Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 256, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
BOUNDS FOR SOLUTIONS TO RETARDED NONLINEAR
DOUBLE INTEGRAL INEQUALITIES
SABIR HUSSAIN, TANZILA RIAZ, QING-HUA MA, JOSIP PEČARIĆ
Abstract. We present bounds for the solution to three types retarded nonlinear integral inequalities in two variables. By doing this, we generalizing
the results presented in [3, 12]. To illustrate our results, we present some
applications.
1. Introduction
In the study of the qualitative behavior for solutions to nonlinear differential and
integral equations, some specific types of inequalities are needed. The Gronwall
inequality [5] and the nonlinear version by Bihari [1] are fundamental tools in the
study of existence, uniqueness, boundedness, stability of solutions of differential,
integral, and integro-differential equations. For this reason, several generalizations
of the Gronwall inequality have been obtained, see [2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15]. Retarded integral inequalities have played an extensive role in the study of
partial differential and integral equations.
In Section 2 of this article, based on the assumptions (A1)–(A3) below, we derive
explicit bounds for the solutions to three types inequalities of retarded nonlinear
integral equations in two variables. In Section 3, the bounds are applied for proving
the global boundedness of solutions to the initial boundary-value problems. We stud
the following three inequalities:
ϕ(u(t, s)) ≤ a(t, s) + b(t, s)
n Z
X
i=1
Z
x
Z
y
+
αi (t0 )
αi (t)
αi (t0 )
Z
βi (s)
h
φ(u(x, y)) fi (x, y)(w(u(x, y))
βi (s0 )
i
gi (m, n)w(u(m, n)) dn dm) + hi (x, y) dy dx
βi (s0 )
2000 Mathematics Subject Classification. 30D05, 26D10.
Key words and phrases. Integral inequalities; Gronwall integral inequality;
integro-differential equation; double integral.
c
2014
Texas State University - San Marcos.
Submitted July 11, 2014. Published December 10, 2014.
1
(1.1)
2
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
ϕ(u(t, s)) ≤ a(t, s) + b(t, s)
n Z
X
αi (t)
x
Z
βi (s)
h
φ(u(x, y)) fi (x, y)φ1 (u(x, y))
βi (s0 )
αi (t0 )
i=1
Z
Z
EJDE-2014/256
y
× (w(u(x, y)) +
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
(1.2)
βi (s0 )
i
+ hi (x, y)φ2 (log(u(x, y))) dy dx
n Z αi (t) Z βi (s)
h
X
φ(u(x, y)) fi (x, y)(w(u(x, y))
ϕ(u(t, s)) ≤ a(t, s) + b(t, s)
αi (t0 )
i=1
x
Z
Z
βi (s0 )
y
(1.3)
gi (m, n)w(u(m, n)) dn dm)
+
βi (s0 )
αi (t0 )
i
+ hi (x, y)L(x, y, w(u(x, y))) dy dx
2. Main Results
Let R be the set of real numbers, R+ : [0, ∞); let t0 , t1 , s0 , s1 be real numbers
such that I := [t0 , t1 ); J := [s0 , s1 ). Denote by C i (M, N ), the class of all i-times
continuously differentiable functions defined on the set M to the set N , 1 ≤ i ≤ n
and C 0 (M, N ) = C(M, N ). The first order partial derivatives of a function z(x, y)
defined on R2 with respect to x and y are denoted by D1 z(x, y)(= zx (x, y)) and
D2 z(x, y)(= zy (x, y)) respectively. To prove our main results, we first list the
following assumptions:
(A1) a, b : I × J → (0, ∞) are nondecreasing in each variable;
(A2) ϕ, w ∈ C(R+ , R+ ), where ϕ and w are strictly increasing and nondecreasing
functions respectively with ϕ(0) = 0; ϕ(t) → ∞ as t → ∞ and w > 0 on
(0, ∞);
(A3) let αi ∈ C 1 (I, I) and βi ∈ C 1 (J, J) be non-decreasing with αi (t) ≤ t on I
and βi (s) ≤ s on J;
(A4) let u, fi , gi , hi ∈ C(I × J, R+ ), 1 ≤ i ≤ n and φ ∈ C(R+ , R+ ) a nondecreasing function such that φ(r) > 0 for r > 0;
(A5) let φ1 , φ2 ∈ C(R+ , R+ ) be nondecreasing functions with φ1 (r) > 0 and
φ2 (r) > 0 for r > 0.
Theorem 2.1. Assume conditions (A1)–(A4) and relation (1.1) hold. Then
u(t, s) ≤ ϕ−1 (G−1 (Ψ−1 (Ψ(c(t, s)) + b(t, s)D(t, s)))),
(2.1)
for all (t, s) ∈ [t0 , T3 ) × [s0 , S3 ) provided that ϕ−1 , G−1 , Ψ−1 are the respective
inverses of ϕ, G, Ψ, and (T3 , S3 ) ∈ I × J is arbitrarily chosen on the boundary of
the planar region: R := {(t, s) ∈ I × J}, provided that the following three relations
hold:
Ψ(c(t, s)) + b(t, s)D(t, s) ∈ Dom(Ψ−1 ),
Ψ−1 (Ψ(c(t, s)) + b(t, s)D(t, s)) ∈ Dom(G−1 ),
−1
G
−1
(Ψ
−1
(Ψ(c(t, s)) + b(t, s)D(t, s))) ∈ Dom(ϕ
where
Z
r
G(r) :=
r0
dp
,
φ(ϕ−1 (p))
r ≥ r0 ≥ 0,
(2.2)
),
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
Z
z
Ψ(z) :=
D(t, s) :=
n Z
X
αi (t)
Z
Z
z ≥ z0 ≥ 0,
,
z
Z
y
gi (m, n) dn dm] dy dz.
fi (z, y)[1 +
βi (s0 )
αi (t0 )
βi (s0 )
αi (t0 )
i=1
z0
βi (s)
dl
w(ϕ−1 (G−1 (l)))
c(t, s) := G(a(t, s)) + b(t, s)
3
n Z
X
αi (t)
βi (s)
hi (u, y) dy du.
αi (t0 )
i=1
Z
βi (s0 )
Proof. By Assumption (A2) and inequality (1.1), we have
n Z αi (t) Z βi (s)
h
X
φ(u(x, y)) fi (x, y)(w(u(x, y))
ϕ(u(t, s)) ≤ a(T, s) + b(T, s)
αi (t0 )
i=1
x
Z
Z
βi (s0 )
(2.3)
y
i
gi (m, n)w(u(m, n)) dn dm) + hi (x, y) dy dx
+
αi (t0 )
βi (s0 )
for all (t, s) ∈ [t0 , T ] × J, T ≤ T3 . Denote the right hand side of (2.3) by η(t, s),
then obviously η(t, s) is positive and non-decreasing function in each variable such
that η(t0 , s) = a(T, s). Then, (2.3) is equivalent to
u(t, s) ≤ ϕ−1 (η(t, s)).
(2.4)
ηt (t, s)
= b(T, s)
Z
n
X
αi0 (t)
βi (s)
h
φ(u(αi (t), y)) fi (αi (t), y)(w(u(αi (t), y))
βi (s0 )
i=1
αi (t) Z y
i
gi (m, n)w(u(m, n)) dn dm) + hi (αi (t), y) dy
+
≤
Z
αi (t0 ) βi (s0 )
Z βi (s)
n
X
αi0 (t)
b(T, s)
βi (s0 )
i=1
Z αi (t) Z y
h
φ(ϕ−1 (η(αi (t), y))) fi (αi (t), y)(w(ϕ−1 (η(αi (t), y)))
i
gi (m, n)w(ϕ−1 (η(m, n))) dn dm) + hi (αi (t), y) dy,
+
αi (t0 )
βi (s0 )
which implies
−1
ηt (t, s) ≤ φ(ϕ
(η(t, s)))b(T, s)
n
X
i=1
Z
αi (t)
Z
y
Z
βi (s)
h
fi (αi (t), y)(w(ϕ−1 (η(αi (t), y)))
βi (s0 )
i
gi (m, n)w(ϕ−1 (η(m, n))) dn dm) + hi (αi (t), y) dy.
+
αi (t0 )
αi0 (t)
βi (s0 )
(2.5)
Then, (2.5) is equivalent to
ηt (t, s)
φ(ϕ−1 (η(t, s)))
Z
n
X
≤ b(T, s)
αi0 (t)
Z
i=1
αi (t) Z y
+
αi (t0 )
βi (s0 )
βi (s)
h
fi (αi (t), y)(w(ϕ−1 (η(αi (t), y)))
βi (s0 )
i
gi (m, n)w(ϕ−1 (η(m, n)))dn dm) + hi (αi (t), y) dy,
4
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
for all (t, s) ∈ [t0 , T ] × J. Replace t by v then integrating from t0 to t with respect
to v and making change of variable on right hand side of the above inequality and
using the definition of G, we have
G(η(t, s)) ≤ G(η(t0 , s)) + b(T, s)
+ b(T, s)
Z
n Z
X
i=1
y
u
αi (t0 )
n Z
X
Z
βi (s)
hi (u, y) dy du
βi (s0 )
(2.6)
−1
(η(u, y)))
βi (s0 )
gi (m, n)w(ϕ−1 (η(m, n))) dn dm) dy du
n Z
X
αi (t)
αi (t0 )
Z
βi (s)
fi (u, y)(w(ϕ−1 (η(u, y)))
βi (s0 )
y
gi (m, n)w(ϕ−1 (η(m, n))) dn dm) dy du.
+
αi (t0 )
Z
βi (s0 )
i=1
u
αi (T )
fi (u, y)(w(ϕ
αi (t0 )
≤ c(T, s) + b(T, s)
Z
n Z
X
i=1 αi (t0 )
αi (t) Z βi (s)
+
αi (t0 )
βi (s0 )
gi (m, n)w(ϕ−1 (η(m, n))) dn dm) dy du
i=1
Z y
u
hi (u, y) dy du
βi (s0 )
βi (s0 )
≤ G(a(T, s)) + b(T, s)
Z
βi (s)
Z
fi (u, y)(w(ϕ−1 (η(u, y)))
+
+ b(T, s)
αi (t)
i=1 αi (t0 )
Z
βi (s)
αi (t)
Z
αi (t0 )
n Z
X
βi (s0 )
Denote the right hand side of (2.6) by Γ(t, s), then obviously Γ(t, s) is positive and
non-decreasing function in each variable such that Γ(t0 , s) = c(T, s). Then, (2.6) is
equivalent to
η(t, s) ≤ G−1 (Γ(t, s)).
(2.7)
By the fact that αi (t) ≤ t and βi (s) ≤ s for (t, s) ∈ I × J, 1 ≤ i ≤ n, and
monotonicity of Γ, w and ϕ−1 , we have
Γt (t, s) = b(T, s)
Z
n
X
αi0 (t)
i=1
αi (t) Z y
Z
βi (s)
fi (αi (t), y)(w(ϕ−1 (η(αi (t), y)))
βi (s0 )
gi (m, n)w(ϕ−1 (η(m, n))) dn dm)dy
αi (t0 ) βi (s0 )
Z βi (s)
n
X
≤ b(T, s)
αi0 (t)
fi (αi (t), y)(w(ϕ−1 (η(t, y)))
β
(s
)
i
0
i=1
Z αi (t) Z y
+
gi (m, n)w(ϕ−1 (η(t, y))) dn dm)dy
αi (t0 ) βi (s0 )
Z βi (s)
n
X
≤ b(T, s)w(ϕ−1 (G−1 (Γ(t, s))))
αi0 (t)
fi (αi (t), y)
βi (s0 )
i=1
Z αi (t) Z y
+
× 1+
gi (m, n) dn dm)dy.
αi (t0 )
βi (s0 )
(2.8)
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
5
Then, (2.8) is written as
Γt (t, s)
w(ϕ−1 (G−1 (Γ(t, s))))
Z βi (s)
Z
n
X
≤ b(T, s)
αi0 (t)
fi (αi (t), y) 1 +
βi (s0 )
i=1
αi (t)
y
Z
gi (m, n) dn dm dy
βi (s0 )
αi (t0 )
Replace t by q then integrating from t0 to t with respect to q and making change
of variable on right hand side of the above inequality and using the definition of Ψ,
we obtain
Ψ(Γ(t, s)) ≤ Ψ(c(T, s)) + b(T, s)D(t, s).
(2.9)
A combination of (2.4), (2.7) and (2.9) yield the desire result (2.1).
Theorem 2.2. Assume conditions (A1)–(A5) and relation (1.2) hold. Then
• if φ1 (r) ≥ φ2 (log(r)), we have
u(t, s) ≤ ϕ−1 (G−1 (H1−1 (J1−1 (J1 (e
c(T, s)) + b(T, s)D(t, s))))),
(2.10)
for (t, s) ∈ [t0 , T1 ) × [s0 , S1 ),
• if φ1 (r) < φ2 (log(r)), we have
u(t, s) ≤ ϕ−1 (G−1 (H2−1 (J2−1 (J2 (e
c(T, s)) + b(T, s)D(t, s))))),
−1
−1
, Hj−1
(2.11)
Jj−1
for all (t, s) ∈ [t0 , T2 ) × [s0 , S2 ), provided that ϕ , G
and
are the
respective inverses of ϕ, G, Hj and Hj ; let (Tj , Sj ) ∈ I × J be arbitrarily chosen on
the boundary of the planar region Rj := {(t, s) ∈ I × J}, j ∈ {1, 2}, provided that
the following four relations are satisfied
Jj (e
c(T, s)) + b(T, s)D(t, s) ∈ Dom(Jj−1 ),
Jj−1 (Jj (e
c(T, s)) + b(T, s)D(t, s)) ∈ Dom(Hj−1 ),
Hj−1 (Jj−1 (Jj (e
c(T, s)) + b(T, s)D(t, s))) ∈ Dom(G−1 ),
(2.12)
G−1 (Hj−1 (Jj−1 (Jj (e
c(T, s)) + b(T, s)D(t, s)))) ∈ Dom(ϕ−1 ),
where
n Z
X
e
c(t, s) = Hj (G(a(T, s))) + b(T, s)
Z
Jj (r) =
r0
r
Z
βi (s)
hi (u, y) dy du,
αi (t0 )
i=1
Hj (r) =
Z r
αi (t)
βi (s0 )
ds
(ϕ−1 (G−1 (s)))
φj
ds
,
−1
−1
w(ϕ (G (Hj−1 (s))))
r0
,
r ≥ r0 ≥ 0.
Proof. By condition (A2) and inequality (1.2), we have
n Z αi (t) Z βi (s)
h
X
ϕ(u(t, s)) ≤ a(T, s) + b(T, s)
φ(u(x, y)) fi (x, y)
i=1
αi (t0 )
βi (s0 )
Z
x
Z
y
× φ1 (u(x, y))(w(u(x, y)) +
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
+ hi (x, y)φ2 log(u(x, y))
i
βi (s0 )
dy dx
(2.13)
6
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
for all (t, s) ∈ [t0 , T ] × J, T ≤ T1 . Denote the right hand side of (2.13) by Θ(t, s),
then obviously Θ(t, s) is positive and non-decreasing function in each variable such
that Θ(t0 , s) = a(T, s). Then (2.13) is equivalent to
u(t, s) ≤ ϕ−1 (Θ(t, s)).
(2.14)
By the fact that αi (t) ≤ t and βi (s) ≤ s for (t, s) ∈ I × J, 1 ≤ i ≤ n, and
monotonicity of φ, ϕ−1 , Θ, we have
Θt (t, s) = b(T, s)
n
X
αi0 (t)
Z
βi (s)
h
φ(u(αi (t), y)) fi (αi (t), y)φ1 (u(αi (t), y))
βi (s0 )
i=1
Z
αi (t)
y
Z
× (w(u(αi (t), y)) +
gi (m, n)w(u(m, n)) dn dm)
βi (s0 )
αi (t0 )
i
+ hi (αi (t), y)φ2 (log(u(αi (t), y))) dy
Z βi (s) h
n
X
≤ b(T, s)φ(ϕ−1 (Θ(t, s)))
αi0 (t)
fi (αi (t), y)φ1 (ϕ−1 (Θ(αi (t), y)))
βi (s0 )
i=1
−1
× (w(ϕ
Z
αi (t)
Z
y
(Θ(αi (t), y))) +
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (αi (t), y)φ2 (log(ϕ−1 (Θ(αi (t), y)))) dy,
(2.15)
for all (t, s) ∈ [t0 , T ] × J. From (2.15), we have
Θt (t, s)
φ(ϕ−1 (Θ(t, s)))
Z
n
X
0
αi (t)
≤ b(T, s)
Z
i=1
αi (t) Z y
+
βi (s)
h
fi (αi (t), y)φ1 (ϕ−1 (Θ(αi (t), y)))(w(ϕ−1 (Θ(αi (t), y)))
βi (s0 )
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (αi (t), y)φ2 (log(ϕ−1 (Θ(αi (t), y)))) dy
Replacing t by v then integrating from t0 to t with respect to v and making change
of variable on right hand side of the above inequality to obtain
G(Θ(t, s)) ≤ G(a(T, s)) + b(T, s)
n Z
X
× (w(ϕ
Z
αi (t0 )
i=1
−1
αi (t)
Z
u
βi (s)
h
fi (u, y)φ1 (ϕ−1 (Θ(u, y)))
βi (s0 )
Z
y
(Θ(u, y))) +
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (u, y)φ2 (log(ϕ−1 (Θ(u, y)))) dy du
(2.16)
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
When φ1 (u) ≥ φ2 (log(u)), by (2.16), we have
n Z αi (t) Z
X
G(Θ(t, s)) ≤ G(a(T, s)) + b(T, s)
φ1 (ϕ−1 (Θ(u, y)))
βi (s0 )
αi (t0 )
i=1
βi (s)
7
h
× fi (u, y)(w(ϕ−1 (Θ(u, y)))
Z u Z y
gi (m, n)w(ϕ−1 (Θ(u, y))) dn dm) + hi (u, y)] dy du.
+
αi (t0 )
βi (s0 )
(2.17)
Denote the right hand side of (2.17) by Λ(t, s), then obviously Λ(t, s) is positive
and non-decreasing function in each variable such that Λ(t0 , s) = G(a(T, s)). Then
(2.17) is equivalent to
Θ(t, s) ≤ G−1 (Λ(t, s)),
(2.18)
Λt (t, s)
= b(T, s)
n
X
i=1
Z αi (t)
βi (s)
Z
αi0 (t)
βi (s0 )
y
Z
i
gi (m, n) dn dm + hi (αi (t), y) dy
× 1+
αi (t0 )
−1
≤ b(T, s)φ1 (ϕ
h
φ1 (ϕ−1 (Θ(αi (t), y))) fi (αi (t), y)w(ϕ−1 (Θ(αi (t), y)))
βi (s0 )
−1
(G
(Λ(t, s))))
n
X
αi0 (t)
Z
βi (s)
h
fi (αi (t), y)
βi (s0 )
i=1
Z
× w(ϕ−1 (G−1 (Λ(αi (t), y)))) 1 +
αi (t)
αi (t0 )
Z
y
gi (m, n) dn dm
βi (s0 )
i
+ hi (αi (t), y) dy.
(2.19)
From (2.19), we have
Λt (t, s)
φ1 (ϕ−1 (G−1 (Λ(t, s))))
Z βi (s) h
n
X
0
≤ b(T, s)
αi (t)
fi (αi (t), y)w(ϕ−1 (G−1 (Λ(αi (t), y))))
i=1
Z αi (t)
βi (s0 )
Z
y
i
gi (m, n) dn dm + hi (αi (t), y) dy,
× 1+
αi (t0 )
βi (s0 )
Replacing t by v then integrating from t0 to t with respect to v and making change
of variable on right hand side of the above inequality and using the definition of
H1 , we obtain
H1 (Λ(t, s))
≤ H1 (G(a(T, s))) + b(T, s)
n Z
X
i=1
Z
× 1+
u
αi (t0 )
Z
y
βi (s0 )
αi (t)
αi (t0 )
Z
βi (s)
h
fi (u, y)w(ϕ−1 (G−1 (Λ(u, y))))
βi (s0 )
i
gi (m, n) dn dm + hi (u, y) dy du
8
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
≤e
c(T, s) + b(T, s)
Z
u
Z
n Z
X
αi (t)
αi (t0 )
i=1
y
βi (s)
Z
EJDE-2014/256
fi (u, y)w(ϕ−1 (G−1 (Λ(u, y))))
βi (s0 )
gi (m, n) dn dm dy du.
× 1+
(2.20)
βi (s0 )
αi (t0 )
e 0 , s) = H1 (G(a(T, s))). Then
Denote the right hand side of (2.20), such that Θ(t
(2.20) is equivalent to
e s)).
Λ(t, s) ≤ H1−1 (Θ(t,
(2.21)
By the fact that αi (t) ≤ t, βi (s) ≤ s for (t, s) ∈ I × J, 1 ≤ i ≤ n, and monotonicity
of w, ϕ−1 and (2.21), we have
Z βi (s)
n
X
0
e
Θt (t, s) = b(T, s)
αi (t)
fi (αi (t), y)w(ϕ−1 (G−1 (Λ(αi (t), y))))
i=1
Z αi (t)
βi (s0 )
y
Z
gi (m, n) dn dm dy
× 1+
αi (t0 )
−1
≤ b(T, s)w(ϕ
βi (s0 )
−1
(G
e s)))))
(H1−1 (Θ(t,
n
X
αi0 (t)
αi (t)
(2.22)
βi (s)
fi (αi (t), y)
βi (s0 )
i=1
Z
Z
y
Z
× (1 +
gi (m, n) dn dm)dy.
αi (t0 )
βi (s0 )
From (2.22), we have
e t (t, s)
Θ
e s)))))
w(ϕ−1 (G−1 (H1−1 (Θ(t,
Z βi (s)
Z
n
X
0
≤ b(T, s)
αi (t)
fi (αi (t), y) 1 +
i=1
βi (s0 )
αi (t)
αi (t0 )
Z
y
gi (m, n) dn dm dy.
βi (s0 )
Replacing t by v then integrating from t0 to t with respect to v and making change
of variable on right hand side of the above inequality and using the definition of
J1 , we obtain
e s)) ≤ J1 (e
J1 (Θ(t,
c(T, s)) + b(T, s)D(t, s)
(2.23)
As T ≤ T1 is arbitrary, a combination of (2.14), (2.18), (2.21) and (2.23) yield
u(t, s) ≤ ϕ−1 (G−1 (H1−1 (J1−1 (J1 (e
c(T, s)) + b(T, s)D(t, s))))).
When φ1 (u) ≤ φ2 (log(u)), by (2.16), we have
G(Θ(t, s))
≤ G(a(T, s)) + b(T, s)
n Z
X
Z
αi (t0 )
i=1
× (w(ϕ−1 (Θ(u, y))) +
αi (t)
Z
u
βi (s)
h
fi (u, y)φ2 (log(ϕ−1 (Θ(u, y))))
βi (s0 )
Z
y
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (u, y)φ2 (log(ϕ−1 (Θ(u, y)))) dy du
n Z αi (t) Z βi (s) h
X
≤ G(a(T, s)) + b(T, s)
fi (u, y)(w(ϕ−1 (Θ(u, y)))
i=1
αi (t0 )
βi (s0 )
EJDE-2014/256
Z
u
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
Z
y
i
gi (m, n)w(u(m, n)) dn dm) + hi (u, y) φ2 (ϕ−1 (Θ(u, y))) dy du
+
αi (t0 )
9
βi (s0 )
Similarly to the above process from (2.17) to (2.23), for T ≤ T2 , and as T is
arbitrary, we have
u(t, s) ≤ ϕ−1 (G−1 (H2−1 (J2−1 (J2 (e
c(T, s)) + b(T, s)D(t, s))))).
Theorem 2.3. Suppose that (A1)–(A5) hold and that L, M ∈ C(R3+ , R+ ) are such
that
0 ≤ L(t, s, u) − L(t, s, v) ≤ M (t, s, v)(u − v),
for u > v. If u(t, s) is a nonnegative and continuous function on I × J satisfying
(1.3), then we have
u(t, s) ≤ ϕ−1 (G−1 (Ψ−1 (Ψ(G(a(t, s))
n Z αi (t) Z βi (s)
X
+ b(t, s)
hi (u, y)L(u, y, 0) dy du)
αi (t0 )
i=1
(2.24)
βi (s0 )
n Z
n
X
+ b(t, s) D(t, s) +
i=1
αi (t)
Z
αi (t0 )
βi (s)
o
M (u, y, 0) dy du ))),
βi (s0 )
for all (t, s) ∈ [t0 , T4 ) × [s0 , S4 ) provided that ϕ−1 , G−1 , Ψ−1 are the respective
inverses of ϕ, G, Ψ, and (T4 , S4 ) ∈ I × J is arbitrarily chosen on the boundary of
the planar region, R4 := {(t, s) ∈ I × J}, provided that the following three relations
are satisfied:
n Z αi (t) Z βi (s)
h
X
e s) := Ψ(G(a(t, s)) + b(t, s)
∆(t,
hi (u, y)L(u, y, 0) dy du)
i=1 αi (t0 ) βi (s0 )
αi (t) Z βi (s)
+ b(t, s){D(t, s) +
n Z
X
i=1
i
M (u, y, 0) dy du} ∈ Dom(Ψ−1 )
αi (t0 )
βi (s0 )
(2.25)
Ψ
−1
−1
e s)) ∈ Dom(G
(∆(t,
),
−1
G
−1
(Ψ
−1
e s))) ∈ Dom(ϕ
(∆(t,
)
(2.26)
Proof. From assumption (A1) and the inequality (1.3), we have
n Z αi (t) Z βi (s)
h
X
ϕ(u(t, s)) ≤ a(T, s) + b(T, s)
φ(u(x, y)) fi (x, y)(w(u(x, y))
i=1
Z
x
Z
αi (t0 )
βi (s0 )
y
+
gi (m, n)w(u(m, n)) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (x, y)L(x, y, w(u(x, y))) dy dx,
(2.27)
for all (t, s) ∈ [t0 , T ] × J, T ≤ T4 . Denote the right hand side of (2.27) by P(t, s),
then obviously P(t, s) is positive and non-decreasing function in each variable,
P(t0 , s) = a(T, s). Then, (2.27) is equivalent to
u(t, s) ≤ ϕ−1 (P(t, s)).
(2.28)
10
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
By the fact that αi (t) ≤ t and βi (s) ≤ s for (t, s) ∈ I × J, 1 ≤ i ≤ n, and
monotonicity of P, ϕ−1 , φ, we have
n Z βi (s)
h
X
Pt (t, s) = b(T, s)
φ(u(αi (t), y)) fi (αi (t), y)(w(u(αi (t), y))
i=1 βi (s0 )
αi (t) Z y
Z
gi (m, n)w(u(m, n)) dn dm)
+
βi (s0 )
αi (t0 )
i
+ hi (αi (t), y)L(αi (t), y, w(u(αi (t), y))) dyαi0 (t)
n Z βi (s) h
X
−1
≤ b(T, s)φ(ϕ (P(t, s)))
fi (αi (t), y)(w(ϕ−1 (P(αi (t), y)))
i=1
αi (t)
Z
Z
gi (m, n)w(ϕ−1 (P(m, n))) dn dm) + hi (αi (t), y)
+
αi (t0 )
βi (s0 )
y
βi (s0 )
i
× L(αi (t), y, w(ϕ−1 (P(αi (t), y)))) dy αi0 (t).
(2.29)
From (2.29), we have
n Z βi (s) h
X
Pt (t, s)
≤
b(T,
s)
fi (αi (t), y)(w(ϕ−1 (P(αi (t), y)))
φ(ϕ−1 (P(t, s)))
β
(s
)
i 0
i=1
Z αi (t) Z y
+
gi (m, n)w(ϕ−1 (P(m, n))) dn dm)
αi (t0 )
βi (s0 )
i
+ hi (αi (t), y)L(αi (t), y, w(ϕ−1 (P(αi (t), y)))) dy αi0 (t),
for all (t, s) ∈ [t0 , T ] × J, T ≤ T4 . Replace t by v then integrating from t0 to t
with respect to v and making change of variable on right hand side of the above
inequality and using the definition of G, we have
n Z αi (t) Z βi (s) h
X
G(P(t, s)) ≤ G(a(T, s)) + b(T, s)
fi (u, y)(w(ϕ−1 (P(u, y)))
i=1
Z
u
Z
βi (s0 )
gi (m, n)w(ϕ−1 (P(m, n))) dn dm) + hi (u, y)
+
αi (t0 )
αi (t0 )
y
βi (s0 )
i
× {L(u, y, 0) + M (u, y, 0)w(ϕ−1 (P(u, y)))} dy du
n Z αi (T ) Z βi (s)
X
≤ G(a(T, s)) + b(T, s)
hi (u, y)L(u, y, 0) dy du
+ b(T, s)
n Z αi (t)
X
i=1
αi (t0 )
i=1 αi (t0 )
βi (s) h
βi (s0 )
Z
Z
u
Z
y
fi (u, y)(1 +
βi (s0 )
gi (m, n) dn dm)
αi (t0 )
βi (s0 )
i
+ M (u, y, 0) w(ϕ−1 (P(u, y))) dy du
(2.30)
Denote the right hand side of (2.30) by Q(t, s), then obviously Q(t, s) is a positive
and nondecreasing function in each variable such that Q(t0 , s) = G(a(T, s)). Then,
(2.30) is equivalent to
P(t, s) ≤ G−1 (Q(t, s)),
(2.31)
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
Q(t0 , s) = G(a(T, s)) + b(T, s)
n Z
X
n Z
X
βi (s)
hi (u, y)L(u, y, 0) dy du,
βi (s0 )
Z
h
fi (αi (t), y)(1 +
βi (s0 )
i=1
βi (s)
Z
αi (t0 )
i=1
Qt (t, s) = b(T, s)
αi (T )
αi (t)
Z
y
gi (m, n)dndm)
αi (t0 )
βi (s0 )
i
+ M (αi (t), y, 0) w(ϕ−1 (P(αi (t), y)))dyαi0 (t)
n Z βi (s) h
X
−1
−1
≤ b(T, s)w(ϕ (G (P(t, s))))
fi (αi (t), y)
αi (t)
Z
y
(2.32)
βi (s0 )
i=1
Z
11
i
gi (m, n)dndm + M (αi (t), y, 0) dyαi0 (t).
× 1+
βi (s0 )
αi (t0 )
Then, (2.32) is written as
Qt (t, s)
−1
w(ϕ (G−1 (P(t, s))))
≤ b(T, s)
n Z
X
βi (s)
Z
h
fi (αi (t), y) 1 +
βi (s0 )
i=1
αi (t)
αi (t0 )
Z
y
gi (m, n)dn dm
(2.33)
βi (s0 )
i
+ M (αi (t), y, 0) dyαi0 (t)
Replacing t by v then integrating from t0 to t with respect to v and making change
of variable on right hand side of (2.33) and using the definition of Ψ, we obtain
n Z αi (T ) Z βi (s)
X
Ψ(Q(t, s)) ≤ Ψ(G(a(T, s)) + b(T, s)
hi (u, y)L(u, y, 0) dy du)
βi (s0 )
i=1 αi (t0 )
αi (t) Z βi (s)
n Z
n
X
+ b(T, s) D(t, s) +
i=1
o
M (u, y, 0) dy du .
αi (t0 )
βi (s0 )
A combination of (2.28), (2.31) and (2.34) yield inequality (2.24).
(2.34)
Corollary 2.4. Suppose (A2)–(A4) are satisfied. If p > q > 0 and c ≥ 0 are
constants such that:
n Z αi (t) Z βi (s)
h
X
p
u (t, s) ≤ c + p
uq (t, s) fi (x, y)(w(u(x, y))
αi (t0 )
i=1
Z
x
Z
y
(2.35)
i
gi (m, n)w(u(m, n)) dn dm) + hi (x, y) dy dx,
+
αi (t0 )
βi (s0 )
βi (s0 )
then
q
Ψ−1
∗ (Ψ∗ (m0 (t, s)) + (p − q)D(t, s)),
n Z αi (t) Z βi (s)
X
p−q
q
+ (p − q)
hi (u, y) dy du,
m0 (t, s) = c
u(t, s) ≤
p−q
i=1
Z
r
Ψ∗ (r) :=
r0
αi (t0 )
du
√ ,
w( p−q u)
βi (s0 )
r ≥ r0 > 0,
(2.36)
12
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
Proof. Denote the right hand side of (2.35) by Ξ(t, s), then obviously Ξ(t, s) is
positive and non-decreasing function in each variable such that Ξ(t0 , s) = c. Then,
(2.35) is equivalent to
p
p
u(t, s) ≤
Ξ(t, s),
(2.37)
Ξt (t, s)
=p
n
X
Z
αi0 (t)
i=1
Z αi (t)
βi (s)
βi (s0 )
y
Z
i
gi (m, n)w(u(m, n)) dn dm) + hi (αi (t), y)uq (αi (t), y) dy
+
βi (s0 )
αi (t0 )
≤p
n
X
Z
αi0 (t)
i=1
Z αi (t)
h
fi (αi (t), y)uq (αi (t), y)(w(u(αi (t), y))
βi (s)
{Ξ(αi (t), y)}q/p
h
(2.38)
p
fi (αi (t), y)(w( p Ξ(αi (t), y))
βi (s0 )
y
Z
i
p
gi (m, n)w( p Ξ(m, n)) dn dm) + hi (αi (t), y) dy,
+
αi (t0 )
βi (s0 )
for (t, s) ∈ [t0 , T ] × J. Then, (2.38) is equivalent to
Z βi (s) h
n
X
p
Ξt (t, s)
p
0
≤
p
α
(t)
f
(α
(t),
y)(w(
Ξ(αi (t), y))
i
i
i
{Ξ(t, s)}q/p
β
(s
)
i
0
i=1
Z αi (t) Z y
i
p
+
gi (m, n)w( p Ξ(m, n)) dn dm) + hi (αi (t), y) dy .
αi (t0 )
βi (s0 )
Replacing t by v then integrating from t0 to t with respect to v, making change
of variable on right hand side of the above inequality and by using that m0 (t, s) is
non-decreasing in each variable, for t ≤ T , we have
(p−q)/p
≤ c(p−q)/q + (p − q)
n Z
X
i=1
Z
u
Z
αi (t)
αi (t0 )
y
+
gi (m, n)w(
αi (t0 )
n Z
X
i=1
u
Z
p
fi (u, y)(w( p Ξ(u, y))
βi (s0 )
i
Ξ(m, n)) dn dm) + hi (u, y) dy du
αi (t)
αi (t0 )
Z
(2.39)
βi (s)
fi (u, y)(w(
p
p
Ξ(u, y))
βi (s0 )
y
+
αi (t0 )
p
p
h
βi (s0 )
≤ m0 (T, s) + (p − q)
Z
βi (s)
Z
p
gi (m, n)w( p Ξ(m, n)) dn dm) dy du
βi (s0 )
Denote the right hand side of (2.39) by γ(t, s), then obviously γ(t, s) is positive
and non-decreasing function in each variable such that γ(t0 , s) = m0 (T, s). Then,
(2.39) is equivalent to
p
Ξ(t, s) ≤ [γ(t, s)] p−q ,
(2.40)
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
γt (t, s) = (p − q)
Z
n
X
αi0 (t)
p
fi (αi (t), y)(w( p Ξ(αi (t), y))
p
gi (m, n)w( p Ξ(m, n)) dn dm) dy
+
βi (s0 )
αi (t0 )
Z
βi (s)
βi (s0 )
i=1
αi (t) Z y
≤ (p − q)
Z
n
X
13
αi0 (t)
Z
βi (s)
(2.41)
p
γ(αi (t), y))
fi (αi (t), y)(w(
p−q
βi (s0 )
i=1
αi (t) Z y
p
gi (m, n)w( p−q γ(m, n)) dn dm) dy.
+
βi (s0 )
αi (t0 )
Then, (2.41) is written as
γt (t, s)
p
w( p−q γ(t, s))
Z
n
X
≤ (p − q)
αi0 (t)
βi (s)
βi (s0 )
i=1
Z
fi (αi (t), y) 1 +
αi (t)
αi (t0 )
Z
y
gi (m, n) dn dm dy.
βi (s0 )
Setting t by l then integrating from t0 to t with respect to l, making change of
variable on right hand side of the above inequality and using γ(t0 , s) = m0 (T, s),
and the definition of Ψ∗ , we have
Ψ∗ (γ(t, s)) ≤ Ψ∗ (γ(t, s)) + (p − q)D(t, s).
A combination of (2.37), (2.40), and (2.42) yield the desire result (2.36).
(2.42)
Remark 2.5.
• For a(t, s) ≡ c, b(t, s) ≡ 1, φ(x) = x, gi ≡ 0 ≡ hi , 1 ≤ i ≤ n.
Then Theorem 2.1 reduces to [3, Theorem 2.2].
• For a(t, s) ≡ c, b(t, s) ≡ 1, φ(x) = x, gi ≡ 0, 1 ≤ i ≤ n. Then Theorem 2.1
reduces to [3, Theorem 2.3].
• For q = 1, gi ≡ 0, 1 ≤ i ≤ n, corollary 2.4 reduces to [3, Corollary 2.4].
• For gi ≡ 0, 1 ≤ i ≤ n, Theorem 2.1 reduces to [12, Theorem 1].
• For gi ≡ 0, 1 ≤ i ≤ n, and w ≡ 1, theorem 2.2 reduces to [12, Theorem 2].
3. Applications
In this section, we apply the inequalities established above to achieve the boundedness of partial integro-differential equations, with several retarded arguments, of
the form
∂ p−1
(z
(t, s)zt (t, s))
∂s h
= F t, s, z(t − l1 (t), s − k1 (s)), . . . , z(t − ln (t), s − kn (s)),
(3.1)
Z tZ s
i
Q(t, s, σ, τ, z(t − l1 (t), s − k1 (s)), . . . , z(t − ln (t), s − kn (s)))dσdτ ,
t0
s0
14
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
and
D2 (D1 ϕ(z(t, s)))
h
= F t, s, z(t − l1 (t), s − k1 (s)), . . . , z(t − ln (t), s − kn (s)),
Z tZ s
i
Q(t, s, σ, τ, z(t − l1 (t), s − k1 (s)), . . . , z(t − ln (t), s − kn (s)))dσ dτ ,
t0
s0
(3.2)
with the given initial boundary conditions
z(t, s0 ) = a1 (t),
z(t0 , s) = a2 (s),
a1 (t0 ) = a2 (s0 ) = 0,
(3.3)
where F ∈ C(I × J × Rn , R), Q ∈ C((I × J) × (I × J) × Rn , R), a1 ∈ C 1 (I, R),
a2 ∈ C 1 (J, R) and li ∈ C 1 (I, R), ki ∈ C 1 (J, R) are nonincreasing and such that
t − li (t) ≥ 0, t − li (t) ∈ C 1 (I, I), s − ki (s) ≥ 0, s − ki (s) ∈ C 1 (J, J), li0 (t) < 1,
ki0 (s) < 1 and li (t0 ) = ki (s0 ) = 0, 1 ≤ i ≤ n, for (t, s) ∈ I × J; let ϕ ∈ C 1 (R, R) be
an increasing function such that ϕ(|u|) ≤ |ϕ(u)|; let ϕ(e(t, s)) = ϕ(a1 (t)) + ϕ(a2 (s))
and
Mi = max
t∈I
1
,
1 − li0 (t)
Ni = max
s∈J
1
,
1 − ki0 (s)
1 ≤ i ≤ n.
(3.4)
The following theorem deals with a boundedness on the solution of (3.2).
Theorem 3.1. Assume that F : I × J × Rn × Rn → R is a continuous function
for which there exist continuous nonnegative functions fi (t, s), gi (t, s) and hi (t, s),
1 ≤ i ≤ n, for (t, s) ∈ I × J such that:
|F (t, s, u1 , . . . , un , j)| ≤ b(t, s)
n
X
φ(|ui |) fi (t, s)w(|ui |) + |j| + hi (t, s) .
i=1
(3.5)
|Q(t, s, v1 , v2 , u1 , u2 , . . . , un )| ≤ gi (t, s)w(|ui |).
If z(t, s) is a solution of (3.2) with conditions (3.3), then
−1
|z(t, s)| ≤ ϕ
−1
G
n Z
X
−1
Ψ
Ψ(c(t, s)) + b(t, s)
i=1
Z
× 1+
δ
φi (t0 )
Z
ψi (s)
φi (t)
Z
f i (δ, η)
φi (t0 )
g i (δ1 , η1 )dη1 dδ1 dη dδ
ψi (s)
ψi (s0 )
(3.6)
,
ψi (s0 )
where,
f i (u, v) = Mi Ni fi (u + li (m), v + ki (p)),
c(t, s) = G(ϕ(e(t, s))) + b(t, s)
g i (u, v) = Mi Ni gi (u
n
X Z φi (t) Z ψi (s)
+ li (σ), v + ki (τ )),
hi (u, v)dv du,
i=1
φi (t0 )
ψi (s0 )
hi (u, v) = Mi Ni hi (u + li (m), v + ki (p))
EJDE-2014/256
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
15
Proof. It is easy to see that the solution z(t, s) of the problem (3.2) with (3.3)
satisfies the equivalent integral equation
ϕ(z(t, s))
s h
F u, v, z(u − l1 (u), v − k1 (v)), . . . , z(u − ln (u), v − kn (v)),
= ϕ(e(t, s)) +
t0 s0
Z uZ v
Q(u, v, σ, τ, z(u − l1 (u), v − k1 (v)), . . . ,
t0 s0
i
z(u − ln (u), v − kn (v)))dσ dτ dv du
Z tZ
(3.7)
By modulus properties and condition (3.5), equation (3.7) has the form
|ϕ(z(t, s))|
Z tZ
s
h
F u, v, z(u − l1 (u), v − k1 (v)), . . . ,
t0 s0
Z uZ v
z(u − ln (u), v − kn (v)),
Q(u, v, σ, τ, z(u − l1 (u), v − k1 (v)), . . . ,
t0 s0
i
z(u − ln (u), v − kn (v)))dσ dτ dv du
Z tZ sX
n
[φ(|z(m − li (m), p − ki (p))|)(fi (m, p)
≤ |ϕ(e(t, s))| + b(t, s)
≤ |ϕ(e(t, s))| + b(t, s)
t0
s0 i=1
× (w(|z(m − li (m), p − ki (p))|)
Z mZ p
+
gi (σ, τ )w(|z(σ − li (σ), τ − ki (τ ))|)dτ dσ)
t0
s0
+ hi (m, p)|φ(z(m − li (m), p − ki (p)))|]dp dm
Z φi (t) Z ψi (s) h
n
X
≤ |ϕ(e(t, s))| + b(t, s)
M i Ni
φ(|z(φi (m), ψi (p))|)
φi (t0 )
i=1
ψi (s0 )
× (fi (φi (m) + li (m), ψi (p) + ki (p))(w(|z(φi (m), ψi (p))|)
Z φi (m) Z ψi (p)
+
Mi Ni gi (φi (σ) + li (σ), ψi (τ ) + ki (τ ))
φi (t0 )
ψi (s0 )
× w(|z(φi (σ), ψi (τ ))|)dψi (τ ) dφi (σ)) + hi (φi (m) + li (m), ψi (p)
i
+ ki (p))φ(|z(φi (m), ψi (p))|)) dψi (p) dφi (m)
which implies
ϕ(|z(t, s)|)
≤ |ϕ(e(t, s))| + b(t, s)
n Z
X
i=1
Z
δ
Z
η
+
φi (t0 )
φi (t)
φi (t0 )
Z
ψi (s)
h
φ(|z(δ, η)|)(f i (δ, η)(w(|z(δ, η)|)
ψi (s0 )
i
g i (δ1 , η1 )w(|z(δ1 , η1 )|)dη1 dδ1 ) + hi (δ, η)φ(|z(δ, η)|)) dη dδ
ψi (s0 )
(3.8)
Now an immediate application of inequality (2.1) to (3.8) yields the desired result
(3.6).
16
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
Theorem 3.2. Assume that F : I × J × Rn × Rn → R is a continuous function
for which there exist continuous nonnegative functions fi (t, s), gi (t, s) and hi (t, s),
1 ≤ i ≤ n, for (t, s) ∈ I × J such that:
n
X
|F (t, s, u1 , . . . , un , j)| ≤
|ui |q [fi (t, s)w(|ui |) + |j| + hi (t, s)],
i=1
|Q(t, s, v1 , v2 , u1 , u2 , . . . , un )| ≤ gi (t, s)w(|ui |),
(3.9)
|ap1 (t) + ap2 (s)| ≤ c
If z(t, s) is a solution of (3.1) with the condition (3.3), then
n Z αi (t) Z βi (s)
X
fei (z, y)
u(t, s) ≤ [Ψ−1
(Ψ
(
m
e
(t,
s))
+
(p
−
q)
∗
0
∗
i=1
Z
z
Z
y
× 1+
αi (t0 )
βi (s0 )
αi (t0 )
βi (s0 )
(3.10)
gei (m, n) dn dm dy dz)]1/(p−q) ,
where
fei (u, v) = Mi Ni fi (u + li (u), v + ki (v)),
gei (u, v) = Mi Ni gi (u + li (σ), v + ki (τ )),
e
hi (u, v) = Mi Ni hi (u + li (u), v + ki (v)),
n Z φi (t) Z ψi (s)
X
(p−q)/p
e
m
e 0 (t, s) = c
+ (p − q)
hi (u, y) dy du
i=1
φi (t0 )
ψi (s0 )
Proof. It is easy to see that the solution z(t, s) of (3.1) with (3.3) satisfies the
equivalent integral equation
Z tZ s
p
= ap1 (t) + ap2 (s) + p
F [u, v, z(u − l1 (u), v − k1 (v)), . . . ,
t0 s0
Z uZ v
(3.11)
z(u − ln (u), v − kn (v)),
Q(u, v, σ, τ, z(u − l1 (u), v − k1 (v)), . . . ,
t0
s0
z(u − ln (u), v − kn (v)))dσdτ ]dv du
By modulus properties and condition (3.9), equation (3.11) has the form
|z p (t, s)|
Z tZ
≤c+p
Z uZ
t0
v
s
s0
h
F u, v, z(u − l1 (u), v − k1 (v)), . . . , z(u − ln (u), v − kn (v)),
Q(u, v, σ, τ, z(u − l1 (u), v − k1 (v)), . . . ,
i
z(u − ln (u), v − kn (v)))dσdτ dv du
Z tZ sX
n h
≤c+p
|z(u − li (u), v − ki (v))|q .fi (u, v)(w(|z(u − li (u), v − ki (v))|)
t0
s0
t0
Z
u
Z
s0 i=1
v
gi (σ, τ )w(|z(σ − li (σ), τ − ki (τ ))|)dτ dσ)
i
+ hi (u, v)|z(u − li (u), v − ki (v))|q dv du
+
t0
s0
EJDE-2014/256
≤c+p
n
X
NONLINEAR DOUBLE INTEGRAL INEQUALITIES
Z
φi (t)
Z
ψi (s)
M i Ni
φi (t0 )
i=1
h
17
|z(φi (u), ψi (v))|q fi (φi (u) + li (u), ψi (v) + ki (v))
ψi (s0 )
Z
φi (u)
Z
ψi (v)
× (w(|z(φi (u), ψi (v))|) +
Mi Ni gi (φi (σ) + li (σ), ψi (τ )
φi (t0 )
ψi (s0 )
+ ki (τ ))w(|z(φi (σ), ψi (τ ))|)dψi (τ )dφi (σ)) + hi (φi (u) + li (u), ψi (v)
i
+ ki (v))|φ(z(φi (u), ψi (v)))| dψi (v)dφi (u)
n Z φi (t) Z ψi (s)
X
[|z(δ, η)|q .fei (δ, η)(w(|z(δ, η)|)
≤c+p
φi (t0 )
i=1
Z
δ
Z
+
φi (t0 )
ψi (s0 )
η
ψi (s0 )
gei (δ1 , η1 )w(|z(δ1 , η1 )|)dη1 dδ1 ) + e
hi (δ, η)|z(δ, η)|q ]dη dδ
Now an immediate application of inequality (2.36) to above inequality yields the
desired result (3.10).
Acknowledgments. The authors are very grateful to Prof. Julio G. Dix and
to the anonymous referees for their helpful comments for improving the original
manuscript. The corresponding author’s research is supported by the University
Scientific and Technological Innovation Project of Guangdong Province of China
(Grant No. 2013KJCX0068).
References
[1] I. Bihari; A generalization of a lemma of Bellman and its application to uniqueness of
differential equations, Acta Math Acad Sci Hunger, 7(1956), 71-94.
[2] W. S. Cheung; Some new nonlinear inequalities and applications to boundary value problems.
Nonlinear Anal. 64 (2006), no. 9, 2112-2128.
[3] Y. J. Cho, Y. H. Kim, J. Peĉarić; New Gronwall-Ou-Lang type integral inequalities and their
applications, J. ANZIAM, 50 (2008), 111-127.
[4] F. Dannan; Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behaviourof
certain second order nonlinear differential equations, J. Math Anal Appl., 108 (1985), 151164.
[5] T. H. Gronwall; Note on the derivative with respect to a parameter of the solutions of a
system of differential equations, Ann. of Math., 20 (1919), 292-296.
[6] O. Lipovan; A retarded Gronwall-like inequality and its applications, J. Math Anal Appl.,
252 (2000), 389-401.
[7] Q. H. Ma, E. H. Yang; Some new Gronwall-Bellman-Bihari type integral inequalties with
delay, Periodica Mathemaica Hungarica, 44 (2002), 225-238.
[8] Q. H. Ma, J. Peĉarić; On some new nonlinear retarded integral inequalities with iterated
integrals and their applications, J. Korean Math. Soc., 45 (2008), No. 2, 331-353.
[9] B. G. Pachpatte; On new inequalties related to certain inequalities in the theory of differential
equations, J. Math Anal. Appl., 189 (1995), 128-144.
[10] B. G. Pachpatte; On some new nonlinear retarded integral inequalities, J. Inequal Pure Appl
Math., 5 (3)(8) (2004).
[11] J. Patil, A. P. Bhadane; Some new nonlinear retarded integral inequalities and applications,
Bulletin of Marathwada Mathematical Society, 14(1) 2013, 56-70.
[12] W. S. Wang; Some generalized nonlinear retarded integral inequalities with applications, J.
Inequal Appl., 2012, 31 (2012).
[13] W. S. Wang; A generlized retarded Gronwall-like inequality in two variables and applicaions
to BVP, Appl Math Comput., 191 (2007), 144-154.
[14] W. S. Wang, C. X. Shen; On generalized retarded inequality with two variables, J. Inequal
Appl., 9(2008).
18
S. HUSSAIN, T. RIAZ, Q.-H. MA, J. PEČARIĆ
EJDE-2014/256
[15] W. S. Wang; Some new nonlinear Gronwall-Bellman-type integral inequalities and applications to BVPs, J. Appl Math Comput., (2010).
Sabir Hussain
Department of Mathematics, University of Engineering and Technology, Lahore Pakistan
E-mail address: sabirhus@gmail.com
Tanzila Riaz
Department of Mathematics, University of Engineering and Technology, Lahore Pakistan
E-mail address: tanzila.ch@hotmail.com
Qing-Hua Ma (corresponding author)
Department of Applied Mathematics, Guangdong University of Foreign Studies,
Guangzhou 510420, China
E-mail address: gdqhma@21cn.com
J. Pečarić
Faculty of Textile Technology, University of Zagreb Pierottijeva 6, 10000 Zagreb,
Croatia
E-mail address: pecaric@element.hr