Xu and Ma Journal of Inequalities and Applications (2017) 2017:187 DOI 10.1186/s13660-017-1460-6
RESEARCH
Open Access
Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications Run Xu*
and Xiangting Ma
*
Correspondence: [email protected] Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
Abstract In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations. MSC: 42B20; 26D07; 26D15 Keywords: Volterra-Fredholm type; integral inequality; integral equations; maxima; iterated integral inequality
1 Introduction Gronwall-Bellman inequality [, ] and Bihari inequality [] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds and other qualitative properties of solutions to differential equations, integral equations and integro-differential equations. In the past few decades, a number of studies have focused on generalizations of the Gronwall-Bellman inequality. For example, in [–], the Gronwall-Bellman-Gamidov type integral inequalities and their generalizations were studied; in [–], the Gronwall-like inequalities and their deformations were investigated; in [–], the Volterra type iterated inequalities were discussed; in [–], the Volterra-Fredholm type inequalities were examined. The Gronwall-Bellman inequality can be stated as follows. If u and f are nonnegative continuous functions on an interval [a, b], and u satisfies the following inequality:
t
u(t) ≤ c +
f (s)u(s) ds,
t ∈ [a, b],
(.)
a
where c ≥ is a constant. Then u(t) ≤ c exp
t
f (s) ds .
(.)
a
© The Author(s) 2017. 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.
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
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In , Pachpatte [] investigated the retarded linear Volterra-Fredholm type integral inequality in two independent variables:
α(x) β(y)
u(x, y) ≤ c +
α(M) β(N)
a(x, y, s, t)u(s, t) dt ds + α(x )
β(y )
b(x, y, s, t)u(s, t) dt ds, α(x )
β(y )
(x, y) ∈ [x , M] × [y , N].
(.)
In , Wang [] investigated a retarded Volterra type integral inequality with two variables: x ψ u(x, y) ≤ a(x, y) + b(x, y) c(s, y)ψ u(s, y) ds x
α (x) β (y)
+ d(x, y) α (x )
α (x) β (y)
+ α (x )
f (s, t)ϕ u(s, t) dt ds
β (y )
f (s, t)ϕ u(s, t) dt ds ,
(x, y) ∈ [x , x ) × [y , y ). (.)
β (y )
In , Lu et al. [] studied the nonlinear retarded Volterra-Fredholm type iterated integral inequality: u(x, y) ≤ k +
α(x) β(y)
α(x )
h (s , t )ω u(s , t )
β(y )
× f (s , t )ω u(s , t ) +
α(x )
s
s
β(y )
t
h (s , t ) f (s , t )ω u(s , t )
t
h (s , t )ω u(s , t ) dt ds dt ds dt ds
+ α(x )
β(y )
α(M) β(N)
+
α(x )
s
β(y )
t
+ α(x )
s
β(y ) t
+ α(x )
h (s , t )ω u(s , t ) f (s , t )ω u(s , t )
h (s , t ) f (s , t )ω u(s , t ) h (s , t )ω u(s , t ) dt ds dt ds dt ds ,
β(y )
(x, y) ∈ .
(.)
In , Huang and Wang [] discussed the retarded nonlinear Volterra-Fredholm type integral inequality with maxima: ϕ v(t) ≤ k +
α(t)
h (s) f (s)φ v(s) +
α(t ) τ
+
h (ξ )φ α(t )
α(T)
+
max v(η) dξ dτ ds
h (ξ )φ α(t )
v(t) ≤ k,
η∈[ξ –h,ξ ]
τ
+
α(t )
h (s) f (s)φ v(s) +
α(t )
h (τ ) f (τ )φ v(τ )
s
t ∈ [t – h, t ].
s
h (τ ) f (τ )φ v(τ )
α(t )
max v(η) dξ dτ ds,
η∈[ξ –h,ξ ]
t ∈ [t , T],
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
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Motivated by the work presented in [, , ], we establish some new retarded nonlinear Volterra-Fredholm type integral inequality with maxima in two independent variables in this paper: ψ u(x, y) ≤ k(x, y) +
∞
l a(s, y)ψ u(s, y) ds +
α(x)
∞
ci (ξ , η, x, y)ϕ
s
αj (M)
j=
∞
∞
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
max u(σ , η) dξ dη ds dt
βj (N)
ej (ξ , η, x, y)ψ
s
dj (s, t, x, y)ψ u(s, t)
∞
+
∞
σ ∈[ξ ,hξ ]
t
l
+
∞
αi (x)
i=
∞
+
t
max u(σ , η) dξ dη ds dt,
σ ∈[ξ ,hξ ]
(x, y) ∈ , (.)
and
∞
u (x, y) ≤ k(x, y) + p
p
a(s, y)u (s, y) ds + α(x)
∞ ∞
+ s
+
i=
∞
∞ ∞
+ s
αi (x)
∞
bi (s, t, x, y)uqi (s, t)
βi (y)
σ ∈[ξ ,hξ ]
αj (M)
j=
∞
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
t
l
l
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η) dξ dη ds dt, σ ∈[ξ ,hξ ]
t
(x, y) ∈ .
(.)
By the amplification method, differential and integration, and the inverse function, we obtain the lower bound estimation of the unknown function. The example is given to illustrate the application of our results.
2 Main results In what follows, R denotes the set of real numbers, R+ = [, +∞), I = [M, +∞), I = [N, +∞) are the given subsets of R, = I × I . C (, S) denotes the class of all continuously differentiable functions defined on set with range in the set S, C(, S) denotes the class of all continuous functions defined on set with range in the set S, and α (t) denotes the derived function of a function α(t). For convenience, we cite some useful lemmas in the discussion of our proof as follows: Lemma . (See []) Let u(t), a(t), b(t) be nonnegative and continuous functions defined for t ∈ R+ . Assume that a(t) is non-increasing function for t ∈ R+ . If
∞
u(t) ≤ a(t) +
b(s)u(s) ds,
t ∈ R+ ,
t
then u(t) ≤ a(t) exp t
∞
b(s) ds ,
t ∈ R+ .
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 4 of 25
From Lemma ., we can get the generalization in two dimensions. Lemma . Let u(x, y), a(x, y), b(x, y) be nonnegative and continuous functions defined for (x, y) ∈ . Assume that a(x, y) is a non-increasing function in the first variable. If
∞
u(x, y) ≤ a(x, y) +
b(s, y)u(s, y) ds,
(x, y) ∈ ,
x
then u(x, y) ≤ a(x, y) exp
∞
b(s, y) ds ,
(x, y) ∈ .
x
Lemma . (See []) Assume that a ≥ , p ≥ q ≥ , and p = . Then, for any K > , q q q–p p – q pq ap ≤ K p a + K . p p
Theorem . Suppose that the following conditions hold: (i) ψ ∈ C(R+ , R+ ) is an increasing function and ψ(u) > , ∀u > , ψ(∞) = ∞. ψ – is the inverse function of ψ. ϕ , ϕ , ϕ /ϕ ∈ C(R+ , R+ ) are increasing functions with ϕi (u) > (i = , ) for u > . ψ – , ϕi (i = , ) are sub-multiplicative and sub-additive, that is, ψ – (αβ) ≤ ψ – (α)ψ – (β), ϕi (αβ) ≤ ϕi (α)ϕi (β),
ψ – (α + β) ≤ ψ – (α) + ψ – (β),
ϕi (α + β) ≤ ϕi (α) + ϕi (β),
α, β ∈ R+ ;
(ii) k(x, y), a(x, y) ∈ C( , R+ ) and k(x, y) is non-increasing in the first variable; (iii) bi (s, t, x, y), ci (s, t, x, y), dj (s, t, x, y), ej (s, t, x, y) ∈ C( , R+ ) for i = , , . . . , l ; j = , , . . . , l ; bi , ci , dj , ej are all non-increasing functions in the last two variables; (iv) α, αi , αj ∈ C(I , I ), βi , βj ∈ C(I , I ) are non-decreasing functions with α(x), αi (x), αj (x) ≥ x, βi (y), βj (y) ≥ y (i = , , . . . , l ; j = , , . . . , l ). h ≥ is a constant; (v) the function u ∈ C( , R+ ) satisfies the inequality ψ u(x, y) ≤ k(x, y) +
∞
a(s, y)ψ u(s, y) ds
α(x)
+
l ∞ i=
αi (x)
∞
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
∞
ci (ξ , η, x, y)ϕ
+ s
+
j=
∞
αj (M)
∞
∞ βj (N)
ej (ξ , η, x, y)ψ
s
max u(σ , η) dξ dη ds dt
dj (s, t, x, y)ψ u(s, t)
∞
+ t
(x, y) ∈ ,
σ ∈[ξ ,hξ ]
t
l
max u(σ , η) dξ dη ds dt,
σ ∈[ξ ,hξ ]
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
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then we have u(x, y) ≤ ψ – k(x, y) + W– W– W W B(M, N) + G– F(M, N) + E(x, y) + F(x, y) A(x, y) ,
(.)
where
z
ds , ϕ (ψ – (s))
W (z) = c
c > , z ∈ (, +∞),
(.)
z
ϕ (ψ – (W– (s))) ds, c > , z ∈ (, +∞), – – c ϕ (ψ (W (s))) u – W W B(M, N) + u + E(M, N) , G(u) = W W D(M, N)
(.)
W (z) =
(.)
on condition that W (+∞) = +∞, W (+∞) = +∞, and G(u) is a strictly increasing function on R+ . We have A(x, y) = exp
a(s, y) ds ,
∞
(.)
α(x) l
B(M, N) =
∞ αi (M)
i=
∞
βi (N)
ci (ξ , η, M, N)ϕ ψ k(ξ , η)A(ξ , η) dξ dη ds dt
∞
+ s
αj (M)
j=
s l
∞
∞
βj (N)
∞ ∞
ej (ξ , η, M, N)A(ξ , η) dξ dη ds dt,
+ s
l
(.)
t
l i=
∞
αi (M) ∞
∞
bi (s, t, M, N)ϕ ψ – A(s, t) ds dt,
∞
∞ ∞
ci (ξ , η, x, y)ϕ ψ
i=
αi (x)
l
∞
βi (y)
αi (x)
i=
∞
s
–
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
∞
s
t
max u(σ , η) dξ dη ds dt
σ ∈[ξ ,hξ ]
A(ξ , η) dξ dη ds dt.
t
ci (ξ , η, x, y)ϕ
+
(.)
βi (N)
Proof Let
z(x, y) =
(.)
dj (s, t, M, N)A(s, t) αj (M)
F(x, y) =
βj (N)
t
j=
E(M, N) =
dj (s, t, M, N)k(s, t)A(s, t)
∞
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
∞ ∞
+
D(M, N) =
–
t
l ∞
+
bi (s, t, M, N)ϕ ψ – k(s, t)A(s, t)
∞
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
αj (M)
j=
∞
∞
∞
dj (s, t, x, y)ψ u(s, t)
βj (N)
∞
ej (ξ , η, x, y)ψ
+ s
Page 6 of 25
t
max u(σ , η) dξ dη ds dt.
σ ∈[ξ ,hξ ]
(.)
Obviously, z(x, y) is non-increasing in each of the variables. From (.), we have ψ u(x, y) ≤ k(x, y) + z(x, y) +
∞
a(s, y)ψ u(s, y) ds.
(.)
α(x)
Applying Lemma ., we obtain ψ u(x, y) ≤ k(x, y) + z(x, y) A(x, y),
(.)
u(x, y) ≤ ψ – k(x, y) + z(x, y) A(x, y) ,
(.)
i.e.
where A(x, y) is defined in (.), and A(x, y) is non-increasing in the first variable. So we have
max u(ξ , y) ≤ max ψ – k(ξ , y) + z(ξ , y) A(ξ , y)
ξ ∈[x,hx]
ξ ∈[x,hx]
≤ ψ – max k(ξ , y)A(ξ , y) + max A(ξ , y)z(ξ , y) ξ ∈[x,hx]
ξ ∈[x,hx]
≤ ψ – k(x, y)A(x, y) + A(x, y)z(x, y) .
(.)
By (.), (.), (.), and condition (i), we deduce
z(x, y) ≤
l
∞
αi (x)
i=
∞
+ +ψ
+
A(ξ , η) ψ – z(ξ , η) dξ dη ds dt
∞ αj (M)
j=
∞ ∞
+ s
≤
∞
dj (s, t, x, y) k(s, t)A(s, t) + A(s, t)z(s, t)
βj (N)
ej (ξ , η, x, y) k(ξ , η)A(ξ , η) + A(ξ , η)z(ξ , η) dξ dη ds dt
t
l
∞
αi (x)
i=
ci (ξ , η, x, y)ϕ ψ – k(ξ , η)A(ξ , η)
t
–
l
bi (s, t, x, y)ϕ ψ – k(s, t)A(s, t) + ψ – A(s, t) ψ – z(s, t)
βi (y)
∞ ∞ s
∞
bi (s, t, x, y) ϕ ψ – k(s, t)A(s, t)
βi (y)
+ ϕ ψ – A(s, t) ϕ ψ – z(s, t) ∞ ∞
+ ci (ξ , η, x, y) ϕ ψ – k(ξ , η)A(ξ , η) s
t
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
–
+ ϕ ψ
+
l
– A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt
∞ αj (M)
j=
∞
∞
dj (s, t, x, y) k(s, t)A(s, t) + A(s, t)z(s, t)
βj (N)
ej (ξ , η, x, y) k(ξ , η)A(ξ , η) + A(ξ , η)z(ξ , η) dξ dη ds dt
∞
+ s
Page 7 of 25
t
≤ B(x, y) + C(x, y) +
l
αi (x)
i=
∞
∞
bi (s, t, x, y)ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
– – ci (ξ , η, x, y)ϕ ψ A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt
∞
+ s
∞
t
≤ B(M, N) + C(M, N) +
l
αi (x)
i=
∞
∞
bi (s, t, x, y)ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
ci (ξ , η, x, y)ϕ ψ – A(ξ , η) ϕ ψ – z(ξ , η) dξ dη ds dt,
∞
+ s
∞
t
∀(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
where B(M, N) is defined in (.), and C(M, N) is defined as follows:
C(M, N) =
l
∞
αj (M)
j=
∞
∞
dj (s, t, M, N)A(s, t)z(s, t)
βj (N)
∞
ej (ξ , η, M, N)A(ξ , η)z(ξ , η) dξ dη ds dt.
+ s
(.)
t
∀X ∈ I , Y ∈ I , for all (x, y) ∈ [X, ∞) × [Y , ∞), we have z(x, y) ≤ B(M, N) + C(M, N) +
l i=
αi (x)
∞
∞
∞
bi (s, t, X, Y )ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
∞
ci (ξ , η, X, Y )ϕ ψ
+ s
–
– A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt. (.)
t
Let z (x, y) denote the function on the right-hand side of (.), which is positive and nonincreasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z(x, y) ≤ z (x, y),
∀(x, y) ∈ [X, ∞) × [Y , ∞),
z (∞, y) = B(M, N) + C(M, N).
(.) (.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
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Differentiating z (x, y) with respect to x, we have ∞ l ∂z (x, y) bi αi (x), t, X, Y ϕ ψ – A αi (x), t ϕ ψ – z αi (x), t =– αi (x) ∂x βi (y) i= ∞ ∞ ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) ϕ ψ – z(ξ , η) dξ dη dt + αi (x)
≥–
l
αi (x)
i=
t
∞
bi αi (x), t, X, Y ϕ ψ – A αi (x), t ϕ ψ – z αi (x), t
βi (y)
∞
+ αi (x)
∞
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) ϕ ψ – z (ξ , η) dξ dη dt,
t
∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
By the monotonicity of ϕ , ϕ , z and the property of αi , βi , from (.), we get (∂/∂x)z (x, y) ϕ (ψ – (z (x, y))) l αi (x) ≥–
bi αi (x), t, X, Y ϕ ψ – A αi (x), t
βi (y)
i=
∞
∞
∞
+
ci (ξ , η, X, Y )ϕ ψ αi (x)
–
A(ξ , η)
ϕ (ψ – (z (ξ , η))) ϕ (ψ – (z (ξ , η)))
t
dξ dη dt,
∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Replacing x with s, and integrating it from x to ∞, we obtain W z (∞, y) – W z (x, y) ≥–
l i=
∞
αi (x)
∞ ∞
+ s
∞
bi (s, t, X, Y )ϕ ψ – A(s, t)
βi (y)
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η)
t
ϕ (ψ (z (ξ , η))) × dξ dη ds dt, ϕ (ψ – (z (ξ , η))) –
∀(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
i.e. l W z (x, y) ≤ W z (∞, y) + i=
∞
αi (x)
∞
bi (s, t, X, Y )ϕ ψ – A(s, t)
βi (y)
ϕ (ψ – (z (ξ , η))) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) dξ dη ds dt ϕ (ψ – (z (ξ , η))) s t l ∞ ∞ ∞ ∞ ci (ξ , η, X, Y ) ≤ W z (∞, y) + E(X, Y ) +
∞ ∞
+
i=
αi (x)
βi (y)
s
t
ϕ (ψ – (z (ξ , η))) × ϕ ψ – A(ξ , η) dξ dη ds dt, ϕ (ψ – (z (ξ , η)))
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 9 of 25
where E(X, Y ) is defined in (.). Let z (x, y) denote the function on the right-hand side of (.), which is positive and non-increasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z (x, y) ≤ W– z (x, y) , ∀(x, y) ∈ [X, ∞) × [Y , ∞), z (∞, y) = W z (∞, y) + E(X, Y ).
(.) (.)
Differentiating z (x, y) with respect to x, we have ∂z (x, y) =– αi (x) ∂x i=
l
∞
βi (y)
∞
αi (x)
∞
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η)
t
ϕ (ψ – (z (ξ , η))) dξ dη dt × ϕ (ψ – (z (ξ , η))) ∞ ∞ ∞ l ≥– αi (x) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) βi (y)
i=
×
αi (x)
t
ϕ (ψ – (W– (z (ξ , η)))) dξ dη dt. ϕ (ψ – (W– (z (ξ , η))))
(.)
By the monotonicity of ϕ /ϕ and z , from (.), we obtain ϕ (ψ – (W– (z (x, y))))(∂/∂x)z (x, y) ϕ (ψ – (W– (z (x, y)))) ∞ ∞ ∞ l αi (x) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) dξ dη dt. ≥– βi (y)
i=
αi (x)
(.)
t
Replace x with s, and integrating it from x to ∞, we get W z (x, y) ≤ W z (∞, y) + F(x, y, X, Y ),
(.)
where F(x, y, X, Y ) =
l i=
∞
αi (x)
∞
βi (y)
∞ ∞
s
ci (ξ , η, X, Y )ϕ ψ A(ξ , η) dξ dη ds dt.
–
t
Obviously, F(x, y, x, y) = F(x, y), which is defined in (.). From (.), (.), (.), (.) and (.), we have z(x, y) ≤ z (x, y) ≤ W– z (x, y) ≤ W– W– W W B(M, N) + C(M, N) + E(X, Y ) + F(x, y, X, Y ) , ∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Since X and Y are chosen arbitrarily, we have z(x, y) ≤ z (x, y) ≤ W– W– W W B(M, N) + C(M, N) + E(x, y) + F(x, y) , ∀(x, y) ∈ [M, ∞) × [N, ∞).
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 10 of 25
By the definition of C(M, N) and (.), we get C(M, N) ≤ z (M, N)D(M, N) ≤ W– W– W W B(M, N) + C(M, N) + E(M, N) + F(M, N) D(M, N),
(.)
or
C(M, N) – W W B(M, N) + C(M, N) + E(M, N) ≤ F(M, N), W W D(M, N)
(.)
where D(M, N) is defined in (.). By (.) and the hypothesis of G, we obtain C(M, N) ≤ G– F(M, N) .
(.)
Combining (.), (.) and (.), we get the desired result.
Corollary . Let the functions k, a, α, bi , ci , αi , βi (i = , , . . . , l ), dj , ej , αj , βj (j = , , . . . , l ) and u be defined as in Theorem ., p is a positive constant and p ≥ . If the function u(x, y) satisfies the inequality,
∞
u (x, y) ≤ k(x, y) + p
p
a(s, y)u (s, y) ds + α(x)
∞ ∞
+ s
+
i=
∞
bi (s, t, x, y)u(s, t)
αi (x)
βi (y)
σ ∈[ξ ,hξ ]
∞ αj (M)
j=
∞
dj (s, t, x, y)up (s, t)
βj (N)
∞ ∞
∞
ci (ξ , η, x, y) max u(σ , η) dξ dη ds dt
t
l
l
p
ej (ξ , η, x, y) max u (σ , η)) dξ dη ds dt,
+ s
σ ∈[ξ ,hξ ]
t
(x, y) ∈ ,
(.)
then: (i) if p > , we have u(x, y) ≤
k(x, y)
+
p p p–
p– p – B(M, N) + G– F (M, N) p + A(x, y) , F (x, y) p
where B(M, N) =
l
∞
∞
+ s
bi (s, t, M, N)k p (s, t)A p (s, t) αi (M)
i=
∞
t
∞
βi (N)
ci (ξ , η, M, N)k p (ξ , η)A p (ξ , η) dξ dη ds dt
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
∞
βj (N)
∞ ∞
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
+ s
F(x, y) =
dj (s, t, M, N)k(s, t)A(s, t)
αj (M)
j=
∞
Page 11 of 25
l
∞ αi (x)
i=
(.)
t
bi (s, t, x, y)A p (s, t)
∞
βi (y)
∞ ∞
+ s
p
ci (ξ , η, x, y)A (ξ , η) dξ dη ds dt,
(.)
p– p , – B(M, N) + u
(.)
t
p G (u) = p–
u D(M, N)
p– p
on condition that G (u) is a strictly increasing function on R+ . (ii) If p = , we have u(x, y) ≤ k(x, y) +
B(M, N) exp (F(x, y)) – D(M, N) exp (F(M, N))
A(x, y),
(.)
where D(M, N) exp F(M, N) <
(.)
and
B(M, N) =
l
∞ αi (M)
i=
∞
βi (N)
ci (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt s
t
l
∞ ∞
+
s l i=
∞
dj (s, t, M, N)k(s, t)A(s, t)
βj (N)
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
(.)
t
∞
∞
bi (s, t, x, y)A(s, t)
αi (x)
βi (y)
∞ ∞
+ s
∞
αj (M)
j=
F(x, y) =
∞
+
+
bi (s, t, M, N)k(s, t)A(s, t)
∞
ci (ξ , η, x, y)A(ξ , η) dξ dη ds dt.
(.)
t
Proof Inequality (.) followed by letting ψ(u(x, y)) = up (x, y), ϕ (u(x, y)) = ϕ (u(x, y)) =
u(x, y) in Theorem .. Then ψ – (u(x, y)) = u p (x, y) and (u + v) p ≤ u p + v p , (uv) p =
up vp .
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 12 of 25
If p > , we have
z
p p– p p– du p – = z c p , /p p– p– c u p p– p– p– – p z+c . W (z) = p
W (z) =
Applying Theorem ., we can easily get (.). If p = , we have
z
du = ln z – ln c, W– (z) = c exp z, u c u u G (u) = W . – W B(M, N) + u = ln D(M, N) D(M, N)(B(M, N) + u)
W (z) =
Obviously, G (u) is a strictly increasing function on R+ , G– (u) is the inverse of G (u), we get
G– (u) =
B(M, N)D(M, N) exp (u) , – D(M, N) exp (u)
D(M, N) exp (u) < ,
where B(M, N) is defined in (.). Applying Theorem ., we can easily get (.). Details
are omitted here. Theorem . Suppose that the following conditions hold: (i) (ii)-(iv) of Theorem . are satisfied; (ii) qi , ri are nonnegative constants with p ≥ qi , p ≥ ri , i = , , . . . , l , and εj , δj are nonnegative constants with p ≥ εj , p ≥ δj , j = , , . . . , l . If (x, y) ∈ , u(x, y) satisfies the following inequality:
∞
up (x, y) ≤ k(x, y) +
a(s, y)up (s, y) ds +
α(x) ∞ ∞
+ s
+
j=
s
αj (M)
t
αi (x)
∞
bi (s, t, x, y)uqi (s, t)
βi (y)
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
∞
+
∞
σ ∈[ξ ,hξ ]
∞ ∞
i=
t
l
l
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η)) dξ dη ds dt, σ ∈[ξ ,hξ ]
(x, y) ∈ ,
(.)
then we have u(x, y) ≤
k(x, y) +
p B (M, N) exp F (x, y) A(x, y) , – D (M, N)
(x, y) ∈ ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 13 of 25
where
B (M, N) =
l
qi qi qip–p p – qi qpi p K K k(s, t) + bi (s, t, M, N)A (s, t) p p βi (N)
∞ αi (M)
i=
∞
∞ ∞
ri
ci (ξ , η, M, N)A p (ξ , η)
+ s
t
p – ri rpi ri rip–p K K dξ dη ds dt k(ξ , η) + × p p l ∞ ∞ ε –p ε εj εj j p – εj pj + K dj (s, t, M, N)A p (s, t) K p k(s, t) + p p j= αj (M) βj (N)
∞ ∞
δj
ej (ξ , η, M, N)A p (ξ , η)
+ s
t
δ –p δ δj jp p – δj pj K k(ξ , η) + K dξ dη ds dt, p p l ∞ ∞ qi qi qip–p F (x, y) = bi (s, t, x, y)A p (s, t) K p i= αi (x) βi (y) ∞ ∞ ri ri rip–p p + ci (ξ , η, x, y)A (ξ , η) K dξ dη ds dt, p s t l ∞ ∞ ε –p εj εj jp D (M, N) = dj (s, t, M, N)A p (s, t) K exp F (s, t) p j= αj (M) βj (N)
×
∞ ∞
+ s
t
(.)
(.)
δ –p δj jp ej (ξ , η, M, N)A (ξ , η) K exp F (ξ , η) dξ dη ds dt p δj p
< .
(.)
Proof Let
z(x, y) =
l
∞
αi (x)
i=
+
∞ ∞
j=
σ ∈[ξ ,hξ ]
∞ αj (M)
∞ ∞
+ s
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
t
l
bi (s, t, x, y)uqi (s, t)
βi (y)
+ s
∞
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η)) dξ dη ds dt. σ ∈[ξ ,hξ ]
t
(.)
Obviously, z(x, y) is non-increasing in every variable. From (.) and (.), we have up (x, y) ≤ k(x, y) + z(x, y) +
∞ α(x)
a(s, y)up (s, y) ds.
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 14 of 25
By Lemma ., we obtain
up (x, y) ≤ k(x, y) + z(x, y) A(x, y),
(x, y) ∈ [M, ∞) × [N, ∞),
(.)
where A(x, y) is defined in (.). Then we get u(x, y) ≤
k(x, y) + z(x, y) A(x, y) p ,
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
By Lemma ., we have
qi k(x, y) + z(x, y) A(x, y) p qi p – qi qpi qi qip–p ≤ A p (x, y) K K k(x, y) + z(x, y) + p p qi –p qi qi p – qi p qi qip–p qi p p = A (x, y) K K + K z(x, y) , k(x, y) + p p p r
i max uri (ξ , y) ≤ max k(ξ , y) + z(ξ , y) A(ξ , y) p
uqi (x, y) ≤
ξ ∈[x,hx]
ξ ∈[x,hx]
≤
max k(ξ , y) + max z(ξ , y)
ξ ∈[x,hx] ri
≤ A p (x, y)
ξ ∈[x,hx]
ri –p p
ri K p
max A(ξ , y)
rpi
ξ ∈[x,hx]
p – ri k(x, y) + z(x, y) + K p
ri p
p – ri rpi ri rip–p ri rip–p = A (x, y) K k(x, y) + K + K z(x, y) , p p p εj –p εj εj –p εj εj p p – εj p εj p εj p K + K z(x, y) , u (x, y) ≤ A (x, y) K k(x, y) + p p p δ –p δ δ –p δj δj jp p – δj pj δj jp δj p max u (ξ , y) ≤ A (x, y) K k(x, y) + K + K z(x, y) . ξ ∈[x,hx] p p p ri p
Combining (.) and (.), we have qi qi qip–p p k(s, t) z(x, y) ≤ bi (s, t, x, y)A (s, t) K p i= αi (x) βi (y) p – qi qpi qi qip–p K + K z(s, t) + p p ∞ ∞ ri ri rip–p p + ci (ξ , η, x, y)A (ξ , η) K k(ξ , η) p s t ri ri –p p – ri p ri p K + K z(ξ , η) dξ dη ds dt + p p l ∞ ∞ ε –p εj εj jp + dj (s, t, x, y)A p (s, t) K k(s, t) p j= αj (M) βj (N) l
∞
ε
+
∞
ε –p
p – εj pj εj jp K + K z(s, t) p p
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
∞ ∞
Page 15 of 25
δ –p
δj jp K k(ξ , η) p s t δ δ –p p – δj pj δj jp + K + K z(ξ , η) dξ dη ds dt p p l ∞ ∞ qi qi qip–p bi (s, t, x, y)A p (s, t) K z(s, t) = B (x, y) + C (x, y) + p i= αi (x) βi (y) ∞ ∞ ri ri rip–p ci (ξ , η, x, y)A p (ξ , η) K z(ξ , η) dξ dη ds dt + p s t δj
+
ej (ξ , η, x, y)A p (ξ , η)
≤ B (M, N) + C (M, N) +
l
αi (x)
i=
∞ ∞
+
s
qi qi qip–p bi (s, t, x, y)A p (s, t) K z(s, t) p βi (y)
∞
t
∞
ri ri rip–p ci (ξ , η, x, y)A p (ξ , η) K z(ξ , η) dξ dη ds dt, p
(x, y) ∈ [M, ∞) × [N, ∞),
(.)
where B (M, N) is defined in (.), C (M, N) is defined as follows:
C (M, N) =
l
∞
αj (M)
j=
ε –p εj εj j dj (s, t, M, N)A p (s, t) K p z(s, t) p βj (N)
∞ ∞
+ s
t
∞
δ –p δj δj jp ej (ξ , η, M, N)A p (ξ , η) K z(ξ , η) dξ dη ds dt. p
(.)
∀X ∈ I , ∀Y ∈ I , for all (x, y) ∈ [X, ∞) × [Y , ∞), we have
z(x, y) ≤ B (M, N) + C (M, N) +
∞
+ s
t
l i=
∞
∞
αi (x)
∞
qi
bi (s, t, X, Y )A p (s, t) βi (y)
qi qip–p K z(s, t) p
ri rip–p ci (ξ , η, X, Y )A (ξ , η) K z(ξ , η) dξ dη ds dt. p ri p
(.)
Let z (x, y) denote the function on the right-hand side of (.), which is positive and nonincreasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z(x, y) ≤ z (x, y),
(x, y) ∈ [X, ∞) × [Y , ∞),
z (∞, y) = B (M, N) + C (M, N). Differentiating z (x, y) with respect to x, we have ∞ l qi qi qi –p ∂z (x, y) =– αi (x) bi αi (x), t, X, Y A p αi (x), t K p z αi (x), t ∂x p βi (y) i= ∞ ∞ ri ri rip–p ci (ξ , η, X, Y )A p (ξ , η) K z(ξ , η) dξ dη dt + p αi (x) t
(.) (.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
l
∞
Page 16 of 25
qi qi qi –p bi αi (x), t, X, Y A p αi (x), t K p z αi (x), t p βi (y) i= ∞ ∞ r –p ri ri ip ci (ξ , η, X, Y )A p (ξ , η) K z (ξ , η) dξ dη dt. + p αi (x) t
≥–
αi (x)
(.)
Dividing both sides of (.) by z (x, y), noticing that z (x, y) is non-increasing in each variable, we have ∞ l qi qi qip–p (∂/∂x)z (x, y) bi αi (x), t, X, Y A p αi (x), t K ≥– αi (x) z (x, y) p βi (y) i= ∞ ∞ ri ri rip–p p + ci (ξ , η, X, Y )A (ξ , η) K dξ dη dt, p αi (x) t (x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Replace x with s, and integrate it from x to ∞, we get z (x, y) ≤ z (∞, y) exp F (x, y, X, Y ) ,
(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
where F (x, y, X, Y ) =
l i=
αi (x)
∞ ∞
+ s
qi qi qi –p bi (s, t, X, Y )A p (s, t) K p p βi (y)
∞
t
∞
ri ri ri –p ci (ξ , η, X, Y )A p (ξ , η) K p dξ dη ds dt. p
(.)
It is obvious that F (x, y, x, y) = F (x, y), which is defined in (.). From (.), (.) and (.), we get
z(x, y) ≤ B (M, N) + C (M, N) exp F (x, y, X, Y ) ,
(x, y) ∈ [X, ∞) × [Y , ∞). (.)
Due to the fact that X, Y are chosen arbitrarily, we have
z(x, y) ≤ B (M, N) + C (M, N) exp F (x, y) ,
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
By the definition of C (M, N), we have B (M, N) + C (M, N) ≤ B (M, N) +
l j=
∞
αj (M)
∞
βj (N)
ε –p
εj εj jp dj (s, t, M, N)A p (s, t) K p
× B (M, N) + C (M, N) exp F (s, t) ∞ ∞ δ –p δj δj jp ej (ξ , η, M, N)A p (ξ , η) K + p s t
× B (M, N) + C (M, N) exp F (ξ , η) dξ dη ds dt
≤ B (M, N) + B (M, N) + C (M, N) D (M, N),
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 17 of 25
where D (M, N) is defined in (.). Then, according to D (M, N) < , we have
B (M, N) + C (M, N) ≤
B (M, N) . – D (M, N)
(.)
From (.) and (.), we get
z(x, y) ≤
B (M, N) exp F (x, y) , – D (M, N)
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
Combining (.) and (.), we obtain the desired result.
Remark . If qi = ri = (i = , , . . . , l ), εj = δj = p (j = , , . . . , l ), the inequality (.) becomes (.), but the proof of Theorem . is different from that of Corollary ..
Corollary . Let k, a, α, αi , βi , bi , ci (i = , , . . . , l ), αj , βj , dj , ej (i = , , . . . , l ) be defined as in Theorem ., then q, r are nonnegative constants with ≤ q ≤ , ≤ r ≤ . For (x, y) ∈ , u(x, y) satisfies the following inequality:
∞
u (x, y) ≤ k(x, y) +
a(s, y)u (s, y) ds
α(x)
+
l i=
∞ αi (x)
∞ ∞
+ s
+
bi (s, t, x, y)uq (s, t)
βi (y)
ci (ξ , η, x, y) max ur (σ , η) dξ dη ds dt σ ∈[ξ ,hξ ]
t
l ∞ j=
+ s
∞
dj (s, t, x, y)u(s, t)
αj (M)
∞ ∞
∞
βj (N)
ej (ξ , η, x, y) max u(σ , η)) dξ dη ds dt, σ ∈[ξ ,hξ ]
t
(x, y) ∈ ,
(.)
then we have B (M, N) exp F (x, y) A(x, y) , – D (M, N)
u(x, y) ≤
k(x, y) +
(.)
where
F (x, y) =
l i=
αi (x)
∞ ∞
+ s
q q q– bi (s, t, x, y)A (s, t) K βi (y)
∞
t
∞
r r– r ci (ξ , η, x, y)A (ξ , η) K dξ dη ds dt,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 18 of 25
B (M, N) =
q q q– – q q bi (s, t, M, N)A (s, t) K k(s, t) + K i= αi (M) βi (N) ∞ ∞ r r– – r r r ci (ξ , η, M, N)A (ξ , η) K k(ξ , η) + K dξ dη ds dt + s t l ∞ ∞ – dj (s, t, M, N)A (s, t) K k(s, t) + K + j= αj (M) βj (N)
l
∞
∞
∞ ∞
+
ej (ξ , η, M, N)A (ξ , η) s
t
D (M, N) =
l
∞ αj (M)
j=
∞
βj (N) ∞
+ s
∞
t
– K k(ξ , η) + K dξ dη ds dt,
(.)
– dj (s, t, M, N)A (s, t) K exp F (s, t)
– ej (ξ , η, M, N)A (ξ , η) K exp F (ξ , η) dξ dη ds dt
< .
(.)
Proof Inequality (.) follows by inequality (.) with p = , qi = q, ri = r (i = , , . . . , l ), εj = δj = (j = , , . . . , l ). Then, applying Theorem ., we can easily get (.). Details are omitted here. Remark . As one can see, the established results above mainly deal with VolterraFredholm type integral inequalities with maxima in two variables. And they are different from the results presented in [, , ]. In Theorem ., in the case of one variable, if we take k(x, y) = k, a(x, y) = , l = l = , b (s, t, x, y) = d (s, t, x, y) = h (s), c (ξ , η, x, y) = e (ξ , η, x, y) = h (ξ ), ψ(u) = ϕ (u), ψ(u) = ϕ (u) in the second iterated integral, orderly, we will get the inequality that is similar to inequality (.). If the above conditions are satisfied in two dimensions and ϕ (maxσ ∈[ξ ,hξ ] u(σ , η)) = ϕ (u(ξ , η)), we get analogs of the inequality (.). And if we take l = , l = , bi (s, t, x, y) = fi (s, t), ci (ξ , η, x, y) = in Theorem ., inequality (.) reduces to (.).
3 Applications in the integral equation In this section, we apply our results in Theorem . and Theorem . to study the retarded Volterra-Fredholm type integral equations with maxima in two variables. Some results on the boundedness of their solutions are presented, which demonstrate that our results can be used to investigate the qualitative properties of solutions of some integral equations. Example We consider the retarded Volterra-Fredholm type integral equation of the form ψ v(x, y) = g (x, y) +
∞
g (s, y)ψ v s + ρ(s), y ds
x
+ s
l
∞ ∞ x
i=
∞
∞ t
Fi s, t, x, y, v s + ρi (s), t + γi (t) ,
y
Fi s, t, x, y,
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
v σ , η + γi (η) dξ dη ds dt
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
∞ ∞ M
j=
∞
Gj s, t, x, y, v s + ρj (s), t + γj (t) ,
N
∞
Gj s, t, x, y, s
Page 19 of 25
t
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
v σ , η + γj (η) dξ dη ds dt,
(x, y) ∈ .
(.)
Suppose that the following conditions hold: (i) g (x, y), g (x, y), v(x, y) ∈ C( , R); (ii) x + ρ(x), x + ρi (x), x + ρj (x) ∈ C (I , I ) and y + γi (y), y + γj (y) ∈ C (I , I ) are strictly increasing with ρ(M) = ρi (M) = ρj (M) = ,
γi (N) = γj (N) = ,
ρ(x) ≥ ,
ρi (x) ≥ ,
γi (y) ≥ ,
γj (y) ≥ for y ≥ N,
ρ (x) > –,
ρi (x) > –,
γi (y) > –,
γj (y) > –
ρj (x) ≥ for x ≥ M,
ρj (x) > –, (i = , , . . . , l ; j = , , . . . , l );
(iii) Fi , Gj ∈ C( × R , R), Fi , Gj ∈ C( × R, R) (i = , , . . . , l ; j = , , . . . , l ). Let α(x) = x + ρ(x), αi (x) = x + ρi (x), αj (x) = x + ρj (x), βi (y) = y + γi (y), βj (y) = y + γj (y). Then α, αi , αj , βi , βj satisfy the condition (iv) of Theorem .. Theorem . In Eq. (.), suppose that the following conditions hold: g (x, y) ≤ a(x, y), g (x, y) ≤ k(x, y), ψ v(x, y) = v(x, y), Fi (s, t, x, y, u, v) ≤ bi (s, t, x, y)ϕ |u| + |v|, Fi (s, t, x, y, u) ≤ ci (s, t, x, y)ϕ |u| , i = , , . . . , l , Gj (s, t, x, y, u, v) ≤ dj (s, t, x, y)|u| + |v|, Gj (s, t, x, y, u) ≤ ej (s, t, x, y)|u|, j = , , . . . , l ,
(.)
where k, a, bi , ci , dj , ej , ϕ , ϕ are defined in Theorem .. Assume that the function u G (u) = W (W ( D (M,N) )) – W (W (B (M, N) + u) + E (M, N)) is increasing. Then we have
the following estimate: v(x, y) ≤ k(x, y) + W – W – W W B (M, N) + G– F (M, N) + E (x, y) + F (x, y) A (x, y), (x, y) ∈ ,
(.)
where
∞
A (x, y) = exp α(x)
M a α – (s), y ds ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 20 of 25
B (M, N) =
l
∞
αi (M)
i=
∞
βi (N)
– – Mi Mi ci αi (ξ ), βi (η), M, N ϕ k(ξ , η)A (ξ , η) dξ dη ds dt
∞
+ s
+
t
l ∞
αj (M)
j=
+ s
t
D (M, N) =
Mj Mj dj αj– (s), βj– (t), M, N k(s, t)A (s, t)
∞ βj (N)
Mj Mj ej αj– (ξ ), βj– (η), M, N k(ξ , η)A (ξ , η) dξ dη ds dt,
∞ ∞
Mi Mi bi αi– (s), βi– (t), M, N ϕ k(s, t)A (s, t)
∞
l
∞
αj (M)
j=
∞
βj (N) ∞
+ s
E (M, N) =
l i=
∞
αi (M)
i=
F (x, y) =
t
l
∞
βi (y)
s
t
∞ ∞
Mi Mi bi αi– (s), βi– (t), M, N ϕ A (s, t) ds dt,
(.)
(.)
Mi Mi
ci αi– (ξ ), βi– (η), x, y ϕ (A (ξ , η) dξ dη ds dt,
(.)
< ∞, α (α – (x))
M = max x∈I
< ∞,
Mi = max
< ∞, αj (αj– (x))
Mj = max
Mi = max
x∈I α (α – (x)) i i
Mj = max x∈I
βi (N) ∞
Mj Mj dj αj– (s), βj– (t), M, N A (s, t)
– – Mj Mj ej αj (ξ ), βj (η), M, N A (ξ , η) dξ dη ds dt, ∞
αi (x)
×
∞
(.)
< ∞,
i = , , . . . , l ;
< ∞, βj (βj– (y))
j = , , . . . , l ;
y∈I β (β – (y)) i i
y∈I
(.)
W , W are defined in Theorem .. Proof By applying the conditions (.) to (.), we have v(x, y) ≤ k(x, y) +
∞
a(s, y) · v s + ρ(s), y ds
x
+
l ∞ ∞
bi (s, t, x, y)ϕ v s + ρi (s), t + γi (t)
i=
x
y
∞
∞
+ s
+
j=
v σ , η + γi (η) dξ dη ds dt
∞ ∞
dj (s, t, x, y)v s + ρj (s), t + γj (t)
M
∞ ∞
+ s
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
t
l
ci (ξ , η, x, y)ϕ
t
N
ej (ξ , η, x, y)
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
v σ , η + γj (η) dξ dη ds dt
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
≤ k(x, y) +
∞
Page 21 of 25
a α – (s), y v(s, y)
α(x)
+
l ∞ i=
αi (x)
∞ ∞
+ s
t
∞
βi (y)
ds α (α – (s))
bi αi– (s), βi– (t), x, y ϕ v(s, t)
ci αi– (ξ ), βi– (η), x, y ϕ max v(σ , η) σ ∈[ξ ,hξ ]
dξ dη – ds dt αi (αi– (ξ )) βi (βi– (η)) αi (αi (s)) βi (βi– (t)) l ∞ ∞ dj αj– (s), βj– (t), x, y v(s, t) + ×
j=
αj (M)
∞ ∞
+ s
t
βj (N)
ej αj– (ξ ), βj– (η), x, y
× max v(σ , η) σ ∈[ξ ,hξ ]
≤ k(x, y) +
∞
αj (αj– (ξ )) βj (βj– (η))
dξ dη
ds dt αj (αj– (s)) βj (βj– (t))
M a α – (s), y v(s, y) ds
α(x)
+
l ∞ i=
αi (x)
∞ ∞
+ s
+
j=
s
t
Mi Mi bi αi– (s), βi– (t), x, y ϕ v(s, t)
Mi Mi ci αi– (ξ ), βi– (η), x, y ϕ max v(σ , η) dξ dη ds dt
αj (M)
+
βi (y)
∞
∞
∞
σ ∈[ξ ,hξ ]
t
l
∞
∞ βj (N)
Mj Mj dj αj– (s), βj– (t), x, y v(s, t)
Mj Mj ej αj– (ξ ), βj– (η), x, y
× max v(σ , η) dξ dη ds dt, σ ∈[ξ ,hξ ]
(.)
for (x, y) ∈ , where M , Mi , Mi (i = , , . . . , l ), Mj , Mj (j = , , . . . , l ) are defined in (.). Applying the results of Theorem . to (.) with ψ(u) = u, a(s, y) = M a(α – (s), y), bi (s, t, x, y) = Mi Mi bi (αi– (s), βi– (t), x, y), ci (ξ , η, x, y) = Mi Mi ci (αi– (ξ ), βi– (η), x, y), ej (αj– (ξ ), βj– (η), x, y), we dj (s, t, x, y) = Mj Mj dj (αj– (s), βj– (t), x, y), ej (ξ , η, x, y) = Mj Mj
obtain the desired estimation (.).
Theorem . In equation (.), suppose that the following conditions hold: g (x, y) ≤ k(x, y), g (x, y) ≤ a(x, y), ψ v(x, y) = vp (x, y), Fi (s, t, x, y, u, v) ≤ bi (s, t, x, y)|u|qi + |v|, Fi (s, t, x, y, u) ≤ ci (s, t, x, y)|u|ri , i = , , . . . , l , Gj (s, t, x, y, u, v) ≤ dj (s, t, x, y)|u|εj + |v|, Gj (s, t, x, y, u) ≤ ej (s, t, x, y)|u|δj , j = , , . . . , l ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 22 of 25
where p, qi , ri , εj , δj , bi , ci , dj , ej (i = , , . . . , l ; j = , , . . . , l ) are defined as in Theorem .. Then we have the following estimate:
v(x, y) ≤
p B (M, N) exp F (x, y) A (x, y) , – D (M, N)
k(x, y) +
(.)
where
B (M, N) =
l i=
∞ αi (M)
∞
βi (N)
qpi Mi Mi bi αi– (s), βi– (t), M, N A (s, t)
qi qip–p p – qi qpi × K K k(s, t) + p p ∞ ∞ + Mi Mi ci αi– (ξ ), βi– (η), M, N s
t
ri rip–p p – ri rpi K dξ dη ds dt × A (ξ , η) K k(ξ , η) + p p l ∞ ∞ εpj + Mj Mj dj αj– (s), βj– (t), M, N A (s, t)
ri p
αj (M)
j=
βj (N)
ε p – εj pj εj k(s, t) + K K × p p ∞ ∞ + Mj Mj ej αj– (ξ ), βj– (η), M, N
εj –p p
s
t
δj p
× A (ξ , η) D (M, N) =
l
∞
(.)
∞
Mj Mj αj (M)
j=
δ –p δ δj jp p – δj pj K k(ξ , η) + K dξ dη ds dt, p p
βj (N)
ε –p εj εj j × dj αj– (s), βj– (t), M, N Ap (s, t) K p exp F (s, t) p ∞ ∞ Mj Mj ej αj– (ξ ), βj– (η), M, N + s
t
δ –p δj jp × A (ξ , η) K exp F (ξ , η) dξ dη ds dt p δj p
< , F (x, y) =
(.)
l i=
∞
αi (x)
s
t
qpi qi qip–p Mi Mi bi αi– (s), βi– (t), x, y A (s, t) K p βi (y)
∞ ∞
+
∞
Mi Mi ci αi– (ξ ), βi– (η), x, y
ri rip–p × A (ξ , η) K dξ dη ds dt. p ri p
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 23 of 25
Proof Applying the conditions of (.) to (.), we have v(x, y)p ≤ k(x, y) +
p a(s, y)v s + ρ(s), y ds
∞
x
+
l ∞ ∞
q bi (s, t, x, y)v s + ρi (s), t + γi (t) i
i=
x
y
∞
∞
+ s
+
j=
ε dj (s, t, x, y)v s + ρj (s), t + γj (t) j
M
N ∞
+
ej (ξ , η, x, y)
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
t
≤ k(x, y) +
p a α – (s), y v(s, y)
∞
α(x)
+
l i=
∞
αi (x)
∞
∞
+ s
t
∞
βi (y)
j=
∞
αj (M)
∞
∞
+ s
t
∞ βj (N)
αi (αi– (ξ ))
βi (βi– (η))
βj (βj– (t))
∞
αj (αj– (ξ )) βj (βj– (η))
αj (αj– (s))
dξ dη
αi (αi– (s))
βi (βi– (t))
ds dt
ej αj– (ξ ), βj– (η), x, y
σ ∈[ξ ,hξ ]
≤ k(x, y) +
ds
ε dj αj– (s), βj– (t), x, y v(s, t) j
δ × max v(σ , η) j ×
α (α – (s))
ci αi– (ξ ), βi– (η), x, y
σ ∈[ξ ,hξ ]
l
δj v σ , η + γj (η) dξ dη ds dt
q bi αi– (s), βi– (t), x, y v(s, t) i
r × max v(σ , η) i
+
ri v σ , η + γi (η) dξ dη ds dt
∞ ∞
∞
s
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
t
l
ci (ξ , η, x, y)
dξ dη
ds dt
p M a α – (s), y v(s, y) ds
α(x)
+
l i=
∞
αi (x)
∞
∞
+ s
+
j=
s
t
q Mi Mi bi αi– (s), βi– (t), x, y v(s, t) i
r Mi Mi ci αi– (ξ ), βi– (η), x, y max v(σ , η) i dξ dη ds dt
αj (M)
+
βi (y)
∞
∞
∞
σ ∈[ξ ,hξ ]
t
l
∞
∞ βj (N)
ε Mj Mj dj αj– (s), βj– (t), x, y v(s, t) j
Mj Mj ej αj– (ξ ), βj– (η), x, y
δ × max v(σ , η) j dξ dη ds dt, σ ∈[ξ ,hξ ]
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 24 of 25
for (x, y) ∈ , where M , Mi , Mi (i = , , . . . , l ), Mj , Mj (j = , , . . . , l ) are defined in (.). Applying the results of Theorem . to (.) with a(s, y) = M a(α – (s), y), bi (s, t, x, y) = Mi Mi bi (αi– (s), βi– (t), x, y), ci (ξ , η, x, y) = Mi Mi ci (αi– (ξ ), βi– (η), x, y), – – – dj (s, t, x, y) = Mj Mj dj (αj (s), βj (t), x, y), ej (ξ , η, x, y) = Mj Mj ej (αj (ξ ), βj– (η), x, y), we obtain the desired estimation (.).
4 Conclusion In this paper, we established several new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables in Theorem . and Theorem ., and gave their specific cases in Corollary . and Corollary ., respectively, which can be used in the analysis of the qualitative properties to solutions of integral equations with maxima. In Theorem . and Theorem ., we also presented the applications to research the boundedness of solutions of retarded nonlinear Volterra-Fredholm type integral equations. Using our method, one can further study the integral inequality with more dimensions. Acknowledgements The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper. This research is supported by National Science Foundation of China (11671227). Competing interests The authors declare that there is no conflict of interest regarding the publication of this paper. Authors’ contributions RX proved parts of the results in Section 2 and participated in Section 3 - Applications. XM carried out the generalized weakly singular integral inequalities and completed part of the proof. All authors read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 21 February 2017 Accepted: 24 July 2017 References 1. Gronwall, TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20(4), 292-296 (1919) 2. Bellman, R: The stability of solutions of linear differential equations. Duke Math. J. 10, 643-647 (1943) 3. Bihari, I: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7, 81-94 (1956) 4. Banov, D, Simeonov, P: Integral Inequalities and Applications. Mathematics and Its Applications, vol. 57. Kluwer Academic, Dordrecht (1992) 5. Pachpatte, BG: Inequalities for Differential and Integral Equations. Mathematics in Science and Engineering, vol. 197. Academic Press, San Diego (1998) 6. Pachpatte, BG: On a certain retarded integral inequality and its applications. J. Inequal. Pure Appl. Math. 5(1), 2-22 (2004) 7. Pachpatte, BG: Explicit bounds on Gamidov type integral inequalities. Tamkang J. Math. 37(1), 1-9 (2006) 8. Ma, QH, Peˇcari´c, J: On certain new nonlinear retarded integral inequalities for functions in two variables and their applications. J. Korean Math. Soc. 45(1), 393-407 (2008) 9. Cheng, KL, Guo, CX, Tang, M: Some nonlinear Gronwall-Bellman-Gamidov integral inequalities and their weakly singular analogues with applications. Abstr. Appl. Anal. 2014, Article ID 562691 (2014) 10. Cheng, KL, Guo, CX: New explicit bounds on Gamidov type integral inequalities for functions in two variables and their applications. Abstr. Appl. Anal. 2014, Article ID 539701 (2014) 11. Lipovan, O: A retarded Gronwall-like inequality and its applications. J. Math. Anal. Appl. 7(1), 7-11 (2000) 12. Agarwal, RP, Deng, S, Zhang, W: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 165(3), 599-612 (2005) 13. Xu, R, Meng, F, Song, C: On some integral inequalities on time scales and their applications. J. Inequal. Appl. 2010, Article ID 464976 (2010) 14. Wang, WS: A general form of retarded integral inequality with two variables and its application. J. Syst. Sci. Math. Sci. 30(3), 425-432 (2010) 15. Agarwal, RP, Ryoo, CS, Kim, YH: New integral inequalities for iterated integrals with applications. J. Inequal. Appl. 2007, Article ID 24385 (2007)
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16. Wang, T, Xu, R: Some integral inequalities in two independent variables on time scales. J. Math. Inequal. 6(1), 107-118 (2012) 17. Wang, T, Xu, R: Bounds for some new integral inequalities with delay on time scales. J. Math. Inequal. 6(1), 1-12 (2012) 18. Abdeldaim, A: On some new integral inequalities of Gronwall-Bellman-Pachpatte type. Appl. Math. Comput. 217(20), 7887-7899 (2011) 19. Ma, QH, Peˇcari´c, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal., Theory Methods Appl. 69(2), 393-407 (2008) 20. Hou, ZY, Wang, WS: A class of nonlinear retarded Volterra-Fredholm type integral inequality and its application. Math. Pract. Theory 44, 21 (2014) 21. Lu, YS, Wang, WS, Zhou, XL, Huang, Y: Generalized nonlinear Volterra-Fredholm type integral inequality with two variables. J. Appl. Math. 2014, Article ID 359280 (2014) 22. Hou, ZY, Wang, WS: A class of nonlinear Volterra-Fredholm type integral inequality with variable lower limit and its application. J. Southwest China Normal Univ. 41, 2 (2016) 23. Huang, CM, Wang, WS: A class of nonlinear Volterra-Fredholm type integral inequalities with maxima. J. Sichuan Normal Univ. 39, 3 (2016) 24. Meng, F, Shao, J: Some new Volterra-Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 223(3), 444-451 (2013) 25. Pachpatte, BG: On some fundamental integral inequalities and their discrete analogues. J. Inequal. Pure Appl. Math. 13, 2 (2001) 26. Jiang, FC, Meng, FW: Explicit bounds on some new nonlinear integral inequality with delay. J. Comput. Appl. Math. 205, 479-487 (2007)
RESEARCH
Open Access
Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications Run Xu*
and Xiangting Ma
*
Correspondence: [email protected] Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
Abstract In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations. MSC: 42B20; 26D07; 26D15 Keywords: Volterra-Fredholm type; integral inequality; integral equations; maxima; iterated integral inequality
1 Introduction Gronwall-Bellman inequality [, ] and Bihari inequality [] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds and other qualitative properties of solutions to differential equations, integral equations and integro-differential equations. In the past few decades, a number of studies have focused on generalizations of the Gronwall-Bellman inequality. For example, in [–], the Gronwall-Bellman-Gamidov type integral inequalities and their generalizations were studied; in [–], the Gronwall-like inequalities and their deformations were investigated; in [–], the Volterra type iterated inequalities were discussed; in [–], the Volterra-Fredholm type inequalities were examined. The Gronwall-Bellman inequality can be stated as follows. If u and f are nonnegative continuous functions on an interval [a, b], and u satisfies the following inequality:
t
u(t) ≤ c +
f (s)u(s) ds,
t ∈ [a, b],
(.)
a
where c ≥ is a constant. Then u(t) ≤ c exp
t
f (s) ds .
(.)
a
© The Author(s) 2017. 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.
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 2 of 25
In , Pachpatte [] investigated the retarded linear Volterra-Fredholm type integral inequality in two independent variables:
α(x) β(y)
u(x, y) ≤ c +
α(M) β(N)
a(x, y, s, t)u(s, t) dt ds + α(x )
β(y )
b(x, y, s, t)u(s, t) dt ds, α(x )
β(y )
(x, y) ∈ [x , M] × [y , N].
(.)
In , Wang [] investigated a retarded Volterra type integral inequality with two variables: x ψ u(x, y) ≤ a(x, y) + b(x, y) c(s, y)ψ u(s, y) ds x
α (x) β (y)
+ d(x, y) α (x )
α (x) β (y)
+ α (x )
f (s, t)ϕ u(s, t) dt ds
β (y )
f (s, t)ϕ u(s, t) dt ds ,
(x, y) ∈ [x , x ) × [y , y ). (.)
β (y )
In , Lu et al. [] studied the nonlinear retarded Volterra-Fredholm type iterated integral inequality: u(x, y) ≤ k +
α(x) β(y)
α(x )
h (s , t )ω u(s , t )
β(y )
× f (s , t )ω u(s , t ) +
α(x )
s
s
β(y )
t
h (s , t ) f (s , t )ω u(s , t )
t
h (s , t )ω u(s , t ) dt ds dt ds dt ds
+ α(x )
β(y )
α(M) β(N)
+
α(x )
s
β(y )
t
+ α(x )
s
β(y ) t
+ α(x )
h (s , t )ω u(s , t ) f (s , t )ω u(s , t )
h (s , t ) f (s , t )ω u(s , t ) h (s , t )ω u(s , t ) dt ds dt ds dt ds ,
β(y )
(x, y) ∈ .
(.)
In , Huang and Wang [] discussed the retarded nonlinear Volterra-Fredholm type integral inequality with maxima: ϕ v(t) ≤ k +
α(t)
h (s) f (s)φ v(s) +
α(t ) τ
+
h (ξ )φ α(t )
α(T)
+
max v(η) dξ dτ ds
h (ξ )φ α(t )
v(t) ≤ k,
η∈[ξ –h,ξ ]
τ
+
α(t )
h (s) f (s)φ v(s) +
α(t )
h (τ ) f (τ )φ v(τ )
s
t ∈ [t – h, t ].
s
h (τ ) f (τ )φ v(τ )
α(t )
max v(η) dξ dτ ds,
η∈[ξ –h,ξ ]
t ∈ [t , T],
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 3 of 25
Motivated by the work presented in [, , ], we establish some new retarded nonlinear Volterra-Fredholm type integral inequality with maxima in two independent variables in this paper: ψ u(x, y) ≤ k(x, y) +
∞
l a(s, y)ψ u(s, y) ds +
α(x)
∞
ci (ξ , η, x, y)ϕ
s
αj (M)
j=
∞
∞
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
max u(σ , η) dξ dη ds dt
βj (N)
ej (ξ , η, x, y)ψ
s
dj (s, t, x, y)ψ u(s, t)
∞
+
∞
σ ∈[ξ ,hξ ]
t
l
+
∞
αi (x)
i=
∞
+
t
max u(σ , η) dξ dη ds dt,
σ ∈[ξ ,hξ ]
(x, y) ∈ , (.)
and
∞
u (x, y) ≤ k(x, y) + p
p
a(s, y)u (s, y) ds + α(x)
∞ ∞
+ s
+
i=
∞
∞ ∞
+ s
αi (x)
∞
bi (s, t, x, y)uqi (s, t)
βi (y)
σ ∈[ξ ,hξ ]
αj (M)
j=
∞
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
t
l
l
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η) dξ dη ds dt, σ ∈[ξ ,hξ ]
t
(x, y) ∈ .
(.)
By the amplification method, differential and integration, and the inverse function, we obtain the lower bound estimation of the unknown function. The example is given to illustrate the application of our results.
2 Main results In what follows, R denotes the set of real numbers, R+ = [, +∞), I = [M, +∞), I = [N, +∞) are the given subsets of R, = I × I . C (, S) denotes the class of all continuously differentiable functions defined on set with range in the set S, C(, S) denotes the class of all continuous functions defined on set with range in the set S, and α (t) denotes the derived function of a function α(t). For convenience, we cite some useful lemmas in the discussion of our proof as follows: Lemma . (See []) Let u(t), a(t), b(t) be nonnegative and continuous functions defined for t ∈ R+ . Assume that a(t) is non-increasing function for t ∈ R+ . If
∞
u(t) ≤ a(t) +
b(s)u(s) ds,
t ∈ R+ ,
t
then u(t) ≤ a(t) exp t
∞
b(s) ds ,
t ∈ R+ .
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
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From Lemma ., we can get the generalization in two dimensions. Lemma . Let u(x, y), a(x, y), b(x, y) be nonnegative and continuous functions defined for (x, y) ∈ . Assume that a(x, y) is a non-increasing function in the first variable. If
∞
u(x, y) ≤ a(x, y) +
b(s, y)u(s, y) ds,
(x, y) ∈ ,
x
then u(x, y) ≤ a(x, y) exp
∞
b(s, y) ds ,
(x, y) ∈ .
x
Lemma . (See []) Assume that a ≥ , p ≥ q ≥ , and p = . Then, for any K > , q q q–p p – q pq ap ≤ K p a + K . p p
Theorem . Suppose that the following conditions hold: (i) ψ ∈ C(R+ , R+ ) is an increasing function and ψ(u) > , ∀u > , ψ(∞) = ∞. ψ – is the inverse function of ψ. ϕ , ϕ , ϕ /ϕ ∈ C(R+ , R+ ) are increasing functions with ϕi (u) > (i = , ) for u > . ψ – , ϕi (i = , ) are sub-multiplicative and sub-additive, that is, ψ – (αβ) ≤ ψ – (α)ψ – (β), ϕi (αβ) ≤ ϕi (α)ϕi (β),
ψ – (α + β) ≤ ψ – (α) + ψ – (β),
ϕi (α + β) ≤ ϕi (α) + ϕi (β),
α, β ∈ R+ ;
(ii) k(x, y), a(x, y) ∈ C( , R+ ) and k(x, y) is non-increasing in the first variable; (iii) bi (s, t, x, y), ci (s, t, x, y), dj (s, t, x, y), ej (s, t, x, y) ∈ C( , R+ ) for i = , , . . . , l ; j = , , . . . , l ; bi , ci , dj , ej are all non-increasing functions in the last two variables; (iv) α, αi , αj ∈ C(I , I ), βi , βj ∈ C(I , I ) are non-decreasing functions with α(x), αi (x), αj (x) ≥ x, βi (y), βj (y) ≥ y (i = , , . . . , l ; j = , , . . . , l ). h ≥ is a constant; (v) the function u ∈ C( , R+ ) satisfies the inequality ψ u(x, y) ≤ k(x, y) +
∞
a(s, y)ψ u(s, y) ds
α(x)
+
l ∞ i=
αi (x)
∞
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
∞
ci (ξ , η, x, y)ϕ
+ s
+
j=
∞
αj (M)
∞
∞ βj (N)
ej (ξ , η, x, y)ψ
s
max u(σ , η) dξ dη ds dt
dj (s, t, x, y)ψ u(s, t)
∞
+ t
(x, y) ∈ ,
σ ∈[ξ ,hξ ]
t
l
max u(σ , η) dξ dη ds dt,
σ ∈[ξ ,hξ ]
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 5 of 25
then we have u(x, y) ≤ ψ – k(x, y) + W– W– W W B(M, N) + G– F(M, N) + E(x, y) + F(x, y) A(x, y) ,
(.)
where
z
ds , ϕ (ψ – (s))
W (z) = c
c > , z ∈ (, +∞),
(.)
z
ϕ (ψ – (W– (s))) ds, c > , z ∈ (, +∞), – – c ϕ (ψ (W (s))) u – W W B(M, N) + u + E(M, N) , G(u) = W W D(M, N)
(.)
W (z) =
(.)
on condition that W (+∞) = +∞, W (+∞) = +∞, and G(u) is a strictly increasing function on R+ . We have A(x, y) = exp
a(s, y) ds ,
∞
(.)
α(x) l
B(M, N) =
∞ αi (M)
i=
∞
βi (N)
ci (ξ , η, M, N)ϕ ψ k(ξ , η)A(ξ , η) dξ dη ds dt
∞
+ s
αj (M)
j=
s l
∞
∞
βj (N)
∞ ∞
ej (ξ , η, M, N)A(ξ , η) dξ dη ds dt,
+ s
l
(.)
t
l i=
∞
αi (M) ∞
∞
bi (s, t, M, N)ϕ ψ – A(s, t) ds dt,
∞
∞ ∞
ci (ξ , η, x, y)ϕ ψ
i=
αi (x)
l
∞
βi (y)
αi (x)
i=
∞
s
–
∞
bi (s, t, x, y)ϕ u(s, t)
βi (y)
∞
s
t
max u(σ , η) dξ dη ds dt
σ ∈[ξ ,hξ ]
A(ξ , η) dξ dη ds dt.
t
ci (ξ , η, x, y)ϕ
+
(.)
βi (N)
Proof Let
z(x, y) =
(.)
dj (s, t, M, N)A(s, t) αj (M)
F(x, y) =
βj (N)
t
j=
E(M, N) =
dj (s, t, M, N)k(s, t)A(s, t)
∞
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
∞ ∞
+
D(M, N) =
–
t
l ∞
+
bi (s, t, M, N)ϕ ψ – k(s, t)A(s, t)
∞
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
αj (M)
j=
∞
∞
∞
dj (s, t, x, y)ψ u(s, t)
βj (N)
∞
ej (ξ , η, x, y)ψ
+ s
Page 6 of 25
t
max u(σ , η) dξ dη ds dt.
σ ∈[ξ ,hξ ]
(.)
Obviously, z(x, y) is non-increasing in each of the variables. From (.), we have ψ u(x, y) ≤ k(x, y) + z(x, y) +
∞
a(s, y)ψ u(s, y) ds.
(.)
α(x)
Applying Lemma ., we obtain ψ u(x, y) ≤ k(x, y) + z(x, y) A(x, y),
(.)
u(x, y) ≤ ψ – k(x, y) + z(x, y) A(x, y) ,
(.)
i.e.
where A(x, y) is defined in (.), and A(x, y) is non-increasing in the first variable. So we have
max u(ξ , y) ≤ max ψ – k(ξ , y) + z(ξ , y) A(ξ , y)
ξ ∈[x,hx]
ξ ∈[x,hx]
≤ ψ – max k(ξ , y)A(ξ , y) + max A(ξ , y)z(ξ , y) ξ ∈[x,hx]
ξ ∈[x,hx]
≤ ψ – k(x, y)A(x, y) + A(x, y)z(x, y) .
(.)
By (.), (.), (.), and condition (i), we deduce
z(x, y) ≤
l
∞
αi (x)
i=
∞
+ +ψ
+
A(ξ , η) ψ – z(ξ , η) dξ dη ds dt
∞ αj (M)
j=
∞ ∞
+ s
≤
∞
dj (s, t, x, y) k(s, t)A(s, t) + A(s, t)z(s, t)
βj (N)
ej (ξ , η, x, y) k(ξ , η)A(ξ , η) + A(ξ , η)z(ξ , η) dξ dη ds dt
t
l
∞
αi (x)
i=
ci (ξ , η, x, y)ϕ ψ – k(ξ , η)A(ξ , η)
t
–
l
bi (s, t, x, y)ϕ ψ – k(s, t)A(s, t) + ψ – A(s, t) ψ – z(s, t)
βi (y)
∞ ∞ s
∞
bi (s, t, x, y) ϕ ψ – k(s, t)A(s, t)
βi (y)
+ ϕ ψ – A(s, t) ϕ ψ – z(s, t) ∞ ∞
+ ci (ξ , η, x, y) ϕ ψ – k(ξ , η)A(ξ , η) s
t
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
–
+ ϕ ψ
+
l
– A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt
∞ αj (M)
j=
∞
∞
dj (s, t, x, y) k(s, t)A(s, t) + A(s, t)z(s, t)
βj (N)
ej (ξ , η, x, y) k(ξ , η)A(ξ , η) + A(ξ , η)z(ξ , η) dξ dη ds dt
∞
+ s
Page 7 of 25
t
≤ B(x, y) + C(x, y) +
l
αi (x)
i=
∞
∞
bi (s, t, x, y)ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
– – ci (ξ , η, x, y)ϕ ψ A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt
∞
+ s
∞
t
≤ B(M, N) + C(M, N) +
l
αi (x)
i=
∞
∞
bi (s, t, x, y)ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
ci (ξ , η, x, y)ϕ ψ – A(ξ , η) ϕ ψ – z(ξ , η) dξ dη ds dt,
∞
+ s
∞
t
∀(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
where B(M, N) is defined in (.), and C(M, N) is defined as follows:
C(M, N) =
l
∞
αj (M)
j=
∞
∞
dj (s, t, M, N)A(s, t)z(s, t)
βj (N)
∞
ej (ξ , η, M, N)A(ξ , η)z(ξ , η) dξ dη ds dt.
+ s
(.)
t
∀X ∈ I , Y ∈ I , for all (x, y) ∈ [X, ∞) × [Y , ∞), we have z(x, y) ≤ B(M, N) + C(M, N) +
l i=
αi (x)
∞
∞
∞
bi (s, t, X, Y )ϕ ψ – A(s, t) ϕ ψ – z(s, t)
βi (y)
∞
ci (ξ , η, X, Y )ϕ ψ
+ s
–
– A(ξ , η) ϕ ψ z(ξ , η) dξ dη ds dt. (.)
t
Let z (x, y) denote the function on the right-hand side of (.), which is positive and nonincreasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z(x, y) ≤ z (x, y),
∀(x, y) ∈ [X, ∞) × [Y , ∞),
z (∞, y) = B(M, N) + C(M, N).
(.) (.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 8 of 25
Differentiating z (x, y) with respect to x, we have ∞ l ∂z (x, y) bi αi (x), t, X, Y ϕ ψ – A αi (x), t ϕ ψ – z αi (x), t =– αi (x) ∂x βi (y) i= ∞ ∞ ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) ϕ ψ – z(ξ , η) dξ dη dt + αi (x)
≥–
l
αi (x)
i=
t
∞
bi αi (x), t, X, Y ϕ ψ – A αi (x), t ϕ ψ – z αi (x), t
βi (y)
∞
+ αi (x)
∞
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) ϕ ψ – z (ξ , η) dξ dη dt,
t
∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
By the monotonicity of ϕ , ϕ , z and the property of αi , βi , from (.), we get (∂/∂x)z (x, y) ϕ (ψ – (z (x, y))) l αi (x) ≥–
bi αi (x), t, X, Y ϕ ψ – A αi (x), t
βi (y)
i=
∞
∞
∞
+
ci (ξ , η, X, Y )ϕ ψ αi (x)
–
A(ξ , η)
ϕ (ψ – (z (ξ , η))) ϕ (ψ – (z (ξ , η)))
t
dξ dη dt,
∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Replacing x with s, and integrating it from x to ∞, we obtain W z (∞, y) – W z (x, y) ≥–
l i=
∞
αi (x)
∞ ∞
+ s
∞
bi (s, t, X, Y )ϕ ψ – A(s, t)
βi (y)
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η)
t
ϕ (ψ (z (ξ , η))) × dξ dη ds dt, ϕ (ψ – (z (ξ , η))) –
∀(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
i.e. l W z (x, y) ≤ W z (∞, y) + i=
∞
αi (x)
∞
bi (s, t, X, Y )ϕ ψ – A(s, t)
βi (y)
ϕ (ψ – (z (ξ , η))) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) dξ dη ds dt ϕ (ψ – (z (ξ , η))) s t l ∞ ∞ ∞ ∞ ci (ξ , η, X, Y ) ≤ W z (∞, y) + E(X, Y ) +
∞ ∞
+
i=
αi (x)
βi (y)
s
t
ϕ (ψ – (z (ξ , η))) × ϕ ψ – A(ξ , η) dξ dη ds dt, ϕ (ψ – (z (ξ , η)))
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 9 of 25
where E(X, Y ) is defined in (.). Let z (x, y) denote the function on the right-hand side of (.), which is positive and non-increasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z (x, y) ≤ W– z (x, y) , ∀(x, y) ∈ [X, ∞) × [Y , ∞), z (∞, y) = W z (∞, y) + E(X, Y ).
(.) (.)
Differentiating z (x, y) with respect to x, we have ∂z (x, y) =– αi (x) ∂x i=
l
∞
βi (y)
∞
αi (x)
∞
ci (ξ , η, X, Y )ϕ ψ – A(ξ , η)
t
ϕ (ψ – (z (ξ , η))) dξ dη dt × ϕ (ψ – (z (ξ , η))) ∞ ∞ ∞ l ≥– αi (x) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) βi (y)
i=
×
αi (x)
t
ϕ (ψ – (W– (z (ξ , η)))) dξ dη dt. ϕ (ψ – (W– (z (ξ , η))))
(.)
By the monotonicity of ϕ /ϕ and z , from (.), we obtain ϕ (ψ – (W– (z (x, y))))(∂/∂x)z (x, y) ϕ (ψ – (W– (z (x, y)))) ∞ ∞ ∞ l αi (x) ci (ξ , η, X, Y )ϕ ψ – A(ξ , η) dξ dη dt. ≥– βi (y)
i=
αi (x)
(.)
t
Replace x with s, and integrating it from x to ∞, we get W z (x, y) ≤ W z (∞, y) + F(x, y, X, Y ),
(.)
where F(x, y, X, Y ) =
l i=
∞
αi (x)
∞
βi (y)
∞ ∞
s
ci (ξ , η, X, Y )ϕ ψ A(ξ , η) dξ dη ds dt.
–
t
Obviously, F(x, y, x, y) = F(x, y), which is defined in (.). From (.), (.), (.), (.) and (.), we have z(x, y) ≤ z (x, y) ≤ W– z (x, y) ≤ W– W– W W B(M, N) + C(M, N) + E(X, Y ) + F(x, y, X, Y ) , ∀(x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Since X and Y are chosen arbitrarily, we have z(x, y) ≤ z (x, y) ≤ W– W– W W B(M, N) + C(M, N) + E(x, y) + F(x, y) , ∀(x, y) ∈ [M, ∞) × [N, ∞).
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 10 of 25
By the definition of C(M, N) and (.), we get C(M, N) ≤ z (M, N)D(M, N) ≤ W– W– W W B(M, N) + C(M, N) + E(M, N) + F(M, N) D(M, N),
(.)
or
C(M, N) – W W B(M, N) + C(M, N) + E(M, N) ≤ F(M, N), W W D(M, N)
(.)
where D(M, N) is defined in (.). By (.) and the hypothesis of G, we obtain C(M, N) ≤ G– F(M, N) .
(.)
Combining (.), (.) and (.), we get the desired result.
Corollary . Let the functions k, a, α, bi , ci , αi , βi (i = , , . . . , l ), dj , ej , αj , βj (j = , , . . . , l ) and u be defined as in Theorem ., p is a positive constant and p ≥ . If the function u(x, y) satisfies the inequality,
∞
u (x, y) ≤ k(x, y) + p
p
a(s, y)u (s, y) ds + α(x)
∞ ∞
+ s
+
i=
∞
bi (s, t, x, y)u(s, t)
αi (x)
βi (y)
σ ∈[ξ ,hξ ]
∞ αj (M)
j=
∞
dj (s, t, x, y)up (s, t)
βj (N)
∞ ∞
∞
ci (ξ , η, x, y) max u(σ , η) dξ dη ds dt
t
l
l
p
ej (ξ , η, x, y) max u (σ , η)) dξ dη ds dt,
+ s
σ ∈[ξ ,hξ ]
t
(x, y) ∈ ,
(.)
then: (i) if p > , we have u(x, y) ≤
k(x, y)
+
p p p–
p– p – B(M, N) + G– F (M, N) p + A(x, y) , F (x, y) p
where B(M, N) =
l
∞
∞
+ s
bi (s, t, M, N)k p (s, t)A p (s, t) αi (M)
i=
∞
t
∞
βi (N)
ci (ξ , η, M, N)k p (ξ , η)A p (ξ , η) dξ dη ds dt
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
∞
βj (N)
∞ ∞
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
+ s
F(x, y) =
dj (s, t, M, N)k(s, t)A(s, t)
αj (M)
j=
∞
Page 11 of 25
l
∞ αi (x)
i=
(.)
t
bi (s, t, x, y)A p (s, t)
∞
βi (y)
∞ ∞
+ s
p
ci (ξ , η, x, y)A (ξ , η) dξ dη ds dt,
(.)
p– p , – B(M, N) + u
(.)
t
p G (u) = p–
u D(M, N)
p– p
on condition that G (u) is a strictly increasing function on R+ . (ii) If p = , we have u(x, y) ≤ k(x, y) +
B(M, N) exp (F(x, y)) – D(M, N) exp (F(M, N))
A(x, y),
(.)
where D(M, N) exp F(M, N) <
(.)
and
B(M, N) =
l
∞ αi (M)
i=
∞
βi (N)
ci (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt s
t
l
∞ ∞
+
s l i=
∞
dj (s, t, M, N)k(s, t)A(s, t)
βj (N)
ej (ξ , η, M, N)k(ξ , η)A(ξ , η) dξ dη ds dt,
(.)
t
∞
∞
bi (s, t, x, y)A(s, t)
αi (x)
βi (y)
∞ ∞
+ s
∞
αj (M)
j=
F(x, y) =
∞
+
+
bi (s, t, M, N)k(s, t)A(s, t)
∞
ci (ξ , η, x, y)A(ξ , η) dξ dη ds dt.
(.)
t
Proof Inequality (.) followed by letting ψ(u(x, y)) = up (x, y), ϕ (u(x, y)) = ϕ (u(x, y)) =
u(x, y) in Theorem .. Then ψ – (u(x, y)) = u p (x, y) and (u + v) p ≤ u p + v p , (uv) p =
up vp .
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 12 of 25
If p > , we have
z
p p– p p– du p – = z c p , /p p– p– c u p p– p– p– – p z+c . W (z) = p
W (z) =
Applying Theorem ., we can easily get (.). If p = , we have
z
du = ln z – ln c, W– (z) = c exp z, u c u u G (u) = W . – W B(M, N) + u = ln D(M, N) D(M, N)(B(M, N) + u)
W (z) =
Obviously, G (u) is a strictly increasing function on R+ , G– (u) is the inverse of G (u), we get
G– (u) =
B(M, N)D(M, N) exp (u) , – D(M, N) exp (u)
D(M, N) exp (u) < ,
where B(M, N) is defined in (.). Applying Theorem ., we can easily get (.). Details
are omitted here. Theorem . Suppose that the following conditions hold: (i) (ii)-(iv) of Theorem . are satisfied; (ii) qi , ri are nonnegative constants with p ≥ qi , p ≥ ri , i = , , . . . , l , and εj , δj are nonnegative constants with p ≥ εj , p ≥ δj , j = , , . . . , l . If (x, y) ∈ , u(x, y) satisfies the following inequality:
∞
up (x, y) ≤ k(x, y) +
a(s, y)up (s, y) ds +
α(x) ∞ ∞
+ s
+
j=
s
αj (M)
t
αi (x)
∞
bi (s, t, x, y)uqi (s, t)
βi (y)
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
∞
+
∞
σ ∈[ξ ,hξ ]
∞ ∞
i=
t
l
l
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η)) dξ dη ds dt, σ ∈[ξ ,hξ ]
(x, y) ∈ ,
(.)
then we have u(x, y) ≤
k(x, y) +
p B (M, N) exp F (x, y) A(x, y) , – D (M, N)
(x, y) ∈ ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 13 of 25
where
B (M, N) =
l
qi qi qip–p p – qi qpi p K K k(s, t) + bi (s, t, M, N)A (s, t) p p βi (N)
∞ αi (M)
i=
∞
∞ ∞
ri
ci (ξ , η, M, N)A p (ξ , η)
+ s
t
p – ri rpi ri rip–p K K dξ dη ds dt k(ξ , η) + × p p l ∞ ∞ ε –p ε εj εj j p – εj pj + K dj (s, t, M, N)A p (s, t) K p k(s, t) + p p j= αj (M) βj (N)
∞ ∞
δj
ej (ξ , η, M, N)A p (ξ , η)
+ s
t
δ –p δ δj jp p – δj pj K k(ξ , η) + K dξ dη ds dt, p p l ∞ ∞ qi qi qip–p F (x, y) = bi (s, t, x, y)A p (s, t) K p i= αi (x) βi (y) ∞ ∞ ri ri rip–p p + ci (ξ , η, x, y)A (ξ , η) K dξ dη ds dt, p s t l ∞ ∞ ε –p εj εj jp D (M, N) = dj (s, t, M, N)A p (s, t) K exp F (s, t) p j= αj (M) βj (N)
×
∞ ∞
+ s
t
(.)
(.)
δ –p δj jp ej (ξ , η, M, N)A (ξ , η) K exp F (ξ , η) dξ dη ds dt p δj p
< .
(.)
Proof Let
z(x, y) =
l
∞
αi (x)
i=
+
∞ ∞
j=
σ ∈[ξ ,hξ ]
∞ αj (M)
∞ ∞
+ s
ci (ξ , η, x, y) max uri (σ , η) dξ dη ds dt
t
l
bi (s, t, x, y)uqi (s, t)
βi (y)
+ s
∞
∞
dj (s, t, x, y)uεj (s, t)
βj (N)
ej (ξ , η, x, y) max uδj (σ , η)) dξ dη ds dt. σ ∈[ξ ,hξ ]
t
(.)
Obviously, z(x, y) is non-increasing in every variable. From (.) and (.), we have up (x, y) ≤ k(x, y) + z(x, y) +
∞ α(x)
a(s, y)up (s, y) ds.
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 14 of 25
By Lemma ., we obtain
up (x, y) ≤ k(x, y) + z(x, y) A(x, y),
(x, y) ∈ [M, ∞) × [N, ∞),
(.)
where A(x, y) is defined in (.). Then we get u(x, y) ≤
k(x, y) + z(x, y) A(x, y) p ,
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
By Lemma ., we have
qi k(x, y) + z(x, y) A(x, y) p qi p – qi qpi qi qip–p ≤ A p (x, y) K K k(x, y) + z(x, y) + p p qi –p qi qi p – qi p qi qip–p qi p p = A (x, y) K K + K z(x, y) , k(x, y) + p p p r
i max uri (ξ , y) ≤ max k(ξ , y) + z(ξ , y) A(ξ , y) p
uqi (x, y) ≤
ξ ∈[x,hx]
ξ ∈[x,hx]
≤
max k(ξ , y) + max z(ξ , y)
ξ ∈[x,hx] ri
≤ A p (x, y)
ξ ∈[x,hx]
ri –p p
ri K p
max A(ξ , y)
rpi
ξ ∈[x,hx]
p – ri k(x, y) + z(x, y) + K p
ri p
p – ri rpi ri rip–p ri rip–p = A (x, y) K k(x, y) + K + K z(x, y) , p p p εj –p εj εj –p εj εj p p – εj p εj p εj p K + K z(x, y) , u (x, y) ≤ A (x, y) K k(x, y) + p p p δ –p δ δ –p δj δj jp p – δj pj δj jp δj p max u (ξ , y) ≤ A (x, y) K k(x, y) + K + K z(x, y) . ξ ∈[x,hx] p p p ri p
Combining (.) and (.), we have qi qi qip–p p k(s, t) z(x, y) ≤ bi (s, t, x, y)A (s, t) K p i= αi (x) βi (y) p – qi qpi qi qip–p K + K z(s, t) + p p ∞ ∞ ri ri rip–p p + ci (ξ , η, x, y)A (ξ , η) K k(ξ , η) p s t ri ri –p p – ri p ri p K + K z(ξ , η) dξ dη ds dt + p p l ∞ ∞ ε –p εj εj jp + dj (s, t, x, y)A p (s, t) K k(s, t) p j= αj (M) βj (N) l
∞
ε
+
∞
ε –p
p – εj pj εj jp K + K z(s, t) p p
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
∞ ∞
Page 15 of 25
δ –p
δj jp K k(ξ , η) p s t δ δ –p p – δj pj δj jp + K + K z(ξ , η) dξ dη ds dt p p l ∞ ∞ qi qi qip–p bi (s, t, x, y)A p (s, t) K z(s, t) = B (x, y) + C (x, y) + p i= αi (x) βi (y) ∞ ∞ ri ri rip–p ci (ξ , η, x, y)A p (ξ , η) K z(ξ , η) dξ dη ds dt + p s t δj
+
ej (ξ , η, x, y)A p (ξ , η)
≤ B (M, N) + C (M, N) +
l
αi (x)
i=
∞ ∞
+
s
qi qi qip–p bi (s, t, x, y)A p (s, t) K z(s, t) p βi (y)
∞
t
∞
ri ri rip–p ci (ξ , η, x, y)A p (ξ , η) K z(ξ , η) dξ dη ds dt, p
(x, y) ∈ [M, ∞) × [N, ∞),
(.)
where B (M, N) is defined in (.), C (M, N) is defined as follows:
C (M, N) =
l
∞
αj (M)
j=
ε –p εj εj j dj (s, t, M, N)A p (s, t) K p z(s, t) p βj (N)
∞ ∞
+ s
t
∞
δ –p δj δj jp ej (ξ , η, M, N)A p (ξ , η) K z(ξ , η) dξ dη ds dt. p
(.)
∀X ∈ I , ∀Y ∈ I , for all (x, y) ∈ [X, ∞) × [Y , ∞), we have
z(x, y) ≤ B (M, N) + C (M, N) +
∞
+ s
t
l i=
∞
∞
αi (x)
∞
qi
bi (s, t, X, Y )A p (s, t) βi (y)
qi qip–p K z(s, t) p
ri rip–p ci (ξ , η, X, Y )A (ξ , η) K z(ξ , η) dξ dη ds dt. p ri p
(.)
Let z (x, y) denote the function on the right-hand side of (.), which is positive and nonincreasing in each of the variables (x, y) ∈ [X, ∞) × [Y , ∞). From (.), we have z(x, y) ≤ z (x, y),
(x, y) ∈ [X, ∞) × [Y , ∞),
z (∞, y) = B (M, N) + C (M, N). Differentiating z (x, y) with respect to x, we have ∞ l qi qi qi –p ∂z (x, y) =– αi (x) bi αi (x), t, X, Y A p αi (x), t K p z αi (x), t ∂x p βi (y) i= ∞ ∞ ri ri rip–p ci (ξ , η, X, Y )A p (ξ , η) K z(ξ , η) dξ dη dt + p αi (x) t
(.) (.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
l
∞
Page 16 of 25
qi qi qi –p bi αi (x), t, X, Y A p αi (x), t K p z αi (x), t p βi (y) i= ∞ ∞ r –p ri ri ip ci (ξ , η, X, Y )A p (ξ , η) K z (ξ , η) dξ dη dt. + p αi (x) t
≥–
αi (x)
(.)
Dividing both sides of (.) by z (x, y), noticing that z (x, y) is non-increasing in each variable, we have ∞ l qi qi qip–p (∂/∂x)z (x, y) bi αi (x), t, X, Y A p αi (x), t K ≥– αi (x) z (x, y) p βi (y) i= ∞ ∞ ri ri rip–p p + ci (ξ , η, X, Y )A (ξ , η) K dξ dη dt, p αi (x) t (x, y) ∈ [X, ∞) × [Y , ∞).
(.)
Replace x with s, and integrate it from x to ∞, we get z (x, y) ≤ z (∞, y) exp F (x, y, X, Y ) ,
(x, y) ∈ [X, ∞) × [Y , ∞),
(.)
where F (x, y, X, Y ) =
l i=
αi (x)
∞ ∞
+ s
qi qi qi –p bi (s, t, X, Y )A p (s, t) K p p βi (y)
∞
t
∞
ri ri ri –p ci (ξ , η, X, Y )A p (ξ , η) K p dξ dη ds dt. p
(.)
It is obvious that F (x, y, x, y) = F (x, y), which is defined in (.). From (.), (.) and (.), we get
z(x, y) ≤ B (M, N) + C (M, N) exp F (x, y, X, Y ) ,
(x, y) ∈ [X, ∞) × [Y , ∞). (.)
Due to the fact that X, Y are chosen arbitrarily, we have
z(x, y) ≤ B (M, N) + C (M, N) exp F (x, y) ,
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
By the definition of C (M, N), we have B (M, N) + C (M, N) ≤ B (M, N) +
l j=
∞
αj (M)
∞
βj (N)
ε –p
εj εj jp dj (s, t, M, N)A p (s, t) K p
× B (M, N) + C (M, N) exp F (s, t) ∞ ∞ δ –p δj δj jp ej (ξ , η, M, N)A p (ξ , η) K + p s t
× B (M, N) + C (M, N) exp F (ξ , η) dξ dη ds dt
≤ B (M, N) + B (M, N) + C (M, N) D (M, N),
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 17 of 25
where D (M, N) is defined in (.). Then, according to D (M, N) < , we have
B (M, N) + C (M, N) ≤
B (M, N) . – D (M, N)
(.)
From (.) and (.), we get
z(x, y) ≤
B (M, N) exp F (x, y) , – D (M, N)
(x, y) ∈ [M, ∞) × [N, ∞).
(.)
Combining (.) and (.), we obtain the desired result.
Remark . If qi = ri = (i = , , . . . , l ), εj = δj = p (j = , , . . . , l ), the inequality (.) becomes (.), but the proof of Theorem . is different from that of Corollary ..
Corollary . Let k, a, α, αi , βi , bi , ci (i = , , . . . , l ), αj , βj , dj , ej (i = , , . . . , l ) be defined as in Theorem ., then q, r are nonnegative constants with ≤ q ≤ , ≤ r ≤ . For (x, y) ∈ , u(x, y) satisfies the following inequality:
∞
u (x, y) ≤ k(x, y) +
a(s, y)u (s, y) ds
α(x)
+
l i=
∞ αi (x)
∞ ∞
+ s
+
bi (s, t, x, y)uq (s, t)
βi (y)
ci (ξ , η, x, y) max ur (σ , η) dξ dη ds dt σ ∈[ξ ,hξ ]
t
l ∞ j=
+ s
∞
dj (s, t, x, y)u(s, t)
αj (M)
∞ ∞
∞
βj (N)
ej (ξ , η, x, y) max u(σ , η)) dξ dη ds dt, σ ∈[ξ ,hξ ]
t
(x, y) ∈ ,
(.)
then we have B (M, N) exp F (x, y) A(x, y) , – D (M, N)
u(x, y) ≤
k(x, y) +
(.)
where
F (x, y) =
l i=
αi (x)
∞ ∞
+ s
q q q– bi (s, t, x, y)A (s, t) K βi (y)
∞
t
∞
r r– r ci (ξ , η, x, y)A (ξ , η) K dξ dη ds dt,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 18 of 25
B (M, N) =
q q q– – q q bi (s, t, M, N)A (s, t) K k(s, t) + K i= αi (M) βi (N) ∞ ∞ r r– – r r r ci (ξ , η, M, N)A (ξ , η) K k(ξ , η) + K dξ dη ds dt + s t l ∞ ∞ – dj (s, t, M, N)A (s, t) K k(s, t) + K + j= αj (M) βj (N)
l
∞
∞
∞ ∞
+
ej (ξ , η, M, N)A (ξ , η) s
t
D (M, N) =
l
∞ αj (M)
j=
∞
βj (N) ∞
+ s
∞
t
– K k(ξ , η) + K dξ dη ds dt,
(.)
– dj (s, t, M, N)A (s, t) K exp F (s, t)
– ej (ξ , η, M, N)A (ξ , η) K exp F (ξ , η) dξ dη ds dt
< .
(.)
Proof Inequality (.) follows by inequality (.) with p = , qi = q, ri = r (i = , , . . . , l ), εj = δj = (j = , , . . . , l ). Then, applying Theorem ., we can easily get (.). Details are omitted here. Remark . As one can see, the established results above mainly deal with VolterraFredholm type integral inequalities with maxima in two variables. And they are different from the results presented in [, , ]. In Theorem ., in the case of one variable, if we take k(x, y) = k, a(x, y) = , l = l = , b (s, t, x, y) = d (s, t, x, y) = h (s), c (ξ , η, x, y) = e (ξ , η, x, y) = h (ξ ), ψ(u) = ϕ (u), ψ(u) = ϕ (u) in the second iterated integral, orderly, we will get the inequality that is similar to inequality (.). If the above conditions are satisfied in two dimensions and ϕ (maxσ ∈[ξ ,hξ ] u(σ , η)) = ϕ (u(ξ , η)), we get analogs of the inequality (.). And if we take l = , l = , bi (s, t, x, y) = fi (s, t), ci (ξ , η, x, y) = in Theorem ., inequality (.) reduces to (.).
3 Applications in the integral equation In this section, we apply our results in Theorem . and Theorem . to study the retarded Volterra-Fredholm type integral equations with maxima in two variables. Some results on the boundedness of their solutions are presented, which demonstrate that our results can be used to investigate the qualitative properties of solutions of some integral equations. Example We consider the retarded Volterra-Fredholm type integral equation of the form ψ v(x, y) = g (x, y) +
∞
g (s, y)ψ v s + ρ(s), y ds
x
+ s
l
∞ ∞ x
i=
∞
∞ t
Fi s, t, x, y, v s + ρi (s), t + γi (t) ,
y
Fi s, t, x, y,
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
v σ , η + γi (η) dξ dη ds dt
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
+
l
∞ ∞ M
j=
∞
Gj s, t, x, y, v s + ρj (s), t + γj (t) ,
N
∞
Gj s, t, x, y, s
Page 19 of 25
t
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
v σ , η + γj (η) dξ dη ds dt,
(x, y) ∈ .
(.)
Suppose that the following conditions hold: (i) g (x, y), g (x, y), v(x, y) ∈ C( , R); (ii) x + ρ(x), x + ρi (x), x + ρj (x) ∈ C (I , I ) and y + γi (y), y + γj (y) ∈ C (I , I ) are strictly increasing with ρ(M) = ρi (M) = ρj (M) = ,
γi (N) = γj (N) = ,
ρ(x) ≥ ,
ρi (x) ≥ ,
γi (y) ≥ ,
γj (y) ≥ for y ≥ N,
ρ (x) > –,
ρi (x) > –,
γi (y) > –,
γj (y) > –
ρj (x) ≥ for x ≥ M,
ρj (x) > –, (i = , , . . . , l ; j = , , . . . , l );
(iii) Fi , Gj ∈ C( × R , R), Fi , Gj ∈ C( × R, R) (i = , , . . . , l ; j = , , . . . , l ). Let α(x) = x + ρ(x), αi (x) = x + ρi (x), αj (x) = x + ρj (x), βi (y) = y + γi (y), βj (y) = y + γj (y). Then α, αi , αj , βi , βj satisfy the condition (iv) of Theorem .. Theorem . In Eq. (.), suppose that the following conditions hold: g (x, y) ≤ a(x, y), g (x, y) ≤ k(x, y), ψ v(x, y) = v(x, y), Fi (s, t, x, y, u, v) ≤ bi (s, t, x, y)ϕ |u| + |v|, Fi (s, t, x, y, u) ≤ ci (s, t, x, y)ϕ |u| , i = , , . . . , l , Gj (s, t, x, y, u, v) ≤ dj (s, t, x, y)|u| + |v|, Gj (s, t, x, y, u) ≤ ej (s, t, x, y)|u|, j = , , . . . , l ,
(.)
where k, a, bi , ci , dj , ej , ϕ , ϕ are defined in Theorem .. Assume that the function u G (u) = W (W ( D (M,N) )) – W (W (B (M, N) + u) + E (M, N)) is increasing. Then we have
the following estimate: v(x, y) ≤ k(x, y) + W – W – W W B (M, N) + G– F (M, N) + E (x, y) + F (x, y) A (x, y), (x, y) ∈ ,
(.)
where
∞
A (x, y) = exp α(x)
M a α – (s), y ds ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 20 of 25
B (M, N) =
l
∞
αi (M)
i=
∞
βi (N)
– – Mi Mi ci αi (ξ ), βi (η), M, N ϕ k(ξ , η)A (ξ , η) dξ dη ds dt
∞
+ s
+
t
l ∞
αj (M)
j=
+ s
t
D (M, N) =
Mj Mj dj αj– (s), βj– (t), M, N k(s, t)A (s, t)
∞ βj (N)
Mj Mj ej αj– (ξ ), βj– (η), M, N k(ξ , η)A (ξ , η) dξ dη ds dt,
∞ ∞
Mi Mi bi αi– (s), βi– (t), M, N ϕ k(s, t)A (s, t)
∞
l
∞
αj (M)
j=
∞
βj (N) ∞
+ s
E (M, N) =
l i=
∞
αi (M)
i=
F (x, y) =
t
l
∞
βi (y)
s
t
∞ ∞
Mi Mi bi αi– (s), βi– (t), M, N ϕ A (s, t) ds dt,
(.)
(.)
Mi Mi
ci αi– (ξ ), βi– (η), x, y ϕ (A (ξ , η) dξ dη ds dt,
(.)
< ∞, α (α – (x))
M = max x∈I
< ∞,
Mi = max
< ∞, αj (αj– (x))
Mj = max
Mi = max
x∈I α (α – (x)) i i
Mj = max x∈I
βi (N) ∞
Mj Mj dj αj– (s), βj– (t), M, N A (s, t)
– – Mj Mj ej αj (ξ ), βj (η), M, N A (ξ , η) dξ dη ds dt, ∞
αi (x)
×
∞
(.)
< ∞,
i = , , . . . , l ;
< ∞, βj (βj– (y))
j = , , . . . , l ;
y∈I β (β – (y)) i i
y∈I
(.)
W , W are defined in Theorem .. Proof By applying the conditions (.) to (.), we have v(x, y) ≤ k(x, y) +
∞
a(s, y) · v s + ρ(s), y ds
x
+
l ∞ ∞
bi (s, t, x, y)ϕ v s + ρi (s), t + γi (t)
i=
x
y
∞
∞
+ s
+
j=
v σ , η + γi (η) dξ dη ds dt
∞ ∞
dj (s, t, x, y)v s + ρj (s), t + γj (t)
M
∞ ∞
+ s
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
t
l
ci (ξ , η, x, y)ϕ
t
N
ej (ξ , η, x, y)
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
v σ , η + γj (η) dξ dη ds dt
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
≤ k(x, y) +
∞
Page 21 of 25
a α – (s), y v(s, y)
α(x)
+
l ∞ i=
αi (x)
∞ ∞
+ s
t
∞
βi (y)
ds α (α – (s))
bi αi– (s), βi– (t), x, y ϕ v(s, t)
ci αi– (ξ ), βi– (η), x, y ϕ max v(σ , η) σ ∈[ξ ,hξ ]
dξ dη – ds dt αi (αi– (ξ )) βi (βi– (η)) αi (αi (s)) βi (βi– (t)) l ∞ ∞ dj αj– (s), βj– (t), x, y v(s, t) + ×
j=
αj (M)
∞ ∞
+ s
t
βj (N)
ej αj– (ξ ), βj– (η), x, y
× max v(σ , η) σ ∈[ξ ,hξ ]
≤ k(x, y) +
∞
αj (αj– (ξ )) βj (βj– (η))
dξ dη
ds dt αj (αj– (s)) βj (βj– (t))
M a α – (s), y v(s, y) ds
α(x)
+
l ∞ i=
αi (x)
∞ ∞
+ s
+
j=
s
t
Mi Mi bi αi– (s), βi– (t), x, y ϕ v(s, t)
Mi Mi ci αi– (ξ ), βi– (η), x, y ϕ max v(σ , η) dξ dη ds dt
αj (M)
+
βi (y)
∞
∞
∞
σ ∈[ξ ,hξ ]
t
l
∞
∞ βj (N)
Mj Mj dj αj– (s), βj– (t), x, y v(s, t)
Mj Mj ej αj– (ξ ), βj– (η), x, y
× max v(σ , η) dξ dη ds dt, σ ∈[ξ ,hξ ]
(.)
for (x, y) ∈ , where M , Mi , Mi (i = , , . . . , l ), Mj , Mj (j = , , . . . , l ) are defined in (.). Applying the results of Theorem . to (.) with ψ(u) = u, a(s, y) = M a(α – (s), y), bi (s, t, x, y) = Mi Mi bi (αi– (s), βi– (t), x, y), ci (ξ , η, x, y) = Mi Mi ci (αi– (ξ ), βi– (η), x, y), ej (αj– (ξ ), βj– (η), x, y), we dj (s, t, x, y) = Mj Mj dj (αj– (s), βj– (t), x, y), ej (ξ , η, x, y) = Mj Mj
obtain the desired estimation (.).
Theorem . In equation (.), suppose that the following conditions hold: g (x, y) ≤ k(x, y), g (x, y) ≤ a(x, y), ψ v(x, y) = vp (x, y), Fi (s, t, x, y, u, v) ≤ bi (s, t, x, y)|u|qi + |v|, Fi (s, t, x, y, u) ≤ ci (s, t, x, y)|u|ri , i = , , . . . , l , Gj (s, t, x, y, u, v) ≤ dj (s, t, x, y)|u|εj + |v|, Gj (s, t, x, y, u) ≤ ej (s, t, x, y)|u|δj , j = , , . . . , l ,
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 22 of 25
where p, qi , ri , εj , δj , bi , ci , dj , ej (i = , , . . . , l ; j = , , . . . , l ) are defined as in Theorem .. Then we have the following estimate:
v(x, y) ≤
p B (M, N) exp F (x, y) A (x, y) , – D (M, N)
k(x, y) +
(.)
where
B (M, N) =
l i=
∞ αi (M)
∞
βi (N)
qpi Mi Mi bi αi– (s), βi– (t), M, N A (s, t)
qi qip–p p – qi qpi × K K k(s, t) + p p ∞ ∞ + Mi Mi ci αi– (ξ ), βi– (η), M, N s
t
ri rip–p p – ri rpi K dξ dη ds dt × A (ξ , η) K k(ξ , η) + p p l ∞ ∞ εpj + Mj Mj dj αj– (s), βj– (t), M, N A (s, t)
ri p
αj (M)
j=
βj (N)
ε p – εj pj εj k(s, t) + K K × p p ∞ ∞ + Mj Mj ej αj– (ξ ), βj– (η), M, N
εj –p p
s
t
δj p
× A (ξ , η) D (M, N) =
l
∞
(.)
∞
Mj Mj αj (M)
j=
δ –p δ δj jp p – δj pj K k(ξ , η) + K dξ dη ds dt, p p
βj (N)
ε –p εj εj j × dj αj– (s), βj– (t), M, N Ap (s, t) K p exp F (s, t) p ∞ ∞ Mj Mj ej αj– (ξ ), βj– (η), M, N + s
t
δ –p δj jp × A (ξ , η) K exp F (ξ , η) dξ dη ds dt p δj p
< , F (x, y) =
(.)
l i=
∞
αi (x)
s
t
qpi qi qip–p Mi Mi bi αi– (s), βi– (t), x, y A (s, t) K p βi (y)
∞ ∞
+
∞
Mi Mi ci αi– (ξ ), βi– (η), x, y
ri rip–p × A (ξ , η) K dξ dη ds dt. p ri p
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 23 of 25
Proof Applying the conditions of (.) to (.), we have v(x, y)p ≤ k(x, y) +
p a(s, y)v s + ρ(s), y ds
∞
x
+
l ∞ ∞
q bi (s, t, x, y)v s + ρi (s), t + γi (t) i
i=
x
y
∞
∞
+ s
+
j=
ε dj (s, t, x, y)v s + ρj (s), t + γj (t) j
M
N ∞
+
ej (ξ , η, x, y)
max
σ ∈[ξ +ρj (ξ ),h(ξ +ρj (ξ ))]
t
≤ k(x, y) +
p a α – (s), y v(s, y)
∞
α(x)
+
l i=
∞
αi (x)
∞
∞
+ s
t
∞
βi (y)
j=
∞
αj (M)
∞
∞
+ s
t
∞ βj (N)
αi (αi– (ξ ))
βi (βi– (η))
βj (βj– (t))
∞
αj (αj– (ξ )) βj (βj– (η))
αj (αj– (s))
dξ dη
αi (αi– (s))
βi (βi– (t))
ds dt
ej αj– (ξ ), βj– (η), x, y
σ ∈[ξ ,hξ ]
≤ k(x, y) +
ds
ε dj αj– (s), βj– (t), x, y v(s, t) j
δ × max v(σ , η) j ×
α (α – (s))
ci αi– (ξ ), βi– (η), x, y
σ ∈[ξ ,hξ ]
l
δj v σ , η + γj (η) dξ dη ds dt
q bi αi– (s), βi– (t), x, y v(s, t) i
r × max v(σ , η) i
+
ri v σ , η + γi (η) dξ dη ds dt
∞ ∞
∞
s
max
σ ∈[ξ +ρi (ξ ),h(ξ +ρi (ξ ))]
t
l
ci (ξ , η, x, y)
dξ dη
ds dt
p M a α – (s), y v(s, y) ds
α(x)
+
l i=
∞
αi (x)
∞
∞
+ s
+
j=
s
t
q Mi Mi bi αi– (s), βi– (t), x, y v(s, t) i
r Mi Mi ci αi– (ξ ), βi– (η), x, y max v(σ , η) i dξ dη ds dt
αj (M)
+
βi (y)
∞
∞
∞
σ ∈[ξ ,hξ ]
t
l
∞
∞ βj (N)
ε Mj Mj dj αj– (s), βj– (t), x, y v(s, t) j
Mj Mj ej αj– (ξ ), βj– (η), x, y
δ × max v(σ , η) j dξ dη ds dt, σ ∈[ξ ,hξ ]
(.)
Xu and Ma Journal of Inequalities and Applications (2017) 2017:187
Page 24 of 25
for (x, y) ∈ , where M , Mi , Mi (i = , , . . . , l ), Mj , Mj (j = , , . . . , l ) are defined in (.). Applying the results of Theorem . to (.) with a(s, y) = M a(α – (s), y), bi (s, t, x, y) = Mi Mi bi (αi– (s), βi– (t), x, y), ci (ξ , η, x, y) = Mi Mi ci (αi– (ξ ), βi– (η), x, y), – – – dj (s, t, x, y) = Mj Mj dj (αj (s), βj (t), x, y), ej (ξ , η, x, y) = Mj Mj ej (αj (ξ ), βj– (η), x, y), we obtain the desired estimation (.).
4 Conclusion In this paper, we established several new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables in Theorem . and Theorem ., and gave their specific cases in Corollary . and Corollary ., respectively, which can be used in the analysis of the qualitative properties to solutions of integral equations with maxima. In Theorem . and Theorem ., we also presented the applications to research the boundedness of solutions of retarded nonlinear Volterra-Fredholm type integral equations. Using our method, one can further study the integral inequality with more dimensions. Acknowledgements The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper. This research is supported by National Science Foundation of China (11671227). Competing interests The authors declare that there is no conflict of interest regarding the publication of this paper. Authors’ contributions RX proved parts of the results in Section 2 and participated in Section 3 - Applications. XM carried out the generalized weakly singular integral inequalities and completed part of the proof. All authors read and approved the final manuscript.
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