Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 147, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS TO DIRICHLET IMPULSIVE
DIFFERENTIAL EQUATIONS THROUGH A LOCAL
MINIMIZATION PRINCIPLE
GHASEM A. AFROUZI, SAEID SHOKOOH, ARMIN HADJIAN
Abstract. A critical point theorem (local minimum result) for differentiable
functionals is used for proving that a Dirichlet impulsive differential equation
admits at least one non-trivial solution. Some particular cases and a concrete
example are also presented.
1. Introduction
In this article, we study the existence of at least one non-trivial classical solution
to the nonlinear Dirichlet boundary-value problem
−(p(t)u0 (t))0 + q(t)u(t) = λf (t, u(t)),
t ∈ [0, T ], t 6= tj ,
u(0) = u(T ) = 0,
∆u0 (tj ) = λIj (u(tj )),
(1.1)
j = 1, 2, . . . , n,
where T > 0, p ∈ C 1 ([0, T ], ]0, +∞[), q ∈ L∞ ([0, T ]), λ ∈]0, +∞[, f : [0, T ] × R → R
is an L1 -Carathéodory function, 0 = t0 < t1 < t2 < · · · < tn < tn+1 = T ,
0 −
∆u0 (tj ) = u0 (t+
u0 (t) − limt→t− u0 (t) and Ij : R → R are
j ) − u (tj ) = limt→t+
j
j
continuous for every j = 1, 2, . . . , n.
The study of impulsive boundary-value problems is important due to its various applications in which abrupt changes at certain times in the evolution process
appear. The dynamics of evolving processes is often subjected to abrupt changes
such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of impulses. Such
problems arise in physics, population dynamics, biotechnology, pharmacokinetics,
industrial robotics.
Recently, many researchers pay their attention to impulsive differential equations
by variational method and critical point theory, and we refer the reader to [12, 15,
17, 18, 19, 20] and references cited therein.
Our analysis is mainly based on the critical point theorem by Bonanno [2],
contained in Theorem 2.1 below. This theorem has been used in several works
2000 Mathematics Subject Classification. 34B37, 34B15, 58E05.
Key words and phrases. Impulsive differential equations; Dirichlet condition;
classical solution; variational methods.
c
2014
Texas State University - San Marcos.
Submitted January 20, 2014. Published June 24, 2014.
1
2
G. A. AFROUZI, S. SHOKOOH, A. HADJIAN
EJDE-2014/147
to obtain existence results for different kinds of problems (see, for instance, [1, 4,
5, 6, 7, 8, 9, 10, 11, 13, 14]).
2. Preliminaries
Our main tool is Ricceri’s variational principle [16, Theorem 2.5] as given in [2,
Theorem 5.1] which is below recalled (see also [2, Proposition 2.1]).
For a given non-empty set X, and two functionals Φ, Ψ : X → R, we define the
functions
supu∈Φ−1 (]r1 ,r2 [) Ψ(u) − Ψ(v)
β(r1 , r2 ) :=
inf
−1
r2 − Φ(v)
v∈Φ (]r1 ,r2 [)
and
Ψ(v) − supu∈Φ−1 (]−∞,r1 ]) Ψ(u)
ρ(r1 , r2 ) :=
sup
Φ(v) − r1
v∈Φ−1 (]r1 ,r2 [)
for all r1 , r2 ∈ R, with r1 < r2 .
Theorem 2.1 ([2, Theorem 5.1]). Let X be a reflexive real Banach space; Φ : X →
R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux
differentiable function whose Gâteaux derivative admits a continuous inverse on
X ∗ ; Ψ : X → R be a continuously Gâteaux differentiable function whose Gâteaux
derivative is compact. Assume that there are r1 , r2 ∈ R, with r1 < r2 , such that
β(r1 , r2 ) < ρ(r1 , r2 ).
Then, setting Iλ := Φ−λΨ, for each
such that Iλ (u0,λ ) ≤ Iλ (u) for all u
λ ∈] ρ(r11,r2 ) , β(r11,r2 ) [ there is u0,λ ∈
∈ Φ−1 (]r1 , r2 [) and Iλ0 (u0,λ ) = 0.
(2.1)
Φ
−1
(]r1 , r2 [)
In the Sobolev space X := H01 (0, T ), consider the inner product
Z T
Z T
0
0
(u, v) :=
p(t)u (t)v (t) dt +
q(t)u(t)v(t) dt ,
0
which induces the norm
Z
kuk :=
0
T
0
2
Z
p(t)(u (t)) dt +
0
T
1/2
.
q(t)(u(t))2 dt
0
Then the following Poincaré-type inequality holds:
Z
i1/2
hZ T
i1/2
Th T 0 2
(u ) (t) dt
.
u2 (t) dt
≤
π 0
0
(2.2)
2 −
Let us introduce some notation that will be used later. Assume that T πq2 > −p− ,
where
p− := ess inf t∈[0,T ] p(t) > 0, q − := ess inf t∈[0,T ] q(t).
p
Moreover, put σ0 := min{T 2 q − /π 2 , 0} and δ := p− + σ0 . Then, we have the
following useful proposition.
Proposition 2.2. Let u ∈ X. Then
1
ku0 kL2 ([0,T ]) ≤ kuk,
√ δ
T
kuk∞ ≤
kuk.
2δ
(2.3)
(2.4)
EJDE-2014/147
DIRICHLET IMPULSIVE DIFFERENTIAL EQUATIONS
Proof. First we prove (2.3). To this end, let q − ≥ 0. Then, σ0 = 0 and δ =
Therefore,
Z T
1
ku0 k2L2 ([0,T ]) ≤ −
p(t)|u0 (t)|2 dt
p
0
Z T
1
1
≤ −
p(t)|u0 (t)|2 + q(t)|u(t)|2 dt = 2 kuk2 .
p
δ
0
3
p
p− .
Thus, the desired inequality
(2.3) follows. On the other hand, if q − < 0, we have
q
2 −
2 −
σ0 = T πq2 and δ = p− + T πq2 . Obviously,
Z T
−
2
q(t)|u(t)|2 dt.
q kukL2 ([0,T ]) ≤
0
Now, applying inequality (2.2) and bearing in mind that q − < 0, one has
T 2 q− 0 2
ku kL2 ([0,T ]) ≤ q − kuk2L2 ([0,T ]) .
π2
By the above inequalities we have
Z T
T 2 q− 0 2
ku
k
≤
q(t)|u(t)|2 dt.
L2 ([0,T ])
π2
0
This inequality together with
−
p
ku0 k2L2 ([0,T ])
Z
≤
T
p(t)|u0 (t)|2 dt,
0
imply (2.3).
In view of Hölder’s inequality and (2.3), one has
√
√
T 0
T
kuk∞ ≤
ku kL2 ([0,T ]) ≤
kuk,
2
2δ
which completes and the proof.
1/2
2
Put k := kpk∞ + Tπ2 kqk∞
. Then, from (2.2) we have
kuk ≤ kku0 kL2 ([0,T ]) .
(2.5)
Here and in the sequel f : [0, T ] × R → R is an L1 -Carathéodory function, namely:
(a) the mapping t 7→ f (t, x) is measurable for every x ∈ R;
(b) the mapping x 7→ f (t, x) is continuous for almost every t ∈ [0, T ];
(c) for every ρ > 0 there exists a function lρ ∈ L1 ([0, T ]) such that
sup |f (t, x)| ≤ lρ (t)
|x|≤ρ
for almost every t ∈ [0, T ].
Corresponding to f we introduce the function F : [0, T ] × R → R as follows
Z x
F (t, x) :=
f (t, ξ) dξ,
0
for all (t, x) ∈ [0, T ] × R.
By a classical solution of problem (1.1), we mean a function
u ∈ w ∈ C([0, T ]) : w|[tj ,tj+1 ] ∈ H 2 ([tj , tj+1 ])
4
G. A. AFROUZI, S. SHOKOOH, A. HADJIAN
EJDE-2014/147
that satisfies the equation in (1.1) a.e. on [0, T ] \ {t1 , . . . , tn }, the limits u0 (t+
j ),
0
u0 (t−
),
j
=
1,
.
.
.
,
n,
exist,
satisfy
the
impulsive
conditions
∆u
(t
)
=
λI
(u(t
j
j
j ))
j
and the boundary condition u(0) = u(T ) = 0.
We say that a function u ∈ X is a weak solution of problem (1.1), if u satisfies
Z T
Z T
0
0
p(t)u (t)v (t) dt +
q(t)u(t)v(t) dt
0
0
T
Z
−λ
f (t, u(t))v(t) dt −
0
n
X
p(tj )Ij (u(tj ))v(tj ) = 0,
j=1
for any v ∈ X.
Lemma 2.3 ([3, Lemma 2.3]). The function u ∈ X is a weak solution of problem
(1.1) if and only if u is a classical solution of (1.1).
Lemma 2.4 ([3, Lemma 3.1]). Assume that
(A1) there exist constants η, θ > 0 and σ ∈ [0, 1[ such that
|Ij (x)| ≤ 2η|x| + θ|x|σ+1
for all x ∈ R, j = 1, 2, . . . , n.
Then, for any u ∈ X, we have
Z u(tj )
n
n
X
X
p(tj )
p(tj ) ηkuk2∞ +
Ij (x) dx ≤
0
j=1
j=1
Also put
p̃ :=
n
X
j=1
√
p(tj ),
µ(τ ) :=
2kτ
,
δ
Γc :=
θ
kukσ+2
.
∞
σ+2
(2.6)
η θ σ−1
+
c
,
c
σ+2
where η, θ, σ are given by (A1) and c, τ are positive constants. We assume throughout, and without further mention, that the assumption (A1) holds.
3. Main results
For a given non-negative constant ν and a positive constant τ with δ 2 ν 2 6= 2k 2 τ 2 ,
put
RT
R 3T /4
max|x|≤ν F (t, x) dt + p̃ν 3 Γν + p̃(µ(τ ))3 Γµ(τ ) − T /4 F (t, τ ) dt
0
aτ (ν) :=
.
δ 2 ν 2 − 2k 2 τ 2
Theorem 3.1. Assume that there
√ exist a non-negative constant ν1 and two positive
constants ν2 and τ , with ν1 < 2τ < δν2 /k, such that
(A2) F (t, ξ) ≥ 0 for all (t, ξ) ∈ ([0, T4 ] ∪ [ 3T
4 , T ]) × [0, τ ];
(A3) aτ (ν2 ) < aτ (ν1 ).
Then, for each λ ∈] T aτ2(ν1 ) , T aτ2(ν2 ) [, problem (1.1) admits at least one non-trivial
classical solution u0 ∈ X such that
2δν1
2δν
√ < ku0 k < √ 2 .
T
T
Proof. The aim is to apply Theorem 2.1 to our problem. To this end, we introduce
the functionals Φ, Ψ : X → R by setting
1
Φ(u) := kuk2 ,
2
EJDE-2014/147
DIRICHLET IMPULSIVE DIFFERENTIAL EQUATIONS
T
Z
F (t, u(t)) dt −
Ψ(u) :=
0
n
X
Z
5
u(tj )
p(tj )
Ij (x) dx,
0
j=1
for every u ∈ X, and put
Iλ (u) := Φ(u) − λΨ(u) ∀ u ∈ X.
Clearly, Φ and Ψ are well defined and continuously Gâteaux differentiable functionals whose Gâteaux derivatives at the point u ∈ X are the functionals Φ0 (u), Ψ0 (u) ∈
X ∗ , given by
Z T
Z T
q(t)u(t)v(t) dt,
p(t)u0 (t)v 0 (t) dt +
Φ0 (u)(v) =
Ψ0 (u)(v) =
0
0
T
Z
f (t, u(t))v(t) dt −
0
n
X
p(tj )Ij (u(tj ))v(tj ),
j=1
for every v ∈ X. Moreover, Φ is coercive and sequentially weakly lower semicontinuous and Ψ is sequentially weakly upper semicontinuous. Also, Φ0 admits
a continuous inverse on X ∗ and Ψ0 is compact. Note that the critical points of
the functional Iλ in X are exactly the weak solutions of problem (1.1). We verify
condition (2.1) of Theorem 2.1. To this end, we put
2δ 2 2
2δ 2 2
r1 :=
ν1 , r2 :=
ν ,
T 2
 T
4τ

if t ∈ [0, T /4[,
 T t,
w(t) := τ,
if t ∈ [T /4, 3T /4],

 4τ
T (T − t), if t ∈]3T /4, T ].
It is easy to verify that w ∈ X and, in particular, taking (2.3) and (2.5) into
account, one has
8k 2 τ 2
8δ 2 τ 2
≤ kwk2 ≤
.
T
T
So, we have
4δ 2 τ 2
4k 2 τ 2
≤ Φ(w) ≤
.
T
T
√
From the condition ν1 < 2τ < δν2 /k, we obtain r1 < Φ(w) < r2 . Moreover, for
all u ∈ X such that u ∈ Φ−1 (] − ∞, r2 [), from (2.4), one has |u(t)| < ν2 for all
t ∈ [0, T ], from which it follows
Z u(tj )
n
Z T
X
sup
Ψ(u) =
sup
F (t, u(t)) dt −
p(tj )
Ij (x) dx
u∈Φ−1 (]−∞,r2 [)
u∈Φ−1 (]−∞,r2 [)
Z
≤
0
0
j=1
0
T
max F (t, x) dt + p̃ν23 Γν2 .
|x|≤ν2
Arguing as before, we obtain
Z
sup
u∈Φ−1 (]−∞,r1 ])
Ψ(u) ≤
0
T
max F (t, x) dt + p̃ν13 Γν1 .
|x|≤ν1
Since 0 ≤ w(t) ≤ τ for each t ∈ [0, T ], assumption (A2) ensures that
Z T /4
Z T
F (t, w(t)) dt +
F (t, w(t)) dt ≥ 0,
0
3T /4
6
G. A. AFROUZI, S. SHOKOOH, A. HADJIAN
EJDE-2014/147
and so
Z
3T /4
Ψ(w) ≥
F (t, τ ) dt −
T /4
Z
n
X
Z
w(tj )
p(tj )
Ij (x) dx
0
j=1
3T /4
≥
F (t, τ ) dt − p̃ (µ(τ ))3 Γµ(τ ) .
T /4
Therefore, we have
β(r1 , r2 ) ≤
≤
supu∈Φ−1 (]−∞,r2 [) Ψ(u) − Ψ(w)
r2 − Φ(w)
RT
R 3T /4
max|x|≤ν2 F (t, x) dt + p̃ν23 Γν2 + p̃(µ(τ ))3 Γµ(τ ) − T /4 F (t, τ ) dt
0
2δ 2 ν22
T
−
4k2 τ 2
T
T
aτ (ν2 ).
2
On the other hand, we have
=
ρ(r1 , r2 ) ≥
≥
Ψ(w) − supu∈Φ−1 (]−∞,r1 ]) Ψ(u)
Φ(w) − r1
R 3T /4
RT
F (t, τ ) dt − p̃(µ(τ ))3 Γµ(τ ) − p̃ν13 Γν1 − 0 max|x|≤ν1 F (t, x) dt
T /4
4k2 τ 2
T
−
2δ 2 ν12
T
T
aτ (ν1 ).
2
So, from assumption (A3), it follows that β(r1 , r2 ) < ρ(r1 , r2 ). Therefore, from
Theorem 2.1, for each λ ∈] T aτ2(ν1 ) , T aτ2(ν2 ) [, the functional Iλ admits at least one
critical point u0 such that r1 < Φ(u0 ) < r2 ; that is,
=
2δν
2δν1
√ < ku0 k < √ 2 ,
T
T
and the proof is complete.
Now, we point out the following consequence of Theorem 3.1.
√
Theorem 3.2. Assume that there are two positive constants ν and τ , with 2kτ <
δν, such that assumption (A2) in Theorem 3.1 holds. Furthermore, suppose that
RT
max
F (t,x) dt+p̃ν 3 Γ
2
R 3T /4
F (t,τ ) dt−p̃ (µ(τ ))3 Γµ(τ )
ν
|x|≤ν
T /4
δ
< 2k
.
(A4) 0
2
ν2
τ2
Then, for each
4k2 τ 2
2δ 2 ν 2
i
h
T
T
λ ∈ R 3T /4
, RT
,
F (t, τ ) dt − p̃ (µ(τ ))3 Γµ(τ ) 0 max|x|≤ν F (t, x) dt + p̃ν 3 Γν
T /4
problem (1.1) admits at least one non-trivial classical solution u0 ∈ X such that
|u0 (t)| < ν for all t ∈ [0, T ].
Proof. The conclusion follows from Theorem 3.1, by taking ν1 = 0 and ν2 = ν.
Indeed, owing to assumption (A4), one has
2 2 RT
τ
1 − 2k
max|x|≤ν F (t, x) dt + p̃ν 3 Γν
ν 2 δ2
0
aτ (ν) <
δ 2 ν 2 − 2k 2 τ 2
EJDE-2014/147
DIRICHLET IMPULSIVE DIFFERENTIAL EQUATIONS
RT
=
0
7
max|x|≤ν F (t, x) dt + p̃ν 3 Γν
.
δ2 ν 2
On the other hand,
R 3T /4
T /4
F (t, τ ) dt − p̃(µ(τ ))3 Γµ(τ )
.
2k 2 τ 2
Now, owing to assumption (A4) and (2.4), it is sufficient to invoke Theorem 3.1 for
concluding the proof.
aτ (0) =
The following result gives the existence of at least one non-trivial classical solution in X to problem (1.1)
R x in the autonomous case. Let f : R → R be a continuous
function. Put F (x) := 0 f (ξ) dξ for all x ∈ R. We have the following result as a
direct consequence of Theorem 3.1.
Theorem 3.3. Assume that there
√ exist a non-negative constant ν1 and two positive
constants ν2 and τ , with ν1 < 2τ < δν2 /k, such that
(A5) f (x) ≥ 0 for all x ∈ [−ν2 , max{ν2 , τ }];
(A6)
T F (ν2 ) + p̃ν2 3 Γν2 + p̃ (µ(τ ))3 Γµ(τ ) −
δ 2 ν22 − 2k 2 τ 2
<
T
2
F (τ )
T F (ν1 ) + p̃ν1 3 Γν1 + p̃(µ(τ ))3 Γµ(τ ) −
δ 2 ν12 − 2k 2 τ 2
T
2
F (τ )
.
Then, for each
λ∈
i
T T F (ν2 ) +
2δ 2 ν22 − 4k 2 τ 2
+ p̃ (µ(τ ))3 Γµ(τ ) −
p̃ ν2 3 Γν2
2δ 2 ν22 − 4k 2 τ 2
T T F (ν1 ) + p̃ ν1 3 Γν1 + p̃ (µ(τ ))3 Γµ(τ ) −
T
2
T
2
F (τ )
,
F (τ )
h
,
the problem
−(p(t)u0 (t))0 + q(t)u(t) = λf (u(t)),
t ∈ [0, T ], t 6= tj ,
u(0) = u(T ) = 0,
0
∆u (tj ) = λIj (u(tj )),
(3.1)
j = 1, 2, . . . , n,
admits at least one non-trivial classical solution u0 ∈ X such that
2δν1
2δν
√ < ku0 k < √ 2 .
T
T
√
Proof. Since δ ≤ k, from the condition ν1 < 2τ < δνk2 we obtain ν1 < ν2 .
Therefore, assumption (A5) means f (x) ≥ 0 for each x ∈ [−ν1 , ν1 ] and f (x) ≥ 0
for each x ∈ [−ν2 , ν2 ], which implies
max
x∈[−ν1 ,ν1 ]
F (x) = F (ν1 )
and
max
x∈[−ν2 ,ν2 ]
F (x) = F (ν2 ).
So, from assumptions (A5) and (A6), we arrive at assumptions (A2) and (A3),
respectively. Hence, Theorem 3.1 yields the conclusion.
Theorem 3.4. Let f : R → R be a non-negative continuous function such that
(A7) limξ→0+
f (ξ)
ξ
= +∞.
8
G. A. AFROUZI, S. SHOKOOH, A. HADJIAN
2
EJDE-2014/147
2
ν
Then, for each λ ∈]0, 2δT supν>0 T F (ν)+
p̃ν 3 Γν [, problem (3.1) admits at least one
non-trivial classical solution u0 ∈ X.
Proof. For fixed λ as in the conclusion, there exists a positive constant ν such that
λ(T F (ν) + p̃ν 3 Γν ) <
2δ 2 ν 2
.
T
(3.2)
= +∞. On the other hand,
Moreover, assumption (A7) implies that limt→0+ Fξ(ξ)
2
( √ 2
(µ(ξ))3 Γµ(ξ)
η δ2 k ,
if 0 < σ < 1,
√
2
=
lim+
2
ξ
ξ→0
Γ1 δ2 k , if σ = 0.
Therefore,
F (ξ) − (µ(ξ))3 Γµ(ξ)
= +∞.
ξ2
ξ→0
√
So, a positive constant τ satisfying 2kτ < δν can be chosen such that
lim+
λ
T F (τ ) − p̃ (µ(τ ))3 Γ
2
µ(τ )
τ2
>
4k 2
.
T
(3.3)
Hence, taking (3.2) and (3.3) into account, Theorem 3.1 ensures the conclusion.
Remark 3.5. Taking (A7) into account, fix ρ > 0 such that f (ξ) > 0 for all
ξ ∈]0, ρ[. Then, put
2δ 2
ν2
λρ :=
sup
.
T ν∈]0,ρ[ T F (ν) + p̃ν 3 Γν
The result of Theorem 3.4 for every λ ∈]0, λρ [ holds with |u0 (t)| < ρ for all t ∈ [0, T ],
where u0 is the ensured non-trivial classical solution in X.
Example 3.6. Let I(u(t1 )) = u(t1 ) for some t1 ∈ (0, 1). Then I : R → R is a
continuous function satisfying the sublinear growth condition (A1) with η = θ = 31
2
ξ
and σ = 0. Now, put p(t) = 1, q(t) = −π
2 for all t ∈ [0, 1] and f (ξ) = (1 + ξ)e for
1
every ξ ∈ R. Clearly, one has δ = √2 . Hence, since
ν2
ν2
2
=
sup
=
,
ν
3Γ
3Γ
νe
+
ν
2e
+1
f
(ξ)
dξ
+
ν
ν
ν∈]0,1[
ν∈]0,1[ 0
ν
2
from Remark 3.5, for every λ ∈ 0, 2e+1
the problem
sup R ν
−u00 (t) −
π2
u(t) = λ(1 + u(t))eu(t) ,
2
u(0) = u(1) = 0,
a.e. in [0, 1],
∆u0 (t1 ) = λu(t1 ),
has at least one non-trivial classical solution u0 ∈ H01 (0, 1) such that |u0 (t)| < 1 for
all t ∈ [0, 1].
Here, we point out a special situation of our main result when the nonlinear
term has separable variables. To be precise, let α ∈ L1 ([0, T ]) such that α(t) ≥ 0
EJDE-2014/147
DIRICHLET IMPULSIVE DIFFERENTIAL EQUATIONS
9
a.e. t ∈ [0, T ], α 6≡ 0, and let g : R → R be a nonnegative continuous function.
Consider the following Dirichlet boundary-value problem
−(p(t)u0 (t))0 + q(t)u(t) = λα(t)g(u(t)),
t ∈ [0, T ], t 6= tj ,
u(0) = u(T ) = 0,
(3.4)
∆u0 (tj ) = λIj (u(tj )),
j = 1, 2, . . . , n.
Rx
RT
Put G(x) := 0 g(ξ) dξ for all x ∈ R, and set kαk1 := 0 α(t) dt and α0 :=
R 3T /4
α(t) dt.
T /4
Corollary 3.7. Let Ij (x) ≤ 0 for all x ∈ R, j = 1, . . . , n. Assume that there
√ exist
a non-negative constant ν1 and two positive constants ν2 and τ , with ν1 < 2τ <
δν2 /k, such that
(A8)
G(ν2 )kαk1 + p̃ν23 Γν2 + p̃(µ(τ ))3 Γµ(τ ) − G(τ )α0
δ 2 ν22 − 2k 2 τ 2
G(ν1 )kαk1 + p̃ν13 Γν1 + p̃(µ(τ ))3 Γµ(τ ) − G(τ )α0
.
<
δ 2 ν12 − 2k 2 τ 2
Then, for each
2δ 2 ν12 − 4k 2 τ 2
,
T G(ν1 )kαk1 + p̃ν13 Γν1 + p̃(µ(τ ))3 Γµ(τ ) − G(τ )α0
h
2δ 2 ν22 − 4k 2 τ 2
,
3
T G(ν2 )kαk1 + p̃ν2 Γν2 + p̃(µ(τ ))3 Γµ(τ ) − G(τ )α0
λ∈
i
problem (3.4) admits at least one positive classical solution u0 ∈ X, such that
2δν1
2δν
√ < ku0 k < √ 2 .
T
T
Proof. Put f (t, ξ) := α(t)g(ξ) for all (t, ξ) ∈ [0, T ] × R. Clearly, F (t, x) = α(t)G(x)
for all (t, x) ∈ [0, T ] × R. Therefore, taking into account that G is a non-decreasing
function, Theorem 3.1 and [3, Lemma 3.6] ensure the conclusion.
An immediate consequence of Corollary 3.7 is the following.
Corollary 3.8. Let Ij (x) ≤ 0 for all√x ∈ R, j = 1, . . . , n. Assume that there exist
two positive constants ν and τ , with 2kτ < δν, such that
(A9)
G(ν)kαk1 +p̃ν 3 Γν
ν2
<
3
δ 2 G(τ )α0 −p̃(µ(τ )) Γµ(τ )
.
2k2
τ2
Then, for each
λ∈
i
h
4k 2 τ 2
2δ 2 ν 2
,
,
3
3
T G(τ )α0 − p̃(µ(τ )) Γµ(τ ) T G(ν)kαk1 + p̃ν Γν
problem (3.4) admits at least one positive classical solution u0 ∈ X, such that
|u0 (t)| < ν for all t ∈ [0, T ].
The above corollary follows directly from Theorem 3.2 and [3, Lemma 3.6].
10
G. A. AFROUZI, S. SHOKOOH, A. HADJIAN
EJDE-2014/147
Now, consider the nonlinear Dirichlet boundary-value problem
−u00 (t) + a(t)u0 (t) + b(t)u(t) = λh(t, u(t)),
t ∈ [0, T ], t 6= tj ,
u(0) = u(T ) = 0,
0
∆u (tj ) = λIj (u(tj )),
(3.5)
j = 1, 2, . . . , n,
where h : [0, T ] × R → R is an L -Carathéodory function and a, b ∈ L∞ ([0, T ])
satisfy the following conditions
π2
ess inf t∈[0,T ] a(t) ≥ 0, ess inf t∈[0,T ] b(t)e−A(t) > − 2 e−A(T ) ,
T
where A(t) be a primitive of a(t).
It is easy to see that the solutions of (1.1) are solutions of (3.5) if
1
p(t) = e−
Let H(t, x) :=
Rt
0
a(ξ) dξ
Rx
0
,
q(t) = b(t)e−
Rt
0
a(ξ) dξ
,
f (t, u) = h(t, u)e−
Rt
0
a(ξ) dξ
.
h(t, ξ) dξ. Then, by a simple computation, we obtain
F (t, x) = e−A(t) H(t, x),
∀(t, x) ∈ [0, T ] × R.
Set
1/2
T2
T2
k̃ := 1 + 2 kbe−A k∞
ess inf t∈[0,T ] b(t)e−A(t) , 0 ,
, σ̃0 := min
2
π
π
p
−A(T
)
δ̃ := e
+ σ̃0 .
For a given non-negative
constant ν and a positive constant τ with δ̃ 2 ν 2 6= 2k̃ 2 τ 2 ,
√
put µ̃(τ ) := 2k̃τ /δ̃ and
Z T
ãτ (v) :=
e−A(t) max H(t, x) dt + p̃ν 3 Γν + p̃ (µ̃(τ ))3 Γµ̃(τ )
Z
0
3T /4
−
|x|≤ν
δ̃ 2 ν 2 − 2k̃ 2 τ 2 .
e−A(t) H(t, τ ) dt
T /4
With the above notation and by Theorem 3.1, we obtain the following existence
property for problem (3.5).
Theorem 3.9. Assume that there
√ exist a non-negative constant ν1 and two positive
constants ν2 and τ , with ν1 < 2τ < δ̃ν2 /k̃, such that
(A10) H(t, ξ) ≥ 0 for all (t, ξ) ∈ ([0, T4 ] ∪ [ 3T
4 , T ]) × [0, τ ];
(A11) ãτ (ν2 ) < ãτ (ν1 ).
Then, for each λ ∈] T ãτ2(ν1 ) , T ãτ2(ν2 ) [, problem (3.5) admits at least one non-trivial
classical solution u0 ∈ X such that
2δ̃ν1
2δ̃ν
√ < ku0 k < √ 2 .
T
T
References
[1] G. A. Afrouzi, A. Hadjian, S. Heidarkhani; Non-trivial solutions for a two-point boundary
value problem, Ann. Polon. Math., 108 (2013), 75–84.
[2] G. Bonanno; A critical point theorem via the Ekeland variational principle, Nonlinear Anal.,
75 (2012), 2992–3007.
[3] G. Bonanno, B. Di Bella, J. Henderson; Existence of solutions to second-order boundaryvalue problems with small perturbations of impulses, Electron. J. Differential Equations, Vol.
2013 (2013), No. 126, pp. 1–14.
EJDE-2014/147
DIRICHLET IMPULSIVE DIFFERENTIAL EQUATIONS
11
[4] G. Bonanno, B. Di Bella, D. O’Regan; Non-trivial solutions for nonlinear fourth-order elastic
beam equations, Comput. Math. Appl., 62 (2011), 1862–1869.
[5] G. Bonanno, S. Heidarkhani, D. O’Regan; Nontrivial solutions for Sturm-Liouville systems
via a local minimum theorem for functionals, Bull. Aust. Math. Soc., 89 (2014), 8–18.
[6] G. Bonanno, G. Molica Bisci, V. Rădulescu; Nonlinear elliptic problems on Riemannian
manifolds and applications to Emden-Fowler type equations, Manuscripta Math., 142 (2013),
157–185.
[7] G. Bonanno, G. Molica Bisci, V. Rădulescu; Weak solutions and energy estimates for a class
of nonlinear elliptic Neumann problems, Adv. Nonlinear Stud., 13 (2013), 373–389.
[8] G. Bonanno, P. F. Pizzimenti; Neumann boundary value problems with not coercive potential,
Mediterr. J. Math., 9 (2012), 601–609.
[9] G. Bonanno, P. F. Pizzimenti; Existence results for nonlinear elliptic problems, Appl. Anal.,
92 (2013), 411–423.
[10] G. Bonanno, A. Sciammetta; An existence result of one nontrivial solution for two point
boundary value problems, Bull. Aust. Math. Soc., 84 (2011), 288–299.
[11] G. Bonanno, A. Sciammetta; Existence and multiplicity results to Neumann problems for
elliptic equations involving the p-Laplacian, J. Math. Anal. Appl., 390 (2012), 59–67.
[12] P. Chen, X. Tang; New existence and multiplicity of solutions for some Dirichlet problems
with impulsive effects, Math. Comput. Modelling, 55 (2012), 723–739.
[13] S. Heidarkhani; Non-trivial solutions for a class of (p1 , . . . , pn )-biharmonic systems with
Navier boundary conditions, Ann. Polon. Math., 105 (2012), 65–76.
[14] S. Heidarkhani; Non-trivial solutions for two-point boundary-value problems of fourth-order
Sturm-Liouville type equations, Electron. J. Differential Equations, Vol. 2012 (2012), No. 27,
pp. 1–9.
[15] J. Nieto, D. O’Regan; Variational approach to impulsive differential equations, Nonlinear
Anal. Real World Appl., 10 (2009), 680–690.
[16] B. Ricceri; A general variational principle and some of its applications, J. Comput. Appl.
Math., 113 (2000), 401–410.
[17] J. Sun, H. Chen, J. Nieto, M. Otero-Novoa; The multiplicity of solutions for perturbed secondorder Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 4575–4586.
[18] Y. Tian, W. Ge, D. Yang; Existence results for second-order system with impulse effects via
variational methods, J. Appl. Math. Comput., 31 (2009), 255–265.
[19] J. Xiao, J. Nieto, Z. Luo; Multiplicity of solutions for nonlinear second order impulsive
differential equations with linear derivative dependence via variational methods, Commun.
Nonlinear Sci. Numer. Simul., 17 (2012), 426–432.
[20] J. Xu, Z. Wei, Y. Ding; Existence of weak solutions for p-Laplacian problem with impulsive
effects, Taiwanese J. Math., 17 (2013), 501–515.
Ghasem A. Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Saeid Shokooh
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: saeid.shokooh@stu.umz.ac.ir
Armin Hadjian
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O.
Box 1339, Bojnord 94531, Iran
E-mail address: hadjian83@gmail.com