(Rn_×Rn_{)\Eare fixed, then the multifunctionG(t,·,·) is}

lower semicontinuous at (x,y), hence (by the lower semicontinuity ofT) the multifunction

(x,y)∈Rn×Rn→(t,x,y) =TG(t,x,y)

is lower semicontinuous at (x,y), as claimed. Now, let: [a,b]×Rn_×Rn_→Rn_{be defined by}

(t,x,y) :=(t,x,y).

It follows by assumption (iv) and by construction that

(t,x,y)⊆

n

i=

αi(t),βi(t)

for all (t,x,y)∈[a,b]×Rn×Rn. ()

By Proposition . and Theorem . of [], the multifunction isL([a,b])⊗B(Rn_)⊗

B(Rn_{)-measurable with nonempty values. Moreover, for allt∈[a,b] one has}

(x,y)∈Rn×Rn:(t,·,·) is not l.s.c. at (x,y)⊆E.

At this point, let us apply Theorem . of [] to the multifunction, choosingk= n,T= [a,b],Xi= R for alli= , . . . , n,S= Rn_{(all the spaces are considered with their Euclidean}

distance and with the usual Lebesgue measure over their Borel family), and, as above,

E:=

n

i=

Ei⊆Rn.

We find that there existQ, . . . ,Qn⊆R, withQi∈B(R) andm(Qi) =  for alli= , . . . , n,

and a functionφ: [a,b]×Rn_×Rn_→Rn_{, such that}

(i) φ(t,x,y)∈(t,x,y)for all(t,x,y)∈[a,b]×Rn_×Rn_;

(ii) for all (x,y)∈(Rn_×Rn_)\[n

i=(P–i (Qi)∪Ei)], the function φ(·,x,y) is L([a,b])

(iii) for a.e.t∈[a,b], one has

(x,y)∈Rn×Rn:φ(t,·,·)is discontinuous at(x,y)⊆

n

i=

Ei∪P–_i (Qi).

Of course, for alli= , . . . , n, one hasm(Pi(Ei∪P–i (Qi))) = .

Now, let us apply Theorem . of [] withf =g=φ, taking into account that for a.e.

t∈[a,b] and for all (x,y)∈Rn_×Rn_{one has}

φ(t,x,y)∈(t,x,y)⊆

n

i=

αi(t),βi(t)

.

We find that there existu∈W,p_{([a,b], R}n_{) and a setU}

∈L([a,b]), withm(U) = , such

that

⎧ ⎨ ⎩

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

u(a) =u(b) = Rn,

and also

u(t),u(t) ∈/

n

i=

Ei∪P–_i (Qi) for allt∈[a,b]\U.

In particular, by assumption (ii) we get

Gt,u(t),u(t) ⊆Ft,u(t),u(t) for allt∈[a,b]\U.

Consequently, taking into account the continuity ofhand the closedness ofX, for allt∈

[a,b]\Uwe get

u(t) =φt,u(t),u(t) ∈t,u(t),u(t) ⊆h–Gt,u(t),u(t) .

In particular, we get

hu(t) ∈Gt,u(t),u(t) ⊆Ft,u(t),u(t)

for allt∈[a,b]\Uand hence the functionusatisfies our conclusion. The proof is now

complete.

Remark We now give two simple examples of application of Theorem .. The first exam-ple concerns with the scalar single-valued case, while the second one deals with the vector multivalued case.

Example . Letn=  and let [a,b] be any compact interval. Let us consider the problem

⎧ ⎨ ⎩

cos(u(t)) =F(t,u(t),u(t)) for a.e.t∈[a,b],

whereF: [a,b]×R×R→Ris the (single-valued) function defined by putting, for each (x,y)∈R_,

F(t,x,y) =

⎧ ⎨ ⎩

 ifx∈Qory∈Q,

 otherwise.

Of course, such a functionF(which does not depend ontexplicitly)is discontinuous at all points(x,y)∈R. We observe that Theorem . can easily be applied by choosing, for anyp∈[, +∞[,

G: [a,b]×R×R→R, G(t,x,y)≡,

E= Q×R,E= R×Q,X= [π, π],α(t)≡π,β(t)≡π,h(t) =cos(t) (restricted to

the interval [π, π]). Indeed, such a functionGisL([a,b])⊗B(R)⊗B(R)-measurable,

P(E) =P(E) = Q, and for allt∈[a,b] one has

(x,y)∈R:G(t,·,·) is discontinuous at (x,y)

∪(x,y)∈R:G(t,x,y)=F(t,x,y)=E∪E.

Moreover, for all (t,x,y)∈[a,b]×R×Rwe have

G(t,x,y) = ∈h(X) and h–G(t,x,y) =h–()⊆[π, π] =α(t),β(t).

Finally, observe that assumption (iii) is satisfied since for allt∈intR(h(X)) = ] – , [ the set

h–_{(t) contains only two points.}

Consequently, by Theorem ., problem () has a solution inu∈W,p_{([a,b], R). Such}

a solutionu also satisfies the conditionu(t)∈[π, π] for a.e.t∈[a,b], hence we get

u∈W,∞_{([a,b], R).}

Moreover, observe that Theorem . can be applied in analogous way also by taking

X= [π+ kπ, π+ kπ] andα(t)≡π+ kπ,β(t)≡π+ kπ, withk∈N. Then, for eachk∈N, we get the existence of a solutionuk∈W,∞_{([a,b], R) such thatu}

k(t)∈[π+

kπ, π+ kπ] for a.e.t∈[a,b]. Hence, problem () has infinitely many solutions. Finally, we note that problem () does not admit the trivial solutionu(t)≡, sinceF(t,x, ) =  for all (t,x)∈[a,b]×Randcos() = .

Example . Letn= , and letψ: R→Rbe defined byψ(v,z) =v+z. In what follows, we denote vectors of Rby the notationsx:= (x,x) andy:= (y,y). LetE⊆Rbe the set

of vectors such that at least one component is rational, that is,

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

.

LetF: [, ]×R_×R_→R_{be defined by}

F(t,x,y) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 if (x,y)∈E,

[ +t,  +t] if (x,y) /∈Eandx< ,

Of course, for any fixedt∈[, ], the multifunctionF(t,·,·)is not lower semicontinuous

at any point (x,y)∈R. By Theorem ., it is easily seen that for any fixedp∈[, +∞[ the problem

⎧ ⎨ ⎩

ψ(u(t))∈F(t,u(t),u(t)) a.e. in [, ],

u() =u() = R

()

has a solutionu∈W,p_{([, ], R}_{). To this aim, chooseX= [, ]×[, ],h:=ψ|}

X,α(t)≡

(, ),β(t)≡(, ),G: [, ]×R×R→R_{, with}

G(t,x,y) =

⎧ ⎨ ⎩

[ +t,  +t] ifx≤,

[ +t,  +t] ifx> 

and

E:= Q×R×R×R, E:= R×Q×R×R,

E:= R×R×Q×R, E:= R×R×R×Q.

Of course, such a multifunctionGisL([, ])⊗B(R_)⊗B(R_{)-measurable with nonempty}

closed values. Moreover, if one fix anyt∈[, ], then the multifunctionG(t,·,·) is lower semicontinuous at each point (x,y)∈R_{, withx}

= . Consequently, for allt∈[, ] we

have

(x,y)∈R×R:G(t,·,·) is not lower semicontinuous at (x,y)

∪(x,y)∈R×R:G(t,x,y)F(t,x,y)=



i=

Ei=E.

Clearly, for alli= , , , , one hasPi(Ei) = Q. Now, observe that for all (t,x,y)∈[, ]×

R×Rwe have

G(t,x,y)⊆[, ]⊆h(X) = [, ]

and

h–G(t,x,y) ⊆X= [, ]×[, ] =α(t),β(t)

×α(t),β(t)

.

Finally, let us show that for all s∈intR(h(X)) = ], [, we haveintX(h–(s)) =∅(though

this fact is quite intuitive - sincehis never locally constant - we shall provide an explicit proof ). To this aim, fixs∈intR(h(X)), and let (v,z)∈h–(s). Therefore, (v,z)∈Xand

h(v,z) =v+z=s. Letbe an open set inXsuch that (v,z)∈. Of course, one can

findr>  such that

Letvbe any point in [v–r,v+r]∩[, ], withv=v. We have

(v,z)∈

and

h(v,z) =v+z=v+z=s.

Hence, the seth–_{(s) has empty interior inX, as claimed. Thus, all the assumptions of}

The-orem . are satisfied. Consequently, problem () has at least a solutionu∈W,p([, ], R). As a matter of fact, sinceu(t)∈Xfor a.e.t∈[, ], we getu∈W,∞_{([, ], R}_{). As before,}

we note that problem () does not admit the trivial solutionu(t)≡R.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics and Computer Science, University of Messina, Viale F. Stagno d’Alcontres 31, Messina,} 98166, Italy.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.

Acknowledgements

The second author was partially supported by the Grant MOST 103-2923-E-039-001-MY3.

Received: 26 February 2015 Accepted: 19 May 2015

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