Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 56, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND TOPOLOGICAL STRUCTURE OF SOLUTION
SETS FOR φ-LAPLACIAN IMPULSIVE DIFFERENTIAL
EQUATIONS
JOHNNY HENDERSON, ABDELGHANI OUAHAB, SAMIA YOUCEFI
Abstract. In this article, we present results on the existence and the topological structure of the solution set for initial-value problems for the first-order
impulsive differential equation
(φ(y 0 ))0 = f (t, y(t)),
a.e. t ∈ [0, b],
−
−
y(t+
k ) − y(tk ) = Ik (y(tk )),
0 +
0 −
0
y (tk ) − y (tk ) = I¯k (y (t−
k )),
0
y(0) = A,
k = 1, . . . , m,
k = 1, . . . , m,
y (0) = B,
where 0 = t0 < t1 < · · · < tm < tm+1 = b, m ∈ N. The functions Ik , I¯k characterize the jump in the solutions at impulse points tk , k = 1, . . . , m. For the
final result of the paper, the hypotheses are modified so that the nonlinearity
f depends on y 0 , but the impulsive conditions and initial conditions remain
the same.
1. Introduction
The dynamics of many processes in physics, population dynamics, biology and
medicine may be subject to abrupt changes such as shocks or perturbations (see
for instance [3, 9] and the references therein). These perturbations may be seen
as impulses. For instance, in the periodic treatment of some diseases, impulses
correspond to the administration of a drug treatment or a missing product. In
environmental sciences, impulses correspond to seasonal changes of the water level
of artificial reservoirs. Their models may be described by impulsive differential
equations. The mathematical study of boundary value problems for differential
equations with impulses was first considered in 1960 by Milman and Myshkis [11]
and then followed by a period of active research which culminated in 1968 with the
monograph by Halanay and Wexler [8].
Various mathematical results (existence, asymptotic behavior, and so on) have
been obtained so far (see [1, 2, 4, 7, 10, 13, 12] and the references therein).
In this paper we shall establish an existence theory for initial-value problems
with impulse effects. We will treat two cases. For the first case, the problem has
2000 Mathematics Subject Classification. 34A37, 34K45.
Key words and phrases. φ-Laplacian; fixed point theorems; impulsive solution;
compactness.
c
2012
Texas State University - San Marcos.
Submitted January 20, 2012. Published April 6, 2012.
1
2
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
the form
(φ(y 0 (t)))0 = f (t, y(t)), t ∈ J := [0, b],
y(t+
k)
0 +
y (tk )
−
−
−
y(t−
k ) = Ik (y(tk )),
−
¯
y 0 (t−
k ) = Ik (y(tk )),
0
y(0) = A,
t 6= tk , k = 1, . . . , m,
(1.1)
k = 1, . . . , m,
(1.2)
k = 1, . . . , m,
(1.3)
y (0) = B,
(1.4)
where f : [0, b] × R → R is a given function, Ik , I¯k ∈ C(R, R), φ : R → R is a
suitable monotone homeomorphism, and A, B ∈ R. For this setting, the proofs of
the two results presented, while involving some cases, are quite straight for word.
The second case is when the second member f may depend on y 0 , and the problem
has the the form
(φ(y 0 (t)))0 = f (t, y(t), y 0 (t)), t ∈ J := [0, b],
t 6= tk , k = 1, . . . , m,
−
−
y(t+
k = 1, . . . , m,
k ) − y(tk ) = Ik (y(tk )),
−
0 +
0 −
¯
y (tk ) − y (tk ) = Ik (y(tk )), k = 1, . . . , m,
0
y(0) = A,
y (0) = B,
(1.5)
(1.6)
(1.7)
(1.8)
where f : [0, b] × R × R → R is a given function, Ik , I¯k , φ and A, B are as in problem
(1.1)–(1.4). Because of the dependency on y 0 , the proof of the result presented is
somewhat more involved. Of course, the second case also covers the first case when
f is independent of y 0 .
The goals of this article are to provide some existence results and to establish
the compactness of solution sets of the above problems.
2. Preliminaries
In this section, we recall from the literature some notation, definitions, and
auxiliary results which will be used throughout this paper. Let J = [0, b] be an
interval of R. C([0, b], R) is the Banach space of all continuous functions from [0, b]
into R with the norm
kyk∞ = sup |y(t)|.
t∈[0,b]
1
L ([0, b], R) denotes the Banach space of Lebesgue integrable functions, with the
norm
Z b
kykL1 =
|y(s)|ds.
0
Definition 2.1. A map f : [p, q] × R → R is said to be L1 -Carathéodory if
(i) t → f (t, y) is measurable for all y ∈ R,
(ii) y → f (t, y) is continuous for almost each t ∈ [p, q],
(iii) for each r > 0, there exists hr ∈ L1 ([p, q], R+ ) such that |f (t, y)| ≤ hr (t)
for almost each t ∈ [p, q] and for all |y| ≤ r.
Lemma 2.2 (Grönwall-Bihari [5]). Let I = [p, q] and let u, g : I → R be positive
continuous functions. Assume there exist c > 0 and a continuous nondecreasing
function h : [0, ∞) → (0, +∞) such that
u(t) ≤ c + g(s)h(u(s))ds,
∀t ∈ I.
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EXISTENCE AND TOPOLOGICAL STRUCTURE
Then
u(t) ≤ H −1
Z
t
g(s)ds ,
3
∀t ∈ I,
p
provided
Z
+∞
c
dy
>
h(y)
Z
q
g(s)ds,
p
where H −1 refers to inverse of the function H(u) =
Ru
dy
c h(y)
for u ≥ c.
3. Main results
Let J0 = [0, t1 ], Jk = (tk , tk+1 ], k = 1, . . . , m, and let yk be the restriction of a
function y to Jk . To define solutions for (1.1) − (1.4), consider the space
P C = y : [0, b] → R, yk ∈ C(Jk , R), k = 0, . . . , m, such that
+
−
y(t−
k ) and y(tk ) exist and satisfy y(tk ) = y(tk ) for k = 1, . . . , m .
Endowed with the norm
kykP C = max{kyk k∞ , k = 0, . . . , m},
kyk k∞ = sup |y(t)|,
t∈Jk
P C is a Banach space.
P C 1 = y ∈ P C : yk0 ∈ C(Jk , R), k = 0, . . . , m, such that
0 +
0 −
0
y 0 (t−
k ) and y (tk ) exist and satisfy y (tk ) = y (tk ) for k = 1, . . . , m .
is a Banach space with the norm
kykP C 1 = max(kykP C , ky 0 kP C ),
or kykP C 1 = kykP C + ky 0 kP C .
Theorem 3.1 (Nonlinear Alternative [6]). Let X be a Banach space with C ⊂ X
closed and convex. Assume U is a relatively open subset of C with 0 ∈ U and
G : U → C is a compact map. Then either,
(i) G has a fixed point in U ; or
(ii) there is a point u ∈ ∂U and λ ∈ (0, 1) with u = λG(u).
Theorem 3.2. Suppose that:
(H1) f : [0, b] × R → R is an Carathéodory function and Ik , I¯k ∈ C(R, R).
(H2) There exist p ∈ L1 (J, R+ ) such that |f (t, u)| ≤ p(t) for a.e. t ∈ J
are satisfied. Then (1.1)-(1.4) has at least one solution and the solutions set
S = {y ∈ P C([0, b], R) : y is a solution of (1.1)-(1.4)}
is compact.
Proof. The proof involves several steps.
Step 1: Consider the problem
(φ(y 0 ))0 = f (t, y) t ∈ [0, t1 ],
y(0) = A,
y 0 (0) = B,
and the map N1 : C([0, t1 ], R) → C([0, t1 ], R),
Z t
Z
y 7→ (N1 y)(t) = A +
φ−1 [φ(B) +
0
s
f (τ, y)dτ ]ds.
0
Clearly the fixed points of N1 are solutions of the problem (3.1).
(3.1)
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J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
To apply the nonlinear alternative of Leray-Schauder type, we first show that
N1 is completely continuous. The proof will be given in several steps.
Claim 1: N sends bounded sets into bounded sets in C([0, t1 ], R). Let
y ∈ D = {y ∈ C([0, t1 ], R) : kyk∞ ≤ q}.
Then for each t ∈ [0, t1 ], we have
t
Z
|φ−1 [φ(B) +
|(N1 y)(t)| ≤ |A| +
s
Z
f (τ, y)]|dτ,
0
0
since
Z
|φ(B) +
s
Z
s
f (τ, y)dτ | ≤ |φ(B)| +
0
|f (τ, y)|dτ
Z0 s
≤ |φ(B)| +
|p(τ )|dτ
0
≤ |φ(B)| + (kpkL1 )t1 ,
it follows that
Z
s
f (τ, y)dτ ] ∈ B(0, l1 ),
[φ(B) +
0
where l1 = |φ(B)| + (kpkL1 )t1 . Since φ−1 is continuous,
|φ−1 (x)| < ∞.
sup
x∈B(0,l1 )
Thus
kN1 (y)k∞ ≤ |A| + t1
sup
|φ−1 (x)| := r
x∈B(0,l1 )
Claim 2: N1 maps bounded sets into equicontinuous sets. Let l1 , l2 ∈ [0, t1 ],
l1 < l2 and D be a bounded set of C([0, t1 ], R) as in Claim 1. Let y ∈ D. Then
Z t
0
−1
|(N1 y) (t)| = |φ [φ(B) +
f (s, y)ds] − φ−1 (φ(B))|
0
Z t
≤ |φ−1 [φ(B) +
f (s, y)ds]| + |B|
0
≤
sup
|φ−1 (x)| + |B| := r0 .
x∈B(0,l1 )
By the mean value theorem, we obtain
|(N1 y)(l2 ) − (N1 y)(l1 )| = |(N1 y)0 (ξ)(l2 − l1 )| ≤ r0 |l2 − l1 |.
As l2 → l1 the right-hand side of the above inequality tends to zero.
Claim 3: N1 is continuous. Let (yn )n∈N be a sequence such that yn → y in
C([0, t1 ], R). Then there is an integer q such that kyn k∞ ≤ q for all n ∈ N and
kyk∞ ≤ q, yn ∈ D and y ∈ D. We have
|(N1 yn )(t) − (N1 y)(t)|
Z t
Z s
Z
−1
−1
≤
|φ [φ(B) +
f (τ, yn )dτ ] − φ [φ(B) +
0
0
s
By the dominated convergence theorem, we have
Z s
Z s
|φ(B) +
f (τ, yn )dτ − φ(B) −
f (τ, y)dτ | → 0
0
f (τ, y)dτ ]|ds.
0
0
as n → ∞,
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EXISTENCE AND TOPOLOGICAL STRUCTURE
5
and since φ−1 is continuous. Then by the dominated convergence theorem, we have
kN1 (yn ) − N1 (y)k∞
Z t1
Z
≤
|φ−1 [φ(B) +
0
s
f (τ, yn )dτ ] − φ−1 [φ(B) +
Z
0
s
f (τ, y)dτ ]|ds → 0,
0
as n → ∞. Thus N1 is continuous.
Claim 4: A priori estimate. Now we show that there exists a constant M0 such
that kyk∞ ≤ M0 where y is a solution if the problem (3.1). Let y a solution of
(3.1):
Z t
Z s
y(t) = A +
φ−1 [φ(B) +
f (τ, y(τ )dτ )]ds.
0
0
Then
Z
t
|φ−1 [φ(B) +
|y(t)| ≤ |A| +
t
|φ−1 [φ(B) +
≤ |A| +
0
Z
s
f (τ, y(τ ))dτ ]|ds
0
0
Z
Z
Z
s
p(τ )dτ ]|ds
0
t
|φ−1 [φ(B) + kpk1 t1 ]|ds
0
Z t
≤ |A| + sup
ds
≤ |A| +
x∈B(0,l1 )
≤ |A| + t1
0
=: M0 .
sup
x∈B(0,l1 )
Thus, kyk∞ = supt∈[0,t1 ] |y(t)| ≤ M0 . Set
U = {y ∈ C([0, t1 ], R) : kyk∞ < M0 + 1}.
As a consequence of Claims 1–4 and the Ascoli-Arzela theorem, we can conclude
that the map N1 : U → C([0, t1 ], R) is compact. From the choice of U there is no
y ∈ ∂U such that y = λN1 y for any λ ∈ (0, 1). As a consequence of the nonlinear
alternative of Leray-Schauder we deduce that N1 has a fixed point denoted by
y0 ∈ U which is solution of the problem (3.1).
Step 2: Consider the problem
(φ(y 0 ))0 = f (t, y) t ∈ (t1 , t2 ],
−
−
y(t+
1 ) = y0 (t1 ) + I1 (y0 (t1 )),
y 0 (t+ ) = y00 (t− ) + I¯1 (y0 (t− )).
1
1
(3.2)
1
It is clear that all solutions of (3.2) are fixed points of the multi-valued operator
N2 : C ∗ → C ∗ , defined by
Z t
Z s
(N2 y)(t) = A1 +
φ−1 [φ(B1 ) +
f (τ, y)dτ ]ds,
t1
t1
where
0 +
C ∗ = {y ∈ C((t1 , t2 ]) : y(t+
1 ), y (t1 ) exist}
and
¯ 1 (t1 )).
A1 = y1 (t1 ) + I1 (y1 (t1 )), B1 = y10 (t1 ) + I(y
As in Step 1, we can prove that N2 at least one fixed point which is a solution of
(3.2).
6
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
Step 3: We continue this process taking into account that ym := y (t ,b] is a
m
solution of the problem
(φ(y 0 ))0 = f (t, y) t ∈ (tm , b],
−
−
y(t+
m ) = ym−1 (tm ) + Im (ym−1 (tm )),
0
−
−
¯
y 0 (t+
m ) = ym−1 (tm ) + Im (ym−1 (tm )).
(3.3)
A solution y of problem (1.1)-(1.4) is ultimately defined by

y0 (t), if t ∈ [0, t1 ],



y (t), if t ∈ (t , t ],
2
1 2
y(t) =

...



ym (t), if t ∈ (tm , tm+1 ].
Step 3: Now we show that the set
S = {y ∈ P C([0, b], R) : y is a solution of (1.1)-(1.4)}
is compact. Let (yn )n∈N be a sequence in S. We put B = {yn : n ∈ N} ⊆
P C([0, b], R). Then from earlier parts of the proof of this theorem, we conclude
that B is bounded and equicontinuous. Then from the Ascoli-Arzela theorem, we
can conclude that B is compact.
Recall that J0 = [0, t1 ] and Jk = (tk , tk+1 ], k = 1, . . . , m. Hence:
• yn |J0 has a subsequence
(ynm )nm ∈N ⊂ S1 = {y ∈ C([0, t1 ], R) : y is a solution of (3.1)}
such that ynm converges to y. Let
Z t
Z
−1
z0 (t) = A +
φ [φ(B) +
0
s
f (τ, y)dτ ]ds,
0
and
|ynm (t) − z0 (t)|
Z t
Z
|φ−1 [φ(B) +
≤
0
s
f (τ, ynm )dτ ] − φ−1 [φ(B) +
s
Z
0
f (τ, y)dτ ]|ds.
0
As nm → +∞, ynm (t) → z0 (t), and then
Z
Z t
y(t) = A +
φ−1 [φ(B) +
0
s
f (τ, y)dτ ]ds.
0
• yn |J1 has a subsequence relabeled as (ynm ) ⊂ S2 converging to y in C ∗ where
S2 = {y ∈ C ∗ : y is a solution of (3.2)}.
Let
Z
t
z1 (t) = A1 +
t1
|ynm (t) − z1 (t)|
Z t
Z
≤
|φ−1 [φ(B1 ) +
t1
φ−1 [φ(B1 ) +
Z
s
f (τ, y)dτ ]ds,
t1
s
t1
f (τ, ynm )dτ ] − φ−1 [φ(B1 ) +
Z
s
f (τ, y)dτ ]|ds.
t1
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EXISTENCE AND TOPOLOGICAL STRUCTURE
As nm → +∞, ynm (t) → z1 (t), and then
Z t
Z
y(t) = A1 +
φ−1 [φ(B1 ) +
t1
7
s
f (τ, y)dτ ]ds.
t1
• We continue this process, and we conclude that {yn | n ∈ N} has subsequence
converging to
Z t
Z s
zm (t) = Am +
φ−1 [φ(Bm ) +
f (τ, y)dτ ]ds, t ∈ (tm , b].
tm
tm
Hence S is compact.
Next we replace (H2) in Theorem 3.2 by
(H3) There exists a continuous nondecreasing function ψ : [0, ∞) → [0, ∞) and
p ∈ L1 (J, R+ ) such that
|f (t, u)| ≤ p(t)ψ(|u|)
a.e. t∈ J and u ∈ R.
Theorem 3.3. Under assumption (H3), problem (1.1)-(1.4) has at least one solution and the solution set is compact.
Proof. As in the proof of Theorem 3.2 we can show that (1.1)-(1.4) has at least one
solution by an application of the nonlinear alternative of Leray-Schauder. We show
only the estimation of a solution y of (1.1)-(1.4).
• For t ∈ [0, t1 ], we have
Z t
Z s
y(t) = A +
φ−1 [φ(B) +
f (τ, y)dτ ]ds.
0
0
We put m(r) = max{|y(r)| : r ∈ [0, t1 ]}, and
Z t
Z s
|y(t)| ≤ |A| +
|φ−1 [φ(B) +
f (τ, y)dτ ]|ds
0
0
Z t
Z s
≤ |A| +
|φ−1 [φ(B) +
p(τ )ψ(|y(τ )|)dτ ]|ds
0
0
Z t
Z s
≤ |A| +
|φ−1 [φ(B) +
p(τ )ψ(m(r))dτ ]|ds
0
0
Z t
≤ |A| +
|φ−1 [φ(B) + t1 kpkL1 ψ(m(r))]|ds.
0
Then
Z
m(t) ≤ |A| +
t
t ∈ [0, t1 ],
ψ1 (m(s))ds,
0
e and ψ(u)
e
where ψ1 = (φ−1 ◦ ψ)
= φ(B)+t1 (kpkL1 )ψ(u). By the nonlinear GrönwallBihari inequality (Lemma 2.2), we infer the bound
m(t) ≤ H −1 (t) ≤ M0 ,
where H(t) =
Rt
dτ
e ).
|A| (φ−1 ◦ψ)(τ
• For t ∈ (t1 , t2 ], we have
Z
t
y(t) = A1 +
t1
φ−1 [φ(B1 ) +
Z
s
f (τ, y)dτ ]ds.
t1
8
J. HENDERSON, A. OUAHAB, S. YOUCEFI
We put m(r) = max{|y(r)| : r ∈ (t1 , t2 ]}, and
Z t
Z
−1
|y(t)| ≤ |A1 | +
|φ [φ(B1 ) +
t1
Z t
|φ−1 [φ(B1 ) +
≤ |A1 | +
t1
Z t
|φ−1 [φ(B1 ) +
≤ |A1 | +
t1
Z t
EJDE-2012/56
s
f (τ, y)dτ ]|ds
t1
Z s
p(τ )ψ(|y(τ )|)dτ ]|ds
t1
Z s
p(τ )ψ(m(r))dτ ]|ds
t1
|φ−1 [φ(B1 ) + t2 (kpkL1 )ψ(m(r))]|ds.
≤ |A1 | +
t1
Then
t
Z
m(t) ≤ |A1 | +
t ∈ [t1 , t2 ],
ψ1 (m(s))ds,
t1
e and ψ(u)
e
where ψ1 = (φ−1 ◦ ψ)
= φ(B1 ) + t2 (kpkL1 )ψ(u).
By the nonlinear Grönwall-Bihari inequality (Lemma 2.2), we infer the bound
m(t) ≤ H −1 (t) ≤ M1 ,
where H(t) =
Rt
dτ
e ).
|A1 | (φ−1 ◦ψ)(τ
• For t ∈ (tm , b], we have
Z
t
y(t) = Am +
−1
φ
Z
s
[φ(Bm ) +
tm
f (τ, y)dτ ]ds.
tm
As in the pattern, there exists Mm > 0 such that
m(t) ≤ H −1 (t) ≤ Mm ,
where H(t) =
Rt
dτ
e ).
|Am | (φ−1 ◦ψ)(τ
kykP C
Hence
≤ max(M0 , M1 , . . . , Mm ) = M.
The proof is complete.
For the next theorem we use the assumptions:
(H4) f : [0, b] × R × R → R is a continuous function.
(H5) There exist a continuous nondecreasing function ψ : R+ × R+ → (0, ∞)
and p ∈ L1 (J, R) such that
|f (t, x, y)| ≤ p(t)ψ(|x|, |y|) f or
with
Z
b
Z
∞
p(s)ds <
0
all
|A|+c|B|
(φ−1
x, y ∈ R,
t∈J
du
.
◦ ψ)(u, u)
Theorem 3.4. Under assumptions (H4), (H5), problem (1.5)-(1.8) has at least one
solution.
Prior to the proof of Theorem 3.4, we present a useful lemma.
Lemma 3.5. The operator L : D → P C(J, R) defined by L(y) = (φ(y 0 ))0 where
0 +
0
¯
D = {y ∈ P C 1 (J, R) : y(t+
k ) = y(tk ) + Ik (y(tk )), y (tk ) = y (tk ) + Ik (y(tk )), y(tk ) =
−
−
y(tk ), y 0 (tk ) = y 0 (tk ), k = 1, . . . , m, y(0) = A, y 0 (0) = B}. Assume that L is well
defined. Then L is bijective and L−1 is completely continuous.
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EXISTENCE AND TOPOLOGICAL STRUCTURE
9
Proof. Step 1: L is bijective.
• L is injective. Let y1 , y2 ∈ D be such that L(y1 ) = L(y2 ). Then
(φ(y10 (t)))0 = (φ(y20 (t)))0 , t ∈ [0, t1 ],
and thus
φ(y10 (t)) − φ(y10 (0)) = φ(y20 (t)) − φ(y20 (0)),
φ(y10 (t))
− φ(B) =
φ(y20 (t))
− φ(B),
t ∈ [0, t1 ],
t ∈ [0, t1 ].
Hence y10 (t) = y20 (t) for t ∈ [0, t1 ]. By integration of this equality, we obtain
Z t
Z t
y10 (s)ds =
y20 (s)ds, t ∈ [0, t1 ]
0
0
which implies y1 (t)−y1 (0) = y2 (t)−y2 (0), t ∈ [0, t1 ]. This implies that y1 (t) = y2 (t),
t ∈ [0, t1 ].
Next,
φ(y10 (t)) − φ(y10 (t1 ) + I¯1 (y1 (t1 ))) = φ(y20 (t)) − φ(y20 (t1 ) + I¯1 (y2 (t1 ))),
implies
y10 (t)
=
y20 (t),
t ∈ (t1 , t2 ], and so
Z t
Z t
0
y1 (s)ds =
y20 (s)ds,
t1
t ∈ (t1 , t2 ]
t ∈ (t1 , t2 ]
t1
implies y1 (t) − (y1 (t1 ) + I1 (y1 (t1 ))) = y2 (t) − (y2 (t1 ) + I1 (y2 (t1 ))), t ∈ (t1 , t2 ], and
then
y1 (t) = y2 (t), t ∈ (t1 , t2 ].
Continuing this pattern,
φ(y10 (t)) − φ(y10 (tm ) + I¯m (y1 (tm ))) = φ(y20 (t)) − φ(y20 (tm ) + I¯m (y2 (tm ))),
implies y10 (t) = y20 (t), t ∈ (tm , b], and so
Z t
Z t
y10 (s)ds =
y20 (s)ds,
tm
t ∈ (tm , b]
t ∈ (tm , b]
tm
implies y1 (t) − (y1 (tm ) + Im (y1 (tm ))) = y2 (t) − (y2 (tm ) + Im (y2 (tm ))), t ∈ (tm , b],
and hence y1 (t) = y2 (t), t ∈ (tm , b]. This implies that y1 = y2 .
• L is surjective. Let h ∈ P C(J, R), then we define

L0 (h)(t),
if t ∈ [0, t1 ],



L (h)(t),
if t ∈ (t1 , t2 ],
1
y(t) =
(3.4)

.
.
.



Lm−1 (h)(t), if t ∈ (tm , b],
where
Z
L0 (h)(t) = A +
0
t
Z
h
φ−1 φ(B) +
s
i
h(τ )dτ ds,
L1 (h)(t) = L0 (h)(t1 ) + I1 (L0 (h)(t1 ))
Z t
Z
h
+
φ−1 φ(L00 (h)(t1 ) + I¯1 (L0 (h)(t1 ))) +
t1
t ∈ [0, t1 ],
0
s
t1
i
h(τ )dτ ds,
t ∈ (t1 , t2 ],
10
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
L2 (h)(t) = L1 (h)(t2 ) + I2 (L1 (h)(t2 ))
Z t
Z
h
+
φ−1 φ(L01 (h)(t2 ) + I¯2 (L1 (h)(t2 ))) +
t2
s
i
h(τ )dτ ds,
t ∈ (t2 , t3 ],
t2
...
Z t
h
φ−1 φ(L0m−1 (h)(tm )
Lm (h)(t) = Lm−1 (h)(tm ) + Im (Lm−1 (h)(tm )) +
tm
Z s
i
+ I¯m (Lm−1 (h)(tm ))) +
h(τ )dτ ds, t ∈ (tm , b].
tm
From (3.4) we can easily check that
X
y(t) = A +
Ik (Lk−1 (h)(tk ))
Z
+
0<tk <t
t
h
−1
φ
X
φ(B +
0
I¯k (Lk−1 (h)(tk ))) +
Z
s
i
h(τ )dτ ds,
Hence
0
−1
y (t) = φ
h
φ(B +
X
t ∈ J.
0
0<tk <t
I¯k (Lk−1 (h)(tk ))) +
Z
s
i
h(s)ds .
0
0<t2 <t
From the definition of y and y we can prove that y(0) = A, y 0 (0) = B, y(t+
k) =
0 +
0
¯
−
y(tk ) + Ik (y(tk )), y (tk ) = y (tk ) + Ik (y(tk )) and y(tk ) = yt , k = j, . . . , m and by
k
using the fact that Ik , I¯k are continuous we can easily prove that y, y 0 ∈ P C(J, R).
Step 2: L−1 is completely continuous.
Claim 1: L−1 is continuous. Let hn ∈ P C(J, R) be such that hn converges
to h in P C(J, R) as n → ∞. We show that L−1 (hn ) converges to L−1 (h). Let
{yn }n∈N ⊂ D such that {L(yn )}n∈N = {hn }n∈N . Then:
• For t ∈ [0, t1 ], we have
Z t
h
i
yn0 (t) = φ−1 φ(B) +
hn (s)ds ,
0
0
and
Z
yn (t) = A +
t
Z
h
φ−1 φ(B) +
0
s
Z t
i
hn (τ )dτ ds = A +
yn0 (s)ds.
0
0
Hence
Z t
h
i
|yn0 (t)| ≤ φ−1 φ(B) +
hn (s)ds ,
0
since
Z
|φ(B) +
t
Z
hn (s)ds| ≤ |φ(B)| +
0
t
|hn (s)|ds
0
≤ |φ(B)| + t1 khn kP C
≤ |φ(B)| + t1 M∗ = K,
Rt
where khn kP C ≤ M∗ for all n ∈ N. Then [φ(B) + 0 hn (s)ds] ∈ B(0, K). Since φ−1
is continuous and B(0, K) is compact,
sup
x∈B(0,K)
|φ−1 (x)| < ∞.
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EXISTENCE AND TOPOLOGICAL STRUCTURE
11
Then
kyn0 k∞ ≤
|φ−1 (x)| := M0 ,
sup
x∈B(0,K)
and
kyn k∞ ≤ |A| + M0 t1 := M0 .
{yn0 }n∈N
So, {yn }n∈N and
are bounded uniformly in C([0, t1 ], R).
We put C = {yn : n ∈ N} ⊆ C([0, t1 ], R). We can easily show that C is bounded
and equicontinuous, and then from the Ascoli-Arzela theorem we conclude that C
is compact. Then yn has a subsequence (ynm ) converging to y. Let
Z t
Z s
h
i
z(t) = A +
φ−1 φ(B) +
h(τ )dτ ds
0
0
so that
Z
t1
|ynm (t) − z(t)| ≤
0
−1
s
Z
h
|φ−1 φ(B) +
s
Z
i
h
hnm (τ )dτ − φ−1 φ(B) +
0
i
h(τ )dτ |ds.
0
is continuous and as nm → ∞, ynm → z(t), then
Z s
Z t
h
i
−1
φ
φ(B) +
h(τ )dτ ds.
y(t) = A +
Since φ
0
0
{yn0 }
By the same technique, we can prove that
converges to y 0 (t) for t ∈ [0, t1 ].
• For t ∈ (t1 , t2 ], we have
Z t
h
i
yn0 (t) = yn0 (t1 ) + φ−1 φ(yn0 (t1 ) + I¯1 (yn (t1 ))) +
hn (s)ds
t1
and
Z
t
yn (t) = yn (t1 ) + I1 (yn (t1 )) +
−1
φ
h
φ(yn0 (t1 )
+ I¯1 (yn (t1 ))) +
t1
t
Z
= yn (t1 ) + I1 (yn (t1 )) +
Z
s
hn (τ )dτ
i
t1
yn0 (s)ds.
t1
Hence
Z t
h
i
|yn0 (t)| ≤ |yn0 (t1 )| + φ−1 φ(yn0 (t1 ) + I¯1 (yn (t1 ))) +
hn (s)ds t1
Z t
h
i
−1
0
¯
≤ M0 + φ
φ(yn (t1 ) + I1 (yn (t1 ))) +
hn (s)ds .
t1
Since
|φ(yn0 (t1 )
+ I¯1 (yn (t1 ))) +
t
Z
hn (s)ds| ≤ |φ(M0 +
t1
|I¯1 (x)|)| + t2 K = K∗ ,
sup
x∈B(0,M0 )
then
|φ(yn0 (t1 ) + I¯1 (yn (t1 ))) +
Z
t
hn (s)ds| ∈ B(0, |φ(M0 +
t1
sup
|I¯1 (x)|)| + t2 K).
x∈B(0,M0 )
Since φ−1 is continuous and B(0, |φ(M0 + supx∈B(0,M0 ) |I¯1 (x)|)| + t2 K) is compact,
kyn0 k∞ ≤ M0 +
|φ−1 (x)| := M1 ,
sup
x∈B(0,|φ(M0 +supx∈B(0,M
0
¯
) |I1 (x)|)|+t2 K)
12
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
and
kyn k∞ ≤ M0 +
|I1 (z)| + M1 t2 := M1 .
sup
x∈B(0,M0 )
Then {yn }n∈N and {yn0 }n∈N are bounded uniformly in C((t1 , t2 ], R). We put C =
{yn : n ∈ N} ⊆ C((t1 , t2 ], R). We can easily show again that C is bounded and
equicontinuous, and then from the Ascoli-Arzela theorem we conclude that C is
compact. Then yn has a subsequence (ynm ) converging to y.
Now, let
Z t
Z s
h
i
0
−1
¯
z(t) = y(t1 ) + I1 (y(t1 )) +
φ(y (t1 ) + I1 (y(t1 ))) +
φ
h(τ )dτ ds.
t1
t1
Then
|ynm (t) − z(t)| ≤ |ynm (t1 ) − y(t1 )| + |I1 (ynm (t1 )) − I1 (y(t1 ))|
Z t
Z s
h
i
|φ−1 φ(yn0 m (t1 ) + I¯1 (ynm (t1 ))) +
hnm (τ )dτ
t1
t1
Z s
h
i
−1
0
−φ
φ(y (t1 ) + I¯1 (y(t1 ))) +
h(τ )dτ |ds,
t1
Since φ−1 is continuous, {yn }n∈N and {yn0 }n∈N converge to y and y 0 , respectively,
for t ∈ [0, t1 ], and as nm → ∞, ynm → z(t), then
Z t
Z s
h
i
y(t) = y(t1 ) + I1 (y(t1 )) +
φ−1 φ(y 0 (t1 ) + I¯1 (y(t1 ))) +
h(τ )dτ ds.
t1
t1
{yn0 }
0
By the same technique, we can prove that
converges to y (t) for t ∈ (t1 , t2 ].
• We continue this process until we get, for every t ∈ (tm , b], that yn (t) converges
to y(t) and yn0 (t) converges to y 0 (t). We conclude that L(y) = h, and this implies
that L−1 is continuous.
Claim 2: L−1 is compact. Let D be a bounded set of P C(J, R) and {yn }n∈N ⊂
−1
L (D). Then there exists {hn }n∈N ⊂ D such that L(yn ) = hn , for all n ∈ N.
−1
We show that |L−1
(hn )(l1 )| tends to zero as l2 → l1 . Since
0 (hn )(l2 ) − L
L0 (yn )(t) = hn (t), t ∈ [0, t1 ], it follows that
Z t
h
i
0
−1
yn (t) = φ
φ(B) +
hn (s)ds , t ∈ [0, t1 ].
0
Using the fact that hn is bounded, thus there exist M0 > 0 such that
kyn k∞ , kyn0 k∞ ≤ M0 ,
for all n ∈ N.
• Let l1 , l2 ∈ [0, t1 ], l1 < l2 . Then, by the mean value theorem,
|yn (l2 ) − yn (l1 )| = |yn0 (ξn )(l2 − l1 )| ≤ M0 |l2 − l1 |,
and
Z
|φ(yn0 )(l2 ) − φ(yn0 )(l1 )| = l2
l1
Z
hn (s)ds ≤
l2
|hn (s)|ds ≤ |l2 − l1 |M∗ ,
l1
where khn kP C ≤ M∗ for all n ∈ N. As l2 → l1 the right hand side of the above inequality tends to zero. Then {yn (·)}n∈N is equicontinuous and {yn }n∈N is bounded.
By the Ascoli-Arzela theorem there exist y0 , z0 ∈ C([0, t1 ], R) such that yn and
φ(yn0 ) converge, respectively, to y0 and z0 . Since φ−1 is a continuous function,
yn0 (t) → φ−1 (z0 )(t), n → ∞.
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EXISTENCE AND TOPOLOGICAL STRUCTURE
13
Set
t
Z
φ−1 (z0 (s))ds, t ∈ [0, t1 ],
y∗ (t) = A +
0
and from
t
Z
yn0 (s)ds, t ∈ [0, t1 ],
yn (t) = A +
0
we have
t1
Z
kyn − y∗ k∞ ≤
|yn0 (s) − φ−1 (z0 )(s)|ds.
0
From the Lebesgue dominated convergence theorem, we deduce that {yn }n∈N converges to y∗ in C([0, t1 ], R), and this implies that
Z t
y0 (t) = A +
φ−1 (z0 (s))ds, t ∈ [0, t1 ] ⇒ y00 (t) = φ−1 (z0 (t)), t ∈ [0, t1 ].
0
Then yn converges to y0 in C 1 ([0, t1 ], R).
For t ∈ (t1 , t2 ],
Z t
h
i
yn0 (t) = φ−1 φ(yn0 (t1 ) + I¯1 (yn (t1 ))) +
hn (s)ds ,
t ∈ (t1 , t2 ].
t1
Using the fact that hn is bounded, there exists M1 > 0 such that
kyn k∞ , kyn0 k∞ ≤ M1 , for all n ∈ N.
• Let l1 , l2 ∈ (t1 , t2 ], l1 < l2 . Then
|yn (l2 ) − yn (l1 )| = |yn0 (ξn )(l2 − l1 )| ≤ M1 |l2 − l1 |,
and
|φ(yn0 )(l2 ) − φ(yn0 )(l1 )| ≤ M∗ |l2 − l1 |.
As l2 → l1 the right hand side of the above inequality tends to zero, then {yn (·)}
and {φ(yn0 )(·)} are equicontinuous. By Ascoli-Arzela theorem there exist y1 , z1 ∈
C([0, t1 ], R) such that yn and φ(yn0 ) converge, respectively, to y1 , z1 . Since φ−1 and
I1 are a continuous functions, then yn0 (t) → φ−1 (z1 )(t) as n → ∞. Set
Z t
y∗∗ (t) = y0 (t1 ) + I1 (y0 (t1 )) +
φ−1 (z1 (s))ds, t ∈ (t1 , t2 ].
t1
From
Z
t
yn (t) = y0 (t1 ) + I1 (y0 (t1 )) +
yn0 (s)ds, t ∈ (t1 , t2 ],
t1
we have
Z
kyn − y∗∗ k∞ ≤
t1
|yn0 (s) − φ−1 (z1 )(s)|ds.
0
From the Lebesgue dominated convergence theorem, we deduce that {yn }n∈N con0 +
verges to y∗∗ in C1 = {y ∈ C 1 ((t1 , t2 ], R) : y(t+
1 ), y (t1 ) exist}. This implies that
Z t
y1 (t) = y0 (t1 ) + I1 (y0 (t1 )) +
φ−1 (z0 (s))ds, t ∈ [0, t1 ]
t1
implies
y10 (t)
−1
=φ
(z1 (t)),
t ∈ (t1 , t2 ]. Then yn converges to y1 in C1 .
14
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
• We continue this process until we have that there exists ym ∈ C 1 ((tm , b], R)
0 +
such that yn converge to ym in Cm = {y ∈ C 1 ((tm , b], R)|y(t+
m ), y (tm ) exist}. We
define

y0 (t), if t ∈ [0, t1 ],



y (t), if t ∈ (t , t ],
1
1 2
(3.5)
y(t) =

.
.
.



ym (t), if t ∈ (tm , b].
It is clear that {yn }n∈N ⊂ P C and yn converges to y in P C. Using the fact that
L−1 is continuous, thus
hn = L−1 (yn ) → L−1 (y) = h
as n → ∞
in P C 1 . Then L(y) = h. Hence L−1 is compact. This completes the proof.
Proof of Theorem 3.4. We consider the following fixed point problem that is equivalent to problem (1.5)–(1.8),
y = (L−1 ◦ F )(y),
where L−1 : P C(J, R) → D and F is the Nemystki operator given by F (t, y) =
f (t, y, y 0 ). From Lemma 3.5 we can prove that (L−1 ◦ F ) is compact. Now we show
that y 6= λ(L−1 ◦ F )(y).
For t ∈ [0, t1 ], we have
Z t
h
i
0
−1
y (t) = φ
φ(B) +
f (s, y(s), y 0 (s))ds ,
0
Z t
y(t) = A +
y 0 (s)ds.
0
Thus
Z
h
|y 0 (t)| ≤ φ−1 φ(B) +
s
0
Z
i
p(s)ψ(|y(s)|, |y 0 (s)|)ds ,
t
|y 0 (s)|ds
|y(t)| ≤ |A| +
0
≤ |A| +
Z t
Z
h
−1
φ(B) +
φ
0
s
0
i
p(τ )ψ(|y(τ )|, |y 0 (τ )|)dτ ds.
Let m(r) = max(supt∈[0,t1 ] |y(t)|, supt∈[0,t1 ] |y 0 (t)|), then
Z t
Z s
h
i
−1
|y(t)| ≤ |A| +
φ(B) +
p(τ )ψ(m(r), m(r))dτ ds
φ
0
0
Z t
−1
≤ |A| +
φ [φ(B) + t1 kpkL1 ψ(m(r), m(r))]ds.
0
Then
Z
m(t) ≤ |A| +
t
ψ1 (m(s))ds,
0
e
with ψ1 = φ−1 ◦ ψe and ψ(u)
= φ(B) + t1 (kpkL1 )ψ(u). By the nonlinear GrönwallBihari inequality (Lemma 2.2), we infer the bound
m(t) ≤ H −1 (t) ≤ M0 ,
EJDE-2012/56
EXISTENCE AND TOPOLOGICAL STRUCTURE
where
t
dτ
|A|
e )
(φ−1 ◦ ψ)(τ
Z
H(t) =
.
For t ∈ (t1 , t2 ], we have
Z t
h
i
y 0 (t) = y 0 (t1 ) + φ−1 φ(y 0 (t1 ) + I¯1 (y(t1 ))) +
f (s, y(s), y 0 (s))ds ,
t1
Z
t
y 0 (s)ds.
y(t) = y(t1 ) + I1 (y(t1 )) +
t1
Thus
Z t
h
i
|y 0 (t)| ≤ |y 0 (t1 )| + φ−1 φ(y 0 (t1 ) + I¯1 (y(t1 ))) +
f (s, y(s), y 0 (s))ds .
t1
Let
m(r) = max( sup |y(t)|, sup |y 0 (t)|),
t∈[0,t1 ]
t∈[0,t1 ]
and then
|y(t)| ≤ |y(t1 )| + |I1 (y(t1 ))| + t2 |y 0 (t1 )|
Z
Z t
h
−1
0
¯
φ(y
(t
)
+
I
(y(t
)))
+
+
φ
1
1
1
t1
t1
≤ M0 + |I1 (y(t1 ))| + t2 |y 0 (t1 )|
Z
Z t
h
−1
0
¯
φ(y (t1 ) + I1 (y(t1 ))) +
φ
t1
s
t1
∗
Then m(t) ≤ M +
s
Rt
i
p(τ )ψ(|y(τ )|, |y 0 (τ )|)dτ ds
i
p(τ )ψ(m(r), m(r))dτ ds.
ψ1 (m(s))ds, where
t1
M ∗ = (1 + t2 )M0 +
|I1 (z)|,
sup
z∈B(0,M0 )
e
ψ1 = φ−1 ◦ ψ,
e
ψ(u)
= φ(y 0 (t1 ) + I¯1 (y(t1 ))) + t2 (kpkL1 )ψ(u, u).
By the nonlinear Grönwall-Bihari inequality (Lemma 2.2), we infer the bound
m(t) ≤ H −1 (t) ≤ M1 ,
where
Z
t
H(t) =
M∗
dτ
(φ−1
e )
◦ ψ)(τ
.
For t ∈ (tm , b], we have
Z
h
y 0 (t) = y 0 (tm ) + φ−1 φ(y 0 (tm ) + I¯m (y(tm ))) +
t
i
f (s, y(s), y 0 (s))ds ,
tm
y(t) = y(tm ) + Im (y(tm ))
Z t
Z
h
−1
0
¯
+
φ
φ(y (tm ) + Im (y(tm ))) +
tm
s
i
f (τ, y(τ ), y 0 (τ ))dτ ds.
tm
So, there exists Mm > 0 such that m(t) ≤ H −1 (t) ≤ Mm , where
Z t
dτ
H(t) =
e )
M ∗∗ (φ−1 ◦ ψ)(τ
15
16
J. HENDERSON, A. OUAHAB, S. YOUCEFI
EJDE-2012/56
and
M ∗∗ = (1 + b)Mm−1 +
sup
|Im (z)|.
z∈B(0,Mm−1 )
Hence
kyk∞ ≤ max(M0 , M1 , . . . , Mm ) := M.
Let
U = {y ∈ P C 1 (J, R) : kykP C 1 < M + 1}.
Then L−1 ◦ F : U → P C 1 (J, R is relatively compact. Assume that there exists
λ ∈ (0, 1) and y ∈ ∂U such that y = λ(L−1 ◦ F )(y). Then kykP C 1 = M + 1, but
kykP C 1 ≤ M . Thus by the nonlinear alternative of Leray-Schauder, we conclude
that L−1 ◦ F has a fixed point which is a solution of (1.5)-(1.8).
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Johnny Henderson
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA
E-mail address: Johnny Henderson@baylor.edu
Abdelghani Ouahab
Laboratory of Mathematics, Sidi-Bel-Abbès University, PoBox 89, 22000 Sidi-Bel-Abbès,
Algeria
E-mail address: agh ouahab@yahoo.fr
Samia Youcefi
Laboratory of Mathematics, Sidi-Bel-Abbès University, PoBox 89, 22000 Sidi-Bel-Abbès,
Algeria
E-mail address: youcefi.samia@yahoo.com