Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 169, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NONLINEAR DELAY INTEGRAL INEQUALITIES FOR
MULTI-VARIABLE FUNCTIONS
HASSANE KHELLAF, MOHAMED EL HADI SMAKDJI
Abstract. In this article, we establish some nonlinear retarded integral inequalities in n independent variables. These inequalities represent a generalization of the results obtained in [1, 9, 12] for function of one and two variables.
Our results can be used in the qualitative theory of delay partial differential
equations and delay integral equations.
1. Introduction
In the study of ordinary differential and integral equations, one often deals with
certain integral inequalities. The Gronwall-Bellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of the existence,
uniqueness, continuation, boundedness, oscillation, stability and other qualitative
properties of the solutions of differential and integral equations. The literature on
such inequalities and their applications is vast; see [3, 4, 7, 8, 15] and references
therein.
During the past few years, investigators have established some useful and interesting delay integral inequalities in order to achieve various goals; see [2, 6, 10, 11,
14] and the references cited therein.
Let us first list the main results of [1, 9, 12], for functions with two variables for
u(x, y) ∈ (∆ ∈ R2+ , R+ ):
Inequality by Ma and Pecaric [9, Theorem 2.1]:
up (x, y) = k +
m Z
X
i=1
+
n Z
X
j=1
α1i (x)
α1i (x0 )
α2j (x)
α2j (x0 )
Z
Z
β1i (y)
ai (s, t)uq (s, t) dt ds
β1i (y0 )
β2j (y)
bj (s, t)uq (s, t)w(u(s, t)) dt ds.
β2j (y0 )
2000 Mathematics Subject Classification. 26D15, 26D20, 26D10.
Key words and phrases. Delay integral inequality; multi-variable function;
delay partial differential equation.
c
2011
Texas State University - San Marcos.
Submitted August 5, 2011. Published December 18, 2011.
1
(1.1)
2
H. KHELLAF, M. SMAKDJI
EJDE-2011/169
Pachpatte’s inequality [12, Theorem 4];
Z xZ y
p
u (x, y) = k +
a(s, t)g1 (u(s, t)) dt ds
x0
y0
α(x)
Z
(1.2)
β(y)
Z
+
b(s, t)g2 (u(s, t)) dt ds.
α(x0 )
β(y0 )
Cheung’s inequality [1, Theorem 2.4]:
Z α(x) Z β(y)
p
a(s, t)uq (s, t) dt ds
up (x, y) = k +
p − q α(x0 ) β(y0 )
Z γ(x) Z δ(y)
+
b(s, t)uq (s, t)ϕ(u(s, t)) dt ds.
γ(x0 )
(1.3)
γ(y0 )
However, sometimes we need to study such inequalities with a function c(x) in
place of the constant term k. Our main result, for functions with n independent
variables, is given in the inequality
Z αej (x)
n1
X
ϕ(u(x)) ≤ c(x) +
dj (x)
aj (x, t)Φ(u(t))w1 (u(t))dt
α
ej (x0 )
j=1
+
n2
X
Z
(1.4)
ek (x)
β
lk (x)
bk (x, t)Φ(u(t))w2 (u(t))dt,
ek (x0 )
β
k=1
where c(x) is a function and all the functions which appear in this inequality are
assumed to be real valued of n variables.
It is interesting to note that the results (1.1)-(1.3) can be deduced from our
inequality (1.4) in some special cases. As applications we give the estimate solution
of retarded partial differential equation.
The main purpose of this article is to establish some nonlinear retarded integral
inequalities for functions of n independent variables which can be used as handy
tools in the theory of partial differential and integral equations with time delays.
These new inequalities represent a generalization of the results obtained by Ma and
Pecaric [9], Pachpatte [12] and by Cheung [1] in case of the functions with one and
two variables. We note that the inequality (1.4) is also a generalization of the main
results in [5, 16].
2. Main results
Rn+
In this article, we denote
= [0, ∞) which is a subset of Rn . All the functions
which appear in the inequalities are assumed to be real valued of n-variables which
are nonnegative and continuous. All integrals are assumed to exist on their domains
of definitions.
For x = (x1 , x2 , . . . , xn ), t = (t1 , t2 , . . . , tn ), x0 = (x01 , x02 , . . . , x0n ) ∈ Rn+ , we shall
denote:
Z αj1 (x1 ) Z αj2 (x2 )
Z αjn (xn )
Z αei (x)
dt =
...
. . . dtn . . . dt1 , j = 1, 2, . . . , n1 ,
αj1 (x01 )
α
ei (x0 )
Z
ek (x)
β
Z
βk1 (x1 )
αj2 (x02 )
Z
Z
βkn (xn )
...
dt =
ek (x0 )
β
αjn (x0n )
βk2 (x2 )
βk1 (x01 )
βk2 (x02 )
. . . dtn . . . dt1 ,
βkn (x0n )
k = 1, 2, . . . , n2 ,
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NONLINEAR DELAY INTEGRAL INEQUALITIES
3
with n1 , n2 ∈ {1, 2, . . . , }. For x, t ∈ Rn+ , we shall write t ≤ x whenever ti ≤ xi ,
i = 1, 2, . . . , n and x ≥ x0 ≥ 0, for x, x0 ∈ Rn+ .
∂
We denote D = D1 D2 . . . Dn , where Di = ∂x
, for i = 1, 2, . . . , n, We use the
i
P
usual convention of writing s∈∅ u(s) = 0 if ∅ is the empty set.
α
ej (t) = αj1 (t1 ), αj2 (t2 ), . . . , αjn (tn ) ∈ Rn+ forj = 1, 2, . . . , n1 ;
βek (t) = αk1 (t1 ), αk2 (t2 ), . . . , αkn (tn ) ∈ Rn+ for k = 1, 2, . . . , n1 .
We denote α
ej (t) ≤ t for j = 1, 2, . . . , n1 whenever αji (ti ) ≤ ti for i = 1, 2, . . . , n
and j = 1, 2, . . . , n1 , and βek (t) ≤ t for k = 1, 2, . . . , n2 whenever βki (ti ) ≤ ti for
i = 1, 2, . . . , n and k = 1, 2, . . . , n2
Our main results read as the follows.
Theorem 2.1. Let c ∈ C(Rn+ , R+ ), w1 , w2 ∈ C(R+ , R+ ) be nondecreasing functions with w1 (u), w2 (u) > 0 on (0, ∞) and let aj (x, t) and bk (x, t) ∈ C(Rn+ ×Rn+ , R+ )
be nondecreasing functions in x for every t fixed for any j = 1, 2, . . . , n1 , k =
1, 2, . . . , n2 . Let αji , βki ∈ C 1 (R+ , R+ ) be nondecreasing functions with αji (ti ) ≤ ti
and βki (ti ) ≤ ti on R+ for i = 1, 2, . . . , n; j = 1, 2, . . . , n1 , k = 1, 2, . . . , n2 and
p > q ≥ 0.
(A1) If u ∈ C(Rn+ , R+ ) and
n1 Z α
ej (x)
X
p
aj (x, t)uq (t)dt
u (x) ≤ c(x) +
j=1
+
n2 Z
X
α
ej (x0 )
ek (x)
β
(2.1)
bk (x, t)uq (t)w1 (u(t))dt,
ek (x0 )
β
k=1
for any x ∈
with x ≤ t ≤ x, then there exists x∗ ∈ Rn+ , such as for all
0
∗
x ≤ t ≤ x , we have
ek (x)
n2 Z β
h
i1/(p−q)
p−q X
b
(x,
t)dt
.
(2.2)
u(x) ≤ Ψ−1
Ψ
(p(x))
+
k
1
1
p
ek (x0 )
β
Rn+
0
k=1
Where
p(x) = c(p−q)/p (x) +
Z
δ
Ψ1 (δ) =
n1 Z α
ej (x)
p−q X
aj (x, t) dt,
p j=1 αej (x0 )
ds
1
δ0
w1 (s p−q )
,
δ > δ0 > 0.
(2.3)
(2.4)
Here, Ψ−1 is the inverse function of Ψ, and the real numbers x∗ are chosen so that
Pn2 R βek (x)
−1
Ψ1 (p(x)) + p−q
ek (x0 ) bk (x, t)dt ∈ dom(Ψ1 ).
k=1 β
p
(A2) If u ∈ C(Rn+ , R+ ) and
n1 Z α
ej (x)
X
p
aj (x, t)uq (t)w1 (u(t))dt
u (x) ≤ c(x) +
j=1
+
n2 Z
X
k=1
α
ej (x0 )
ek (x)
β
ek (x0 )
β
bk (x, t)uq (t)w2 (u(t))dt.
(2.5)
4
H. KHELLAF, M. SMAKDJI
EJDE-2011/169
(i) In the case w2 (u) ≤ w1 (u), for any x ∈ Rn+ with x0 ≤ t ≤ x, there exists
ξ1 ∈ Rn+ , such as for all x0 ≤ t ≤ ξ1 , we have
1/(p−q)
(p−q)/p
u(x) ≤ Ψ−1
Ψ
(c
(x))
+
e(x)
.
1
1
(ii) In the case w1 (u) ≤ w2 (u), for any x ∈ Rn+ with x0 ≤ t ≤ x, there exists
ξ2 ∈ Rn+ , such as for all x0 ≤ t ≤ ξ2 , we have
1/(p−q)
u(x) ≤ Ψ−1
Ψ2 (c(p−q)/p (x)) + e(x)
,
2
where
n
1
p − qhX
e(x) =
p
j=1
Z δ
Ψi (δ) =
δ0
Z
ek (x)
β
aj (x, t)dt +
ek (x0 )
β
ds
1
n2 Z
X
k=1
,
ek (x)
β
i
bk (x, t)dt ,
ek (x0 )
β
δ > δ0 > 0, for i = 1, 2.
wi (s p−q )
Here, Ψ−1
is the inverse function of Ψi and the real numbers ξi are chosen so that
i
Ψ2 (c(p−q)/p (x)) + e(x) ∈ dom(Ψ−1
i ) for i = 1, 2 respectively.
The proof of the above theorem will be given in the next section.
Corollary 2.2. Let the functions u, c, w1 , aj , bk (j = 1, 2, . . . , n1 ; k = 1, 2, . . . , n1 )
and the constants p, q be defined as in Theorem 2.1 and
n1 Z αj (x) Z αj (y)
X
up (x, y) ≤ c(x, y) +
aj (x, y, s, t)uq (s, t) ds dt
+
j=1 αj (x0 ) αj (y0 )
Z
Z βk (y)
n
2
β
k (x)
X
(2.6)
q
bk (x, y, s, t)u (t)w1 (u(t))dt,
k=1
βk (x0 )
βk (y0 )
for any (x, y) ∈ R2+ with x0 ≤ s ≤ x and y0 ≤ t ≤ y, then there exists (x∗ , y ∗ ) ∈ Rn+ ,
such as for all x0 ≤ s ≤ x∗ and y0 ≤ s ≤ y ∗ , then
1/(p−q)
p−q
B1 (x, y)
,
(2.7)
u(x, y) ≤ Ψ−1 Ψ(p1 (x, y)) +
p
where
p1 (x, y) = c(p−q)/p (x, y) +
A1 (x, y) =
B1 (x, y) =
n1 Z
X
αj (x)
αj (y)
Z
aj (x, y, s, t) ds dt,
j=1 αj (x0 )
n2 Z βk (x)
X
k=1
p−q
A1 (x, y),
p
αj (y0 )
Z
βk (x0 )
βk (y)
bk (x, y, s, t) ds dt,
βk (y0 )
and
Z
δ
Ψ(δ) =
δ0
ds
w1
(s1/(p−q) )
,
δ > δ0 > 0.
(2.8)
Here, Ψ−1 is the inverse function of Ψ, and the real numbers (x∗ , y ∗ ) are chosen
−1
so that Ψ(p1 (x, y)) + p−q
).
p B1 (x, y) ∈ dom(Ψ
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NONLINEAR DELAY INTEGRAL INEQUALITIES
5
Remark 2.3. Setting aj (x, y, s, t) = aj (s, t), bk (x, y, s, t) = bk (s, t) and c(x, y) = k
≥ 0 in Corollary 2.2, we obtain Ma and Pecaric’s result [9, Theorem 2.1].
p
p
Remark 2.4. Defining aj (x, y, s, t) = p−q
aj (s, t), bk (x, y, s, t) = p−q
bk (s, t)
c(x, y) = k > 0 (Constant) and j = k = 1 in Corollary 2.2, we obtain Cheung’s
result [1, Theorem 2.4].
Obviously, (1.1)–(1.3) are special cases of Theorem 2.1. So our result includes
the main results in [9, 12, 1].
Using Theorem 2.1, we can get some more generalized results as follow:
Theorem 2.5. Let the functions u, c, wi , aj , bk (i = 1, 2, j = 1, 2, . . . , n1 , k =
1, 2, . . . , n1 ) be defined as in Theorem 2.1. Moreover, let ϕ ∈ C(R+ , R+ ) be a
strictly increasing function so that limx→∞ ϕ(x) = ∞, and let Φ ∈ C(R+ , R+ ) be
nondecreasing function with Φ(x) > 0 for all x ∈ Rn+ .
(B1) If u ∈ C(Rn+ , R+ ) and
n1 Z α
ej (x)
X
ϕ(u(x)) ≤ c(x) +
aj (x, t)Φ(u(t))dt
j=1
+
n2 Z
X
α
ej (x0 )
(2.9)
ek (x)
β
bk (x, t)Φ(u(t))w1 (u(t))dt,
ek (x0 )
β
k=1
for any x ∈ Rn+ with x0 ≤ t ≤ x, then there exists x∗ ∈ Rn+ , so that for all
x0 ≤ t ≤ x∗ , we have
u(x) ≤ ϕ−1 G−1 [Ψ−1
(2.10)
1 (Ψ1 (π(x)) + B(x))] ,
where
π(x) = G(c(x)) + A(x),
n1 Z α
ej (x)
X
A(x) =
aj (x, t)dt,
j=1
B(x) =
x
Z
G(x) =
x0
Z
δ
Ψi (δ) =
δ0
ek (x)
β
bk (x, t)dt,
ds
,
wi (ϕ−1 (G−1 (s)))
x > x0 > 0,
(2.14)
δ > δ0 > 0, i = 1, 2.
(2.15)
The real number x∗ is chosen so that Ψ1 (π(x)) + B(x) ∈ dom(Ψ−1
1 ).
(B2) If u ∈ C(Rn+ , R+ ) and
n1 Z α
ej (x)
X
ϕ(u(x)) ≤ c(x) +
aj (x, t)Φ(u(t))w1 (u(t))dt
+
n2 Z
X
k=1
α
ej (x0 )
ek (x)
β
ek (x0 )
β
(2.13)
ek (x0 )
β
ds
,
Φ(ϕ−1 (s))
j=1
(2.12)
α
ej (x0 )
n2 Z
X
k=1
(2.11)
bk (x, t)Φ(u(t))w2 (u(t))dt.
6
H. KHELLAF, M. SMAKDJI
EJDE-2011/169
(i) When w2 (u) ≤ w1 (u), for any x ∈ Rn+ with x0 ≤ t ≤ x, there exists ξ1 ∈ Rn+ , so
that for all x0 ≤ t ≤ ξ1 , we have
Ψ1 (G(c(x))) + A(x) + B(x) ] .
u(x) ≤ ϕ−1 G−1 [Ψ−1
1
(ii) When w1 (u) ≤ w2 (u), for any x ∈ Rn+ with x0 ≤ t ≤ x, there exists ξ2 ∈ Rn+ ,
so that for all x0 ≤ t ≤ ξ2 , we have
Ψ2 (G(c(x))) + A(x) + B(x) ] .
u(x) ≤ ϕ−1 G−1 [Ψ−1
2
Where A, B, G and Ψi (i = 1, 2) are defined in (2.12)-(2.15), Ψ−1
is the inverse
i
function of Ψi and the real numbers ξi are chosen so that Ψi (G(c(x))) + A(x) +
B(x) ∈ dom(Ψ−1
i ) for i = 1, 2 respectively.
Many interesting corollaries can also be obtained from the above theorems (in
the case of one or n independent variables).
Corollary 2.6 (Inequality in one variable). Let p > q ≥ 0, c > 0 be constant and
w1 , w2 be defined as in Theorem 2.1. Moreover, let aj (x, t) and bk (x, t) ∈ C(R+ ×
R+ , R+ ) be nondecreasing functions in x for every t fixed and αj , βk ∈ C 1 (R+ , R+ )
be nondecreasing functions with αj (t) ≤ t and βk (t) ≤ ti on R+ for j = 1, 2, . . . , n1 ,
k = 1, 2, . . . , n2 for any j = 1, 2, . . . , n1 , k = 1, 2, . . . , n2 .
(C1) Let u ∈ C(R+ , R+ ) and
n1 Z αj (x)
p X
aj (x, t)u(t)q dt
u(x)p ≤ cp/(p−q) +
p − q j=1 0
n2 Z βk (x)
p X
bk (x, t)u(t)q w1 (u(t))dt,
+
p−q
0
k=1
for any x ∈ R+ with 0 ≤ t ≤ x. Then there exists (x∗ ) ∈ R+ , so that for all
0 ≤ t ≤ x∗ , we have
1/(p−q)
u(x) ≤ [Ψ−1
Ψ1 (π(x)) + B(x) ]
.
(2.16)
1
Where π(x) = c + A(x) and
A(x) =
B(x) =
n1 Z
X
j=1 0
n2 Z βk (x)
X
k=1
Z
δ
Ψi (δ) =
aj (x, t)dt,
(2.17)
bk (x, t)dt,
(2.18)
0
ds
1
δ0
αj (x)
wi (s p−q )
δ > δ0 > 0, i = 1, 2.
Where the real number x∗ is chosen so that Ψ1 (π(x)) + B(x) ∈ dom(Ψ−1
1 ).
(C2) If u ∈ C(R+ , R+ ) and
n1 Z αj (x)
p X
u(x)p ≤ cp/(p−q) +
aj (x, t)u(t)q w1 (u(t))dt
p − q j=1 0
n2 Z βk (x)
p X
+
bk (x, t)u(t)q w2 (u(t))dt.
p−q
0
k=1
(2.19)
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NONLINEAR DELAY INTEGRAL INEQUALITIES
7
(i) In the case w2 (u) ≤ w1 (u), for any x, t ∈ R+ with 0 ≤ t ≤ x, we have
1/(p−q)
u(x) ≤ u(x) ≤ [Ψ−1
.
1 (Ψ1 (c) + A(x) + B(x))]
(ii) In the case w1 (u) ≤ w2 (u), for any x, t ∈ R+ with 0 ≤ t ≤ x, we have
1/(p−q)
u(x) ≤ u(x) ≤ [Ψ−1
.
2 (Ψ2 (c) + A(x) + B(x))]
Where Ψi , A, B (i = 1, 2) are defined in (2.17)-(2.19).
Remark 2.7. (i) Corollary 2.6 (C1) reduces to Sun’s inequality [16, Theorem 2.1]
in case of one variable (n = 1) when aj (x, t) = aj (t), bk (x, t) = bk (t), βk (x) = αj (x)
and j = k = 1.
(ii) Corollary 2.6 (C2) reduces to Sun’s inequality [16, Theorem 2.2] in case of
one variable (n = 1) when aj (x, t) = aj (t),bk (x, t) = bk (t), βk (x) = x and j = k = 1
and w1 = w2 .
Remark 2.8. Under some suitable conditions in (B1), the inequality (2.9) gives a
new estimate for the inequality (2.1) in (A1).
Theorem 2.9. Let the functions u, c, , ϕ, Φ, wi , aj , bk (i = 1, 2, j = 1, 2, . . . , n1 ,
k = 1, 2, . . . , n1 ) be defined as in Theorem 2.5 and If
Z αej (x)
n1
X
ϕ(u(x)) ≤ c(x) +
dj (x)
aj (x, t)Φ(u(t))w1 (u(t))dt
α
ej (x0 )
j=1
+
n2
X
Z
ek (x)
β
lk (x)
bk (x, t)Φ(u(t))w2 (u(t))dt,
ek (x0 )
β
k=1
for any x ∈ Rn+ , we have
e
e
u(x) ≤ ϕ−1 G−1 [Ψ−1 Ψ(G(c(x))) + A(x)
+ B(x)
] ,
where
e
A(x)
=
n1
X
Z
dj (x)
aj (x, t)dt,
α
ej (x0 )
j=1
e
B(x)
=
α
ej (x)
n2
X
Z
ek (x)
β
bk (x, t)dt.
lk (x)
ek (x0 )
β
k=1
Corollary 2.10. If
up (x) ≤ c(x) +
Z
α
e(x)
a(t)uq (t) + b(t)up (t)dt
0
for any x ∈ Rn+ with x0 ≤ t ≤ x, then there exists x∗ ∈ Rn+ , so that for all
x0 ≤ t ≤ x∗ , we have
h p Z αe(x)
i
p−q
p
p
c
(x) exp
a(t) + b(t)dt
u(x) ≤
p−q
p−q 0
Remark 2.11. (i) Theorem 2.9 reduced to [5, Theorem 2.2] in the case of one
variable, when ϕ(x) = x, bk (x, t) = 0, w1 (t) = 1, j = 1 and n = 1
(ii) Theorem 2.9 is also a generalization of the main result in Lipovan [5, Theorem
2.1] in case of one variable, when ϕ(x) = x, bk (x, t) = 0, w1 (t) = 1, Φ(t) = 1, for
any x, t ∈ R+ (n = 1) and for j = 1.
8
H. KHELLAF, M. SMAKDJI
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Remark 2.12. (i) Under some suitable conditions, Theorem 2.9 reduced to Theorem 2.3 and Theorem 2-4 in case of two variables of the main results in Zhang and
Meng [17].
(ii) Under some suitable conditions in Theorem 2.9, we can also obtain other
estimations of the Ma and Pecaric’s inequality (1.1) and the main results in [9].
Remark 2.13. Theorem 2.9 further reduces to the main results in [1, Theorem
2.1, 2.2, 2.4] and the results in [13].
3. Proof of theorems
Since the proofs resemble each other, we give the details for (A1) in Theorem
2.9 only; the proofs of the remaining inequalities can be completed by following the
proofs of the above-mentioned inequalities.
Proof of Theorem 2.1 (A1). Fixing arbitrary numbers y = (y1 , . . . , yn ) ∈ Rn+ with
x0 < y ≤ x∗ , we define on [x0 ; y] a function z(x) by
z(x) = c(y) +
n1 Z
X
j=1
α
ej (x)
aj (y, t)uq (t)dt +
α
ej (x0 )
n2 Z
X
k=1
ek (x)
β
bk (y, t)uq (t)w1 (u(t))dt. (3.1)
ek (x0 )
β
Then z(x) is a positive and nondecreasing function with z(x0 ) = c(y), and
u(x) ≤ z(x)1/p ,
x ∈ [x0 ; y].
(3.2)
We know that
D1 D2 . . . Dn z(x) =
n1
X
0
0
0
aj (y, α
ej (x))uq (e
αj (x))αj1
αj2
. . . αjn
j=1
n2
X
+
0
0
0
bj (y, βej (x))uq (βej (x))w1 (u(βej (x)))βk1
βk2
. . . βkn
k=1
≤z
q/p
n1
hX
0
0
0
aj (y, α
ej (x))αj1
(x)
(x1 )αj2
(x2 ) . . . αjn
(xn )
(3.3)
j=1
+
n2
X
i
0
0
0
bj (y, βej (x))w1 (z 1/p (βej (x)))βk1
βk2
. . . βkn
.
k=1
Using the above inequality, we have
n
1
D1 D2 . . . Dn z(x) h X
0
0
0
aj (y, α
ej (x))αj1
≤
(x1 )αj2
(x2 ) . . . αjn
(xn )
z q/p (x)
j=1
+
n2
X
k=1
i
0
0
0
bj (y, βej (x))w1 (z 1/p (βej (x)))βk1
βk2
. . . βkn
.
(3.4)
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NONLINEAR DELAY INTEGRAL INEQUALITIES
9
Using D1 D2 · · · Dn−1 z(x) ≥ 0, pq z (q−p)/p (x) ≥ 0, Dn (x) ≥ 0 and(3.4), we have
D D . . . D
1 2
n−1 z(x)
q/p
z (x)
D1 D2 · · · Dn z(x)
≤
z q/p (x)
n1
X
0
0
0
≤
aj (y, α
ej (x))αj1
(x1 )αj2
(x2 ) . . . αjn
(xn )
Dn
j=1
n2
X
+
(3.5)
0
0
0
bk (y, βek (x))w1 (z 1/p (βek (x)))βk1
βk2
· · · βkn
.
k=1
Fixing x1 , x2 , . . . , xn−1 , setting xn = tn and integrating (3.5) from x0n to xn , we
obtain
D1 D2 · · · Dn−1 z(x)
z q/p (x)
n
1
X Z αjn (xn )
≤
aj y, αj1 (x1 ), αj2 (x2 ), · · · , αjn−1 (xn−1 ), αjn (tn )
αjn (x0n )
j=1
0
0
0
× αj1
αj2
· · · αjn−1
dtn
Z
n
2
β
(x
)
n
kn
X
+
bk (y, βk1 (x1 ), βk2 (x2 ), · · · , βkn−1 (xn−1 ), tn )
βkn (x0n )
k=1
× w1 (z
1/p
0
0
0
(βk1 , βk2 , · · · , βkn−1 , tn ))βk1
(x1 )βk2
(x2 ) · · · βkn−1
(xn−1 ) dtn .
Using the same method, we obtain
D1 z(x)
z q/p (x)
n1 h Z
X
≤
αjn (xn )
Z
αjn (x0n )
j=1
Z βjn (xn )
n2
X
+
[
k=1
αjn (xn )
...
βjn (x0n )
αjn (x0n )
Z
0
aj (y, αj1 (x1 ), t2 , . . . , tn )αj1
(x1 )dtn . . . dt2
(3.6)
βjn (xn )
...
bk (y, βk1 (x1 ), t2 , . . . , tn )
βjn (x0n )
i
0
× w1 (z 1/p (βk1 (x1 ), t2 , . . . , tn ))βk1
(x1 )dtn . . . dt2 .
Integrating (3.6) form x01 to x1 , we obtain
n1 Z α
ej (y)
X
p (p−q)/p
p (p−q)/p
z
(x) ≤
c
(y) +
aj (y, t)dt
p−q
p−q
ej (x0 )
j=1 α
+
i
n2 Z
X
k=1
ek (x)
β
ek (x0 )
β
bk (y, t)w1 (z 1/p (t))dt,
10
H. KHELLAF, M. SMAKDJI
EJDE-2011/169
for all x ∈ [x0 ; y], which implies that
z (p−q)/p (x) ≤ c(p−q)/p (y) +
n1 Z α
ej (y)
p−q X
aj (y, t)dt
p j=1 αej (x0 )
ek (x)
n2 Z β
p−q X
bk (y, t)w1 (z 1/p (t))dt.
+
p
ek (x0 )
β
(3.7)
k=1
Setting r1 (x) = z
(p−q)/p
(x), (3.7) can be rewritten as
ek (x)
n2 Z β
p−q X
1/(p−q)
r1 (x) ≤ p(y) +
bk (y, t)w1 (r1
(t))dt.
p
0
e
βk (x )
k=1
0
Defining v(x) on [x ; y], by
v(x) = p(y) +
ek (x)
n2 Z β
p−q X
1/(p−q)
bk (y, t)w1 (r1
(t))dt,
p
0
e
βk (x )
(3.8)
k=1
by (3.8), we have v(x0 ) = p(y) and
z (p−q)/p (x) ≤ v(x),
(3.9)
and
D1 D2 · · · Dn v(x) =
n2
p−q X
1/(p−q) e
0
0
0
bk (y, βek (x))w1 (r1
(βk (x)))βk1
βk2
. . . βkn
p
k=1
n2
p−q X
0
0
0
≤
bk (y, βek (x))w1 (v 1/(p−q) (βek (x)))βk1
βk2
. . . βkn
.
p
k=1
Using the same method as above, we obtain
D1 v(x)
w1 (v(x)1/p−q )
Z βjn (xn )
n2 h Z βjn (xn )
i
p−q X
0
≤
...
bk (y, βk1 (x1 ), t2 , . . . , tn )βk1
(x1 )dtn . . . dt2 .
p
βjn (x0n )
βjn (x0n )
k=1
Integrating form x01 to x1 , we obtain
Ψ1 (v(x)) ≤ Ψ1 (p(y)) +
ek (x)
n2 Z β
p−q X
bk (y, t)dt,
p
ek (x0 )
β
(3.10)
k=1
from (3.10) and for any arbitrary y, we obtain
ek (y)
n2 Z β
h
i
p−q X
v(y) ≤ Ψ−1
Ψ
(p(y))
+
b
(y,
t)dt
.
1
k
1
p
ek (x0 )
β
(3.11)
k=1
From (3.11) and (3.9),
ek (y)
n2 Z β
h
ip/(p−q)
p−q X
−1
z(y) ≤ Ψ1 Ψ1 (p(y)) +
bk (y, t)dt
.
p
ek (x0 )
β
k=1
By (3.12) and (3.2),
ek (y)
n2 Z β
i1/(p−q)
h
p−q X
−1
u(y) ≤ Ψ1 Ψ1 (p(y)) +
bk (y, t)dt
.
p
ek (x0 )
β
k=1
(3.12)
EJDE-2011/169
NONLINEAR DELAY INTEGRAL INEQUALITIES
Since y ≤ x∗ is arbitrary, the proof is complete.
11
Proof of Theorem 2.9. Fixing arbitrary numbers τ = (τ1 , . . . , τn ) ∈ Rn+ with x0 <
τ ≤ ξ, we define on [x0 ; τ ] a function z(x) by
Z αej (x)
n1
X
z(x) = c(τ ) +
dj (τ )
aj (τ, t)Φ(u(t))w1 (u(t))dt
α
ej (x0 )
j=1
+
n2
X
ek (x)
β
Z
lk (τ )
bk (τ, t)Φ(u(t))w2 (u(t))dt.
ek (x0 )
β
k=1
Then z(x) is a positive and nondecreasing function with z(x0 ) = c(τ ), and
u(x) ≤ ϕ−1 (z(x));
x ∈ [x0 ; τ ].
We know that
D1 D2 · · · Dn z(x)
=
n1
X
0
0
0
dj (τ )aj (τ, α
ej (x))Φ(u(e
αj (x)))w1 (u(e
αj (x)))αj1
αj2
. . . αjn
j=1
n2
X
+
0
0
0
lk (τ )bk (τ, βek (x))Φ(u(βek (x)))w2 (u(βek (x)))βk1
βk2
. . . βkn
,
k=1
n1
hX
0
0
0
≤ Φ(ϕ−1 (z(x))
dj (τ )aj (τ, α
ej (x))w1 (ϕ−1 (z(e
αj (x)))αj1
αj2
. . . αjn
j=1
+
n2
X
i
0
0
0
lk (τ )bk (τ, βek (x))w2 (ϕ−1 (z(βek (x)))βk1
βk2
. . . βkn
.
k=1
Using the same method in proof of the Theorem 2.1, and for all x ∈ [x0 ; τ ], which
implies that
Z αej (x)
n1
h
X
z(x) ≤ G−1 G(c(τ )) +
dj (τ )
aj (τ, t)w1 (u(t))dt
α
ej (x0 )
j=1
+
n2
X
Z
ek (x)
β
lk (τ )
i
bk (τ, t)w2 (u(t))dt .
ek (x0 )
β
k=1
Defining v(x) on [x0 ; τ ] by
v(x) = G(c(τ )) +
n1
X
Z
dj (τ )
j=1
+
n2
X
Z
α
ej (x)
aj (τ, t)w1 (u(t))dt
α
ej (x0 )
ek (x)
β
lk (τ )
bk (τ, t)w2 (u(t))dt.
ek (x0 )
β
k=1
We have v(x0 ) = G(c(τ )), z(x) ≤ G−1 (v(x)), and
u(x) ≤ ϕ−1 (G−1 (v(x))).
Then we obtain
n
1
hX
D1 D2 · · · Dn v(x)
0
0
0
≤
dj (τ )aj (τ, α
ej (x))αj1
(x1 )αj2
(x2 ) . . . αjn
(xn )
−1
−1
w1 (ϕ (G (v(x))))
j=1
(3.13)
12
H. KHELLAF, M. SMAKDJI
+
n2
X
EJDE-2011/169
i
0
0
0
lk (τ )bk (τ, βek (x))βk1
(x1 )βk2
(x2 ) . . . βkn
(xn ) .
k=1
Using the same method as above, we obtain
Z αej (x)
Z
n1
n2
X
X
Ψ1 (v(x)) ≤ Ψ1 (G(c(τ ))) +
dj (τ )
aj (τ, t)dt +
lk (τ )
α
ej (x0 )
j=1
k=1
ek (x)
β
bk (τ, t)dt.
ek (x0 )
β
From which we have
Z
n1
h
X
v(τ ) ≤ Ψ−1
Ψ
(G(c(τ
)))
+
d
(τ
)
1
j
1
+
Z
ek (τ )
β
lk (τ )
aj (τ, t)dt
α
ej (x0 )
j=1
n2
X
α
ej (τ )
(3.14)
i
bk (τ, t)dt ,
ek (x0 )
β
k=1
for any arbitrary numbers τ ∈ Rn+ , with x0 < τ ≤ ξ. From (3.13) and (3.14), we
obtain
Z αej (τ )
n1
n
h
X
u(τ ) ≤ ϕ−1 G−1 Ψ−1
Ψ
(G(c(τ
)))
+
d
(τ
)
aj (τ, t)dt
1
j
1
j=1
+
n2
X
Z
ek (τ )
β
lk (τ )
bk (τ, t)dt
io
α
ej (x0 )
.
ek (x0 )
β
k=1
Since τ is arbitrary and τ ≤ ξ, we obtain the result in the Theorem 2.9.
4. An application
In this section we present an immediate application of our results (Theorem 2.1
and Corollary 2.10) to study boundedness of solutions of delay partial differential
equations. First we consider the nonlinear partial delay differential equation in Rn :
Dup (x) = h(x, u(x), u(x − α
e(x)),
up (0, x2 , x3 , . . . , xn ) = c1 (x1 ),
up (0, x2 , x3 , . . . xn−1 , xn ) = cn (xn )
(4.1)
p
u (. . . , xi−1 , 0, xi+1 , . . . ) = ci (xi ) for i = 2, 3, . . . , n − 1,
ci (0) = 0
for i = 1, 2, . . . , n.
For x = (x1 ; x2 , . . . , xn ) ∈ Rn+ and α
e(x) = α1 (x1 ), α2 (x2 ), . . . , αn (xn ) ∈ Rn+
for αi , ci ∈ C 1 (R+ , R+ ) for i = 1, 2, . . . , n. Where h : Rn+ × R × R → R, is a
continuous function. Assume that these functions are defined and continuous on
their respective domains of definition such that
α
e(x) ≤ x,
for all x = (x1 ; x2 , . . . , xn ) ∈ Rn+ ,
q
p
|h(x, u, v)| ≤ a(x)|v(x)| + b(x)|v(x)| ,
(4.2)
(4.3)
for x ∈ Rn+ , where p > q ≥ 0 is a constants and a(x), b(x) are nonnegative, continuous functions defined for x ∈ Rn+ . For any solution u(x) of the boundary value
problem (4.1),
Z x
n
X
p
u (x) =
ci (xi ) +
h(t, u(t), u(t − α
e(t))dt,
(4.4)
i=1
0
EJDE-2011/169
NONLINEAR DELAY INTEGRAL INEQUALITIES
13
For all x, t ∈ Rn+ with 0 ≤ t ≤ x. Using (4.1), (4.3) and a suitable change of
variables in (4.4), we have
Z αe(x)
q
p
|up (x)| ≤ c(x) +
e
a(t) |u(t)| + eb(t) |u(t)| dt,
(4.5)
0
Pn
with c(x) = i=1 |ci (xi )|, e
a, eb ∈ C 1 (Rn+ , R+ ).
(E1) Applying (A1) in Theorem 2.1 to (4.5), when α
ej = βek , aj (x, t) = e
a(t),
p−q
e
bk (x, t) = b(t) with j = k = 1 and w1 (u) = u , we obtain a bound for the
solution u(x):
Z
1/(p−q)
1 Z αe(x)
p − q αe(x)
eb(t)dt .
e
a(t)dt
exp
(4.6)
u(x) ≤ c(p−q)/p (x) +
p
p 0
0
(E2) Or by a direct application of Corollary 2.10 to (4.5),
h p Z αe(x)
i
p−q
p
c p exp
u(x) ≤
[e
a(t) + eb(t)]dt .
p−q
p−q 0
(4.7)
Remark 4.1. In the special case (p = 2 and q = 1) in the boundary value problem
(4.1), we have
(i) By (4.6), we obtain
Z
1 Z αe(x)
p
1 αe(x)
eb(t)dt .
c(x) +
e
a(t)dt exp
u(x) ≤
2 0
2 0
(ii) Or by using (4.7),
h Z
p
u(x) ≤ 2 c(x) exp 2
α
e(x)
i
[e
a(t) + eb(t)]dt .
0
Remark 4.2. Note that the results given here can be very easily generalized to
obtain explicit bounds on integral inequalities involving several retarded arguments.
Using similar method of those in the proof of the Theorems above, we can also
obtain a new reversed inequalities of our results.
References
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[2] W. S. Cheung, Q. H. Ma; Nonlinear retarded integral inequalities for functions in two variables. J. Concr. Appl. Math. 2(2004) 119-134.
[3] M. Denche, H. Khellaf; Integral inequalities similar to Gronwall inequality, Electron. J. Diff.
Equ. , Vol 2007 (2007), No. 176, 1-14.
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Math. Sin. (Engl. Ser.) 21 (2005) 423–434.
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Hassane Khellaf
Department of Mathematics, Faculty of Exact Sciences, University of Mentouri, Constantine 25000, Algeria
E-mail address: khellafhassane@umc.edu.dz, khellaf1973@gmail.com
Mohamed el hadi Smakdji
Department of Mathematics, Faculty of Exact Sciences, University of Mentouri, Constantine 25000, Algeria
E-mail address: smakelhadi71@gmail.com