# Periodic Solutions of Second-Order Differential Inclusions Systems withp(t)-Laplacian

Authors: Liang Zhang; Peng Zhang
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

The existence of periodic solutions for nonautonomous second-order differential inclusion systems withp(t)-Laplacian is considered. We get some existence results of periodic solutions for system, (d/dt)(|u˙(t)|p(t)−2u˙(t))∈∂F(t,u(t)) a.e. t∈[0,T], u(0)−u(T)=u˙(0)−u˙(T)=0, by using nonsmooth critical point theory. Our results generalize and improve some theorems in the literature.


---

## Body

## 1. Introduction

Consider the second-order system withp(t)-Laplacian
(1.1)ddt(|u˙(t)|p(t)-2u˙(t))∈∂F(t,u(t))a.e.t∈[0,T],u(0)-u(T)=u˙(0)-u˙(T)=0,
where T>0, ∂ denotes the Clarke subdifferential, and p(t)∈C([0,T],ℝ+) satisfies the following assumption:(A)p(0)=p(T) and p-:=min0≤t≤Tp(t)>1, where q+>1 which satisfies 1/p-+1/q+=1.Moreover, we suppose thatF:[0,T]×ℝN→ℝ satisfies the following assumption:(A’)F(t,x) is measurable in t for every x∈ℝN and locally Lipschitz in x for a.e. t∈[0,T], F(t,0)∈L1(0,T) and there exist positive constants C, C0, and α∈[0,∞) such that
(1.2)ζ∈∂F(t,x)⇒|ζ|≤C|x|α+C0,for a.e.t∈[0,T] and all x∈ℝN.IfF∈C1 and p(t)≡p>1, system (1.1) reduces to the ordinary p-Laplacian system:
(1.3)ddt(|u˙(t)|p-2u˙(t))=∇F(t,u(t))a.e.t∈[0,T],u(0)-u(T)=u˙(0)-u˙(T)=0.Especially, whenp=2, then system (1.3) reduces to
(1.4)u¨(t)=∇F(t,u(t))a.e.t∈[0,T],u(0)-u(T)=u˙(0)-u˙(T)=0.The corresponding functionalψ on HT1 given by
(1.5)ψ(u):=12∫0T‍|u˙(t)|2dt+∫0T‍F(t,u(t))dt
is continuously differentiable and weakly lower semicontinuous on HT1 (see [1]), where
(1.6)HT1≔{u:[0,T]→ℝℕ,uisabsolutelycontinuous,u(0)=u(T),u˙∈L2([0,T];ℝℕ)}
is a Hilbert space with a norm defined by
(1.7)∥u∥HT1:=(∫0T‍|u(t)|2dt+∫0T|u˙(t)|2dt‍)1/2,
for u∈HT1.Since Mawhin and Willem studied the periodic solutions of Hamilton system and obtained a series of results (see [2]). Considerable attention has been paid to the existence of periodic and subharmonic solutions for system (1.3) and (1.4) by the use of critical point theory in variational methods. Many solvability conditions are given, such as Ambrosetti-Rabinowitz conditions, coercivity condition, the convexity condition, the boundedness condition, the subadditive condition, the sublinear condition, and the periodicity condition, see [3–7] and the references therein.The classical critical point theory was developed in the sixties and seventies forC1 functionals. The needs of specific applications (such as nonsmooth mechanics and nonsmooth gradient systems) and the impressive progress in nonsmooth analysis and multivalued analysis led to extensions of the critical point theory to nondifferentiable functions, in particular locally Lipschitz functions. The nonsmooth critical point theory for locally Lipschitz functions started with the work of Chang (see [8]). He was able to construct a substitute for the pseudogradient vector field of the smooth theory and use it to obtain nonsmooth versions of the Mountain Pass Theorem of Ambrosetti and Rabinowitz (see [9]) and of the Saddle Point Theorem of Rabinowitz (see [10]). Chang used his theory to study semilinear elliptic boundary value problems with a discontinuous nonlinearity. Later, in 2000, Kourogenis and Papageorgiou (see [11]) obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. Subsequently, many authors also studied the nonsmooth critical point theory (see [2, 11–17]), then the nonsmooth critical point theory is also widely used to deal with the nonlinear boundary value problems (see [11, 14, 15, 17–26]). A good survey for nonsmooth critical point theory and nonlinear boundary value problems is the book of  Gasinski and Papageorgiou [22].The operator(d/dt)(|u˙(t)|p(t)-2u˙(t)) is said to be p(t)-Laplacian and becomes p-Laplacian when p(t)≡p (a constant). The p(t)-Laplacian possesses more complicated nonlinearity than p-Laplacian, for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field. Another field of application of equations with variable exponent growth conditions is image processing (see [12, 27]). The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [12, 28–33] for an overview on this subject.In 2003, X. L. Fan and X. Fan (see [34]) studied the ordinary p(t)-Laplacian system and introduced a generalized Orlicz-Sobolev space WT1,p(t), which is different from the usual space WT1,p, then Wang and Yuan (see [35]) obtained the existence and mulplicity of periodic solutions for ordinary p(t)-Laplacian system (1.1) with a smooth potential F under the generalized Ambrosetti-Rabinowitz conditions. In recent years, there are some papers discussing existence and multiplicity of periodic solutions and subharmonic solutions for problem (1.3) and (1.4) when the potential F is just locally Lipschitz in the second variable x not continuously differentiable. Some results were obtained based on various hypotheses on the potential F. Here we only mention [7, 20, 21, 25, 26], and it should be noted that the abstract result of Clarke (see [1, Theorem 2.7.5]) plays an important role in the establishment of corresponding variational structure. However, to the best of our knowledge, few papers investigated the existence of solutions for problem (1.1), because the main difficulty is the abstract result of Clarke cannot be applied to system (1.1) directly. So we have to find a new approach to solve this problem, and our main idea of the new approach comes from the inspiration of the Theorem 2.7.3 and Theorem 2.7.5 in [1].The main purpose of this paper is to establish the corresponding variational structure for system (1.1), and we get some existence results of periodic solutions for system (1.1) by using nonsmooth critical point theory. Our results are extensions of the results presented in [4, 26], and our results are new even in the case F∈C1 for system (1.1). The paper is divided into four sections. Basic definitions and preliminary results are collected in Section 2. We give the main results and proofs in Section 3. In Section 4, three examples are presented to illustrate our results.In this paper, we denote byp+:=max0≤t≤Tp(t)>1 throughout this paper, and we use 〈·,·〉 and |·| to denote the usual inner product and norm in ℝN, respectively.


## 2. Basic Definitions and Preliminary Results

In this section, we recall some known results in nonsmooth critical point theory, and the properties of spaceWT1,p(t) are listed for the convenience of readers.Definition 2.1 (see [35]).
Letp(t) satisfy the condition (A), define
(2.1)Lp(t)([0,T],ℝN)={u∈L1([0,T],ℝN);∫0T‍|u|p(t)dt<∞},
with the norm
(2.2)|u|p(t):=inf{λ>0;∫0T‍|uλ|p(t)dt≤1}.
Foru∈Lloc1([0,T],ℝN), let u' denote the weak derivative of u, if u'∈Lloc1([0,T],ℝN) and satisfies
(2.3)∫0T‍u′ϕdt=-∫0T‍uϕ′dt,∀ϕ∈C0∞([0,T],ℝN).
Define(2.4)W1,p(t)([0,T],ℝN)={u∈Lp(t)([0,T],ℝN);u′∈Lp(t)([0,T],ℝN)},
with the norm ∥u∥W1,p(t):=|u|p(t)+|u'|p(t).Remark 2.2.
Ifp(t)=p, where p∈[1,∞) is a constant, by the definition of |u|p(t), it is easy to get |u|p=(∫0T‍|u(t)|pdt)1/p, which is the same with the usual norm in space Lp.The spaceLp(t) is a generalized Lebesgue space, and the space W1,p(t) is a generalized Sobolev space. Because most of the following Lemmas have appeared in [1, 7, 24, 34], we omitted their proofs.Lemma 2.3 (see [34]).
L
p
(
t
) and W1,p(t) are both Banach spaces with the norms defined above, when p->1, they are reflexive.Lemma 2.4 (see [34]).
The spaceLp(t) is a separable, uniform convex Banach space, its conjugate space is Lq(t), for any u∈Lp(t) and v∈Lq(t), we have
(2.5)|∫0T‍uvdt|≤2|u|p(t)|v|q(t),
where 1/p(t)+1/q(t)=1.Lemma 2.5 (see [35]).
If we denoteρ(u)=∫0T‍|u|p(t)dt, forallu∈Lp(t), then(i)
|u|p(t)<1(=1;>1)⇔ρ(u)<1(=1;>1)(ii)
|u|p(t)>1⇒|u|p(t)p-≤ρ(u)≤|u|p(t)p+, |u|p(t)<1⇒|u|p(t)p+≤ρ(u)≤|u|p(t)p-(iii)
|u|p(t)→0⇔ρ(u)→0; |u|p(t)→∞⇔ρ(u)→∞(iv)
Foru∈Lp(t) and u≠0, |u|p(t)=λ⇔ρ(u/λ)=1.Definition 2.6 (see [2]).
(2.6)
C
T
∞
=
C
T
∞
(
ℝ
,
ℝ
N
)
:
=
{
u
∈
C
∞
(
ℝ
,
ℝ
N
)
;
u
is
T
-periodic
}
,
with the norm ∥u∥∞:=maxt∈[0,T]|u(t)|.For a constantp∈[1,∞), using another conception of weak derivative which is called T-weak derivative, Mawhin and Willem gave the definition of the space WT1,p by the following way.Definition 2.7 (see [2]).
Letu∈L1([0,T],ℝN) and v∈L1([0,T],ℝN), if
(2.7)∫0T‍vϕdt=-∫0T‍uϕ′dt,∀ϕ∈CT∞,
then v is called a T-weak derivative of u and is denoted by u˙.Definition 2.8 (see [2]).
Define(2.8)WT1,p([0,T],ℝN)={u∈Lp([0,T],ℝN);u˙∈Lp([0,T],ℝN)},
with the norm ∥u∥WT1,p=(|u|pp+|u˙|pp)1/p.Definition 2.9 (see [34]).
Define(2.9)WT1,p(t)([0,T],ℝN)={u∈Lp(t)([0,T],ℝN);u˙∈Lp(t)([0,T],ℝN)},
and let HT1,p(t)([0,T],ℝN) be the closure of CT∞ in W1,p(t)([0,T],ℝN).Remark 2.10.
From Definition2.8, if u∈WT1,p(t)([0,T],ℝN), it is easy to conclude that u∈WT1,p-([0,T],ℝN).Lemma 2.11 (see [34]).
(i)
CT∞([0,T],ℝN) is dense in WT1,p(t)([0,T],ℝN),(ii)
WT1,p(t)([0,T],ℝN)=HT1,p(t)([0,T],ℝN)≔{u∈W1,p(t)([0,T],ℝN);u(0)=u(T)},(iii)
Ifu∈HT1,1, then the derivative u' is also the T-weak derivative u˙, that is, u'=u˙.Remark 2.12.
In the following paper, we use∥u∥ instead of ∥u∥WT1,p(t) for convenience without clear indications.Lemma 2.13 (see [24]).
Assume thatu∈WT1,1, then(i)
∫0T‍u˙dt=0,(ii)
u has its continuous representation, which is still denoted by u(t)=∫0t‍u˙(s)ds+u(0), u(0)=u(T),(iii)
u˙ is the classical derivative of u, if u˙∈C([0,T],ℝN).Since every closed linear subspace of a reflexive Banach space is also reflexive, we haveLemma 2.14 (see [34]).
H
T
1
,
p
(
t
)
(
[
0
,
T
]
,
ℝ
N
) is a reflexive Banach space if p->1.Obviously, there are continuous embeddingsLp(t)↪Lp-, W1,p(t)↪W1,p-, and HT1,p(t)↪HT1,p-. By the classical Sobolev embedding theorem we obtain.Lemma 2.15 (see [34]).
There is a continuous embedding(2.10)W1,p(t)(orHT1,p(t))↪C([0,T],ℝN),
when p->1, the embedding is compact.In order to establish the variational structure for system (1.1), it is necessary to construct some appropriate function spaces. The Cartesian product space W is defined by
(2.11)W=Lp(t)([0,T],ℝN)×Lp(t)([0,T],ℝN)
and is also a reflexive and separable Banach space with respect to the norm
(2.12)∥v∥W=|v1|p(t)+|v2|p(t),
where v=(v1,v2)∈W.Lemma 2.16.
Define the operatorA: WT1,p(t)([0,T],ℝN)→W as follows:
(2.13)Au=(u,u˙),∀u∈WT1,p(t),
then Wp(t):={(u,u˙):∀u∈WT1,p(t)} is also a reflexive and separable Banach space with respect to the norm defined in (2.12).Proof.
Let(un,u˙n) be a Cauchy sequence in Wp(t), then there exists (u,v) in W such that (un,u˙n) converge to (u,v) in W. We have
(2.14)∫0T‍u˙nφdt=-∫0T‍unφ'dt,∀φ∈CT∞,
by Definition 2.7, then by Lemma 2.4, we conclude
(2.15)∫0T‍vφdt=-∫0T‍uφ′dt,∀φ∈CT∞,
as n→∞ in (2.14). In view of (2.15), v is the T-weak derivative of u, that is, (u,v) is also in Wp(t), so Wp(t) is a complete subspace of W, which implies Wp(t) is also a reflexive and separable Banach space.Remark 2.17.
We use∥(u,u˙)∥Wp(t) to denote the norm in Wp(t) defined by (2.12).Definition 2.18.
LetL∞([0,T],ℝN×ℝN) denote the space of essentially bounded measurable functions from [0,T] into ℝN×ℝN under the norm
(2.16)∥(u,v)∥L∞:=esssup{|u(t)|+|v(t)|:t∈[0,T]},
it is obvious that L∞([0,T],ℝN×ℝN) is a Banach space under the norm defined above.Remark 2.19.
We useW∞ and ∥(u,v)∥W∞ to denote L∞([0,T],ℝN×ℝN) and ∥(u,v)∥L∞, respectively.Lemma 2.20.
W
p
(
t
)
∩
W
∞ is a closed subspace of W∞.Proof.
Let(un,u˙n) be a Cauchy sequence in Wp(t)∩W∞ with respect to the norm defined in (2.16). Then there exists (u,v) in W∞ such that (un,u˙n) converge to (u,v) in W∞. By Definition 2.7, we have
(2.17)∫0T‍u˙nφdt=-∫0T‍unφ′dt,∀φ∈CT∞,
we conclude that
(2.18)∫0T‍vφdt=-∫0T‍uφ′dt,∀φ∈CT∞,
by Lemma 2.4 as n→∞ in (2.17). In view of (2.18), v is the T-weak derivative of u, that is, (u,v) is also in Wp(t)∩W∞, so Wp(t)∩W∞ is a complete subspace of W∞, which implies Wp(t)∩W∞ is a closed subspace of W∞.Lemma 2.21.
Supposef is a bounded linear functional on Wp(t), if restricted to the space Wp(t)∩W∞, denoted by f', that is,
(2.19)f′(u,u˙)=f(u,u˙),∀(u,u˙)∈Wp(t)∩W∞,
then f' is a bounded linear functional on Wp(t)∩W∞.Proof.
It is obvious thatf' is a linear functional on Wp(t)∩W∞, so we only to show the f' is continuous on Wp(t)∩W∞.
Let|u|p(t)=λ and |u˙|p(t)=μ, that is,
(2.20)∫0T‍|u(t)λ|p(t)dt=1,∫0T‍|u˙(t)μ|p(t)dt=1,
where (u,u˙)∈Wp(t)∩W∞ and (u,u˙)≠(0,0), then
(2.21)∫0T‍|u(t)C1∥(u,u˙)∥W∞|p(t)dt≤1,∫0T‍|u˙(t)C1∥(u,u˙)∥W∞|p(t)dt≤1,
by Definition 2.1 and (2.16), where C1:=|1|p(t). Then we conclude
(2.22)|u|p(t)≤C1∥(u,u˙)∥W∞,|u˙|p(t)≤C1∥(u,u˙)∥W∞,
by (2.21) and Definition 2.1.
Furthermore, the norm off in Wp(t) is
(2.23)∥f∥=supWp(t)∖0|f(u,u˙)|∥(u,u˙)∥Wp(t)=supWp(t)∖0|f(u,u˙)|(|u|p(t)+|u˙|p(t)),
therefore, combining (2.22) and (2.23), we get the bound
(2.24)∥f'∥≤2C1∥f∥,
and this completes the proof.Lemma 2.22.
The spaceWT1,p(t)=W~T1,p(t)⊕ℝN, where
(2.25)W~T1,p(t):={u∈WT1,p(t);∫0T‍u(t)dt=0},
there exists C2>0, if u∈W~T1,p(t), such that
(2.26)∥u∥∞≤2C2(∫0T|u˙(t)|p(t)‍dt)1/p-+2C2T1/p-.Proof.
LetE={t∈[0,T]∣|u˙(t)|≥1}, from Remark 2.10, u∈WT1,p-, from the inequality in classical Sobolev space, there exists a positive constant C2>0, such that
(2.27)∥u∥∞≤C2(∫0T|u˙(t)|p-dt)1/p-=C2(∫E‍|u˙(t)|p-dt+∫[0,T]∖E‍|u˙(t)|p-dt)1/p-≤C2(∫E‍|u˙(t)|p(t)dt+meas[0,T]∖E)1/p-≤C2(∫0T|u˙(t)|p(t)dt+T)1/p-≤2C2(∫0T|u˙(t)|p(t)dt)1/p-+2C2T1/p-,
and this completes the proof.Lemma 2.23 (see [34]).
Each of the following two norms is equivalent to the norm inWT1,p(t):(i)
|u˙|p(t)+|u|q, 1≤q≤∞,(ii)
|u˙|p(t)+|u-|, where u-=(1/T)∫0T‍u(t)dt.Proposition 2.24.
In spaceWT1,p(t), ∥u∥→∞⇒∫0T‍|u˙|p(t)dt+|u-|→∞.Proof.
From Lemma2.23, there exists a positive constant C3, such that
(2.28)∥u∥≤C3(|u˙|p(t)+|u-|),
if |u˙|p(t)<1, it is easy to get
(2.29)|u˙|p(t)<∫0T‍|u˙|p(t)dt+1.
When |u˙|p(t)≥1, we conclude that
(2.30)|u˙|p(t)≤(∫0T|u˙|p(t)‍dt)1/p-,
by Lemma 2.5, it follows (2.29) and (2.30) that
(2.31)∥u∥≤C3((∫0T‍|u˙|p(t)dt)1/p-+1+|u-|),
which implies that
(2.32)∥u∥→∞⇒∫0T‍|u˙|p(t)dt+|u-|→∞.
The proof is completed.Lemma 2.25 (see [34]).
Ifu,un∈Lp(t)(n=1,2,…), then the following statements are equivalent to each other(i)
limn→∞|un-u|p(t)=0,(ii)
limn→∞ρ(un-u)=0,(iii)
un→u in measure in [0,T] and limn→∞ρ(un)=ρ(u).Definition 2.26 (see [1]).
Letf be Lipschitz near a given point x in a Banach space X, and let v be any other vector in X. The generalized directional derivative of f at x in the direction v, denoted by f0(x;v), is defined as follows:
(2.33)f0(x;v)=limsupy→x,λ↓0f(y+λυ)-f(y)λ,
where y is also a vector in X and λ is a positive scalar, and we denote by
(2.34)∂f(x):={x*∈X*:f0(x;v)≥〈x*,v〉,∀vinX},
the generalized gradient of f at x (the Clarke subdifferential).Lemma 2.27 (see [1]).
Letx and y be points in a Banach space X, and suppose that f is Lipschitz on open set containing the line segment [x,y]. Then there exists a point u in (x,y) such that
(2.35)f(y)-f(x)∈〈∂f(u),y-x〉.Definition 2.28 (see [8]).
A pointu∈X is said to be a critical point of a locally Lipschitz f if θ∈∂f(u), namely, f0(u;v)≥0 for all every v∈X. A real number c is called a critical value of f if there is a critical point u∈X such that f(u)=c.Definition 2.29 (see [8]).
Iff is a locally Lipschitz function, we say that f satisfies the Palais-Smale condition if each sequence (xn) in X such that (f(xn)) is bounded and limn→∞λ(xn)=0 has a convergent subsequence. We define λ(x)=minx*∈∂f(x)∥x*∥, where the minimum exists from the fact that ∂f(x) is aw*-weakly compact convex subset.Lemma 2.30 (see [8]).
LetX be a real Banach space, and let f be a locally Lipschitz function defined on X satisfying the (PS) condition. Suppose X=X1⊕X2 with a finite dimensional subspace X1, and there exist constants b1<b2 and a bounded neighborhood N of θ in X1 such that
(2.36)f|X2≥b2,f|∂N≤b1.
Then f has a critical point.Lemma 2.31.
The functional given by(2.37)φ(u)=∫0T‍1p(t)|u˙(t)|p(t)dt+∫0T‍F(t,u(t))dt
is weakly lower semicontinuous on WT1,p(t).Proof.
We divideφ into two parts, φ(u):=J(u)+H(u), where
(2.38)J(u):=∫0T‍1p(t)|u˙(t)|p(t)dt,H(u):=∫0T‍F(t,u(t))dt,
it is obvious that J is convex and continuous by Lemma 2.25, then J is weakly lower semicontinuous by Theorem 1.2  in [2], and H is weakly continuous, that is, φ is the sum of two weakly lower semicontinuous functionals, which implies that φ is weakly lower semicontinuous.Lemma 2.32 (see [35]).
The functionalJ defined in Lemma 2.31 is continuously differentiable on WT1,p(t) and J' is given by
(2.39)〈J′(u),v〉=∫0T‍(|u˙(t)|p(t)-2u˙(t),v˙(t))dt,
and J' is a mapping of (S+), that is, if un⇀u weakly in WT1,p(t) and
(2.40)limsupn→∞(J′(un)-J′(u),un-u))≤0,
then un has a convergent subsequence on WT1,p(t).Clarke considered the following abstract framework in [1]:(i)
let(T,𝒯,μ) be a σ-finite positive measure space, and let Y be a separable Banach space,(ii)
letZ be a closed subspace of L∞(T,Y), where L∞(T,Y) denotes the space of measure essentially bounded functions mapping T to Y, equipped with the usual supremum norm,(iii)
define a functionalf on Z via
(2.41)f(x)=∫T‍ft(x(t))μ(dt),
where ft:Y→ℝ(t∈T) is a given family of functions,(iv)
suppose that the mappingt→ft(v) is measurable for each v in Y, and that x is a point at which f(x) is defined (finitely),(v)
suppose that there existε>0 and a function k(t) in L1(T,ℝ) such that
(2.42)|ft(v1)-ft(v2)|≤k(t)∥v1-v2∥Y,
for all t∈T and all v1 and v2 in x(t)+εBY.Under this conditions described above,f is Lipschitz in a neighborhood of x and one has
(2.43)∂f(x)⊂∫T‍∂ft(x(t))μ(dt).
Further, if each ft is regular at x(t) for each t, then f is regular at x and equality holds.We give an example to illustrate Clarke’s abstract framework with the following cast of characters:(i)
(T,𝒯,μ):=[0,T] with Lebesgue measure, and let Y:=ℝN, which is a separable Banach space,(ii)
letZ:=C([0,T],ℝN), which is a closed subspace of L∞([0,T],ℝN),(iii)
define a functionalf on Z via
(2.44)f(x)=∫0T‍F(t,x(t))dt,(iv)
F(t,x) satisfies the condition (A’).Under the hypothesis above, we only need to justify the condition (2.42), in fact, by Lebourg’s mean value theorem,
(2.45)|F(t,x1)-F(t,x2)|=|〈ζt,x1-x2〉|,
where λt∈(x1,x2) and ζt∈∂F(t,λt) for a.e. t∈[0,T] and all x1 and x2 in x(t)+εBY, where x(t)∈Z and ε is a positive constant.In view of (A’), we get(2.46)|F(t,x1)-F(t,x2)|≤[2αC(|x1|α+|x2|α)+C0]|x1-x2|≤[22α+1C(|x(t)|α+εα)+C0]|x1-x2|,
for a.e. t∈[0,T] and all x1, x2 in x(t)+εBY.We can apply Clarke’s abstract framework to our example, that is, for anyζ∈∂f(u) such that
(2.47)〈ζ,v〉=∫0T‍(q(t),v(t))dt,
for all v∈Z, where q(t) is a measurable selection of ∂F(t,x(t)).Now we can prove the following result which is fundamental in our paper.Lemma 2.33.
SupposeL:[0,T]×ℝN×ℝN→ℝ is given by
(2.48)L(t,x,y)=L1(t,x,y)+L2(t,x,y),
where
(2.49)L1(t,x,y)=1p(t)|y|p(t),L2(t,x,y)=F(t,x),
and F(t,x) satisfies the condition (A’). The corresponding functionals f1 and f2 on Wp(t) are given by
(2.50)f1(u,u˙)=∫0T‍L1(t,u(t),u˙(t))dt=∫0T‍1p(t)|u˙(t)|p(t)dt,f2(u,u˙)=∫0T‍L2(t,u(t),u˙(t))dt=∫0T‍F(t,u(t))dt.
Then f=f1+f2 is Lipschitz on Wp(t) and one has
(2.51)∂f(u,u˙)⊂∫0T‍{|u˙(t)|p(t)-2u˙(t)}×{∂F(t,u(t))}dt.Proof.
Take an arbitrary element(u0,u˙0) in Wp(t), and it suffices to prove f is Lipschitz on (u0,u˙0) and (2.51) holds for (u0,u˙0).
f
1 is continuously differentiable on (u0,u˙0), that is,
(2.52)〈f1′(u0,u˙0),(v,v˙)〉=∫0T‍(|u˙0(t)|p(t)-2u˙0(t),v˙(t))dt,
for any (v,v˙) in Wp(t), so f1 is Lipschitz on (u0,u˙0).
When∥(ui,u˙i)-(u0,u˙0)∥Wp(t)≤ε(i=1,2), we conclude
(2.53)∥ui(t)-u0(t)∥∞≤C4ε,
by Lemma 2.15, where C4 is a positive constant. Arguing as in (2.46),
(2.54)|F(t,u1(t))-F(t,u2(t))|≤k′(t)|u1(t)-u2(t)|,
for a.e. t∈[0,T], where k'(t)∈L1([0,T];ℝ+).
By Lemma2.15 and (2.54), we have
(2.55)|f2(u1,u˙1)-f2(u2,u˙2)|=|∫0T‍F(t,u1(t))-F(t,u2(t))dt|≤∫0T‍k′(t)dt∥u1(t)-u2(t)∥∞≤C4∫0T‍k′(t)dt∥(u1,u˙1)-(u2,u˙2)∥Wp(t),
so f2 is also Lipschitz on (u0,u˙0), which implies that f as the sum of two Lipschitz functionals is also Lipschitz on (u0,u˙0).
For anyζ in ∂f2(u0,u˙0), one has
(2.56)∫0T‍F0(t,u0(t);v(t))dt≥f20((u0,u˙0);(v,v˙))≥〈ζ,(v,v˙)〉,
for any (v,v˙) in Wp(t) by Fatou Lemma and it is obvious that
(2.57)L20(t,u0(t),u˙0(t);v1,v2)=F0(t,u0(t);v1),
for a.e. t∈[0,T] and all (v1,v2) in ℝN×ℝN, then we conclude
(2.58)∫0T‍L20(t,u0(t),u˙0(t);v(t),v˙(t))dt≥f20((u0,u˙0);(v,v˙))≥〈ζ,(v,v˙)〉,
by (2.56) for any (v,v˙) in Wp(t) and (2.58) remains true if we restrict (v,v˙) to Wp(t)∩W∞, which is a closed subspace of W∞ by Lemma 2.20.
By Lemma2.21, we conclude the bounded linear functional ζ on Wp(t) restricted to Wp(t)∩W∞ is also a bounded linear functional, and we use ζ' to denote the functional restricted on Wp(t)∩W∞.
We interpret (2.58) as saying that ζ' belongs to the subgradient at (0, 0) of the convex functional
(2.59)f^2(v,v˙):=∫0T‍f^t(v(t),v˙(t))dt,
which is defined in Wp(t)∩W∞, where f^t(v1,v2):=L20(t,u0(t),u˙0(t);v1,v2) for all (v1,v2) in ℝN×ℝN.
In view of condition (A’) and (2.57), we have
(2.60)|L20(t,u0(t),u˙0(t);v1,v2)-L20(t,u0(t),u˙0(t);v3,v4)|=|F0(t,u0(t);v1)-F0(t,u0(t);v3)|≤(C(|u0(t)|α+C0)|v1-v3|≤(C(|u0(t)|α+C0)(|v1-v3|+|v2-v4|),
for a.e. t∈[0,T] and all (v1,v2), (v3,v4) in ℝN×ℝN.
Now we can apply Clarke’s abstract framework tof^2 with the following cast of characters:(i)
(T,𝒯,μ):=[0,T] with Lebesgue measure, and let Y:=ℝN×ℝN, which is a separable Banach space with the norm |·|+|·|,(ii)
letZ:=Wp(t)∩W∞, which is a closed subspace of W∞, and W∞ denotes the space of measure essentially bounded functions mapping T to Y, equipped with the usual supremum norm by Definition 2.18,(iii)
define a functionalf^2 on Z by (2.59),(iv)
the mappingt→L20(t,u0(t),u˙0(t);v1,v2) is measurable for each (v1,v2) in ℝN×ℝN by  (2.57), see details in [1], and that (0,0) is a point at which f^2 is defined (finitely),(v)
the condition (2.42) in Clarke’s abstract framework is satisfied by (2.60).By (2.57), We get
(2.61)∂f^t(0,0)=∂L2(t,u0(t),u˙0(t))⊂∂F(t,u0(t))×{0},
thus, every ζ'∈∂f^2(0,0) can be written as
(2.62)〈ζ′,(v,v˙)〉=∫0T‍(q(t),v(t))+(0,v˙(t))dt=∫0T‍(q(t),v(t))dt,
for any (v,v˙) in Wp(t)∩W∞, where q(t)∈∂F(t,u0(t)) for a.e. t∈[0,T].Whenv∈CT∞([0,T],ℝN), it is obvious that (v,v˙)∈Wp(t)∩W∞ and (v,v˙) is dense in Wp(t) by Lemma 2.11. So for each (v,v˙)∈Wp(t), we can choose (vn,v˙n)∈Wp(t)∩W∞ such that
(2.63)∥(vn,v˙n)-(v,v˙)∥Wp(t)→0,〈ζ,(vn,v˙n)〉→〈ζ,(v,v˙)〉.Combining (2.62) and (2.63), we have
(2.64)〈ζ,(v,v˙)〉=∫0T‍(q(t),v(t))dt,
for all (v,v˙)∈Wp(t).We conclude(2.65)∂f(u0,u˙0)⊂∂f1(u0,u˙0)+∂f2(u0,u˙0)=∫0T‍{|u˙0(t)|p(t)-2u˙0(t)}×{∂F(t,u0(t))}dt,
and this completes the proof.


## 3. Main Results and Proofs of Theorems

Theorem 3.1.
LetF(t,x) satisfy the condition (A’) with α∈[0,p--1), and we suppose the following condition holds
(A1)|x|-q+α∫0T‍F(t,x)dt→+∞as|x|→∞,
where q+ is the same in condition (A).Then system (1.1) has at least one solution which minimizes φ in WT1,p(t).If we replace the (A1) in Theorem 3.1 by the following condition:
(A2)|x|-q+α∫0T‍F(t,x)dt→-∞as|x|→∞,
we obtain the following theorem.Theorem 3.2.
LetF(t,x) satisfy the condition (A’) with α∈[0,p--1) and (A2). Then system (1.1) has at least one solution in WT1,p(t).Remark 3.3.
Theorems3.1 and 3.2  generalize Theorems 1 and 2, respectively in [3].Proof of Theorem3.1.
Foru∈WT1,p(t), let u-=(1/T)∫0T‍u(t)dt and u~=u-u-. From Lemma 2.27, it follows that there exist z(t) in (u-,u(t)) such that
(3.1)F(t,u(t))-F(t,u-)=〈ζt,u~(t)〉,
for a.e. t∈[0,T], where ζt∈∂F(t,z(t)).
It follows from (A’), Young inequality and Lemma2.22 that
(3.2)|∫0T‍[F(t,u(t))-F(t,u-)]dt|≤∫0T‍|F(t,u(t))-F(t,u-)|dt≤∫0T‍|ζt||u~(t)|dt≤∫0T‍[2αC(|u-|α+|u~(t)|α)+C0]|u~(t)|dt≤2αCT∥u~∥∞α+1+2αCT∥u~∥∞|u-|α+C0T∥u~∥∞≤12p+∫0T‍|u˙(t)|p(t)dt+C5(∫0T‍|u˙(t)|p(t)dt)(α+1)/p-+C6(∫0T‍|u˙(t)|p(t)dt)1/p-+C7|u-|q+α+C8,
for all u∈WT1,p(t), and some positive constants C5, C6, C7, and C8.
Hence we have(3.3)φ(u)≥1p+∫0T‍|u˙(t)|p(t)dt+∫0T‍F(t,u-)dt+∫0T‍[F(t,u(t))-F(t,u-)]dt≥12p+∫0T‍|u˙(t)|p(t)dt-C5(∫0T‍|u˙(t)|p(t)dt)(α+1)/p--C6(∫0T‍|u˙(t)|p(t)dt)1/p--C7|u-|q+α-C8+∫0T‍F(t,u-)dt≥12p+∫0T‍|u˙(t)|p(t)dt-C5(∫0T‍|u˙(t)|p(t)dt)(α+1)/p--C6(∫0T‍|u˙(t)|p(t)dt)1/p-+|u-|q+α(|u-|-q+α∫0T‍F(t,u-)dt-C7)-C8,
for all u∈WT1,p(t), which implies that
(3.4)φ(u)→∞as∥u∥→∞,
because of α<p--1 and the Proposition 2.24.
By Lemma2.31, the functional φ is weakly lower semicontinuous on WT1,p(t), and it follows that φ has a minimum u0 on WT1,p(t) by Theorem 1.1 in [2]. Proposition  2.3.2 in [1] implies that 0∈∂φ(u0), that is, u0 is a critical point for φ. So, problem (1.1) has at least one solution u0∈WT1,p(t).Proof of Theorem3.2.
We will show thatφ defined in Lemma 2.31 satisfies the (PS) condition. Let {un} be a sequence in WT1,p(t) such that φ(un) is bounded and λ(un)→0 as n→∞. Using the definition of λ(un), it results that for each n≥n0 there exists un*∈∂φ(un) with
(3.5)|〈un*,h〉|≤∥h∥,∀h∈WT1,p(t).
In view of Lemma2.33, if un*∈∂φ(un), it results that there exist qn(t)∈∂F(t,un(t)) such that
(3.6)|〈un*,u~n〉|=|∫0T‍|u˙n(t)|p(t)dt+∫0T‍(qn(t),u~n(t))dt|≤∥u~n∥,∀n≥n0.
It follows Lemma2.22 and Young inequality that
(3.7)|∫0T‍(qn(t),u~n(t))|≤C∫0T‍|u-n+u~n(t)|α|u~n(t)|dt+C0∫0T‍|u~n(t)|dt≤2p--1C∫0T‍(|u-n|α+|u~n(t)|α)|u~n(t)|dt+C0∫0T‍|u~n(t)|dt≤2p--1CT(|u-n|α+∥u~n∥∞α)∥u~n∥∞+C0T∥u~n∥∞=((12)1/p-∥u~n∥∞4C2)((2p-+1)(2)1/p-CC2T|u-n|α)+2p--1CT∥u~n∥∞1+α+C0T∥u~n∥∞≤12∫0T‍|u˙n(t)|p(t)dt+C9|u-n|q+α+C10(∫0T‍|u˙n(t)|p(t)dt)(α+1)/p-+C11(∫0T‍|u˙n(t)|p(t)dt)1/p-+C12,
for all n and some positive constants C9, C10, C11, and C12, where C2 is the same as in Lemma 2.22.
Hence, we have(3.8)∥u~n∥≥|〈un*,u~n〉|≥∫0T‍|u˙n(t)|p(t)dt+∫0T‍(qn(t),u~n(t))dt≥12∫0T‍|u˙n(t)|p(t)dt-C9|u-n|q+α-C10(∫0T‍|u˙n(t)|p(t)dt)(α+1)/p--C11(∫0T‍|u˙n(t)|p(t)dt)1/p--C12,
for all n≥n0.
It follows from (2.31) that
(3.9)∥u~n∥≤C3((∫0T‍|u˙n(t)|p(t)dt)1/p-+1),
by (3.8) and (3.9), we have
(3.10)(∫0T‍|u˙n(t)|p(t)dt)1/q+≤C13|u-n|α+C14,
for some positive constants C13, C14, and all n≥n0.
By the proof of (3.2) we have
(3.11)∫0T‍[F(t,un(t))-F(t,u-n)]dt≤12p+∫0T‍|u˙n(t)|p(t)dt+C5(∫0T‍|u˙(t)|np(t)dt)(α+1)/p-+C6(∫0T‍|u˙n(t)|p(t)dt)1/p-+C7|u-n|q+α+C8,
for all n.
It follows from the boundness of{φ(un)}, (3.10) and (3.11) that
(3.12)C9≤φ(un)=∫0T‍1p(t)|u˙n(t)|p(t)dt+∫0T‍[F(t,un(t))-F(t,u-n)]dt+∫0T‍F(t,u-n)dt≤(1p-+12p+)∫0T‍|u˙n(t)|p(t)dt+C7|u-n|q+α+C5(∫0T‍|u˙n(t)|p(t)dt)(α+1)/p-+C6(∫0T‍|u˙n(t)|p(t)dt)1/p-+∫0T‍F(t,u-n)dt+C8≤|u-n|q+α(|u-n|-q+α∫0T‍F(t,u-n)dt+C7)+C15,
for all n≥n0 and some positive constant C15.
It follows (A2) and (3.12) that (|u-n|) is bounded, hence {∥un∥} is bounded by (2.31) and (3.10).
The sequence{un} has a subsequence, also denoted by {un}, such that
(3.13)un⇀uweaklyinWT1,p(t),un→ustronglyinC([0,T];ℝN),
and ∥un∥∞≤C16 is bounded by Lemma 2.15, where C16 is a positive constant.
Therefore we haveun*∈∂φ(un), where un* is the function from the Palais-Smale condition, and u*∈∂φ(u) such that
(3.14)〈un*-u*,un-u〉→0,
as n→∞, so
(3.15)〈un*-u*,un-u〉=∫0T‍(qn(t)-q(t),un(t)-u(t))dt+∫0T‍(|u˙n(t)|p(t)-2u˙n(t)-|u˙(t)|p(t)-2u˙(t),u˙n(t)-u˙(t))dt,
where qn(t)∈∂(F(t,un(t)) and q(t)∈∂(F(t,u(t)).
By (3.14) and (3.15), we get 〈J'(u)-J'(un),u-un〉→0, that is,
(3.16)∫0T‍(|u˙n(t)|p(t)-2u˙n(t)-|u˙(t)|p(t)-2u˙(t),u˙n(t)-u˙(t))dt→0,
so it follows from Lemma 2.32 that {un} admits a convergent subsequence.
We now proveφ satisfies the other conditions of Lemma 2.30. Let W~T1,p(t) be the subspace of WT1,p(t) defined in Lemma 2.22, then we have
(3.17)φ(u)→+∞,
as ∥u∥→∞ in W~T1,p(t). In fact it follows from (3.2) that
(3.18)|∫0T‍[F(t,u(t))-F(t,0)]dt|≤∫0T‍|ζ||u(t)|dt≤C17(∫0T‍|u˙(t)|p(t)dt)(α+1)/p-+C18(∫0T‍|u˙(t)|p(t)dt)1/p-+C19,
for all u∈W~T1,p(t) and some positive constants C17, C18, and C19. 							(3.19)φ(u)-∫0T‍F(t,0)dt=∫0T‍1p(t)|u˙(t)|p(t)dt+∫0T‍[F(t,u(t))-F(t,0)]dt≥1p+∫0T‍|u˙(t)|p(t)dt-C17(∫0T‍|u˙(t)|p(t)dt)(α+1)/p--C18(∫0T‍|u˙(t)|p(t)dt)1/p--C19,
for all u∈W~T1,p(t), which implies (3.17) by Proposition 2.24.
Moreover, we have(3.20)φ(x)→-∞,
as |x|→∞ in ℝN, which follows (A2).
We have proved the functionalφ satisfies all the conditions of Lemma 2.30, so we know that φ has at least one critical point by Lemma 2.30, which is a periodic solution for system (1.1). The proof is complete.


## 4. Example

In this section, we give three examples to illustrate our results.Example 4.1.
In system (1.1), let F(t,x)=|x| and
(4.1)p(t)={3+t,0≤t≤T2,3-t+T,T2<t≤T,
it is easy to verify that |ζ|≤1, where ζ∈∂F(t,x) for every t∈[0,T] and all x∈ℝN.
By Theorem3.1, system (1.1) has at least one solution u∈WT1,p(t), but it is obvious that the results in the reference cannot be applied to our example.Example 4.2.
In system (1.1), let
(4.2)p(t)={2t+1,0≤t≤T2,2(1-2T/2)T(t-T2)+2T/2+1T2<t≤T,
and F(t,x)=-|x|, it is easy to verify that |ζ|≤1, where ζ∈∂F(t,x) for every t∈[0,T] and all x∈ℝN.
By Theorem3.2, system (1.1) has at least one solution u∈WT1,p(t), but it is obvious that the results in the reference cannot be applied to our example.Example 4.3.
In system (1.1), let p(t)=sinωt+5, and
(4.3)F(t,x)=(sinωt-12)|x|3+x1,
where ω denotes the positive constant 2π/T.
It is obvious thatF is continuously differentiable, then the the Clarke subdifferential set ∂F(t,x) reduces to one element ∇F(t,x), then
(4.4)|∇F(t,x)|≤6(|x|2+1),|x|-8/3∫0T‍F(t,x)dt→-∞as|x|→+∞.
These show that all conditions of Theorem3.2 are satisfied, where
(4.5)α=2,p-=4,q+=43,
and by Theorem 3.2, system (1.1) has at least one periodic solution on WT1,p(t). But the results in [35] cannot be applied to our example, so our results are new even in the case F∈C1 for system (1.1).



---

*Source: 475956-2012-11-06.xml*