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A new approach to interval-valued inequalities
Advances in Difference Equations volume 2020, Article number: 319 (2020)
Abstract
The objective of this work is to advance and simplify the notion of Gronwall’s inequality. By using an efficient partial order and concept of gH-differentiability on interval-valued functions, we investigate some new variants of Gronwall type inequalities on time scales.
1 Introduction
Many problems in real life involve Gronwall’s inequality [23]. It has had an important role in the research of differential and integral equations for nearly 100 years. Its first generalization proved by Richard Bellman [7] motivated many researchers to obtain various generalizations and extensions [2, 3, 6, 31, 34, 35]. The Gronwall–Bellman type inequalities enable critical insight into the uniqueness of solutions, a priori and error estimate in the Galerkin method [41, Ch. 3].
Several research papers in the interval analysis (IA) are based on the demonstration of an uncertain variable as an interval [22, 30, 32, 38]. The relevant formulations of interval calculus on time scales, including some general approaches to differential theory, have been systematized in recent paper [29]. The interval-valued functions and sequences have been recently studied by many authors in various aspects (see [16–21]).
Inequalities are used as a tool for almost all mathematical branches and other subjects of applied and engineering sciences. A detailed study of various inequalities is found in [4, 24, 28, 42]. Some of the differential integral inequalities have been prolonged into set-valued function [5, 10, 14, 15, 36]. Among the more recent investigations on interval-valued Gronwall type inequalities, let us mention the work of Younus et al. [39, 40], where the authors obtain Gronwall inequalities for the interval-valued functions under the notion of Kulish–Mirankor partial order on a set of compact intervals. However, there are many other partial orders, which cannot be covered by Kulish–Mirankor partial order.
In the study of Gronwall type inequalities, an important notion is an exponential function on time scales. A difficult situation has accrued in the case of trigonometric, exponential, hyperbolic, and parabolic functions, where Hilger’s technique [25, 26] differs from Bohner and Peterson’s technique [8, 9]. A newly improved trigonometric, hyperbolic, and parabolic functions base on Cayley transformation has been defined by Cieśliński [12, 13].
In the main part of the proposed study, we firstly discuss some new variants of Gronwall type inequalities on time scale by using the concept of Cayley exponential function, which is the generalization of some inequalities from [1, 11, 27]. Also, by defining an efficient partial order on a set of compact intervals, we obtain new variants of Gronwall type inequality for interval-valued functions, which gives more general than existing results of [39, 40].
2 Preliminary notation
For time scales calculus, we refer to [8, 29].
In order to define Cayley-exponential (shortly, C-exponential) function, Cieśliński [13], redefined a notion of regressivity as follows:
and
Under the binary operation ⊕, defined by \(\alpha \oplus \beta = \frac{\alpha +\beta }{1+\frac{1}{4}\mu ^{2}\alpha \beta }\), \(\mathcal{R}^{+}\) is an Abelian group [13, Theorem 3.14]. However, the set \(\mathcal{R}\) is not closed with respect to ⊕.
For \(f\in \mathcal{R}\) and \(s\in \mathbb{T}\), consider the subsequent initial value problem (IVP)
where
For \(h\in \mathbb{R}^{+}\), the Cayley transformation \(\xi _{h}\) is defined as
and the Cayley-exponential function for \(f\in \mathcal{R}\) is defined by
It is easy to see that \(E_{f}(\cdot ,s)\) on \(\mathbb{T}\) is the unique solution of IVP (2.1).
Lemma 2.1
([13])
If\(\alpha ,\beta \in \mathcal{R}\), then the subsequent properties hold:
-
1.
\(E_{\alpha }(t^{\sigma },t_{0})= \frac{1+\frac{1}{2}\alpha ( t ) \mu ( t ) }{1-\frac{1}{2}\alpha ( t ) \mu ( t ) }E_{ \alpha }(t,t_{0})\),
-
2.
\(( E_{\alpha }(t,t_{0}) ) ^{-1}=E_{-\alpha }(t,t_{0})= \frac{1}{E_{\alpha }(t,t_{0})}\),
-
3.
\(\overline{E_{\alpha }(t,t_{0})}=E_{\bar{\alpha }}(t,t_{0})\),
-
4.
\(E_{\alpha }(t,t_{0})E_{\alpha }(t_{0},t_{1})=E_{\alpha }(t,t_{1})\),
-
5.
\(E_{\alpha }(t,t_{0})E_{\beta }(t,t_{0})=E_{\alpha \oplus \beta }(t,t_{0})\).
Lemma 2.2
([13])
If\(\alpha \in \mathcal{R}^{+}\), then\(E_{\alpha }>0\).
Lemma 2.3
([13])
\(E_{\alpha }(t,t_{0})=e_{\beta }(t,t_{0})\)if\(\alpha ( t ) = \frac{\beta ( t ) }{1+\frac{1}{2}\beta ( t ) \mu ( t ) }\), \(\beta ( t ) = \frac{\alpha ( t ) }{1-\frac{1}{2}\alpha ( t ) \mu ( t ) }\), with\(\alpha \mu \neq \pm 2\)and\(\beta \mu \neq -1\).
These lemmas are Theorem 3.10 and 3.13 in [13] and Theorem 3.2 in [12], respectively.
Let
For \([\bar{x},\underline {x}]\), \([\bar{y},\underline {y}]\in \mathcal{K}_{C}\),
and
respectively. By definition, we have \(\lambda X=X\lambda \) ∀ \(\lambda \in \mathbb{R}\).
Moreover,
where “\(\ominus _{g}\)” is called gH-difference [30, 37].
For \(X=[\bar{x},\underline {x}]\in \mathcal{K}_{C}\), width of X is defined as \(w(X)=\underline {x}-\bar{x}\). By using \(w ( \cdot ) \), we can write
More explicitly, for \(X,Y,C\in \mathcal{K}_{C}\), we have
Since \(\mathcal{K}_{C}\) is not totally order set (e.g., see [10, 30, 33, 39]). To compare the images of IVFs in the context of inequalities, several partial order relations exist over \(\mathcal{K}_{C}\), which are summarized as follows.
For \(X,Y\in \mathcal{K}_{C}\), such that \(X= [ \bar{x},x ] \), \(Y= [ \bar{y},y ] \), we say that:
-
1.
“\(X\preceq _{\mathrm{LU}}Y\) (or \(X\preceq _{\mathrm{LR}}Y\)), ⇔ \(\bar{x}\leq \bar{y}\) and \(\underline {x}\leq \underline {y}\), \(X\prec _{\mathrm{LU}}Y\) if \(X\preceq _{\mathrm{LU}}Y\) and \(X\neq Y\)”.
-
2.
“\(X\preceq _{\mathrm{LC}}Y\) ⇔ \(\bar{x}\leq \bar{y}\) and \(m ( X ) \leq m ( Y ) \), \(X\prec _{\mathrm{LC}}Y\) if \(X\preceq _{\mathrm{LC}}Y\) and \(X\neq Y\), where \(m ( X ) =\frac{\bar{x}+x}{2}\)”.
-
3.
“\(X\preceq _{\mathrm{UC}}Y\) ⇔ \(\underline {x}\leq \underline {y}\) and \(m ( X ) \leq m ( Y ) \), \(X\prec _{\mathrm{UC}}Y\) if \(X\preceq _{\mathrm{UC}}Y\) and \(X\neq Y\)”.
-
4.
“\(X\preceq _{\mathrm{CW}}Y\) ⇔ \(m ( X ) \leq m ( Y ) \) and \(w ( X ) \leq w ( Y ) \), \(X\prec _{\mathrm{CW}}Y\) if \(X\preceq _{\mathrm{CW}}Y\) and \(X\neq Y\), where \(w ( X ) =\underline {x}- \bar{x}\)”.
-
5.
“\(X\preceq _{\mathrm{LW}}Y\) ⇔ \(\bar{x}\leq \bar{y}\) and \(w ( X ) \leq w ( Y ) \), \(X\prec _{\mathrm{LW}}Y\) if \(X\preceq _{\mathrm{LW}}Y\) and \(X\neq Y\)”.
-
6.
“\(X\preceq _{\mathrm{UW}}Y\) ⇔ \(\underline {x}\leq \underline {y}\) and \(w ( X ) \leq w ( Y ) \), \(X\prec _{\mathrm{UW}}Y\) if \(X\preceq _{\mathrm{UW}}Y\) and \(X\neq Y\)”.
Let \(\mathbb{P}= \{ \preceq _{\mathrm{LU}},\preceq _{\mathrm{LC}},\preceq _{\mathrm{UC}}, \preceq _{\mathrm{CW}},\preceq _{\mathrm{LW}},\preceq _{\mathrm{UW}} \} \) be the set of these partial orders on \(\mathcal{K}_{C}\).
Some properties of these partial orders are examined in the following results.
Lemma 2.4
Let\(\mathbb{P}_{1}:\mathbb{=} \{ \preceq _{\mathrm{LU}},\preceq _{\mathrm{LC}}, \preceq _{\mathrm{UC}},\preceq _{\mathrm{CW}},\preceq _{\mathrm{UW}} \} \). If\(X\preceq _{\mathrm{LW}}Y\), then\(X\preceq _{\ast }Y\) ∀ \(\preceq _{\ast }\in \mathbb{P}_{1}\).
Proof
For \(X,Y\in K_{C}\), with \(X= [ \bar{x},x ] \), \(Y= [ \bar{y},y ] \), it implies that \(\bar{x}\leq \bar{y}\) and \(\underline {x}-\bar{x}\leq \underline {y}-\bar{y}\). By adding these two inequalities, we have \(\underline {x}\leq \underline {y}\) and, furthermore, \(m ( X ) \leq m ( Y ) \). Hence \(X\preceq _{\ast }Y\), ∀ \(\preceq _{\ast }\in \mathbb{P}_{1}\). □
Lemma 2.5
Let\(\mathbb{P}_{2}:\mathbb{=} \{ \preceq _{\mathrm{UC}},\preceq _{\mathrm{UW}} \} \). If\(X\preceq _{\mathrm{CW}}Y\), then\(X\preceq _{\ast }Y\) ∀ \(\preceq _{\ast }\in \mathbb{P}_{2}\).
Proof
For \(X,Y\in K_{C}\), with \(X= [ \bar{x},x ] \), \(Y= [ \bar{y},y ] \), we have \(\bar{x}+\underline {x}\leq \bar{y}+\underline {y}\) and \(\underline {x}-\bar{x}\leq \underline {y}-\bar{y}\). By adding these two inequalities, we have \(\underline {x}\leq \underline {y}\). Hence \(X\preceq _{\ast }Y\), ∀ \(\preceq _{\ast }\in \mathbb{P}_{2}\). □
Lemma 2.6
Let\(X,Y,C\in \mathcal{K}_{C}\). If\(X\preceq _{\mathrm{LW}}Y\)and\(w ( X ) \geq w ( C ) \), then\(X\ominus _{g}C\preceq _{\mathrm{LW}}Y\ominus _{g}C\).
Proof
For \(X,Y,C\in I_{c}\) with \(X= [ \bar{x},x ] \), \(Y= [ \bar{y},y ] \) and \(C= [ \bar{c},c^{+} ] \), LW partial order implies that \(\bar{x}\leq \bar{y}\) and \(\underline {x}-\bar{x}\leq \underline {y}-\bar{y}\). Since \(w ( X ) \geq w ( C ) \), moreover \(w ( Y ) \geq w ( X ) \geq w ( C ) \), it follows that \(X\ominus _{g}C= [ \bar{x}-\bar{c},x-c^{+} ] \) and \(Y\ominus _{g}C= [ \bar{y}-\bar{c},y-c^{+} ] \). By using the fact \(\bar{x}\leq \bar{y}\) and \(\underline {x}-\bar{x}\leq \underline {y}-\bar{y}\) implies that \(\bar{x}-\bar{c}\leq \bar{y}-\bar{c}\) and \(\underline {x}-\bar{x}- ( \underline {c} -\bar{c} ) \leq \underline {y}-\bar{y}- ( \underline {c}-\bar{c} ) \). Hence, we obtain that \(X\ominus _{g}C\preceq _{\mathrm{LW}}Y\ominus _{g}C\). □
The subsequent corollaries are direct implications of Lemma 2.4 and 2.5.
Corollary 2.7
If\(X\preceq _{\mathrm{LU}}Y\), then\(X\preceq _{\mathrm{LC}}Y\)and\(X\preceq _{\mathrm{UC}}Y\).
Corollary 2.8
If\(X\preceq _{\mathrm{CW}}Y\), then\(X\preceq _{\mathrm{UC}}Y\)and\(X\preceq _{\mathrm{UW}}Y\).
Corollary 2.9
If\(X\preceq _{\mathrm{UW}}Y\), then\(X\preceq _{\mathrm{UC}}Y\).
However, the converse of the above implications may not be true. To demonstrate this, we provide the following examples.
Example 2.10
For \(X= [ 1,4 ] \) and \(Y= [ 3,5 ] \), \(X\preceq _{\mathrm{LU}}Y\), but \(X\npreceq _{\mathrm{CW}}Y,X\npreceq _{\mathrm{LW}}Y\) and \(X\npreceq _{\mathrm{UW}}Y\).
If \(X= [ 1,4 ] \) and \(Y= [ 3,3.5 ] \), then \(X\preceq _{\mathrm{LC}}Y \), but \(X\npreceq _{\ast }Y\) for all \(\{ \preceq _{\mathrm{LU}},\preceq _{\mathrm{LW}},\preceq _{\mathrm{UC}}, [4] \preceq _{\mathrm{CW}}, \preceq _{\mathrm{UW}} \} \).
\([ 1,2 ] \preceq _{\mathrm{UC}} [ \frac{1}{2},4 ] \), but \([ 1,2 ] \npreceq _{\mathrm{LU}} [ \frac{1}{2},4 ] \) and \([ 1,2 ] \npreceq _{\mathrm{LC}} [ \frac{1}{2},4 ] \), furthermore, \([ 2,\frac{7}{2} ] \preceq _{\mathrm{UC}} [ 3,4 ] \), \([ 2,\frac{7}{2} ] \npreceq _{\ast } [ 3,4 ] \), ∀ \(\{ \preceq _{\mathrm{LW}},\preceq _{\mathrm{CW}},\preceq _{\mathrm{UW}} \} \).
Moreover, for \(X= [ 1,2 ] \) and \(Y= [ \frac{1}{2},5 ] \), \(X\preceq _{\mathrm{CW}}Y,X\npreceq _{\mathrm{LU}}Y,X\npreceq _{\mathrm{LC}}Y\), and \(X\npreceq _{\mathrm{LW}}Y\).
Finally, let \(X= [ 3,4 ] \) and \(Y= [ \frac{1}{2},5 ] \), then \(X\preceq _{\mathrm{UW}}Y,X\npreceq _{\mathrm{LU}}Y,X\npreceq _{\mathrm{LC}}Y\), \(X\npreceq _{\mathrm{LW}}Y\), and \(X\npreceq _{\mathrm{CW}}Y\).
It is noted that the partial order \(\preceq _{\mathrm{LC}}\) does not imply other partial orders as shown in Example 2.10.
For the interval-valued calculus on time scales, we refer to [29].
3 Main results
Throughout this section, assume that \(\varsigma _{0}\in \mathbb{T}\), \(\mathbb{T}_{0}=[\varsigma _{0},\infty )\cap \mathbb{T}\) and \(\mathbb{T}_{0}^{-}=(-\infty ,\varsigma _{0}]\cap \mathbb{T}\).
Lemma 3.1
([40])
Let\(f,x\in C_{rd}\)and\(a\in \mathcal{R}^{+}\). Then
implies
∀ \(\varsigma \in \mathbb{T}_{0}\).
Lemma 3.2
([40])
Let\(f,x\in C_{rd}\)and\(a\in \mathcal{R}^{+}\). Then
implies
and
implies
Theorem 3.3
([40])
Suppose that\(f,x\in C_{rd}\), \(a\in \mathcal{R}^{+}\), and\(a\geq 0\). Then
implies
Corollary 3.4
([40])
Suppose that\(x\in C_{rd}\), \(a\in \mathcal{R}^{+}\), and\(a\geq 0\). Then
implies
Corollary 3.5
([40])
Suppose that\(x\in C_{rd}\), \(f_{0}\in \mathcal{R}\), \(a\in \mathcal{R}^{+}\), and\(a\geq 0\). Then
implies
Corollary 3.6
([40])
If\(a,q\in \mathcal{R}^{+}\)with\(a ( \varsigma ) \leq q ( \varsigma ) \) ∀ \(\varsigma \in \mathbb{T}\), then
Moreover,
Similar to Theorem 3.3, one can get the following results.
Theorem 3.7
([40])
Suppose that\(f,g,x\in C_{rd}\), \(\alpha _{0}\in \mathcal{R} \), \(q\in \mathcal{R}^{+}\), and\(q\geq 0\). Then
implies
An important consequence of Lemma 3.2 is as follows.
Theorem 3.8
([40])
Suppose that\(f,g,x\in C_{rd}\), \(\alpha _{0}\in \mathcal{R} \), \(q\in \mathcal{R}^{+}\), and\(q\geq 0\). Then
implies
3.1 Interval-valued case
For IVF \(F:\mathbb{T}\rightarrow \mathcal{K}_{C}\), define
If \(F:\mathbb{T}\rightarrow \mathcal{K}_{C}\) such that \(F ( \varsigma ) = [ f^{-} ( \varsigma ) ,f^{+} ( \varsigma ) ] \), then (3.19) implies that
By the definition of midpoint function, we can get
By using “\(\preceq _{\mathrm{LC}}\)” and (3.19), one can easily get the following result.
Lemma 3.9
Let\(F,G:\mathbb{T}\rightarrow \mathcal{K}_{C}\). If\(F ( \varsigma ) \preceq _{\mathrm{LC}}G ( \varsigma ) \) ∀ \(\varsigma \in \mathbb{T}\), then\(\langle F ( \varsigma ) \rangle \preceq _{\mathrm{LC}} \langle G ( \varsigma ) \rangle \).
Let us start this section with comparison results for IVFs under LC-partial order. For further discussion, let us consider some function classes:
Lemma 3.10
Let\(F,X\in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\). \(( a ) \)If\(X\in C_{gH}^{1,1st}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
\(( b ) \)If\(X\in C_{gH}^{1,2nd}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Proof
Let \(F,X\in C_{rd}^{\mathcal{K}_{C}}\) with \(X ( \varsigma ) = [ \bar{x} ( \varsigma ) ,x ( \varsigma ) ] \) and \(F ( \varsigma ) = [ f^{-} ( \varsigma ) ,f^{+} ( \varsigma ) ] \).
\((a)\) If \(X\in C_{gH}^{1,1st}\), then \(X^{\Delta } ( \varsigma ) = [ ( \bar{x} ) ^{\Delta } ( \varsigma ) , ( x ) ^{\Delta } ( \varsigma ) ] \). First, we consider the case if \(a(\varsigma )\geq 0\) on \(\mathbb{T}_{0}\), we have \(a ( \varsigma ) \langle X ( \varsigma ) \rangle = [ a ( \varsigma ) \langle \bar{x} ( \varsigma ) \rangle ,a ( \varsigma ) \langle x ( \varsigma ) \rangle ] \). By using inequality (3.22), we obtain
Applying LC-order, we obtain
and
By using Lemma 3.1 on (3.26) and (3.27) respectively, we obtain
and
Inequalities (3.28) and (3.29) yield (3.23). \(a(\varsigma )<0\) on \(\mathbb{T}_{0}\) implies that \(a ( \varsigma ) X ( \varsigma ) = [4] [ a ( \varsigma ) \langle x ( \varsigma ) \rangle ,a ( \varsigma ) \langle \bar{x} ( \varsigma ) \rangle ] \). By using inequality (3.22), we obtain
Applying LC-order, we have
and
By using Lemma 3.1 on (3.30) and (3.31) respectively, we obtain
and
Since \(a(\varsigma )<0\) and \(a\in \mathcal{R}^{+}\), it follows that \(( -a ) \in \mathcal{R}^{+}\) and \(a\leq -a\). Therefore, Lemma 2.1 and Corollary 3.6 imply that
Combining (3.33) and (3.34), we get
\((b)\) If \(X\in C_{gH}^{1,2nd}\), then \(X^{\Delta } ( \varsigma ) = [ ( x ) ^{ \Delta } ( \varsigma ) , ( \bar{x} ) ^{\Delta } ( \varsigma ) ] \) and for \(a ( \varsigma ) \geq 0\), so we have \(a ( \varsigma ) X ( \varsigma ) = [ a ( \varsigma ) \langle \bar{x} ( \varsigma ) \rangle ,a ( \varsigma ) \langle x ( \varsigma ) \rangle ] \). Inequality (3.24) implies that
and
It follows that
and
By using (3.35) and (3.36) in LC order, we can get (3.25). For \(a ( \varsigma ) <0\), similar to the second inequality of part (a), we can obtain (3.25). □
One of the consequences of Lemma 3.9 and Lemma 3.10 is as follows.
Lemma 3.11
Let\(F,X\in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\).
\(( a ) \)If\(X\in C_{gH}^{1,1st}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
\(( b ) \)If\(X\in C_{gH}^{1,2nd}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Similar to Lemma 3.10, by applying Lemma 3.2, we get the subsequent result.
Lemma 3.12
Let\(F,X\in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\).
\(( a ) \)If\(X\in C_{gH}^{1,1st}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
\(( b ) \)If\(X\in C_{gH}^{1,2nd}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Lemma 3.13
Let\(F,X\in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\).
\(( a ) \)If\(X\in C_{gH}^{1,1st}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
\(( b ) \)If\(X\in C_{gH}^{1,2nd}\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Remark 3.14
It is noted that in Lemmas 3.10 and 3.12 we get a more simple and relaxed condition as compared to the main results in [40].
Now onward, we are assuming that all functions are bounded.
Theorem 3.15
Let\(F,X\in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\), \(a ( \varsigma ) \geq 0\) ∀ \(\varsigma \in \mathbb{T}_{0}\),
holds ∀ \(\varsigma \in \mathbb{T}_{0}\). Then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Proof
Consider \(Z ( \varsigma ) =\int _{\varsigma _{0}}^{\varsigma }a ( \tau ) \langle X ( \tau ) \rangle \Delta \tau \). Since \(a ( \tau )\), \(\langle X ( \tau ) \rangle \) are bounded and belong to \(C_{rd}\) class, therefore it follows that \(Z\in C_{gH}^{1,1st}\) and \(Z^{\Delta } ( \varsigma ) =a ( \varsigma ) \langle X ( \varsigma ) \rangle \), \(\varsigma \in \mathbb{T}_{0}\). From inequality (3.41), we can see that \(\langle X ( \varsigma ) \rangle \preceq _{\mathrm{LC}} \langle F ( \varsigma ) \rangle + \langle Z ( \varsigma ) \rangle \). Clearly,
Part \((a)\) in Lemma 3.10 and \(Z ( \varsigma _{0} ) = \{ 0 \} \) implies that
and hence assertion (3.42) follows by inequality (3.41). □
Corollary 3.16
Let\(X\in C_{rd}^{\mathcal{K}_{C}}\), \(a\in \mathcal{R}^{+}\), \(a\geq 0\), and\(X_{0}\in \mathcal{K}_{C}\). If
then
Proof
In Theorem 3.15, if we take \(F ( \varsigma ) =X_{0}\), we can get (3.44). □
Corollary 3.17
Let\(X\in C_{rd}^{\mathcal{K}_{C}}\), \(a\in \mathcal{R}^{+}\), \(a\geq 0\) ∋
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Similar to Theorem 3.15, we derive the subsequent theorem.
Theorem 3.18
Let\(F,Q,X\in C_{rd}^{\mathcal{K}_{C}}\), \(a\in \mathcal{R}^{+}\), \(a\geq 0\)\(b_{0}\in \mathcal{R}^{+}\)such that
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
If we take \(F ( \varsigma ) =Q ( \varsigma ) =0\) in Theorem 3.18, then one can get the following.
Corollary 3.19
Suppose\(X ( \varsigma ) \in C_{rd}^{\mathcal{K}_{C}}\)and\(a\in \mathcal{R}^{+}\), \(a\geq 0\)\(b_{0}\in \mathcal{R}^{+}\)such that
then
∀ \(\varsigma \in \mathbb{T}_{0}\).
Remark 3.20
If \(b_{0}=1\) in Corollary 3.19, then we get Corollary 3.17.
4 Conclusions
In this paper, we presented certain results of Gronwall type inequalities concerning interval-valued functions under \(\preceq _{\mathrm{LC}}\). These inequalities render explicit bounds of unknown functions. By using \(\preceq _{\mathrm{LC}}\)the assumptions in the main results become more relaxed compared to the main results in [40]. The results can be more beneficial in the subject of the uniqueness of solution for interval-valued differential equations or interval-valued integrodifferential equations. Moreover, we will extend these inequalities to fuzzy-interval-valued functions in our forthcoming work. This research also points out that Gronwall’s inequality for interval-valued functions can be reduced to a family of classical Gronwall’s inequality for real-valued functions. The interval versions of Gronwall’s inequality exhibited in this study are tools to work in an uncertain environment. Furthermore, as these inequalities are given by applying different assumptions than those used in the earlier research articles, our results are new.
References
Akin-Bohner, E., Bohner, M., Akin, F.: Pachpatte inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math. 6(1), Article 6, 23 pp. (2005)
Alzabut, J., Abdeljawad, T.: A generalized discrete fractional Gronwall’s inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system. Appl. Anal. Discrete Math. 12, 036 (2018)
Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, 101 (2019)
An, Y., Ye, G., Zhao, D., Liu, W.: Hermite–Hadamard type inequalities for interval (h1, h2)-convex functions. Mathematics 7, 436 (2019)
Anastassiou, G.A.: Advanced Inequalities. World Scientific, New Jersey (2011)
Bainov, D., Simeonov, P.: Integral Inequalities and Applications. Kluwer Academic, Dordrecht (1992)
Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales, an Introduction with Applications. Birkhäuser, Basel (2001)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Chalco-Cano, Y., Flores-Franulič, A., Román-Flores, H.: Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput. Appl. Math. 31(2), 457–472 (2012)
Choi, S.K., Koo, N.: On a Gronwall-type inequality on time scales. J. Chungcheong Math. Soc. 23(1), 137–147 (2010)
Cieśliński, J.L.: Some implications of a new approach to exponential functions on time scales. In: Discrete Contin. Dyn. Syst. 2011, in: Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference. Suppl., vol. I, pp. 302–311 (2011)
Cieśliński, J.L.: New definitions of exponential, hyperbolic and trigonometric functions on time scales. J. Math. Anal. Appl. 388(1), 8–22 (2012)
Costa, T.M., Román-Flores, H., Chalco-Cano, Y.: Opial-type inequalities for interval-valued functions. Fuzzy Sets Syst. 358, 48–63 (2019)
Costa, T.M., Silva, G.N., Chalco-Cano, Y., Román-Flores, H.: Gauss-type integral inequalities for interval and fuzzy-interval-valued functions. Comput. Appl. Math. 38(2), Art. 58, 13 pp. (2019)
Esi, A.: A new class of interval numbers. J. Qafqaz Univ. Math. Comput. Sci. 31, 98–102 (2011)
Esi, A.: Strongly almost-convergence and statistically almost-convergence of interval numbers. Sci. Magna 7(2), 117–122 (2011)
Esi, A.: Lacunary sequence spaces of interval numbers. Thai J. Math. 10(2), 445–451 (2012)
Esi, A.: Double lacunary sequence spaces of double sequence of interval numbers. Proyecciones 31(1), 297–306 (2012)
Esi, A.: Sequence spaces of interval numbers. Appl. Math. Inf. Sci. 8(3), 1099–1102 (2014)
Esi, A.: Statistical and lacunary statistical convergence of interval numbers in topological groups. Acta Sci., Technol. 36(3), 491–495 (2014)
Gallego-Posada, J.D., Puerta-Yepes, E.: Interval analysis and optimization applied to parameter estimation under uncertainty. Bol. Soc. Parana. Mat. 36(2), 107–121 (2018)
Gronwall, T.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 292–296 (1919)
Guo, Y., Ye, G., Zhao, D., Liu, W.: Some integral inequalities for log-h-convex interval-valued functions. IEEE Access 7, 86739–86745 (2019)
Hilger, S.: Analysis on measure chains: a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)
Hilger, S.: Special functions, Laplace and Fourier transform on measure chains. Dyn. Syst. Appl. 8(3–4), 471–488 (1999)
Li, W.N., Sheng, W.: Some Gronwall type inequalities on time scales. J. Math. Inequal. 4(1), 67–76 (2010)
Liu, X., Ye, G., Zhao, D., Liu, W.: Fractional Hermite–Hadamard type inequalities for interval-valued functions. J. Inequal. Appl. 2019, 266 (2019)
Lupulescu, V.: Hukuhara differentiability of interval-valued functions and interval differential equations on time scales. Inf. Sci. 248, 50–67 (2013)
Markov, S.: Calculus for interval functions of a real variables. Computing 22(4), 325–337 (1979)
Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Dordrecht (1991)
Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets Syst. 135(1), 5–9 (2003)
Moore, R.E.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1979)
Pachpatte, B.G.: Inequalities for Differential and Integral Equations. Mathematics in Science and Engineering, vol. 197. Academic Press, San Diego (1998)
Pachpatte, B.G.: Inequalities for Finite Difference Equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 247. Dekker, New York (2002)
Roman-Flores, H., Chalco-Cano, Y., Silva, G.N.: A note on Gronwall type inequality for interval valued functions. In: IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (2013)
Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 161(11), 1564–1584 (2010)
Yadav, V., Bhurjee, A.K., Karmaker, S., Dikshit, A.K.: A facility location model for municipal solid waste management system under uncertain environment. Sci. Total Environ. 603–604, 760–771 (2017)
Younus, A., Asif, M., Farhad, K.: On Gronwall type inequalities for interval-valued functions on time scales. J. Inequal. Appl. 2015, 271, 18 pp. (2015)
Younus, A., Asif, M., Farhad, K., Nisar, O.: Some new variants of interval-valued Gronwall type inequalities on time scales. Iran. J. Fuzzy Syst. 16(5), 187–198 (2019)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications I, Fixed-Point Theorems. Springer, New York (1986)
Zhao, D., Ye, G., Liu, W., Torres, D.F.M.: Some inequalities for interval-valued functions on time scales. Soft Comput. 23(15), 6005–6015 (2019)
Acknowledgements
J. Alzabut would like to thank Prince Sultan University for supporting this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
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Younus, A., Asif, M., Alzabut, J. et al. A new approach to interval-valued inequalities. Adv Differ Equ 2020, 319 (2020). https://doi.org/10.1186/s13662-020-02781-z
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DOI: https://doi.org/10.1186/s13662-020-02781-z
MSC
- 26D15
- 26E25
- 39A12
- 34N05
Keywords
- Gronwall inequality
- Interval-valued functions
- Generalized Hukuhara difference
- Time scales