Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 106-115.
EXISTENCE RESULTS FOR THREE-POINT
BOUNDARY VALUE PROBLEMS FOR SECOND ORDER
IMPULSIVE DIFFERENTIAL EQUATIONS
(COMMUNICATED BY FIORALBA CAKONI)
MUSTAPHA LAKRIB AND RAHMA GUEN
Abstract. In this paper, under weak conditions on the impulse functions,
we prove existence results for three-point boundary value problems for second
order impulsive differential equations. The approach is based on fixed point
theorems.
1. Introduction
The purpose of this paper is to establish existence of solutions for the following
problem
x′′ (t) + f (t, x(t), x′ (t)) = 0, a.e. t ∈ J := [0, 1], t ̸= t1 ,
′
∆x(t1 ) = I1 (x(t1 ), x (t1 )),
x(0) = 0,
′
′
∆x (t1 ) = I2 (x(t1 ), x (t1 )),
x(1) = αx(η),
(1.1)
(1.2)
(1.3)
where the function f : J × R × R → R and the impulse functions I1 , I2 :
Rn × Rn → Rn are given, the impulsive moment t1 is such that 0 < t1 < 1,
−
′
′ +
′ −
∆x(t1 ) = x(t+
1 ) − x(t1 ), ∆x (t1 ) = x (t1 ) − x (t1 ) and α ∈ R, 0 < η < 1 are such
that αη ̸= 1.
We note that we could consider three-point boundary value problems for second
order impulsive differential equations with an arbitrary finite number of impulses.
However, for clarity and brevity, we restrict our attention to the case of one impulse.
In addition, the difference between the theory of one or an arbitrary finite number
of impulses is quite minimal.
Impulsive differential equations are important mathematical tools for providing a
better understanding of many real-world models. Relative to the theory of impulsive
differential equations and its applications, we refer the interested reader to [14] and
references therein, and the monographs [1, 12, 16].
n
n
n
2000 Mathematics Subject Classification. 34K30, 34K45, 34G20.
Key words and phrases. Impulsive differential equations, three-point boundary value problems,
fixed point theorems, existence results.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted September 17, 2010. Published October 1, 2011.
106
THREE-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS
107
In the literature, many existence results for impulsive differential equations are
proved under restrictive conditions on the impulse functions (see for instance [2,
3, 4, 5, 7, 8, 9, 13, 18]). In references [10] and [11], the first author proved some
existence results under weak conditions on the impulse functions. Here, we give
some existence results of solutions for the initial value problem (1.1)-(1.3) using
some fixed points theorems. In our main results (Theorem 3.1 and Theorem 3.2),
only the continuity of the impulse functions, I1 and I2 , is required. The proofs are
based on the following well-known fixed points theorems.
Theorem 1.1 (Schaefer Fixed Point Theorem [17]). Let E be a normed space and
let Γ : E → E be a completely continuous map, that is, it is a continuous mapping
which is relatively compact on each bounded subset of E. If the set E = {x ∈ E :
λx = Γx for some λ > 1} is bounded, then Γ has a fixed point.
Theorem 1.2 (Sadovskii Fixed Point Theorem [15]). Let E be a Banach space
and let Γ : E → E be a completely continuous map. If Γ(B) ⊂ B for a nonempty
closed, convex and bounded set B of E, then Γ has a fixed point in B.
Theorem 1.3 (Banach Fixed Point Theorem [6]). Let E be a Banach space and let
Γ : E → E be a contraction map, that is, for all x, y ∈ E, |Γ(x) − Γ(y)| ≤ L|x − y|,
with L < 1, then Γ has a unique fixed point.
The paper is formulated as follows. In Section 2, some definitions and lemmas
are given and hypotheses on data f , I1 and I2 are stated. In Section 3, we establish
our existence theorems for the initial value problem (1.1)-(1.3).
2. Preliminaries
Let L1 (J, Rn ) be the Banach space of measurable functions x : J → Rn which
are Lebesgue integrable, normed by
∫ 1
∥x∥L1 =
|x(t)|dt, x ∈ L1 (J, Rn ).
0
By P C(J, Rn ) we denote the Banach space of functions x : J → Rn which are
continuous at t ̸= t1 , left continuous at t = t1 and right-hand limit at t = t1 exists,
equipped with the norm
∥x∥ = sup{|x(t)| : t ∈ J},
x ∈ P C(J, Rn ).
In a similar fashion to above, we define P C 1 (J, Rn ) as the Banach space of
functions x : J → Rn , x ∈ P C(J, Rn ), which are continuously differentiable at
t ̸= t1 , with x′ left continuous at t = t1 and right-hand limit at t = t1 exists,
equipped with the norm
∥x∥0 = max{∥x∥, ∥x′ ∥},
x ∈ P C 1 (J, Rn ).
A function x ∈ P C 1 (J, Rn ) is called a solution of (1.1)-(1.3) if x satisfies the differential equation (1.1) almost everywhere on J \{t1 } and the conditions (1.2)-(1.3).
To establish our main results, we need the following assumptions for the initial
value problem (1.1)-(1.3).
(H1) The function f : J × Rn × Rn → Rn is Carathéodory, that is,
(i) for every x, y ∈ Rn , the function f (·, x, y) : J → Rn is measurable,
(ii) for almost every t ∈ J, the function f (t, ·, ·) : Rn × Rn → Rn is
continuous.
108
M. LAKRIB AND R. GUEN
(H2) There exist functions p, q, r ∈ L1 (J, R+ ) such that
|f (t, x, y)| ≤ p(t)|x| + q(t)|y| + r(t) for almost every t ∈ J and all x, y ∈ Rn .
(H2)∗ There exist a function q ∈ L1 (J, R+ ) and a continuous nondecreasing function ψ : R+ → R+ such that
|f (t, x, y)| ≤ q(t)ψ(max{|x|, |y|}) for almost every t ∈ J and all x, y ∈ Rn .
(H3) The impulse functions I1 , I2 : Rn × Rn → Rn are continuous.
Let us now give some lemmas which are needed in the sequel.
Lemma 2.1. Let a, b, α ∈ R. Suppose that 0 < t1 < η < 1, αη ̸= 1 and f : J → Rn
is in L1 (J, Rn ). Then the boundary value problem
x′′ (t) + f (t) = 0, a.e. t ∈ J := [0, 1], t ̸= t1 ,
∆x(t1 ) = a,
x(0) = 0,
(2.1)
′
∆x (t1 ) = b,
(2.2)
x(1) = αx(η)
(2.3)
has a unique solution
∫
1
x(t) =
G(t, s)f (s)ds + aV (t) + bW (t),
t ∈ J,
(2.4)
(t, s) ∈ J × J,
(2.5)
0
where

(1 − t) + α(t − η)


,
s


1 − αη



(1 − s) + α(s − η)


t
,
− αη
G(t, s) = s(1 − t)1 +
αη(t − s)


,



1
−
αη



(1 − s)


t
,
1 − αη
V (t) =
s ≤ min{t, η}
t ≤ s ≤ η,
η ≤ s ≤ t,
s ≥ max{t, η},
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
,
1 − αη
t ∈ J,
(2.6)
and
W (t) = −(1 − H(t − t1 ))t −
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
t1 ,
1 − αη
t ∈ J. (2.7)
Remark. The function H : R → {0, 1} in (2.6) and (2.7) is the Heaviside function
which is defined by H(s) = 0 if s ≤ 0 and H(s) = 1 if s > 0.
Proof. Without boundary value conditions (2.3), problem (2.1)-(2.2) has solutions
in the form
∫ t
x(t) = A + Bt −
(t − s)f (s)ds + H(t − t1 )b(t − t1 ) + H(t − t1 )a.
0
By the boundary conditions (2.3) and standard calculation we get A = 0 and
∫ 1
∫ η
1−s
α(η − s)
(1 − t1 ) + α(t1 − η)
1−α
B=
f (s)ds −
f (s)ds −
b−
a
1
−
αη
1
−
αη
1
−
αη
1
− αη
0
0
THREE-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS
so that
∫
x(t) =
∫
t
1
(t − s)f (s)ds + t
−
0
0
(1 − s)
f (s)ds − t
1 − αη
∫
η
0
109
α(η − s)
f (s)ds
1 − αη
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
a
1 − αη
{
}
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
t1 b.
− (1 − H(t − t1 ))t +
1 − αη
+
(2.8)
Now, if t ≤ η, (2.8) can be rewritten as
{∫ t
}
∫ η
tα
x(t) = −
(η − s)f (s)ds +
(η − s)f (s)ds
1 − αη
0
t
}
{∫ t
∫ η
∫ 1
t
(1 − s)f (s)ds +
(1 − s)f (s)ds +
(1 − s)f (s)ds
+
1 − αη
0
t
η
∫ t
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
−
(t − s)f (s)ds +
a
1 − αη
0
{
}
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
− (1 − H(t − t1 ))t +
t1 b
1 − αη
∫ 1
∫ η
1−s
(1 − s) + α(s − η)
=
t
f (s)ds +
f (s)ds
t
1 − αη
1 − αη
η
t
∫ t
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
(1 − t) + α(t − η)
+
f (s)ds +
a
s
1
−
αη
1 − αη
0
}
{
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
t1 b.
− (1 − H(t − t1 ))t +
1 − αη
Similarly, if η ≤ t, (2.8) can be expressed
∫ η
∫ t
x(t) = −
(t − s)f (s)ds −
(t − s)f (s)ds −
0
t
+
1 − αη
{∫
η
η
∫
∫
η
(η − s)f (s)ds
0
t
(1 − s)f (s)ds +
0
tα
1 − αη
∫
(1 − s)f (s)ds +
η
1
}
(1 − s)f (s)ds
t
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
+
a
1 − αη
{
}
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
t1 b
− (1 − H(t − t1 ))t +
1 − αη
∫ η
∫ t
(1 − t) + α(t − η)
s(1 − t) + αη(t − s)
=
s
f (s)ds +
f (s)ds
1
−
αη
1 − αη
0
η
∫ 1
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
t(1 − s)
+
f (s)ds +
a
1
−
αη
1 − αη
t
{
}
(H(t − t1 ) − t) + α(t − ηH(t − t1 ))
− (1 − H(t − t1 ))t +
t1 b.
1 − αη
The lemma is proved.
110
M. LAKRIB AND R. GUEN
In view of Lemma 2.1 a useful operator will now be introduced so that fixed
points of the operator will be solutions of the initial value problem (1.1)-(1.3).
Lemma 2.2. Consider the initial value problem (1.1)-(1.3). Suppose that the function f : J × Rn × Rn → Rn satisfies conditions (H1) and (H2) or (H1) and (H2)∗ ,
and the impulse functions I1 , I2 : Rn × Rn → Rn satisfy condition (H3). Consider
the operator Γ : P C 1 (J, Rn ) → P C 1 (J, Rn ) defined, for x ∈ P C 1 (J, R) and t ∈ J,
by
∫ 1
(Γx)(t) =
G(t, s)f (s, x(s), x′ (s))ds
(2.9)
0
+V (t)I1 (x(t1 ), x′ (t1 )) + W (t)I2 (x(t1 ), x′ (t1 )),
where G, V and W are given in (2.5)-(2.7), with 0 < t1 < η < 1 and αη ̸= 1.
If x is a fixed point of Γ, then x is a solution of problem (1.1)-(1.3).
Proof. Let x ∈ P C 1 (J, Rn ) be a fixed point of Γ. To prove that x satisfies the
problem (1.1)-(1.3), just differentiate
∫ 1
x(t) =
G(t, s)f (s, x(s), x′ (s))ds + V (t)I1 (x(t1 ), x′ (t1 )) + W (t)I2 (x(t1 ), x′ (t1 ))
0
to obtain (1.1) and also show that (1.2)-(1.3) hold by direct computation.
The result below is needed to prove our main theorems (Theorem 3.1 and Theorem 3.2).
Lemma 2.3. Suppose that the function f : J × Rn × Rn → Rn satisfies conditions
(H1) and (H2) or (H1) and (H2)∗ , and the impulse functions I1 , I2 : Rn ×Rn → Rn
satisfy condition (H3). Then the operator Γ : P C 1 (J, Rn ) → P C 1 (J, Rn ) given by
(2.9) is completely continuous.
Proof. This follows in a standard step-by-step process and so omitted.
Remark. In the rest of this paper, M0 , M1 , V0 and W0 are the constants defined
by
∂G
M0 := sup |G(t, s)|, M1 := sup |
(t, s)|,
(t,s)∈J×J
(t,s)∈J×J ∂t
V0 := sup |V (t)|, V1 := sup |V ′ (t)|,
t∈J
t∈J
W0 := sup |W (t)| and W1 := sup |W ′ (t)|,
t∈J
t∈J
where G, V and W are given in (2.5)-(2.7), with 0 < t1 < η < 1 and αη ̸= 1.
3. Existence results
In this section we state and prove our existence results for problem (1.1)-(1.3).
Theorem 3.1. Assume that (H1), (H2) and (H3) hold. Further if
[
]
M1 ∥q∥L1
< 1,
M0 ∥q∥L1 < 1, M1 ∥q∥L1 < 1 and M0 ∥p∥L1 1 +
1 − M1 ∥q∥L1
then the initial value problem (1.1)-(1.3) has a solution on J.
(3.1)
THREE-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS
111
Proof. We will apply Theorem 1.1 to obtain the existence of a solution for x = Γx,
where Γ : P C 1 (J, Rn ) → P C 1 (J, Rn ) is given by (2.9).
Note that, by Lemma 2.3, Γ is a completely continuous operator. Then it suffices
to verify that all possible solutions of the family of problems
λx = Γx,
λ>1
(3.2)
are bounded a priori in P C 1 (J, Rn ) by a constant independent of λ.
Let x be a solution to (3.2) and let λ > 1 be such that λx = Γx. Then x|[0,t1 ]
satisfies, for each t ∈ [0, t1 ],
(∫ 1
)
x(t) = λ−1
G(t, s)f (s, x(s), x′ (s))ds .
0
It is straightforward to verify that
(∫
)
1
′
|x(t)| ≤ M0
[p(s) sup |x(u)| + q(s) sup |x (u)| + r(s)]ds .
u∈[0,t1 ]
0
(3.3)
u∈[0,t1 ]
Introduce the constants ξ = sup{|x(s)| : s ∈ [0, t1 ]} and ξ = sup{|x′ (s)| : s ∈ [0, t1 ]}
in (3.3) to obtain
|x(t)| ≤ M0 (∥p∥L1 ξ + ∥q∥L1 ξ + ∥r∥L1 )
from which we deduce that
ξ ≤ M0 (∥p∥L1 ξ + ∥q∥L1 ξ + ∥r∥L1 ).
(3.4)
In the other hand, x|[0,t1 ] is such that, for each t ∈ [0, t1 ],
(∫ 1
)
∂G
x′ (t) = λ−1
(t, s)f (s, x(s), x′ (s))ds .
0 ∂t
It is easy to verify that
(∫
)
1
′
′
|x (t)| ≤ M1
[p(s) sup |x(u)| + q(s) sup |x (u)| + r(s)]ds
u∈[0,t1 ]
0
(
)
≤ M1 ∥p∥L1 ξ + ∥q∥L1 ξ + ∥r∥L1
from which we get
u∈[0,t1 ]
)
(
ξ ≤ M1 ∥p∥L1 ξ + ∥q∥L1 ξ + ∥r∥L1
and this gives
M1 (∥p∥L1 ξ + ∥r∥L1 )
.
(1 − M1 ∥q∥L1 )
Replace (3.5) into (3.4) to obtain after some calculation
)
(
M1 ∥q∥L1
∥r∥L1
M0 1 + 1−M
1 ∥q∥L1
(
) := ξ1
ξ≤
M1 ∥q∥L1
1 − M0 ∥p∥L1 1 + 1−M
∥q∥
1
1
ξ≤
(3.5)
L
then, from (3.5), we get
ξ≤
M1 (∥p∥L1 ξ1 + ∥r∥L1 )
:= ξ 1 .
(1 − M1 ∥q∥L1 )
(3.6)
112
M. LAKRIB AND R. GUEN
Now, consider x|J . It satisfies, for each t ∈ J,
(∫ 1
x(t) = λ−1
G(t, s)f (s, x(s), x′ (s))ds
0
)
+V (t)I1 (x(t1 ), x′ (t1 )) + W (t)I2 (x(t1 ), x′ (t1 )) .
Therefore,
(∫
)
1
′
[p(s) sup |x(u)| + q(s) sup |x (u)| + r(s)]ds
|x(t)| ≤ M0
u∈[0,1]
0
+V0
sup
u∈[0,1]
|I1 (u, v)| + W0
|u|≤ξ1 ,|v|≤ξ 1
sup
|I2 (u, v)|.
(3.7)
|u|≤ξ1 ,|v|≤ξ 1
′
Denote ρ = sup{|x(t)| : t ∈ J}, ρ = sup{|x (t)| : t ∈ J}, I 1 = sup{|I1 (u, v)| : |u| ≤
ξ1 , |v| ≤ ξ 1 } and I 2 = sup{|I2 (u, v)| : |u| ≤ ξ1 , |v| ≤ ξ 1 }. From (3.7) we obtain
|x(t)| ≤ M0 (∥p∥L1 ρ + ∥q∥L1 ρ + ∥r∥L1 ) + V0 I 1 + W0 I 2
from which we get
ρ ≤ M0 (∥p∥L1 ρ + ∥q∥L1 ρ + ∥r∥L1 ) + V0 I 1 + W0 I 2 .
(3.8)
In the other hand, x|J is such that, for each t ∈ J,
( ∫ 1 ∂G
x′ (t) = λ−1
(t, s)f (s, x(s), x′ (s))ds
0 ∂t
)
+V ′ (t)I1 (x(t1 ), x′ (t1 )) + W ′ (t)I2 (x(t1 ), x′ (t1 ))
from which one can get
ρ ≤ M1 (∥p∥L1 ρ + ∥q∥L1 ρ + ∥r∥L1 ) + V1 I 1 + W1 I 2
and then
ρ≤
M1 (∥p∥L1 ρ + ∥r∥L1 ) + V1 I 1 + W1 I 2
.
1 − M1 ∥q∥L1
Replace (3.9) into (3.8) and compute to obtain
(
)
M1 ∥q∥L1
V1 I 1 +W1 I 2
∥r∥L1 + M0 1−M
+ V0 I 1 + W0 I 2
M0 1 + 1−M
∥q∥
1
1
1 ∥q∥L1
L
(
)
:= ρ1
ρ≤
M1 ∥q∥L1
1 − M0 ∥p∥L1 1 + 1−M
∥q∥
1
1
(3.9)
(3.10)
L
and then, from (3.9), we get
ρ≤
M1 (∥p∥L1 ρ1 + ∥r∥L1 ) + V1 I 1 + W1 I 2
:= ρ1 .
1 − M1 ∥q∥L1
(3.11)
Hence, there exists a constant ϱ = max{ρ1 , ρ1 } such that
∥x∥0 = max{sup |x(t)|, sup |x′ (t)|} ≤ ϱ.
t∈J
t∈J
This finish to show that all possible solutions of (3.2) are bounded in P C 1 (J, Rn )
by the constant ϱ.
As a result the conclusion of Theorem 1.1 holds and consequently the initial
value problem (1.1)-(1.3) has a solution x on J. This completes the proof.
THREE-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS
113
If hypothesis (H2) and condition (3.1) in Theorem 3.5 are replaced by hypothesis
(H2)∗ and condition (3.12) below, we obtain a new existence result. This result is
not a consequence of Theorem 3.5.
Theorem 3.2. Under conditions (H1), (H2)∗ and (H3), the initial value problem
(1.1)-(1.3) has a solution on J, provided that the function ψ in (H2)∗ satisfies
M0 ∥q∥L1 lim inf
r→+∞
+V0 lim inf
ψ(r)
r
sup |I1 (u, v)|
|u|,|v|≤r
+ W0 lim inf
r
r→+∞
sup |I2 (u, v)|
|u|,|v|≤r
< 1.
r
r→+∞
(3.12)
Proof. We claim that there exists r > 0 such that Γ(Br ) ⊆ Br where the operator Γ
is defined by (2.9) and Br is the closed ball in P C 1 (J, Rn ) with center 0 and radius
r. If this property is false, then for each r > 0 there exist xr ∈ Br and tr ∈ J such
that |(Γxr )(tr )| > r. From this it follows that
r < |(Γxr )(tr )|
∫ 1
=
G(tr , s)f (s, xr (s), xr ′ (s))ds
+ V (t)I1 (xr (t1 ), xr ′ (t1 )) + W (t)I2 (xr (t1 ), xr ′ (t1 ))
0
∫
1
≤ M0
0
q(s)ψ(max{|xr (s)|, |xr ′ (s)|})ds
+ V0 I1 (xr (t1 ), xr ′ (t1 )) + W0 I2 (xr (t1 ), xr ′ (t1 ))
≤ M0 ∥q∥L1 ψ(r) + V0
sup |I1 (u, v)| + W0
|u|,|v|≤r
sup |I2 (u, v)|.
|u|,|v|≤r
Hence, we obtain
1 ≤ M0 ∥q∥L1 lim inf
r→+∞
ψ(r)
r
sup |I1 (u, v)|
+ V0 lim inf
r→+∞
|u|,|v|≤r
r
sup |I2 (u, v)|
+ W0 lim inf
|u|,|v|≤r
r→+∞
r
,
which contradicts (3.12).
Let r > 0 be such that Γ : Br → Br . By Lemma 2.3, Γ is completely continuous,
and from Theorem 1.2 we conclude that the initial value problem (1.1)-(1.3) has a
solution x on J. The proof is finished.
From the proof of Theorem 3.2, we immediately obtain the following corollaries.
Corollary 3.3. Assume that (H1) and (H2)∗ hold. In addition, assume that the
following condition is satisfied.
(H3)∗ The impulse functions I1 , I2 : Rn × Rn → Rn are continuous and there exist
constants ai , bi ∈ R+ , i = 1, 2, 3, such that |I1 (x, y)| ≤ a1 |x| + a2 |y| + a3
and |I2 (x, y)| ≤ b1 |x| + b2 |y| + b3 |, for all x, y ∈ Rn .
Then the initial value problem (1.1)-(1.3) has a solution on J, provided that
M0 ∥q∥L1 lim inf
r→+∞
ψ(r)
+ V0 (a1 + a2 ) + W0 (b1 + b2 ) < 1.
r
114
M. LAKRIB AND R. GUEN
Corollary 3.4. Assume that (H1), (H2)∗ and the following condition hold.
(H3)∗∗ The impulse functions I1 , I2 : Rn → Rn are continuous and there exist constants ai , bi ∈ R+ , i = 1, 2, 3, αi , βi ∈ [0, 1), i = 1, 2, such that
|I1 (x, y)| ≤ a1 |x|α1 + a2 |y|α2 + a3 and |I2 (x, y)| ≤ b1 |x|β1 + b2 |y|β2 + b3 , for
all x, y ∈ Rn .
Then the initial value problem (1.1)-(1.3) has a solution on J, provided that
M0 ∥q∥L1 lim inf
r→+∞
ψ(r)
< 1.
r
We finish this paper with the following uniqueness result for solutions of problem
(1.1)-(1.3) which involves Lipschitz condition on the functions f , I1 and I2 .
Theorem 3.5. Assume that, for every x, y ∈ Rn , the function f (·, x, y) : J → Rn
is measurable and the following conditions hold.
(H2)∗∗ There exist functions p, q ∈ L1 (J, R+ ) such that, for almost every t ∈ J
and all x1 , x2 , y1 , y2 ∈ Rn ,
|f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ p(t)|x1 − x2 | + q(t)|y1 − y2 |.
(H3)∗∗∗ There exist constants αi , βi ∈ R+ , i = 1, 2, such that, for almost every
t ∈ J and all x1 , x2 , y1 , y2 ∈ Rn ,
|I1 (x1 , y1 ) − I1 (x2 , y2 )| ≤ α1 |x1 − x2 | + α2 |y1 − y2 |,
|I2 (x1 , y1 ) − I2 (x2 , y2 )| ≤ β1 |x1 − x2 | + β2 |y1 − y2 |.
Further if
M [∥p∥L1 + ∥q∥L1 ] + V [α1 + α2 ] + W [β1 + β2 ] < 1,
where M = max{M0 , M1 }, V = max{V0 , V1 } and V = max{W0 , W1 }, then the
initial value problem (1.1)-(1.3) has a unique solution on J.
Proof. Let x, y ∈ P C 1 (J, Rn ). We have, for each t ∈ J,
∫ 1
|(Γx)(t) − (Γy)(t)| ≤
|G(t, s)||f (s, x(s), x′ (s)) − f (s, y(s), y ′ (s))|ds
0
+|V (t)||I1 (x(t1 ), x′ (t1 )) − I1 (y(t1 ), y ′ (t1 ))|
+|W (t)||I2 (x(t1 ), x′ (t1 )) − I2 (y(t1 ), y ′ (t1 ))|
from which it is easy to deduce the following inequality
(
)
|(Γx)(t) − (Γy)(t)| ≤ M (∥p∥L1 + ∥q∥L1 ) + V (α1 + α2 ) + W (β1 + β2 ) ∥x − y∥0 .
Moreover, we have on the other hand
∫ 1
∂G
|f (s, x(s), x′ (s)) − f (s, y(s), y ′ (s))|ds
|(Γx)′ (t) − (Γy)′ (t)| ≤
(t,
s)
∂t
0
′
+ |V (t)| |I1 (x(t1 ), x′ (t1 )) − I1 (y(t1 ), y ′ (t1 ))|
′
′
′
( + |W (t)| |I2 (x(t1 ), x (t1 )) − I2 (y(t1 ), y (t1 ))|
≤ M (∥p∥L1 + ∥q∥L1 ) + V (α1 + α2 )
)
+W (β1 + β2 ) ∥x − y∥0 .
Hence,
(
)
∥Γx − Γy∥0 ≤ M [∥p∥L1 + ∥q∥L1 ] + V [α1 + α2 ] + W [β1 + β2 ] ∥x − y∥0 ,
that is, Γ is a contraction operator.
THREE-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS
115
Thus, Theorem 1.3 applies yielding the existence of the unique fixed point of Γ
and thus problem (1.1)-(1.3) has a unique solution on J.
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Mustapha Lakrib and Rahma Guen,
Laboratoire de Mathématiques, Université Djillali Liabès, BP 89 Sidi Bel Abbès, 22000,
Algérie
E-mail address: m.lakrib@univ-sba.dz, rah.guen@gmail.com