MICHAEL E. FILIPPAKIS AND NIKOLAOS S. PAPAGEORGIOU
Received 7 October 2003 and in revised form 9 March 2004
We examinep-Lienard systems driven by the vectorp-Laplacian differential operator and having a multivalued nonlinearity. We consider Dirichlet systems. Using a fixed point principle for set-valued maps and a nonuniform nonresonance condition, we establish the existence of solutions.
1. Introduction
In this paper, we use fixed point theory to study the following multivalued p-Lienard system:
x(t)p−2
x(t)+ d dt∇G
x(t)+Ft,x(t),x(t)0 a.e. onT=[0,b],
x(0)=x(b)=0, 1< p <∞.
(1.1)
In the last decade, there have been many papers dealing with second-order multival-ued boundary value problems. We mention the works of Erbe and Krawcewicz [5,6], Frigon [7,8], Halidias and Papageorgiou [9], Kandilakis and Papageorgiou [11], Kyritsi et al. [12], Palmucci and Papalini [17], and Pruszko [19]. In all the above works, with the exception of Kyritsi et al. [12], p=2 (linear differential operator),G=0, andg=0. Moreover, in Frigon [7,8] and Palmucci and Papalini [17], the inclusions are scalar (i.e., N=1). Finally we should mention that recently single-valued p-Lienard systems were studied by Mawhin [14] and Man´asevich and Mawhin [13].
In this work, for problem (1.1), we prove an existence theorem under conditions of nonuniform nonresonance with respect to the first weighted eigenvalue of the negative vector ordinaryp-Laplacian with Dirichlet boundary conditions [15,20]. Our approach is based on the multivalued version of the Leray-Schauder alternative principle due to Bader [1] (seeSection 2).
2. Mathematical background
In this section, we recall some basic definitions and facts from multivalued analysis, the spectral properties of the negative vector p-Laplacian, and the multivalued fixed point principles mentioned in the introduction. For details, we refer to Denkowski et al. [3] and Hu and Papageorgiou [10] (for multivalued analysis), to Denkowski et al. [2] and Zhang [20] (for the spectral properties of thep-Laplacian), and to Bader [1] (for the multivalued fixed point principle; similar results can also be found in O’Regan and Precup [16] and Precup [18]).
Let (Ω,Σ) be a measurable space andX a separable Banach space. We introduce the following notations:
Pf(c)(X)=A⊆X: nonempty, closed (and convex), P(w)k(c)(X)=
A⊆X: nonempty, (weakly) compact (and convex). (2.1)
A multifunction F:Ω→Pf(X) is said to be measurable if, for allx∈X,ω→d(x,
F(ω))=inf [x−y:y∈F(ω)] is measurable. A multifunctionF:Ω→2X\{∅}is said to be “graph measurable” if GrF= {(ω,x)∈Ω×X:x∈F(ω)} ∈Σ×B(X), withB(X) being the Borel σ-field ofX. ForPf(X)-valued multifunctions, measurability implies graph measurability and the converse is true ifΣis complete (i.e.,Σ=Σˆ=the universalσ -field). Letµbe a finite measure on (Ω,Σ), 1≤p≤ ∞, andF:Ω→2X\{∅}. We introduce the setSFp= {f ∈Lp(Ω,X) : f(ω)∈F(ω)µ-a.e.}. This set may be empty. For a graph-measurable multifunction, it is nonempty if and only if inf [y:y∈F(ω)]≤ϕ(ω)µ-a.e. onΩ, withϕ∈Lp(Ω)
+.
Let Y,Z be Hausdorfftopological spaces. A multifunctionG:Y →2Z\{∅}is said to be “upper semicontinuous” (usc for short) if, for allC⊆Zclosed,G−(C)= {y∈Y: G(y)∩C= ∅}is closed or equivalently for allU⊆Z open,G+{y∈Y:G(y)⊆U} is open. IfZis a regular space, then aPf(Z)-valued multifunction which is usc has a closed graph. The converse is true if the multifunctionGis locally compact (i.e., for everyy∈Y, there exists a neighborhoodU of y such thatG(U) is compact inZ). APk(Z)-valued multifunction which is usc maps compact sets to compact sets.
Consider the following weighted nonlinear eigenvalue problem inRN:
−x(t)p−2x(t)=λθ(t)x(t)p−2x(t) a.e. onT=[0,b], x(0)=x(b)=0, 1< p <∞,θ∈L∞(T), {θ >0}1>0,λ∈R.
(2.2)
Here by| · |1 we denote the 1-dimensional Lebesgue measure. The real parameters λ, for which problem (2.3) has a nontrivial solution, are called eigenvalues of the neg-ative vectorp-Laplacian with Dirichlet boundary conditions denoted by (−p,W01,p(T,
RN)), with weight θ∈L∞(T). The corresponding nontrivial solutions are known as eigenfunctions. We know that the eigenvalues of problem (2.3) are the same as those of the corresponding scalar problem [13]. Then from Denkowski et al. [2] and Zhang [20], we know that there exist two sequences{λn(θ)}n≥1and{λ−n(θ)}n≥1such thatλn(θ)>0,
sequence{λn(θ)}n≥1. Also, forλ1(θ)>0, we have the following variational characteriza-tion:
λ1(θ)=inf
xp p b
0θ(t)x(t)
p
dt:x∈W 1,p
0
T,RN,x=0 . (2.3)
The infimum is attained at the normalized principal eigenfunctionu1 (λ1(θ)>0 is simple) andu1(t)=0 a.e. onT. Also,λ1(θ) is strictly monotone with respect toθ, namely, ifθ1(t)≤θ2(t) a.e. onTwith strict inequality on a set of positive measure, thenλ1(θ2)< λ1(θ1) (see (3.2)).
Finally we state the multivalued fixed point principle that we will use in the study of problem (1.1). So letY,Zbe two Banach spaces andC⊆Y,D⊆Ztwo nonempty closed and convex sets. We consider multifunctionsG:C→2C\{∅}which have a decomposi-tionG=K◦N, satisfying the following:K:D→Cis completely continuous, namely, if zn−→w zinD, thenK(zn)→K(z) inCandN:C→Pwkc(D) is usc fromC, furnished with the strong topology intoD, furnished with the weak topology.
Theorem2.1. IfC,D, andG=K◦Nare as above,0∈C, andGis compact (namely,G maps bounded subsets ofCinto relatively compact subsets ofD),thenone of the following alternatives holds:
(a)S= {y∈C:y∈µG(y)for someµ∈(0, 1)}is unbounded or (b)Ghas a fixed point, that is, there existsy∈Csuch thaty∈G(y).
Remark 2.2. Evidently this is a multivalued version of the classical Leray-Schauder al-ternative principle [2, page 206]. In contrast to previous multivalued extensions of the Leray-Schauder alternative principal [4, page 61], Theorem 2.1 does not requireG to have convex values, which is important when dealing with nonlinear problems such as (1.1).
3. Nonuniform nonresonance
In this section, we deal with problem (1.1) using a condition of nonuniform nonreso-nance with respect to the first eigenvalueλ1(θ)>0. Our hypotheses on the multivalued nonlinearityF(t,x,y) are as follows.
(H(F)1)F:T×RN×RN→Pkc(RN) is a multifunction such that (i) for allx,y∈RN,t→F(t,x,y) is graph measurable; (ii) for almost allt∈T, (x,y)→F(t,x,y) is usc;
(iii) for everyM >0, there existsγM∈L1(T)+such that, for almost allt∈T, all x,y ≤M, and allu∈F(t,x,y), we haveu ≤γM(t);
(iv) there existsθ∈L∞(T),θ(t)≥0 a.e. onT, with strict inequality on a set of positive measure and
lim sup x→+∞
sup(u,x)RN :u∈F(t,x,y), y∈RN
xp ≤θ(t) (3.1)
Remark 3.1. Hypothesis (H(F)1)(iv) is the nonuniform nonresonance condition. In the literature [15,20], we encounter the conditionθ(t)≤λ1a.e. onTwith strict inequality on a set of positive measure. Hereλ1>0 is the principal eigenvalue corresponding to the unit weightθ=1 (i.e.,λ1=λ1(1)). Then by virtue of the strict monotonicity property, we haveλ1(λ1)=1< λ1(θ), which is the condition assumed in hypothesis (H(F)1)(iv). (H(G)1)G∈C2(RN,R).
Givenh∈L1(T,RN), we consider the following Dirichlet problem:
−x(t)p−2x(t)=h(t) a.e. onT=[0,b],
x(0)=x(b)=0. (3.2)
From Man´asevich and Mawhin [13, Lemma 4.1], we know that problem (3.3) has a unique solutionK(h)∈C1
0(T,RN)= {x∈C1(TRN) :x(0)=x(b)=0}. So we can define the solution mapK:L1(T,RN)→C1
0(T,RN). Proposition3.2. K:L1(T,RN)→C1
0(T,RN)is completely continuous, that is, ifhn−→w h
inL1(T,RN), thenK(h
n)→K(h)inC01(T,RN).
Proof. Lethn−→w hinL1(T,RN) and setxn=K(hn),n≥1. We have
−xn(t) p−2
xn(t)
=hn(t) a.e. onT,xn(0)=xn(b)=0,n≥1. (3.3)
Taking the inner product withxn(t), integrating overT, and performing integration by parts, we obtain
x
n p
p≤hn1xn∞≤c1xnpfor somec1>0 and alln≥1. (3.4) Here we have used H¨older and Poincare inequalities. It follows that
xn
n≥1⊆Lp
T,RNis bounded (sincep >1) =⇒xn
n≥1⊆W 1,p
0
T,RNis bounded (by the Poincare inequality). (3.5)
So from (3.22) we infer that
x
n p−2
xn
n≥1⊆W1,q
T,RN1 p+
1 q=1
is bounded
=⇒xnp−2xnn≥1⊆CT,RNis relatively compact
(3.6)
(recall thatW1,q(T,RN) is embedded compactly inC(T,RN)). The mapϕ
p:RN→RN, defined byϕp(y)= yp−2y, y∈RN\{∅}, and ϕp(0)=0, is a homeomorphism and so ˆϕp−1:C(T,RN)→C(T,RN), defined by ˆϕp−1(y)(·)=ϕ−p1(y(·)), is continuous and bounded. Thus it follows that
xnn≥1⊆CT,RNis relatively compact =⇒xn
n≥1⊆C
1 0
Therefore we may assume thatxn→xinC01(T,RN). Also{xnp−2xn}n≥1⊆W1,q(T,
RN) is bounded and so we may assume that x
np−2xn w
−→u in W1,q(T,RN) and
x
np−2xn→uinC(T,RN) (becauseW1,q(T,RN) is embedded compactly inC(T,RN)). It follows thatu= xp−2x. Hence if in (3.22) we pass to the limit asn→ ∞, we obtain
−x(t)p−2x(t)=h(t) a.e. onT=[0,b],x(0)=x(b)=0
=⇒K(h)=x. (3.8)
Since every subsequence of{xn}n≥1has a further subsequence which converges toxin C1
0(T,RN), we conclude that the original sequence converges too. This proves the
com-plete continuity ofK.
LetNF:C01(T,RN)→2L 1(T,RN)
be the multivalued Nemitsky operator corresponding toF, that is,
NF(x)=
u∈L1T,RN:u(t)∈Ft,x(t),x(t)a.e. onT. (3.9)
Also letN:C10(T,RN)→2L 1(T,RN)
be defined by
N(x)= d dx∇G
x(·)+NF(x). (3.10)
This multifunction has the following structure.
Proposition3.3. Ifhypotheses(H(F)1)and(H(G)1)hold,thenNhas values inPwkc(L1(T,
RN))and it is usc fromC1
0(T,RN)with the norm topology intoL1(T,RN)with the weak topology.
Proof. ClearlyN has closed, convex values which are uniformly integrable (see hypoth-esis (H(F)1)(iii)). Therefore for everyx∈C10(T,RN),N(x) is convex andw-compact in L1(T,RN). What is not immediately clear is thatN(x)= ∅, since hypotheses (H(F)
1)(i) and (ii) in general do not imply the graph measurability of (t,x,y)→F(t,x,y) [10, page 227]. To see thatN(x)= ∅, we proceed as follows. Let {sn}n≥1,{rn}n≥1 be step func-tions such thatsn→xandrn→xa.e. onTandsn(t) ≤ x(t),rn(t) ≤ x(t)a.e. onT,n≥1. Then by virtue of hypothesis (H(F)1)(i), for everyn≥1, the multifunc-tiont→F(t,sn(t),rn(t)) is measurable and so by the Yankon-von Neumann-Aumann se-lection theorem [10, page 158], we can findun:T→RN a measurable map such that
un(t)∈F(t,sn(t),rn(t)) for allt∈T. Note thatsn∞,rn∞≤M1for someM1>0 and alln≥1. Soun(t) ≤γM1(t) a.e. onT, withγM1∈L
1(T)
+(see hypothesis (H(F)1)(iii)). Thus by virtue of the Dunford-Pettis theorem, we may assume thatun−→w uinL1(T,RN) asn→ ∞. From Hu and Papageorgiou [10, page 694], we have
u(t)∈conv lim sup n→∞ F
t,sn(t),rn(t)⊆Ft,x(t),x(t) a.e. onT, (3.11)
Next we check the upper semicontinuity of N intoL1(T,RN)
w (L1(T,RN)w equals the Banach spaceL1(T,RN) furnished with the weak topology). Because of hypothesis (H(F)1)(iii),Nis locally compact intoL1(T,RN)w(recall that uniformly integrable sets are relatively compact in L1(T,RN)
w). Also on weakly compact subsets of L1(T,RN), the relative weak topology is metrizable. Therefore to check the upper semicontinuity ofN, it suffices to show that GrN is sequentially closed inC1
0(T,RN)×L1(T,RN)w (see Section 2). To this end, let (xn,fn)∈GrN,n≥1, and suppose thatxn→xinC10(T,RN) and fn−→w f inL1(T,RN). For everyn≥1, we have
fn(t)=dtd∇G
xn(t)
+un(t) a.e. onT, withun∈S1F(·,xn(·),xn(·)). (3.12)
Because of hypothesis (H(F)1)(iii), we may assume (at least for a subsequence) that un−→w uinL1(T,RN). As before, from Hu and Papageorgiou [10, page 694], we have
u(t)∈conv lim sup n→∞ F
t,xn(t),xn(t)
⊆Ft,x(t),x(t) a.e. onT (3.13)
(again the last inclusion follows from hypothesis (H(F)1)(ii)). Sou∈S1F(·,x(·),x(·)). Also by virtue of hypothesis (H(G)1), we have
d dt∇G
xn(t)
=Gxn(t)
xn(t)−→Gx(t)x(t)= d dt∇G
x(t), ∀t∈T
=⇒ d dt∇G
xn(·)
−→ d dt∇G
x(·) inL1T,RN
(by the dominated convergence theorem).
(3.14)
So in the limit asn→ ∞, we have
f = d dt∇G
x(·)+u withu∈NF(x)
=⇒(x,f)∈GrN.
(3.15)
This proves the desired upper semicontinuity ofN.
Proposition3.4. There existsξ >0such that, for allx∈W01,p(T,RN),
xp p−
b
0θ(t)
x(t)p
dt≥ξxp
p. (3.16)
Proof. Letη:W01,p(T,RN)→Rbe the functional defined by
η(x)= xp p−
b
0θ(t)
x(t)p
dt. (3.17)
From the variational characterization of λ1(θ)>1, we see that η(x)>0 for all x∈ W01,p(T,RN),x=0. Suppose that the proposition was not true. Then by virtue of thep -homogeneity ofη, we can find{xn}n≥1⊆W
1,p
By the Poincare inequality, the sequence{xn}n≥1⊆W01,p(T,RN) is bounded and so we may assume that
xn−−→w x inW01,p
T,RN, xn−→x inC0
T,RN. (3.18)
Also exploiting the weak lower semicontinuity of the norm functional in a Banach space, we obtain
xp p≤
b
0θ(t)
x(t)p
dt=⇒λ1(θ)≤1, (3.19)
a contradiction to our hypothesis thatλ1(θ)>1.
We introduce the set
S=x∈C1 0
T,RN:x∈λKN(x), 0< λ <1. (3.20)
Proposition3.5. Ifhypotheses(H(F)1)and(H(G)1)hold,thenS⊆C01(T,RN)is bounded. Proof. Letx∈S. We have
1
λx∈KN(x) with 0< λ <1 =⇒λp1−1x(t)
p−2
x(t)+ d dt∇G
x(t)+u(t)=0 a.e. onT, withu∈S1F(·,x(·),x(·))
=⇒x(t)p−2x(t)+λp−1d dt∇G
x(t)+λp−1u(t)=0 a.e. onT.
(3.21)
Taking the inner product withx(t), integrate overT, and perform integration by parts, we obtain
−xp p−λp−1
b
0
∇Gx(t),x(t)RNdt+λp−1 b
0
u(t),x(t)RNdt=0. (3.22)
Remark that
b
0
∇Gx(t),x(t)RNdt= b
0 d dtG
x(t)dt=Gx(b)−Gx(0)=0. (3.23)
By virtue of hypotheses (H(F)1)(iii) and (iv), givenε >0, we can findγε∈L1(T)+such that for almost allt∈T, allx,y∈RN, and allu∈F(t,x,y), we have
(u,x)RN ≤
θ(t) +εxp+γ
ε(t). (3.24)
So we have
b
0
u(t),x(t)RNdt≤ b
0θ(t)
x(t)p
Using (3.24) and (3.27) in (3.23), we obtain
xp p≤
b
0θ(t)
x(t)p
dt+εxpp+γε1
=⇒ξxp p−λε
1x p
p≤γε1
(3.26)
(seeProposition 3.5and recall thatλ1xpp≤ xpp,λ1=λ1(1)).
Chooseε >0 so thatε < λ1ξ. Then from the last inequality, we infer that
{x}
x∈S⊆LpT,RNis bounded
=⇒S⊆W01,p
T,RNis bounded (by Poincare’s inequality) =⇒S⊆C0
T,RNis relatively compact.
(3.27)
Also we have
x(t)p−2 x(t)
≤Gx(t)ᏸx(t)+u(t)a.e. onT
≤M2x(t)+θ(t) +ε+γε(t)a.e. onTfor someM2>0 (see (3.25)) =⇒xp−2x
x∈S⊆W1,1
T,RNis bounded
=⇒xp−2x
x∈S⊆C
T,RNis bounded
sinceW1,1T,RNis embedded continuously but not compactly inCT,RN
=⇒ {x}
x∈S⊆C
T,RNis bounded.
(3.28)
From (3.28) and (3.29), we conclude thatS⊆C01(T,RN) is bounded. Propositions3.2,3.3, and3.5permit the use ofTheorem 2.1. So we obtain the follow-ing existence result for problem (1.1).
Theorem3.6. Ifhypotheses(H(F)1)and(H(G)1)hold,thenproblem (1.1) has a solution x∈C1
0(T,RN)withxp−2x∈W1,1(T,RN).
As an application of this theorem, we consider the following system:
x(t)p−2
x(t)+x(t)p−2Ax(t) +Ft,x(t)e(t) a.e. onT=[0,b],
x(0)=x(b)=0,e∈L1T,RN. (3.29)
Our hypotheses on the data of problem (3.29) are the following. (H(A))Ais anN×Nmatrix such that for allx∈RN we have (Ax,x)
RN ≤θx2with
θ <(πρ/b)p.
Remark 3.7. The quantityπp is defined byπp=2(p−1)1/ p 1
0(1/(1−t)1/ p)dt=2(p− 1)1/ p((π/ p)/sin(π/ p)). Ifp=2, thenπ
2=π. Recall that the eigenvalues of (−p,W
1,p
0 (T,
RN)) areλ
(H(F)1)F:T×RN→Pkc(RN) is a multifunction such that (i) for allx∈RN,t→F(t,x) is graph measurable; (ii) for almost allt∈T,x→F(t,x) is usc;
(iii) for every M >0, there exists γM ∈L1(T)+ such that for almost allt∈T, all x ≤M, and allu∈F(t,x), we haveu ≤γM(t);
(iv) limx→∞((u,x)RN/xp)=0 uniformly for almost allt∈Tand allu∈F(t,x).
InvokingTheorem 3.6, we obtain the following existence result for problem (3.29).
Theorem3.8. Ifhypotheses(H(A))and(H(F)1)hold,thenfor everye∈L1(T,RN), prob-lem (3.29) has a solutionx∈C10(T,RN)withxp−2x∈W1,1(T,RN).
Remark 3.9. Theorem 3.8extends Theorem 7.1 of Man´asevich and Mawhin [13].
Acknowledgments
The authors wish to thank a very knowledgeable referee for pointing out an error in the first version of the paper and for constructive remarks. Michael E. Filippakis was supported by a grant from the National Scholarship Foundation of Greece (IKY).
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Michael E. Filippakis: Department of Mathematics, National Technical University, Zografou Cam-pus, 15780 Athens, Greece
E-mail address:[email protected]
Nikolaos S. Papageorgiou: Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
