Some generalized nonlinear retarded integral inequalities with applications

Abstract

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions.

Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.

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 [3–23]. Recently, Pachpatte [19] provided the explicit estimations of following integral inequalities:

$$\begin{array}{ll}\hfill u^p\left(t\right)& \le c+p\sum _{i=1}^n\underset{{\alpha }_i\left(t_0\right)}{\overset{{\alpha }_i\left(t\right)}{\int }}\left[a_i\left(s\right)u^p\left(s\right)+b_i\left(s\right)u\left(s\right)\right]ds,\\ \hfill u^p\left(t\right)& \le c+p\sum _{i=1}^n\underset{{\alpha }_i\left(t_0\right)}{\overset{{\alpha }_i\left(t\right)}{\int }}\left[a_i\left(s\right)u\left(s\right)w\left(u\left(s\right)\right)+b_i\left(s\right)u\left(s\right)\right]ds,\end{array}$$

and

$$\begin{array}{ll}\hfill u^p\left(x,y\right)& \le c+p\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[a_i\left(s,t\right)u^p\left(s,t\right)+b_i\left(s,t\right)u\left(s,t\right)\right]dtds,\\ \hfill u^p\left(x,y\right)& \le c+p\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[a_i\left(s,t\right)u\left(s,t\right)w\left(u\left(s,t\right)\right)+b_i\left(s,t\right)u\left(s,t\right)\right]dtds,\end{array}$$

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

$$\begin{array}{ll}\hfill u^p\left(x,y\right)& \le a+\frac{p}{p-q}\underset{b_1\left(x_0\right)}{\overset{b_1\left(x\right)}{\int }}\underset{c_1\left(y_0\right)}{\overset{c_1\left(y\right)}{\int }}{g}_1\left(s,t\right)u^q\left(s,t\right)dtds\\ +\frac{p}{p-q}\underset{b_2\left(x_0\right)}{\overset{b_2\left(x\right)}{\int }}\underset{c_2\left(y_0\right)}{\overset{c_2\left(y\right)}{\int }}{g}_2\left(s,t\right)u^q\left(s,t\right)\psi \left(u\left(s,t\right)\right)dtds.\end{array}$$

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

$$\begin{array}{ll}\hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^n\underset{{\alpha }_i\left(t_0\right)}{\overset{{\alpha }_i\left(t\right)}{\int }}{u}^q\left(s\right)\left[f_i\left(s\right)\phi \left(u\left(s\right)\right)+g_i\left(s\right)\right]ds,\\ \hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^n\underset{{\alpha }_i\left(t_0\right)}{\overset{{\alpha }_i\left(t\right)}{\int }}{u}^q\left(s\right)\left[f_i\left(s\right){\phi }_1\left(u\left(s\right)\right)+g_i\left(s\right){\phi }_2\left(\text{log}\left(u\left(s\right)\right)\right)\right]ds,\\ \hfill \phi \left(u\left(t\right)\right)& \le c+\sum _{i=1}^n\underset{{\alpha }_i\left(t_0\right)}{\overset{{\alpha }_i\left(t\right)}{\int }}{u}^q\left(s\right)\left[f_i\left(s\right)L\left(s,u\left(s\right)\right)+g_i\left(s\right)u\left(s\right)\right]ds,\end{array}$$

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

$$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\gamma \left(x_0\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left(y_0\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)\phi \left(u\left(s,t\right)\right)dtds,\\ \hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\alpha \left(x_0\right)}{\overset{\alpha \left(x\right)}{\int }}\underset{\beta \left(y_0\right)}{\overset{\beta \left(y\right)}{\int }}g\left(s,t\right)u\left(u,s\right)dtds\\ +\underset{\gamma \left(x_0\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left(y_0\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)u\left(s,t\right)\phi \left(u\left(s,t\right)\right)dtds,\\ \hfill \psi \left(u\left(x,y\right)\right)& \le c+\underset{\alpha \left(x_0\right)}{\overset{\alpha \left(x\right)}{\int }}\underset{\beta \left(y_0\right)}{\overset{\beta \left(y\right)}{\int }}g\left(s,t\right)w\left(u\left(s,t\right)\right)dtds\\ +\underset{\gamma \left(x_0\right)}{\overset{\gamma \left(x\right)}{\int }}\underset{\delta \left(y_0\right)}{\overset{\delta \left(y\right)}{\int }}f\left(s,t\right)w\left(u\left(s,t\right)\right)\phi \left(u\left(s,t\right)\right)dtds,\end{array}$$

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:

$$\begin{array}{c}\psi (u(x,y))\le a(x,y)+b(x,y){\displaystyle \sum _{i=1}^n{\displaystyle \underset{{\alpha }_i(x_0)}{\overset{{\alpha }_i(x)}{\int }}{\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}w(u(s,t))[f_i(s,t)\phi (u(s,t))}}}\\ +g_i(s,t)]dtds,\end{array}$$

(1.1)

$$\begin{array}{c}\psi (u(x,y))\le a(x,y)+b(x,y){\displaystyle \sum _{i=1}^n{\displaystyle \underset{{\alpha }_i(x_0)}{\overset{{\alpha }_i(x)}{\int }}{\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}w(u(s,t))[f_i(s,t){\phi }_1(u(s,t))}}}\\ +g_i(s,t){\phi }_2(\log (u(s,t)))]dtds,\end{array}$$

(1.2)

$$\begin{array}{c}\psi (u(x,y))\le a(x,y)+b(x,y){\displaystyle \sum _{i=1}^n{\displaystyle \underset{{\alpha }_i(x_0)}{\overset{{\alpha }_i(x)}{\int }}{\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}w(u(s,t))[f_i(s,t)L(s,t,u(s,t))}}}\\ +g_i(s,t)u(s,t)]dtds,\end{array}$$

(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) ψ ∈ C(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, ϕ₂ ∈ C(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

$$W\left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{w\left({\psi }^{-1}\left(s\right)\right)},r>0,W\left(0\right):=\underset{r\to 0^{+}}{\text{lim}}W\left(r\right),$$

(2.2)

$$\Phi \left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{\phi \left({\psi }^{-1}\left(W^{-1}\left(s\right)\right)\right)},r>0,\Phi \left(0\right):=\underset{r\to 0^{+}}{\text{lim}}\Phi \left(r\right),$$

(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 x₀ ≤ X ≤ X₁ 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),

$$\begin{array}{c}{\eta }_x(x,y)=b(X,y){\displaystyle \sum _{i=1}^n{{\alpha }^{\prime }}_i}(x){\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}w(u({\alpha }_i(x),t))[f_i({\alpha }_i(x),t)\phi (u({\alpha }_i(x),t))+g_i({\alpha }_i(x),t)]dt}\\ \le w({\psi }^{-1}(\eta (x,y)))b(x,y){\displaystyle \sum _{i=1}^n{{\alpha }^{\prime }}_i}(x){\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}[f_i({\alpha }_i(x),t)\phi ({\psi }^{-1}(\eta ({\alpha }_i(x),t)))]}\\ +g_i({\alpha }_i(x),t)]dt.\end{array}$$

(2.9)

From (2.9), we get

$$\begin{array}{c}\frac{{\eta }_x(x,y)}{w({\psi }^{-1}(\eta (x,y)))}\le b(X,y){\displaystyle \sum _{i=1}^n{{\alpha }^{\prime }}_i}(x){\displaystyle \underset{{\beta }_i(y_0)}{\overset{{\beta }_i(y)}{\int }}[f_i({\alpha }_i(x),t)\phi ({\psi }^{-1}(\eta ({\alpha }_i(x),t)))}\\ +g_i({\alpha }_i(x),t)]dt,\end{array}$$

(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),

$$\begin{array}{ll}\hfill W\left(\eta \left(x,y\right)\right)& \le W\left(\eta \left(x_0,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)+g_i\left(s,t\right)\right]dtds\\ =W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)+g_i\left(s,t\right)\right]dtds\\ \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{g}_i\left(s,t\right)dtds\\ +b\left(X,Y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)dtds\\ =c\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)\phi \left({\psi }^{-1}\left(\eta \left(s,t\right)\right)\right)dtds,\end{array}$$

(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),

$$\begin{array}{ll}\hfill {\Gamma }_x\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left({\alpha }_i\left(x\right),t\right)\phi \left({\psi }^{-1}\left(\eta \left({\alpha }_i\left(x\right),t\right)\right)\right)dt\\ \le b\left(X,y\right)\phi \left({\psi }^{-1}\left(W^{-1}\left(\Gamma \left(x,y\right)\right)\right)\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left({\alpha }_i\left(x\right),t\right)dt.\end{array}$$

(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

$$\begin{array}{ll}\hfill \Phi \left(\Gamma \left(x,y\right)\right)& \le \Phi \left(\Gamma \left(x_{0}y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\\ =\Phi \left(c\left(X,Y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds,\end{array}$$

(2.16)

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

$$\begin{array}{ll}\hfill \Gamma \left(x,y\right)& \le {\Phi }^{-1}\left(\Phi \left(c\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\right)\\ ={\Phi }^{-1}\left(\Phi \left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{g}_i\left(s,t\right)dtds\right)\right.\\ \left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\right),\end{array}$$

(2.17)

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

$$\begin{array}{ll}\hfill u\left(x,y\right)& \le {\psi }^{-1}\left(\eta \left(x,y\right)\right)\le {\psi }^{-1}\left(W^{-1}\left(\Gamma \left(x,y\right)\right)\right)\\ \le {\psi }^{-1}\left(W^{-1}\left({\Phi }^{-1}\left(\Phi \right.\right.\right.\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{g}_i\left(s,t\right)dtds\right)\\ \left.\left.\left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\right)\right)\right),\end{array}$$

(2.18)

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

$$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left(W^{-1}\left({\Phi }^{-1}\left(\Phi \right.\right.\right.\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{g}_i\left(s,t\right)dtds|\right)\\ \left.\left.\left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\right)\right)\right),\end{array}$$

(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

$$\begin{array}{ll}\hfill u\left(x,y\right)& \le {\psi }^{-1}\left(W^{-1}\left({\Phi }^{-1}\left(\Phi \right.\right.\right.\left(W\left(a\left(x,y\right)\right)+b\left(x,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{g}_i\left(s,t\right)dtds\right)\\ \left.\left.\left.+b\left(x,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)dtds\right)\right)\right),\end{array}$$

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,

$$\begin{array}{ll}\hfill D_j\left(x,y\right)& :={\Psi }_j\left(W\left(a\left(x,y\right)\right)\right)+b\left(x,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds,\\ \hfill {\Psi }_j\left(r\right)& :=\underset{1}{\overset{r}{\int }}\frac{ds}{{\phi }_j\left({\psi }^{-1}\left(W^{-1}\left(s\right)\right)\right)},{\Psi }_j\left(0\right):=\underset{r\to 0+}{\text{lim}}{\Psi }_j\left(r\right),\end{array}$$

(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

$$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le a\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[f_i\left(s,t\right){\phi }_1\left(u\left(s,t\right)\right)\right.\\ \left.+g_i\left(s,t\right){\phi }_2\left(\text{log}\left(u\left(s,t\right)\right)\right)\right]dtds,\end{array}$$

(2.25)

for all (x,y) ∈ [x₀,X] × [y₀,y₁), where x₀ ≤ X ≤ X₂ 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),

$$\begin{array}{ll}\hfill {\Xi }_x\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}w\left(u\left({\alpha }_i\left(s\right),t\right)\right)\left[f_i\left({\alpha }_i\left(x\right),t\right)\right.{\phi }_1\left(u\left({\alpha }_i\left(x\right),t\right)\right)\\ \left.+g_i\left({\alpha }_i\left(x\right),t\right){\phi }_2\left(\text{log}\left(u\left({\alpha }_i\left(x\right),t\right)\right)\right)\right]dt\\ \le b\left(X,y\right)w\left({\psi }^{-1}\left(\Xi \left(x,y\right)\right)\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left({\alpha }_i\left(x\right),t\right){\phi }_1\left({\psi }^{-1}\left(\Xi \left({\alpha }_i\left(x\right),t\right)\right)\right)\right.\\ \left.+g_i\left({\alpha }_i\left(x\right),t\right){\phi }_2\left(\text{log}\left({\psi }^{-1}\left(\Xi \left({\alpha }_i\left(x\right),t\right)\right)\right)\right)\right]dt,\end{array}$$

(2.26)

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

$$\begin{array}{ll}\hfill \frac{{\Xi }_x\left(x,y\right)}{w\left({\psi }^{-1}\left(\Xi \left(x,y\right)\right)\right)}& \le b\left(X,y\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left({\alpha }_i\left(x\right),t\right){\phi }_1\left({\psi }^{-1}\left(\Xi \left({\alpha }_i\left(x\right),t\right)\right)\right)\right.\\ \left.+g_i\left({\alpha }_i\left(x\right),t\right){\phi }_2\left(\text{log}\left({\psi }^{-1}\left(\Xi \left({\alpha }_i\left(x\right),t\right)\right)\right)\right)\right]dt,\end{array}$$

(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

$$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(\Xi \left(x_0,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right){\phi }_1\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right.\\ \left.+g_i\left(s,t\right){\phi }_2\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)\right]dtds\\ =W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\right.{\phi }_1\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\\ \left.+g_i\left(s,t\right){\phi }_2\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)\right]dtds,\end{array}$$

(2.28)

for all (x,y) ∈ [x₀,X] × [y₀,y₁).

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

$$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\right.\\ \left.+g_i\left(s,t\right)\right]{\phi }_1\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)dtds,\end{array}$$

(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

$$\Xi \left(x,y\right)\le W^{-1}\left(\Theta \left(x,y\right)\right),\forall \left(x,y\right)\in \left[x_0,X\right]\times \left[y_0,Y_2\right),$$

(2.30)

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

$$\begin{array}{ll}\hfill {\Theta }_x\left(x,y\right)& =b\left(X,y\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left({\alpha }_i\left(x\right),t\right)+g_i\left({\alpha }_i\left(x\right),t\right)\right]{\phi }_1\left({\psi }^{-1}\left(\Xi \left({\alpha }_i\left(x\right),t\right)\right)\right)dt\\ \le b\left(X,y\right){\phi }_1\left({\psi }^{-1}\left(W^{-1}\left(\Theta \left(x,y\right)\right)\right)\right)\sum _{i=1}^n{{\alpha }_i}^{\prime }\left(x\right)\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left({\alpha }_i\left(x\right),t\right)+g_i\left({\alpha }_i\left(x\right),t\right)\right]dt,\end{array}$$

(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

$$\begin{array}{ll}\hfill {\Psi }_1\left(\Theta \left(x,y\right)\right)& \le {\Psi }_1\left(\Theta \left(x_0,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds\\ ={\Psi }_1\left(W\left(a\left(X,y\right)\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds.\end{array}$$

(2.33)

From (2.30) and (2.33), we conclude

$$\begin{array}{c}u\left(x,y\right)\le {\psi }^{-1}\left(\Xi \left(x,y\right)\right)\le {\psi }^{-1}\left(W^{-1}\left(\Theta \left(x,y\right)\right)\right)\le {\psi }^{-1}\left[W^{-1}\left({\Psi }_1^{-1}\left(\right.\right.\right.\\ \left.\left.\left.{\Psi }_1\left(W\left(a\left(X,y\right)\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i{(}_x)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds\right)\right)\right],\end{array}$$

(2.34)

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

$$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[W^{-1}\left({\Psi }_1^{-1}\left({\Psi }_1\left(W\left(a\left(X,y\right)\right)\right)\right.\right.\right.\\ \left.\left.\left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds\right)\right)\right].\end{array}$$

(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

$$\begin{array}{ll}\hfill W\left(\Xi \left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\right.\\ \left.+g_i\left(s,t\right)\right]{\phi }_2\left(\text{log}\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)\right)dtds,\\ \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)\right.\\ \left.+g_i\left(s,t\right)\right]{\phi }_2\left({\psi }^{-1}\left(\Xi \left(s,t\right)\right)\right)dtds,\end{array}$$

(2.36)

for all (x,y) ∈ [x₀,X] × [y₀,y₁), where x₀ ≤ X ≤ X₃. Similarly to the above process from (2.29) to (2.35), from (2.36), we obtain

$$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[W^{-1}\left({\Psi }_2^{-1}\left({\Psi }_2\left(W\left(a\left(X,y\right)\right)\right)\right.\right.\right.\\ \left.\left.\left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)+g_i\left(s,t\right)\right]dtds\right)\right)\right].\end{array}$$

(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),

$${\Psi }_3\left(r\right):=\underset{1}{\overset{r}{\int }}\frac{ds}{{\psi }^{-1}\left(W^{-1}\left(s\right)\right)},r>0,{\Psi }_3\left(0\right):=\underset{r\to 0+}{\text{lim}}{\Psi }_3\left(r\right),$$

(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

$$\begin{array}{ll}\hfill \psi \left(u\left(x,y\right)\right)& \le a\left(X,y\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[f_i\left(s,t\right)L\left(s,t,u\left(s,t\right)\right)\right.\\ \left.+g_i\left(s,t\right)u\left(s,t\right)\right]dtds,\end{array}$$

(2.42)

for all (x,y) ∈ [x₀,X] × [y₀,y₁), where x₀ ≤ X ≤ X₄ 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

$$\begin{array}{ll}\hfill W\left(P\left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)L\left(s,t,{\psi }^{-1}\left(P\left(s,t\right)\right)\right)\right.\\ \left.+g_i\left(s,t\right){\psi }^{-1}\left(P\left(s,t\right)\right)\right]dtds,\forall \left(x,y\right)\in \left[x_{0}X\right]\times \left[y_0,y_1\right).\end{array}$$

(2.43)

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

$$\begin{array}{ll}\hfill W\left(P\left(x,y\right)\right)& \le W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)L\left(s,t,0\right)dtds\\ +b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)M\left(s,t,0\right)+g_i\left(s,t\right)\right]{\psi }^{-1}\left(P\left(s,t\right)\right)dtds,\end{array}$$

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

$$\begin{array}{ll}\hfill u\left(X,y\right)& \le {\psi }^{-1}\left[W^{-1}\left({\Psi }_3^{-1}\left({\Psi }_3\right.\left(W\left(a\left(X,y\right)\right)+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}{f}_i\left(s,t\right)L\left(s,t,0\right)dtds\right)\right.\right.\\ \left.\left.\left.+b\left(X,y\right)\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(X\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\left[f_i\left(s,t\right)M\left(s,t,0\right)+g_i\left(s,t\right)\right]dtds\right)\right)\right],\end{array}$$

(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):

$$\left\{\begin{array}{c}\frac{{\partial }^2\psi \left(z\left(x,y\right)\right)}{\partial x\partial y}=F\left(x,y,z\left({\alpha }_1\left(x\right),{\beta }_1\left(y\right)\right),z\left({\alpha }_2\left(x\right),{\beta }_2\left(y\right)\right),...,z\left({\alpha }_n\left(x\right),{\beta }_n\left(y\right)\right)\right),\hfill \\ z\left(x,y_0\right)=a_1\left(x\right),z\left(x_0,y\right)=a_2\left(y\right),a_1\left(x_0\right)=a_2\left(y_0\right)=0,\hfill \end{array}\right.$$

(3.1)

for x ∈ I,y ∈ J, where x₀,y₀,x₁,y₁ ∈ ℝ₊ are constants, I := [x₀,x₁), J := [y₀,y₁), F ∈ C(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 F ∈ C(I × J × ℝⁿ,ℝ) satisfies

$$\left|F\left(x,y,u_1,u_2,...,u_n\right)\right|\le \sum _{i=1}^{n}w\left(\left|u_i\right|\right)\left[f_i\left(x,y\right)\phi \left(\left|u_i\right|\right)+g_i\left(x,y\right)\right],\left(x,y\right)\in I\times J,$$

(3.2)

where f _i ,g _i ∈ C(I × J,ℝ₊) and w,ϕ ∈ C(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

$$\begin{array}{ll}\hfill B\left(x,y\right)& :=\Phi \left(A\left(x,y\right)\right)+\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\frac{f_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds,\\ \hfill A\left(x,y\right)& :=W\left(\psi \left(\left|a_1\left(x\right)\right|\right)+\psi \left(\left|a_2\left(y\right)\right|\right)\right)+\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i^{-1}}{\int }}\frac{g_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds,\end{array}$$

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

$$\begin{array}{ll}\hfill \psi \left(z\left(x,y\right)\right)& =\psi \left(a_1\left(x\right)\right)+\psi \left(a_2\left(y\right)\right)+\underset{x_0}{\overset{x}{\int }}\underset{y_0}{\overset{y}{\int }}F\left(s,t,z\left({\alpha }_1\left(s\right),{\beta }_1\left(t\right)\right),z\left({\alpha }_2\left(s\right),{\beta }_2\left(t\right)\right)\right.,...,\\ \left.z\left({\alpha }_n\left(s\right),{\beta }_n\left(t\right)\right)\right)dtds.\end{array}$$

(3.4)

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

$$\begin{array}{l}\psi \left(\left|z\left(x,y\right)\right|\right)\\ \le \psi \left(\left|a_1\left(x\right)\right|\right)+\psi \left(\left|a_2\left(y\right)\right|\right)\\ +\underset{x_0}{\overset{x}{\int }}\underset{y_0}{\overset{y}{\int }}\left|F\left(s,t,z\left({\alpha }_1\left(s\right),{\beta }_1\left(t\right)\right),z\left({\alpha }_2\left(s\right),{\beta }_2\left(t\right)\right),...,z\left({\alpha }_2\left(s\right),{\beta }_n\left(t\right)\right)\right)\right|dtds\\ \le \psi \left(\left|a_1\left(x\right)\right|\right)+\psi \left(\left|a_2\left(y\right)\right|\right)\\ +\underset{x_0}{\overset{x}{\int }}\underset{y_0}{\overset{y}{\int }}\sum _{i=1}^{n}w\left(\left|z\left({\alpha }_i\left(s\right),{\beta }_i\left(t\right)\right)\right|\right)\left[f_i\left(s,t\right)\phi \left(\left|z\left({\alpha }_i\left(s\right),{\beta }_i\left(t\right)\right)\right|\right)+g_i\left(s,t\right)\right]dtds\\ =\psi \left(\left|a_1\left(x\right)\right|\right)+\psi \left(\left|a_2\left(y\right)\right|\right)+\sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\\ \frac{w\left(\left|z\left(s_1,t_1\right)\right|\right)\left[f_i\left({\alpha }_i^{-1}\left(s_1\right),{\beta }_i^{-1}\left(t_1\right)\right)\phi \left(\left|z\left(s_1,t_1\right)\right|\right)+g_i\left({\alpha }_i^{-1}\left(s_1\right),{\beta }_i^{-1}\left(t_1\right)\right)\right]}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s_1\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t_1\right)\right)}d{t}_{1}d{s}_1,\end{array}$$

(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

$$\begin{array}{c}\psi \left(\left|a_1\left(x\right)\right|\right)+\psi \left(\left|a_2\left(y\right)\right|\right)<\infty ,\\ \sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\frac{f_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds<\infty ,\\ \sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\frac{g_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds<\infty ,\end{array}$$

on [x₀,X₁) × [y₀,Y₁), then every solution z(x,y) of BVP (3.1) is bounded on [x₀,X₁) × [y₀,Y₁).

Next, we discuss the uniqueness of solutions for BVP (3.1).

Corollary 2. Consider BVP (3.1) and suppose that F ∈ C(I × J × ℝⁿ,ℝ) satisfies

$$\left|F\left(x,y,u_1,u_2,\dots ,u_n\right)-F\left(x,y,v_1,v_2,\dots ,v_n\right)\right|\le \sum _{i=1}^n{f}_i\left(x,y\right)\left|\psi \left(u_i\right)-\psi \left(v_i\right)\right|,$$

(3.6)

for all (x,y) ∈ I × J and u _i , v _i ∈ ℝ, i = 1, 2,..., n, where I = [x₀, x₁], J = [y₀, y₁] are two finite intervals, and f _i ∈ C(I × J, ℝ₊),i = 1,2,...,n. Then BVP (3.1) has at most one solution on I × J.

Proof. Assume that both z(x,y) and $\tilde{z}\left(x,y\right)$ are solutions of BVP (3.1). From the equivalent integral Equations (3.4) and (3.6), we have

$$\begin{array}{l}\left|\psi \left(z\left(x,y\right)\right)-\psi \left(\tilde{z}\left(x,y\right)\right)\right|\\ \le \underset{x_0}{\overset{x}{\int }}\underset{y_0}{\overset{y}{\int }}\left|F\left(s,t,z\left({\alpha }_1\left(s\right),{\beta }_1\left(t\right)\right),z\left({\alpha }_2\left(s\right),{\beta }_2\left(t\right)\right),\dots ,z\left({\alpha }_n\left(s\right),{\beta }_n\left(t\right)\right)\right)\right.\\ \left.-F\left(s,t,\tilde{z}\left({\alpha }_1\left(s\right),{\beta }_1\left(t\right)\right),\tilde{z}\left({\alpha }_2\left(s\right),{\beta }_2\left(t\right)\right),\dots ,\tilde{z}\left({\alpha }_n\left(s\right),{\beta }_n\left(t\right)\right)\right)\right|dtds\end{array}$$

$$\begin{array}{l}\le \underset{x_0}{\overset{x}{\int }}\underset{y_0}{\overset{y}{\int }}\sum _{i=1}^n{f}_i\left(s,t\right)\left|\psi \left(z\left({\alpha }_i\left(s\right),{\beta }_i\left(t\right)\right)\right)-\psi \left(\tilde{z}\left({\alpha }_i\left(s\right),{\beta }_i\left(t\right)\right)\right)\right|dtds\\ \le \epsilon +\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\sum _{i=1}^n\frac{f_i\left({\alpha }_i^{-1}\left(s_1\right),{\beta }_i^{-1}\left(t_1\right)\right)\left|\psi \left(z\left(s_1,t_1\right)\right)-\psi \left(\tilde{z}\left(s_1,t_1\right)\right)\right|}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s_1\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t_1\right)\right)}d{t}_{1}d{s}_1,\end{array}$$

(3.7)

for all (x,y) ∈ I × J, where changes of variables s₁ = α _i (s), t₁ = β _i (t) are made, ε > 0 is an arbitrary small number. Clearly, the inequality (3.7) is in the form of (1.1). Suitably applying our Theorem 1 to (3.7), we get an estimate of the form (2.1) for all (x,y) ∈ I × J,

$$\left|\psi \left(z\left(x,y\right)\right)-\psi \left(\tilde{z}\left(x,y\right)\right)\right|\le \epsilon \text{exp}\left(\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}\sum _{i=1}^n\frac{f_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds\right).$$

(3.8)

Letting ε → 0₊, since ${\int }_{{\alpha }_i\left(x_0\right)}^{{\alpha }_i\left(x\right)}{\int }_{{\beta }_i\left(y_0\right)}^{{\beta }_i\left(y\right)}{\sum }_{i=1}^n\frac{f_i\left({\alpha }_i^{-1}\left(s\right),{\beta }_i^{-1}\left(t\right)\right)}{{{\alpha }_i}^{\prime }\left({\alpha }_i^{-1}\left(s\right)\right){{\beta }_i}^{\prime }\left({\beta }_i^{-1}\left(t\right)\right)}dtds$ is finite on finite intervals I and J, ψ is a strictly increasing function, from (3.8), we conclude that $\left|\psi \left(z\left(x,y\right)\right)-\psi \left(\tilde{z}\left(x,y\right)\right)\right|\le 0$, implying that $z\left(x,y\right)=\tilde{z}\left(x,y\right)$ for all (x,y) ∈ I × J. The uniqueness is proved.

Remark Suppose that F ∈ C(I × J × ℝⁿ,ℝ) in BVP (3.1) satisfies

$$\begin{array}{ll}\hfill \left|F\left(x,y,u_1,u_2,\dots ,u_n\right)\right|& \le \sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[f_i\left(s,t\right){\phi }_1\left(u\left(s,t\right)\right)\right.\\ \left.+g_i\left(s,t\right){\phi }_2\left(\text{log}\left(u\left(s,t\right)\right)\right)\right]dtds.\end{array}$$

By using Theorem 2, we can give an estimate for solutions of the BVP (3.1).

Suppose that F ∈ C(I × J × ℝⁿ,ℝ) in BVP (3.1) satisfies

$$\begin{array}{ll}\hfill \left|F\left(x,y,u_1,u_2,\dots ,u_n\right)\right|& \le \sum _{i=1}^n\underset{{\alpha }_i\left(x_0\right)}{\overset{{\alpha }_i\left(x\right)}{\int }}\underset{{\beta }_i\left(y_0\right)}{\overset{{\beta }_i\left(y\right)}{\int }}w\left(u\left(s,t\right)\right)\left[f_i\left(s,t\right)L\left(s,t,u\left(s,t\right)\right)\right.\\ \left.+g_i\left(s,t\right)u\left(s,t\right)\right]dtds.\end{array}$$

By using Theorem 3, we can give an estimate for solutions of the BVP (3.1) too.