Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay

Zenab Alrikaby; Xia Liu; Tonghua Zhang; Federico Frascoli; Zenab Alrikaby; Xia Liu; Tonghua Zhang; Federico Frascoli

doi:10.3934/mbe.2019083

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 1729-1749. doi: 10.3934/mbe.2019083

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Research article

Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay

1.
Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
2.
Department of Mathematics, University of Thi-Qar, Nasiriyah, Iraq
3.
College of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, P.R., China

Received: 19 November 2018 Accepted: 02 February 2019 Published: 04 March 2019

In this paper, we construct a discrete time delay Lac operon model with nonlinear degradation rate for mRNA, resulting from the interaction among several identical mRNA pieces. By taking a discrete time delay as bifurcation parameter, we investigate the nonlinear dynamical behaviour arising from the model, using mathematical tools such as stability and bifurcation theory. Firstly, we discuss the existence and uniqueness of the equilibrium for this system and investigate the effect of discrete delay on its dynamical behaviour. Absence or limited delay causes the system to have a stable equilibrium, which changes into a Hopf point producing oscillations if time delay is increased. These sustained oscillation are shown to be present only if the nonlinear degradation rate for mRNA satisfies specific conditions. The direction of the Hopf bifurcation giving rise to such oscillations is also determined, via the use of the so-called multiple time scales technique. Finally, numerical simulations are shown to validate and expand the theoretical analysis. Overall, our findings suggest that the degree of nonlinearity of the model can be used as a control parameter for the stabilisation of the system.

Keywords:

Citation: Zenab Alrikaby, Xia Liu, Tonghua Zhang, Federico Frascoli. Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 1729-1749. doi: 10.3934/mbe.2019083

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Abstract

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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