Weakly Singular Wendroff-Type Integral Inequalities of Multiple Variables with Multiple Nonlinear Terms and Their Applications

Abstract

This paper systematically studies a class of weakly singular Wendroff-type integral inequalities with multiple variables and multiple nonlinear terms. We establish explicit bounds for the unknown functions by utilizing the method of characteristic functions and mathematical induction. These results generalize and improve existing inequalities found in the literature. Furthermore, we apply them to fractional partial differential equations to study the uniqueness, boundedness, and continuous dependence of solutions. Even in the presence of singularities, the proposed method proves effective. An application example is provided to illustrate the validity of the main results.

1. Introduction

The research significance of singular kernel integral inequality is that it provides a theoretical framework for dealing with integral equations with singular kernels, which is widely used in mathematical analysis, physics, engineering and other fields. Such inequalities can help to analyze and solve systems with singular properties, especially in describing nonlinear and non-smooth problems. Its practical value is reflected in several aspects: in signal processing, image reconstruction, quantum mechanics, fluid dynamics and other fields, singular kernel integral inequalities can be used to describe complex physical phenomena, such as electromagnetic fields, particle propagation, etc. In addition, it also plays an important role in numerical analysis and optimization algorithms, helping to improve the solution efficiency and accuracy. Through in-depth study of these inequalities, we can promote the modeling, analysis and solution of practical problems. Singular integral inequality is a very important part in the theory of differential equations, and a series of results have been obtained in this field.

For example, in 1981, Henry [1] proposed a linear integral inequality with a singular kernel, shown as follows:

$$u\left(t\right)\le a+b{\int }_0^t{(t-s)}^{\beta -1}u\left(s\right)ds.$$

The results can be used to study the qualitative property equations of parabolic differentials.

In 1997, $Medve\check{d}$ [2] gave a new method to study linear and nonlinear singular integral Gronwall–Bellman–Bihari-type inequality and Henry-type inequality, which is shown as follows:

$$\begin{array}{c}u\left(t\right)\le a\left(t\right)+{\int }_0^t{(t-s)}^{\beta -1}{s}^{\gamma -1}F\left(s\right)u\left(s\right)ds,\hfill \\ u\left(t\right)\le a\left(t\right)+{\int }_0^t{(t-s)}^{\beta -1}F\left(s\right)w\left(u\left(s\right)\right)ds,\hfill \end{array}$$

and the estimates of the solutions are given separately.

In 2002, Ma and Yang [3] studied more generalized weakly singular Volterra-type integral inequalities using the improved $Medve\check{d}$ method. It looks like this:

$$\begin{array}{cc}\hfill u\left(t\right)& \le a\left(t\right)+b\left(t\right){\int }_0^t{(t^{\alpha }-s^{\alpha })}^{\beta -1}{s}^{\gamma -1}g\left(s\right)u\left(s\right)ds\hfill \\ & +c\left(t\right){\int }_0^t{(t^{\alpha }-s^{\alpha })}^{\beta -1}{s}^{\gamma -1}g\left(s\right)\omega \left(u\left(s\right)\right)ds.\hfill \end{array}$$

In 2008, Wu [4] studied a class of weakly odd Volterra-type integral inequalities with n nonlinear terms, which are shown as follows:

$$u\left(t\right)\le a\left(t\right)+\sum _{i=1}^n{\int }_{b_i\left(0\right)}^{b_i\left(t\right)}{(t^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}{f}_i(t,s)w_i\left(u\left(s\right)\right)ds.$$

Thereafter, in 2008, Cheung et al. [5] respectively studied the following binary weakly singular Wendroff-type integral inequalities by applying the method in reference [3], which are shown as follows:

$$\begin{array}{cc}\hfill u^p(x,y)& \le a(x,y)+b(x,y){\int }_0^x{\int }_0^y{(x^{\alpha }-s^{\alpha })}^{\beta -1}{s}^{\gamma -1}\hfill \\ & \times {(y^{\alpha }-t^{\alpha })}^{\beta -1}{t}^{\gamma -1}f(s,t)u^q(s,t)dtds.\hfill \end{array}$$

On this basis, in 2010, Wang and Zheng [6] discussed a more general form of nonlinear Wendroff-type weakly singular integral inequality for a class of binary functions, which is shown as follows:

$$\begin{array}{cc}\hfill u(x,y)& \le a(x,y)+{\int }_0^x{\int }_0^y{(x^{\alpha }-s^{\alpha })}^{\beta -1}{s}^{\gamma -1}\hfill \\ & \times {(y^{\alpha }-t^{\alpha })}^{\beta -1}{t}^{\gamma -1}f(x,y,s,t)w\left(u(s,t)\right)dsdt.\hfill \end{array}$$

In 2013, Wu et al. [7] studied a new class of weakly singular Wendroff-type integral inequalities, which are shown as follows:

$$\begin{array}{cc}\hfill u(x,y)& \le a(x,y)\hfill \\ & +{\int }_0^x{\int }_0^y{(x^{{\alpha }_1}-s^{{\alpha }_1})}^{{\beta }_1-1}{s}^{{\gamma }_1-1}{(y^{{\alpha }_2}-t^{{\alpha }_2})}^{{\beta }_2-1}\hfill \\ & \times t^{{\gamma }_2-1}{f}_1(x,y,s,t)w\left(u(s,t)\right)dsdt\hfill \\ & +{\int }_0^x{\int }_0^y{(x^{{\alpha }_3}-s^{{\alpha }_3})}^{{\beta }_3-1}{s}^{{\gamma }_3-1}{(y^{{\alpha }_4}-t^{{\alpha }_4})}^{{\beta }_4-1}\hfill \\ & \times t^{{\gamma }_4-1}{f}_2(x,y,s,t)u(s,t)dsdt.\hfill \end{array}$$

In 2014, the weakly singular Wendroff-type integral inequalities of two nonlinear terms were studied in [8], which are shown as follows:

$$\begin{array}{cc}\hfill u(x,y)& \le a(x,y)+\sum _{i=1}^2{\int }_0^x{\int }_0^y{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}{(y^{{\sigma }_i}-t^{{\sigma }_i})}^{{\mu }_i-1}\hfill \\ & \times t^{{\tau }_i-1}{f}_i(x,y,s,t)w_i\left(u(s,t)\right)dsdt.\hfill \end{array}$$

On the basis of Zheng [6] and Wu [8], in 2017, Liu [9] studied a class of weakly singular Wendroff-type integral inequalities with n nonlinear terms, which are shown as follows:

$$\begin{array}{cc}\hfill u(x,y)& \le a(x,y)+\sum _{i=1}^n{\int }_0^x{\int }_0^y{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}{(y^{{\sigma }_i}-t^{{\sigma }_i})}^{{\mu }_i-1}\hfill \\ & \times t^{{\tau }_i-1}{f}_i(x,y,s,t)w_i\left(u(s,t)\right)dsdt.\hfill \end{array}$$

In 2024, Yang [10] studied a class of generalized Wendroff-type weakly singular integral inequalities, which are shown as follows:

$$u(x,y)\le a(x,y)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(x_0\right)}^{c_i\left(x\right)}{\alpha }_i(x,y,s,t)f_i(x,y,s,t)w_i\left(u(s,t)\right)dsdt.$$

More work on singular integral inequalities is emerging, see Refs. [11,12,13]. It can be seen that with the development of integral and differential equations, some new integral inequalities need to be considered in order to meet the wider application of differential and integral equations. In this paper we focus on Wendroff-type integral inequalities and their application to the study of well-posed solutions of differential and integral equations.

Although References [9,10] discussed Wendroff-type inequalities, they primarily focused on continuous kernels or single-variable cases. In contrast to [9,10], this paper introduces multiple nonlinear terms and specifically addresses the difficulties caused by weakly singular kernels in a multi-variable environment.

On the basis of References [3,4,5,6,7,8,9,10,11,12,13,14,15], this paper considers the weakly singular Wendroff-type integral inequality of multiple variables with n nonlinear terms as follows.

$$\begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}\hfill \\ & \times {(y^{{\alpha }_i}-t^{{\alpha }_i})}^{{\beta }_i-1}{t}^{{\gamma }_i-1}{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{{\beta }_i-1}{\tau }^{{\gamma }_i-1}{f}_i(x,y,z,s,t,\tau )\hfill \\ & \times W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)& \le a(x_1,x_2,\dots ,x_n)+\sum _{i=1}^n{\int }_{b_{1,i}\left(x_{1,0}\right)}^{b_{1,i}\left(x_1\right)}{\int }_{b_{2,i}\left(x_{2,0}\right)}^{b_{2,i}\left(x_2\right)}\dots {\int }_{b_{n,i}\left(x_{n,0}\right)}^{b_{n,i}\left(x_n\right)}\hfill \\ & \times {\left(x_1^{{\alpha }_i}-s_1^{{\alpha }_i}\right)}^{{\beta }_i-1}{s}_1^{{\gamma }_i-1}{(x_2^{{\alpha }_i}-s_2^{{\alpha }_i})}^{{\beta }_i-1}{s}_2^{{\gamma }_i-1}\dots {(x_n^{{\alpha }_i}-s_n^{{\alpha }_i})}^{{\beta }_i-1}{s}_n^{{\gamma }_i-1}\hfill \\ & \times f_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)W_i\left(u(s_1,s_2,\dots ,s_n)\right)d{s}_{1}d{s}_2\dots d{s}_n.\hfill \end{array}$$

Before this, we need to study the estimation of the solution of a class of Wendroff-type integral inequalities as a theorem that can be used to refer to. Secondly, combined with the modified Medved method, we can obtain the estimation of unknown functions of Wendroff-type weakly singular integral inequalities. Finally, it is applied to study the boundedness, uniqueness and continuous dependence of solutions of fractional partial differential equations.

The remainder of this paper is organized as follows: Section 2 presents the necessary lemmas and definitions. Section 3 establishes the main results, including the estimation of solutions for the proposed inequalities. Section 4 discusses the applications to fractional partial differential equations, focusing on uniqueness and continuous dependence.

2. Definitions and Lemmas

Definition 1.

The Riemann–Liouville integral of order α of a function $f\left(t\right)$ with respect to t is defined as

$$\begin{array}{ccc}& & I_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds,t\ge 0.\hfill \end{array}$$

Definition 2.

The derivative of a function $f\left(t\right)$ with respect to t of order γ is defined as

$$\begin{array}{ccc}& & D_t^{\gamma }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\gamma )}}{\displaystyle \frac{d}{dt}}{\int }_0^t{(t-s)}^{-\gamma }f\left(s\right)dt,t\ge 0.\hfill \end{array}$$

Lemma 1

(see [3]). Let α, β, γ and p be positive constants. Then,

$$\begin{array}{ccc}& & {\int }_0^t{(t^{\alpha }-s^{\alpha })}^{p(\beta -1)}{s}^{p(\gamma -1)}ds={\displaystyle \frac{t^{\theta }}{\alpha }}\mathrm{B}\left[{\displaystyle \frac{p(\gamma -1)+1}{\alpha }},p(\beta -1)+1\right],t\ge 0.\hfill \end{array}$$

where $\mathrm{B}[\xi ,\eta ]={\int }_0^1{s}^{\xi -1}{(1-s)}^{\eta -1}ds(Re\xi >0,Re\eta >0)$ is the well-known Beta function and $\theta =p\left[\alpha (\beta -1)+\gamma -1\right]+1$.

Lemma 2

(see [3]). (Discrete Jensen’s inequality) Let $A_1,A_2,\dots ,A_n$ be non-negative real numbers, $r>1$ be a real number and n be a natural number, then

$$\begin{array}{ccc}& & {(A_1,A_2,\dots ,A_n)}^r\le n^{r-1}(A_1^r,A_2^r,\dots ,A_n^r).\hfill \end{array}$$

Lemma 3

(see [12]). (Holder integral inequality) Suppose the function $f\left(x\right)\in L^p$$\left([t_0,+\infty ],\mathbb{R}\right)$, the function $g\left(x\right)\in L^q\left([t_0,+\infty ],\mathbb{R}\right)$, the constant p, $q>1$ and $1/p+1/q=1$. Then

$$\begin{array}{ccc}& & {\int }_{t_0}^t\left|f\left(s\right)g\left(s\right)\right|ds\le {\left({\int }_{t_0}^t{\left|f\left(s\right)\right|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{t_0}^t{\left|g\left(s\right)\right|}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Lemma 4

(see [12]). Suppose the function $f\left(x\right)\in L^p$$\left([t_0,+\infty ],\mathbb{R}\right)$, the function $g\left(x\right)\in L^q\left([t_0,+\infty ],\mathbb{R}\right)$, the constant p, $q>1$ and $1/p+1/q=1$, $\alpha \left(t\right)$ is a continuously differentiable non-decreasing function, $\alpha \left(t\right)\le t$, $\alpha \left(t_0\right)=t_0$. Then

$$\begin{array}{ccc}& & {\int }_{\alpha \left(t_0\right)}^{\alpha \left(t\right)}|f\left(s\right)g\left(s\right)|ds\le {\left({\int }_{\alpha \left(t_0\right)}^{\alpha \left(t\right)}{|f\left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_{\alpha \left(t_0\right)}^{\alpha \left(t\right)}{|g\left(s\right)|}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

3. Main Results

In this section, we will prove estimates of solutions of weakly singular integral inequalities of Wendroff-type for multiple variables with n nonlinear terms. In this chapter, let $\mathbb{R}$ denote the set of real numbers, ${\mathbb{R}}_{+}$ denotes the set of non-negative real numbers. $x_0$, $y_0$ is a constant real number, $I=(x_0,+\infty )$, $J=(y_0,+\infty )$, $T=(z_0,+\infty )$, $\mathsf{\Lambda }=I\times J\times T\subset {\mathbb{R}}^3$, define the following set of functions:

$$\mathsf{\Omega }({\mathbb{R}}_{+},{\mathbb{R}}_{+})=\{\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}|\phi \text{is a non-decreasing function}\},$$

$$C^1(I,I)=\{\phi :I\to I|\phi \mathrm{is}\mathrm{a}\mathrm{continuously}\mathrm{differentiable}\mathrm{function}\},$$

$$C(\mathsf{\Lambda },{\mathbb{R}}_{+})=\{\phi :\mathsf{\Lambda }\to {\mathbb{R}}_{+}|\phi \mathrm{is}\mathrm{a}\mathrm{continuous}\mathrm{function}\},$$

$$C_{\mathsf{\Omega }}(\mathsf{\Lambda },{\mathbb{R}}_{+})=\left\{\phi :\mathsf{\Lambda }\to {\mathbb{R}}_{+}\left|\begin{array}{c}\phi \mathrm{is}\mathrm{continuous}\mathrm{and}\text{non-decreasing}\\ \mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{each}\mathrm{variable}\end{array}\right.\right\}.$$

Assume that

(K₁)

${\alpha }_i\in \left(0,1\right],{\beta }_i\in \left(0,1\right),p>1$ and satisfy $p({\beta }_i-1)+1>0,$ ${\gamma }_i>1-{\displaystyle \frac{1}{p}}$ and ${\displaystyle \frac{1}{p}}+{\alpha }_i({\beta }_i-1)+{\gamma }_i-1\ge 0,$ where $i=1,2,\dots ,n.$

(K₂)

$a\in C_{\mathsf{\Omega }}(\mathsf{\Lambda },{\mathbb{R}}_{+})$.

(K₃)

$b_i\in C^1(I,I)\cap \mathsf{\Omega }(I,I),$ $c_i\in C^1(J,J)\cap \mathsf{\Omega }(J,J),$ $d_i\in C^1(T,T)\cap \mathsf{\Omega }(T,T),$ and for all $x\in I,y\in J,z\in T$ there exists $0\le b_i\left(x\right)\le x,0\le c_i\left(y\right)\le y,0\le d_i\left(z\right)\le z.$ where i = 1, 2, …, n.

(K₄)

$f_i(·,·,·,s,t,\tau )\in C_{\mathsf{\Omega }}(\mathsf{\Lambda },{\mathbb{R}}_{+}),f_i(x,y,z,·,·,·)\in C(\mathsf{\Lambda },{\mathbb{R}}_{+})$ where i = 1, 2, …, n.

(K₅)

$W_i\left(\eta \right)\in \mathsf{\Omega }({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ is monotonically non-decreasing, when $\eta >0$ has $W_i\left(\eta \right)>0,$ and ${\displaystyle \frac{W_2\left(\eta \right)}{W_1\left(\eta \right)}},{\displaystyle \frac{W_3\left(\eta \right)}{W_2\left(\eta \right)}},\dots ,{\displaystyle \frac{W_n\left(\eta \right)}{W_{n-1}\left(\eta \right)}}$ is monotonically non-decreasing, where $i=1,2,\dots ,n.$

(K₆)

$I_i:=\left(x_i,+\infty \right),{\mathsf{\Lambda }}^1={\displaystyle \prod _{i=1}^n}{I}_i\subset {\mathbb{R}}^n,a\in C_{\mathsf{\Omega }}({\mathsf{\Lambda }}^1,{\mathbb{R}}_{+}),b_i\in C^1(I_i,I_i)\cap \mathsf{\Omega }(I_i,I_i),$ and for all $x_i\in I_i,$ there exists $0\le b_{i,j}\left(x_i\right)\le x_i.$ $f_i(·,·,\dots ,s_1,s_2,\dots ,s_n)\in C_{\mathsf{\Omega }}({\mathsf{\Lambda }}^1,{\mathbb{R}}_{+}),f_i(x_1,x_2,\dots ,x_n,·,·,\dots )\in C({\mathsf{\Lambda }}^1,{\mathbb{R}}_{+}).$ where i, j = 1, 2, …, n.

Theorem 1.

Under the condition that Assumptions $\left(K_2\right)$–$\left(K_5\right)$ hold, if $u\in C(\mathsf{\Lambda },{\mathbb{R}}_{+})$, and $u(x,y,z)$ satisfies the weakly singular integral inequality

$$\begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times f_i(x,y,z,s,t,\tau )W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}$$

(1)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Then

$$\begin{array}{cc}\hfill u(x,y,z)& \le G_n^{-1}[G_n\left(G_{n-1}^{-1}\left(a_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}$$

(2)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Where

$$a_1(x,y,z)=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}a(\sigma ,\mu ,\zeta ),$$

(3)

$$\begin{array}{cc}\hfill a_i(x,y,z)& =G_{i-1}\left(G_{i-2}^{-1}\left(a_{i-1}(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_{i-1}\left(x_0\right)}^{b_{i-1}\left(x\right)}{\int }_{c_{i-1}\left(y_0\right)}^{c_{i-1}\left(y\right)}{\int }_{d_{i-1}\left(z_0\right)}^{d_{i-1}\left(z\right)}{\widehat{f}}_{i-1}(x,y,z,s,t,\tau )dsdtd\tau ,\hfill \end{array}$$

(4)

for $i=2,3,\dots ,n.$

$${\widehat{f}}_i(x,y,z,s,t,\tau )=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}{f}_i(\sigma ,\mu ,\zeta ,s,t,\tau ),$$

(5)

$G_i^{-1}\left(u\right)$ is the inverse function of $G_i\left(u\right)$,

$$G_i\left(u\right)={\int }_{u_i}^u{\displaystyle \frac{d\xi }{W_i\left(\xi \right)}},u\ge u_i>0,i=1,2,\dots ,n.$$

(6)

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{cc}& G_i\left(G_{i-1}^{-1}\left(a_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}\hfill \\ & \times {\widehat{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left(G_i^{-1}\right).\hfill \end{array}$$

(7)

The induction hypothesis will be used next to prove Theorem 1.

Proof. 

Let $u(x,y,z)$ be a solution of Inequality (1). Since the functions $f_i(x,y,z,s,t,\tau )$ and $W_i\left(u\right)$ are non-negative, and the limits satisfy $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$, we have the following inequality:

$$\begin{array}{cc}\hfill u(x,y,z)& =a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times f_i(x,y,z,s,t,\tau )W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \\ & \le a(X,Y,Z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times f_i(X,Y,Z,s,t,\tau )W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}$$

(8)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

(i)

When $n=1$, define an auxiliary function $v_1(x,y,z)$,

$$\begin{array}{cc}\hfill v_1(x,y,z)& ={\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{f}_1(X,Y,Z,s,t,\tau )\hfill \\ & \times W_1\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}$$

then Equation (8) can be transformed into

$$u(x,y,z)\le a(X,Y,Z)+v_1(x,y,z),$$

(9)

Since $W_1\left(u\right)>0$ is a non-decreasing function, the following inequality relationship holds

$$\begin{array}{cc}\hfill 0<W_1\left(u(s,t,\tau )\right)& \le W_1(a_1(X,Y,Z)+v_1(s,t,\tau ))\hfill \\ & \le W_1(a_1(X,Y,Z)+v_1(b_1\left(x\right),c_1\left(y\right),d_1\left(z\right)))\hfill \\ & \le W_1(a_1(X,Y,Z)+v_1(x,y,z)).\hfill \end{array}$$

Taking the partial derivative of the left and right sides of (9) with respect to x, using the function $W_1\left(u\right)>0$ and monotonically non-decreasing with respect to u can obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x}}u(x,y,z)& \le {\displaystyle \frac{\partial }{\partial x}}(a_1(X,Y,Z)+v_1(x,y,z))\le {\displaystyle \frac{\partial }{\partial x}}{v}_1(x,y,z)\hfill \\ & \le b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_1\left(x\right),t,\tau )W_1\left(u(b_1\left(x\right),t,\tau )\right)dtd\tau \hfill \\ & \le b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_1\left(x\right),t,\tau )\hfill \\ & \times W_1(a_1(X,Y,Z)+v_1(b_1\left(x\right),y,z))dtd\tau \hfill \\ & \le b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_1\left(x\right),t,\tau )dtd\tau \hfill \\ & \times W_1(a_1(X,Y,Z)+v_1(x,y,z))\hfill \end{array}$$

(10)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Equation (10) shows that

$$\begin{array}{cc}& {\displaystyle \frac{\partial u(x,y,z)}{W_1(a_1(X,Y,Z)+v_1(x,y,z))}}\hfill \\ & \le \left[b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_1\left(x\right),t,\tau )dtd\tau \right]\partial x,\hfill \end{array}$$

(11)

it follows from $W_1\left(u\right)>0$, integrating both sides of (11) from $x_0$ to x, and having the definition of $G_1\left(u\right)$ in Equation (6)

$$\begin{array}{cc}& G_1\left(a_1(X,Y,Z)+v_1(x,y,z)\right)-G_1\left(a_1(X,Y,Z)+v_1(x_0,y,z)\right)\hfill \\ & \le {\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau ,\hfill \end{array}$$

(12)

since $v_1(x_0,y,z)=0$, the arrangement of Equation (12) can be obtained.

$$\begin{array}{cc}& G_1\left(a_1(X,Y,Z)+v_1(x,y,z)\right)\hfill \\ & \le G_1\left(a_1(X,Y,Z)\right){\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau ,\hfill \end{array}$$

(13)

From the properties of the function $G_i$, it can be known that both $G_i$ and $G_i^{-1}$ are non-decreasing functions on ${\mathbb{R}}_{+}$. Therefore, $G_1(+\infty )=+\infty$. Then, from Equation (13), we can obtain

$$\begin{array}{cc}\hfill G_1\left(0\right)& \le G_i\left(G_{i-1}^{-1}\left(a_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}\hfill \\ & \times {\widehat{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau <+\infty =G_1(+\infty ),\hfill \end{array}$$

that is

$$\begin{array}{cc}& G_1\left(a_1(X,Y,Z)\right)+{\int }_{b_1\left(x_0\right)}^{b_1\left(X\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(Y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(Z\right)}\hfill \\ & \times {\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left(G_i^{-1}\right).\hfill \end{array}$$

Combining Equations (9) and (13) and the monotonicity of $G_i^{-1}$, we can obtain

$$\begin{array}{cc}\hfill u(x,y,z)& \le G_1^{-1}[G_1\left(a_1(X,Y,Z)\right)+{\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}\hfill \\ & \times {\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau ],\hfill \end{array}$$

(14)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

In Equation (14), let $x=X$, $y=Y$ and $z=Z$. Considering the arbitrariness of the constants X, Y and Z, we have

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le G_1^{-1}[G_1\left(a_1(x,y,z)\right)+{\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}\hfill \\ & \times {\widehat{f}}_1(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

So when $n=1$, Equation (2) holds.

(ii)

Suppose that when $n=k$, Equation (2) holds. The proof is that when $n=k+1$, Equation (2) also holds. Define an auxiliary function $v_2(x,y,z)$,

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill v_2(x,y,z)& =\sum _{i=1}^{k+1}{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times {\widehat{f}}_i(X,Y,Z,s,t,\tau )W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(15)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. then Equation (8) can be transformed into

$$\begin{array}{ccc}& & u(x,y,z)\le a(X,Y,Z)+v_2(x,y,z).\hfill \end{array}$$

(16)

Since $W_i\left(u\right)>0(i=1,2,\dots ,k+1)$ is a non-decreasing function, the following inequality relationship holds

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0<W_i\left(u(s,t,\tau )\right)& \le W_i(a(X,Y,Z)+v_2(s,t,\tau ))\hfill \\ & \le W_i(a(X,Y,Z)+v_2(b_i\left(x\right),c_i\left(y\right),d_i\left(z\right)))\hfill \\ & \le W_i(a(X,Y,Z)+v_2(x,y,z)).\hfill \end{array}\end{array}$$

(17)

Taking the partial derivative of the left and right sides of (16) with respect to x, using Equation (17) and monotonically non-decreasing with respect to u can obtain

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x}}u(x,y,z)& \le {\displaystyle \frac{\partial }{\partial x}}(a(X,Y,Z)+v_2(x,y,z))\le {\displaystyle \frac{\partial }{\partial x}}{v}_2(x,y,z)\hfill \\ & \le \sum _{i=1}^{k+1}{b}_i^{\prime }\left(x\right){\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\widehat{f}}_i(X,Y,Z,b_i\left(x\right),t,\tau )W_i\left(u(b_i\left(x\right),t,\tau )\right)dtd\tau \hfill \\ & \le b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_1\left(x\right),t,\tau )dtd\tau \hfill \\ & \times W_1(a(X,Y,Z)+v_2(b_1\left(x\right),y,z))dtd\tau \hfill \\ & +\sum _{i=2}^{k+1}{b}_i^{\prime }\left(x\right){\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\widehat{f}}_i(X,Y,Z,b_i\left(x\right),t,\tau )\hfill \\ & \times W_i(a(X,Y,Z)+v_2(b_i\left(x\right),y,z))dtd\tau ,\hfill \end{array}\hfill \end{array}$$

we have

$$\begin{array}{ccc}& & \begin{array}{cc}& {\displaystyle \frac{\partial (a(X,Y,Z)+v_2(x,y,z))}{W_1(a(X,Y,Z)+v_2(x,y,z))}}\hfill \\ & \le \left[b_1^{\prime }\left(x\right){\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,b_i\left(x\right),t,\tau )dtd\tau \right]\partial x\hfill \\ & +\sum _{i=1}^k[b_{i+1}^{\prime }\left(x\right){\int }_{c_{i+1}\left(y_0\right)}^{c_{i+1}\left(y\right)}{\int }_{d_{i+1}\left(z_0\right)}^{d_{i+1}\left(z\right)}{\widehat{f}}_{i+1}(X,Y,Z,b_i\left(x\right),t,\tau )\hfill \\ & \times {\displaystyle \frac{W_{i+1}\left(a(X,Y,Z)+v_2(b_i\left(x\right),y,z)\right)}{W_1\left(a(X,Y,Z)+v_2(b_i\left(x\right),c_i\left(y\right),d_i\left(z\right))\right)}}dtd\tau ]\partial x.\hfill \end{array}\hfill \end{array}$$

(18)

Let $M_{i+1}\left(u\right)={\displaystyle \frac{W_{i+1}\left(u\right)}{W_1\left(u\right)}}(i=1,2,\dots ,k)$. It follows from the definition of $W_1\left(u\right)$ that $M_{i+1}\left(u\right)$ is a monotone and non-negative function, and ${\displaystyle \frac{M_3\left(u\right)}{M_2\left(u\right)}}$, ${\displaystyle \frac{M_4\left(u\right)}{M_3\left(u\right)}}$, ${\displaystyle \frac{M_{k+1}\left(u\right)}{M_k\left(u\right)}}$ is monotone and non-decreasing, integrating from $x_0$ to x on both sides of Equation (18), and it follows from the definition of $G_1\left(u\right)$ that

$$\begin{array}{ccc}& & \begin{array}{cc}& G_1\left(a(X,Y,Z)+v_2(x,y,z)\right)-G_1\left(a(X,Y,Z)+v_2(x_0,y,z)\right)\hfill \\ & \le {\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau \hfill \\ & +\sum _{i=1}^k{\int }_{b_{i+1}\left(x_0\right)}^{b_{i+1}\left(x\right)}{\int }_{c_{i+1}\left(y_0\right)}^{c_{i+1}\left(y\right)}{\int }_{d_{i+1}\left(z_0\right)}^{d_{i+1}\left(z\right)}{\widehat{f}}_{i+1}(X,Y,Z,s,t,\tau )\hfill \\ & \times M_{i+1}[a(X,Y,Z)+v_2(s,t,\tau )]dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(19)

since $v_2(x_0,y,z)=0$, the arrangement of Equation (18) can be obtained.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill G_1(a(X,Y,Z)+v_2(x,y,z))& \le G_1\left(a(X,Y,Z)\right)+{\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}\hfill \\ & \times {\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau \hfill \\ & +\sum _{i=1}^k{\int }_{b_{i+1}\left(x_0\right)}^{b_{i+1}\left(x\right)}{\int }_{c_{i+1}\left(y_0\right)}^{c_{i+1}\left(y\right)}{\int }_{d_{i+1}\left(z_0\right)}^{d_{i+1}\left(z\right)}{\widehat{f}}_{i+1}(X,Y,Z,s,t,\tau )\hfill \\ & \times M_{i+1}[a(X,Y,Z)+v_2(s,t,\tau )]dsdtd\tau .\hfill \end{array}\hfill \end{array}$$

(20)

According to Equation (4), the definition of $a_2(x,y,z)$ is as follows.

$$a_2(X,Y,Z)=G_1\left(a_1(X,Y,Z)\right)+{\int }_{b_1\left(x_0\right)}^{b_1\left(x\right)}{\int }_{c_1\left(y_0\right)}^{c_1\left(y\right)}{\int }_{d_1\left(z_0\right)}^{d_1\left(z\right)}{\widehat{f}}_1(X,Y,Z,s,t,\tau )dsdtd\tau .$$

Let $g(x,y,z)=G_1(a(X,Y,Z)+v_2(x,y,z))$, then Equation (20) can be reduced to

$$\begin{array}{cc}\hfill g(x,y,z)& \le a_2\left(X,Y,Z\right)+\sum _{i=1}^k{\int }_{b_{i+1}\left(x_0\right)}^{b_{i+1}\left(x\right)}{\int }_{c_{i+1}\left(y_0\right)}^{c_{i+1}\left(y\right)}{\int }_{d_{i+1}\left(z_0\right)}^{d_{i+1}\left(z\right)}\hfill \\ & \times {\widehat{f}}_{i+1}(X,Y,Z,s,t,\tau )M_{i+1}\left[G_1^{-1}\left(g(s,t,\tau )\right)\right]dsdtd\tau ,\hfill \end{array}$$

(21)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Equation (21) has the same form as Equation (1). From functions $a_2(X,Y,Z)$ and $M_{i+1}\left(u\right)$ satisfying the conditions of functions $a(x,y,z)$ and $W_i\left(u\right)$ in Equation (1) respectively, and from the result of induction hypothesis $n=k$, we obtain

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill g(x,y,z)& \le {\mathrm{\Phi }}_{k+1}^{-1}[{\mathrm{\Phi }}_{k+1}\left({\mathrm{\Phi }}_k^{-1}\left(a_{k+1}(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_{k+1}\left(x_0\right)}^{b_{k+1}\left(x\right)}{\int }_{c_{k+1}\left(y_0\right)}^{c_{k+1}\left(y\right)}{\int }_{d_{k+1}\left(z_0\right)}^{d_{k+1}\left(z\right)}{\widehat{f}}_{k+1}(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(22)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Where

$$\begin{array}{ccc}& & {\mathrm{\Phi }}_{k+1}\left(u\right)={\int }_{{\tilde{u}}_{k+1}}^u{\displaystyle \frac{d\xi }{M_{k+1}\left(G_1^{-1}\left(\xi \right)\right)}},\hfill \end{array}$$

where

$$\begin{array}{ccc}& & {\tilde{u}}_{k+1}=G_1\left(u_{k+1}\right).\hfill \end{array}$$

${\mathrm{\Phi }}_{k+1}^{-1}\left(u\right)$ is the inverse function of ${\mathrm{\Phi }}_{k+1}\left(u\right)$, and notice that

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill {\mathrm{\Phi }}_{k+1}\left(u\right)& ={\int }_{{\tilde{u}}_{k+1}}^u{\displaystyle \frac{d\xi }{M_{k+1}\left(G_1^{-1}\left(\xi \right)\right)}}={\int }_{G_1\left(u_{k+1}\right)}^u{\displaystyle \frac{W_1\left(G_1^{-1}\left(\xi \right)\right)}{W_{k+1}\left(G_1^{-1}\left(\xi \right)\right)}}d\xi \hfill \\ & ={\int }_{u_{k+1}}^{G_1^{-1}\left(u\right)}{\displaystyle \frac{d\xi }{W_{k+1}\left(\xi \right)}}=G_{k+1}\left(G_1^{-1}\left(u\right)\right).\hfill \end{array}\hfill \end{array}$$

So

$$\begin{array}{ccc}& & {\mathrm{\Phi }}_{k+1}^{-1}\left(u\right)=G_1\left(G_{k+1}^{-1}\left(u\right)\right),\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& & {\mathrm{\Phi }}_k\left(u\right)=G_k\left(G_1^{-1}\left(u\right)\right).\hfill \end{array}$$

(24)

Then Equation (22) can be arranged by Equations (23) and (24) as follows.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill g(x,y,z)& \le G_1\left\{G_{k+1}^{-1}\right[G_{k+1}\left(G_k^{-1}\left(a_{k+1}(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_{k+1}\left(x_0\right)}^{b_{k+1}\left(x\right)}{\int }_{c_{k+1}\left(y_0\right)}^{c_{k+1}\left(y\right)}{\int }_{d_{k+1}\left(z_0\right)}^{d_{k+1}\left(z\right)}{\widehat{f}}_{k+1}(x,y,z,s,t,\tau )dsdtd\tau \left]\right\},\hfill \end{array}\hfill \end{array}$$

(25)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Furthermore, Equation (25) can be transformed from Equations (21) and (9) to

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le G_{k+1}^{-1}[G_{k+1}\left(G_k^{-1}\left(a_{k+1}(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_{k+1}\left(x_0\right)}^{b_{k+1}\left(x\right)}{\int }_{c_{k+1}\left(y_0\right)}^{c_{k+1}\left(y\right)}{\int }_{d_{k+1}\left(z_0\right)}^{d_{k+1}\left(z\right)}{\widehat{f}}_{k+1}(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Thus when $n=k+1$, the proposition also holds.

Finally, Theorem 1 is proved by (i) and (ii). □

Establishing the boundedness of solutions of integral inequalities leads to the boundedness of the solutions of the corresponding integral or differential equations. Next we can discuss the boundedness of Equation (1). To analyze the boundedness of Equation (1) with recursive form requires the following assumption:

(K₇)

$a\in C_{\mathsf{\Omega }}(\mathsf{\Lambda },{\mathbb{R}}_{+})$ is bounded.

(K₈)

For all ${\widehat{f}}_i$, $b_i$, $c_i$, $d_i$ such that

$$\begin{array}{ccc}& & \underset{(x,y,z)\to (+\infty ,+\infty ,+\infty )}{lim}{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\widehat{f}}_i(x,y,z,s,t,\tau )dsdtd\tau <+\infty .\hfill \end{array}$$

Following from Assumption $\left(K_7\right)$, we know that function $a_1(x,y,z)$ is bounded, and combining with Equations (4) and (6) in Theorem 1, we can show that function $a_i(x,y,z)(i=1,2,\dots ,n)$ is bounded. Additionally, Assumption $\left(K_8\right)$ implies that the function $u(x,y,z)$ is bounded with respect to $(x,y,z)\in \mathsf{\Lambda }$ in Inequality (1).

Corollary 1.

Under the condition that Assumptions $\left(K_2\right)$–$\left(K_4\right)$ hold, if $u\in C(\mathsf{\Lambda },{\mathbb{R}}_{+})$, and $u(x,y,z)$ satisfies the weakly singular integral inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times f_i(x,y,z,s,t,\tau ){\left(u(s,t,\tau )\right)}^{k}dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(26)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Then

(i) 

If $k=1$,

$$\begin{array}{ccc}& & u(x,y,z)\le {\tilde{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\exp \left[{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau \right],\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Where ${\tilde{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\tilde{G}}_i\left(u\right)$,

$$\begin{array}{ccc}& & {\tilde{G}}_i\left(u\right)={\int }_{u_i}^u{\displaystyle \frac{d\xi }{\xi }}=ln{\displaystyle \frac{u}{u_i}},u\ge u_i>0,i=1,2,\dots ,n.\hfill \end{array}$$

(27)

$$\begin{array}{ccc}& & {\tilde{G}}_i^{-1}\left(u\right)=u_i{e}^u,Dom\left({\tilde{G}}_i^{-1}\right)=\left[0,\infty \right),i=1,2,\dots ,n.\hfill \end{array}$$

(28)

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{ccc}& & \begin{array}{cc}& {\tilde{G}}_i\left({\tilde{G}}_{i-1}^{-1}\left(a_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}\hfill \\ & \times {\widehat{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left({\tilde{G}}_i^{-1}\right).\hfill \end{array}\hfill \end{array}$$

(ii) 

If $0<k<1$,

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le [{\left({\overline{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\right)}^{1-k}+(1-k)\hfill \\ & \times {\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau {]}^{1/(1-k)},\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Where ${\overline{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\overline{G}}_i\left(u\right)$,

$$\begin{array}{ccc}& & {\overline{G}}_i\left(u\right)={\int }_{u_i}^u{\displaystyle \frac{d\xi }{{\xi }^k}}={\displaystyle \frac{1}{1-k}}\left(u^{1-k}-u_i^{1-k}\right),u\ge u_i>0,i=1,2,\dots ,n.\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & \begin{array}{cc}& {\overline{G}}_i^{-1}\left(u\right)={\left(u_i^{1-k}+(1-k)u\right)}^{1/(1-k)},\hfill \\ & Dom\left({\overline{G}}_i^{-1}\right)=\left[0,\infty \right),i=1,2,\dots ,n.\hfill \end{array}\hfill \end{array}$$

(30)

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{ccc}& & \begin{array}{cc}& {\overline{G}}_i\left({\overline{G}}_{i-1}^{-1}\left(a_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}\hfill \\ & \times {\widehat{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left({\overline{G}}_i^{-1}\right).\hfill \end{array}\hfill \end{array}$$

The functions $a_i(x,y,z)$, ${\widehat{f}}_i(x,y,z,s,t,\tau )$ are all defined in Theorem 1.

Proof. 

It is not difficult to see that Inequality (26) is a special case of Inequality (1) in Theorem 1, and taking $W\left[\eta \right]={\eta }^k$ yields Inequality (26). When applying Theorem 1 to Inequality (26), since

(i)

If $k=1$, from Equation (2) combined with Equation (27), we can obtain the following.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le {\tilde{G}}_n^{-1}[ln\left({\displaystyle \frac{1}{v_n}}\left({\tilde{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(31)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Equation (31) is combined with Equation (28) to obtain

$$\begin{array}{ccc}& & u(x,y,z)\le {\tilde{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\exp \left[{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau \right],\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

(ii)

If $0<k<1$, from Inequality (2) combined with Equation (29), we can obtain the following.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le {\overline{G}}_n^{-1}[{\displaystyle \frac{1}{1-k}}\left({\left({\overline{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\right)}^{1-k}-u_n^{1-k}\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(32)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Inequality (32) is combined with Equation (30) to obtain

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le [{\left({\overline{G}}_{n-1}^{-1}\left(a_n(x,y,z)\right)\right)}^{1-k}+(1-k)\hfill \\ & \times {\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\widehat{f}}_n(x,y,z,s,t,\tau )dsdtd\tau {]}^{1/(1-k)},\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

It follows from (i) and (ii) that Corollary 1 is proved. □

Theorem 2.

Under the condition that Assumptions $\left(K_1\right)$–$\left(K_5\right)$ hold, if $u\in C(\mathsf{\Lambda },{\mathbb{R}}_{+})$, and $u(x,y,z)$ satisfies the weakly singular integral inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}\hfill \\ & \times {(y^{{\alpha }_i}-t^{{\alpha }_i})}^{{\beta }_i-1}{t}^{{\gamma }_i-1}{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{{\beta }_i-1}{\tau }^{{\gamma }_i-1}\hfill \\ & \times f_i(x,y,z,s,t,\tau )W_i\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(33)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Then

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le \left\{{\widehat{G}}_n^{-1}\right[{\widehat{G}}_n\left({\widehat{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q},\hfill \end{array}\hfill \end{array}$$

(34)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Where

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,\hfill \end{array}$$

$$\begin{array}{ccc}& & \tilde{a}(x,y,z)=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}a(\sigma ,\mu ,\zeta ),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & A_1(x,y,z)={(n+1)}^{q-1}{\tilde{a}}^q(x,y,z),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill A_i(x,y,z)& ={\widehat{G}}_{i-1}\left({\widehat{G}}_{i-2}^{-1}\left(A_{i-1}(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_{i-1}\left(x_0\right)}^{b_{i-1}\left(x\right)}{\int }_{c_{i-1}\left(y_0\right)}^{c_{i-1}\left(y\right)}{\int }_{d_{i-1}\left(z_0\right)}^{d_{i-1}\left(z\right)}{\tilde{f}}_{i-1}(x,y,z,s,t,\tau )dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(37)

for $i=1,2,\dots ,n$.

$$\begin{array}{ccc}& & {\widehat{f}}_i(x,y,z,s,t,\tau )=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}{f}_i(\sigma ,\mu ,\zeta ,s,t,\tau ),\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& & {\tilde{f}}_i(x,y,z,s,t,\tau )={(n+1)}^{q-1}{H}_i^q{\widehat{f}}_i^q(x,y,z,s,t,\tau ),\hfill \end{array}$$

(39)

for $i=1,2,\dots ,n$.

$$\begin{array}{ccc}& & H_i={\alpha }_i^{-3/p}{\left(xyz\right)}^{(1/p)+{\alpha }_i({\beta }_i-1)+({\gamma }_i-1)}\mathrm{B}{\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p({\beta }_i-1)+1\right]}^{3/p},\hfill \end{array}$$

for $i=1,2,\dots ,n$.

${\widehat{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\widehat{G}}_i\left(u\right)$,

$$\begin{array}{ccc}& & {\widehat{G}}_i\left(u\right)={\int }_{u_i}^u{\displaystyle \frac{d\xi }{{\tilde{W}}_i\left(\xi \right)}},u\ge u_i>0,i=1,2,\dots ,n.\hfill \end{array}$$

(40)

where

$$\begin{array}{ccc}& & {\tilde{W}}_i\left(u\right)=W_i^q\left(u^{1/q}\right).\hfill \end{array}$$

(41)

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{ccc}& & \begin{array}{cc}& {\widehat{G}}_i\left({\widehat{G}}_{i-1}^{-1}\left(a_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}\hfill \\ & \times {\tilde{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left({\widehat{G}}_i^{-1}\right),\hfill \end{array}\hfill \end{array}$$

for $i=1,2,\dots ,n$.

Proof. 

Let ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, and $p>1$, $q>0$, apply Lemma 4 to Equation (33), and it follows

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n({\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{p({\beta }_i-1)}{s}^{p({\gamma }_i-1)}\hfill \\ & \times {(y^{{\alpha }_i}-t^{{\alpha }_i})}^{p({\beta }_i-1)}{t}^{p({\gamma }_i-1)}{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{p({\beta }_i-1)}{\tau }^{p({\gamma }_i-1)}{)}^{1/p}\hfill \\ & \times {\left({\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{f}_i^q(x,y,z,s,t,\tau )W_i^q\left(u(s,t,\tau )\right)dsdtd\tau \right)}^{1/q},\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Since $0\le b_i\left(x\right)\le x$, $0\le c_i\left(y\right)\le y$, $0\le d_i\left(z\right)\le z$ can be obtained

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n({\int }_0^x{\int }_0^y{\int }_0^z{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{p({\beta }_i-1)}{s}^{p({\gamma }_i-1)}\hfill \\ & \times {(y^{{\alpha }_i}-t^{{\alpha }_i})}^{p({\beta }_i-1)}{t}^{p({\gamma }_i-1)}{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{p({\beta }_i-1)}{\tau }^{p({\gamma }_i-1)}{)}^{1/p}\hfill \\ & \times {\left({\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{f}_i^q(x,y,z,s,t,\tau )W_i^q\left(u(s,t,\tau )\right)dsdtd\tau \right)}^{1/q},\hfill \end{array}\hfill \end{array}$$

(42)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Defining the function $N_1$, $N_2$, $N_3$, and using Lemma 1, the following can be obtained.

$$\begin{array}{ccc}& & N_1={\int }_0^x{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{p({\beta }_i-1)}{s}^{p({\gamma }_i-1)}ds={\displaystyle \frac{x^{{\theta }_i}}{{\alpha }_i}}\mathrm{B}\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p({\beta }_i-1)+1\right],\hfill \end{array}$$

$$\begin{array}{ccc}& & N_2={\int }_0^y{(y^{{\alpha }_i}-t^{{\alpha }_i})}^{p({\beta }_i-1)}{t}^{p({\gamma }_i-1)}ds={\displaystyle \frac{y^{{\theta }_i}}{{\alpha }_i}}\mathrm{B}\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p({\beta }_i-1)+1\right],\hfill \end{array}$$

$$\begin{array}{ccc}& & N_3={\int }_0^z{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{p({\beta }_i-1)}{\tau }^{p({\gamma }_i-1)}ds={\displaystyle \frac{z^{{\theta }_i}}{{\alpha }_i}}\mathrm{B}\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p({\beta }_i-1)+1\right],\hfill \end{array}$$

where ${\theta }_i=p({\alpha }_i({\beta }_i-1)+{\gamma }_i-1)+1$, $i=1,2,\dots ,n$.

Define the function $H_i$ as

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill H_i& ={\left(N_1{N}_2{N}_3\right)}^{1/p}\hfill \\ & ={\alpha }_i^{-3/p}{\left(xyz\right)}^{(1/p)+{\alpha }_i({\beta }_i-1)+({\gamma }_i-1)}\mathrm{B}{\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p({\beta }_i-1)+1\right]}^{3/p},\hfill \end{array}\hfill \end{array}$$

(43)

for $i=1,2,\dots ,n$.

From Equation (43), Equation (42) is changed to

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{H}_i({\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{f}_i^q\left(x,y,z,s,t,\tau \right)\hfill \\ & \times W_i^q\left(u(s,t,\tau )\right)dsdtd\tau {)}^{1/q},\hfill \end{array}\hfill \end{array}$$

(44)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Taking a constant q to the second power on both sides of Equation (44) and applying Lemma 2, we obtain the following result.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u^q(x,y,z)& \le {(n+1)}^{q-1}[a^q(x,y,z)+\sum _{i=1}^n{H}_i^q{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times f_i^q(x,y,z,s,t,\tau )W_i^q\left(u(s,t,\tau )\right)dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(45)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

It follows from Equations (35) and (38) that $a(x,y,z)\le \tilde{a}(x,y,z)$, $f_i(x,y,z,s,t,\tau )\le {\widehat{f}}_i(x,y,z,s,t,\tau )$, and $\tilde{a}(x,y,z)$ are monotonically non-decreasing with respect to x, y, z, and ${\widehat{f}}_i(x,y,z,s,t,\tau )$ is continuous and monotonically non-decreasing with respect to x, y, z. Equation (45) can then be transformed into

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u^q(x,y,z)& \le {(n+1)}^{q-1}[{\tilde{a}}^q(x,y,z)+\sum _{i=1}^n{H}_i^q{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}\hfill \\ & \times {\widehat{f}}_i^q(x,y,z,s,t,\tau )W_i^q\left(u(s,t,\tau )\right)dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(46)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

For any fixed value of $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$, and Equation (36) and Equation (39) make Equation (46) become

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u^q(x,y,z)& \le A_1(X,Y,Z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\tilde{f}}_i^q(X,Y,Z,s,t,\tau )\hfill \\ & \times W_i^q\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Let the function $V_i(x,y,z)$ be

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& =A_1(X,Y,Z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\tilde{f}}_i^q(X,Y,Z,s,t,\tau )\hfill \\ & \times W_i^q\left(u(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(47)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

And from Equation (47) we can get

$$\begin{array}{ccc}& & V_i(x_0,y,z)=A_1(X,Y,Z),\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & u^q(x,y,z)\le V_i(x,y,z)\Rightarrow u(x,y,z)\le V_i^{1/q}(x,y,z),\hfill \end{array}$$

(49)

from Equation (49), Equation (48) can be changed into

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& =A_1(X,Y,Z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\tilde{f}}_i^q(X,Y,Z,s,t,\tau )\hfill \\ & \times W_i^q\left(V_i^{1/q}(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(50)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Define ${\tilde{G}}_i\left(u\right)$ in Equation (40), and using Equation (41) can make Equation (50) become

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& =A_1(X,Y,Z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{\tilde{f}}_i^q(X,Y,Z,s,t,\tau )\hfill \\ & \times {\tilde{W}}_i\left(V(s,t,\tau )\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(51)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

From the definitions of $A_1(X,Y,Z)$, ${\tilde{f}}_i(X,Y,Z,s,t,\tau )$, and ${\tilde{W}}_i\left(V(s,t,\tau )\right)$, we know that $A_1(X,Y,Z)$, ${\tilde{f}}_i(X,Y,Z,s,t,\tau )$, and ${\tilde{W}}_i\left(V(s,t,\tau )\right)$ satisfy the conditions of functions $a(x,y,z)$, $f_i(x,y,z,s,t,\tau )$, and $W_i\left(v\right)$ in Theorem 1, respectively, then Equation (51) can be obtained by applying Theorem 1 as follows.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& \le {\widehat{G}}_n^{-1}[{\widehat{G}}_n\left({\widehat{G}}_{n-1}^{-1}\left(A_n\left(X,Y,Z\right)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(X,Y,Z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Let $x=X$, $y=Y$ and $z=Z$. Considering the arbitrariness of the constants X, Y and Z, we have

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& \le {\widehat{G}}_n^{-1}[{\widehat{G}}_n\left({\widehat{G}}_{n-1}^{-1}\left(A_n\left(x,y,z\right)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(52)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Finally, Equation (49) makes Equation (52) become

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill V_i(x,y,z)& \le \left\{{\widehat{G}}_n^{-1}\right[{\widehat{G}}_n\left({\widehat{G}}_{n-1}^{-1}\left(A_n\left(x,y,z\right)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q},\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Thus Theorem 2 is proved. □

Corollary 2.

Under the condition that Assumptions $\left(K_1\right)$–$\left(K_4\right)$ hold, if $u\in C(\mathsf{\Lambda },{\mathbb{R}}_{+})$, and $u(x,y,z)$ satisfies the weakly singular integral inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a(x,y,z)+\sum _{i=1}^n{\int }_{b_i\left(x_0\right)}^{b_i\left(x\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(z\right)}{(x^{{\alpha }_i}-s^{{\alpha }_i})}^{{\beta }_i-1}{s}^{{\gamma }_i-1}\hfill \\ & \times {(y^{{\alpha }_i}-t^{{\alpha }_i})}^{{\beta }_i-1}{t}^{{\gamma }_i-1}{(z^{{\alpha }_i}-{\tau }^{{\alpha }_i})}^{{\beta }_i-1}{\tau }^{{\gamma }_i-1}\hfill \\ & \times f_i(x,y,z,s,t,\tau ){\left(u(s,t,\tau )\right)}^{k}dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(53)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$. Then

(i) 

If $k=1$,

$$\begin{array}{c}\hfill u(x,y,z)\le {\left({\tilde{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)}^{1/q}exp\left[{\displaystyle \frac{1}{q}}{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau \right],\end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Where ${\tilde{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\tilde{G}}_i\left(u\right)$, ${\tilde{G}}_i\left(u\right)$ is defined in Equation (27), $A_i(x,y,z)$ in Equation (37), and ${\tilde{f}}_i(x,y,z,s,t,\tau )$ in Equation (39).

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{c}\hfill {\tilde{G}}_i\left({\tilde{G}}_{i-1}^{-1}\left(A_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}{\tilde{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left({\tilde{G}}_i^{-1}\right),\end{array}$$

for $i=1,2,\dots ,n$.

(ii) 

If $0<k<1$,

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le [{\left({\overline{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)}^{1-k}+(1-k)\hfill \\ & \times {\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau {]}^{1/\left[q\right(1-k\left)\right]},\hfill \end{array}\hfill \end{array}$$

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Where ${\overline{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\overline{G}}_i\left(u\right)$, ${\overline{G}}_i\left(u\right)$ is defined in Equation (29), $A_i(x,y,z)$ in Equation (37), and ${\tilde{f}}_i(x,y,z,s,t,\tau )$ in Equation (39).

And $X,Y,Z\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{c}\hfill {\overline{G}}_i\left({\overline{G}}_{i-1}^{-1}\left(A_i(X,Y,Z)\right)\right)+{\int }_{b_i\left(x_0\right)}^{b_i\left(X\right)}{\int }_{c_i\left(y_0\right)}^{c_i\left(Y\right)}{\int }_{d_i\left(z_0\right)}^{d_i\left(Z\right)}{\tilde{f}}_i(X,Y,Z,s,t,\tau )dsdtd\tau \in Dom\left({\overline{G}}_i^{-1}\right),\end{array}$$

for $i=1,2,\dots ,n$.

Proof. 

It is not difficult to see that Inequality (53) is a special case of Inequality (33) in Theorem 2, and taking $W\left[\eta \right]={\eta }^k$ yields Inequality (53). When applying Theorem 2 to Inequality (53),

(i)

If $k=1$, by replacing ${\widehat{G}}_i\left(u\right)$ by ${\tilde{G}}_i\left(u\right)$ in Equation (34) according to the change of proposition, the following can be obtained.

$$\begin{array}{cc}& \begin{array}{cc}\hfill u(x,y,z)\le & \left\{{\tilde{G}}_n^{-1}\right[{\tilde{G}}_n\left({\tilde{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q},\hfill \end{array}\end{array}$$

(54)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Using Equations (27) and (28) in Corollary 1, Equation (54) can be obtained.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)\le & \left\{{\tilde{G}}_n^{-1}\right[{\tilde{G}}_n\left({\tilde{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q}\hfill \\ \hfill =& \left\{{\tilde{G}}_n^{-1}\right[ln\left({\displaystyle \frac{1}{u_n}}\left({\tilde{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q}\hfill \\ \hfill =& {\left({\tilde{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)}^{1/q}exp[{\displaystyle \frac{1}{q}}{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}\hfill \\ & \times {\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ].\hfill \end{array}\hfill \end{array}$$

(ii)

If $0<k<1$, by replacing ${\widehat{G}}_i\left(u\right)$ by ${\overline{G}}_i\left(u\right)$ in Equation (34) according to the change of proposition, the following can be obtained

$$\begin{array}{cc}& \begin{array}{cc}\hfill u(x,y,z)\le & \left\{{\overline{G}}_n^{-1}\right[{\overline{G}}_n\left({\overline{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q},\hfill \end{array}\end{array}$$

(55)

for $x_0\le x\le X$, $y_0\le y\le Y$, $z_0\le z\le Z$.

Using Equations (29) and (30) in Corollary 1, Equation (55) can be obtained

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)\le & \left\{{\overline{G}}_n^{-1}\right[{\overline{G}}_n\left({\overline{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau ]{\}}^{1/q}\hfill \\ \hfill =& \{{\overline{G}}_n^{-1}\left[{\displaystyle \frac{1}{1-k}}{\left({\overline{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)}^{1-k}-u_n^{1-k}\right]\hfill \\ & +{\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}{\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau {\}}^{1/q}\hfill \\ \hfill =& [{\left({\overline{G}}_{n-1}^{-1}\left(A_n(x,y,z)\right)\right)}^{1-k}+(1-k){\int }_{b_n\left(x_0\right)}^{b_n\left(x\right)}{\int }_{c_n\left(y_0\right)}^{c_n\left(y\right)}{\int }_{d_n\left(z_0\right)}^{d_n\left(z\right)}\hfill \\ & \times {\tilde{f}}_n(x,y,z,s,t,\tau )dsdtd\tau {]}^{1/\left[q(1-k)\right]}.\hfill \end{array}\hfill \end{array}$$

Thus by (i) and (ii), Corollary 2 is proved. □

Theorem 3.

Under the condition that Assumptions $\left(K_1\right)$, $\left(K_5\right)$ and $\left(K_6\right)$ hold, if $u\in C({\mathsf{\Lambda }}^1,{\mathbb{R}}_{+})$, and $u(x_1,x_2,\dots ,x_n)$ satisfies the weakly singular integral inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)\le & a(x_1,x_2,\dots ,x_n)+\sum _{i=1}^n{\int }_{b_{1,i}\left(x_{1,0}\right)}^{b_{1,i}\left(x_1\right)}{\int }_{b_{2,i}\left(x_{2,0}\right)}^{b_{2,i}\left(x_2\right)}\dots {\int }_{b_{n,i}\left(x_{n,0}\right)}^{b_{n,i}\left(x_n\right)}\hfill \\ & \times {\left(x_1^{{\alpha }_i}-s_1^{{\alpha }_i}\right)}^{{\beta }_i-1}{s}_1^{{\gamma }_i-1}{(x_2^{{\alpha }_i}-s_2^{{\alpha }_i})}^{{\beta }_i-1}{s}_2^{{\gamma }_i-1}\dots {(x_n^{{\alpha }_i}-s_n^{{\alpha }_i})}^{{\beta }_i-1}{s}_n^{{\gamma }_i-1}\hfill \\ & \times f_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)W_i\left(u(s_1,s_2,\dots ,s_n)\right)d{s}_{1}d{s}_2\dots d{s}_n,\hfill \end{array}\hfill \end{array}$$

for $x_{1,0}\le x_1\le X_1$, $x_{2,0}\le x_2\le X_2$,…, $x_{n,0}\le x_n\le X_n$. Then

$$\begin{array}{cc}& \begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)\le & \left\{{\widehat{G}}_n^{-1}\right[{\widehat{G}}_n\left({\widehat{G}}_{n-1}^{-1}\left({\tilde{A}}_n(x_1,x_2,\dots ,x_n)\right)\right)\hfill \\ & +{\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(x_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(x_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(x_n\right)}{\overline{f}}_n(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)\hfill \\ & \times d{s}_{1}d{s}_2\dots d{s}_n]{\}}^{1/q},\hfill \end{array}\end{array}$$

for $x_{1,0}\le x_1\le X_1$, $x_{2,0}\le x_2\le X_2$,…, $x_{n,0}\le x_n\le X_n$. Where

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,\end{array}$$

$$\begin{array}{c}\hfill \overline{a}(x_1,x_2,\dots ,x_n)=\underset{x_{1,0}\le {\sigma }_1\le x_1,x_{2,0}\le {\sigma }_2\le x_2,\dots ,x_{n,0}\le {\sigma }_n\le x_n}{max}a({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n),\end{array}$$

$$\begin{array}{c}\hfill {\tilde{A}}_1(x_1,x_2,\dots ,x_n)={(n+1)}^{q-1}{\overline{a}}^q(x_1,x_2,\dots ,x_n),\end{array}$$

$$\begin{array}{cc}& \begin{array}{cc}\hfill {\tilde{A}}_i(x_1,x_2,\dots ,x_n)\le & {\widehat{G}}_{i-1}\left({\widehat{G}}_{i-2}^{-1}\left({\tilde{A}}_{i-1}(x_1,x_2,\dots ,x_n)\right)\right)\hfill \\ & +{\int }_{b_{1,i-1}\left(x_{1,0}\right)}^{b_{1,i-1}\left(x_1\right)}{\int }_{b_{2,i-1}\left(x_{2,0}\right)}^{b_{2,i-1}\left(x_2\right)}\dots {\int }_{b_{n,i-1}\left(x_{n,0}\right)}^{b_{n,i-1}\left(x_n\right)}\hfill \\ & \times {\tilde{f}}_{i-1}(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n,\hfill \end{array}\end{array}$$

(56)

for $i=1,2,\dots ,n$.

$$\begin{array}{ccc}& & {\widehat{f}}_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}{f}_i({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n,s_1,s_2,\dots ,s_n),\hfill \end{array}$$

$$\begin{array}{cc}& \begin{array}{cc}\hfill {\overline{f}}_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)=& {(n+1)}^{q-1}{\tilde{H}}_i^q\hfill \\ & \times {\widehat{f}}_i^q(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n),\hfill \end{array}\end{array}$$

(57)

for $i=1,2,\dots ,n$.

$$\begin{array}{cc}& \begin{array}{cc}\hfill {\tilde{H}}_i=& {\alpha }_i^{-n/p}{\left(\prod _{i=1}^n{x}_i\right)}^{(1/p)+{\alpha }_i({\beta }_i-1)+({\gamma }_i-1)}\hfill \\ & \times \mathrm{B}{\left[{\displaystyle \frac{p({\gamma }_i-1)+1}{{\alpha }_i}},p_i({\beta }_i-1)+1\right]}^{n/p},\hfill \end{array}\end{array}$$

(58)

for $i=1,2,\dots ,n$.

${\widehat{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\widehat{G}}_i\left(u\right)$; ${\widehat{G}}_i\left(u\right)$ is defined in Equation (40) of Theorem 2.

And $X_1,X_2,\dots ,X_n\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{cc}& \begin{array}{cc}& {\widehat{G}}_i\left({\widehat{G}}_{i-1}^{-1}\left(A_i(X_1,X_2,\dots ,X_n)\right)\right)+{\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(X_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(X_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(X_n\right)}\hfill \\ & \times {\overline{f}}_i(X_1,X_2,\dots ,X_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n\in Dom\left({\widehat{G}}_i^{-1}\right),\hfill \end{array}\end{array}$$

for $i=1,2,\dots ,n$.

Corollary 3.

Under the condition that Assumptions $\left(K_1\right)$ and $\left(K_6\right)$ hold, if $u\in C({\mathsf{\Lambda }}^1,{\mathbb{R}}_{+})$, and $u(x_1,x_2,\dots ,x_n)$ satisfies the weakly singular integral inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)& \le a(x_1,x_2,\dots ,x_n)+\sum _{i=1}^n{\int }_{b_{1,i}\left(x_{1,0}\right)}^{b_{1,i}\left(x_1\right)}{\int }_{b_{2,i}\left(x_{2,0}\right)}^{b_{2,i}\left(x_2\right)}\dots {\int }_{b_{n,i}\left(x_{n,0}\right)}^{b_{n,i}\left(x_n\right)}\hfill \\ & \times {\left(x_1^{{\alpha }_i}-s_1^{{\alpha }_i}\right)}^{{\beta }_i-1}{s}_1^{{\gamma }_i-1}{(x_2^{{\alpha }_i}-s_2^{{\alpha }_i})}^{{\beta }_i-1}{s}_2^{{\gamma }_i-1}\dots {(x_n^{{\alpha }_i}-s_n^{{\alpha }_i})}^{{\beta }_i-1}{s}_n^{{\gamma }_i-1}\hfill \\ & \times f_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n){\left(u(s_1,s_2,\dots ,s_n)\right)}^{k}d{s}_{1}d{s}_2\dots d{s}_n,\hfill \end{array}\hfill \end{array}$$

for $x_{1,0}\le x_1\le X_1$, $x_{2,0}\le x_2\le X_2$,…, $x_{n,0}\le x_n\le X_n$. Then

(i) 

If $k=1$,

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)& \le {\left({\tilde{G}}_{n-1}^{-1}\left({\tilde{A}}_n(x_1,x_2,\dots ,x_n)\right)\right)}^{1/q}\hfill \\ & \times exp[{\displaystyle \frac{1}{q}}{\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(x_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(x_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(x_n\right)}\hfill \\ & \times {\overline{f}}_n(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n],\hfill \end{array}\hfill \end{array}$$

for $x_{1,0}\le x_1\le X_1$, $x_{2,0}\le x_2\le X_2$,…, $x_{n,0}\le x_n\le X_n$. Where ${\tilde{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\tilde{G}}_i\left(u\right)$, ${\tilde{G}}_i\left(u\right)$ is defined in Equation (27), ${\tilde{A}}_i(x_1,x_2,\dots ,x_n)$ in Equation (56), and ${\overline{f}}_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)$ in Equation (57).

And $X_1,X_2,\dots ,X_n\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{cc}& \begin{array}{cc}& {\tilde{G}}_i\left({\tilde{G}}_{i-1}^{-1}\left(A_i(X_1,X_2,\dots ,X_n)\right)\right)+{\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(X_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(X_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(X_n\right)}\hfill \\ & \times {\overline{f}}_i(X_1,X_2,\dots ,X_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n\in Dom\left({\tilde{G}}_i^{-1}\right),\hfill \end{array}\end{array}$$

for $i=1,2,\dots ,n$.

(ii) 

If $0<k<1$,

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x_1,x_2,\dots ,x_n)& \le [{\left({\overline{G}}_{n-1}^{-1}\left({\tilde{A}}_n(x_1,x_2,\dots ,x_n)\right)\right)}^{1-k}\hfill \\ & +(1-k){\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(x_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(x_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(x_n\right)}\hfill \\ & \times {\overline{f}}_n(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n{]}^{1/\left[q\right(1-k\left)\right]},\hfill \end{array}\hfill \end{array}$$

for $x_{1,0}\le x_1\le X_1$, $x_{2,0}\le x_2\le X_2$,…, $x_{n,0}\le x_n\le X_n$.

Where ${\overline{G}}_i^{-1}\left(u\right)$ is the inverse function of ${\overline{G}}_i\left(u\right)$, ${\overline{G}}_i\left(u\right)$ is defined in Equation (29), ${\tilde{A}}_i(x_1,x_2,\dots ,x_n)$ in Equation (56), and ${\overline{f}}_i(x_1,x_2,\dots ,x_n,s_1,s_2,\dots ,s_n)$ in Equation (57).

And $X_1,X_2,\dots ,X_n\in {\mathbb{R}}_{+}$ are chosen such that

$$\begin{array}{ccc}& & \begin{array}{cc}& {\overline{G}}_i\left({\overline{G}}_{i-1}^{-1}\left(A_i(X_1,X_2,\dots ,X_n)\right)\right)+{\int }_{b_{1,n}\left(x_{1,0}\right)}^{b_{1,n}\left(X_1\right)}{\int }_{b_{2,n}\left(x_{2,0}\right)}^{b_{2,n}\left(X_2\right)}\dots {\int }_{b_{n,n}\left(x_{n,0}\right)}^{b_{n,n}\left(X_n\right)}\hfill \\ & \times {\overline{f}}_i(X_1,X_2,\dots ,X_n,s_1,s_2,\dots ,s_n)d{s}_{1}d{s}_2\dots d{s}_n\in Dom\left({\overline{G}}_i^{-1}\right),\hfill \end{array}\hfill \end{array}$$

for $i=1,2,\dots ,n$.

4. Applications

[i] In this section, we consider the following boundary value problem (BVP) for delayed fractional partial differential equations:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}\left({}^{c}D_{{\chi }_0}^{\beta }u\right)(x,y,z)=U(x,y,z,u(x,y,z),u(b\left(x\right),c\left(y\right),d\left(z\right))),\hfill \\ u(x,y_0,z_0)=\phi \left(x\right),u(x_0,y,z_0)=\psi \left(y\right),u(x_0,y_0,z)=\varphi \left(z\right),\hfill \end{array}\right.\hfill \end{array}$$

(59)

for $x,y,z\in \mathsf{\Lambda }$. Where ${\chi }_0=(x_0,y_0,z_0)$, $U=C(\mathsf{\Lambda }\times {\mathbb{R}}^3,\mathbb{R})$, the functions $\phi =C_{\mathsf{\Omega }}(I,\mathbb{R})$, $\psi =C_{\mathsf{\Omega }}(J,\mathbb{R})$, $\varphi =C_{\mathsf{\Omega }}(T,\mathbb{R})$ and satisfy the condition $\phi \left(x_0\right)=\psi \left(y_0\right)=\varphi \left(z_0\right)$, $b=C_{\mathsf{\Omega }}^1(I,I)$, $c=C_{\mathsf{\Omega }}^1(J,J)$, $d=C_{\mathsf{\Omega }}^1(T,T)$, $0\le b\left(x\right)\le x$, $0\le c\left(y\right)\le y$, $0\le d\left(z\right)\le z$.

The operator ${}^{c}D_{{\chi }_0}^{\beta }$ denotes the Caputo fractional function and $\beta =({\beta }_1,{\beta }_2,{\beta }_3)\in {(0,1]}^3$ is its order. With Definitions 1 and 2, we know that

$$\begin{array}{ccc}& & \left({}^{c}D_{{\chi }_0}^{\beta }u\right)(x,y,z)=I_{{\chi }_0}^{1-\beta }{\displaystyle \frac{{\partial }^{3}u(x,y,z)}{\partial x\partial y\partial z}},\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill \left(I_{{\chi }_0}^{\beta }u\right)(x,y,z)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)\mathsf{\Gamma }\left({\beta }_2\right)\mathsf{\Gamma }\left({\beta }_3\right)}}{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z{(x-s)}^{{\beta }_1-1}\hfill \\ & \times {(y-t)}^{{\beta }_2-1}{(z-\tau )}^{{\beta }_3-1}u(s,t,\tau )dsdtd\tau .\hfill \end{array}\hfill \end{array}$$

In the following, we will discuss the boundedness, uniqueness and continuity of the solution of Equation (1) in turn. To obtain the conditions for boundedness, we first present the estimated solution of Equation (1).

Considering that the functions $\phi$, $\psi$ and $\varphi$ in Equation BVP (59) are all bounded, suppose that

$$\begin{array}{ccc}& & \begin{array}{cc}& \left|U(x,y,z,u(x,y,z),u(b\left(x\right),c\left(y\right),d\left(z\right))\right|\hfill \\ & \le h_1(x,y,z)W_1(|u(x,y,z)|)+h_2(x,y,z)W_2\left(|u(b\left(x\right),c\left(y\right),d\left(z\right))|\right),\hfill \end{array}\hfill \end{array}$$

(60)

where $h_i\in C(\mathsf{\Lambda },R_{+})$, $W_i$ satisfies assumption condition $\left(K_5\right)$$(i=1,2)$. The estimation of $u(x,y,z)$ in Equation BVP (59) can be obtained as follows.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill \left|u(x,y,z)\right|& \le G_2^{-1}[G_2\left(G_1^{-1}\left(a_2(x,y,z)\right)\right)\hfill \\ & +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(61)

where

$$\begin{array}{ccc}& & G_i\left(v\right)={\int }_{v_i}^v{\displaystyle \frac{d\xi }{W_i\left(\xi \right)}},v>v_i>0,i=1,2.\hfill \end{array}$$

(62)

$$\begin{array}{ccc}& & a_1(x,y,z)=\phi \left(x\right)+\psi \left(y\right)+\varphi \left(z\right)-\phi \left(x_0\right),\hfill \end{array}$$

(63)

$$\begin{array}{ccc}& & a_2(x,y,z)=G_1\left(a_1(x,y,z)\right)+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z{F}_1(x,y,z,s,t,\tau )dsdtd\tau ,\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & F_1(x,y,z,s,t,\tau )=\alpha (x,y,z,s,t,\tau )h_1(x,y,z),\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& & F_2(x,y,z,s,t,\tau )=\alpha (x,y,z,s,t,\tau )\left[{\displaystyle \frac{h_2\left(b^{-1}\left(s\right),c^{-1}\left(t\right),d^{-1}\left(\tau \right)\right)}{\left|b^{\prime }\left(b^{-1}\left(s\right)\right)c^{\prime }\left(c^{-1}\left(t\right)\right)d^{\prime }\left(d^{-1}\left(\tau \right)\right)\right|}}\right],\hfill \end{array}$$

(66)

$$\begin{array}{ccc}& & \alpha (x,y,z,s,t,\tau )={\displaystyle \frac{{(x-s)}^{{\beta }_1-1}{(y-t)}^{{\beta }_2-1}{(z-\tau )}^{{\beta }_3-1}}{\mathsf{\Gamma }\left({\beta }_1\right)\mathsf{\Gamma }\left({\beta }_2\right)\mathsf{\Gamma }\left({\beta }_3\right)}},\hfill \end{array}$$

(67)

$$\begin{array}{ccc}& & \underset{(x,y,z)\to (+\infty ,+\infty ,+\infty )}{lim}{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z{F}_1(x,y,z,s,t,\tau )dsdtd\tau <+\infty ,\hfill \end{array}$$

(68)

$$\begin{array}{ccc}& & \underset{(x,y,z)\to (+\infty ,+\infty ,+\infty )}{lim}{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau <+\infty .\hfill \end{array}$$

(69)

Proof. 

According to the definition of the Caputo fractional partial derivative in Reference [13], Equation BVP (59) can be transformed into the following equivalent integral equation.

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le a_1(x,y,z)+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )\hfill \\ & \times U(s,t,\tau ,u(s,t,\tau ),u(b\left(s\right),c\left(t\right),d\left(\tau \right)\left)\right)dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(70)

for $x,y,z\in \mathsf{\Lambda }$, where function $a_1(x,y,z)$ is defined in Equation (63) and function $\alpha (x,y,z,s,t,\tau )$ is defined in Equation (67). Substituting Equation (60) into Equation (70) and setting $s_1=b\left(s\right)$, $t_1=c\left(t\right)$, ${\tau }_1=d\left(\tau \right)$, we can deduce

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill \left|u(x,y,z)\right|& \le \left|a_1(x,y,z)\right|+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_1(s,t,\tau )\hfill \\ & \times W_1\left(\left|u(s,t,\tau )\right|\right)dsdtd\tau +{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )\hfill \\ & \times h_2(s,t,\tau )W_2\left(\left|u\left(b\right(s),c(t),d(\tau \left)\right)\right|\right)dsdtd\tau \hfill \\ & \le \left|a_1(x,y,z)\right|+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_1(s,t,\tau )\hfill \\ & \times W_1\left(\left|u(s,t,\tau )\right|\right)dsdtd\tau +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}\hfill \\ & \times \alpha (x,y,z,b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))\hfill \\ & \times {\displaystyle \frac{h_2(b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))}{\left|b^{\prime }\left(b^{-1}\left(s_1\right)\right)c^{\prime }\left(c^{-1}\left(t_1\right)\right)d^{\prime }\left(d^{-1}\left({\tau }_1\right)\right)\right|}}{W}_2\left(\left|u(s_1,t_1,{\tau }_1)\right|\right)d{s}_{1}d{t}_{1}d{\tau }_1.\hfill \end{array}\hfill \end{array}$$

(71)

Clearly, Inequality (71) satisfies the form of Equation (1) in Theorem 1, and the conditions $\left(K_2\right)$–$\left(K_5\right)$ are all assumed to hold. Therefore, the estimated Equation (61) for $u(x,y,z)$ can be obtained from Theorem 1. Note that since the functions $\phi$, $\psi$, $\varphi$ are all bounded functions, the holding of Equations (68) and (69) implies that the conditions $\left(K_7\right)$, $\left(K_8\right)$ are assumed to be satisfied, thus, reusing the boundedness proof of the estimated solution $u(x,y,z)$ in Theorem 1, we can show that $u(x,y,z)$ is bounded. □

Next, we will discuss the uniqueness of the solution of Equation BVP (59). For all $(x,y,z)\in \mathsf{\Lambda }$, suppose

$$\begin{array}{cc}& \begin{array}{cc}& \left|U(x,y,z,u(x,y,z),u(b\left(x\right),c\left(y\right),d\left(z\right)))-U(x,y,z,\tilde{u}(x,y,z),\tilde{u}(b\left(x\right),c\left(y\right),d\left(z\right)))\right|\hfill \\ & \le h_1(x,y,z)W_1(|u(x,y,z)-\tilde{u}(x,y,z)|)\hfill \\ & +h_2(x,y,z)W_2(|u(b\left(x\right),c\left(y\right),d\left(z\right))-\tilde{u}(b\left(x\right),c\left(y\right),d\left(z\right))|),\hfill \end{array}\end{array}$$

(72)

where $h_i\in C(\mathsf{\Lambda },R_{+})$, $W_i$ satisfies assumption condition $\left(K_5\right)$$(i=1,2)$. Suppose that both $u(x,y,z)$ and $\tilde{u}(x,y,z)$ are solutions of Equation BVP (59); then there is at most one solution of Equation BVP (59) on $\mathsf{\Lambda }$.

Proof. 

By equivalent integral Equation (70) and setting $s_1=b\left(s\right)$, $t_1=c\left(t\right)$, ${\tau }_1=d\left(\tau \right)$, we can deduce

$$\begin{array}{ccc}& & \begin{array}{cc}& \left|u(x,y,z)-\tilde{u}(x,y,z)\right|\hfill \\ & \le {\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_1(s,t,\tau )W_1\left(\left|u(s,t,\tau )-\tilde{u}(s,t,\tau )\right|\right)dsdtd\tau \hfill \\ & +{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_2(s,t,\tau )\hfill \\ & \times W_2(|u(b\left(s\right),c\left(t\right),d\left(\tau \right))-\tilde{u}(b\left(s\right),c\left(t\right),d\left(\tau \right))|)dsdtd\tau \hfill \\ & \le {\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_1(s,t,\tau )W_1\left(\left|u(s,t,\tau )-\tilde{u}(s,t,\tau )\right|\right)dsdtd\tau \hfill \\ & +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}\alpha (x,y,z,b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))\hfill \\ & \times {\displaystyle \frac{h_2(b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))}{\left|b^{\prime }\left(b^{-1}\left(s_1\right)\right)c^{\prime }\left(c^{-1}\left(t_1\right)\right)d^{\prime }\left(d^{-1}\left({\tau }_1\right)\right)\right|}}{W}_2\left(\left|u(s_1,t_1,{\tau }_1)-\tilde{u}(s_1,t_1,{\tau }_1)\right|\right)d{s}_{1}d{t}_{1}d{\tau }_1.\hfill \end{array}\hfill \end{array}$$

(73)

Clearly, Inequality (73) satisfies the form of Equation (1) in Theorem 1; similar to the proof of the boundedness of solutions, it can be verified that the Assumptions $\left(K_2\right)$–$\left(K_5\right)$ are satisfied. □

Using Theorem 1 for Equation (70), the following can be obtained

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill \left|u(x,y,z)-\tilde{u}(x,y,z)\right|& \le G_2^{-1}[G_2\left(G_1^{-1}\left(a_2(x,y,z)\right)\right)\hfill \\ & +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(74)

where $G_i$, $F_i$$(i=1,2)$, $a_2(x,y,z)$ are respectively defined in Equation (62), Equation (65), Equation (66), and Equation (67). It can be found that $a(x,y,z)=0$, such that $a_1(x,y,z)=0$. Combining the definition of $G_i$ and the properties of $W_i$, the following can be obtained.

$$\begin{array}{ccc}& & \underset{v\to 0}{lim}{G}_i\left(v\right)=-\infty ,\underset{v\to -\infty }{lim}{G}_i^{-1}\left(v\right)=0,i=1,2.\hfill \end{array}$$

(75)

Since Equations (67) and (68) both hold, then we have

$$\begin{array}{ccc}& & a_2(x,y,z)=G_1\left(a_1(x,y,z)\right)+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z{F}_1(x,y,z,s,t,\tau )dsdtd\tau =-\infty ,\hfill \end{array}$$

$$\begin{array}{ccc}& & G_1^{-1}\left(a_2(x,y,z)\right)=0,\hfill \end{array}$$

$$\begin{array}{ccc}& & G_2\left(G_1^{-1}\left(a_2(x,y,z)\right)\right)+{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau =-\infty ,\hfill \end{array}$$

$$\begin{array}{ccc}& & G_2^{-1}\left[G_2\left(G_1^{-1}\left(a_2(x,y,z)\right)\right)+{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau \right]=0.\hfill \end{array}$$

Therefore, the conclusion $\left|u(x,y,z)-\tilde{u}(x,y,z)\right|\le 0$ is obtained from Equation (74), that is, there is $u(x,y,z)=\tilde{u}(x,y,z)$ on $(x,y,z)\in \mathsf{\Lambda }$. Therefore, the uniqueness of the solution to Equation BVP (59) is proved.

Finally, we will discuss the continuous dependence of the solution of Equation BVP (59) on the given functions U, $\phi$, $\psi$, $\varphi$. Consider another delayed fractional partial differential equation BVP:

$$\begin{array}{ccc}& & \left\{\begin{array}{c}\left({}^{c}D_{{\chi }_0}^{\beta }u\right)(x,y,z)=\tilde{U}(x,y,z,\tilde{u}(x,y,z),\tilde{u}(b\left(x\right),c\left(y\right),d\left(z\right))),\hfill \\ \tilde{u}(x,y_0,z_0)=\tilde{\phi }\left(x\right),\tilde{u}(x_0,y,z_0)=\tilde{\psi }\left(y\right),\tilde{u}(x_0,y_0,z)=\tilde{\varphi }\left(z\right),\hfill \\ \tilde{\phi }\left(x_0\right)=\tilde{\psi }\left(y_0\right)=\tilde{\varphi }\left(z_0\right),\hfill \end{array}\right.\hfill \end{array}$$

(76)

for $x,y,z\in \mathsf{\Lambda }$. Where $U=C(\mathsf{\Lambda }\times {\mathbb{R}}^3,\mathbb{R})$, $\phi =C_{\mathsf{\Omega }}(I,\mathbb{R})$, $\psi =C_{\mathsf{\Omega }}(J,\mathbb{R})$, $\varphi =C_{\mathsf{\Omega }}(T,\mathbb{R})$. Suppose that the function U satisfies Equation (68) on a finite interval $\widehat{I}=\left[\begin{array}{c}{x}_0,x_1\end{array}\right]$, $\widehat{J}=\left[\begin{array}{c}{y}_0,y_1\end{array}\right]$, $\widehat{T}=\left[\begin{array}{c}{z}_0,z_1\end{array}\right]$, and $\tilde{\mathsf{\Lambda }}=\widehat{I}\times \widehat{J}\times \widehat{T}\subset {\mathbb{R}}^3$, $\left(|\tilde{U}-U|,|\tilde{\phi }-\phi |,|\tilde{\psi }-\psi |,|\tilde{\varphi }-\varphi |\right)\to 0$. Then the solution $\tilde{u}(x,y,z)$ in BVP (72) is sufficiently close to the solution $u(x,y,z)$ in BVP (59).

Proof. 

Using the uniqueness of the estimated solution $u(x,y,z)$, for the arbitrarily small constant ${\chi }_0$, assume

$$\begin{array}{c}\hfill \underset{x\in \widehat{I}}{max}|\tilde{\phi }-\phi |\le {\chi }_0,\underset{y\in \widehat{J}}{max}|\tilde{\psi }-\psi |\le {\chi }_0,\underset{z\in \widehat{T}}{max}|\tilde{\varphi }-\varphi |\le {\chi }_0,\end{array}$$

$$\begin{array}{c}\hfill \underset{(x,y,z,s,t,\tau )\in \tilde{\mathsf{\Lambda }}}{max}|\tilde{U}-U|\le {\chi }_0.\end{array}$$

By the equivalent integral Equation (70) as well as Equations (63) and (67), and setting $s_1=b\left(s\right)$, $t_1=c\left(t\right)$, ${\tau }_1=d\left(\tau \right)$, we can deduce

$$\begin{array}{cc}& \left|\begin{array}{c}\tilde{u}(x,y,z)-u(x,y,z)\end{array}\right|\hfill \\ & \le \left|\tilde{\phi }\left(x\right)-\phi \left(x\right)+\tilde{\psi }\left(y\right)-\psi \left(y\right)+\tilde{\varphi }\left(z\right)-\varphi \left(z\right)+\phi \left(x_0\right)-\tilde{\phi }\left(x_0\right)\right|\hfill \\ & +{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )|\tilde{U}(s,t,\tau ,\tilde{u}(s,t,\tau ),\tilde{u}(b\left(s\right),c\left(t\right),d\left(\tau \right)))\hfill \\ & -U(s,t,\tau ,u(s,t,\tau ),u(b\left(s\right),c\left(t\right),d\left(\tau \right)\left)\right)|dsdtd\tau \hfill \\ & \le 4{\chi }_0+{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )|\tilde{U}(s,t,\tau ,\tilde{u}(s,t,\tau ),\tilde{u}(b\left(s\right),c\left(t\right),d\left(\tau \right)))\hfill \\ & -U(s,t,\tau ,\tilde{u}(s,t,\tau ),\tilde{u}(b\left(s\right),c\left(t\right),d\left(\tau \right)))|dsdtd\tau \hfill \\ & +{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )|U\left(s,t,\tau ,\tilde{u}(s,t,\tau ),\tilde{u}(b\left(s\right),c\left(t\right),d\left(\tau \right))\right)\hfill \\ & -U(s,t,\tau ,u(s,t,\tau ),u(b\left(s\right),c\left(t\right),d\left(\tau \right)\left)\right)|dsdtd\tau \hfill \\ & \le \left[4+{\displaystyle \frac{{(x_1-x_0)}^{{\beta }_1}{(y_1-y_0)}^{{\beta }_2}{(z_1-z_0)}^{{\beta }_3}}{\mathsf{\Gamma }({\beta }_1+1)\mathsf{\Gamma }({\beta }_2+1)\mathsf{\Gamma }({\beta }_3+1)}}\right]{\chi }_0\hfill \\ & +{\int }_{x_0}^x{\int }_{y_0}^y{\int }_{z_0}^z\alpha (x,y,z,s,t,\tau )h_1(s,t,\tau )W_1\left(\left|\tilde{u}(s,t,\tau )-u(s,t,\tau )\right|\right)dsdtd\tau \hfill \\ & +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}\alpha (x,y,z,b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))\hfill \\ & \times {\displaystyle \frac{h_2(b^{-1}\left(s_1\right),c^{-1}\left(t_1\right),d^{-1}\left({\tau }_1\right))}{\left|b^{\prime }\left(b^{-1}\left(s_1\right)\right)c^{\prime }\left(c^{-1}\left(t_1\right)\right)d^{\prime }\left(d^{-1}\left({\tau }_1\right)\right)\right|}}{W}_2\left(\left|\tilde{u}\left(s_1,t_1,{\tau }_1\right)-u\left(s_1,t_1,{\tau }_1\right)\right|\right)d{s}_{1}d{t}_{1}d{\tau }_1.\hfill \end{array}$$

(77)

Clearly, Inequality (77) satisfies the form of Equation (1) in Theorem 1; applying Equation (77), we can determine that

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill \left|\tilde{u}(x,y,z)-u(x,y,z)\right|& \le G_2^{-1}[G_2\left(G_1^{-1}\left(a_2(x,y,z)\right)\right)\hfill \\ & +{\int }_{b\left(x_0\right)}^{b\left(x\right)}{\int }_{c\left(y_0\right)}^{c\left(y\right)}{\int }_{d\left(z_0\right)}^{d\left(z\right)}{F}_2(x,y,z,s,t,\tau )dsdtd\tau ],\hfill \end{array}\hfill \end{array}$$

(78)

where

$$\begin{array}{ccc}& & a_1(x,y,z)=\left[4+{\displaystyle \frac{{(x_1-x_0)}^{{\beta }_1}{(y_1-y_0)}^{{\beta }_2}{(z_1-z_0)}^{{\beta }_3}}{\mathsf{\Gamma }({\beta }_1+1)\mathsf{\Gamma }({\beta }_2+1)\mathsf{\Gamma }({\beta }_3+1)}}\right]{\chi }_0,\hfill \end{array}$$

$G_i$, $F_i$$(i=1,2)$, $a_2(x,y,z)$ are respectively defined in Equation (62), Equation (65), Equation (66), and Equation (67). It can be concluded that when ${\chi }_0\to 0$, $a_i(x,y,z)\to 0$$(i=1,2)$. Finally, through Equations (75) and (78), we can obtain:

$$\begin{array}{ccc}& & \underset{{\chi }_0\to 0}{lim}|\tilde{u}(x,y,z)-u(x,y,z)|=0.\hfill \end{array}$$

Thus, the continuity of the solution of the equation has been proven.

[ii] Consider the following inequality

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill u(x,y,z)& \le {\displaystyle \frac{1}{x+y+z+1}}+{\int }_0^{\sqrt{x}}{\int }_0^{\sqrt{y}}{\int }_0^{\sqrt{z}}{(x-s)}^{-1/5}{s}^{-1/2}\hfill \\ & \times {(y-t)}^{-1/5}{t}^{-1/2}{(z-\tau )}^{-1/5}{\tau }^{-1/2}xyz{e}^{-s-2t-3\tau }\hfill \\ & \times \sqrt[3]{u(s,t,\tau )}dsdtd\tau +{\int }_0^x{\int }_0^y{\int }_0^z{(x-s)}^{-1/5}{s}^{-1/2}\hfill \\ & \times {(y-t)}^{-1/5}{t}^{-1/2}{(z-\tau )}^{-1/5}{\tau }^{-1/2}\sqrt{u(s,t,\tau )}dsdtd\tau ,\hfill \end{array}\hfill \end{array}$$

(79)

for $x\ge 0$, $y\ge 0$, $z\ge 0$. Let ${\alpha }_i=1$$(i=1,2)$, ${\beta }_1={\displaystyle \frac{4}{5}}$, ${\gamma }_1={\displaystyle \frac{1}{2}}$, ${\beta }_2={\displaystyle \frac{2}{3}}$, ${\gamma }_2={\displaystyle \frac{2}{3}}$. From ${\gamma }_i>1-{\displaystyle \frac{1}{p}}$ and $p>1$, we get $1<p<2$; then from ${\displaystyle \frac{1}{p}}+{\alpha }_i({\beta }_i-1)+{\gamma }_i-1\ge 0$, we can take $p={\displaystyle \frac{4}{3}}$. It is easy to compute $q=4$ from ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$. Let $a(x,y,z)={\displaystyle \frac{1}{x+y+z+1}}$, $f_1(x,y,z,s,t,\tau )=xyz{e}^{-s-2t-3\tau }$, $f_2(x,y,z,s,t,\tau )=1$, $W_1\left(u\right)=\sqrt[3]{u}$, $W_2\left(u\right)=\sqrt{u}$ again, it follows that ${\displaystyle \frac{W_2\left(u\right)}{W_1\left(u\right)}}=u^{1/6}$ is monotonically non-decreasing.

Obviously, Equation (79) satisfies the conditions in Theorem 2, then

$$\begin{array}{ccc}& & \tilde{a}(x,y,z)=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}a(\sigma ,\mu ,\zeta )=1,\hfill \end{array}$$

$$\begin{array}{ccc}& & A_1(x,y,z)={(2+1)}^{4-1}{\tilde{a}}^4(x,y,z)=27,\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill H_1& ={\alpha }_1^{-3/p}{\left(xyz\right)}^{(1/p)+{\alpha }_1({\beta }_1-1)+({\gamma }_1-1)}\mathrm{B}{\left[{\displaystyle \frac{p({\gamma }_1-1)+1}{{\alpha }_1}},p({\beta }_1-1)+1\right]}^{3/p}\hfill \\ & ={\left(xyz\right)}^{1/20}\mathrm{B}{\left[{\displaystyle \frac{1}{3}},{\displaystyle \frac{11}{15}}\right]}^{9/4},\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill H_2& ={\alpha }_2^{-3/p}{\left(xyz\right)}^{(1/p)+{\alpha }_2({\beta }_2-1)+({\gamma }_2-1)}\mathrm{B}{\left[{\displaystyle \frac{p({\gamma }_2-1)+1}{{\alpha }_2}},p({\beta }_2-1)+1\right]}^{3/p}\hfill \\ & ={\left(xyz\right)}^{1/2}\mathrm{B}{\left[{\displaystyle \frac{5}{9}},{\displaystyle \frac{5}{9}}\right]}^{9/4},\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{f}}_1(x,y,z,s,t,\tau )=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}{f}_1(\sigma ,\mu ,\zeta ,s,t,\tau )=xyz{e}^{-s-2t-3\tau },\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{f}}_2(x,y,z,s,t,\tau )=\underset{0\le \sigma \le x,0\le \mu \le y,0\le \zeta \le z}{max}{f}_2(\sigma ,\mu ,\zeta ,s,t,\tau )=1,\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill {\tilde{f}}_1(x,y,z,s,t,\tau )& ={(2+1)}^{4-1}{H}_1^4{\widehat{f}}_1^4(x,y,z,s,t,\tau )\hfill \\ & =27{\left(xyz\right)}^{21/5}\mathrm{B}{\left[{\displaystyle \frac{1}{3}},{\displaystyle \frac{11}{15}}\right]}^9{e}^{-4s-8t-12\tau },\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill {\tilde{f}}_2(x,y,z,s,t,\tau )& ={(2+1)}^{4-1}{H}_2^4{\widehat{f}}_2^4(x,y,z,s,t,\tau )\hfill \\ & =27{\left(xyz\right)}^{1/3}\mathrm{B}{\left[{\displaystyle \frac{5}{9}},{\displaystyle \frac{5}{9}}\right]}^9,\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{G}}_1\left(v\right)={\int }_{v_1}^v{\displaystyle \frac{1}{\sqrt[3]{\xi }}}d\xi ={\displaystyle \frac{3}{2}}\left(v^{2/3}-v_1^{2/3}\right),v\ge v_1>0.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{G}}_1^{-1}\left(v\right)={\left({\displaystyle \frac{2}{3}}v-v_1^{2/3}\right)}^{3/2}.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{G}}_2\left(v\right)={\int }_{v_2}^v{\displaystyle \frac{1}{\sqrt{\xi }}}d\xi =2\left(\sqrt{v}-\sqrt{v_2}\right),v\ge v_2>0.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\widehat{G}}_2^{-1}\left(v\right)={\left({\displaystyle \frac{v}{2}}+\sqrt{v_2}\right)}^2.\hfill \end{array}$$

$$\begin{array}{ccc}& & \begin{array}{cc}\hfill A_2(x,y,z)& ={\widehat{G}}_1\left(A_1(x,y,z)\right)+{\int }_0^{\sqrt{x}}{\int }_0^{\sqrt{y}}{\int }_0^{\sqrt{z}}{\tilde{f}}_1(x,y,z,s,t,\tau )dsdtd\tau \hfill \\ & ={\displaystyle \frac{3}{2}}\left(9-v_1^{2/3}\right)-{\displaystyle \frac{9}{128}}{\left(xyz\right)}^{21/5}\mathrm{B}{\left[{\displaystyle \frac{1}{3}},{\displaystyle \frac{11}{15}}\right]}^9\hfill \\ & \times \left(e^{-4\sqrt{x}}-1\right)\left(e^{-8\sqrt{y}}-1\right)\left(e^{-12\sqrt{z}}-1\right).\hfill \end{array}\hfill \end{array}$$

Since ${\int }_{v_1}^{\infty }{\displaystyle \frac{1}{\sqrt[3]{\xi }}}d\xi =\infty$, ${\int }_{v_2}^{\infty }{\displaystyle \frac{1}{\sqrt{\xi }}}d\xi =\infty$, so $X=\infty$, $Y=\infty$, $Z=\infty$. Therefore, from the conclusion of Theorem 2,we can obtain

$$\begin{array}{cc}& u(x,y,z)\hfill \\ & \le {\left\{{\widehat{G}}_2^{-1}\left[{\widehat{G}}_2\left({\widehat{G}}_1^{-1}\left(A_2(x,y,z)\right)\right)+{\int }_0^x{\int }_0^y{\int }_0^z{\tilde{f}}_2(x,y,z,s,t,\tau )dsdtd\tau \right]\right\}}^{1/4}\hfill \\ & =\left\{\right[9-{\displaystyle \frac{3}{63}}{\left(xyz\right)}^{21/5}\mathrm{B}{\left[{\displaystyle \frac{1}{3}},{\displaystyle \frac{11}{15}}\right]}^9\left(e^{-4\sqrt{x}}-1\right)\left(e^{-8\sqrt{y}}-1\right)\left(e^{-12\sqrt{z}}-1\right){]}^{3/4}\hfill \\ & +{\displaystyle \frac{27}{2}}{\left(xyz\right)}^{4/3}\mathrm{B}{\left[{\displaystyle \frac{5}{9}},{\displaystyle \frac{5}{9}}\right]}^9{\}}^{1/2},\hfill \end{array}$$

for $x\ge 0$, $y\ge 0$, $z\ge 0$. This implies that the solution $u(x,y,z)$ of Equation (79) is bounded. □