## 1 Introduction

Various generalizations of Gronwall inequality [1, 2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations, integral equations, and differential-integral equations. There are a lot of articles investigating its generalizations such as [323]. Recently, Pachpatte [19] provided the explicit estimations of following integral inequalities:

$\begin{array}{ll}\hfill {u}^{p}\left(t\right)& \le c+p\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({t}_{0}\right)}{\overset{{\alpha }_{i}\left(t\right)}{\int }}\left[{a}_{i}\left(s\right){u}^{p}\left(s\right)+{b}_{i}\left(s\right)u\left(s\right)\right]ds,\phantom{\rule{2em}{0ex}}\\ \hfill {u}^{p}\left(t\right)& \le c+p\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({t}_{0}\right)}{\overset{{\alpha }_{i}\left(t\right)}{\int }}\left[{a}_{i}\left(s\right)u\left(s\right)w\left(u\left(s\right)\right)+{b}_{i}\left(s\right)u\left(s\right)\right]ds,\phantom{\rule{2em}{0ex}}\end{array}$

and

$\begin{array}{ll}\hfill {u}^{p}\left(x,y\right)& \le c+p\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{a}_{i}\left(s,t\right){u}^{p}\left(s,t\right)+{b}_{i}\left(s,t\right)u\left(s,t\right)\right]dtds,\phantom{\rule{2em}{0ex}}\\ \hfill {u}^{p}\left(x,y\right)& \le c+p\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{a}_{i}\left(s,t\right)u\left(s,t\right)w\left(u\left(s,t\right)\right)+{b}_{i}\left(s,t\right)u\left(s,t\right)\right]dtds,\phantom{\rule{2em}{0ex}}\end{array}$

where c is a constant. Cheung [7] investigated the inequality

$\begin{array}{ll}\hfill {u}^{p}\left(x,y\right)& \le a+\frac{p}{p-q}\underset{{b}_{1}\left({x}_{0}\right)}{\overset{{b}_{1}\left(x\right)}{\int }}\underset{{c}_{1}\left({y}_{0}\right)}{\overset{{c}_{1}\left(y\right)}{\int }}{g}_{1}\left(s,t\right){u}^{q}\left(s,t\right)dtds\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\frac{p}{p-q}\underset{{b}_{2}\left({x}_{0}\right)}{\overset{{b}_{2}\left(x\right)}{\int }}\underset{{c}_{2}\left({y}_{0}\right)}{\overset{{c}_{2}\left(y\right)}{\int }}{g}_{2}\left(s,t\right){u}^{q}\left(s,t\right)\psi \left(u\left(s,t\right)\right)dtds.\phantom{\rule{2em}{0ex}}\end{array}$

Agarwal et al. [3] obtained the explicit bounds to the solutions of the following retarded integral inequalities:

$\begin{array}{ll}\hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({t}_{0}\right)}{\overset{{\alpha }_{i}\left(t\right)}{\int }}{u}^{q}\left(s\right)\left[{f}_{i}\left(s\right)\phi \left(u\left(s\right)\right)+{g}_{i}\left(s\right)\right]ds,\phantom{\rule{2em}{0ex}}\\ \hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({t}_{0}\right)}{\overset{{\alpha }_{i}\left(t\right)}{\int }}{u}^{q}\left(s\right)\left[{f}_{i}\left(s\right){\phi }_{1}\left(u\left(s\right)\right)+{g}_{i}\left(s\right){\phi }_{2}\left(\text{log}\left(u\left(s\right)\right)\right)\right]ds,\phantom{\rule{2em}{0ex}}\\ \hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({t}_{0}\right)}{\overset{{\alpha }_{i}\left(t\right)}{\int }}{u}^{q}\left(s\right)\left[{f}_{i}\left(s\right)L\left(s,u\left(s\right)\right)+{g}_{i}\left(s\right)u\left(s\right)\right]ds,\phantom{\rule{2em}{0ex}}\end{array}$

where c is a constant. Chen et al. [6] discussed the following inequalities:

$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\gamma \left({x}_{0}\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left({y}_{0}\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)\phi \left(u\left(s,t\right)\right)dtds,\phantom{\rule{2em}{0ex}}\\ \hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\alpha \left({x}_{0}\right)}{\overset{\alpha \left(x\right)}{\int }}\underset{\beta \left({y}_{0}\right)}{\overset{\beta \left(y\right)}{\int }}g\left(s,t\right)u\left(u,s\right)dtds\phantom{\rule{2em}{0ex}}\\ +\underset{\gamma \left({x}_{0}\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left({y}_{0}\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)u\left(s,t\right)\phi \left(u\left(s,t\right)\right)dtds,\phantom{\rule{2em}{0ex}}\\ \hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\alpha \left({x}_{0}\right)}{\overset{\alpha \left(x\right)}{\int }}\underset{\beta \left({y}_{0}\right)}{\overset{\beta \left(y\right)}{\int }}g\left(s,t\right)w\left(u\left(s,t\right)\right)dtds\phantom{\rule{2em}{0ex}}\\ +\underset{\gamma \left({x}_{0}\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left({y}_{0}\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)w\left(u\left(s,t\right)\right)\phi \left(u\left(s,t\right)\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$

where c is a constant.

In this article, motivated mainly by the works of Agarwal et al. [3] and Chen et al. [6], Cheung [7], Pachpatte [19], we discuss more general forms of following integral inequalities:

$\begin{array}{c}\psi \left(u\left(x,y\right)\right)\le a\left(x,y\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right)\phi \left(u\left(s,t\right)\right)\\ +{g}_{i}\left(s,t\right)\right]dtds,\end{array}$
(1.1)
$\begin{array}{c}\psi \left(u\left(x,y\right)\right)\le a\left(x,y\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right){\phi }_{1}\left(u\left(s,t\right)\right)\\ +{g}_{i}\left(s,t\right){\phi }_{2}\left(\mathrm{log}\left(u\left(s,t\right)\right)\right)\right]dtds,\end{array}$
(1.2)
$\begin{array}{c}\psi \left(u\left(x,y\right)\right)\le a\left(x,y\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right)L\left(s,t,u\left(s,t\right)\right)\\ +{g}_{i}\left(s,t\right)u\left(s,t\right)\right]dtds,\end{array}$
(1.3)

for (x, y) ∈ [x0, x1) × [y0, y1), where a(x, y), b(x, y) are nonnegative and nondecreasing functions in each variable. In inequalities (1.1)-(1.3), we generalized the constant c in [1, 5] to the function a(x,y), the function u(x) in [1] to the u(x,y) with two variables.

## 2 Main result

Throughout this article, x0, x1, y0, y1 ∈ ℝ are given numbers. I := [x0,x1), J := [y0,y1), Δ:= [x0,x1) × [y0,y1), ℝ+ := [0,∞). Consider (1.1)-(1.3), and suppose that

(H1) ψC(ℝ+, ℝ+) is a strictly increasing function with ψ(0) = 0 and ψ(t) → ∞ as t → ∞;

(H2) a, b: Δ → (0, ∞) are nondecreasing in each variable;

(H3) w, ϕ, ϕ1, ϕ2C(ℝ+,ℝ+) are nondecreasing with w(0) > 0, ϕ(r) > 0, ϕ1(r) > 0 and ϕ2(r) > 0 for r > 0;

(H4) α i C1(I,I) and β i C1(J,J) are nondecreasing such that α i (x) ≤ x, α i (x0) = x0, β i (y) ≤ y and β i (y0) = y0, i = 1, 2,..., n;

(H5) f i , g i C(Δ,ℝ+), i = 1,2,...,n.

Theorem 1. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.1). Then we have

$u\left(x,y\right)\le {\psi }^{-1}\left({W}^{-1}\left({\Phi }^{-1}\left(B\left(x,y\right)\right)\right)\right),$
(2.1)

for all (x,y) ∈ [x0,X1) × [y0,Y1), where

$W\left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{w\left({\psi }^{-1}\left(s\right)\right)},\phantom{\rule{1em}{0ex}}r>0,\phantom{\rule{1em}{0ex}}W\left(0\right):=\underset{r\to {0}^{+}}{\text{lim}}W\left(r\right),$
(2.2)
$\Phi \left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{\phi \left({\psi }^{-1}\left({W}^{-1}\left(s\right)\right)\right)},\phantom{\rule{1em}{0ex}}r>0,\phantom{\rule{1em}{0ex}}\Phi \left(0\right):=\underset{r\to {0}^{+}}{\text{lim}}\Phi \left(r\right),$
(2.3)
$B\left(x,y\right):=\Phi \left(A\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds,$
(2.4)
$A\left(x,y\right):=W\left(a\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds,$
(2.5)

ψ-1, W-1 and Φ-1 denote the inverse function of ψ, W and Φ, respectively, and (X1,Y1) ∈ Δ is arbitrarily given on the boundary of the planar region

$ℛ:=\left\{\left(x,y\right)\in \Delta :B\left(x,y\right)\in \text{Dom}\left({\Phi }^{-1}\right),{\Phi }^{-1}\left(B\left(x,y\right)\right)\in \text{Dom}\left({W}^{-1}\right)\right\}.$
(2.6)

Proof. From assumption H2 and the inequality (1.1), we have

$\psi \left(u\left(x,y\right)\right)\le a\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right)\phi \left(u\left(s,t\right)\right)+{g}_{i}\left(s,t\right)\right]dtds,$
(2.7)

for all (x,y) ∈ [x0,X] × [y0,y1), where x0XX1 is chosen arbitrarily. Define a function η(x, y) by the right-hand side of (2.7). Clearly, η(x, y) is a positive and nondecreasing function in each variable, η(x0,y) = a(X,y) > 0. Then, (2.7) is equivalent to

$u\left(x,y\right)\le {\psi }^{-1}\left(\eta \left(x,y\right)\right),$
(2.8)

for all (x,y) ∈ [x0,X] × [y0,y1). By the fact that α i (x) ≤ x for x ∈ [x0,x1), β i (y) ≤ y for y ∈ [y0,y1),i = 1,2,...,n, and the monotonicity of w,ψ-1,η, we have for all (x,y) ∈ [x0,X] × [y0,y1),

$\begin{array}{c}{\eta }_{x}\left(x,y\right)=b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }^{\prime }}_{i}\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left({\alpha }_{i}\left(x\right),t\right)\right)\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)\phi \left(u\left({\alpha }_{i}\left(x\right),t\right)\right)+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]dt\\ \le w\left({\psi }^{-1}\left(\eta \left(x,y\right)\right)\right)b\left(x,y\right)\sum _{i=1}^{n}{{\alpha }^{\prime }}_{i}\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)\phi \left({\psi }^{-1}\left(\eta \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\right]\\ +{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]dt.\end{array}$
(2.9)

From (2.9), we get

$\begin{array}{c}\frac{{\eta }_{x}\left(x,y\right)}{w\left({\psi }^{-1}\left(\eta \left(x,y\right)\right)\right)}\le b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }^{\prime }}_{i}\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)\phi \left({\psi }^{-1}\left(\eta \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\\ +{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]dt,\end{array}$
(2.10)

for all (x,y) ∈ [x0,X] × [y0,y1). Integrating (2.10) from x0 to x, by the definition of W in (2.2), we get for all (x,y) ∈ [x0,X] × [y0,y1),

$\begin{array}{ll}\hfill W\left(\eta \left(x,y\right)\right)& \le W\left(\eta \left({x}_{0},y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)+{g}_{i}\left(s,t\right)\right]dtds\phantom{\rule{2em}{0ex}}\\ =W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)+{g}_{i}\left(s,t\right)\right]dtds\phantom{\rule{2em}{0ex}}\\ \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,Y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)dtds\phantom{\rule{2em}{0ex}}\\ =c\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.11)

where

$c\left(X,y\right)=W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds.$
(2.12)

Now, define a function Γ(x,y) by the right-hand side of (2.11). Clearly, Γ(x,y) is a positive and nondecreasing function in each variable, Γ(x0,y) = c(X, y) > 0. then, (2.11) is equivalent to

$\eta \left(x,y\right)\le {W}^{-1}\left(\Gamma \left(x,y\right)\right),$
(2.13)

for all (x,y) ∈ [x0,X] × [y0,Y1), where Y1 is defined in (2.6). By the fact that α i (x) ≤ x for x ∈ [x0,x1), β i (y) ≤ y for y ∈ [y0,y1), i = 1, 2,...,n, and the monotonicity of ϕ, ψ-1, W-1, Γ, we have for all (x,y) ∈ [x0,X] × [y0,Y1),

$\begin{array}{ll}\hfill {\Gamma }_{x}\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)\phi \left({\psi }^{-1}\left(\eta \left({\alpha }_{i}\left(x\right),t\right)\right)\right)dt\phantom{\rule{2em}{0ex}}\\ \le b\left(X,y\right)\phi \left({\psi }^{-1}\left({W}^{-1}\left(\Gamma \left(x,y\right)\right)\right)\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)dt.\phantom{\rule{2em}{0ex}}\end{array}$
(2.14)

From (2.14), we have for all (x,y) ∈ [x0,X] × [y0,Y1),

$\frac{{\Gamma }_{x}\left(x,y\right)}{\phi \left({\psi }^{-1}\left({W}^{-1}\left(\Gamma \left(x,y\right)\right)\right)\right)}\le b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)dt.$
(2.15)

Integrating (2.15) from x0 to x, by the definition of Φ in (2.3), we get

$\begin{array}{ll}\hfill \Phi \left(\Gamma \left(x,y\right)\right)& \le \Phi \left(\Gamma \left({x}_{0}y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds\phantom{\rule{2em}{0ex}}\\ =\Phi \left(c\left(X,Y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.16)

for all (x,y) ∈ [x0,X] × [y0,Y1). From (2.12) and (2.16), we find

$\begin{array}{ll}\hfill \Gamma \left(x,y\right)& \le {\Phi }^{-1}\left(\Phi \left(c\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds\right)\phantom{\rule{2em}{0ex}}\\ ={\Phi }^{-1}\left(\Phi \left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds),\phantom{\rule{2em}{0ex}}\end{array}$
(2.17)

for all (x, y) ∈ [x0, X] × [y0, Y1). From (2.8), (2.13), and (2.17), we get

$\begin{array}{ll}\hfill u\left(x,y\right)& \le {\psi }^{-1}\left(\eta \left(x,y\right)\right)\le {\psi }^{-1}\left({W}^{-1}\left(\Gamma \left(x,y\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ \le {\psi }^{-1}\left({W}^{-1}\left({\Phi }^{-1}\left(\Phi \right\right\right\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds))),\phantom{\rule{2em}{0ex}}\end{array}$
(2.18)

for all (x, y) ∈ [x0,X] × [y0,Y1). Let x = X, from (2.18), we observe that

$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left({W}^{-1}\left({\Phi }^{-1}\left(\Phi \right\right\right\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds|\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds))),\phantom{\rule{2em}{0ex}}\end{array}$
(2.19)

for all (X, y) ∈ [x0, X1) × [y0, Y1), where X1 is defined by (2.6). Since X ∈ [x0, X1) is arbitrary, from (2.19), we get the required estimations

$\begin{array}{ll}\hfill u\left(x,y\right)& \le {\psi }^{-1}\left({W}^{-1}\left({\Phi }^{-1}\left(\Phi \right\right\right\left(W\left(a\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{g}_{i}\left(s,t\right)dtds\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)dtds))),\phantom{\rule{2em}{0ex}}\end{array}$

for all (x,y) ∈ [x0,X1) × [y0,Y1). Theorem 1 is proved.

Remark that Theorem 1 generalizes Theorem 2.1 in [3].

Theorem 2. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.2). Then

(i) if ϕ1(u) ≥ ϕ2(log(u)), we have

$u\left(x,y\right)\le {\psi }^{-1}\left[{W}^{-1}\left({\psi }_{1}^{-1}\left({D}_{1}\left(x,y\right)\right)\right)\right],$
(2.20)

for all (x,y) ∈ [x0,X2) × [y0,Y2),

(ii) if ϕ1(u) < ϕ2(log(u)), we have

$u\left(x,y\right)\le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{2}^{-1}\left({D}_{2}\left(x,y\right)\right)\right)\right],$
(2.21)

for all (x,y) ∈ [x0,X3) × [y0,Y3), where W is defined by (2.2) in Theorem 1,

$\begin{array}{ll}\hfill {D}_{j}\left(x,y\right)& :={\Psi }_{j}\left(W\left(a\left(x,y\right)\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds,\phantom{\rule{2em}{0ex}}\\ \hfill {\Psi }_{j}\left(r\right)& :=\underset{1}{\overset{r}{\int }}\frac{ds}{{\phi }_{j}\left({\psi }^{-1}\left({W}^{-1}\left(s\right)\right)\right)},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\Psi }_{j}\left(0\right):=\underset{r\to 0+}{\text{lim}}{\Psi }_{j}\left(r\right),\phantom{\rule{2em}{0ex}}\end{array}$
(2.22)

j = 1, 2, ψ-1, W-1, ${\Psi }_{1}^{-1}$ and ${\Psi }_{2}^{-1}$ denote the inverse function of ψ, W, Ψ1 and Ψ2, respectively, (X2,Y2) is arbitrarily given on the boundary of the planar region

${ℛ}_{1}:=\left\{\left(x,y\right)\in \Delta :{D}_{1}\left(x,y\right)\in \text{Dom}\left({\Psi }_{1}^{-1}\right),{\Psi }_{1}^{-1}\left({D}_{1}\left(x,y\right)\right)\in \text{Dom}\left({W}^{-1}\right)\right\},$
(2.23)

and (X3,Y3) is arbitrarily given on the boundary of the planar region

${ℛ}_{2}:=\left\{\left(x,y\right)\in \Delta :{D}_{2}\left(x,y\right)\in \text{Dom}\left({\Psi }_{2}^{-1}\right),{\Psi }_{2}^{-1}\left({D}_{2}\left(x,y\right)\right)\in \text{Dom}\left({W}^{-1}\right)\right\}.$
(2.24)

Proof. From the inequality (1.2), we have

$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le a\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right){\phi }_{1}\left(u\left(s,t\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right){\phi }_{2}\left(\text{log}\left(u\left(s,t\right)\right)\right)]dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.25)

for all (x,y) ∈ [x0,X] × [y0,y1), where x0XX2 is chosen arbitrarily. Let Ξ(x,y) denote the right-hand side of (2.25), which is a positive and nondecreasing function in each variable, Ξ(x0,y) = a(X,y). Then, (2.25) is equivalent to u(x,y) ≤ ψ-1(Ξ(x,y)). By the fact that α i (x) ≤ x for x ∈ [x0, x1), β i (y) ≤ y for y ∈ [y0, y1), i = 1, 2,..., n, and the monotonicity of w,ψ-1,Ξ, we have for all (x,y) ∈ [x0,X] × [y0,y1),

$\begin{array}{ll}\hfill {\Xi }_{x}\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left({\alpha }_{i}\left(s\right),t\right)\right)\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right{\phi }_{1}\left(u\left({\alpha }_{i}\left(x\right),t\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right){\phi }_{2}\left(\text{log}\left(u\left({\alpha }_{i}\left(x\right),t\right)\right)\right)]dt\phantom{\rule{2em}{0ex}}\\ \le b\left(X,y\right)w\left({\psi }^{-1}\left(\Xi \left(x,y\right)\right)\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right){\phi }_{1}\left({\psi }^{-1}\left(\Xi \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right){\phi }_{2}\left(\text{log}\left({\psi }^{-1}\left(\Xi \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\right)]dt,\phantom{\rule{2em}{0ex}}\end{array}$
(2.26)

for all (x,y) ∈ [x0,X] × [y0,y1). From (2.26), we have

$\begin{array}{ll}\hfill \frac{{\Xi }_{x}\left(x,y\right)}{w\left({\psi }^{-1}\left(\Xi \left(x,y\right)\right)\right)}& \le b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right){\phi }_{1}\left({\psi }^{-1}\left(\Xi \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right){\phi }_{2}\left(\text{log}\left({\psi }^{-1}\left(\Xi \left({\alpha }_{i}\left(x\right),t\right)\right)\right)\right)]dt,\phantom{\rule{2em}{0ex}}\end{array}$
(2.27)

for all (x,y) ∈ [x0,X] × [y0,y1). Integrating (2.27) from x0 to x, by the definition of W in (2.2), we get

$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(\Xi \left({x}_{0},y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right){\phi }_{1}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right){\phi }_{2}\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)]dtds\phantom{\rule{2em}{0ex}}\\ =W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\right{\phi }_{1}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right){\phi }_{2}\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)]dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.28)

for all (x,y) ∈ [x0,X] × [y0,y1).

When ϕ1(u) ≥ ϕ2(log(u)), from the inequality (2.28), we have

$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right)]{\phi }_{1}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.29)

for all (x,y) ∈ [x0,X] × [y0,y1). Now, define a function Θ(x,y) by the right-hand side of (2.29). Clearly, Θ(x,y) is a positive and nondecreasing function in each variable, Θ(x0,y) = W(a(X,y)) > 0. Then, (2.29) is equivalent to

$\Xi \left(x,y\right)\le {W}^{-1}\left(\Theta \left(x,y\right)\right),\phantom{\rule{1em}{0ex}}\forall \left(x,y\right)\in \left[{x}_{0},X\right]×\left[{y}_{0},{Y}_{2}\right),$
(2.30)

where Y2 is defined by (2.23). Differentiating Θ(x,y) in x for any fixed y ∈ [y0,Y2), we have

$\begin{array}{ll}\hfill {\Theta }_{x}\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]{\phi }_{1}\left({\psi }^{-1}\left(\Xi \left({\alpha }_{i}\left(x\right),t\right)\right)\right)dt\phantom{\rule{2em}{0ex}}\\ \le b\left(X,y\right){\phi }_{1}\left({\psi }^{-1}\left({W}^{-1}\left(\Theta \left(x,y\right)\right)\right)\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]dt,\phantom{\rule{2em}{0ex}}\end{array}$
(2.31)

for all (x,y) ∈ [x0,X] × [y0,Y2). From (2.31), we have

$\frac{{\Theta }_{x}\left(x,y\right)}{{\phi }_{1}\left({\psi }^{-1}\left({W}^{-1}\left(\Theta \left(x,y\right)\right)\right)\right)}\le b\left(X,y\right)\sum _{i=1}^{n}{{\alpha }_{i}}^{\prime }\left(x\right)\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left({\alpha }_{i}\left(x\right),t\right)+{g}_{i}\left({\alpha }_{i}\left(x\right),t\right)\right]dt,$
(2.32)

for all (x,y) ∈ [x0,X] × [y0,Y2). Integrating (2.32) from x0 to x, by the definition of Ψ1 in (2.22), we obtain

$\begin{array}{ll}\hfill {\Psi }_{1}\left(\Theta \left(x,y\right)\right)& \le {\Psi }_{1}\left(\Theta \left({x}_{0},y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds\phantom{\rule{2em}{0ex}}\\ ={\Psi }_{1}\left(W\left(a\left(X,y\right)\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds.\phantom{\rule{2em}{0ex}}\end{array}$
(2.33)

From (2.30) and (2.33), we conclude

$\begin{array}{c}u\left(x,y\right)\le {\psi }^{-1}\left(\Xi \left(x,y\right)\right)\le {\psi }^{-1}\left({W}^{-1}\left(\Theta \left(x,y\right)\right)\right)\le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{1}^{-1}\left(\right\right\right\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\Psi }_{1}\left(W\left(a\left(X,y\right)\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}{\left(}_{x}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds))],\end{array}$
(2.34)

for all (x,y) ∈ [x0,X] × [y0,Y2). Let x = X, from (2.34), we get

$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{1}^{-1}\left({\Psi }_{1}\left(W\left(a\left(X,y\right)\right)\right)\right\right\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds))].\phantom{\rule{2em}{0ex}}\end{array}$
(2.35)

Since X ∈ [x0,X2) is arbitrary, from the inequality (2.35), we obtain the required inequality in (2.20).

When ϕ1(u) ≤ ϕ2(log(u)), from the inequality (2.28), we have

$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right)]{\phi }_{2}\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)dtds,\phantom{\rule{2em}{0ex}}\\ \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right)]{\phi }_{2}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.36)

for all (x,y) ∈ [x0,X] × [y0,y1), where x0XX3. Similarly to the above process from (2.29) to (2.35), from (2.36), we obtain

$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{2}^{-1}\left({\Psi }_{2}\left(W\left(a\left(X,y\right)\right)\right)\right\right\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)+{g}_{i}\left(s,t\right)\right]dtds))].\phantom{\rule{2em}{0ex}}\end{array}$
(2.37)

Since X ∈ [x0,X3) is arbitrary, where X3 is defined by (2.24), from the inequality (2.37), we obtain the required inequality in (2.21).

Theorem 3. Suppose that (H1-H5) hold and that L,$M\in C\left({ℝ}_{+}^{3},{ℝ}_{+}\right)$ satisfy

$0\le L\left(s,t,u\right)-L\left(s,t,v\right)\le M\left(s,t,v\right)\left(u-v\right),$
(2.38)

for s, t, u, v ∈ ℝ+ with u > v ≥ 0. If u(x,y) is a nonnegative and continuous function on Δ satisfying (1.3), then we have

$u\left(x,y\right)\le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{3}^{-1}\left(E\left(x,y\right)\right)\right)\right],$
(2.39)

for all (x,y) ∈ [x0,X4) × [y0,Y4), where W is defined by (2.2),

${\Psi }_{3}\left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{{\psi }^{-1}\left({W}^{-1}\left(s\right)\right)},\phantom{\rule{1em}{0ex}}r>0,\phantom{\rule{1em}{0ex}}{\Psi }_{3}\left(0\right):=\underset{r\to 0+}{\text{lim}}{\Psi }_{3}\left(r\right),$
(2.40)
$E\left(x,y\right):={\Psi }_{3}\left(F\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{i=1}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)M\left(s,t,0\right)+{g}_{i}\left(s,t\right)\right]dtds,$
$F\left(x,y\right):=W\left(a\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)L\left(s,t,0\right)dtds,$

ψ-1,W-1 and ${\Psi }_{3}^{-1}$ denote the inverse function of ψ, W and Ψ3, respectively, and (X4,Y4) ∈ Δ is arbitrarily given on the boundary of the planar region

$ℛ:=\left\{\left(x,y\right)\in \Delta :E\left(x,y\right)\in \text{Dom}\left({\Psi }_{3}^{-1}\right),{\Psi }_{3}^{-1}\left(E\left(x,y\right)\right)\in \text{Dom}\left({W}^{-1}\right)\right\}.$
(2.41)

Proof. From the inequality (1.3), we have

$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le a\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left(x0\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[{f}_{i}\left(s,t\right)L\left(s,t,u\left(s,t\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right)u\left(s,t\right)]dtds,\phantom{\rule{2em}{0ex}}\end{array}$
(2.42)

for all (x,y) ∈ [x0,X] × [y0,y1), where x0XX4 is chosen arbitrarily. Let P(x,y) denote the right-hand side of (2.42), which is a positive and nondecreasing function in each variable, P(x0,y) = a(X,y). Similarly to the process in the proof of Theorem 2 from (2.25) to (2.28), we obtain

$\begin{array}{ll}\hfill W\left(P\left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)L\left(s,t,{\psi }^{-1}\left(P\left(s,t\right)\right)\right)\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{g}_{i}\left(s,t\right){\psi }^{-1}\left(P\left(s,t\right)\right)]dtds,\phantom{\rule{1em}{0ex}}\forall \left(x,y\right)\in \left[{x}_{0}X\right]×\left[{y}_{0},{y}_{1}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(2.43)

From the inequality (2.38) and (2.43), we get

$\begin{array}{ll}\hfill W\left(P\left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)L\left(s,t,0\right)dtds\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)M\left(s,t,0\right)+{g}_{i}\left(s,t\right)\right]{\psi }^{-1}\left(P\left(s,t\right)\right)dtds,\phantom{\rule{2em}{0ex}}\end{array}$

for all (x, y) ∈ [x0,X] × [y0,y1). Similarly to the process in the proof of Theorem 2 from (2.29) to (2.35), we obtain

$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[{W}^{-1}\left({\Psi }_{3}^{-1}\left({\Psi }_{3}\right\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}{f}_{i}\left(s,t\right)L\left(s,t,0\right)dtds\right)\right\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+b\left(X,y\right)\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(X\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\left[{f}_{i}\left(s,t\right)M\left(s,t,0\right)+{g}_{i}\left(s,t\right)\right]dtds))],\phantom{\rule{2em}{0ex}}\end{array}$
(2.44)

where Ψ3 is defined by (2.40). Since X ∈ [x0,X4) is arbitrary, where X4 is defined by (2.41), from the inequality (2.44), we obtain the required inequality in (2.39).

## 3 Applications to BVP

In this section we use our result to study certain properties of solution of the following boundary value problem (simply called BVP):

$\left\{\begin{array}{c}\frac{{\partial }^{2}\psi \left(z\left(x,y\right)\right)}{\partial x\partial y}=F\left(x,y,z\left({\alpha }_{1}\left(x\right),{\beta }_{1}\left(y\right)\right),z\left({\alpha }_{2}\left(x\right),{\beta }_{2}\left(y\right)\right),...,z\left({\alpha }_{n}\left(x\right),{\beta }_{n}\left(y\right)\right)\right),\hfill \\ z\left(x,{y}_{0}\right)={a}_{1}\left(x\right),\phantom{\rule{1em}{0ex}}z\left({x}_{0},y\right)={a}_{2}\left(y\right),{a}_{1}\left({x}_{0}\right)={a}_{2}\left({y}_{0}\right)=0,\hfill \end{array}\right\$
(3.1)

for xI,yJ, where x0,y0,x1,y1 ∈ ℝ+ are constants, I := [x0,x1), J := [y0,y1), FC(I × J × ℝn,ℝ), ψ: ℝ → ℝ is strictly increasing on ℝ+ with ψ(0) = 0, |ψ(r)| = ψ(|r|) > 0, for |r| > 0 and ψ(t) → ∞ as t → ∞; functions α i C1(I,I);β i C1(J,J),i = 1,2,...,n are nondecreasing such that α i (x) ≤ x, β i (y) ≤ y,α i (x0) = x0, β i (y0) = y0; |a1| ∈ C1(I,ℝ+), |a2| ∈ C1(J,ℝ+) are both nondecreasing. Though this equation is similar to the equation discussed in Section 3 in [3], our results are more general than the results obtained in [3].

We first give an estimate for solutions of the BVP (3.1) so as to obtain a condition for boundedness.

Corollary 1. Consider BVP (3.1) and suppose that FC(I × J × ℝn,ℝ) satisfies

$\left|F\left(x,y,{u}_{1},{u}_{2},...,{u}_{n}\right)\right|\le \sum _{i=1}^{n}w\left(\left|{u}_{i}\right|\right)\left[{f}_{i}\left(x,y\right)\phi \left(\left|{u}_{i}\right|\right)+{g}_{i}\left(x,y\right)\right],\phantom{\rule{1em}{0ex}}\left(x,y\right)\in I×J,$
(3.2)

where f i ,g i C(I × J,ℝ+) and w,ϕC(ℝ+,ℝ+) are nondecreasing such that w(u) > 0,ϕ(u) > 0 for u > 0. Then all solutions z(x,y) of BVP (3.1) have the estimate

$\left|z\left(x,y\right)\right|\le {\psi }^{-1}\left({W}^{-1}\left({\Phi }^{-1}\left(B\left(x,y\right)\right)\right)\right),$
(3.3)

for all (x,y) ∈ [x0,X1) × [y0,Y1), where

$\begin{array}{ll}\hfill B\left(x,y\right)& :=\Phi \left(A\left(x,y\right)\right)+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\frac{{f}_{i}\left({\alpha }_{i}^{-1}\left(s\right),{\beta }_{i}^{-1}\left(t\right)\right)}{{{\alpha }_{i}}^{\prime }\left({\alpha }_{i}^{-1}\left(s\right)\right){{\beta }_{i}}^{\prime }\left({\beta }_{i}^{-1}\left(t\right)\right)}dtds,\phantom{\rule{2em}{0ex}}\\ \hfill A\left(x,y\right)& :=W\left(\psi \left(\left|{a}_{1}\left(x\right)\right|\right)+\psi \left(\left|{a}_{2}\left(y\right)\right|\right)\right)+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}^{-1}}{\int }}\frac{{g}_{i}\left({\alpha }_{i}^{-1}\left(s\right),{\beta }_{i}^{-1}\left(t\right)\right)}{{{\alpha }_{i}}^{\prime }\left({\alpha }_{i}^{-1}\left(s\right)\right){{\beta }_{i}}^{\prime }\left({\beta }_{i}^{-1}\left(t\right)\right)}dtds,\phantom{\rule{2em}{0ex}}\end{array}$

for all (x,y) ∈ [x0,X1) × [y0,Y1), where functions W, W-1, Φ, Φ-1 and real numbers X1, Y1 are given as in Theorem 1.

Proof. The equivalent integral equation of BVP (3.1) is

$\begin{array}{ll}\hfill \psi \left(z\left(x,y\right)\right)& =\psi \left({a}_{1}\left(x\right)\right)+\psi \left({a}_{2}\left(y\right)\right)+\underset{{x}_{0}}{\overset{x}{\int }}\underset{{y}_{0}}{\overset{y}{\int }}F\left(s,t,z\left({\alpha }_{1}\left(s\right),{\beta }_{1}\left(t\right)\right),z\left({\alpha }_{2}\left(s\right),{\beta }_{2}\left(t\right)\right)\right,...,\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}z\left({\alpha }_{n}\left(s\right),{\beta }_{n}\left(t\right)\right))dtds.\phantom{\rule{2em}{0ex}}\end{array}$
(3.4)

By (3.2) and (3.4), we get that

$\begin{array}{l}\psi \left(\left|z\left(x,y\right)\right|\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\le \psi \left(\left|{a}_{1}\left(x\right)\right|\right)+\psi \left(\left|{a}_{2}\left(y\right)\right|\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\underset{{x}_{0}}{\overset{x}{\int }}\underset{{y}_{0}}{\overset{y}{\int }}\left|F\left(s,t,z\left({\alpha }_{1}\left(s\right),{\beta }_{1}\left(t\right)\right),z\left({\alpha }_{2}\left(s\right),{\beta }_{2}\left(t\right)\right),...,z\left({\alpha }_{2}\left(s\right),{\beta }_{n}\left(t\right)\right)\right)\right|dtds\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\le \psi \left(\left|{a}_{1}\left(x\right)\right|\right)+\psi \left(\left|{a}_{2}\left(y\right)\right|\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\underset{{x}_{0}}{\overset{x}{\int }}\underset{{y}_{0}}{\overset{y}{\int }}\sum _{i=1}^{n}w\left(\left|z\left({\alpha }_{i}\left(s\right),{\beta }_{i}\left(t\right)\right)\right|\right)\left[{f}_{i}\left(s,t\right)\phi \left(\left|z\left({\alpha }_{i}\left(s\right),{\beta }_{i}\left(t\right)\right)\right|\right)+{g}_{i}\left(s,t\right)\right]dtds\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}=\psi \left(\left|{a}_{1}\left(x\right)\right|\right)+\psi \left(\left|{a}_{2}\left(y\right)\right|\right)+\sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\underset{{\beta }_{i}\left({y}_{0}\right)}{\overset{{\beta }_{i}\left(y\right)}{\int }}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\frac{w\left(\left|z\left({s}_{1},{t}_{1}\right)\right|\right)\left[{f}_{i}\left({\alpha }_{i}^{-1}\left({s}_{1}\right),{\beta }_{i}^{-1}\left({t}_{1}\right)\right)\phi \left(\left|z\left({s}_{1},{t}_{1}\right)\right|\right)+{g}_{i}\left({\alpha }_{i}^{-1}\left({s}_{1}\right),{\beta }_{i}^{-1}\left({t}_{1}\right)\right)\right]}{{{\alpha }_{i}}^{\prime }\left({\alpha }_{i}^{-1}\left({s}_{1}\right)\right){{\beta }_{i}}^{\prime }\left({\beta }_{i}^{-1}\left({t}_{1}\right)\right)}d{t}_{1}d{s}_{1},\phantom{\rule{2em}{0ex}}\end{array}$
(3.5)

where a change of variables s1 = α i (s), t1 = β i (t),i = 1,2,...,n are made. Clearly, the inequality (3.5) is in the form of (1.1). Thus the estimate (3.3) of the solution z(x,y) in this corollary is obtained immediately by our Theorem 1.

Our Corollary 1 actually gives a condition of boundedness for solutions. Concretely, if

$\begin{array}{c}\psi \left(\left|{a}_{1}\left(x\right)\right|\right)+\psi \left(\left|{a}_{2}\left(y\right)\right|\right)<\infty ,\\ \sum _{i=1}^{n}\underset{{\alpha }_{i}\left({x}_{0}\right)}{\overset{{\alpha }_{i}\left(x\right)}{\int }}\end{array}$