Boudeliou and Khellaf Journal of Inequalities and Applications (2015) 2015:313 DOI 10.1186/s13660-015-0837-7
RESEARCH
Open Access
On some delay nonlinear integral inequalities in two independent variables Ammar Boudeliou* and Hassane Khellaf *
Correspondence:
[email protected] Department of Mathematics, Mentouri University, Ain El Bey Road, Constantine, 25000, Algeria
Abstract The purpose of this paper is to generalize some integral inequalities in two independent variables with delay which can be used as handy tools in the study of certain partial differential equations and integral equations with delay. An application is given to illustrate the usefulness of our results. Keywords: retarded integral inequalities; two independent variables; differential and integral equations with time delay; nondecreasing functions
1 Introduction The integral inequalities which provide explicit bounds on unknown functions play an important role in the development of the theory of differential and integral equations. The Gronwall-Bellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, continuation, boundedness, oscillation and stability, and other qualitative properties of solutions of differential and integral equations. The literature on such inequalities and their applications is vast; see [–] and the references given therein. In [] Ferreira and Torres, have discussed the following useful nonlinear retarded integral inequality: φ u(t) ≤ c(t) +
α(t)
f (t, s)η u(s) ω u(s) + g(t, s)η u(s) ds.
Motivated by the results obtained in [, ] and [] we establish a general two independent variables retarded version which can be used as a tool to study the boundedness of solutions of differential and integral equations.
2 Main results In what follows, R denotes the set of real numbers, R+ = [, +∞), I = [, M], I = [, N] are the given subsets of R, and = I × I . C i (A, B) denotes the class of all i times continuously differentiable functions defined on a set A with range in the set B (i = , , . . .) and C (A, B) = C(A, B). Lemma . Let u(x, y), f (x, y), σ (x, y) ∈ C(, R+ ) and a(x, y) ∈ C(, R+ ) be nondecreasing with respect to (x, y) ∈ , let α ∈ C (I , I ), β ∈ C (I , I ) be nondecreasing with α(x) ≤ x on © 2015 Boudeliou and Khellaf. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Boudeliou and Khellaf Journal of Inequalities and Applications (2015) 2015:313
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I , β(y) ≤ y on I . Further let ψ, ω ∈ C(R+ , R+ ) be nondecreasing functions with {ψ, ω}(u) > for u > , and limu→+∞ ψ(u) = +∞. If u(x, y) satisfies ψ u(x, y) ≤ a(x, y) +
α(x) β(y)
σ (s, t)f (s, t)ω u(s, t) dt ds
(.)
for (x, y) ∈ , then – u(x, y) ≤ ψ G G a(x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y , where
v
G(v) = v
ds , ω(ψ – (s))
v ≥ v > ,
+∞
G(+∞) = v
and (x , y ) ∈ is chosen so that (G(a(x, y)) +
α(x) β(y)
ds = +∞ ω(ψ – (s))
(.)
σ (s, t)f (s, t) dt ds) ∈ Dom(G– ).
Theorem . Let u, a, f , α, and β be as in Lemma .. Let σ (x, y), σ (x, y) ∈ C(, R+ ). Further ψ, ω, η ∈ C(R+ , R+ ) be nondecreasing functions with {ψ, ω, η}(u) > for u > , and limu→+∞ ψ(u) = +∞. (A ) If u(x, y) satisfies ψ u(x, y) ≤ a(x, y) +
α(x) β(y)
s
+
σ (s, t) f (s, t)ω u(s, t)
σ (τ , t)ω u(τ , t) dτ dt ds
(.)
for (x, y) ∈ , then u(x, y) ≤ ψ – G– p(x, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y , where G is defined by (.) and
α(x) β(y)
σ (s, t)
p(x, y) = G a(x, y) +
α(x) β(y)
s
+
α(x) β(y)
(.)
σ (s, t)f (s, t) dt ds) ∈ Dom(G– ).
σ (s, t) f (s, t)ω u(s, t) η u(s, t)
σ (τ , t) dτ dt ds
and (x , y ) ∈ is chosen so that (p(x, y) + (A ) If u(x, y) satisfies ψ u(x, y) ≤ a(x, y) +
s
σ (τ , t)ω u(τ , t) dτ dt ds
(.)
for (x, y) ∈ , then u(x, y) ≤ ψ – G– F – F p(x, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds
(.)
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for ≤ x ≤ x , ≤ y ≤ y , where G and p are as in (A ), and
v
F(v) = v
ds , η(ψ – (G– (s)))
v ≥ v > ,
F(+∞) = +∞,
and (x , y ) ∈ is chosen so that [F(p(x, y)) + (A ) If u(x, y)satisfies ψ u(x, y) ≤ a(x, y) +
α(x) β(y)
s
+
α(x) β(y)
(.)
σ (s, t)f (s, t) dt ds] ∈ Dom(F – ).
σ (s, t) f (s, t)ω u(s, t) η u(s, t)
σ (τ , t)ω u(τ , t) η u(τ , t) dτ dt ds
(.)
for (x, y) ∈ , then – – u(x, y) ≤ ψ G F p (x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y where p (x, y) = F G a(x, y) +
α(x) β(y)
σ (s, t)
σ (τ , t) dτ dt ds
and (x , y ) ∈ is chosen so that [p (x, y) +
s
α(x) β(y)
σ (s, t)f (s, t) dt ds] ∈ Dom(F – ).
The proof of the theorem will be given in the next section. Remark . If we take σ (x, y) = , then Theorem .(A ) reduces to Lemma .. Corollary . Let the functions u, f , σ , σ , a, α, and β be as in Theorem .. Further q > p > are constants. (B ) If u(x, y) satisfies up (x, y) ≤ a(x, y) +
α(x) β(y)
s
σ (s, t) f (s, t)up (s, t)
p
σ (τ , t)u (τ , t) dτ dt ds
+
(.)
for (x, y) ∈ , then α(x) β(y)
s p u(x, y) ≤ a(x, y) exp σ (s, t) f (s, t) + σ (τ , t) dτ dt ds . p
(.)
(B ) If u(x, y) satisfies α(x) β(y) q σ (s, t) f (s, t)up (s, t) u (x, y) ≤ a(x, y) + q–p s σ (τ , t)up (τ , t) dτ dt ds + q
(.)
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for (x, y) ∈ , then u(x, y) ≤ p(x, y) +
q–p
α(x) β(y)
σ (s, t)f (s, t) dt ds
,
(.)
where
p(x, y) = a(x, y)
q–p q
α(x) β(y)
+
σ (s, t)
s
σ (τ , t) dτ dt ds.
Corollary . Let the functions u, a, f , σ , σ , α, and β be as in Theorem .. Further q, p, and r are constants with p > , r > and q > p + r. (C ) If u(x, y) satisfies α(x) β(y)
u (x, y) ≤ a(x, y) + q
s
+
σ (s, t) f (s, t)up (s, t)ur (s, t)
σ (τ , t)up (τ , t) dτ dt ds
(.)
for (x, y) ∈ , then u(x, y) ≤
p(x, y)
q–p–r q–p
q–p–r + q
q–p–r
α(x) β(y)
σ (s, t)f (s, t) dt ds
,
(.)
where
p(x, y) = a(x, y)
q–p q
q–p + q
α(x) β(y)
σ (s, t)
s
σ (τ , t) dτ dt ds.
(C ) If u(x, y) satisfies α(x) β(y)
u (x, y) ≤ a(x, y) + q
s
+
σ (s, t) f (s, t)up (s, t)ur (s, t)
σ (τ , t)up (τ , t)ur (τ , t) dτ dt ds
(.)
for (x, y) ∈ , then
q–p–r q – p – r α(x) β(y) σ (s, t)f (s, t) dt ds , u(x, y) ≤ p (x, y) + q
(.)
where
q–p–r q – p – r p (x, y) = a(x, y) q + q
α(x) β(y)
σ (s, t)
s
σ (τ , t) dτ dt ds.
Theorem . Let u, f , σ , σ , a, α, β, ψ, ω, and η be as in Theorem .. If u(x, y) satisfies
ψ u(x, y) ≤ a(x, y) +
α(x) β(y)
σ (s, t)η u(s, t)
× f (s, t)ω u(s, t) +
s
σ (τ , t) dτ dt ds
(.)
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for (x, y) ∈ , then – u(x, y) ≤ ψ G F– F p (x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y , where
v
G (v) = v
F (v) =
ds , η(ψ – (s))
v
v ≥ v > ,
+∞
G (+∞) = v
ds
– – v ω[ψ (G (s))]
p (x, y) = G a(x, y) +
v ≥ v > ,
,
α(x) β(y)
σ (s, t)
s
ds = +∞, η(ψ – (s))
(.)
F (+∞) = +∞,
(.)
σ (τ , t) dτ dt ds,
(.)
and (x , y ) ∈ is chosen so that [F (p (x, y)) +
α(x) β(y)
σ (s, t)f (s, t) dt ds] ∈ Dom(F– ).
Theorem . Let u, f , σ , σ , a, α, β, ψ, and ω be as in Theorem ., and p > a constant. If u(x, y) satisfies ψ u(x, y) ≤ a(x, y) +
α(x) β(y)
σ (s, t)up (s, t)
s × f (s, t)ω u(s, t) + σ (τ , t) dτ dt ds
(.)
for (x, y) ∈ , then u(x, y) ≤ ψ – G– F– F p (x, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y , where
v
G (v) = v
ds , – [ψ (s)]p
v ≥ v > ,
+∞
G (+∞) = v
ds [ψ – (s)]p
= +∞
(.)
and F , p are as in Theorem . and (x , y ) ∈ is chosen so that
F p (x, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds ∈ Dom F– .
Remark . The inequality established in Theorem . generalizes Theorem of [] s (with p = , a(x, y) = b(x) + c(x), σ (s, t)f (s, t) = h(s, t), and σ (s, t)( σ (τ , t) dτ ) = g(s, t)). Corollary . Let u, f , σ , σ , a, α, β, and ω be as in Theorem . and q > p > be constants. If u(x, y) satisfies α(x) β(y) p u (x, y) ≤ a(x, y) + σ (s, t)up (s, t) p–q s × f (s, t)ω u(s, t) + σ (τ , t) dτ dt ds q
(.)
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for (x, y) ∈ , then u(x, y) ≤ F– F p (x, y) +
q–p
α(x) β(y)
σ (s, t)f (s, t) dt ds
(.)
for ≤ x ≤ x , ≤ y ≤ y , where q–p p (x, y) = a(x, y) q +
α(x) β(y)
σ (s, t)
s
σ (τ , t) dτ dt ds
and F is defined in Theorem .. Remark . Setting a(x, y) = b(x) + c(x), σ (s, t)f (s, t) = h(s, t), and σ (s, t)( g(s, t) in Corollary . we obtain Theorem of []. p
Remark . Setting a(x, y) = c p–q , σ (s, t)f (s, t) = h(t), and σ (s, t)( keeping y fixed in Corollary ., we obtain Theorem . of [].
s
s
σ (τ , t) dτ ) =
σ (τ , t) dτ ) = g(t) and
3 Proof of theorems Proof of Lemma . First we assume that a(x, y) > . Fixing an arbitrary (x , y ) ∈ , we define a positive and nondecreasing function z(x, y) by
α(x) β(y)
z(x, y) = a(x , y ) +
σ (s, t)f (s, t)ω u(s, t) dt ds
for ≤ x ≤ x ≤ x , ≤ y ≤ y ≤ y , then z(, y) = z(x, ) = a(x , y ) and u(x, y) ≤ ψ – z(x, y) ,
(.)
and then we have ∂z(x, y) = α (x) ∂x ≤ α (x)
β(y)
σ α(x), t f α(x), t ω u α(x), t dt
β(y)
σ α(x), t f α(x), t ω ψ – z α(x), t dt
≤ ω ψ – z α(x), β(y) α (x)
β(y)
σ α(x), t f α(x), t dt
or ∂z(x,y) ∂x ω(ψ – (z(x, y)))
≤ α (x)
β(y)
σ α(x), t f α(x), t dt.
Keeping y fixed, setting x = s, integrating the last inequality with respect to s from to x, and making the change of variable s = α(x) we get
G z(x, y) ≤ G z(, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds
≤ G a(x , y ) +
α(x) β(y)
σ (s, t)f (s, t) dt ds.
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Since (x , y ) ∈ is chosen arbitrary, z(x, y) ≤ G G a(x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds .
So from the last inequality and (.) we obtain (.). If a(x, y) = , we carry out the above procedure with > instead of a(x, y) and subsequently let → . Proof of Theorem . (A ) By the same steps of the proof of Lemma . we can obtain (.), with suitable changes. (A ) Assume that a(x, y) > . Fixing an arbitrary (x , y ) ∈ , we define a positive and nondecreasing function z(x, y) by α(x) β(y)
z(x, y) = a(x , y ) +
σ (s, t) f (s, t)ω u(s, t) η u(s, t)
σ (τ , t)ω u(τ , t) dτ dt ds
s
+
for ≤ x ≤ x ≤ x , ≤ y ≤ y ≤ y , then z(, y) = z(x, ) = a(x , y ) and u(x, y) ≤ ψ – z(x, y) , β(y) ∂z(x, y) = α (x) σ α(x), t f α(x), t ω u α(x), t η u α(x), t ∂x α(x) σ (τ , t)ω u(τ , t) dτ dt +
≤ α (x)
β(y)
α(x)
+
(.)
σ α(x), t f α(x), t ω ψ – z α(x), t η ψ – z α(x), t
σ (τ , t)ω ψ – z(τ , t) dτ dt
≤ α (x) · ω ψ – z α(x), β(y) β(y) – σ α(x), t f α(x), t η ψ z α(x), t + ×
α(x)
σ (τ , t) dτ dt
then ∂z(x,y) ∂x ω(ψ – (z(x, y)))
β(y)
≤ α (x)
σ α(x), t f α(x), t η ψ – z α(x), t
α(x)
σ (τ , t) dτ dt.
+
Keeping y fixed, setting x = s integrating the last inequality with respect to s from to x, and making the change of variable s = α(x) we get
G z(x, y) ≤ G z(, y) +
s
α(x) β(y)
σ (τ , t) dτ dt ds
+
σ (s, t) f (s, t)η ψ – z(s, t)
Boudeliou and Khellaf Journal of Inequalities and Applications (2015) 2015:313
≤ G a(x , y ) +
s
σ (s, t) f (s, t)η ψ – z(s, t)
α(x) β(y)
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σ (τ , t) dτ dt ds.
+
Since (x , y ) ∈ is chosen arbitrarily, the last inequality can be rewritten as G z(x, y) ≤ p(x, y) +
α(x) β(y)
σ (s, t)f (s, t)η ψ – z(s, t) dt ds.
(.)
Since p(x, y) is a nondecreasing function, an application of Lemma . to (.) gives us z(x, y) ≤ G F – F p(x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds
.
(.)
From (.) and (.) we obtain the desired inequality (.). Now we take the case a(x, y) = for some (x, y) ∈ . Let a (x, y) = a(x, y) + , for all (x, y) ∈ , where > is arbitrary, then a (x, y) > and a (x, y) ∈ C(, R+ ) be nondecreasing with respect to (x, y) ∈ . We carry out the above procedure with a (x, y) > instead of a(x, y), and we get – – u(x, y) ≤ ψ G F F p (x, y) +
α(x) β(y)
–
σ (s, t)f (s, t) dt ds ,
where
α(x) β(y)
p (x, y) = G a (x, y) +
σ (s, t)
s
σ (τ , t) dτ dt ds.
Letting → + , we obtain (.). (A ) Assume that a(x, y) > . Fixing an arbitrary (x , y ) ∈ , we define a positive and nondecreasing function z(x, y) by
α(x) β(y)
z(x, y) = a(x , y ) +
s
+
σ (s, t) f (s, t)ω u(s, t) η u(s, t)
σ (τ , t)ω u(τ , t) η u(τ , t) dτ dt ds
for ≤ x ≤ x ≤ x , ≤ y ≤ y ≤ y , then z(, y) = z(x, ) = a(x , y ), and u(x, y) ≤ ψ – z(x, y) .
(.)
By the same steps as the proof of Theorem .(A ), we obtain z(x, y) ≤ G G a(x , y ) +
α(x) β(y)
–
+
s
σ (s, t) f (s, t)η ψ – z(s, t)
σ (τ , t)η ψ – z(τ , t) dτ dt ds .
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We define a nonnegative and nondecreasing function v(x, y) by α(x) β(y)
v(x, y) = G a(x , y ) + +
s
σ (s, t) f (s, t)η ψ – z(s, t)
σ (τ , t)η ψ – z(τ , t) dτ dt ds;
then v(, y) = v(x, ) = G(a(x , y )), z(x, y) ≤ G– v(x, y) ,
(.)
and then β(y) ∂v(x, y) ≤ α (x) σ α(x), t f α(x), t η ψ – G– v α(x), y ∂x α(x) σ (τ , t)η ψ – G– v(τ , y) dτ dt +
≤ α (x) · η ψ – G– v α(x), β(y)
σ α(x), t f α(x), t
α(x)
β(y)
σ (τ , t) dτ dt
+
or ∂v(x,y) ∂x η(ψ – (G– (v(x, y))))
≤ α (x)
β(y)
α(x)
+
σ α(x), t f α(x), t
σ (τ , t) dτ dt.
Fixing y and integrating the last inequality with respect to s = x from to x and using a change of variables yield the inequality F v(x, y) ≤ F v(, y) +
α(x) β(y)
s σ (s, t) f (s, t) + σ (τ , t) dτ dt ds
or v(x, y) ≤ F – F G a(x , y ) +
α(x) β(y)
σ (s, t)
s
× f (s, t) +
σ (τ , t) dτ dt ds .
(.)
From (.)-(.), and since (x , y ) ∈ is chosen arbitrarily, we obtain the desired inequality (.). If a(x, y) = , we carry out the above procedure with > instead of a(x, y) and subsequently let → . Proof of Corollary . (B ) In Theorem .(A ), by letting ψ(u) = ω(u) = up , we obtain
v
G(v) = v
ds = ω(ψ – (s))
v
v
v ds = ln , s v
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and hence v ≥ v > .
G– (v) = v exp(v),
From equation (.), we obtain the inequality (.). (B ) In Theorem .(A ), by letting ψ(u) = uq , ω(u) = up we have
v
G(v) = v
ds = ω(ψ – (s))
v
ds s
v
p q
=
q–p q q–p v q – v q , q–p
v ≥ v >
and
q–p q – p q–p q v G– (v) = v + q
we obtain the inequality (.).
Proof of Corollary . (C ) An application of Theorem .(A ) with ψ(u) = uq , ω(u) = up , and η(u) = ur yields the desired inequality (.). (C ) An application of Theorem .(A ) with ψ(u) = uq , ω(u) = up , and η(u) = ur yields the desired inequality (.). Proof of Theorem . Suppose that a(x, y) > . Fixing an arbitrary (x , y ) ∈ , we define a positive and nondecreasing function z(x, y) by
α(x) β(y)
z(x, y) = a(x , y ) +
s σ (s, t)η u(s, t) f (s, t)ω u(s, t) + σ (τ , t) dτ dt ds
for ≤ x ≤ x ≤ x , ≤ y ≤ y ≤ y , then z(, y) = z(x, ) = a(x , y ), u(x, y) ≤ ψ – z(x, y)
(.)
and β(y) ∂z(x, y) f α(x), t ω ψ – z α(x), t ≤ α (x) σ α(x), t η ψ – z α(x), t ∂x α(x) σ (τ , t) dτ dt +
≤ α (x)η ψ – z α(x), β(y)
α(x)
β(y)
σ α(x), t f α(x), t ω ψ – z α(x), t
σ (τ , t) dτ dt,
+
then ∂z(x,y) ∂x η[ψ – (z(x, y))]
≤ α (x) +
β(y)
α(x)
σ α(x), t f α(x), t ω ψ – z α(x), t
σ (τ , t) dτ dt.
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Keeping y fixed, setting x = s and integrating the last inequality with respect to s from to x and making the change of variable, we obtain G z(x, y) ≤ G z(, y) +
α(x) β(y)
s
+
σ (s, t) f (s, t)ω ψ – z(s, t)
σ (τ , t) dτ dt ds;
then
G z(x, y) ≤ G a(x , y ) +
α(x) β(y)
s
σ (s, t) f (s, t)ω ψ – z(s, t)
σ (τ , t) dτ dt ds.
+
Since (x , y ) ∈ is chosen arbitrary, the last inequality can be restated as G z(x, y) ≤ p (x, y) +
α(x) β(y)
σ (s, t)f (s, t)ω ψ – z(s, t) dt ds.
(.)
It is easy to observe that p (x, y) is positive and nondecreasing function for all (x, y) ∈ , then an application of Lemma . to (.) yields the inequality z(x, y) ≤ G– F– F p (x, y) +
α(x) β(y)
σ (s, t)f (s, t) dt ds
.
(.)
From (.) and (.) we get the desired inequality (.). If a(x, y) = , we carry out the above procedure with > instead of a(x, y) and subsequently let → . Proof of Theorem . An application of Theorem ., with η(u) = up yields the desired inequality (.). Proof of Corollary . An application of Theorem . with ψ(u(x, y)) = up to (.) yields the inequality (.); to save space we omit the details.
4 An application In this section, we present an application of our results to the qualitative analysis of solutions to the retarded integro differential equations. We study the boundedness of the solutions of the initial boundary value problem for partial delay integro differential equations of the form
x B s, y, z s – h (s), y ds , (.) D D zq (x, y) = A x, y, z x – h (x), y – h (y) ,
z(x, ) = a (x),
z(, y) = a (y),
a () = a () =
for (x, y) ∈ , where z, b ∈ C(, R+ ), A ∈ C( × R , R), B ∈ C( × R, R) and h ∈ C (I , R+ ), h ∈ C (I , R+ ) are nondecreasing functions such that h (x) ≤ x on I , h (y) ≤ y on I , and h (x) < , h (y) < .
Boudeliou and Khellaf Journal of Inequalities and Applications (2015) 2015:313
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Theorem . Assume that the functions b, A, B in (.) satisfy the conditions a (x) + a (y) ≤ a(x, y), A(s, t, z, u) ≤ q σ (s, t) f (s, t)|z|p + |u| , q–p B(τ , t, z) ≤ σ (τ , t)|z|p ,
(.) (.) (.)
where a(x, y), σ (s, t), f (s, t), and σ (τ , t) are as in Theorem ., q > p > are constants. If z(x, y) satisfies (.), then z(x, y) ≤ p(x, y) + M M
q–p
α(x) β(y)
σ (s, t)f (s, t) dt ds
,
(.)
where q–p p(x, y) = a(x, y) q + M M
α(x) β(y)
s
σ (s, t) M σ (τ , t) dτ dt ds
and M = Max x∈I
, – h (x)
M = Max y∈I
– h (y)
and σ (γ , ξ ) = σ (γ + h (s), ξ + h (t)), σ (μ, ξ ) = σ (μ, ξ + h (t)), f (γ , ξ ) = f (γ + h (s), ξ + h (t)). Proof If z(x, y) is any solution of (.), then zq (x, y) = a (x) + a (y),
x y s A s, t, z s – h (s), t – h (t) , B τ , t, z τ – h (τ ), t dτ dt ds.
(.)
Using the conditions (.)-(.) in (.) we obtain x y p z(x, y)q ≤ a(x, y) + q – p σ (s, t) f (s, t)z s – h (s), t – h (t) q s p + σ (τ , t) z(τ , t) dτ dt ds.
(.)
Now making a change of variables on the right side of (.), s – h (s) = γ , t – h (t) = ξ , x – h (x) = α(x) for x ∈ I , y – h (y) = β(y) for y ∈ I we obtain the inequality α(x) β(y) p z(x, y)q ≤ a(x, y) + q – p M M σ (γ , ξ ) f (γ , ξ )z(γ , ξ ) q γ p + M σ (μ, ξ )z(μ, t) dμ dξ dγ .
(.)
Boudeliou and Khellaf Journal of Inequalities and Applications (2015) 2015:313
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We can rewrite the inequality (.) as follows: α(x) β(y) p z(x, y)q ≤ a(x, y) + q – p M M σ (s, t) f (s, t)z(s, t) q s p + M σ (τ , t) z(τ , t) dτ dt ds.
(.)
As an application of Corollary .(B ) to (.) with u(x, y) = |z(x, y)| we obtain the desired inequality (.). Corollary . If z(x, y) satisfies the equation
x D D zp (x, y) = A x, y, z x – h (x), y – h (y) , B s, y, z s – h (s), y ds ,
(.)
z(x, ) = a (x),
z(, y) = a (y),
a () = a () =
and we suppose that the conditions (.)-(.) are satisfied, then we have the inequality z(x, y)p ≤ a(x, y) + M M + M
α(x) β(y)
s
p σ (s, t) f (s, t)z(s, t)
p σ (τ , t) z(τ , t) dτ dt ds,
(.)
then we obtain α(x) β(y) z(x, y) ≤ a(x, y) p exp M M σ (s, t) p
s σ (τ , t) dτ dt ds , × f (s, t) + M
(.)
where σ , f , σ , M , and M are as in Theorem .. Proof By an application of Corollary .(B ) to (.) we obtain the desired inequality (.). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Acknowledgements The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. Received: 27 July 2015 Accepted: 16 September 2015 References 1. Bainov, D, Simeonov, P: Integral Inequalities and Applications. Kluwer Academic, Dordrecht (1992) 2. Pachpatte, BG: Inequalities for Differential and Integral Equations. Academic Press, San Diego (1998) 3. Denche, M, Khellaf, H, Smakdji, M: Some new generalized nonlinear integral inequalities for functions of two independent variables. Int. J. Math. Anal. 7(40), 1961-1976 (2013) 4. Denche, M, Khellaf, H, Smakdji, M: Integral inequalities with time delay in two independent variables. Electron. J. Differ. Equ. 2014, 117 (2014)
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5. Khan, ZA: On nonlinear integral inequalities of Gronwall type in two independent variables. Appl. Math. Sci. 7(55), 2745-2757 (2013) 6. Pachpatte, BG: On a certain retarded integral inequality and its applications. JIPAM. J. Inequal. Pure Appl. Math. 5, Article 19 (2004) 7. Persson, LE, Ragusa, MA, Samko, N, Wall, P: Commutators of Hardy operators in vanishing Morrey spaces. In: 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA 2012). AIP Conference Proceedings, vol. 1493, pp. 859-866 (2012). doi:10.1063/1.4765588 8. Ferreira, RAC, Torres, DFM: Generalized retarded integral inequalities. Appl. Math. Lett. 22, 876-881 (2009) 9. Peˇcari´c, J, Ma, Q-H: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69, 393-407 (2008) 10. Fan, M, Meng, F, Tian, Y: A generalization of retarded integral inequalities in two independent variables and their applications. Appl. Math. Comput. 221, 239-248 (2013) 11. Sun, YG, Xu, R: On retarded integral inequalities in two independent variables and their applications. Appl. Math. Comput. 182, 1260-1266 (2006) 12. Sun, YG: On retarded integral inequalities and their applications. J. Math. Anal. Appl. 301, 265-275 (2005)
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