\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 256, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/256\hfil Nonlinear double integral inequalities] {Bounds for solutions to retarded nonlinear double integral inequalities} \author[S. Hussain, T. Riaz, Q.-H. Ma, J. Pe\v{c}ari\'c \hfil EJDE-2014/256\hfilneg] {Sabir Hussain, Tanzila Riaz, Qing-Hua Ma, Josip Pe\v{c}ari\'c} % in alphabetical order \address{Sabir Hussain \newline Department of Mathematics, University of Engineering and Technology, Lahore Pakistan} \email{sabirhus@gmail.com} \address{Tanzila Riaz \newline Department of Mathematics, University of Engineering and Technology, Lahore Pakistan} \email{tanzila.ch@hotmail.com} \address{Qing-Hua Ma (corresponding author)\newline Department of Applied Mathematics, Guangdong University of Foreign Studies, \newline Guangzhou 510420, China} \email{gdqhma@21cn.com} \address{J. Pe\v{c}ari\'c \newline Faculty of Textile Technology, University of Zagreb Pierottijeva 6, 10000 Zagreb, Croatia} \email{pecaric@element.hr} \thanks{Submitted July 11, 2014. Published December 10, 2014.} \subjclass[2000]{30D05, 26D10} \keywords{Integral inequalities; Gronwall integral inequality; \hfill\break\indent integro-differential equation; double integral} \begin{abstract} We present bounds for the solution to three types retarded nonlinear integral inequalities in two variables. By doing this, we generalizing the results presented in \cite{c2,w1}. To illustrate our results, we present some applications. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In the study of the qualitative behavior for solutions to nonlinear differential and integral equations, some specific types of inequalities are needed. The Gronwall inequality \cite{g1} and the nonlinear version by Bihari \cite{b1} are fundamental tools in the study of existence, uniqueness, boundedness, stability of solutions of differential, integral, and integro-differential equations. For this reason, several generalizations of the Gronwall inequality have been obtained, see \cite{c1,c2,d1,l1,m1,m2,p1,p2,p3,w1,w2,w3,w4}. Retarded integral inequalities have played an extensive role in the study of partial differential and integral equations. In Section \ref{Sec2} of this article, based on the assumptions (A1)--(A3) below, we derive explicit bounds for the solutions to three types inequalities of retarded nonlinear integral equations in two variables. In Section \ref{Sec3}, the bounds are applied for proving the global boundedness of solutions to the initial boundary-value problems. We stud the following three inequalities: \begin{gather} \label{eq13} \begin{aligned} \varphi(u(t,s)) &\leq a(t,s)+b(t,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y)(w(u(x,y)) \\ &\quad +\int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n)) \,dn\, dm)+h_i(x,y)\Big]\,dy\, dx \end{aligned}\\ \label{eq14} \begin{aligned} \varphi(u(t,s)) &\leq a(t,s)+b(t,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y) \phi_1(u(x,y))\\ &\quad\times (w(u(x,y))+\int_{\alpha_i (t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm) \\ &\quad +h_i(x,y)\phi_{2}(\log(u(x,y)))\Big]\,dy\, dx \end{aligned} \\ \label{eq15} \begin{aligned} \varphi(u(t,s)) &\leq a(t,s)+b(t,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y) (w(u(x,y))\\ &\quad +\int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i (m,n)w(u(m,n))\,dn\, dm)\\ &\quad +h_i(x,y)L(x,y,w(u(x,y)))\Big]\,dy\, dx \end{aligned} \end{gather} \section{Main Results}\label{Sec2} Let $\mathbb{R}$ be the set of real numbers, $\mathbb{R}_{+}:[0,\infty)$; let $t_0,t_1,s_0,s_1$ be real numbers such that $I:=[t_0,t_1)$; $J:=[s_0,s_1)$. Denote by $C^{i}(M,N)$, the class of all $i$-times continuously differentiable functions defined on the set $M$ to the set $N$, $1\leq i\leq n$ and $C^{0}(M,N)=C(M,N)$. The first order partial derivatives of a function $z(x,y)$ defined on $\mathbb{R}^{2}$ with respect to $x$ and $y$ are denoted by $D_1z(x,y)(=z_{x}(x,y))$ and $D_{2}z(x,y)(=z_{y}(x,y))$ respectively. To prove our main results, we first list the following assumptions: \begin{itemize} \item[(A1)] $a,b:I\times J\to(0,\infty)$ are nondecreasing in each variable; \item[(A2)] $\varphi,w\in C(\mathbb{R}_{+},\mathbb{R}_{+})$, where $\varphi$ and $w$ are strictly increasing and nondecreasing functions respectively with $\varphi(0)=0$; $\varphi(t)\to\infty$ as $t\to\infty$ and $w>0$ on $(0,\infty)$; \item[(A3)] let $\alpha_i\in C^{1}(I,I)$ and $\beta_i\in C^{1}(J,J)$ be non-decreasing with $\alpha_i(t)\leq t$ on $I$ and $\beta_i(s)\leq s$ on $J$; \item[(A4)] let $u,f_i,g_i,h_i\in C(I\times J,\mathbb{R}_{+})$, $1\leq i\leq n$ and $\phi\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ a non-decreasing function such that $\phi(r)>0$ for $r>0$; \item[(A5)] let $\phi_1,\phi_{2}\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ be nondecreasing functions with $ \phi_1(r)>0$ and $\phi_{2}(r)>0$ for $r>0$. \end{itemize} \begin{theorem}\label{thm1} Assume conditions {\rm (A1)--(A4)} and relation \eqref{eq13} hold. Then \begin{equation}\label{eq1} u(t,s)\leq\varphi^{-1}(G^{-1}(\Psi^{-1}(\Psi(c(t,s))+b(t,s)D(t,s)))), \end{equation} for all $(t,s)\in [t_0,T_3)\times[s_0,S_3)$ provided that $\varphi^{-1},G^{-1},\Psi^{-1}$ are the respective inverses of $\varphi,G,\Psi$, and $(T_3,S_3)\in I\times J$ is arbitrarily chosen on the boundary of the planar region: $\mathfrak{R}:=\{(t,s)\in I\times J\}$, provided that the following three relations hold: \begin{equation}\label{A01} \begin{gathered} \Psi(c(t,s))+b(t,s)D(t,s)\in \operatorname{Dom}(\Psi^{-1}),\\ \Psi^{-1}(\Psi(c(t,s))+b(t,s)D(t,s))\in \operatorname{Dom}(G^{-1}),\\ G^{-1}(\Psi^{-1}(\Psi(c(t,s))+b(t,s)D(t,s)))\in \operatorname{Dom}(\varphi^{-1}), \end{gathered} \end{equation} where \begin{gather*} G(r):=\int_{r_0}^{r}\frac{dp}{\phi(\varphi^{-1}(p))},\quad r\geq r_0\geq0, \\ \Psi(z):=\int_{z_0}^{z}\frac{dl}{w(\varphi^{-1}(G^{-1}(l)))}, \quad z\geq z_0\geq0, \\ D(t,s):=\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(z,y)[1+\int_{\alpha_i(t_0)}^{z} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm]\,dy\, dz. \\ c(t,s):=G(a(t,s))+b(t,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)\,dy\, du. \end{gather*} \end{theorem} \begin{proof} By Assumption (A2) and inequality \eqref{eq13}, we have \begin{equation} \label{eq2} \begin{aligned} \varphi(u(t,s)) &\leq a(T,s)+b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y)(w(u(x,y))\\ &\quad + \int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\, dm)+h_i(x,y)\Big]\,dy\, dx \end{aligned} \end{equation} for all $(t,s)\in[t_0,T]\times J$, $T\leq T_3$. Denote the right hand side of \eqref{eq2} by $\eta(t,s)$, then obviously $\eta(t,s)$ is positive and non-decreasing function in each variable such that $\eta(t_0,s)=a(T,s)$. Then, \eqref{eq2} is equivalent to \begin{equation}\label{A4} u(t,s)\leq \varphi^{-1}(\eta(t,s)). \end{equation} \begin{align*} &\eta_{t}(t,s)\\ &= b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(\alpha_i(t),y)) \Big[f_i(\alpha_i(t),y)(w(u(\alpha_i(t),y)) \\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\, dm)+h_i(\alpha_i(t),y)\Big]dy \\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(\varphi^{-1} (\eta(\alpha_i(t),y)))\Big[f_i(\alpha_i(t),y) (w(\varphi^{-1}(\eta(\alpha_i(t),y))) \\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\varphi^{-1}(\eta(m,n)))\,dn\, dm)+h_i(\alpha_i(t),y)\Big]dy, \end{align*} which implies \begin{equation} \label{eq52} \begin{aligned} \eta_{t}(t,s) &\leq \phi(\varphi^{-1}(\eta(t,s)))b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (w(\varphi^{-1}(\eta(\alpha_i(t),y))) \\ &+\quad \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\varphi^{-1}(\eta(m,n)))\,dn\, dm)+h_i(\alpha_i(t),y)\Big]dy. \end{aligned} \end{equation} Then, \eqref{eq52} is equivalent to \begin{align*} &\frac{\eta_{t}(t,s)}{\phi(\varphi^{-1}(\eta(t,s)))}\\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (w(\varphi^{-1}(\eta(\alpha_i(t),y))) \\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y}g_i(m,n) w(\varphi^{-1}(\eta(m,n)))dn\,dm)+h_i(\alpha_i(t),y)\Big]dy, \end{align*} for all $(t,s)\in[t_0,T]\times J$. Replace $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of the above inequality and using the definition of $G$, we have \begin{equation} \label{eq3} \begin{aligned} G(\eta(t,s)) &\leq G(\eta(t_0,s))+b(T,s) \sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)\,dy\,du\\ &\quad +b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(u,y)(w(\varphi^{-1}(\eta(u,y)))\\ &\quad + \int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y}g_i(m,n) w(\varphi^{-1}(\eta(m,n)))\,dn\,dm)\,dy\, du \\ &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(T)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)\,dy\, du \\ &\quad +b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(u,y)(w(\varphi^{-1}(\eta(u,y))) \\ &\quad + \int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y}g_i(m,n) w(\varphi^{-1}(\eta(m,n)))\,dn\, dm)\,dy\, du \\ &\leq c(T,s)+b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(u,y)(w(\varphi^{-1}(\eta(u,y)))\\ &\quad + \int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y}g_i (m,n)w(\varphi^{-1}(\eta(m,n)))\,dn\,dm)\,dy\, du. \end{aligned} \end{equation} Denote the right hand side of \eqref{eq3} by $\Gamma(t,s)$, then obviously $\Gamma(t,s)$ is positive and non-decreasing function in each variable such that $\Gamma(t_0,s)=c(T,s)$. Then, \eqref{eq3} is equivalent to \begin{equation}\label{A5} \eta(t,s)\leq G^{-1}(\Gamma(t,s)). \end{equation} By the fact that $\alpha_i(t)\leq t$ and $\beta_i(s)\leq s$ for $(t,s)\in I\times J$, $1\leq i\leq n$, and monotonicity of $\Gamma$, $w$ and $\varphi^{-1}$, we have \begin{equation} \label{eq4} \begin{aligned} \Gamma_{t}(t,s) &= b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) (w(\varphi^{-1}(\eta(\alpha_i(t),y)))\\ &\quad + \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y}g_i (m,n)w(\varphi^{-1}(\eta(m,n)))\,dn\, dm)dy \\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) (w(\varphi^{-1}(\eta(t,y))) \\ &\quad + \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)w(\varphi^{-1}(\eta(t,y)))\,dn\, dm)dy\\ &\leq b(T,s)w(\varphi^{-1}(G^{-1}(\Gamma(t,s))))\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) \\ &\times\Big(1+ \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\Big)\,dn\,dm)dy. \end{aligned} \end{equation} Then, \eqref{eq4} is written as \begin{align*} &\frac{\Gamma_{t}(t,s)}{w(\varphi^{-1}(G^{-1}(\Gamma(t,s))))}\\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) \Big(1+ \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y}g_i(m,n) \,dn\,dm\Big)\,dy \end{align*} Replace $t$ by $q$ then integrating from $t_0$ to $t$ with respect to $q$ and making change of variable on right hand side of the above inequality and using the definition of $\Psi$, we obtain \begin{equation}\label{eq5} \Psi(\Gamma(t,s))\leq \Psi(c(T,s))+b(T,s)D(t,s). \end{equation} A combination of \eqref{A4}$, \eqref{A5}$ and \eqref{eq5} yield the desire result \eqref{eq1}. \end{proof} \begin{theorem}\label{thm2} Assume conditions {\rm (A1)--(A5)} and relation \eqref{eq14} hold. Then \begin{itemize} \item if $\phi_1(r)\geq \phi_{2}(\log(r))$, we have \begin{equation}\label{eq16} u(t,s)\leq\varphi^{-1}(G^{-1}(H_1^{-1}(J_1^{-1}(J_1(\widetilde{c}(T,s)) +b(T,s)D(t,s))))), \end{equation} for $(t,s)\in[t_0,T_1)\times[s_0,S_1)$, \item if $\phi_1(r)< \phi_{2}(\log(r))$, we have \begin{equation}\label{eq17} u(t,s)\leq\varphi^{-1}(G^{-1}(H_{2}^{-1}(J_{2}^{-1}(J_{2}(\widetilde{c}(T,s)) +b(T,s)D(t,s))))), \end{equation} \end{itemize} for all $(t,s)\in [t_0,T_{2})\times[s_0,S_{2})$, provided that $\varphi^{-1},G^{-1},H_{j}^{-1}$ and $J_{j}^{-1}$ are the respective inverses of $\varphi,G,H_{j}$ and $H_{j}$; let $(T_{j},S_{j})\in I\times J$ be arbitrarily chosen on the boundary of the planar region $ \mathfrak{R}_{j}:=\{(t,s)\in I\times J\}$, $j\in\{1,2\}$, provided that the following four relations are satisfied \begin{equation}\label{A02} \begin{gathered} J_{j}(\widetilde{c}(T,s)) +b(T,s)D(t,s)\in \operatorname{Dom}(J_{j}^{-1}),\\ J_{j}^{-1}(J_{j}(\widetilde{c}(T,s)) +b(T,s)D(t,s))\in \operatorname{Dom}(H_{j}^{-1}),\\ H_{j}^{-1}(J_{j}^{-1}(J_{j}(\widetilde{c}(T,s)) +b(T,s)D(t,s)))\in \operatorname{Dom}(G^{-1}),\\ G^{-1}(H_{j}^{-1}(J_{j}^{-1}(J_{j}(\widetilde{c}(T,s)) +b(T,s)D(t,s))))\in \operatorname{Dom}(\varphi^{-1}), \end{gathered} \end{equation} where \begin{gather*} %\label{eq9} \widetilde{c}(t,s) = H_{j}(G(a(T,s)))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} h_i(u,y)\,dy\, du, \\ H_{j}(r)=\int_{r_0}^{r}\frac{ds}{\phi_{j}(\varphi^{-1}(G^{-1}(s)))},\\ J_{j}(r)=\int_{r_0}^{r}\frac{ds}{w(\varphi^{-1}(G^{-1}(H_{j}^{-1}(s))))}, \quad r\geq r_0\geq0. \end{gather*} \end{theorem} \begin{proof} By condition (A2) and inequality \eqref{eq14}, we have \begin{equation} \label{eq6} \begin{aligned} \varphi(u(t,s)) &\leq a(T,s)+b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y)\\ &\quad\times \phi_1(u(x,y))(w(u(x,y)) +\int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\, dm)\\ &\quad +h_i(x,y)\phi_{2}\big(\log(u(x,y))\big)\Big]\,dy\, dx \end{aligned} \end{equation} for all $(t,s)\in[t_0,T]\times J$, $T\leq T_1$. Denote the right hand side of \eqref{eq6} by $\Theta(t,s)$, then obviously $\Theta(t,s)$ is positive and non-decreasing function in each variable such that $\Theta(t_0,s)=a(T,s)$. Then \eqref{eq6} is equivalent to \begin{equation}\label{A9} u(t,s)\leq \varphi^{-1}(\Theta(t,s)). \end{equation} By the fact that $\alpha_i(t)\leq t$ and $\beta_i(s)\leq s$ for $(t,s)\in I\times J$, $1\leq i\leq n$, and monotonicity of $\phi$, $\varphi^{-1}$, $\Theta$, we have \begin{equation} \label{eq41} \begin{aligned} \Theta_{t}(t,s) &= b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(\alpha_i(t),y)) \Big[f_i(\alpha_i(t),y)\phi_1(u(\alpha_i(t),y)) \\ &\quad \times (w(u(\alpha_i(t),y))+ \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm) \\ &\quad +h_i(\alpha_i(t),y) \phi_{2}(\log(u(\alpha_i(t),y)))\Big]dy \\ &\leq b(T,s)\phi(\varphi^{-1}(\Theta(t,s)))\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y)\phi_1 (\varphi^{-1}(\Theta(\alpha_i(t),y))) \\ &\quad \times (w(\varphi^{-1}(\Theta(\alpha_i(t),y)))+ \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)w(u(m,n))\,dn\,dm) \\ &\quad +h_i(\alpha_i(t),y)\phi_{2} (\log(\varphi^{-1}(\Theta(\alpha_i(t),y))))\Big]dy, \end{aligned} \end{equation} for all $(t,s)\in[t_0,T]\times J$. From \eqref{eq41}, we have \begin{align*} &\frac{\Theta_{t}(t,s)}{\phi(\varphi^{-1}(\Theta(t,s)))}\\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y)\phi_1 (\varphi^{-1}(\Theta(\alpha_i(t),y))) (w(\varphi^{-1}(\Theta(\alpha_i(t),y)))\\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\, dm)\\ &\quad +h_i(\alpha_i(t),y) \phi_{2}(\log(\varphi^{-1}(\Theta(\alpha_i(t),y))))\Big]dy \end{align*} Replacing $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of the above inequality to obtain \begin{equation} \label{eq10} \begin{aligned} G(\Theta(t,s)) &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y)\phi_1(\varphi^{-1}(\Theta(u,y))) \\ &\quad \times (w(\varphi^{-1}(\Theta(u,y)))+\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm)\\ &\quad +h_i(u,y)\phi_{2}(\log(\varphi^{-1}(\Theta(u,y))))\Big]\,dy\, du \end{aligned} \end{equation} When $\phi_1(u)\geq \phi_{2}(\log(u))$, by \eqref{eq10}, we have \begin{equation} \label{eq8} \begin{aligned} G(\Theta(t,s)) &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \phi_1(\varphi^{-1}(\Theta(u,y))) \\ &\quad\times \Big[f_i(u,y)(w(\varphi^{-1}(\Theta(u,y)))\\ &\quad +\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\varphi^{-1}(\Theta(u,y)))\,dn\, dm)+h_i(u,y)]\,dy\, du. \end{aligned} \end{equation} Denote the right hand side of \eqref{eq8} by $\Lambda(t,s)$, then obviously $\Lambda(t,s)$ is positive and non-decreasing function in each variable such that $\Lambda(t_0,s)=G(a(T,s))$. Then \eqref{eq8} is equivalent to \begin{gather}\label{A10} \Theta(t,s)\leq G^{-1}(\Lambda(t,s)), \\ \label{eq42} \begin{aligned} &\Lambda_{t}(t,s)\\ &= b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\phi_1(\varphi^{-1} (\Theta(\alpha_i(t),y))) \Big[f_i(\alpha_i(t),y)w(\varphi^{-1}(\Theta(\alpha_i(t),y))) \\ &\quad\times \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)+h_i(\alpha_i(t),y)\Big]dy \\ &\leq b(T,s)\phi_1(\varphi^{-1}(G^{-1}(\Lambda(t,s))))\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) \\ &\quad\times w(\varphi^{-1}(G^{-1}(\Lambda(\alpha_i(t),y)))) \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)\\ &\quad +h_i(\alpha_i(t),y)\Big]dy. \end{aligned} \end{gather} From \eqref{eq42}, we have \begin{align*} &\frac{\Lambda_{t}(t,s)}{\phi_1(\varphi^{-1}(G^{-1}(\Lambda(t,s))))}\\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) w(\varphi^{-1}(G^{-1}(\Lambda(\alpha_i(t),y))))\\ &\quad\times\Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big) +h_i(\alpha_i(t),y)\Big]dy, \end{align*} Replacing $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of the above inequality and using the definition of $H_1$, we obtain \begin{align} &H_1(\Lambda(t,s)) \nonumber \\ &\leq H_1(G(a(T,s)))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y)w(\varphi^{-1}(G^{-1}(\Lambda(u,y)))) \nonumber\\ &\quad \times\Big(1+\int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)+h_i(u,y)\Big]\,dy\, du \nonumber \\ &\leq \widetilde{c}(T,s)+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} f_i(u,y)w(\varphi^{-1}(G^{-1}(\Lambda(u,y)))) \nonumber \\ &\quad \times\Big(1+\int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)\,dy\, du. \label{eq55} \end{align} Denote the right hand side of \eqref{eq55}, such that $\widetilde{\Theta}(t_0,s)=H_1(G(a(T,s)))$. Then \eqref{eq55} is equivalent to \begin{equation}\label{eq56} \Lambda(t,s)\leq H_1^{-1}(\widetilde{\Theta}(t,s)). \end{equation} By the fact that $\alpha_i(t)\leq t,\beta_i(s)\leq s$ for $(t,s)\in I\times J$, $1\leq i\leq n$, and monotonicity of $w,\varphi^{-1}$ and \eqref{eq56}, we have \begin{equation} \label{eq05} \begin{aligned} \widetilde{\Theta}_{t}(t,s) &= b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)} f_i(\alpha_i(t),y)w(\varphi^{-1}(G^{-1}(\Lambda(\alpha_i(t),y)))) \\ &\quad \times\Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)dy \\ &\leq b(T,s)w(\varphi^{-1}(G^{-1}(H_1^{-1}(\widetilde{\Theta}(t,s)))))\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)} f_i(\alpha_i(t),y) \\ &\quad \times(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm)dy. \end{aligned} \end{equation} From \eqref{eq05}, we have \begin{align*} &\frac{\widetilde{\Theta}_{t}(t,s)}{w(\varphi^{-1}(G^{-1}(H_1^{-1} (\widetilde{\Theta}(t,s)))))}\\ &\leq b(T,s)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)} f_i(\alpha_i(t),y) \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)dy. \end{align*} Replacing $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of the above inequality and using the definition of $J_1$, we obtain \begin{equation}\label{eq005} J_1(\widetilde{\Theta}(t,s))\leq J_1(\widetilde{c}(T,s)) +b(T,s)D(t,s) \end{equation} As $T\leq T_1$ is arbitrary, a combination of \eqref{A9}, \eqref{A10}, \eqref{eq56} and \eqref{eq005} yield \[ u(t,s)\leq\varphi^{-1}(G^{-1}(H_1^{-1}(J_1^{-1}(J_1(\widetilde{c}(T,s)) +b(T,s)D(t,s))))). \] When $\phi_1(u)\leq \phi_{2}(\log(u))$, by \eqref{eq10}, we have \begin{align*} &G(\Theta(t,s))\\ &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y)\phi_{2}(\log(\varphi^{-1}(\Theta(u,y))))\\ &\quad\times (w(\varphi^{-1}(\Theta(u,y))) +\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm)\\ &\quad +h_i(u,y)\phi_{2}(\log(\varphi^{-1}(\Theta(u,y))))\Big]\,dy\, du\\ &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(u,y) (w(\varphi^{-1}(\Theta(u,y)))\\ &\quad +\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n) w(u(m,n))\,dn\,dm) +h_i(u,y)\Big]\phi_{2}(\varphi^{-1}(\Theta(u,y)))\,dy\, du \end{align*} Similarly to the above process from \eqref{eq8} to \eqref{eq005}, for $T\leq T_{2}$, and as $T$ is arbitrary, we have \[ u(t,s)\leq\varphi^{-1}(G^{-1}(H_{2}^{-1}(J_{2}^{-1}(J_{2}(\widetilde{c}(T,s)) +b(T,s)D(t,s))))). \] \end{proof} \begin{theorem}\label{thm3} Suppose that {\rm (A1)--(A5)} hold and that $L,M\in C(\mathbb{R}_{+}^{3},\mathbb{R}_{+})$ are such that \[ 0\leq L(t,s,u)-L(t,s,v)\leq M(t,s,v)(u-v), \] for $u> v$. If $u(t,s)$ is a nonnegative and continuous function on $I\times J$ satisfying \eqref{eq15}, then we have \begin{equation}\label{eq36} \begin{aligned} u(t,s) &\leq \varphi^{-1}(G^{-1}(\Psi^{-1}(\Psi(G(a(t,s))\\ &\quad +b(t,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)L(u,y,0)\,dy\, du)\\ &\quad +b(t,s)\Big\{D(t,s)+\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}M(u,y,0)\,dy\, du\Big\}))), \end{aligned} \end{equation} for all $(t,s)\in [t_0,T_4)\times[s_0,S_4)$ provided that $\varphi^{-1},G^{-1},\Psi^{-1}$ are the respective inverses of $\varphi,G,\Psi$, and $(T_4,S_4)\in I\times J$ is arbitrarily chosen on the boundary of the planar region, $\mathfrak{R}_4:=\{(t,s)\in I\times J\}$, provided that the following three relations are satisfied: \begin{gather}\label{eq37} \begin{aligned} \widetilde{\Delta}(t,s) &:=\Big[\Psi(G(a(t,s))+b(t,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)L(u,y,0)\,dy\, du)\\ &\quad +b(t,s)\{D(t,s)+\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}M(u,y,0)\,dy\, du\}\Big]\in \operatorname{Dom}(\Psi^{-1}) \end{aligned} \\ \label{eq38} \Psi^{-1}(\widetilde{\Delta}(t,s))\in \operatorname{Dom}(G^{-1}),\quad G^{-1}(\Psi^{-1}(\widetilde{\Delta}(t,s)))\in \operatorname{Dom}(\varphi^{-1}) \end{gather} \end{theorem} \begin{proof} From assumption (A1) and the inequality \eqref{eq15}, we have \begin{equation} \label{eq33} \begin{aligned} \varphi(u(t,s)) &\leq a(T,s)+b(T,s)\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(x,y))\Big[f_i(x,y) (w(u(x,y))\\ &\quad +\int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm)\\ &\quad +h_i(x,y)L(x,y,w(u(x,y)))\Big]\,dy\, dx, \end{aligned} \end{equation} for all $(t,s)\in [t_0,T]\times J$, $T\leq T_4$. Denote the right hand side of \eqref{eq33} by $\mathfrak{P}(t,s)$, then obviously $\mathfrak{P}(t,s)$ is positive and non-decreasing function in each variable, $\mathfrak{P}(t_0,s)=a(T,s)$. Then, \eqref{eq33} is equivalent to \begin{equation}\label{eq34} u(t,s)\leq \varphi^{-1}(\mathfrak{P}(t,s)). \end{equation} By the fact that $\alpha_i(t)\leq t$ and $\beta_i(s)\leq s$ for $(t,s)\in I\times J$, $1\leq i\leq n$, and monotonicity of $\mathfrak{P}$, $\varphi^{-1}$, $\phi$, we have \begin{equation} \label{eq43} \begin{aligned} \mathfrak{P}_{t}(t,s) &= b(T,s)\sum_{i=1}^{n} \int_{\beta_i(s_0)}^{\beta_i(s)}\phi(u(\alpha_i(t),y))\Big[f_i(\alpha_i(t),y) (w(u(\alpha_i(t),y)) \\&\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\,dm) \\ &\quad +h_i(\alpha_i(t),y) L(\alpha_i(t),y,w(u(\alpha_i(t),y)))\Big]dy\alpha'_i(t) \\ &\leq b(T,s)\phi(\varphi^{-1}(\mathfrak{P}(t,s)))\sum_{i=1}^{n} \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (w(\varphi^{-1}(\mathfrak{P}(\alpha_i(t),y))) \\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)w(\varphi^{-1}(\mathfrak{P}(m,n)))\,dn\, dm)+h_i(\alpha_i(t),y) \\ &\quad \times L(\alpha_i(t),y,w(\varphi^{-1} (\mathfrak{P}(\alpha_i(t),y))))\Big]\,dy\, \alpha'_i(t). \end{aligned} \end{equation} From \eqref{eq43}, we have \begin{align*} \frac{\mathfrak{P}_{t}(t,s) }{\phi(\varphi^{-1}(\mathfrak{P}(t,s)))} &\leq b(T,s)\sum_{i=1}^{n} \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (w(\varphi^{-1}(\mathfrak{P}(\alpha_i(t),y)))\\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)w(\varphi^{-1}(\mathfrak{P}(m,n)))\,dn\, dm)\\ &\quad +h_i(\alpha_i(t),y)L(\alpha_i(t),y,w(\varphi^{-1} (\mathfrak{P}(\alpha_i(t),y))))\Big]\,dy\,\alpha'_i(t), \end{align*} for all $(t,s)\in [t_0,T]\times J$, $T\leq T_4$. Replace $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of the above inequality and using the definition of $G$, we have \begin{equation} \label{eq30} \begin{aligned} G(\mathfrak{P}(t,s)) &\leq G(a(T,s))+ b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y) (w(\varphi^{-1}(\mathfrak{P}(u,y))) \\ &\quad +\int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y} g_i(m,n)w(\varphi^{-1}(\mathfrak{P}(m,n)))\,dn\, dm)+h_i(u,y) \\ &\quad\times \{L(u,y,0)+M(u,y,0)w(\varphi^{-1} (\mathfrak{P}(u,y)))\}\Big]\,dy\, du \\ &\leq G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(T)}\int_{\beta_i(s_0)}^{\beta_i(s)} h_i(u,y)L(u,y,0)\,dy\, du \\ &\quad + b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y) (1+\int_{\alpha_i(t_0)}^{u}\int_{\beta_i(s_0)}^{y} g_i(m,n)\,dn\, dm) \\ &\quad +M(u,y,0)\Big]w(\varphi^{-1}(\mathfrak{P}(u,y)))\,dy\, du \end{aligned} \end{equation} Denote the right hand side of \eqref{eq30} by $\mathfrak{Q}(t,s)$, then obviously $\mathfrak{Q}(t,s)$ is a positive and nondecreasing function in each variable such that $\mathfrak{Q}(t_0,s)=G(a(T,s))$. Then, \eqref{eq30} is equivalent to \begin{gather}\label{eq44} \mathfrak{P}(t,s)\leq G^{-1}(\mathfrak{Q}(t,s)),\\ \mathfrak{Q}(t_0,s)=G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(T)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)L(u,y,0)\,dy\, du, \nonumber\\ \label{eq45} \begin{aligned} \mathfrak{Q}_{t}(t,s) &= b(T,s)\sum_{i=1}^{n} \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (1+\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)dn dm) \\ &\quad +M(\alpha_i(t),y,0)\Big]w(\varphi^{-1} (\mathfrak{P}(\alpha_i(t),y)))dy\alpha'_i(t) \\ &\leq b(T,s)w(\varphi^{-1}(G^{-1}(\mathfrak{P}(t,s))))\sum_{i=1}^{n} \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) \\ &\quad\times \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)dn dm\Big)+M(\alpha_i(t),y,0)\Big]dy\alpha'_i(t). \end{aligned} \end{gather} Then, \eqref{eq45} is written as \begin{equation} \label{eq31} \begin{aligned} &\frac{\mathfrak{Q}_{t}(t,s)}{w(\varphi^{-1}(G^{-1}(\mathfrak{P}(t,s))))}\\ &\leq b(T,s)\sum_{i=1}^{n}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(\alpha_i(t),y) \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{y} g_i(m,n)dn\, dm\Big)\\ &\quad +M(\alpha_i(t),y,0)\Big]dy\alpha'_i(t) \end{aligned} \end{equation} Replacing $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$ and making change of variable on right hand side of \eqref{eq31} and using the definition of $\Psi$, we obtain \begin{equation} \label{eq32} \begin{aligned} \Psi(\mathfrak{Q}(t,s)) &\leq \Psi(G(a(T,s))+b(T,s)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(T)}\int_{\beta_i(s_0)}^{\beta_i(s)} h_i(u,y)L(u,y,0)\,dy\, du) \\ &\quad +b(T,s)\Big\{D(t,s)+\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}M(u,y,0)\,dy\, du\Big\}. \end{aligned} \end{equation} A combination of \eqref{eq34}, \eqref{eq44} and \eqref{eq32} yield inequality \eqref{eq36}. \end{proof} \begin{corollary}\label{cor1} Suppose {\rm (A2)--(A4)} are satisfied. If $p>q>0$ and $c\geq0$ are constants such that: \begin{equation} \label{eq11} \begin{aligned} u^{p}(t,s) &\leq c+p\sum_{i=1}^{n}\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}u^{q}(t,s)\Big[f_i(x,y)(w(u(x,y))\\ &\quad +\int_{\alpha_i(t_0)}^{x}\int_{\beta_i(s_0)}^{y}g_i(m,n) w(u(m,n))\,dn\, dm)+h_i(x,y)\Big]\,dy\, dx, \end{aligned} \end{equation} then \begin{gather}\label{eq29} u(t,s)\leq \sqrt[p-q]{\Psi_{*}^{-1}(\Psi_{*}(m_0(t,s))+(p-q)D(t,s))},\\ m_0(t,s)=c^\frac{p-q}{q}+(p-q)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}h_i(u,y)\,dy\, du, \nonumber\\ \Psi_{*}(r):=\int_{r_0}^{r}\frac{du}{w(\sqrt[p-q]{u})},\quad r\geq r_0>0, \nonumber \end{gather} \end{corollary} \begin{proof} Denote the right hand side of \eqref{eq11} by $\Xi(t,s)$, then obviously $\Xi(t,s)$ is positive and non-decreasing function in each variable such that $\Xi(t_0,s)=c$. Then, \eqref{eq11} is equivalent to \begin{gather}\label{eq46} u(t,s)\leq\sqrt[p]{\Xi(t,s)}, \\ \label{eq47} \begin{aligned} &\Xi_{t}(t,s)\\ &= p\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) u^{q}(\alpha_i(t),y) (w(u(\alpha_i(t),y)) \\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(u(m,n))\,dn\, dm)+h_i(\alpha_i(t),y)u^{q}(\alpha_i(t),y)\Big]dy \\ &\leq p\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\{\Xi(\alpha_i(t),y)\}^{q/p} \Big[f_i(\alpha_i(t),y) (w(\sqrt[p]{\Xi(\alpha_i(t),y)}) \\ &\quad + \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n) w(\sqrt[p]{\Xi(m,n)})\,dn\, dm)+h_i(\alpha_i(t),y)\Big]dy, \end{aligned} \end{gather} for $(t,s)\in[t_0,T]\times J$. Then, \eqref{eq47} is equivalent to \begin{align*} \frac{\Xi_{t}(t,s)}{\{\Xi(t,s)\}^{q/p}} &\leq p\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}\Big[f_i(\alpha_i(t),y) (w(\sqrt[p]{\Xi(\alpha_i(t),y)})\\ &\quad +\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\sqrt[p]{\Xi(m,n)})\,dn\, dm)+h_i(\alpha_i(t),y)\Big]dy\,. \end{align*} Replacing $t$ by $v$ then integrating from $t_0$ to $t$ with respect to $v$, making change of variable on right hand side of the above inequality and by using that $m_0(t,s)$ is non-decreasing in each variable, for $t\leq T$, we have \begin{equation} \label{eq12} \begin{aligned} [\Xi(t,s)]^{(p-q)/p} &\leq c^{(p-q)/q}+(p-q)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} \Big[f_i(u,y) (w(\sqrt[p]{\Xi(u,y)}) \\ &\quad +\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\sqrt[p]{\Xi(m,n)})\,dn\, dm)+h_i(u,y)\Big]\,dy\,du \\ &\leq m_0(T,s)+(p-q)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)}\int_{\beta_i(s_0)}^{\beta_i(s)} f_i(u,y) (w(\sqrt[p]{\Xi(u,y)}) \\ &\quad +\int_{\alpha_i(t_0)}^{u} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\sqrt[p]{\Xi(m,n)})\,dn\, dm)\,dy\, du \end{aligned} \end{equation} Denote the right hand side of \eqref{eq12} by $\gamma(t,s)$, then obviously $\gamma(t,s)$ is positive and non-decreasing function in each variable such that $\gamma(t_0,s)=m_0(T,s)$. Then, \eqref{eq12} is equivalent to \begin{gather}\label{eq48} \Xi(t,s)\leq [\gamma(t,s)]^\frac{p}{p-q}, \\ \label{eq49} \begin{aligned} \gamma_{t}(t,s) &= (p-q)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) (w(\sqrt[p]{\Xi(\alpha_i(t),y)})\\ &\quad + \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\sqrt[p]{\Xi(m,n)})\,dn\, dm)\,dy \\ &\leq (p-q)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) (w(\sqrt[p-q]{\gamma(\alpha_i(t),y)})\\ &\quad + \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)w(\sqrt[p-q]{\gamma(m,n)})\,dn\, dm)\,dy. \end{aligned} \end{gather} Then, \eqref{eq49} is written as \begin{align*} & \frac{\gamma_{t}(t,s)}{w(\sqrt[p-q]{\gamma(t,s)})} \\ &\leq (p-q)\sum_{i=1}^{n}\alpha'_i(t) \int_{\beta_i(s_0)}^{\beta_i(s)}f_i(\alpha_i(t),y) \Big(1+\int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{y}g_i(m,n)\,dn\, dm\Big)dy. \end{align*} Setting $t$ by $l$ then integrating from $t_0$ to $t$ with respect to $l$, making change of variable on right hand side of the above inequality and using $\gamma(t_0,s)=m_0(T,s)$, and the definition of $\Psi_{*}$, we have \begin{equation} \label{eq50} \Psi_{*}(\gamma(t,s)) \leq \Psi_{*}(\gamma(t,s))+(p-q)D(t,s). \end{equation} A combination of \eqref{eq46}, \eqref{eq48}, and \eqref{eq50} yield the desire result \eqref{eq29}. \end{proof} \begin{remark} \label{rmk1}\rm \begin{itemize} \item For $a(t,s)\equiv c$, $b(t,s)\equiv1$, $\phi(x)= x$, $g_i\equiv0\equiv h_i$, $1\leq i\leq n$. Then Theorem \ref{thm1} reduces to \cite[Theorem 2.2]{c2}. \item For $a(t,s)\equiv c$, $b(t,s)\equiv1$, $\phi(x)= x$, $g_i\equiv0$, $1\leq i\leq n$. Then Theorem \ref{thm1} reduces to \cite[Theorem 2.3]{c2}. \item For $q=1$, $g_i\equiv0$, $1\leq i\leq n$, corollary \ref{cor1} reduces to \cite[Corollary 2.4]{c2}. \item For $g_i\equiv0$, $1\leq i\leq n$, Theorem \ref{thm1} reduces to \cite[Theorem 1]{w1}. \item For $g_i\equiv0$, $1\leq i\leq n$, and $w\equiv1$, theorem \ref{thm2} reduces to \cite[Theorem 2]{w1}. \end{itemize} \end{remark} \section{Applications}\label{Sec3} In this section, we apply the inequalities established above to achieve the boundedness of partial integro-differential equations, with several retarded arguments, of the form \begin{equation} \label{eq18} \begin{aligned} &\frac{\partial}{\partial s}(z^{p-1}(t,s)z_{t}(t,s))\\ &=F\Big[t,s,z(t-l_1(t),s-k_1(s)),\dots ,z(t-l_{n}(t),s-k_{n}(s)),\\ &\int_{t_0}^{t}\int_{s_0}^{s}Q(t,s,\sigma,\tau,z(t-l_1(t),s-k_1(s)), \dots ,z(t-l_{n}(t),s-k_{n}(s)))d\sigma d\tau\Big], \end{aligned} \end{equation} and \begin{equation} \label{eq21} \begin{aligned} &D_{2}(D_1\varphi(z(t,s)))\\ &=F\Big[t,s,z(t-l_1(t),s-k_1(s)),\dots , z(t-l_{n}(t),s-k_{n}(s)),\\ &\quad \int_{t_0}^{t}\int_{s_0}^{s}Q(t,s,\sigma,\tau,z(t-l_1(t),s-k_1(s)), \dots , z(t-l_{n}(t),s-k_{n}(s)))d\sigma\, d\tau\Big], \end{aligned} \end{equation} with the given initial boundary conditions \begin{equation}\label{eq19} z(t,s_0)=a_1(t),\quad z(t_0,s)=a_{2}(s),\quad a_1(t_0)=a_{2}(s_0)=0, \end{equation} where $F\in C(I\times J\times\mathbb{R}^{n},\mathbb{R})$, $Q\in C((I\times J)\times (I\times J)\times\mathbb{R}^{n},\mathbb{R})$, $a_1\in C^{1}(I,\mathbb{R})$, $a_{2}\in C^{1}(J,\mathbb{R})$ and $l_i\in C^{1}(I,\mathbb{R})$, $k_i\in C^{1}(J,\mathbb{R})$ are nonincreasing and such that $t-l_i(t)\geq0$, $t-l_i(t)\in C^{1}(I,I)$, $s-k_i(s)\geq0$, $s-k_i(s)\in C^{1}(J,J)$, $l'_i(t)<1$, $k'_i(s)<1$ and $l_i(t_0)=k_i(s_0)=0$, $1\leq i\leq n$, for $(t,s)\in I\times J$; let $\varphi\in C^{1}(\mathbb{R},\mathbb{R})$ be an increasing function such that $\varphi(|u|)\leq|\varphi(u)|$; let $\varphi(e(t,s))=\varphi(a_1(t))+\varphi(a_{2}(s))$ and \begin{equation}\label{eq20} M_i=\max_{t\in I}\frac{1}{1-l'_i(t)},\quad N_i=\max_{s\in J}\frac{1}{1-k'_i(s)},\quad 1\leq i\leq n. \end{equation} The following theorem deals with a boundedness on the solution of \eqref{eq21}. \begin{theorem}\label{thm4} Assume that $F:I\times J\times\mathbb{R}^{n}\times \mathbb{R}^{n}\to\mathbb{R}$ is a continuous function for which there exist continuous nonnegative functions $f_i(t,s), g_i(t,s)$ and $h_i(t,s)$, $1\leq i\leq n$, for $(t,s)\in I\times J$ such that: \begin{equation}\label{eq22} \begin{gathered} |F(t,s,u_1,\dots ,u_{n},j)|\leq b(t,s)\sum_{i=1}^{n}\phi(|u_i|)\big[f_i(t,s)w(|u_i|)+|j|+h_i(t,s)\big].\\ |Q(t,s,v_1,v_{2},u_1,u_{2},\dots ,u_{n})|\leq g_i(t,s)w(|u_i|). \end{gathered} \end{equation} If $z(t,s)$ is a solution of \eqref{eq21} with conditions \eqref{eq19}, then \begin{equation} \label{eq25} \begin{aligned} |z(t,s)| &\leq\varphi^{-1}\Big(G^{-1}\Big(\Psi^{-1}\Big(\Psi(\overline{c}(t,s))+b(t,s) \sum_{i=1}^{n}\int_{\phi_i(t_0)}^{\phi_i(t)}\int_{\psi_i(s_0)}^{\psi_i(s)} \overline{f}_i(\delta,\eta)\\ &\quad \times\Big(1+\int_{\phi_i(t_0)}^{\delta} \int_{\psi_i(s_0)}^{\psi_i(s)} \overline{g}_i(\delta_1,\eta_1)d\eta_1d\delta_1\Big)d\eta\,d\delta\Big)\Big)\Big), \end{aligned} \end{equation} where, \begin{gather*} \overline{f}_i(u,v)=M_iN_if_i(u+l_i(m),v+k_i(p)),\quad \overline{g}_i(u,v)=M_iN_ig_i(u+l_i(\sigma),v+k_i(\tau)),\\ \overline{c}(t,s)=G(\varphi(e(t,s)))+b(t,s)\sum_{i=1}^{n} \int_{\phi_i(t_0)}^{\phi_i(t)}\int_{\psi_i(s_0)}^{\psi_i(s)} \overline{h}_i(u,v)dv\, du, \\ \overline{h}_i(u,v)=M_iN_ih_i(u+l_i(m),v+k_i(p)) \end{gather*} \end{theorem} \begin{proof} It is easy to see that the solution $z(t,s)$ of the problem \eqref{eq21} with \eqref{eq19} satisfies the equivalent integral equation \begin{equation} \label{eq23} \begin{aligned} &\varphi(z(t,s))\\ &= \varphi(e(t,s))+\int_{t_0}^{t} \int_{s_0}^{s} F\Big[u,v,z(u-l_1(u),v-k_1(v)),\dots ,z(u-l_{n}(u),v-k_{n}(v)),\\ &\quad \int_{t_0}^{u}\int_{s_0}^{v}Q(u,v,\sigma, \tau,z(u-l_1(u),v-k_1(v)),\dots ,\\ &\quad z(u-l_{n}(u),v-k_{n}(v)))d\sigma\,d\tau\Big]dv\, du \end{aligned} \end{equation} By modulus properties and condition \eqref{eq22}, equation \eqref{eq23} has the form \begin{align*} &|\varphi(z(t,s))| \\ &\leq |\varphi(e(t,s))|+b(t,s)\int_{t_0}^{t}\int_{s_0}^{s} \Big|F\Big[u,v,z(u-l_1(u),v-k_1(v)),\dots , \\ &\quad z(u-l_{n}(u),v-k_{n}(v)), \int_{t_0}^{u}\int_{s_0}^{v}Q(u,v,\sigma, \tau,z(u-l_1(u),v-k_1(v)),\dots ,\\ &\quad z(u-l_{n}(u),v-k_{n}(v)))d\sigma\, d\tau\Big]\Big|dv\, du\\ & \leq |\varphi(e(t,s))|+b(t,s)\int_{t_0}^{t}\int_{s_0}^{s}\sum_{i=1}^{n} [\phi(|z(m-l_i(m),p-k_i(p))|)(f_i(m,p)\\ &\quad\times (w(|z(m-l_i(m),p-k_i(p))|)\\ &\quad +\int_{t_0}^{m}\int_{s_0}^{p}g_i(\sigma,\tau) w(|z(\sigma-l_i(\sigma),\tau-k_i(\tau))|)d\tau\, d\sigma)\\ &\quad +h_i(m,p)|\phi(z(m-l_i(m),p-k_i(p)))|]dp\, dm \\ &\leq |\varphi(e(t,s))|+b(t,s)\sum_{i=1}^{n}M_iN_i\int_{\phi_i(t_0)}^{\phi_i(t)} \int_{\psi_i(s_0)}^{\psi_i(s)}\Big[\phi(|z(\phi_i(m),\psi_i(p))|) \\ &\times (f_i(\phi_i(m)+l_i(m),\psi_i(p)+k_i(p)) (w(|z(\phi_i(m),\psi_i(p))|)\\ &\quad +\int_{\phi_i(t_0)}^{\phi_i(m)}\int_{\psi_i(s_0)}^{\psi_i(p)}M_iN_i g_i(\phi_i(\sigma)+l_i(\sigma),\psi_i(\tau)+k_i(\tau))\\ &\quad\times w(|z(\phi_i(\sigma),\psi_i(\tau))|)d\psi_i(\tau)\, d\phi_i(\sigma)) +h_i(\phi_i(m)+l_i(m),\psi_i(p)\\ &\quad +k_i(p)) \phi(|z(\phi_i(m),\psi_i(p))|))\Big]d\psi_i(p)\, d\phi_i(m) \end{align*} which implies \begin{equation} \label{eq24} \begin{aligned} &\varphi(|z(t,s)|)\\ &\leq|\varphi(e(t,s))|+b(t,s)\sum_{i=1}^{n} \int_{\phi_i(t_0)}^{\phi_i(t)}\int_{\psi_i(s_0)}^{\psi_i(s)} \Big[\phi(|z(\delta,\eta)|) (\overline{f}_i(\delta,\eta) (w(|z(\delta,\eta)|)\\ &\quad +\int_{\phi_i(t_0)}^{\delta}\int_{\psi_i(s_0)}^{\eta} \overline{g}_i(\delta_1,\eta_1)w(|z(\delta_1,\eta_1)|)d\eta_1\, d\delta_1) +\overline{h}_i(\delta,\eta)\phi(|z(\delta,\eta)|))\Big]d\eta\, d\delta \end{aligned} \end{equation} Now an immediate application of inequality \eqref{eq1} to \eqref{eq24} yields the desired result \eqref{eq25}. \end{proof} \begin{theorem}\label{thm5} Assume that $F:I\times J\times\mathbb{R}^{n} \times\mathbb{R}^{n}\to\mathbb{R}$ is a continuous function for which there exist continuous nonnegative functions $f_i(t,s), g_i(t,s)$ and $h_i(t,s)$, $1\leq i\leq n$, for $(t,s)\in I\times J$ such that: \begin{equation}\label{eq26} \begin{gathered} |F(t,s,u_1,\dots ,u_{n},j)|\leq \sum_{i=1}^{n}|u_i|^{q}[f_i(t,s)w(|u_i|)+|j|+h_i(t,s)],\\ |Q(t,s,v_1,v_{2},u_1,u_{2},\dots ,u_{n})|\leq g_i(t,s)w(|u_i|),\\ |a_1^{p}(t)+a_{2}^{p}(s)|\leq c \end{gathered} \end{equation} If $z(t,s)$ is a solution of \eqref{eq18} with the condition \eqref{eq19}, then \begin{equation} \label{eq53} \begin{aligned} u(t,s)&\leq [\Psi_{*}^{-1}(\Psi_{*}(\widetilde{m}_0(t,s))+(p-q)\sum_{i=1}^{n} \int_{\alpha_i(t_0)}^{\alpha_i(t)} \int_{\beta_i(s_0)}^{\beta_i(s)}\widetilde{f}_i(z,y)\\ &\quad\times \Big(1+\int_{\alpha_i(t_0)}^{z} \int_{\beta_i(s_0)}^{y}\widetilde{g}_i(m,n)\,dn\, dm\Big)\,dy\, dz)]^{1/(p-q)}, \end{aligned} \end{equation} where \begin{gather*} \widetilde{f}_i(\mathfrak{u},\mathfrak{v})= M_iN_if_i(\mathfrak{u}+l_i(u),\mathfrak{v}+k_i(v)),\quad \widetilde{g}_i(\mathfrak{u},\mathfrak{v})= M_iN_ig_i(\mathfrak{u}+l_i(\sigma),\mathfrak{v}+k_i(\tau)), \\ \widetilde{h}_i(\mathfrak{u},\mathfrak{v})= M_iN_ih_i(\mathfrak{u}+l_i(u),\mathfrak{v}+k_i(v)), \\ \widetilde{m}_0(t,s)=c^{(p-q)/p}+(p-q)\sum_{i=1}^{n} \int_{\phi_i(t_0)}^{\phi_i(t)}\int_{\psi_i(s_0)}^{\psi_i(s)} \widetilde{h}_i(u,y)\,dy\, du \end{gather*} \end{theorem} \begin{proof} It is easy to see that the solution $z(t,s)$ of \eqref{eq18} with \eqref{eq19} satisfies the equivalent integral equation \begin{equation} \label{eq27} \begin{aligned} [z(t,s)]^{p} &=a_1^{p}(t)+a_{2}^{p}(s)+p\int_{t_0}^{t}\int_{s_0}^{s}F[u,v,z(u-l_1(u),v-k_1(v)), \dots ,\\ &\quad z(u-l_{n}(u),v-k_{n}(v)), \int_{t_0}^{u}\int_{s_0}^{v}Q(u,v,\sigma,\tau,z(u-l_1(u),v-k_1(v)),\dots ,\\ &\quad z(u-l_{n}(u),v-k_{n}(v)))d\sigma d\tau]dv\, du \end{aligned} \end{equation} By modulus properties and condition \eqref{eq26}, equation \eqref{eq27} has the form % \label{eq28} \begin{align*} & |z^{p}(t,s)| \\ &\leq c+p\int_{t_0}^{t}\int_{s_0}^{s} \Big|F\Big[u,v,z(u-l_1(u),v-k_1(v)),\dots , z(u-l_{n}(u),v-k_{n}(v)), \\ &\quad \int_{t_0}^{u}\int_{s_0}^{v}Q(u,v,\sigma,\tau,z(u-l_1(u),v-k_1(v)),\dots ,\\ &\quad z(u-l_{n}(u),v-k_{n}(v)))d\sigma d\tau\Big]\Big|\,dv\, du \\ &\leq c+p\int_{t_0}^{t}\int_{s_0}^{s} \sum_{i=1}^{n}\Big[|z(u-l_i(u),v-k_i(v))|^{q}.f_i(u,v) (w(|z(u-l_i(u),v-k_i(v))|) \\ &\quad +\int_{t_0}^{u}\int_{s_0}^{v}g_i (\sigma,\tau)w(|z(\sigma-l_i(\sigma),\tau-k_i(\tau))|)d\tau\, d\sigma)\\ &\quad +h_i(u,v)|z(u-l_i(u),v-k_i(v))|^{q}\Big]dv\, du \\ &\leq c+p\sum_{i=1}^{n}M_iN_i\int_{\phi_i(t_0)}^{\phi_i(t)} \int_{\psi_i(s_0)}^{\psi_i(s)} \Big[|z(\phi_i(u),\psi_i(v))|^{q} f_i(\phi_i(u)+l_i(u),\psi_i(v)+k_i(v)) \\ &\quad\times (w(|z(\phi_i(u),\psi_i(v))|)+ \int_{\phi_i(t_0)}^{\phi_i(u)}\int_{\psi_i(s_0)}^{\psi_i(v)} M_iN_ig_i(\phi_i(\sigma)+l_i(\sigma),\psi_i(\tau)\\ &\quad +k_i(\tau)) w(|z(\phi_i(\sigma),\psi_i(\tau))|)d\psi_i(\tau) d\phi_i(\sigma)) +h_i(\phi_i(u)+l_i(u),\psi_i(v)\\ &\quad +k_i(v))| \phi(z(\phi_i(u),\psi_i(v)))|\Big]d\psi_i(v) d\phi_i(u) \\ &\leq c+p\sum_{i=1}^{n}\int_{\phi_i(t_0)}^{\phi_i(t)} \int_{\psi_i(s_0)}^{\psi_i(s)} [|z(\delta,\eta)|^{q} .\widetilde{f}_i(\delta,\eta) (w(|z(\delta,\eta)|) \\ &\quad +\int_{\phi_i(t_0)}^{\delta}\int_{\psi_i(s_0)}^{\eta} \widetilde{g}_i(\delta_1,\eta_1) w(|z(\delta_1,\eta_1)|)d\eta_1 d\delta_1) +\widetilde{h}_i(\delta,\eta)|z(\delta,\eta)|^{q}]d\eta\, d\delta \end{align*} Now an immediate application of inequality \eqref{eq29} to above inequality yields the desired result \eqref{eq53}. \end{proof} \subsection*{Acknowledgments} The authors are very grateful to Prof. Julio G. 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