Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 97, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF INFINITELY MANY ANTI-PERIODIC
SOLUTIONS FOR SECOND-ORDER IMPULSIVE
DIFFERENTIAL INCLUSIONS
SHAPOUR HEIDARKHANI, GHASEM A. AFROUZI,
ARMIN HADJIAN, JOHNNY HENDERSON
Abstract. In this article, we establish the existence of infinitely many antiperiodic solutions for a second-order impulsive differential inclusion with a
perturbed nonlinearity and two parameters. The technical approach is mainly
based on a critical point theorem for non-smooth functionals.
1. Introduction
The aim of this article is to show the existence of infinitely many solutions for
the following two parameter second-order impulsive differential inclusion subject to
anti-periodic boundary conditions
−(φp (u0 (x)))0 + M φp (u(x)) ∈ λF (u(x)) + µG(x, u(x))
0
−∆φp (u (xk )) = Ik (u(xk )),
u(0) = −u(T ),
in [0, T ] \ Q,
k = 1, 2, . . . , m,
(1.1)
0
u (0) = −u0 (T ),
where Q = {x1 , x2 , . . . , xm }, p > 1, T > 0, M ≥ 0, φp (x) := |x|p−2 x, 0 = x0 <
0 −
0 +
x1 < · · · < xm < xm+1 = T , ∆φp (u0 (xk )) := φp (u0 (x+
k )) − φp (u (xk )), with u (xk )
−
and u0 (xk ) denoting the right and left limits, respectively, of u0 (x) at x = xk , Ik ∈
C(R, R), k = 1, 2, . . . , m, λ is a positive parameter, µ is a nonnegative parameter,
and F is a multifunction defined on R, satisfying
(F1) F : R → 2R is upper semicontinuous with compact convex values;
(F2) min F, max F : R → R are Borel measurable;
(F3) |ξ| ≤ a(1 + |s|r−1 ) for all s ∈ R, ξ ∈ F (s), r > 1 (a > 0).
Also, G is a multifunction defined on [0, T ] × R, satisfying
(G1) G(x, ·) : R → 2R is upper semicontinuous with compact convex values for
a.e. x ∈ [0, T ] \ Q;
(G2) min G, max G : ([0, T ] \ Q) × R → R are Borel measurable;
(G3) |ξ| ≤ a(1 + |s|r−1 ) for a.e. x ∈ [0, T ], s ∈ R, ξ ∈ G(x, s), r > 1 (a > 0).
2000 Mathematics Subject Classification. 58E05, 49J52, 34A60.
Key words and phrases. Differential inclusions; impulsive; anti-periodic solution;
non-smooth critical point theory.
c
2013
Texas State University - San Marcos.
Submitted January 29, 2013. Published April 16, 2013.
1
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S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
EJDE-2013/97
Impulsive differential equations are used to describe various models of real-world
processes that are subject to a sudden change. These models are studied in physics,
population dynamics, ecology, industrial robotics, biotechnology, economics, optimal control, and so forth. Associated with this development, a theory of impulsive
differential equations has been given extensive attention. Differential inclusions
arise in models for control systems, mechanical systems, economical systems, game
theory, and biological systems to name a few. It is very important to study antiperiodic boundary value problems because they can be applied to interpolation
problems [5], antiperiodic wavelets [3], the Hill differential operator [6], and so on.
It is natural from both a physical standpoint as well as a theoretical view to give
considerable attention to a synthesis involving problems for impulsive differential
inclusion with anti-periodic boundary conditions.
Recently, multiplicity of solutions for differential inclusions via non-smooth variational methods and critical point theory has been considered and here we cite the
papers [9, 10, 11, 12, 16]. For instance, in [11], the author, employing a non-smooth
Ricceri-type variational principle [15], developed by Marano and Motreanu [13], has
established the existence of infinitely many, radially symmetric solutions for a differential inclusion problem in RN . Also, in [12], the authors extended a recent result
of Ricceri concerning the existence of three critical points of certain non-smooth
functionals. Two applications have been given, both in the theory of differential
inclusions; the first one concerns a non-homogeneous Neumann boundary value
problem, the second one treats a quasilinear elliptic inclusion problem in the whole
RN . In [9], the author, under convenient assumptions, has investigated the existence of at least three positive solutions for a differential inclusion involving the
p-Laplacian operator on a bounded domain, with homogeneous Dirichlet boundary
conditions and a perturbed nonlinearity depending on two positive parameters; his
result also ensured an estimate on the norms of the solutions independent of both
the perturbation and the parameters. Very recently, Tian and Henderson in [16],
based on a non-smooth version of critical point theory of Ricceri due to Iannizzotto
[9], have established the existence of at least three solutions for the problem (1.1)
whenever λ is large enough and µ is small enough.
In the present paper, motivated by [16], employing an abstract critical point
result (see Theorem 2.6 below), we are interested in ensuring the existence of infinitely many anti-periodic solutions for the problem (1.1); see Theorem 3.1 below.
We refer to [2], in which related variational methods are used for non-homogeneous
problems.
To the best of our knowledge, no investigation has been devoted to establishing
the existence of infinitely many solutions to such a problem as (1.1). For a couple
of references on impulsive differential inclusions, we refer to [7] and [8].
A special case of our main result is the following theorem.
Theorem 1.1. Assume that (F1)–(F3) hold, and Ii (0) = 0, Ii (s)s < 0, s ∈ R,
i = 1, 2, . . . , m. Furthermore, suppose that
Rt
sup|t|≤ξ min 0 F (s)ds
= 0,
lim inf
ξ→+∞
ξp
RT
R ξ( T −x)
min 0 2
F (s) ds dx
0
P
lim sup
= +∞.
m R ξ( T2 −xi )
1
2M
T
p T +
p+1 −
ξ→+∞
Ii (s)ds
i=1 0
pξ
p+1 ( 2 )
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3
Then, the problem (1.1), for λ = 1 and µ = 0, admits a sequence of pairwise distinct
solutions.
2. Basic definitions and preliminary results
Let (X, k · kX ) be a real Banach space. We denote by X ∗ the dual space of X,
while h·, ·i stands for the duality pairing between X ∗ and X. A function ϕ : X → R
is called locally Lipschitz if, for all u ∈ X, there exist a neighborhood U of u and
a real number L > 0 such that
|ϕ(v) − ϕ(w)| ≤ Lkv − wkX
for all v, w ∈ U.
If ϕ is locally Lipschitz and u ∈ X, the generalized directional derivative of ϕ at u
along the direction v ∈ X is
ϕ◦ (u; v) := lim sup
w→u, τ →0+
ϕ(w + τ v) − ϕ(w)
.
τ
The generalized gradient of ϕ at u is the set
∂ϕ(u) := {u∗ ∈ X ∗ : hu∗ , vi ≤ ϕ◦ (u; v) for all v ∈ X}.
∗
So ∂ϕ : X → 2X is a multifunction. We say that ϕ has compact gradient if ∂ϕ
maps bounded subsets of X into relatively compact subsets of X ∗ .
Lemma 2.1 ([14, Proposition 1.1]). Let ϕ ∈ C 1 (X) be a functional. Then ϕ is
locally Lipschitz and
ϕ◦ (u; v) = hϕ0 (u), vi
0
∂ϕ(u) = {ϕ (u)}
for all u, v ∈ X;
for all u ∈ X.
Lemma 2.2 ([14, Proposition 1.3]). Let ϕ : X → R be a locally Lipschitz functional.
Then ϕ◦ (u; ·) is subadditive and positively homogeneous for all u ∈ X, and
ϕ◦ (u; v) ≤ Lkvk
for all u, v ∈ X,
with L > 0 being a Lipschitz constant for ϕ around u.
Lemma 2.3 ([4]). Let ϕ : X → R be a locally Lipschitz functional. Then ϕ◦ :
X × X → R is upper semicontinuous and for all λ ≥ 0, u, v ∈ X,
(λϕ)◦ (u; v) = λϕ◦ (u; v).
Moreover, if ϕ, ψ : X → R are locally Lipschitz functionals, then
(ϕ + ψ)◦ (u; v) ≤ ϕ◦ (u; v) + ψ ◦ (u; v)
for all u, v ∈ X.
Lemma 2.4 ([14, Proposition 1.6]). Let ϕ, ψ : X → R be locally Lipschitz functionals. Then
∂(λϕ)(u) = λ∂ϕ(u)
for all u ∈ X, λ ∈ R,
∂(ϕ + ψ)(u) ⊆ ∂ϕ(u) + ∂ψ(u)
for all u ∈ X.
Lemma 2.5 ([9, Proposition 1.6]). Let ϕ : X → R be a locally Lipschitz functional
with a compact gradient. Then ϕ is sequentially weakly continuous.
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S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
EJDE-2013/97
We say that u ∈ X is a (generalized) critical point of a locally Lipschitz functional
ϕ if 0 ∈ ∂ϕ(u); i.e.,
ϕ◦ (u; v) ≥ 0 for all v ∈ X.
When a non-smooth functional, g : X → (−∞, +∞), is expressed as a sum of a
locally Lipschitz function, ϕ : X → R, and a convex, proper, and lower semicontinuous function, j : X → (−∞, +∞); that is, g := ϕ + j, a (generalized) critical
point of g is every u ∈ X such that
ϕ◦ (u; v − u) + j(v) − j(u) ≥ 0
for all v ∈ X (see [14, Chapter 3]).
Hereafter, we assume that X is a reflexive real Banach space, N : X → R is
a sequentially weakly lower semicontinuous functional, Υ : X → R is a sequentially weakly upper semicontinuous functional, λ is a positive parameter, j : X →
(−∞, +∞) is a convex, proper, and lower semicontinuous functional, and D(j) is
the effective domain of j. Write
M := Υ − j,
Iλ := N − λM = (N − λΥ) + λj.
We also assume that N is coercive and
D(j) ∩ N −1 ((−∞, r)) 6= ∅
(2.1)
for all r > inf X N . Moreover, owing to (2.1) and provided r > inf X N , we can
define
supv∈N −1 ((−∞,r)) M(v) − M(u)
,
ϕ(r) :=
inf
r − N (u)
u∈N −1 ((−∞,r))
γ := lim inf ϕ(r), δ := lim inf + ϕ(r).
r→+∞
r→(inf X N )
If N and Υ are locally Lipschitz functionals, in [1, Theorem 2.1] the following result
is proved; it is a more precise version of [13, Theorem 1.1] (see also [15]).
Theorem 2.6. Under the above assumption on X, N and M, one has
(a) For every r > inf X N and every λ ∈ (0, 1/ϕ(r)), the restriction of the
functional Iλ = N −λM to N −1 ((−∞, r)) admits a global minimum, which
is a critical point (local minimum) of Iλ in X.
(b) If γ < +∞, then for each λ ∈ (0, 1/γ), the following alternative holds:
either
(b1) Iλ possesses a global minimum, or
(b2) there is a sequence {un } of critical points (local minima) of Iλ such
that limn→+∞ N (un ) = +∞.
(c) If δ < +∞, then for each λ ∈ (0, 1/δ), the following alternative holds:
either
(c1) there is a global minimum of N which is a local minimum of Iλ , or
(c2) there is a sequence {un } of pairwise distinct critical points (local minima) of Iλ , with limn→+∞ N (un ) = inf X N , which converges weakly
to a global minimum of N .
Now we recall some basic definitions and notation. On the reflexive Banach
space X := {u ∈ W 1,p ([0, T ]) : u(0) = −u(T )} we consider the norm
Z T
1/p
kukX :=
|u0 (x)|p + M |u(x)|p dx
0
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EXISTENCE OF INFINITELY MANY ANTI-PERIODIC SOLUTIONS
5
for all u ∈ X, which is equivalent to the usual norm (note that M ≥ 0). We
recall that X is compactly embedded into the space C 0 ([0, T ]) endowed with the
maximum norm k · kC 0 .
Lemma 2.7 ([16, Lemma 3.3]). Let u ∈ X. Then
1
kukC 0 ≤ T 1/q kukX ,
2
where 1/p + 1/q = 1.
(2.2)
Obviously, X is compactly embedded into Lγ ([0, T ]) endowed with the usual
norm k · kLγ , for all γ ≥ 1.
Definition 2.8. A function u ∈ X is a weak solution of the problem (1.1) if there
exists u∗ ∈ Lγ ([0, T ]) (for some γ > 1) such that
Z Th
m
i
X
φp (u0 (x))v 0 (x) + M φp (u(x))v(x) − u∗ (x)v(x) dx −
Ii (u(xi ))v(xi ) = 0
0
i=1
for all v ∈ X and u∗ ∈ λF (u(x)) + µG(x, u(x)) for a.e. x ∈ [0, T ].
Definition 2.9. By a solution of the impulsive differential inclusion (1.1) we will
understand a function u : [0, T ] \ Q → R is of class C 1 with φp (u0 ) absolutely
continuous, satisfying
−(φp (u0 (x)))0 + M φp (u(x)) = u∗
0
−∆φp (u (xk )) = Ik (u(xk )),
u(0) = −u(T ),
in [0, T ] \ Q,
k = 1, 2, . . . , m,
0
u (0) = −u0 (T ),
where u∗ ∈ λF (u(x)) + µG(x, u(x)) and u∗ ∈ Lγ ([0, T ]) (for some γ > 1).
Lemma 2.10 ([16, Lemma 3.5]). If a function u ∈ X is a weak solution of (1.1),
then u is a classical solution of (1.1).
We introduce for a.e. x ∈ [0, T ] and all s ∈ R, the Aumann-type set-valued
integral
Z s
nZ s
o
F (t)dt =
f (t)dt : f : R → R is a measurable selection of F
0
0
RT
Ru
and set F(u) = 0 min 0 F (s) ds dx for all u ∈ Lp ([0, T ]); the Aumann-type setvalued integral
Z s
nZ s
o
G(x, t)dt =
g(x, t)dt : g : [0, T ] × R → R is a measurable selection of G
0
0
and set G(u) =
RT
0
min
Ru
0
G(x, s) ds dx for all u ∈ Lp ([0, T ]).
Lemma 2.11 ([10, Lemma 3.1]). The functionals F, G : Lp ([0, T ]) → R are well
defined and Lipschitz on any bounded subset of Lp ([0, T ]). Moreover, for all u ∈
Lp ([0, T ]) and all u∗ ∈ ∂(F(u) + G(u)),
u∗ (x) ∈ F (u(x)) + G(x, u(x))
for a.e. x ∈ [0, T ].
We define an energy functional for the problem (1.1) by setting
m Z u(xi )
X
1
p
Ii (s)ds
Iλ (u) = kukX − λF(u) − µG(u) −
p
i=1 0
for all u ∈ X.
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S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
EJDE-2013/97
Lemma 2.12 ([16, Lemma 4.4]). The functional Iλ : X → R is locally Lipschitz.
Moreover, for each critical point u ∈ X of Iλ , u is a weak solution of (1.1).
3. Main results
We formulate our main result using the following assumptions:
(F4)
Rt
sup|t|≤ξ min 0 F (s)ds
lim inf
ξ→+∞
ξp
RT
R ξ( T −x)
F (s) ds dx
min 0 2
1 2 p
0
;
< ( ) lim sup
Pm R ξ( T2 −xi )
2M
T
1
p T
p
p+1 −
ξ→+∞
Ii (s)ds
i=1 0
p ξ T + p+1 ( 2 )
(I1) Ii (0) = 0, Ii (s)s < 0, s ∈ R, i = 1, 2, . . . , m.
Theorem 3.1. Assume that (F1)–(F4), (I1) hold. Let
RT
R ξ( T −x)
.
min 0 2
F (s) ds dx
0
P
λ1 := 1 lim sup
,
m R ξ( T2 −xi )
ξ→+∞ 1 ξ p T + 2M ( T )p+1 −
Ii (s)ds
i=1 0
p
p+1 2
Rt
.
sup|t|≤ξ min 0 F (s)ds
p
.
λ2 := 1 lim inf
ξ→+∞
1
p
2ξ
T
Then, for every λ ∈ (λ1 , λ2 ), and every multifunction G satisfying
RT
Rt
(G4) 0 min 0 G(x, s) ds dx ≥ 0 for all t ∈ R, and
(G5) G∞ := limξ→+∞
if we put
R
sup|t|≤ξ min 0t G(x,s)ds
ξp
< +∞,
Rt
sup|t|≤ξ min 0 F (s)ds 1 2p
pT p
µG,λ :=
1 − λ p lim inf
,
pG∞ T p
2 ξ→+∞
ξp
where µG,λ = +∞ when G∞ = 0, problem (1.1) admits an unbounded sequence of
solutions for every µ ∈ [0, µG,λ ) in X.
Proof. Our aim is to apply Theorem 2.6(b) to (1.1). To this end, we fix λ ∈ (λ1 , λ2 )
and let G be a multifunction satisfying (G1)–(G5). Since λ < λ2 , we have
Rt
sup|t|≤ξ min 0 F (s)ds 1 2p
pT p
µG,λ =
1 − λ p lim inf
> 0.
pG∞ T p
2 ξ→+∞
ξp
Now fix µ ∈ (0, µg,λ ), put ν1 := λ1 , and
ν2 :=
1+
λ2
pT p µ
2p λ λ2
G∞
.
If G∞ = 0, then ν1 = λ1 , ν2 = λ2 and λ ∈ (ν1 , ν2 ). If G∞ 6= 0, since µ < µG,λ , we
have
λ
pT p
+ p µ G∞ < 1,
λ2
2
and so
λ2
> λ,
pT p µ
1 + 2p λ λ2 G∞
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EXISTENCE OF INFINITELY MANY ANTI-PERIODIC SOLUTIONS
7
namely, λ < ν2 . Hence, taking into account that λ > λ1 = ν1 , one has λ ∈ (ν1 , ν2 ).
Now, set
µ
J(x, s) := F (s) + G(x, s)
λ
for all (x, s) ∈ [0, T ] × R. Assume j identically zero in X and for each u ∈ X put
Z u
Z T
m Z u(xi )
X
1
p
J(x, s) ds dx,
N (u) := kukX −
min
Ii (s)ds, Υ(u) :=
p
0
0
i=1 0
M(u) := Υ(u) − j(u) = Υ(u),
Iλ (u) := N (u) − λM(u) = N (u) − λΥ(u).
It is a simple matter to verify that N is sequentially weakly lower semicontinuous
on X. Clearly, N ∈ C 1 (X). By Lemma 2.1, N is locally Lipschitz on X. By
Lemma 2.11, F and G are locally Lipschitz on Lp ([0, T ]). So, Υ is locally Lipschitz
on Lp ([0, T ]). Moreover, X is compactly embedded into Lp ([0, T ]). So Υ is locally
Lipschitz on X. Furthermore, Υ is sequentially weakly upper semicontinuous. For
all u ∈ X, by (I1 ),
Z u(xi )
Ii (s)ds < 0, i = 1, 2, . . . , m.
0
So, we have
m
X
1
N (u) = kukpX −
p
i=1
Z
u(xi )
Ii (s)ds >
0
1
kukpX
p
for all u ∈ X. Hence, N is coercive and inf X N = N (0) = 0. We want to prove
that, under our hypotheses, there exists a sequence {un } ⊂ X of critical points for
the functional Iλ , that is, every element un satisfies
Iλ◦ (un , v − un ) ≥ 0,
for every v ∈ X.
Now, we claim that γ < +∞. To see this, let {ξn } be a sequence of positive numbers
such that limn→+∞ ξn = +∞ and
Rt
Rt
sup|t|≤ξn min 0 J(x, s)ds
sup|t|≤ξ min 0 J(x, s)ds
lim
= lim inf
.
(3.1)
n→+∞
ξ→+∞
ξnp
ξp
Put
p
1 2ξn
rn :=
, for all n ∈ N.
p T 1/q
Then, for all v ∈ X with N (v) < rn , taking into account that kvkpX < prn and
kvkC 0 ≤ 12 T 1/q kvkX , one has |v(x)| ≤ ξn for every x ∈ [0, T ]. Therefore, for all
n ∈ N,
supv∈N −1 ((−∞,r)) M(v) − M(u)
ϕ(rn ) =
inf
r − N (u)
u∈N −1 ((−∞,r))
µ
supkvkpX <prn F(v) + λ G(v)
≤
rn
Rt
RT
Rt
RT
sup|t|≤ξn 0 min 0 F (s) ds dx + µλ 0 min 0 G(x, s) ds dx
≤
rn
Rt
Rt
h
T p sup|t|≤ξn min 0 F (s)ds µ sup|t|≤ξn min 0 G(x, s)ds i
≤p
+
.
2
ξnp
ξnp
λ
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S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
EJDE-2013/97
Moreover, from Assumptions (F4) and (G5), we have
Rt
Rt
sup|t|≤ξn min 0 F (s)ds
µ sup|t|≤ξn min 0 G(x, s)ds
lim
+ lim
< +∞,
n→+∞
n→+∞ λ
ξnp
ξnp
which follows
Rt
sup|t|≤ξn min 0 J(x, s)ds
lim
< +∞.
n→+∞
ξnp
Therefore,
Rt
sup|t|≤ξ min 0 J(x, s)ds
T p
lim inf
< +∞.
(3.2)
γ ≤ lim inf ϕ(rn ) ≤ p
n→+∞
2 ξ→+∞
ξp
Since
Rt
Rt
Rt
sup|t|≤ξ min 0 J(x, s)ds
sup|t|≤ξ min 0 F (s)ds µ sup|t|≤ξ min 0 G(x, s)ds
≤
+
,
ξp
ξp
ξp
λ
and taking (G5) into account, we get
Rt
Rt
sup|t|≤ξ min 0 J(x, s)ds
sup|t|≤ξ min 0 F (s)ds µ
lim inf
≤ lim inf
+ G∞ . (3.3)
ξ→+∞
ξ→+∞
ξp
ξp
λ
Moreover, from Assumption (G4) we obtain
RT
R ξ( T −x)
J(x, s) ds dx
min 0 2
0
lim sup
P
m R ξ( T2 −xi )
ξ→+∞ 1 ξ p T + 2M ( T )p+1 −
Ii (s)ds
i=1 0
p
p+1 2
R ξ( T2 −x)
F (s) ds dx
0 .
Pm R ξ( T2 −xi )
2M T p+1
T + p+1 ( 2 )
− i=1 0
Ii (s)ds
RT
≥ lim sup
0
ξ→+∞ 1 ξ p
p
min
Therefore, from (3.3) and (3.4), we observe that
1
λ ∈ (ν1 , ν2 ) ⊆
R
lim supξ→+∞
T
0
1 p
pξ
min
2M T p+1
T + p+1
(2)
−
1
lim inf ξ→+∞
,
R ξ( T2 −x)
0
(3.4)
J(x,s) ds dx
Pm R ξ( T2 −xi )
i=1
0
Ii (s)ds
Rt
J(x,s)ds
0
1 2ξ p
(
)
p T
sup|t|≤ξ min
⊆ (0, 1/γ).
For the fixed λ, the inequality (3.2) ensures that the condition (b) of Theorem 2.6
can be applied and either Iλ has a global minimum or there exists a sequence {un }
of weak solutions of the problem (1.1) such that limn→∞ kun k = +∞, The other
step is to show that for the fixed λ the functional Iλ has no global minimum. Let
us verify that the functional Iλ is unbounded from below. Since
RT
R ξ( T −x)
min 0 2
F (s) ds dx
1
0
< lim sup
Pm R ξ( T2 −xi )
1
2M
T
p
p+1
λ
ξ→+∞
Ii (s)ds
− i=1 0
p ξ T + p+1 ( 2 )
RT
R ξ( T2 −x)
min 0
J(x, s) ds dx
0
≤ lim sup
,
P
m R ξ( T2 −xi )
ξ→+∞ 1 ξ p T + 2M ( T )p+1 −
I
(s)ds
i
i=1 0
p
p+1 2
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9
there exists a sequence {ηn } of positive numbers and a constant τ such that
limn→+∞ ηn = +∞ and
RT
R η ( T −x)
min 0 n 2
J(x, s) ds dx
1
0
<τ <
(3.5)
P
m R ηn ( T2 −xi )
1
λ
ηn p T + 2M ( T )p+1 −
Ii (s)ds
p
p+1
i=1 0
2
for each n ∈ N large enough. For all n ∈ N, set
T
−x .
2
For any fixed n ∈ N, it is easy to see that wn ∈ X and, in particular, one has
2M T p+1 ,
kwn kpX = ηnp T +
( )
p+1 2
wn (x) := ηn
and so
T
m Z
1 p
2M T p+1 X ηn ( 2 −xi )
Ii (s)ds.
ηn T +
( )
−
p
p+1 2
i=1 0
N (wn ) =
(3.6)
By (3.5) and (3.6), we see that
Iλ (wn ) = N (wn ) − λM(wn )
T
m Z
2M T p+1 X ηn ( 2 −xi )
1 ( )
−
Ii (s)ds
= ηnp T +
p
p+1 2
i=1 0
Z T
Z ηn ( T2 −x)
−λ
J(x, s) ds dx
min
0
0
<
1
p
ηnp
T
m Z
2M T p+1 X ηn ( 2 −xi )
( )
T+
−
Ii (s)ds (1 − λτ )
p+1 2
i=1 0
for every n ∈ N large enough. Since λτ > 1 and limn→+∞ ηn = +∞, we have
lim I (wn )
n→+∞ λ
= −∞.
Then, the functional Iλ is unbounded from below, and it follows that Iλ has no
global minimum. Therefore, from part (b) of Theorem 2.6, the functional Iλ admits
a sequence of critical points {un } ⊂ X such that limn→+∞ N (un ) = +∞. Since N
is bounded on bounded sets, and taking into account that limn→+∞ N (un ) = +∞,
then {un } has to be unbounded, i.e.,
lim kun kX = +∞.
n→+∞
Moreover, if un ∈ X is a critical point of Iλ , clearly, by definition, one has
Iλ◦ (un , v − un ) ≥ 0,
for every v ∈ X.
Finally, by Lemma 2.12, the critical points of Iλ are weak solutions for the problem
(1.1), and by Lemma 2.10, every weak solution of (1.1) is a solution of (1.1). Hence,
the assertion follows.
Remark 3.2. Under the conditions
sup|t|≤ξ min
lim inf
ξ→+∞
ξp
Rt
0
F (s)ds
= 0,
10
S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
RT
lim sup
ξ→+∞ 1 ξ p
p
R ξ( T2 −x)
min
0
0
2M T p+1
p+1 ( 2 )
T+
EJDE-2013/97
−
F (s) ds dx
Pm R ξ( T2 −xi )
i=1 0
= +∞,
Ii (s)ds
p
from Theorem 3.1, we see that for every λ > 0 and for each µ ∈ 0, pT 2p G∞ ,
problem (1.1) admits a sequence of solutions which is unbounded in X. Moreover,
if G∞ = 0, the result holds for every λ > 0 and µ ≥ 0.
The following result is a special case of Theorem 3.1 with µ = 0.
Theorem 3.3. Assume that (F1)–(F4), (I1) hold. Then, for each
1
λ∈
RT
lim supξ→+∞
min
0
0
2M T p+1
T + p+1
(2)
−
1 p
pξ
F (s) ds dx
Pm R ξ( T2 −xi )
i=1
0
Ii (s)ds
!
1
lim inf ξ→+∞
,
R ξ( T2 −x)
sup|t|≤ξ min
Rt
0
1 2ξ p
p( T )
F (s)ds
,
the problem
−(φp (u0 (x)))0 + M φp (u(x)) ∈ λF (u(x))
0
−∆φp (u (xk )) = Ik (u(xk )),
in [0, T ] \ Q,
k = 1, 2, . . . , m,
0
u (0) = −u0 (T )
u(0) = −u(T ),
has an unbounded sequence of solutions in X.
Now, we present the following example to illustrate our results.
Example 3.4. Consider the problem
−(φ3 (u0 (x)))0 + φ3 (u(x)) ∈ λF (u(x))
0
−∆φ3 (u (x1 )) = I1 (u(x1 )),
0
u(0) = −u(2),
in [0, 2] \ {1},
x1 = 1,
0
u (0) = −u (2),
where, for s ∈ R,


{0},

[0, 1],
F (s) =

{s − 2−1/3 + 1},



{s + 2−1/3 + 1},
if
if
if
if
|s| < 2−1/3 ,
|s| = 2−1/3 ,
s > 2−1/3 ,
s < −2−1/3 .
Simple calculations show that
Z
sup
min
|t|≤2−1/3
and
R2
min
R ξ(1−x)
F (s) ds dx
0
R ξ(1−x1 )
− 0
I1 (s)ds
Z 1
Z ξx
6 1
=
min
F (s) ds dx
5 ξ 3 −1
0
0
5 3
6ξ
t
F (s)ds = 0
0
(3.7)
EJDE-2013/97
EXISTENCE OF INFINITELY MANY ANTI-PERIODIC SOLUTIONS
11
Z ξx
Z 0
Z −1/3 Z ξx
6 1 −2
max F (s) ds dx
max
F
(s)
ds
dx
+
=
5 ξ 3 −1
−2−1/3 0
0
Z 2−1/3 Z ξx
Z 1 Z ξx
+
max F (s) ds dx +
max F (s) ds dx > 0
2−1/3
0
0
0
for some ξ ∈ R. So,
sup|t|≤ξ min
Rt
F (s)ds
0
= 0,
1 3
ξ→+∞
ξ
3
R2
R ξ(1−x)
min 0
F (s) ds dx
0
> 0.
lim sup
R ξ(1−x1 )
5
3
ξ→+∞
I1 (s)ds
6ξ − 0
lim inf
Hence, using Theorem 3.3, problem (3.7), for λ lying in a convenient interval, has
an unbounded sequence of solutions in X := {u ∈ W 1,3 ([0, 2]) : u(0) = −u(2)}.
Here we point out the following consequences of Theorem 3.3, using the assumptions
R
p
sup|t|≤ξ min 0t F (s)ds
(F5) lim inf ξ→+∞
< p1 T2 ;
ξp
RT
(F6) lim supξ→+∞
0
1 p
pξ
min
2M T
T + p+1
(2
R ξ( T2 −x)
0
)p+1
−
F (s) ds dx
Pm R ξ( T2 −xi )
i=1
0
> 1.
Ii (s)ds
Corollary 3.5. Assume that (F1)–(F3), (F5)–(F6), (I1) hold. Then, the problem
−(φp (u0 (x)))0 + M φp (u(x)) ∈ F (u(x))
0
−∆φp (u (xk )) = Ik (u(xk )),
in [0, T ] \ Q,
k = 1, 2, . . . , m,
0
u (0) = −u0 (T )
u(0) = −u(T ),
has an unbounded sequence of solutions in X.
Remark 3.6. Theorem 1.1 in the Introduction is an immediate consequence of
Corollary 3.5.
Now, we give the following consequence of the main result.
Corollary 3.7. Let F1 : R → 2R be an upper semicontinuous multifunction with
compact convex values, such that min F1 , max F1 : R → R are Borel measurable and
|ξ| ≤ a(1 + |s|r1 −1 ) for all s ∈ R, ξ ∈ F1 (s), r1 > 1 (a > 0). Furthermore, suppose
that
R
(C1) lim inf ξ→+∞
sup|t|≤ξ min
ξp
RT
(C2) lim supξ→+∞
0
1 p
pξ
t
0
F1 (s)ds
min
< +∞;
R ξ( T2 −x)
0
2M T p+1
T + p+1
(2)
−
F1 (s) ds dx
Pm R ξ( T2 −xi )
i=1
0
= +∞.
Ii (s)ds
Then, for every multifunction F2 : R → 2R which is upper semicontinuous with
compact convex values, min F2 , max F2 : R → R are Borel measurable and |ξ| ≤
b(1 + |s|r2 −1 ) for all s ∈ R, ξ ∈ F2 (s), r2 > 1 (b > 0), and satisfies the conditions
Z t
sup min
F2 (s)ds ≤ 0
t∈R
and
RT
0
lim inf
ξ→+∞ 1 p
pξ
T+
min
0
R ξ( T2 −x)
0
2M T p+1
p+1 ( 2 )
−
F2 (s) ds dx
Pm R ξ( T2 −xi )
i=1 0
> −∞,
Ii (s)ds
12
S. HEIDARKHANI, G. A. AFROUZI, A. HADJIAN, J. HENDERSON
EJDE-2013/97
for each
1
λ ∈ 0,
lim inf ξ→+∞
sup|t|≤ξ min
Rt
0
1 2ξ p
p( T )
F1 (s)ds
,
and the problem
−(φp (u0 (x)))0 + M φp (u(x)) ∈ λ(F1 (u(x)) + F2 (u(x)))
0
−∆φp (u (xk )) = Ik (u(xk )),
in [0, T ] \ Q,
k = 1, 2, . . . , m,
0
u (0) = −u0 (T )
u(0) = −u(T ),
has an unbounded sequence of solutions in X.
Proof. Set F (t) = F1 (t) + F2 (t) for all t ∈ R. Assumption (C2) along with the
condition
RT
R ξ( T −x)
min 0 2
F2 (s) ds dx
0
P
lim inf
> −∞
m R ξ( T2 −xi )
ξ→+∞ 1 p
2M T p+1
ξ
T
+
(
)
−
I
(s)ds
i
i=1 0
p
p+1 2
yield
RT
0
lim sup
ξ→+∞ 1 ξ p
p
T+
RT
= lim sup
ξ→+∞
0
min
R ξ( T2 −x)
0
2M T p+1
p+1 ( 2 )
−
F (s) ds dx
Pm R ξ( T2 −xi )
Ii (s)ds
R ξ( T −x)
F1 (s) ds dx + 0 min 0 2
F2 (s) ds dx
min 0
= +∞.
Pm R ξ( T2 −xi )
1 p
2M T p+1
− i=1 0
Ii (s)ds
p ξ T + p+1 ( 2 )
i=1 0
R ξ( T2 −x)
RT
Moreover, Assumption (C1) and the condition
Z t
sup min
F2 (s)ds ≤ 0
t∈R
0
ensure that
sup|t|≤ξ min
lim inf
ξ→+∞
ξp
Rt
0
F (s)ds
Rt
sup|t|≤ξ min 0 F1 (s)ds
≤ lim inf
< +∞.
ξ→+∞
ξp
Since
1
lim inf ξ→+∞
sup|t|≤ξ min
Rt
0
1 2ξ p
p( T )
F (s)ds
1
≥
lim inf ξ→+∞
sup|t|≤ξ min
by applying Theorem 3.3, we have the desired conclusion.
Rt
0
1 2ξ p
p( T )
F1 (s)ds
,
Remark 3.8. We observe that in Theorem 3.1 we can replace ξ → +∞ with
ξ → 0+ , and then by the same argument as in the proof of Theorem 3.1, but
using conclusion (c) of Theorem 2.6 instead of (b), problem (1.1) has a sequence of
solutions, which strongly converges to 0 in X.
Acknowledgments. Shapour Heidarkhani was supported by a grant 91470046
from IPM.
EJDE-2013/97
EXISTENCE OF INFINITELY MANY ANTI-PERIODIC SOLUTIONS
13
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Shapour Heidarkhani
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box: 19395-5746, Tehran, Iran
E-mail address: s.heidarkhani@razi.ac.ir
Ghasem A. Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Armin Hadjian
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: a.hadjian@umz.ac.ir
Johnny Henderson
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA
E-mail address: Johnny Henderson@baylor.edu