Generalized integral inequalities on time scales
https://doi.org/10.1186/s13660-016-1170-5
© Fayyaz et al. 2016
Received: 4 March 2016
Accepted: 7 September 2016
Published: 26 September 2016
Abstract
The theory of dynamic equations on time scales which was formulated by Hilger is an area of mathematics which is currently receiving profuse attention. Despite the fact that the basic objective of times scales is to bring together the study of difference and differential equations, it also extends these classical cases to ‘in-between’. In the present article we present a version of Feng Qi integral inequalities on time scales which are in fact generalizations of results given in different articles.
Keywords
MSC
1 Introduction
In the past few years, time scales calculus has fascinated scientists by virtue of its tremendous application in many branches of sciences, e.g. population dynamics, finance, physics, etc. [1, 2]. During the last decade, with the evolution of integral and differential equations as in including difference equations, voluminous integral and differential inequalities have been disclosed [3–8]. These handy discoveries concern a significant part in the research of boundedness, comprehensive existence, stability of solutions of integral and differential equations, and likewise of difference equations. Subsequently, Hilger [9] put forward the time scales’ theory as a theory that combines differential and difference calculus in a sophisticated manner. Afterwards several researchers have talked about various facets of the dynamic equations on time scales including various inequalities in detail [8, 10]. However, the Feng Qi integral inequalities on time scales have been given limited attention.
In 2000, Feng Qi explored some new types of integral inequalities and presented his results through an analytic approach and mathematical induction. His work motivated many researchers and a group of people at different times continued the work of Feng Qi. Mohamed Akkouchi, [11] studied a general version of a problem posed by Feng Qi [12] in the context of a measured space equipped with a positive finite measure. Hong Yong [13] in his short note introduced some parameters and imposed sufficient conditions so that Qi type integral inequality holds.
Moreover, Kamel Brahim et al. [14] introduced some Feng Qi type q-integral inequalities, in quantum calculus.
Motivated by the afore-mentioned discussion, the present article is concerned with Feng Qi integral inequalities on time scales. To the best of the authors’ knowledge, no contribution is available in the literature about Feng Qi type integral inequalities on time scales.
2 Preliminaries
For the convenience of the reader, we provide some significant and main characteristics of time scale calculus, basic definitions, and results. By a time scale we understand any non-empty and closed subset of real numbers with the ordering inherited from the set of real numbers. Although [2] and [15] are good sources to study the basics of time scale and quantum calculus, we just state some definitions, important results, and notations necessary for our article. Throughout this paper we will denote a time scale by the symbol \(\mathbb {T}\) and we mean \([a,b]\cap\mathbb{T}\) by \([a,b]_{\mathbb{T}}\), where \([a,b]\subset\mathbb{R}\). From [2], we have extracted some definitions and results which are as follows.
Definition 2.1
Proposition 2.2
- (i)If \(\sigma(u)>u\), then ϕ is Δ-differentiable at \(u\in\mathbb{T}^{k}\) with$$\phi^{\Delta}(u)=\frac{\phi(\sigma(u))-\phi(u)}{\mu(u)}. $$
- (ii)If \(\rho(u)< t\), then ϕ is ∇-differentiable at \(u\in\mathbb{T}_{k}\) with$$\phi^{\nabla}(u)=\frac{\phi(u)-\phi(\rho(u))}{\nu(u)}. $$
Proposition 2.3
- (i)The sum \(\phi+f:\mathbb{T}\rightarrow\mathbb{R}\) is Δ-differentiable at u with$$(\phi+f)^{\Delta}(u)=\phi^{\Delta}(u)+f^{\Delta}(u). $$
- (ii)For any constant \(c, c \phi:\mathbb{T}\rightarrow\mathbb {R}\) is Δ-differentiable at u with$$(c \phi)^{\Delta}(u)=c \phi^{\Delta}(u). $$
Proposition 2.4
- (i)The sum \(\phi+f:\mathbb{T}\rightarrow\mathbb{R}\) is ∇-differentiable at u with$$(\phi+f)^{\nabla}(u)=\phi^{\nabla}(u)+f^{\nabla}(u). $$
- (ii)For any constant \(c, c f:\mathbb{T}\rightarrow\mathbb{R}\) is ∇-differentiable at u with$$(c f)^{\nabla}(u)=c f^{\nabla}(u). $$
Proposition 2.5
Proposition 2.6
Proposition 2.7
Proposition 2.8
- (i)Assume that \(\phi:\mathbb{T}\rightarrow\mathbb{R}\) is Δ-differentiable on \(\mathbb{T}^{k}\). Then ϕ is ∇-differentiable at u andfor \(u \in\mathbb{T}_{k}\) such that \(\sigma(\rho(u))=u\).$$\phi^{\nabla}(u)=\phi^{\Delta}\bigl(\rho(u)\bigr) $$
- (ii)Assume that \(\phi:\mathbb{T}\rightarrow\mathbb{R}\) is ∇-differentiable on \(\mathbb{T}_{k}\), then ϕ is Δ-differentiable at u andfor \(u \in\mathbb{T}^{k}\) such that \(\rho(\sigma(u))=u\).$$\phi^{\nabla}(u)=\phi^{\Delta}\bigl(\rho(u)\bigr) $$
This article has been organized into five sections. In the first section we give an introduction of the article, the second section contains preliminaries which will be used in the sequel, and the third section is reserved for results related to Fenq Qi type Δ-integral inequalities defined on a discrete time scale, which generalizes some results given in [17] and [14]. In [17] and [14], there are given the q-analog and the h-analog, respectively, of some Feng Qi inequalities given in [18, 19], and [20]; therefore, the fourth section will explore the results related to Feng Qi type ∇-integral inequalities defined on a discrete time scale. The last section highlights the important results of present article.
3 Feng Qi type Δ-integral inequalities
This section is devoted to the results related to Fenq Qi type Δ-integral inequalities defined on discrete time scales, which generalize some results given in [17] and [14].
Lemma 3.1
Proof
Remark 3.2
If we set \(\sigma(v)=qv+h\) in the preceding lemma defined on \(\mathbb {T}^{a}_{(q,h)}\), then we get the following result.
Theorem 3.3
Proof
If we choose \(\sigma(v)=qv+h\) defined on \(\mathbb{T}^{a}_{(q,h)}\) then we get the following corollary.
Corollary 3.4
Remark 3.5
Observation shows that Theorem 3 of [17] is a special case of Corollary 3.4 if we simply set \(q=1\). Further we note that our result is also a refinement of Proposition 3.2 of [14], and if we set \(h=0\) in the above theorem then we get a more refined result as compared to Proposition 3.2 of [14].
Theorem 3.6
Proof
We end this section with a corollary and some remarks.
By setting \(\sigma(v)=qv+h\) defined on \(\mathbb{T}^{a}_{(q,h)}\), then we get the following corollary.
Corollary 3.7
Remark 3.8
It is worth noting that Theorem 4 of [17] becomes a special case of Corollary 3.7 if we simply set \(q=1\). Furthermore, it can be noted that this result is a refinement of Proposition 3.5 of [14] and if we set \(h=0\) in the above corollary then we get a more refined result than Proposition 3.5 of [14].
4 Feng Qi type ∇-integral inequalities
In [17] and [14] there are given the q-analog and the h-analog, respectively, of some Feng Qi ∇-integral inequalities, which are given in [18, 19], and [20]. Hence, this part of the article is concerned with the results related to Fenq Qi type ∇-integral inequalities defined on a discrete time scale.
Lemma 4.1
Proof
Remark 4.2
If in the preceding lemma we set \(\rho(v)=q^{-1}(v-h)\) defined on \(\mathbb{T}^{a}_{(q,h)}\), then we obtain the following result.
Theorem 4.3
Proof
If we choose \(\rho(v)=q^{-1}(v-h)\) defined on \(\mathbb{T}^{a}_{(q,h)}\), then we get the following corollary.
Corollary 4.4
Theorem 4.5
Proof
If we choose \(\rho(v)=q^{-1}(v-h)\) defined on \(\mathbb{T}^{a}_{(q,h)}\) then we get the following corollary.
Corollary 4.6
5 Conclusion
The article is aimed to explore Feng Qi integral inequalities on time scales while considering Δ- and ∇-integrals. The present article generalizes some results of various articles, Lemma 2.1 of [14, 17] is a special case of Lemma 3.1. Moreover, observations show that Theorem 3 of [17] is a special case of Corollary 3.4, in addition to results of the present article there is a refinement of Proposition 3.2 in [14]. It can also be noted that Corollary 3.7 gives us Theorem 4 of [14] as a particular case and refines Proposition 3.5 of [17].
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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