Electronic Journal of Differential Equations, Vol. 2003(2003), No. 52, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
MULTIPLE SOLUTIONS FOR NONRESONANCE IMPULSIVE
FUNCTIONAL DIFFERENTIAL EQUATIONS
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
Abstract. In this paper we investigate the existence of multiple solutions for
first and second order impulsive functional differential equations with boundary conditions. Our main tool is the Leggett and Williams fixed point theorem.
1. Introduction
This paper is concerned with the existence of three nonnegative solutions for
initial value problems for first and second order impulsive functional differential
equations with boundary conditions. Initially we consider the first order impulsive
functional differential equation,
y 0 (t) − λy(t) = f (t, yt ),
∆y|t=tk =
a.e. t ∈ [0, T ], t 6= tk , k = 1, . . . , m,
Ik (y(t−
k )),
y(t) = φ(t),
k = 1, . . . , m,
(1.2)
t ∈ [−r, 0], y(0) = y(T ),
+
(1.1)
(1.3)
+
where λ > 0, f : [0, T ] × D → R , Ik ∈ C(R, R ), 0 < r < ∞, 0 = t0 < t1 < · · · <
−
−
+
tm < tm+1 = T , ∆y|t=tk = y(t+
k ) − y(tk ), y(tk ) and y(tk ) represent the left and
right limits of y(t) at t = tk , respectively. D = {Ψ : [−r, 0] → R+ ; Ψ is continuous
everywhere except for a finite number of points t̄ at which Ψ(t̄) and Ψ(t̄+ ) exist
and Ψ(t̄− ) = Ψ(t̄)}, φ ∈ D. For any function y defined on [−r, T ] and any t ∈ J,
we denote by yt the element of D defined by
yt (θ) = y(t + θ),
θ ∈ [−r, 0].
Here yt (·) represents the history of the state from time t − r, to the present time t.
Later, we study the second order impulsive functional differential equations with
boundary conditions and fixed moments of the form
y 00 (t) − λy(t) = f (t, yt ),
a.e. t ∈ [0, T ], t 6= tk , k = 1, . . . , m,
∆y|t=tk = Ik (y(t−
k )),
0
∆y |t=tk = I k (y(t−
k )),
y(t) = φ(t),
t ∈ [−r, 0],
k = 1, . . . , m,
(1.5)
k = 1, . . . , m,
y(0) − y(T ) = µ0 ,
0
(1.6)
0
y (0) − y (T ) = µ1 ,
2000 Mathematics Subject Classification. 34A37, 34K25.
Key words and phrases. Nonresonance impulsive functional differential equations,
boundary conditions, fixed point, multiple solutions, cone, concave functional.
c
2003
Southwest Texas State University.
Submitted January 8, 2003. Published May 3, 2003.
1
(1.4)
(1.7)
2
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
EJDE–2003/52
where F , Ik , and φ are as in problem (1.1)-(1.3), I k ∈ C(R, R+ ) k = 1, . . . , m, and
µ0 , µ1 ∈ R.
Note that when µ0 = µ1 = 0 we have periodic boundary conditions. Differential
equations with impulses are a basic tool to study evolution processes that are
subjected to abrupt changes in their state. Such equations arise naturally from
a wide variety of applications, such as space-craft control, inspection processes in
operations research, drug administration, and threshold theory in biology. There
has been a significant development in impulse theory, especially in the area of
impulsive differential equations with fixed moments; see the monographs of Bainov
and Simeonov [7], Lakshmikantham et al. [15], and Samoilenko and Perestyuk [17]
and the papers of Benchohra et al [8], Franco et al [11] and the references cited
therein.
The existence of multiple solutions for boundary value problems for impulsive
differential equations was studied by Guo and Liu [13] and Agarwal and O’Regan [2].
Notice that when the impulses are absent (i.e. Ik = 0, k = 1, . . . , m) the existence
of three solutions and multiple solutions for ordinary differential equations was
studied in [1, 3, 4, 5, 6, 14].
The main theorems of this note extend some existence results in the above literature to the impulsive case. Our approach here is based on the Leggett and Williams
fixed point theorem in cones [16].
2. Preliminaries
In this section, we introduce notation, definitions, and preliminary facts which
are used for the rest of this paper.
By C(J, R) we denote the Banach space of all continuous functions from J into
R with the norm
kyk∞ := sup{|y(t)| : t ∈ J}.
L1 (J, R) denotes the Banach space of measurable functions y : J → R which are
Lebesgue integrable, with
Z T
kykL1 =
|y(t)|dt.
0
Let (a, b) be an open interval and AC i (a, b), R) be the space of i-times differentiable
functions y : (a, b) → R, whose ith derivative, y (i) , is absolutely continuous.
Let (E, k · k) be a Banach space and C be a cone in E. By concave nonnegative
continuous functional ψ on C we mean a continuous mapping ψ : C → [0, ∞) with
ψ(λx + (1 − λ)y) ≥ λψ(x) + (1 − λ)ψ(y)
for all x, y ∈ C λ ∈ [0, 1].
To define solutions of (1.1)-(1.3) we shall consider the space
P C = y : [0, T ] → R : yk ∈ C(Jk , R), k = 0, . . . , m and there exist y(t−
k)
+
−
and y(tk ) with y(tk ) = y(tk ), k = 1, . . . , m
which is a Banach space with the norm
kykP C = max{kyk kJk , k = 0, . . . , m},
where yk is the restriction of y to Jk = (tk , tk+1 ], k = 0, . . . , m. Set Ω = {y :
[−r, T ] → R : y ∈ D ∩ P C}. Then Ω is a Banach space with norm
kykΩ = sup{|y(t)| : t ∈ [−r, T ]}.
EJDE–2003/52
MULTIPLE SOLUTIONS FOR NONRESONANCE
3
Definition 2.1. A map f : J × D → R+ is said to be L1 -Carathéodory if
(i) t 7→ f (t, u) is measurable for each y ∈ D;
(ii) u →
7 f (t, u) is continuous for almost all t ∈ J;
(iii) For each q > 0, there exists hq ∈ L1 (J, R+ ) such that |f (t, u)| ≤ hq (t) for
all kukD ≤ q and for almost all t ∈ J.
Our consideration is based on the following fixed point theorem given by Leggett
and Williams in 1979 [16](see also Guo and Lakshmikantham [12]).
Theorem 2.2. Let E be a Banach space, C ⊂ E a cone of E and R > 0 a constant.
Let CR = {y ∈ C : kyk < R}. Suppose a concave nonnegative continuous functional
ψ exists on the cone C with ψ(y) ≤ kyk for y ∈ C R , and let N : C R → C R be a
continuous compact map. Assume there are numbers r, L and K with 0 < r < L <
K ≤ R such that
(A1) {y ∈ C(ψ, L, K) : ψ(y) > L} =
6 ∅ and ψ(N (y)) > L for all y ∈ C(ψ, L, K);
(A2) kN (y)k < r for all y ∈ C r ;
(A3) ψ(N (y)) > L for all y ∈ C(ψ, L, R) with kN (y)k > K, where CK = {y ∈
C : kyk ≤ K} and
C(ψ, L, K) = {y ∈ C : ψ(y) ≥ L and kyk ≤ K}.
Then N has at least three fixed points y1 , y2 , y3 in C R . Furthermore, we have
y1 ∈ Cr ,
y2 ∈ {y ∈ C(ψ, L, R) : ψ(y) > L}
y3 ∈ C R − {C(ψ, L, R) ∪ C r }.
3. First Order Impulsive FDEs
Let us start by defining what we mean by a solution of problem (1.1)–(1.3).
Definition 3.1. A function y ∈ Ω ∩ ∪m
k=0 AC((tk , tk+1 ), R) is said to be a solution
of (1.1)–(1.3) if y satisfies y 0 (t) − λy(t) = f (t, yt ) a.e. on J\{t1 , . . . , tm }, and
∆y|t=tk = Ik (y(t−
k )), k = 1, . . . , m, y(t) = φ(t), t ∈ [−r, 0], and y(0) = y(T ).
For the next theorem we need the following assumptions:
(H1) There exist constants ck such that |Ik (x)| ≤ ck , k = 1, . . . , m for each x ∈ R
(H2) There exist a function g : [0, ∞) → [0, ∞) continuous and non-decreasing,
a function p ∈ L1 (J, R+ ), r > 0, and a constant 0 < M ≤ 1 such that
M p(t)g(kuk) ≤ |H(t, s)f (t, u)| ≤ p(t)g(kuk)
for each (t, s, u) ∈ J × J × D, and
Z T
m
X
1
ck + g(r)
p(t)dt < r;
1 − e−λT
0
k=1
(H3) There exist L > r and an interval [a, b] ⊂ (0, T ) such that
min
t∈[a,b]
M
m
X
m
X
H(t, tk )Ik (y(tk )) ≥ M
ck ,
k=1
k=1
m
X
k=1
Z
ck + g(L)
T
p(s)ds > L;
0
4
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
EJDE–2003/52
(H4) There exist R, K, 0 < M1 < M, M1−1 L < K ≤ R such that
Z T
m
X
H(t, tk )Ik (y(tk )) +
H(t, s)f (s, ys )ds
min
t∈[a,b]
0
k=1
≥ M1 M∗
m
X
Z
T
ck + g(R)
p(s)ds
0
k=1
where M∗ = sup(t,s)∈[0,T ]×[0,T ] |H(t, s)| and
Z T
m
X
1
ck + g(R)
p(t)dt < R .
1 − e−λT
0
k=1
Theorem 3.2. Assume (H1)–(H4) are satisfied. Then problem (1.1)–(1.3) has at
least three solutions.
Proof. We transform the problem into a fixed point problem. Consider the operator,
N : Ω → Ω defined by
(
φ(t),
if t ∈ [−r, 0];
N (y)(t) = R T
Pm
−
H(t,
s)f
(s,
y
)ds
+
H(t,
t
)I
(y(t
)),
if t ∈ [0, T ],
s
k k
k=1
k
0
where
−λT
H(t, s) = (e
−1
− 1)
(
e−λ(T +s−t) , 0 ≤ s ≤ t ≤ T,
e−λ(s−t) ,
0 ≤ t < s ≤ T.
Remark 3.3. It is easy to show that the fixed points of N are solutions to the
problem (1.1)-(1.3); see [9].
We shall show that N satisfies the assumptions of Theorem 2.2. This will be
done in several steps.
Step 1: N is continuous. Let {yn } be a sequence such that yn → y in Ω. Then
|N (yn (t)) − N (y(t))|
Z T
m
X
|H(t, s)||f (s, yns ) − f (s, ys )|ds +
|H(t, tk )||Ik (yn (tk )) − Ik (y(tk ))|
≤
0
k=1
1
≤
1 − e−λT
Z
T
|f (s, yns ) − f (s, ys )|ds +
0
m
X
1
|Ik (yn (tk )) − Ik (y(tk ))|.
1 − e−λT
k=1
Since the functions H, Ik , k = 1, . . . , m are continuous and f is an L1 -Carathéodory,
kN (yn ) − N (y)kΩ
≤
m
X
1
1
1 +
kf
(.,
y
)
−
f
(.,
y)k
|Ik (yn (tk )) − Ik (y(tk ))|
n
L
1 − e−λT
1 − e−λT
k=1
which approaches zero as n → ∞.
Step 2: N maps bounded sets into bounded sets in Ω. Indeed, it is sufficient
to show that for any q > 0 there exists a positive constant ` such that for each
y ∈ Bq = {y ∈ Ω : kykΩ ≤ q} one has kN (y)kΩ ≤ `. Let y ∈ Bq . Then for t ∈ [0, T ]
we have
Z T
m
X
N (y)(t) =
H(t, s)f (s, ys )ds +
H(t, tk )Ik (y(tk )).
0
k=1
EJDE–2003/52
MULTIPLE SOLUTIONS FOR NONRESONANCE
5
By (H2) we have for each t ∈ [0, T ]
Z T
m
X
|H(t, tk )||Ik (y(tk ))|
|N (y)(t)| ≤
|H(t, s)||f (s, ys )|d s +
0
k=1
T
Z
|H(t, s)|hq (s)d s +
≤
0
m
X
|H(t, tk )|ck .
k=1
Then for each h ∈ N (Bq ) we have
1
kN (y)kΩ ≤
1 − e−λT
Z
T
hq (s)d s +
0
m
X
ck := `.
k=1
Step 3: N maps bounded set into equicontinuous sets of Ω. Let τ1 , τ2 ∈ [0, T ],
τ1 < τ2 and Bq be a bounded set of Ω as in Step 2. Let y ∈ Bq and t ∈ [0, T ] we
have
Z T
m
X
N (y)(t) =
H(t, s)f (s, ys )d s +
H(t, tk )Ik (y(tk )).
0
k=1
Then
|N (y)(τ2 ) − N (y)(τ1 )|
Z T
m
X
≤
|H(τ2 , s) − H(τ1 , s)|hq (s)ds +
|H(τ2 , tk ) − H(τ1 , tk )|ck .
0
k=1
As τ2 → τ1 the right-hand side of the above inequality tends to zero.
As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we
can conclude that N : Ω → Ω is completely continuous.
Let C = {y ∈ Ω : y(t) ≥ 0 for t ∈ [−r, T ]} be a cone in Ω. Since f, H, Ik ,
k = 1, . . . , m are positive functions, then N (C) ⊂ C and N : C R → C R is compact.
By (H1), (H2), (H4) we can show that if y ∈ C R then N (y) ⊂ C R . Let ψ : C →
[0, ∞) defined by ψ(y) = mint∈[a,b] y(t). It is clear that ψ is a nonnegative concave
continuous functional and ψ(y) ≤ kykΩ for y ∈ C R . Now it remains to show that
the hypotheses of Theorem 2.2 are satisfied. First notice that condition (A2) of
Theorem 2.2 holds since for y ∈ C r , and from (H1) and (H2) we have
Z T
m
X
|N (y)(t)| ≤
|H(t, s)||f (s, ys )|ds +
|H(t, tk )|Ik (y(tk ))|
0
Z
k=1
T
≤
g(kys k)p(s)ds +
0
≤ g(r)kpkL1 +
m
X
|H(t, tk )|Ik (y(tk ))|
k=1
m
X
1
1 − e−λT
ck < r.
k=1
for t ∈ [−r, T ]. By the definition of C(ψ, L, K), y
Let K ≥ L and y(t) = L+K
2
belongs to C(ψ, L, K). Then y ∈ {y ∈ C(ψ, L, K) : ψ(y) > L}. Also if y ∈
C(ψ, L, K) then
Z T
m
X
ψ(N (y)) = min
H(t, tk )Ik (y(tk )) +
H(t, s)f (s, ys )ds .
t∈[a,b]
k=1
0
6
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
EJDE–2003/52
Then from (H3) we have
ψ(N (y)) = min
t∈[a,b]
≥ M
m
X
H(t, tk )Ik (y(tk )) +
≥M
T
Z
g(kys k)p(s)ds
ck + M
0
k=1
m
X
H(t, s)f (s, ys )ds
0
k=1
m
X
!
T
Z
Z
T
ck + g(L)
p(s)ds > L.
0
k=1
So the conditions (A1) and (A2) of Theorem 2.2 are satisfied.
Finally we will be prove that (A3) of Theorem 2.2 holds. Let y ∈ C(ψ, L, R)
with kN (y)kΩ > K Thus
Z T
m
X
ψ(N (y)) = min
H(t, tk )Ik (y(tk )) +
H(t, s)f (s, ys )ds
t∈[a,b]
≥ M1 M∗
0
k=1
m
X
Z
T
ck + g(R)
k=1
p(s)ds
0
≥ M1 kN (y)kΩ > M1 K > L.
Thus condition (A3) holds. Then Leggett and Williams fixed point theorem implies
that N has at least three fixed points y1 , y2 , y3 which are solutions to problem (1.1)–
(1.3). Furthermore, we have
y1 ∈ Cr , y2 ∈ {y ∈ C(ψ, L, R) : ψ(y) > L},
y3 ∈ CR − {C(ψ, L, R) ∪ (Cr )}.
4. Second Order Impulsive FDEs
In this section we give an existence result for the boundary-value problem (1.4)–
(1.7).
1
Definition 4.1. A function y ∈ Ω ∩ ∪m
k=0 AC ((tk , tk+1 ), R) is said to be a solution
00
of (1.4)–(1.7) if y satisfies y (t) − λy(t) = f (t, yt ) a.e. on J\{t1 , . . . , tm } and
−
0
the conditions ∆y|t=tk = Ik (y(t−
k )), ∆y |t=tk = I k (y(tk )), k = 1, . . . , m, y(t) =
0
0
φ(t), t ∈ [−r, 0], y(0) − y(T ) = µ0 , y (0) − y (T ) = µ1 .
We now consider the “linear problem”
y 00 (t) − λy(t) = g(t), t 6= tk , k = 1, . . . , m,
(4.1)
1
subjected to the conditions (1.5), (1.6), (1.7), and where g ∈ L ([tk , tk+1 ], R). Note
that (1.5)–(1.7), (4.1) is not really a linear problem since the impulsive functions are
not necessarily linear. However, if Ik , I¯k , k = 1, . . . , m are linear, then (1.5)–(1.7),
(4.1) is a linear impulsive problem.
We need the following auxiliary result:
1
Lemma 4.2. y ∈ Ω ∩ ∪m
k=0 AC ((tk , tk+1 ), R) is a solution of (1.5)–(1.7), (4.1), if
and only if y ∈ Ω is a solution of the impulsive integral functional equation,

t ∈ [−r, 0],

φ(t),
RT
y(t) =
(4.2)
M (t, s)h(s)ds + M (t, 0)µ1 + N (t, 0)µ0

 0 Pm
¯
+ k=1 [M (t, tk )Ik (y(tk )) + N (t, tk )Ik (y(tk ))], t ∈ [0, T ],
EJDE–2003/52
MULTIPLE SOLUTIONS FOR NONRESONANCE
7
where
−1
M (t, s) = √ √
2 λ(e λT
( √
√
e λ(T +s−t) + e λ(t−s) ,
√
√
− 1) e λ(T +t−s) + e λ(s−t) ,
0 ≤ s ≤ t ≤ T,
0 ≤ t < s ≤ T,
and
∂
1
√
N (t, s) = M (t, s) =
λT
∂t
− 1)
2(e
( √
√
e√λ(T +s−t) −
e λ(t−s) ,
√
e λ(s−t) − e λ(T +t−s) ,
0 ≤ s ≤ t ≤ T,
0 ≤ t < s ≤ T.
We omit the proof of this lemma since it is similar to the proof of results in [10].
We are now in a position to state and prove our existence result for problem
(1.4)-(1.7). We first list the following hypotheses:
(H5) There exist constants dk such that |I k (x)| ≤ dk , k = 1, . . . , m for each
x∈R
(H6) There exist a function g ∗ : [0, ∞) → [0, ∞) continuous and non-decreasing,
a function p ∈ L1 (J, R+ ), r∗ > 0, and 0 < M ∗ ≤ 1 such that
M ∗ p(t)g ∗ (kuk) ≤ |M (t, s)f (t, u)| ≤ p(t)g ∗ (kuk)
for each (t, s, u) ∈ J × J × D, and
C
m
X
(ck + dk ) + C∗ [|µ1 | + |µ0 |] +
∗
∗
p(s)ds < r∗
|M (t, s)|g (r )
sup
(t,s)∈[0,T ]×[0,T ]
k=1
T
Z
0
where
C = max(
|M (t, s)|,
sup
(t,s)∈[0,T ]×[0,T ]
|N (t, s)|),
sup
(t,s)∈[0,T ]×[0,T ]
C∗ = max( sup |M (t, 0)|, sup |N (t, 0)|)
t∈[0,T ]]
∗
∗
t∈[0,T ]
∗
(H7) There exist L > r , 0 < M ≤ 1 and an interval [a, b] ⊂ (0, T ) such that
Z T
m
X
M (t, s)f (s, ys )ds
min
[M (t, tk )Ik (y(tk )) + N (t, tk )I k (y(tk ))] +
t∈[a,b]
0
k=1
≥ M ∗ min
t∈[a,b]
M (t, 0)µ1 + N (t, 0)µ0 + g ∗ (L∗ )
Z
T
p(s)d s +
0
m
X
[ck + dk ]
k=1
∗
>L
(H8) There exist R∗ , K, 0 < M1∗ ≤ M ∗ such that M1∗−1 L < K ≤ R∗ ,
Z T
m
X
C
[ck + dk ] + C∗ (|µ0 | + |µ1 |) +
sup
|M (t, s)|g ∗ (R∗ )
p(s)ds < R∗
(t,s)∈[0,T ]×[0,T ]
k=1
and
min
t∈[a,b]
M (t, 0)µ1 + N (t, 0)µ0 +
m
X
M (t, tk )Ik (y(tk ))
k=1
Z T
+ N (t, tk )I k (y(tk )) +
M (t, s)f (s, ys )ds
0
∗
≥ M1 C∗ (|µ1 | + |µ0 |)
0
8
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
+
∗
∗
Z
T
|M (t, s)|g (R )
sup
(t,s)∈[0,T ]×[0,T ]
p(s)ds + C
0
EJDE–2003/52
m
X
[ck + dk ] .
k=1
Theorem 4.3. Suppose that hypotheses (H1), (H5) − (H8) are satisfied. Then the
problem (1.4)–(1.7) has at least three positive solutions.
Proof. We transform the problem into a fixed point problem. Consider the operator
N1 : Ω → Ω defined by

t ∈ [−r, 0],

φ(t),
RT
N1 (y)(t) =
M (t, s)f (s, ys )ds + M (t, 0)µ1 + N (t, 0)µ0

 0 Pm
+ k=1 [M (t, tk )Ik (y(tk )) + N (t, tk )I k (y(tk ))], t ∈ [0, T ],
As in Theorem 3.2 we can show that N1 is compact. Now we prove only that the
hypotheses of Theorem 2.2 are satisfied.
Let C = {y ∈ Ω : y(t) ≥ 0 for t ∈ [−r, T ]} be a cone in Ω. It is clear that
N1 (C) ⊂ C and N1 : C R∗ → C R∗ is compact. By (H1), (H5), (H6), (H8) we can
show that if y ∈ C R∗ then N1 (y) ∈ C R∗ . Let ψ : C → [0, ∞) defined by ψ(y) =
mint∈[a,b] y(t). It is clear that ψ is a nonnegative concave continuous functional and
ψ(y) ≤ kykΩ for y ∈ C R∗ . Now it remains to show that the hypotheses of Theorem
2.2 are satisfied. First notice that condition (A2) holds since for y ∈ C r∗ , we have
from (H1), (H5), (H6),
Z T
|N1 (y)(t)| ≤
|M (t, s)||f (s, ys )|d s + |M (t, 0)||µ1 | + |N (t, 0)||µ0 |
0
+
Z
≤
m
X
[|M (t, tk )|ck + |N (t, tk )|dk ]
k=1
T
∗
g (kys k)p(s)d s + C∗ (|µ1 | + |µ0 |) + C
0
≤
(ck + dk )
k=1
∗
t∈[0,T ]×[0,T ]
m
X
∗
Z
|M (t, s)|g (L )
sup
+C
m
X
T
p(s)ds + C∗ (|µ1 | + |µ0 |)
0
(ck + dk ) < r∗ .
k=1
∗
∗
∗
∗
Let K ≥ L and y(t) = L +K
for t ∈ [−r, T ]. By the definition of C(ψ, L∗ , K ∗ ), y
2
∗
∗
is in C(ψ, L , K ). Then y ∈ {y ∈ C(ψ, L∗ , K ∗ ) : ψ(y) > L∗ }. Also if y ∈
C(ψ, L∗ , K ∗ ) we have
Z T
ψ(N1 (y)) = min
M (t, s)f (s, ys )ds + M (t, 0)µ1 + N (t, 0)µ0
t∈[a,b]
0
+
m
X
[M (t, tk )Ik (y(tk )) + N (t, tk )I k (y(tk )) .
k=1
Then from (H7) we get
Z
m
X
∗
∗
∗
ψ(N1 (y)) ≥ M min M (t, 0)µ1 + N (t, 0)µ0 +
(ck + dk ) + g (L )
t∈[a,b]
∗
>L .
k=1
0
T
p(s)ds
EJDE–2003/52
MULTIPLE SOLUTIONS FOR NONRESONANCE
9
So conditions (A1) and (A2) of Theorem 2.2 are satisfied.
Finally to see that (A3) holds let y ∈ C(ψ, L∗ , R∗ ) with kN1 (y)kΩ > K ∗ then
from (H8) we have
ψ(N1 (y))
m
X
≥ M1∗
(ck + dk ) + C∗ |µ1 | + C|µ0 | +
t∈[0,T ]×[0,T ]
k=1
|M (t, s)|g ∗ (R∗ )
sup
≥
M1∗
≥
M1∗ kN1 (y)kΩ
C∗ |µ1 | + C|µ0 | + C
>
M1∗ K
∗
m
X
k=1
∗
(ck + dk ) +
Z
T
p(s)ds
0
∗
∗
Z
|M (t, s)|g (R )
sup
t∈[0,T ]×[0,T ]
T
p(s)ds
0
>L .
Thus condition (A3) for Theorem 2.2 holds. As consequence of Leggett and Williams
theorem we deduce that N1 has at least three fixed points y1 , y2 , y3 which are solutions to problem (1.4)–(1.7). Furthermore, we have
y1 ∈ Cr∗ ,
y2 ∈ {y ∈ C(ψ, L∗ , R∗ ) : ψ(y) > L∗ },
y3 ∈ CR∗ −{C(ψ, L∗ , R∗ )∪Cr∗ }.
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10
MOUFFAK BENCHOHRA & ABDELGHANI OUAHAB
EJDE–2003/52
Mouffak Benchohra
Laboratoire de Mathématiques, Université de Sidi Bel Abbes, BP 89 2000 Sidi Bel Abbes,
Algérie
E-mail address: benchohra@univ-sba.dz
Abdelghani Ouahab
Laboratoire de Mathématiques, Université de Sidi Bel Abbes, BP 89 2000 Sidi Bel Abbes,
Algérie
E-mail address: ouahabi abdelghani@yahoo.fr