R E S E A R C H
Open Access
Structure of the solution set for a partial
differential inclusion
Yi Cheng
1*, Ravi P Agarwal
2,3, Afif Ben Amar
4and Donal O’Regan
5*Correspondence: [email protected]
1Department of Mathematics, Bohai University, Jinzhou, 121013, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compactRδif the perturbation term of the
related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex.
MSC: 34B15; 34B16; 37J40
Keywords: biharmonic problem; differential inclusion; set-valued mapping; path-connected; compactRδ
1 Introduction
In this paper, we examine the following biharmonic problem of the partial differential inclusion:
⎧ ⎪ ⎨ ⎪ ⎩
u∈H(x,u,∇u,u) a.e. in,
u= on∂, ∂u
∂n= on∂.
(.)
Hereis a bounded domain inRNwith a smooth boundary∂, andH:×R×RN×R→
R\ {∅}is a set-valued map. Biharmonic equations with Dirichlet boundary conditions
were studied by Lions-Magenes [, ], Mozolevski-Süli [, ], Amrouche-Fontes [], and Amrouche-Raudin [, ]. Boundary value problems involving partial differential equations with discontinuous nonlinearities which may be reduced to boundary value problems for partial differential inclusions were studied by Carl-Heikkilä [, ] and Chang [, ] (we refer the reader also to the work of Marano [, ]). In [], Xue-Cheng studied periodic problems for a nonlinear evolution inclusion, defined on an evolution triple of spaces, driven by a monotone operator, and with a perturbation term which is multivalued. They established existence theorems for periodic solutions, extremal periodic solutions and a strong relaxation theorem in Banach spaces, which are similar to those in Xue-Yu [] in infinite dimensional spaces. In [], Cheng-Cong-Xue considered the following boundary
value problem:
u∈G(x,u,∇u) a.e. in,
u= on∂, (.)
and they established the existence of solutions for inclusions with convex- and nonconvex-valued perturbations, extremal solutions, and a strong relaxation theorem in a strong so-lution sense. The multivalued term in problem (.) that we consider contains not only the gradient but also a Laplacian item, and we obtain results in a weak solution sense. The Lipschitz condition (H(F)(iv) in []) with respect to the second variableuof the multifunctionGis essential to get a strong relaxation theorem in Cheng-Cong-Xue []. However, we only need a one-sided Lipschitz condition to get this result. Furthermore, the topological structure of the solution set is discussed (this is not considered in [] and []).
Inspired by Cheng-Cong-Xue [], in this paper we prove existence theorems for both ‘convex’ and ‘nonconvex’ cases by using techniques from multivalued analysis and fixed point theory. For related works on this subject, we refer the reader to [, –] and the references therein. Based on the Baire category method, De Blasi and Pianigiani in [] gave an existence result for the following problem:
∇u∈extF(x,u) a.e.x∈,
u(x) =ϕ(x), x∈∂. (.) In this paper, we will also consider the differential inclusion in whichH(x,u,∇u,u) will be replaced by its extreme point setextH(x,u,∇u,u). We show that the resulting problem always has a solution (‘extremal solutions’) and the solution set is dense in the solution set of the convexified version of the problem (‘strong relaxation theorem’). Also we address the structural properties of the solution sets for this type of biharmonic inclu-sion problem. In [], Himmelberg and Van Vleck studied the topological structure of the solution set inRN for the ordinary differential inclusions:
˙
x(t)∈Ft,x(t), x() = ,
and proved that the solution set is anRδ-set. For Cauchy problems the topological struc-ture of the solution set of evolution inclusions was examined by Bothe [], Andres-Pavlackova [], Gabor-Grudzka [], and Chen-Wang-Zhou [] in a Banach space, Bakowska-Gabor [], and O’Regan [] in Fréchet spaces.
We also refer the reader to the works of Papageorgiou-Shahzad [] for the first-order evolution inclusion and Papageorgiou-Yannakakis [] for the second-order evolution in-clusion where the structure of solution sets was discussed. Following their lead, in this paper, we obtain theRδ-structure of the solution set for a biharmonic differential inclu-sion based on the space variablex∈. We prove that the solution set of the biharmonic inclusion problem in the convex-valued case is compactRδinC(), and the solution set is path-connected in the case of a nonconvex-valued orientor field.
existence theorems for the both convex and nonconvex multivalued terms. Here, our re-sults are based on the Leray-Schauder alternative. In Section , a relaxation theorem is established. Finally the properties of the solution set is given in Section .
2 Preliminaries
In this section, we introduce some basic definitions and facts which are essential tools in the later sections; see Hu-Papageorgiou [] for details.
Let RN (N ≥) be the N-dimensional real Euclidean space. Throughout this paper the symboldenotes a nonempty, bounded, open set ofRN, with a smooth boundary
∂. Moreover, from now on, ‘measurable’ simply means Lebesgue measurable. Given two nonnegative constants k,p≥, we denote by Wk,p() the space of all real-valued functions defined on whose weak partial derivatives up to the order k lie inLp(), equipped withWk,p() the usual norm ·
k,p. Ifu∈W,p(), we setu= ni= ∂u
∂xi,
∇u=gradu= (∂u
∂xi) N
i=. For any real numberp> , we denote byqthe dual exponent ofp (and throughout the paper we assumep> ).
LetVbe a Hausdorff topological space and a multifunctionF:→V\ {∅}. We
intro-duce the following notations:
Pk(V) ={D⊂V:Dis a nonempty compact subset ofV},
Pc(V) ={D⊂V:Dis a nonempty and convex subset ofV},
Pfc(V) ={D⊂V:Dis a nonempty, closed. and convex subset ofV},
P(w)kc(V) =
D⊂V:Dis a nonempty, (weakly) compact, and convex subset ofV.
Definition . LetX be a Banach space. A multifunctionF:→Pf(X) is said to be
‘measurable’, if for ally∈X, theR+-valued functionx→d(y,F(x)) =inf{y–v,v∈F(x)} is measurable.
The above definition of measurability is equivalent to saying that
GrF=(x,v)∈×X:v∈F(x)∈×B(X),
withB(X) being the Borel σ-field ofX, is Lebesgueσ-field of, that is,x→F(x) is graph measurable. In general, however, we can only say that measurability implies graph measurability.
Definition . A generalized metric known in the literature as the ‘Hausdorff metric’, is obtained by setting
h(A,B) =maxsup a∈A
d(a,B),sup b∈B
d(b,A)
for allA,B∈Pkc(V).
Definition . LetY,Zbe Hausdorff topological spaces andβ:Y→Z\{∅}.β(·) is called
A multifunction which is both USC and LSC is said to be continuous. From Theorem . and Remark . of [], we note the following result.
Theorem . Let X be a Banach Space with the weak topology, and D⊆X a weakly compact,convex subset of X,Then any weakly sequentially upper semicontinuous map F:D→Pwkc(D)has a fixed point,i.e.,there exists x∈D,such that x∈F(x).
Remark . RecallF:D→Pwkc(D) is weakly sequentially upper semicontinuous if for
any weakly closed setAofD,F–(A) is sequentially closed for the weak topology onD.
We now use Theorem . to obtain the following result.
Lemma . Letbe a nonempty,closed,convex subset of a Banach space X.Suppose F:
→Pc()has a weakly sequentially closed graph and F()is weakly relatively compact.
Then F has a fixed point.
Proof LetD=co(F()). It follows from the Krein-Šmulian theorem thatDis a weakly compact convex set. Note F()⊆D⊆. Notice also thatT =F|D :D→Pc(D). First
we prove thatGrTis weakly compact. Since (X×X)w=Xw×Xw(Xwis the spaceX
en-dowed its weak topology), it follows that D×Dis a weakly compact subset ofX×X. Also,GrT ={(x,y)∈D×X:y∈T(x)} ⊂D×D, so,GrT is weakly relatively compact. Let (x,y)∈D×Dbe weakly adherent toGrT. Then from the Eberlein-Šmulian theorem we can find{({xn},{yn})}n⊆GrTsuch thatyn∈T(xn),xn→xweakly andyn→yweakly
inX. BecauseFhas weakly sequentially closed graph,y∈T(x) and so (x,y)∈GrT. There-fore,GrT is a weakly closed subset ofD×Dand so weakly compact. ConsequentlyT(x) is weakly closed and so a weakly compact subset ofDfor everyx∈D. In view of Theo-rem . it suffices to show thatT is weakly sequentially upper semicontinuous. First we note that GrT is weakly closed and therefore is sequentially weakly closed. Let A⊂D
be a weakly closed set and letxn∈T–(A) withxn→xweakly. Now sinceT(xn)∩A=∅
andT(xn)⊂D, then foryn∈T(xn)∩Awe may assumeyn→yweakly for somey∈A.
Since (xn,yn)∈GrTandGrTis sequentially weakly closed, we havey∈T(x)∩Aand so
x∈T–(A). ThusT–(A) is sequentially weakly closed. Applying Theorem ., we see that
T has a fixed pointx∈D⊂. ThereforeFhas a fixed point. LetXbe a Banach space andLp(,X) be the Banach space of all functionsu:→X
which are Bochner integrable. LetD(Lp(,X)) be the collection of nonempty
decompos-able subsets ofLp(,X). Next is the Bressan-Colombo continuous selection theorem.
Lemma .(see,e.g., []) Let X be a separable metric space and let F:X→D(Lp(,X))
be a LSC multifunction with closed decomposable values.Then F has a continuous selec-tion.
LetXbe a separable Banach Space andC(,X) be the Banach space of all continuous functions. A multifunction F:×X→Pwkc(X) is said to be of Carathéodory type, if
compact subsetsηksuch thatη=
k≥ηk. Letη⊂η, such thatηis dense inηandσ -compact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [].
Lemma .(see,e.g., []) Let the multifunction F:×X→Pwkc(X)be of Carathéodory
type and be integrably bounded.Then there exists a continuous function g:η→Lp(,X)
such that for almost x∈,if u(·)∈η,then g(u)(x)∈extF(x,u(x)),and if u(·)∈η\η,then
g(u)(x)∈extF(x,u(x)).
Definition . A nonempty subsetDofY is said to be contractible if there exist a point
y∈Dand a continuous functionh: [, ]×D→Dsuch thath(,y) =yandh(,y) =y for everyy∈D.
Definition . A subsetDof a metric space is called anRδ-set if there exists a decreasing sequence{Dn}of compact and contractible sets such that
D= ∞
n=
Dn.
Note that a compactRδ setDis nonempty, compact, and connected. However, in con-trast to contractible sets, a compactRδ setDneed not be path-connected. We also need the following approximation result that can be proved from Proposition . of [] with minor modifications to accommodate the presence ofx∈.
Lemma . Let G:×R×RN→P
fc(R)be a multifunction such that (i) ∀(u,s)∈R×RN,x→G(x,u,s)is measurable;
(ii) ∀x∈,(u,s)→G(x,u,s)is USC;
(iii) ∀(x,u,s)∈×R×RN,|G(x,u,s)| ≤ϕ(x)a.e.withϕ(x)∈Lq+().
Then there exists a sequence of multifunctions Gn:×R×RN →Pfc(R),n≥with the
following properties:
(a) For everyx∈,and(u,s)∈R×RNthere existμ
n(u,s) > andεn> such that if
u,u∈Bεn(u) ={y∈R:|u–y| ≤εn},s,s∈Bεn(s),then
h(Gn(x,u,s),Gn(x,u,s))≤μn(x,u,s)ϕ(x)(|u–u|+s–s)a.e. (i.e.,Gn(x,u,s)
is locallyh-Lipschitz with respect to(u,s)).
(b) G(x,u,s)⊆ · · · ⊆Gn(x,u,s)⊆Gn–(x,u,s)⊆ · · ·,|G(x,u,s)| ≤ϕ(x)a.e.n≥,
Gn(x,u,s)→G(x,u,s)asn≥for every(x,u,s)∈×R×RN,and finally there
existsgn:×R×RN →R,measurable inx,locally Lipschitz in(u,s)and
gn(x,u,s)∈Gn(x,u,s)for every(x,u,s)∈×R×RN.Moreover,ifG(x,·,·)is
h-continuous,thenx→Gn(x,u,s)is measurable(hence(x,u,s)→Gn(x,u,s)is
measurable too;see[]).
Theorem .(see,e.g., []) Let X and Y be two normed spaces.If T:X→Y is a compact linear operator and{xn}nis a sequence in X such that xn→x weakly then T(xn)→T(x)
strongly.
3 Existence theorems of solutions
Definition . A functionu∈W,p() is called a solution of problem (.) if
wheref(x)∈H(x,u,∇u,u) and·,·denotes (here and in the sequel) the duality pairing betweenW–,p() andW,q().
Consider the kernelDp() of a biharmonic operator with Dirichlet boundary condi-tions:
Dp() :=
z∈W,p() :z= ,z=∂z
∂n= on∂
, (.)
where n is the unit vector normal to∂pointing outside . ThenWp() :=W,p()/
Dp() is a reflexive Banach space, with norm · W, which is equivalent to the quotient
norm · W,p. From Theorem . of [],Dp() = whenp<N. We prove an existence
theorem for nonconvex problems under the following assumptions:
H(F): H:×R×RN×R→Pk(R)is a multifunction satisfying the following properties: (a) (x,u,s,t)→H(x,u,s,t)is graph measurable.
(b) For almost allx∈,(u,s,t)→H(x,u,s,t)is LSC. (c) For every(u,s,t)∈R×RN×R, there existω
(x)∈Lp(),ω(x)∈L
p
–α(),
ω(x)∈L
p
–β(),ω (x)∈L
p
–γ()such that H(x,u,s,t)=|v|:v∈H(x,u,s,t)
≤ω(x) +ω(x)|u|α+ω(x)sβ+ω(x)|t|γ a.e.x∈,
where≤α,β,γ < .
Theorem . If assumption H(F)holds,then the partial differential inclusion(.)has a
solution u∈Wp().
Proof Following [], we define the biharmonic operatorL:=:W
p()→W–,p(), and thenL:Wp()→W–,p() is a linear mapping. According to Theorem . of [], for
eachf ∈W–,p() the following problem:
u=f(x) a.e. in,
u=∂∂nu= on∂, (.)
has only one solutionu∈Wp(), and
uW≤Cf–,p, (.)
where the constantCdepends only onN,p, and. ThusL:Wp()→W–,p() is one to one and surjective, which impliesL–:W–,p
()→Wp() is well defined. Thus,u=
L–(f), and from (.), we haveL–(f)
W≤Cf–,p. Then we see thatL–:W–,p()→
Wp() is a bounded linear operator, which impliesL–is continuous fromW–,p() to
Wp(). LetK ⊂W–,p() be a bounded subset, and then we find that L–(K) is also bounded inWp(). From the Sobolev embedding theorem, we see that the embedding
Now letSpH:Wp()→W
–,p
()(Lp()⊂W–,p
()) be the multivalued Nemytskii op-erator corresponding toHdefined by
SpH(u) =v∈Lp() :v∈H(x,u,∇u,u) a.e. on.
We show thatSpH(·) has nonempty, closed, decomposable values inLp() and is LSC. The
closedness and decomposability of the values ofSHp(·) are easy to check. For the nonempti-ness, note that ifu∈Wp(), by hypothesisH(F)(i), (x,u,s,t)→H(x,u,s,t) is graph mea-surable, so we can apply Aumann’s selection theorem and get a measurable mapv:→R
and then
g(u)p≤ω(x)p+ω(x)|u|αp+ω(x)|∇u|βp+ω(x)|u|γp
≤ ωp+ωp –α
|u|pα α
/p
+ωp –β
|∇u|pβ β
/p
+ωp –γ
|u|pγ γ
/p
=ωp+ω–αp uαp+ω–βp ∇uβp +ω–γp uγp.
Thus by (.), it follows that
uW≤CLu–,p
≤Cg(u)p
≤Cωp+Cω–αp uαp+Cω–βp ∇uβp+Cω–γp uγp
≤Cωp+CuθW (.)
for some θ ∈(, ). Now since < θ < , we can find a constant M > such that
uW≤M. Thus in view of the continuity of the embeddingWp()→W–,p(), it fol-lows that is bounded inW–,p(). Apply the Leray-Schauder alternative, and we find that there existsu∈Wp() such thatu=L–◦g(u),i.e.,uis a solution of problem (.).
This completes the proof.
The assumptions we need for convex problems are as follows:
H(F): H:×R×RN×R→Pkc(R)is a multifunction satisfying the following properties: (a) (x,u,s,t)→H(x,u,s,t)is graph measurable.
(b) For almost allx∈,(u,s,t)→H(x,u,s,t)is USC; andH(F)(iii) holds.
Theorem . If assumption H(F)holds,then the solution set of the partial differential
in-clusion(.)is nonempty in Wp().Moreover,the solution set is weakly compact in Wp().
Proof In view of the proof of Theorem ., we only need to emphasis those steps where the proofs differ.
In this case the multivalued Nemytskii operatorSpH:Wp()→L p()
has nonempty con-vex values inLp(). The convexity of the values ofSpH(·) are clear. To prove the nonempti-ness, letu∈Wp(), and let{sn}n≥,{rn}n≥,{tn}n≥be three sequences of step functions such that
sn→u, rn→ ∇u, tn→u
and
|sn| ≤ |u|, rn ≤ ∇u, tn ≤ |u| a.e. on.
Then by virtue of hypothesisH(F)(i), for everyn≥,x→H(x,sn,rn,tn) admits a
mea-surable selectorgn(x). From hypothesisH(F)(iii) and (.), we have
sup gn∈H(x,sn,rn,tn)
Now p> , and the set {gn,n≥}is bounded in the reflexive spaceLp(), so it is
rela-tively weakly compact inLp(). Consequently by Eberlein-Šmulian’s theorem and passing
to a subsequence if necessary, we may assume thatgn→g weakly inLp(). Then from
Theorem . in [], we find
g(x)∈conv limgn(x)
n≥
⊆conv limH(x,sn,rn,tn)
⊆H(x,u,∇u,u) a.e. on, (.)
here, the last inclusion being a consequence of hypothesisH(F)(ii). Thusg∈SpH(u), which
shows thatSHp(·) is nonempty.
As in Theorem ., we obtain thea prioribound for the solution set of problem (.). Let
D=x∈Wp() :xW≤M
,
whereM> is a constant (constructed as in Theorem .). Note that the setDis closed, convex, and bounded inWp() (which is reflexive), so it is compact in a weak topology.
Due to the convexity of the values of SHp(·) andL– being a linear map, it follows that
L–◦Sp
H(u) is convex for everyu∈D. Using an argument similar to that in (.), we obtain
L–◦Sp
H(D)⊆D. Next we show thatL–◦S p
H:D→Pc(D) has weakly sequentially closed
graph. Let (un,vn)∈D×D,n≥, be in the graph ofL–◦SpH with (un,vn) converging
weakly to (u,v). Note
vn∈L–◦SHp(un), n≥, (.)
and passing to a subsequence, we note thatun(x)→u(x) a.e. on. Now for everyn∈N
there existsfn∈SpH(un) such thatvn=L–(fn). Again from hypothesisH(F)(iii) and (.), we have
sup
n
fnp≤Cωp+CuθW.
Nowp> , and the set{fn,n≥}is bounded in the reflexive spaceLp(), so it is relatively
weakly compact inLp(). Consequently by the Eberlein-Šmulian theorem and passing to a subsequence if necessary, we may assume thatfn→f weakly inLp(). As above we have
f(x)∈conv limfn(x)
n≥
⊆conv limH(x,un,∇un,un)
⊆H(x,u,∇u,u) a.e. on, (.)
and thusf ∈SpH(u). Letw=L–(f), and notew∈L–◦SpH(u). From Theorem ., we know that L–:W–,p()→Wp() is continuous. Since the embeddingLp()⊆W–,p() is
compact, we get (see Theorem .) thatfn→f inW–,p(). Thus,vn=L–(fn)→w=
L–(f). Therefore,v∈L–◦Sp
H(u), and soL–◦S p
H:D→Pc(D) has weakly sequentially
Invoking Lemma ., there existsu∈Dsuch that u∈L–◦Sp
H(u). Evidently this is a
solution of problem (.). LetSdenote the solution set of problem (.). Again, as in The-orem ., we have
|S|=supuW:u∈S
≤M,
whereM> . SinceSis bounded andWp() is reflexive, it follows thatS w
is weakly com-pact. Letu∈Wp() be weakly adherent toS. SinceS
w
is weakly compact, by Eberlein-Šmulian’s theorem there exists a sequence{un}n⊆Ssuch thatun→uweakly inu∈D⊂
Wp(). SinceL–◦SpHhas weakly sequentially closed graph, it follows thatu∈L–◦S p H(u)
which implies thatu∈S. HenceSw=SandSis weakly closed. ThereforeSis weakly com-pact inWp().
4 Relation theorem of solutions
In this section, we are concerned with the following extremal problem:
⎧ ⎪ ⎨ ⎪ ⎩
u∈extH(x,u,∇u,u) a.e. in,
u= on∂, ∂u
∂n= on∂,
(.)
whereextH(x,u,∇u,u) denotes the extremal point set ofH(x,u,∇u,u). The precise assumptions on the data of problem (.) are the following:
H(F): H:×R×RN ×R→Pkc(R)is multifunction such that for almost allx∈,
(u,s,t)→H(x,u,s,t)ish-continuous, andH(F)(i), (iii) holds, wherep>N ≥. In the following let S denote the solution set of (.), andSe denote the solution set
of (.).
Theorem . If assumption H(F)holds,then the partial differential inclusion(.)has a
solution u∈Wp()∩C().
Proof AsHis replaced byextH, we also obtain ana prioribound forSe. Let
|Se|=sup
uW:u∈Se
≤M
for some constantM> . By virtue of hypothesisH(F)(iii) and (.), there existsa(x)∈
Lp+() such that for everyu∈Se,|H(x,u,∇u,u)| ≤a(x). Let
V=v∈Lp() :v(x)≤a(x) a.e. on,
and we now show thatQ=L–(V) is a compact convex subset inC(). The closedness and convexity ofQare clear. We only need to show its compactness. To this aim, since
Q⊆W,p() and p>N, the embeddingW,p()→C() is compact, and hence Q⊆ C() is compact. From Lemma ., we can find a continuous map:g:Q→Lp() such
u=L–◦g(u), which is a solution of (.). ThereforeS
e=∅inWp()∩C(). This complete
the proof.
To prove our next result, we need the following definition.
Definition .(see []) The multifunctionH:×R×RN ×R→P
k(R) is called
‘one-sided Lipschitz (OSL)’ continuous if there is an integrable functionL:→Rsuch that for everyu,u∈R,x∈,s∈RN,t∈R, andv∈H(x,u,s,t) there existsv∈H(x,u,s,t) such that (v–v)·(u–u)≤L(x)|u–u|.
We also recall Poincaré’s inequality: there exists a constantλ> depending onN,p,
such that
|u|dx≤λ
|u|dx (.)
for everyu∈Wp().
Theorem . If the hypothesis H(F)holds,and
(i) His one-sided Lipschitz(OSL)continuous;
(ii) L(x)≤α<λ,whereLis from Definition.andλfrom(.); then SCe()=S,the closure is taken in C().
Proof Letug∈S, then there existg∈Lp() andg(x)∈H(x,u,∇u,u) a.e. on, such that
⎧ ⎪ ⎨ ⎪ ⎩
u
g=g(x) a.e. in,
ug= on∂,
∂ug
∂n = on∂.
(.)
As earlier, we letV={v∈Lp() :|v(x)| ≤a(x) a.e. on}andQ=L–(V), and thenQis a compact convex subset ofC(). For anyu∈Qandε> , we define the multifunction
Qε(x) =
v∈H(x,u,∇u,u) : (g–v)·(ug–u)≤L(x)|ug–u|+ε
.
Clearly, for everyx∈, Qε(x)=∅, and it is graph measurable. On applying Aumann’s selection theorem, we get a measurable functionv:x→Rsuch thatv(x)∈Qε(x) almost everywhere on. We define the multifunction
Rε(u) =
v∈SHp(·,u(·),∇u(·),u(·)): (g–v)·(ug–u)≤L(x)|ug–u|+ε
.
It is clear thatRε:Q→L
p()
has nonempty and decomposable values. Moreover, from Theorem of [] Rε(·) is LSC. Thereforeu→Rε(u) is LSC whereRε(u) is the closed hull ofRε(u), and has closed and decomposable values. Thus in view of Lemma . there exists a continuous mapfε:Q→Lp() such thatfε(u)∈Rε(u) for allu∈Q. Now invok-ing the Relaxation Theorem in [], we can find a continuous mapgε:Q→Lp() such thatgε(u)(x)∈extH(x,u,∇u,u) almost everywhere on, andfε(u) –gε(u)p≤εfor all
≤
α|un–ug|+
n
dx
→α
|ˆu–ug|dx (.)
asn→ ∞. Hence, from (.), (.), and (.), we find
λ
|ˆu–ug|dx≤α
|ˆu–ug|dx. (.)
Finally since λ >α, it follows thatug=uˆ,i.e.,un→ug andun∈Seforn≥, and hence
S⊆Se. Also,Sis closed inC() (see the proof of Theorem .), and thusS=S C()
e .
5 Properties of the solutions set
In this section, keeping our above hypotheses on the orientor fieldH(x,u,s), we will estab-lish the topological regularity of the solution setSfor the following biharmonic problem:
⎧ ⎪ ⎨ ⎪ ⎩
u∈H(x,u,∇u) a.e. in,
u= on∂, ∂u
∂n= on∂.
(.)
Remark . The results obtained above for the problem (.) hold for the partial differ-ential inclusion (.).
From Theorem ., it is easy to show that for everyf ∈H(x,u,∇u)⊆Lp(), problem
(.) has at least one weak solutionu=L–(f)∈W
p() anduW ≤Cfp, whereCis
a constant independent ofuandf. LetLp()
wdenote the Lebesgue-Bochner space
fur-nished with the weak topology. From the proof of Theorem ., it follows that the map
P=L–:Lp()
w→Wp() is sequentially continuous.
Remark . Since the embedding of Wp()→C() is compact when p≥ N, P :
Lp()
w→C() is completely continuous.
Next we introduce the following hypothesis:
H(F): H:×R×RN→Pkc(R)is a multifunction satisfying the following properties: (i) (x,s,t)→H(x,s,t)is graph measurable.
(ii) For almost allx∈,(s,t)→H(x,s,t)is USC. (iii) For every(s,t)∈R×RN, there existξ
(x)∈Lp(),ξ(x)∈L
p
–γ(),
ξ(x)∈L
p
–η(), such that
H(x,s,t)=sup|v|:v∈H(x,s,t)≤ξ(x) +ξ(x)|u|γ+ξ(x)|s|η
for a.e.x∈, where≤γ,η< .
Theorem . If hypothesis H(F)holds and p≥N,then the solution set S of problem(.)
Proof Similar reasoning as in the proof of Theorem ., for almost allx∈, allu∈S, and allv∈H(x,u,∇u), we may assume that|v| ≤a(x) wherea(x)∈Lp+(). From Lemma ., we obtain a sequence of multifunctionsHn:×R×RN →Pfc(R). For everyn≥, consider
the following biharmonic problem of the partial differential inclusion:
⎧ ⎪ ⎨ ⎪ ⎩
u∈H
n(x,u,∇u) a.e. in,
u= on∂, ∂u
∂n= on∂.
(.)
Now from the proof of Theorem . and Remark ., we find that problem (.) has a nonempty solution setSn⊆Wp() which is compact inC().
First, we prove that the setSnis contractible. Letgn(x,u,∇u) be locally Lipschitz with
respect to u, measurable selector of Hn(x,u,∇u) postulated from Lemma .. Let d= dist(,∂) wheredist(,∂) denotes the distance from to ∂ and ⊆, then
d∈I:= [,τ] whereτ=maxx∈{dist(x,∂)}. Let
δ=
x∈|dist(x,∂)≥δ.
Forδ∈[,τ], givenz∈Sn, there exists agn∈Hn(x,u,∇u) such thatz=P(gn). For each
u∈Sn, letzu(δ)(x)∈Wp() be the unique solution of
⎧ ⎪ ⎨ ⎪ ⎩
z(x) =gn(x,z(x),∇z) a.e. inδ,
z(x) =u(x) on∂δ, ∂z
∂n=
∂u
∂n on∂δ.
(.)
Defineμ:I×Sn→Snby
μ(δ,u)(x) =
u(x) forx∈\δ,
zu(δ)(x) forx∈δ.
(.)
Evidently μ(,u) =z, andμ(τ,u) =ufor every u∈Sn. In the following we show that μ(δ,u)∈Sn(u) for each (δ,u)∈I×Sn. Note that for each u∈Sn(u) there exists fn∈
Hn(x,u,∇u), such thatu=P(fn). Setgn(x) =fnχ[,δ](x) +gnχ(δ,τ](x) for eachx∈whereχ is the characteristic function. It is easy to see thatgn(x)∈Hn(x,u,∇u). ThusP(gn) =u(x)
for all\δ,P(gn) =zu(x) for allx∈δ, which implies thatP(gn) =μ(δ,u)(x), and thus
μ(δ,u)(x)∈Sn. In order to prove thatSnis contractible inC(), we only need to show
thatμ(δ,u) is continuous inI×C(). To this aim, we let (δm,um)→(δ,u) inI×Sn, and
consider the following two distinct cases.
Case :δm≥δ for every m≥,and thenδm ⊆δ. Letvm(x) =μ(δm,um)(x),x∈.
Evidently, vm(x)∈Sn, m≥, and hence passing to a subsequence if necessary, we may
assume thatvm→vinC() asm→ ∞. Clearly,v(x) =u(x) for a.e.\δ. Also letw∈
Wp() be the unique solution of
w=g
n(t,v,∇v) a.e. onδ,
w(x) =u(x), ∂w
∂n=
∂u
∂n on∂δ.
LetN≥,m≥N, then formlarge enoughvm(·) satisfiesvm=gn(x,vm,∇vm) a.e. on
Remark . If hypothesis H(F) holds, then for everyx∈,S(x) ={u(x)|u∈S} (the reachable set atx∈) is compact and connected inR.
WhenH(x,u,∇u) has nonconvex values, an analogous result for the topological struc-ture can be obtained if we modify our hypothesis on the continuity ofH(x,u,∇u). There-fore, in this case we can prove that the solution set is path-connected. In our next result we prove that the solution set of (.) is path-connected under the following assumption:
H(F): H:×R×RN→Pk(R)is a multifunction such that for almost allx∈,(u,s)→
H(x,u,s)is LSC, andH(F)(i), (iii) holds.
Theorem . If hypothesis H(F)holds and there exists a function b(x)∈L∞+(),such that
hH(x,u,∇u),H(x,v,∇v)≤b(x)u(x) –v(x) a.e. on,
where Cb∞< with C in(.),then S⊆C()is nonempty and path-connected.
Proof As in Theorem .,a prioriestimates for the problem (.) can be obtained easily. Thus we may assume that|H(x,u,∇u)| ≤a(x) a.e.witha(x)∈Lp+() for allu∈S. Let
Oα=
g∈Lp() :g(x)≤a(x) a.e. on
and consider the multifunctionN:Oα→Pf(Lp()) defined byN(g) =SpH(·,P(g),∇P(g))(here
P(g) is the solution map as before). Letf,g∈Oαand letv∈N(g). Letε> and define
Dε(x) =
u∈Hx,P(f),∇P(f):u(x) –v(x)≤dv(x),Hx,P(f),∇P(f)+ε.
Setχ(x,u) =d(v(x),H(x,P(f),∇P(f))) –|v(x) –u(x)|+ε. Clearly, for everyx∈,Dε(x)=∅ and GrDε ={(x,u)∈ GrH(·,P(f),∇P(f)) : ≤ χ(x,u)}. By H(F), (x,u,s)→H(x,u,s) is measurable, thus x → H(x,P(f),∇P(f)) is measurable. Since (x,u) → χ(x,u) is a Carathéodory function, it is jointly measurable. Hence,GrDε∈×B(R). Applying Au-mann’s selection theorem we findu:→Rmeasurable andu(x)∈Dε(x) a.e. on. Thus it follows that
dv,N(f)≤ v–up
=
|v–u|pdx
p
≤
dv(x),Hx,P(f),∇P(f)+εpdx
p
≤
hHx,P(f),∇P(f),Hx,P(g),∇P(g)pdx
p
+||pε
≤
b(x)P(f)(x) –P(g)(x)pdx
p
+||pε
≤ b∞Cf(x) –g(x)p+ε||
Asε→, we findd(v,N(f))≤ b∞Cf(x) –g(x)p. Exchanging the roles off andgwe
also haved(u,N(g))≤ b∞Cf(x) –g(x)p. Thush(N(f),N(g))≤ b∞Cf(x) –g(x)p
withb∞C< . Let:={g∈Oα:g∈N(g)}. By Nadler’s fixed point theorem [] we find that=∅and from [] we know that is an absolute retract inLp(). Since an absolute
retract is path-connected,is path-connected. ThusP() is path-connected inC(). SinceP() =S, we see thatSis nonempty and path-connected inC(). This completes
the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript. Each of the authors YC, RPA, ABA, and DO contributed to each part of this work equally and read and approved the final version of the manuscript.
Author details
1Department of Mathematics, Bohai University, Jinzhou, 121013, P.R. China.2Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA.3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.4Department of Mathematics, Faculty of Sciences, Sfax University, LA 1171, Sfax, 3000, Tunisia.5School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.
Acknowledgements
This work is partially supported by National Natural Science Foundation of China (No. 11401042).
Received: 17 November 2015 Accepted: 8 December 2015
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