Abstract
In the article, we present the explicit bounds for three generalized delay dynamic Gronwall–Bellman type integral inequalities on time scales, which are the unification of continuous and discrete results. As applications, the boundedness for the solutions of delay dynamic integro-differential equations with initial conditions is discussed.
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1 Introduction
Theory of time scales is the unification of both continuous and discrete analysis due to Stephen Hilger [1] in his PhD thesis, and it has wide applications in quantum calculus and difference and differential calculus. Due to its vast contributions in different branches of mathematics, it attracts the researchers and mathematicians to work on it. The role of inequalities cannot be forgot because they have huge contributions in the theory of differential equations [2–22], bivariate means [23–31], calculation and optimization [32–49], special functions [50–69], probability and statistics [70–75], and so on.
It is well known that the explicit bounds for integral inequalities on unknown functions are very useful in the qualitative and quantitative analysis for the solutions of differential, integral, and integro-differential equations [76–100]. One of the most important inequalities in mathematics is the Gronwall inequality, it is an indispensable tool to obtain various estimates in the theory of ordinary and stochastic differential equations. The differential form of this inequality was proved by Gronwall [101] and its integral form was proved by Bellman [102]. Gronwall–Bellman integral inequalities have a lot of contributions to analyzing the behavior of solutions of the differential and integral equations. Recently, to dealt with the difficulties encountered in the solutions of differential and difference equations, the Gronwall–Bellman integral inequality on time scales has become one of the most important topics in mathematics research. A lot of work has been done in this area [103–106]. To discuss the abstract analysis of the solutions of certain types of dynamic equations, Gronwall–Bellman delay type inequality on time scales can play a significant role, but it still cannot deal with the abstract analysis for the solutions of some more general differential and difference equations. This gives us the motivation to discuss some generalized delay integral inequalities on time scales. The main purpose of the article is to provide the explicit bounds for some delay double integral inequalities on time scales, present the characteristics of the solutions of certain integro-differential equations, and establish their discrete inequalities with initial conditions by use of the analytical and numerical methods in our obtained results.
2 Preliminaries
The time scale \(\mathbb{T}\) is a nonempty closed subset of the real numbers set \(\mathbb{R}\), its forward jump operator \(\sigma : \mathbb{T}\rightarrow \mathbb{T}\) for \(t\in \mathbb{T}\) is defined by \(\sigma (t):=\inf \{ r\in \mathbb{T}: r>t\}\) and its backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) for \(t\in \mathbb{T}\) is defined by \(\rho (t):=\sup \{ r\in \mathbb{T}: r< t\}\). The derived set \(\mathbb{T}^{k}\) is defined as follows: if \(\mathbb{T}\) has a left-scattered maximum m, then \(\mathbb{T}^{k}= \mathbb{T}-\{m\}\); otherwise, \(\mathbb{T}^{k}=\mathbb{T}\).
Lemma 2.1
(see [107])
Let \(a, x\in \mathbb{T}^{k}\)with \(x>a\), \(f:\mathbb{T} \times \mathbb{T}^{k}\rightarrow \mathbb{R}\)be continuous at \((x,x)\), \(f^{\Delta }(x,\cdot )\)be the derivative offwith respect to its first variable such that it is right-dense continuous on \([a,\sigma (x)]\), and \(g(x)=\int _{a}^{x}f(x,\tau )\Delta \tau \). If, for each \(\epsilon >0\), there exists a neighborhoodUofxindependent of \(\tau \in [a,\sigma (x)]\)such that
for all \(y\in U\), then
Theorem 2.2
(see [108])
Let \({f}_{1}: \mathbb{R\rightarrow \mathbb{R}}\)be continuous and differentiable and \({f}_{2}:\mathbb{T}\rightarrow \mathbb{R}\)be delta differentiable. Then \({f}_{1}^{\circ }{f}_{2}: \mathbb{T}\rightarrow \mathbb{R}\)is delta differentiable and
Theorem 2.3
(see [108])
Let \({h}_{1}:\mathbb{T}\rightarrow \mathbb{R}\)be strictly increasing such that \(\overline{\mathbb{T}}={h}_{1}(\mathbb{T})\)is a time scale, and \({h}_{2}:\overline{{\mathbb{T}}}\rightarrow \mathbb{R}\). Then
if \({h}_{1}^{\Delta }({x})\)and \({h}_{2}^{\overline{\Delta }}({h}_{1}( {x}))\)exist for \({x}\in \mathbb{T}^{k}\).
3 Main results
Throughout the section, \(\mathbb{R}\) represents the set of real numbers, \(\mathbb{R}_{0}^{+}=[0,\infty )\), \(\mathbb{R}_{1}^{+}=[1,\infty )\), \(\mathbb{Z}\) represents the set of integers, \(\mathbb{N}_{0}\) represents the set of nonnegative integers, \(\mathbb{A}_{j}\subseteq \mathbb{N} _{0}\) \((1\leq j\leq 2)\), \({x}_{0j}\in \mathbb{T}\), \(\mathbb{T}_{j}=[ {x}_{0j},\infty )_{\mathbb{T}}\subseteq {\mathbb{T}}^{k}\) is the time scale, \(\rho _{ji}\) is the backward jump operator, \({X}^{\Delta y_{i}}(y _{1},y_{2},\ldots ,y_{n})\) \((1\leq i\leq n)\) is the partial delta-derivative of X with respect to its ith variable and \({\Delta y_{i}}{X}(y_{1},y_{2},\ldots ,y_{n})\) is the forward difference of X with respect to its ith variable.
Theorem 3.1
Let \(u, r_{i}, a_{j}:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}_{0}^{+}\)and \(f_{i}, g_{i}, f^{\Delta {x}_{1}}_{i}: \mathbb{T}_{1}^{2}\times \mathbb{T}_{2}^{2}\rightarrow \mathbb{R}_{0} ^{+}\)be nonnegative and right-dense continuous such that \(a_{j}\)is nondecreasing with respect to its each variable, let \(\gamma _{ji}: \mathbb{T}_{j}\rightarrow \mathbb{R}_{0}^{+}\)be nonnegative, nondecreasing, and right-dense continuous such that \(\gamma _{ji}( {x}_{j})\leq {x}_{j}\)and \(\gamma ^{\Delta }_{ji}({x}_{j})>0\), \(\mu _{ji}:\mathbb{T}_{j}\rightarrow \mathbb{T}\)such that \(\mu _{ji}( {x}_{j})\leq {x}_{j}\)and \(-\infty <\mathfrak{p}_{j}=\inf \{\min (\mu _{ji}({x}_{j}) ),{x}_{j}\in \mathbb{T}_{j}\}\leq {x}_{0j}\), \(\mathfrak{a}:([\mathfrak{p_{1}},{x}_{01}]\times [ \mathfrak{p_{2}},{x}_{02}])_{\mathbb{T}^{2}}\rightarrow \mathbb{R} _{0}^{+}\)be nonnegative and right-dense continuous, \(w, w_{j}: \mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}\)be nonnegative, nondecreasing, and continuous such that \(w(p)>0\), \(w_{j}(p)>0\)for \(p>0\)and
for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with the initial condition
Then
for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\)if
where \({G}_{1}^{-1}\)and \({G}_{2}^{-1}\)are respectively the inverse functions of \({G}_{1}\)and \({G}_{2}\), and \(\tilde{x}_{1}\), \(\tilde{x} _{2}\)are chosen such that
Proof
For fixed numbers \(\bar{x}_{1}\in \mathbb{T}_{1}\), \(\bar{x}_{2}\in \mathbb{T}_{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\), inequality (3.1) can be rewritten as follows:
Let
for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\). Then
and
If \(\mu _{1i}({x}_{1})\geq {x}_{01}\) and \(\mu _{2i}({x}_{2})\geq {x} _{02}\) for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02},\bar{x}_{2}]_{\mathbb{T}}\), then \(\mu _{1i}({x}_{1}) \in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\), \(\mu _{2i}({x}_{1})\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\), and
On the other hand, if \(\mu _{1i}({x}_{1})\leq {x}_{01}\) or \(\mu _{2i}( {x}_{2})\leq {x}_{02}\), then from (3.2) we have
It follows from (3.7) and (3.8) that
for \(x_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \(x_{2}\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\).
From Lemma 2.1, (3.4), and (3.9), one has
and
Using the fact that \(w_{1}\), \(w^{-1}\), and \(\xi _{1}\) are nondecreasing, we have
Dividing both sides by \({w_{1}(w^{-1}(\xi _{1}({x}_{1},{x}_{2})))}\), we obtain
Integrating over \([{x}_{01},{{x}_{1}}]\), then using the definition of \({G}_{1}\) and (3.5), we get
Let
Then we have
and
It follows from Lemma 2.1 and (3.11) that
Making use of (3.13) and the fact that \(w_{2}\), \(w^{-1}\), \(G_{1}^{-1}\), and \(\zeta _{1}\) are nondecreasing, we get
Integrating over \([{x}_{01},{{x}_{1}}]\), then using the definition of \({G}_{2}\) and (3.12), we get
A combination of (3.6), (3.13), and (3.15) gives the desired result (3.3). □
Remark 3.2
Let \(\mathbb{T}=\mathbb{Z}\), \(a_{1}({x}_{1},{x}_{2})=c\), \(a_{2}({x} _{1},{x}_{2})=w_{2}=1\), \(\gamma _{ji}=\mu _{ji}=I\), \(n=1\), \(f_{i}({x} _{1},{t}_{1},{x}_{2},{t}_{2})=f({t}_{1},{t}_{2})\), \(g_{i}=r_{i}=0\). Then Theorem 3.1 coincides with Theorem 2.1 in [109]. Moreover, if \(w(u)=u^{p}\), then it coincides with Theorem 2.1 in [110].
Remark 3.3
Let \(\mathbb{T}=\mathbb{R}\), \(a_{1}({x}_{1},{x}_{2})=c\), \(a_{2}({x} _{1},{x}_{2})=1\), \(w_{1}=\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=f_{i}({t}_{1},{t}_{2})\), \(g_{i}=0\) (\(1\leq i\leq n\)). Then Theorem 3.1 leads to Theorem 2.3 in [111]. Moreover, if \(r_{i}=0\), then it reduces to Theorem 2.2 in [111].
Remark 3.4
Let \(\mathbb{T}=\mathbb{R}\), \(w_{1}(u)=u^{q}\), \(\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), \(g_{i}=0\) \((1\leq i\leq n)\). Then Theorem 3.1 becomes Theorem 2.1 in [112].
Remark 3.5
Let \(n=1\), \(\gamma _{ji}=I\), \(w_{2}=1\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=f({t}_{1},{t}_{2})\), and \(g_{i}=r_{i}=0\). Theorem 3.1 coincides with Theorem 1 in [113].
Remark 3.6
Let \(\mathbb{T}=\mathbb{Z}\), \(w_{2}=1\), \(\gamma _{ji}=\mu _{ji}=I\), \(n=1\), \(f_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=f({t}_{1},{t}_{2})\), and \(g_{i}=r_{i}=0\). Then Theorem 3.1 leads to Theorem 1 in [114].
Remark 3.7
Let \(\mathbb{T}=\mathbb{R}\), \(\mu _{ji}=I\), \(f_{i}({x}_{1},{t}_{1}, {x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.1 reduces to Theorem 1 in [115].
Theorem 3.8
Let \(a_{j}\), \(r_{i}\), w, \(w_{j}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), \(\mu _{ji}\), and \(\mathfrak{a}\)be defined as in Theorem 3.1such that \(w_{j+2}\)has the same conditions of \(w_{j}\), and let \(u:\mathbb{T} _{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}_{1}^{+}\)be a right-dense continuous function such that
for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with initial condition (3.2). Then the following statements are true:
(1) If \(w_{2}({u})\geq w_{4}(\log ({u}))\), then
for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\);
(2) If \(w_{2}({u})< w_{4}(\log ({u}))\), then
for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{3}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{4}\), where
c, \({G}_{1}\), and \({G}_{2}\)are defined as in Theorem 3.1, \({G}_{1}^{-1}\), \({G}_{2}^{-1}\), \({G}_{3}^{-1}\), and \({G}_{4}^{-1}\)are respectively the inverse functions of \({G}_{1}\), \({G}_{2}\), \({G}_{3}\), and \({G}_{4}\), and \(\tilde{x}_{1}\), \(\tilde{x}_{2}\), \(\tilde{x}_{3}\), and \(\tilde{x}_{4}\)are chosen such that
Proof
Let \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\). Then inequality (3.16) can be rewritten as
for \({x}_{01}\leq {x}_{1}\leq \bar{x}_{1}\) and \({x}_{02}\leq {x}_{2} \leq \bar{x}_{2}\).
Let
Then one has
and
On taking the identical steps as in (3.7)–(3.10), we get
For \(w_{2}(\mathfrak{u})\geq w_{4}(\log (\mathfrak{u}))\), inequality (3.20) can be rewritten as follows:
Let
Then
and
On taking the identical steps as those in (3.13)–(3.14), one has
Let
Then
and
It follows from Lemma 2.1 and (3.23) that
From (3.25) and the fact that \(w_{3}\), \(w^{-1}\), \(G_{1}^{-1}\), \(G_{2}^{-1}\), and ϑ are nondecreasing, we have
Using the definition of \({Q}_{1}\) and (3.24), integrating over \([{x}_{01},{{x}_{1}}]\), we get
It follows from (3.19), (3.22), (3.25), and (3.26) that
Let \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\). Then
Since \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02} \leq \bar{x}_{2}\leq \tilde{x}_{2}\) are chosen arbitrarily, so after substituting \(\bar{x}_{1}\) and respectively \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.17).
For \(w_{2}(\mathfrak{u})< w_{4}(\log (\mathfrak{u}))\), inequality (3.20) can be rewritten as follows:
On taking the identical steps as those (3.21)–(3.27), then after substituting \(\bar{x}_{1}\) and \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.18). □
Remark 3.9
Let \(\mathbb{T}=\mathrm{R}\), \(w_{3}=1\), \(\mu _{ji}=I\), \(f_{i}({x}_{1}, {t}_{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.8 leads to Theorem 2 in [115].
Theorem 3.10
Let \(a_{j}\), \(r_{i}\), w, \(w_{j}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), \(\mu _{ji}\), and \(\mathfrak{a}\)be defined as in Theorem 3.1, and \(\mathrm{L}, \mathrm{M}:\mathbb{T}_{1}\times \mathbb{T}_{2}\times \mathbb{R}_{0}^{+}\rightarrow \mathbb{R}_{0}^{+}\)be right-dense continuous on \(\mathbb{T}_{1}\times \mathbb{T}_{2}\)and continuous on \(\mathbb{R}_{0}^{+}\)such that
for \(\mathfrak{u}>\mathfrak{v}\geq 0\)and \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\), and
for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with initial condition (3.2). Then
for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\), where
c, \({G}_{1}\), and \({G}_{2}\)are as defined in Theorem 3.1, and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that
Proof
Let \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\). Then inequality (3.28) can be rewritten as follows:
for \({x}_{01}\leq {x}_{1}\leq \bar{x}_{1}\) and \({x}_{02}\leq {x}_{2} \leq \bar{x}_{2}\). Let
Then
and
On taking the identical steps as in (3.17)–(3.10), one has
Let
Then
and
On taking the identical steps as in (3.13) to (3.14), one has
It follows from (3.30)–(3.32) that
Let \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\). Then we have
Since \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x}_{1}\) and \({x}_{02} \leq \bar{x}_{2}\leq \tilde{x}_{2}\) are chosen arbitrarily, we obtain
□
Remark 3.11
Let \(\mathbb{T}=\mathbb{R}\), \(w_{2}=\mu _{ji}=I\), \(f_{i}({x}_{1},{t} _{1},{x}_{2},{t}_{2})=f_{i}({t}_{1},{t}_{2})\), and \(g_{i}=0\). Then Theorem 3.10 coincides with Theorem 3 in [115].
Corollary 3.12
Letu, \(r_{i}\), \(w_{2}\), \(f_{i}\), \(g_{i}\), \(\gamma _{ji}\), and \(\mu _{ji}\)be defined as in Theorem 3.1, \(\bar{\mathfrak{a}}({x}_{1}, {x}_{2})\)be a nonnegative and right-dense continuous function defined on \(([\overline{\mathfrak{p}}_{1},{x}_{01}]\times [\overline{ \mathfrak{p}}_{2},{x}_{02}])_{\mathbb{T}^{2}}\), and \(q_{1}>q_{2}>0\)and \(\mathfrak{C}\geq 0\)be constants such that
for \(({x}_{1},{x}_{2})\in \mathbb{T}_{1}\times \mathbb{T}_{2}\)with the initial condition
Then
for all \({x}_{01}\leq {x}_{1}\leq \tilde{x}_{1}\)and \({x}_{02}\leq {x}_{2}\leq \tilde{x}_{2}\), where
cis defined as in Theorem 3.1, \({H}_{1}^{-1}\)is the inverse function of \({H}_{1}\), and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that
Proof
Let
Then
and
where \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T} _{2}\) are fixed numbers with \({x}_{01}\leq \bar{x}_{1}\leq \tilde{x} _{1}\) and \({x}_{02}\leq \bar{x}_{2}\leq \tilde{x}_{2}\).
For \({x}_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \({x}_{2}\in [x _{02},\bar{x}_{2}]_{\mathbb{T}}\), if \(\mu _{1i}({x}_{1})\geq {x}_{01}\) and \(\mu _{2i}({x}_{2})\geq {x}_{02}\), then \(\mu _{1i}({x}_{1})\in [x _{01},\bar{x}_{1}]_{\mathbb{T}}\), \(\mu _{2i}({x}_{1})\in [x_{02}, \bar{x}_{2}]_{\mathbb{T}}\), and
If \(\mu _{1i}({x}_{1})\leq {x}_{01}\) or \(\mu _{2i}({x}_{2})\leq {x}_{02}\), then from (3.34) we obtain
It follows from (3.39) and (3.40) that
for \({x}_{1}\in [x_{01},\bar{x}_{1}]_{\mathbb{T}}\) and \({x}_{2}\in [x _{02},\bar{x}_{2}]_{\mathbb{T}}\).
From Lemma 2.1 and (3.36) we know that
Making use of (3.41) and the fact that \({\overline{\xi }_{1}}\) is nondecreasing, we have
Theorem 2.2 and \({\overline{\xi }_{1}}^{\Delta _{{x}_{1}}}({x}_{1}, {x}_{2})\geq 0\) lead to the conclusion that
From (3.42) and (3.43) we have
Making use of (3.37) and integrating over \([{x}_{01},{{x}_{1}}]\), we get
Let
Then we have
and
It follows from Lemma 2.1 and (3.44) that
It follows from (3.46) and the fact that \(w_{2}\) and \({\overline{ \zeta }_{1}}\) are nondecreasing that
Making use of (3.45) and the definition of \(H_{1}\), integrating over \([{x}_{01},{{x}_{1}}]\) gives
Combining (3.38), (3.46), and (3.47), we get
Letting \({x}_{1}=\bar{x}_{1}\) and \({x}_{2}=\bar{x}_{2}\) in (3.3), and considering \(\bar{x}_{1}\in \mathbb{T}_{1}\) and \(\bar{x}_{2}\in \mathbb{T}_{2}\) are arbitrary, after substituting \(\bar{x}_{1}\) and \(\bar{x}_{2}\) with \({x}_{1}\) and \({x}_{2}\), we obtain the desired result (3.35). □
Let \(\mathbb{T}=\mathrm{Z}\). Then Theorem 3.1 leads to Corollary 3.13 immediately.
Corollary 3.13
Let \(u, r_{i}, a_{j}:A_{1}\times A_{2}\rightarrow \mathbb{R}_{0}^{+}\)and \(f_{i}, g_{i}, f_{i}^{{\Delta {x_{1}}}}:A_{1}^{2}\times A_{2}^{2} \rightarrow \mathbb{R}_{0}^{+}\)be nonnegative real-valued functions such that \(a_{j}\)is nondecreasing with respect to its each variable, let \(\gamma _{ji}:A_{j}\rightarrow \mathbb{R}_{0}^{+}\)be a nonnegative and nondecreasing function, \(\tilde{\mathfrak{a}}: \{-\rho _{1i}, \ldots ,-1,0\}\times \{-\rho _{2i},\ldots ,-1,0\}\rightarrow \mathbb{R}_{0}^{+}\)be a nonnegative function, \(-\infty < \tilde{\mathfrak{p}}_{j}=\inf \{\min ({x}_{j}-\rho _{ji} ), {x}_{j}\in {A}_{j}\}\leq {0}\), wand \(w_{j}\)be as defined in Theorem 3.1. If \(u({x}_{1},{x}_{2})\)satisfies the following discrete inequality
with the following initial condition
then
for all \(0\leq {x}_{1}\leq \tilde{x}_{1}\)and \(0\leq {x}_{2}\leq \tilde{x}_{2}\), where
\({H}_{2}^{-1}\)and \({H}_{3}^{-1}\)are respectively the inverse functions of \({H}_{2}\)and \({H}_{3}\), and \(\tilde{x}_{1}\)and \(\tilde{x}_{2}\)are chosen such that
4 Applications
Example 4.1
Consider the following integro-differential equation with several arguments:
with the initial condition
for \(\mathrm{F}:\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\times \mathrm{R}^{n+1}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n+1}\), \(\mathbf{Q}:\mathbb{T}_{1}^{2}\times \mathbb{T} _{2}^{2}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n}\), \(u:\mathbb{T}_{1}\times \mathbb{T} _{2}\rightarrow \mathbb{R} /\{0\}\), \(\mathfrak{a}_{j}:\mathbb{T}_{j} \rightarrow \mathbb{R}\), \(\mathfrak{a}: ([\mathfrak{p}_{1},{x}_{01}] \times [\mathfrak{p}_{2},{x}_{02}])_{{\mathbb{T}}^{2}}\rightarrow \mathbb{R}\), \(a_{1}:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}\) are right-dense continuous functions and \(\mu _{ji}\) is as defined in Theorem 3.1.
Theorem 4.2
Assume that
where \(f_{i}\), \(g_{i}\), \(r_{i}\), \(a_{2}\)are as defined in Theorem 3.1, \(a_{1}({x}_{1},{x}_{2})=\sum_{j=1}^{2}\mathfrak{a}_{j}({x}_{j})\), \(w_{1}(\eta )=\sqrt[3]{{\sigma ^{2}}(\eta )}+\sqrt[3]{{\sigma }(\eta ) \eta }+\sqrt[3]{\eta ^{2}}\), \(w_{2}(\eta )=\sqrt{{\sigma }(\sqrt[3]{ \eta })}+\sqrt[6]{\eta }\)for \(\eta \in \mathbb{R}_{0}^{+}\), and \(u({x}_{1},{x}_{2})\)is a solution of equation (4.1) with initial condition (4.2), then
where
Proof
Let \(\tilde{\mathbb{T}}=\varrho (\mathbb{T}_{1},{x}_{2})\) and \(\tilde{G}_{j}(\eta )=\sqrt[4-j]{\eta }\), where \(\varrho ({x}_{1}, {x}_{2})\) is strictly increasing with respect to \({x}_{1}\in \mathbb{T}_{1}\). Then it follows from Theorem 2.3 that
From (4.3) we know that (4.5) has the form
Therefore, the desired result follows easily from (3.3) and (4.6). □
Example 4.3
Consider another integro-differential equation with several arguments:
with the initial condition
for \(\mathrm{F}:\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\times \mathbb{R}^{n+1}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n+1}\), \(\mathbf{Q}:\mathbb{T}_{1}^{2}\times \mathbb{T} _{2}^{2}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is right-dense continuous on \(\mathbb{T}^{2}_{1}\times \mathbb{T}^{2}_{2}\) and continuous on \(\mathbb{R}^{n}\), \(u: \mathbb{T}_{1}\times \mathbb{T} _{2}\rightarrow \mathbb{R} / \{0\}\), \(\mathfrak{f}_{j}:\mathbb{T}_{j} \rightarrow \mathbb{R}\), \(\mathfrak{a}:([\mathfrak{p}_{1},{x}_{01}] \times [\mathfrak{p}_{2},{x}_{02}])_{{\mathbb{T}}^{2}}\rightarrow \mathbb{R}\) are right-dense continuous functions, \(\mathfrak{C}\) is a nonzero constant such that \(\mathfrak{C}\geq \sum_{j=1}^{2}| \mathfrak{f}_{j}({x}_{j})|\) and \(\mu _{ji}\) is as defined in Theorem 3.1.
Theorem 4.4
Assume that
where \(f_{i}\), \(g_{i}\), and \(r_{i}\)are as defined in Theorem 3.1, \(w_{2}(\eta )=\sqrt[3]{{\sigma ^{2}}(\eta ^{3})}+\sqrt[3]{{\sigma }(\eta ^{3})\eta ^{3}}+{\eta ^{2}}\)for \(\eta \in \mathbb{R}_{0}^{+}\). If \(u({x}_{1},{x}_{2})\)is a solution of equation (4.7) satisfying initial condition (4.8), then
where \(\mathfrak{b}_{4}({x}_{1},{x}_{2})\)is defined by (4.4) and
Proof
Let \(\overline{\mathbb{T}}=\varpi (\mathbb{T}_{1},{x}_{2})\) and \(\overline{G}_{2}(\eta )=\sqrt[3]{\eta }\), where ϖ is strictly increasing with respect to \({x}_{1}\in \mathbb{T}_{1}\). Then by Theorem 2.3 we have
Integrating equation (4.7) over \([{x}_{02},{{x}_{2}}]\) gives
It follows from Theorem 2.2 and \(\frac{u^{\Delta {{x}_{1}}}({x}_{1}, {x}_{2})}{u({x}_{1},{x}_{2})}\geq 0\) that
From (4.10) and (4.11) one has
Integrating over \([\mathfrak{x}_{01},{\mathfrak{x}_{1}}]\) leads to
From (4.9) we know that inequality (4.12) has the form
Therefore, the desired result follows from (3.5) and (4.13). □
Example 4.5
Consider the delay discrete inequality (4.8) satisfying initial condition (4.9) with \(u({x}_{1},{x}_{2})=27^{{x}_{1}{x}_{2}}\), \({\rho }_{ji}=ji\), \(a_{1}({x}_{1},{x}_{2})=27^{8}\), \(a_{2}({x}_{1}, {x}_{2})=\sqrt[10]{{x}_{1}{x}_{2}}\), \(f_{i}({x}_{1},{t}_{1},{x}_{2}, {t}_{2})=\arctan (\sqrt[i+1]{{x}_{1}+{t}_{1}+{x}_{2}+{t}_{2}})\), \(r_{i}({x}_{1},{x}_{2})=\sqrt[i+1]{\exp ({{x}_{1}}{{x}_{2}})}\), \(\gamma _{ji}=w=I\), \(g_{i}({x}_{1},{t}_{1},{x}_{2},{t}_{2})=10^{-i-1} \sqrt[i+1]{ {x}_{1}+{t}_{1}+{x}_{2}+{t}_{2}}\) (\(1\leq i\leq 2\)), and \({w}_{j}\) is defined as in Theorem 4.2.
We find that the numerical solution agrees with the analytical solution for some discrete inequalities by calculating the value of \(u(x,y)\) from (4.8) and (4.10) (see Table 1).
5 Conclusion
In the article, we have presented several explicit bounds for the delay double integral inequalities on time scales and have given their applications to the solutions of certain integro-differential equations. Our results are the improvements and generalizations of some previously known results. Furthermore, we have found some new estimates for the integral inequalities in the form of exponential function.
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Rafeeq, S., Kalsoom, H., Hussain, S. et al. Delay dynamic double integral inequalities on time scales with applications. Adv Differ Equ 2020, 40 (2020). https://doi.org/10.1186/s13662-020-2516-3
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DOI: https://doi.org/10.1186/s13662-020-2516-3