A mathematical model to study the 2014–2015 large-scale dengue epidemics in Kaohsiung and Tainan cities in Taiwan, China

Salihu Sabiu Musa; Shi Zhao; Hei-Shen Chan; Zhen Jin; Daihai He; Salihu Sabiu Musa; Shi Zhao; Hei-Shen Chan; Zhen Jin; Daihai He

doi:10.3934/mbe.2019190

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3841-3863. doi: 10.3934/mbe.2019190

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Research article Special Issues

A mathematical model to study the 2014–2015 large-scale dengue epidemics in Kaohsiung and Tainan cities in Taiwan, China

1.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
2.
School of Nursing, Hong Kong Polytechnic University, Hong Kong, China
3.
Complex System Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
4.
Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, China
5.
Key Laboratory of Computational Intelligence and Chinese Information, Processing of Ministry of Education, Taiyuan 030006, China

Received: 18 January 2019 Accepted: 19 April 2019 Published: 29 April 2019

Dengue virus (DENV) infection is endemic in many places of the tropical and subtropical regions, which poses serious public health threat globally. We develop and analyze a mathematical model to study the transmission dynamics of the dengue epidemics. Our qualitative analyzes show that the model has two equilibria, namely the disease-free equilibrium (DFE) which is locally asymp- totically stable when the basic reproduction number (

$\mathcal{R}_0$ ) is less than one and unstable if

$\mathcal{R}_0$ > 1, and endemic equilibrium (EE) which is globally asymp-totically stable when

$\mathcal{R}_0$ > 1. Further analyzes reveals that the model exhibit the phenomena of backward bifurcation (BB) (a situation where a stable DFE co-exists with a stable EE even when the

$\mathcal{R}_0$ < 1), which makes the disease control more diffi-cult. The model is applied to the real dengue epidemic data in Kaohsiung and Tainan cities in Taiwan, China to evaluate the fitting performance. We propose two reconstruction approaches to estimate the time-dependent

$\mathcal{R}_0$ , and we find a consistent fitting results and equivalent goodness-of-fit. Our findings highlight the similarity of the dengue outbreaks in the two cities. We find that despite the proximity in Kaohsiung and Tainan cities, the estimated transmission rates are neither completely synchronized, nor periodically in-phase perfectly in the two cities. We also show the time lags between the seasonal waves in the two cities likely occurred. It is further shown via sensitivity analysis result that proper sanitation of the mosquito breeding sites and avoiding the mosquito bites are the key control measures to future dengue outbreaks in Taiwan.

Keywords:

Citation: Salihu Sabiu Musa, Shi Zhao, Hei-Shen Chan, Zhen Jin, Daihai He. A mathematical model to study the 2014–2015 large-scale dengue epidemics in Kaohsiung and Tainan cities in Taiwan, China[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3841-3863. doi: 10.3934/mbe.2019190

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Abstract

$\mathcal{R}_0$ ) is less than one and unstable if

$\mathcal{R}_0$ > 1, and endemic equilibrium (EE) which is globally asymp-totically stable when

$\mathcal{R}_0$ > 1. Further analyzes reveals that the model exhibit the phenomena of backward bifurcation (BB) (a situation where a stable DFE co-exists with a stable EE even when the

1. Introduction

In 1960, Opial ^[12] established the following inequality:

Theorem A Suppose $f\in C^{1}[0, h]$ satisfies $f(0) = f(h) = 0$ and $f(x) > 0$ for all $x\in (0, h).$ Then the inequality holds

$\begin{align} \int_{0}^h\left|f(x)f'(x)\right|dx\leq\frac{h}{4}\int_{0}^{h}(f'(x))^{2}dx, \end{align}$

(1.1)

where this constant $h/4$ is best possible.

Many generalizations and extensions of Opial's inequality were established ^{[2,4,5,6,7,8,9,10,11,15,16,17,18,19]}. For an extensive survey on these inequalities, see ^[13]. Opial's inequality and its generalizations and extensions play a fundamental role in the ordinary and partial differential equations as well as difference equation ^{[2,3,4,6,7,9,10,11,17]}. In particular, Agarwal and Pang ^[3] proved the following Opial-Wirtinger's type inequalities.

Theorem B Let $\lambda\geq 1$ be a given real number, and let $p(t)$ be a nonnegative and continuous function on $[0, a]$ . Further, let $x(t)$ be an absolutely continuous function on $[0, a]$ , with $x(0) = x(a) = 0$ . Then

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}dt\leq\frac{1}{2}\int_{0}^{a}[t(a-t)]^{(\lambda-1)/2}p(t)dt\int_{0}^{a}\left|x'(t) \right|^{\lambda}dt. \end{align}$

(1.2)

The first aim of the present paper is to establish Opial-Wirtinger's type inequalities involving Katugampola conformable partial derivatives and $\alpha$ -conformable integrals (see Section 2). Our result is given in the following theorem, which is a generalization of (1.2).

Theorem 1.1 Let $\lambda\geq 1$ be a real number and $\alpha\in (0, 1]$ , and let $p(s, t)$ be a nonnegative and continuous functions on $[0, a]\times [0, b]$ . Further, let $x(s, t)$ be an absolutely continuous function and Katugampola partial derivable on $[0, a]\times [0, b]$ , with $x(s, 0) = x(0, t) = x(0, 0) = 0$ and $x(a, b) = x(a, t) = x(s, b) = 0$ . If $p > 1,$ $\frac{1}{p}+\frac{1}{q} = 1$ Then

$\int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\leq\frac{p+q}{pq} \left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\left(\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)d_{\alpha}sd_{\alpha}t\right)$

$\begin{align} \times\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$

(1.3)

where

$\Gamma_{abpq\lambda\alpha}(s, t) = \left\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\right\}^{\alpha(\lambda-1)}.$

Remark 1.1 Let $x(s, t)$ reduce to $s(t)$ and with suitable modifications, and $p = q = 2$ and $\alpha = 1$ , (1.3) become (1.2).

Theorem C Let $\lambda\geq 1$ be a given real number, and let $p(t)$ be a nonnegative and continuous function on $[0, a]$ . Further, let $x(t)$ be an absolutely continuous function on $[0, a]$ , with $x(0) = x(a) = 0$ . Then

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}dt\leq\frac{1}{2}\left(\frac{a}{2} \right)^{\lambda-1}\left(\int_{0}^{a}p(t)dt\right)\int_{0}^{a}\left|x'(t)\right|^{\lambda}dt. \end{align}$

(1.4)

Another aim of this paper is to establish the following inequality involving Katugampola conformable partial derivatives and $\alpha$ -conformable integrals. Our result is given in the following theorem.

Theorem 1.2 Let $j = 1, 2$ and $\lambda\geq 1$ be a real number, and let $p_{j}(s, t)$ be a nonnegative and continuous functions on $[0, a]\times [0, b]$ . Further, let $x_{j}(s, t)$ be an absolutely continuous function and Katugampola partial derivable on $[0, a]\times [0, b]$ , with $x_{j}(s, 0) = x_{j}(0, t) = x_{j}(0, 0) = 0$ and $x_{j}(a, b) = x_{j}(a, t) = x_{j}(s, b) = 0$ . Then for $\alpha\in(0, 1]$

$\int_{0}^{a}\int_{0}^{b}\left(p_{1}(s, t)|x_{1}(s, t)|^{\lambda}+p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\right)d_{\alpha}sd_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{1}(s, t) d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda} d_{\alpha}sd_{\alpha}t$

$\begin{align} +\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{2}(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg]. \end{align}$

(1.5)

2. Katugampola conformable partial derivatives

Here, let's recall the well-known Katugampola derivative formulation of conformable derivative of order for $\alpha\in(0, 1]$ and $t\in[0, \infty)$ , given by

$\begin{align} {\cal D}_{\alpha}(f)(t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{f(te^{\varepsilon t^{-\alpha}})-f(t)}{\varepsilon}, \end{align}$

(2.1)

and

$\begin{align} {\cal D}_{\alpha}(f)(0) = \lim\limits_{t\rightarrow 0}D_{\alpha}(f)(t), \end{align}$

(2.2)

provided the limits exist. If $f$ is fully differentiable at $t$ , then

${\cal D}_{\alpha}(f)(t) = t^{1-\alpha}\frac{df}{dt}(t).$

A function $f$ is $\alpha$ -differentiable at a point $t\geq 0$ , if the limits in (2.1) and (2.2) exist and are finite. Inspired by this, we propose a new concept of $\alpha$ -conformable partial derivative. In the way of (1.4), $\alpha$ -conformable partial derivative is defined in as follows:

Definition 2.1 ^[20] ( $\alpha$ -conformable partial derivative) Let $\alpha\in (0, 1]$ and $s, t\in[0, \infty)$ . Suppose $f(s, t)$ is a continuous function and partial derivable, the $\alpha$ -conformable partial derivative at a point $s\geq 0$ , denoted by $\frac{\partial}{\partial s}(f)_{\alpha}(s, t)$ , defined by

$\begin{align} \frac{\partial}{\partial s}(f)_{\alpha}(s, t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{f(se^{\varepsilon s^{-\alpha}}, t)-f(s, t)} {\varepsilon}, \end{align}$

(2.3)

provided the limits exist, and call $\alpha$ -conformable partial derivable.

Recently, Katugampola conformable partial derivative is defined in as follows:

Definition 2.2 ^[20] (Katugampola conformable partial derivatives) Let $\alpha\in (0, 1]$ and $s, t\in[0, \infty)$ . Suppose $f(s, t)$ and $\frac{\partial}{\partial s}(f)_{\alpha}(s, t)$ are continuous functions and partial derivable, the Katugampola conformable partial derivative, denoted by $\frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s, t)$ , defined by

$\begin{align} \frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s, t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{\frac{\partial}{\partial s}(f)_{\alpha}(s, te^{\varepsilon t^{-\alpha}})-\frac{\partial}{\partial s}(f)_{\alpha}(s, t)}{\varepsilon}, \end{align}$

(2.4)

provided the limits exist, and call Katugampola conformable partial derivable.

Definition 2.3 ^[20] ( $\alpha$ -conformable integral) Let $\alpha\in (0, 1]$ , $0\leq a < b$ and $0\leq c < d$ . A function $f(x, y):[a, b]\times[c, d]\rightarrow {\Bbb R}$ is $\alpha$ -conformable integrable, if the integral

$\begin{align} \int_{a}^{b}\int_{c}^{d}f(x, y)d_{\alpha}xd_{\alpha}y: = \int_{a}^{b}\int_{c}^{d} (xy)^{\alpha-1}f(x, y)dxdy \end{align}$

(2.5)

exists and is finite.

3. Main results

Theorem 3.1 Let $\lambda\geq 1$ be a real number and $\alpha\in (0, 1]$ , and let $p(s, t)$ be a nonnegative and continuous functions on $[0, a]\times [0, b]$ . Further, let $x(s, t)$ be an absolutely continuous function and Katugampola partial derivable on $[0, a]\times [0, b]$ , with $x(s, 0) = x(0, t) = x(0, 0) = 0$ and $x(a, b) = x(a, t) = x(s, b) = 0$ . If $p > 1,$ $\frac{1}{p}+\frac{1}{q} = 1$ Then

$\begin{align} \times\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$

(3.1)

where

$\Gamma_{abpq\lambda\alpha}(s, t) = \left\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\right\}^{\alpha(\lambda-1)}.$

Proof From (2.4) and (2.5), we have

$x(s, t) = \int_{0}^{s}\int_{0}^{t}\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t.$

By using Hölder's inequality with indices $\lambda$ and $\lambda/(\lambda-1)$ , we have

$|x(s, t)|^{\lambda/p}\leq \left[\left(\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t\right)^{\lambda}\right]^{1/p}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\begin{align} \leq\left(\frac{1}{\alpha^{2}}(st)^{\alpha}\right)^{(\lambda-1)/p}\left(\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/p}. \end{align}$

(3.2)

Similarly, from

$x(s, t) = \int_{s}^{a}\int_{t}^{b}\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t,$

we obtain

$\begin{align} |x(s, t)|^{\lambda/q}\leq \left(\frac{1}{\alpha^{2}}[(a-s)(b-t)]^{\alpha}\right)^{(\lambda-1)/q} \left(\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{q}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/q}. \end{align}$

(3.3)

Now a multiplication of (3.2) and (3.3), and by using the well-known Young inequality gives

$\begin{eqnarray*} |x(s, t)|^{\lambda}&\leq &\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\cdot\left(\int_{0}^{s}\int_{0}^{t} \left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/p}\\ &\times&\left(\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/q}\\ &\leq &\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\cdot\Bigg(\frac{1}{p}\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ &+&\frac{1}{q}\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg) \end{eqnarray*}$

$\begin{align} \; \; \; \; \; \; \; \; \; \; = \frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$

(3.4)

where

$\Gamma_{abpq\lambda\alpha}(s, t) = \Big\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\Big\} ^{\alpha(\lambda-1)}.$

Multiplying the both sides of (3.4) by $p(s, t)$ and $\alpha$ –conformable integrating both sides over $t$ from $0$ to $b$ first and then integrating the resulting inequality over $s$ from $0$ to $a$ , we obtain

$\int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\leq\frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)\left(\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t$

$= \frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\left(\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t.$

This completes the proof.

Remark 3.1 Let $x(s, t)$ reduce to $s(t)$ and with suitable modifications, (3.1) becomes the following result.

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}d_{\alpha}t\leq\frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\int_{0}^{a} \Gamma_{apq\lambda\alpha}(t)p(t)d_{\alpha}t\int_{0}^{a}\left|{\cal D}_{\alpha}(x)(t) \right|^{\lambda}d_{\alpha}t, \end{align}$

(3.5)

where ${\cal D}_{\alpha}(x)(t)$ is Katugampola derivative (2.1) stated in the introduction, and

$\Gamma_{apq\lambda\alpha}(t) = \Big\{t^{1/p}(a-t)^{1/q}\Big\} ^{\alpha(\lambda-1)}.$

Putting $p = q = 2$ and $\alpha = 1$ in (3.5), (3.5) becomes inequality (1.2) established by Agarwal and Pang ^[3] stated in the introduction.

Taking for $\alpha = 1$ , $p = q = 2$ and $p(s, t) = {\rm constant}$ in (3.1), we have the following interesting result.

$\int_{0}^{a}\int_{0}^{b}|x(s, t)|^{\lambda}dsdt\leq \frac{1}{2}(ab)^{\lambda}\left[B\left(\frac{\lambda+1}{2}, \frac{\lambda+1}{2}\right)\right] ^{2}\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}x(s, t)\right|^{\lambda}dsdt,$

where $B$ is the Beta function.

Theorem 3.2 Let $j = 1, 2$ and $\lambda\geq 1$ be a real number, and let $p_{j}(s, t)$ be a nonnegative and continuous functions on $[0, a]\times [0, b]$ . Further, let $x_{j}(s, t)$ be an absolutely continuous function and Katugampola partial derivable on $[0, a]\times [0, b]$ , with $x_{j}(s, 0) = x_{j}(0, t) = x_{j}(0, 0) = 0$ and $x_{j}(a, b) = x_{j}(a, t) = x_{j}(s, b) = 0$ . Then for $\alpha\in(0, 1]$

(3.6)

Proof Because

$x_{1}(s, t) = \int_{0}^{s}\int_{0}^{t}\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t = \int_{s}^{a}\int_{t}^{b}\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t.$

Hence

$|x_{1}(s, t)|\leq\frac{1}{2}\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t.$

By Hölder's inequality with indices $\lambda$ and $\lambda/(\lambda-1)$ , it follows that

$p_{1}(s, t)|x_{1}(s, t)|^{\lambda}\leq\frac{1}{2^{\lambda}}p_{1}(s, t)\left(\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} (x_{1})_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t \right)^{\lambda}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\begin{align} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}(st)^{\alpha(\lambda-1)}p_{1}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$

(3.7)

Similarly

$\begin{align} p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}(st)^{\alpha(\lambda-1)}p_{2}(s, t)\int_{0}^{a}\int_{0}^{b} \left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$

(3.8)

Taking the sum of (3.7) and (3.8) and $\alpha$ -integrating the resulting inequalities over $t$ from $0$ to $b$ first and then over $s$ from $0$ to $a$ , we obtain

$\int_{0}^{a}\int_{0}^{b}\left(p_{1}(s, t)|x_{1}(s, t)|^{\lambda}+p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\right)d_{\alpha}s d_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\begin{eqnarray*} &\leq&\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg\{\int_{0}^{a}\int_{0}^{b}\left((st)^{\alpha(\lambda-1)}p_{1}(s, t) \int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t\\ &+&\int_{0}^{a}\int_{0}^{b}\left((st)^{\alpha(\lambda-1)}p_{2}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t\Bigg\}\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{1}(s, t) d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda} d_{\alpha}sd_{\alpha}t\\ &+&\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{2}(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg]. \end{eqnarray*}$

Remark 3.2 Taking for $x_{1}(s, t) = x_{2}(s, t) = x(s, t)$ and $p_{1}(s, t) = p_{2}(s, t) = p(s, t)$ in (3.6), (3.6) changes to the following inequality.

$\int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\leq\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\begin{align} \times\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p(s, t) d_{\alpha}sd_{\alpha}t\right) \int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t. \end{align}$

(3.9)

Putting $\alpha = 1$ in (3.9), we have

$\begin{align} \int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}dsdt\leq\frac{1}{2^{\lambda}}\left(\int_{0}^{a}\int_{0}^{b}(st)^{\lambda-1}p(s, t) dsdt\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} x(s, t)\right|^{\lambda}dsdt. \end{align}$

(3.10)

Let $x(s, t)$ reduce to $s(t)$ and with suitable modifications, and $\lambda = 1$ , (2.10) becomes the following result.

$\begin{align} \int_{0}^{a}p(t)|x(t)|dt\leq\frac{1}{2}\left(\int_{0}^{a}p(t)dt\right)\int_{0}^{a}\left|x'(t)\right|dt. \end{align}$

(3.11)

This is just a new inequality established by Agarwal and Pang ^[4]. For $\lambda = 2$ the inequality (3.11) has appear in the work of Traple ^[14], Pachpatte ^[13] proved it for $\lambda = 2m$ ( $m\geq 1$ an integer).

Remark 3.3 Let $x_{j}(s, t)$ reduce to $x_{j}(t)$ ( $j = 1, 2$ ) and $p_{j}(s, t)$ reduce to $p_{j}(t)$ ( $j = 1, 2$ ) with suitable modifications, (3.6) becomes the following interesting result.

$\int_{0}^{a}\left(p_{1}(t)|x_{1}(t)|^{\lambda}+p_{2}(t)|x_{2}(t)|^{\lambda}\right)d_{\alpha}t\leq\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}t^{\alpha(\lambda-1)}p_{1}(t)d_{\alpha}t\right)\int_{0}^{a}\left|{\cal D }_{\alpha}(x'_{1})(t)\right|^{\lambda} d_{\alpha}t$

$\begin{align} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; +\left(\int_{0}^{a}t^{\alpha(\lambda-1)}p_{2}(t)d_{\alpha}t\right)\int_{0}^{a}\left|{\cal D}_{\alpha}(x'_{2})(t)\right|^{\lambda}d_{\alpha}t\Bigg]. \end{align}$

(3.12)

Putting $\lambda = 1$ and $\alpha = 1$ in (3.12), we have the following interesting result.

$\int_{0}^{a}\left(p_{1}(t)|x_{1}(t)|+p_{2}(t)|x_{2}(t)|\right)dt\leq\frac{1}{2}\Bigg(\int_{0}^{a}p_{1}(t)dt\int_{0}^{a}\left|x'_{1}(t)\right| dt+\int_{0}^{a}p_{2}(t)dt\int_{0}^{a}\left|x'_{2}(t)\right|dt\Bigg).$

Finally, we give an example to verify the effectiveness of the new inequalities. Estimate the following double integrals:

$\int_{0}^{1}\int_{0}^{1}\Big[st(s-1)(t-1)\Big]^{\lambda}dsdt,$

where $\lambda\geq 1$ .

Let $x_{1}(s,t)=x_{2}(s,t)= x(s, t) = st(s-1)(t-1)$ , $p_{1}(s,t)=p_{2}(s,t)= p(s, t) = (st)^{1-\alpha}$ , $a = b = 1$ and $0 < \alpha\leq 1$ , and by using Theorem 3.2, we obtain

$\int_{0}^{1}\int_{0}^{1}\Big[st(s-1)(t-1)\Big]^{\lambda}dsdt\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$

$\begin{eqnarray*} & = &\int_{0}^{1}\int_{0}^{1}p(s, t) \left|x(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ &\leq&\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\int_{0}^{1}\int_{0}^{1}(st)^{\alpha(\lambda-1)}p(s, t) d_{\alpha}sd_{\alpha}t\right) \int_{0}^{1}\int_{0}^{1}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \int_{0}^{1}\int_{0}^{1}\Big[(2s-1)(2t-1)\Big]^{\lambda}(st)^{\alpha-1}dsdt\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \left(\frac{1}{2^{\alpha-1}}\int_{-1}^{1}t^{\lambda}\frac{1}{(t+1)^{1-\alpha}}dt\right)^{2}\\ &\leq&\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \left(\frac{1}{2^{\alpha-1}}\frac{2^{\alpha}}{\alpha}\right)^{2}\\ & = &\frac{2^{2-\lambda}}{\alpha^{2\lambda}\left(\alpha(\lambda-1)+1\right)^{2}}. \end{eqnarray*}$

4. Conclusions

We have introduced a general version of Opial-Wirtinger's type integral inequality for the Katugampola partial derivatives. The established results are generalization of some existing Opial type integral inequalities in the previous published studies. For further investigations we propose to consider the Opial-Wirtinger's type inequalities for other partial derivatives.

Acknowledgments

I would like to thank that research is supported by National Natural Science Foundation of China(11471334, 10971205).

Conflict of interest

The author declares no conflicts of interest.

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