- Research
- Open Access
New aspects of Opial-type integral inequalities
- Yasemin Başcı^{1}Email author and
- Dumitru Baleanu^{2, 3}
https://doi.org/10.1186/s13662-018-1912-4
© The Author(s) 2018
- Received: 9 August 2018
- Accepted: 2 December 2018
- Published: 6 December 2018
Abstract
In this manuscript, we prove new aspects for several Opial-type integral inequalities for the left and right Caputo–Fabrizio operators with nonsingular kernel. For this purpose we use the inequalities obtained by Andrić et al. (Integral Transforms Spec. Funct. 25(4):324–335, 2014), which is the generalization of an inequality of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, 1995). Besides, examples are presented to validate the reported results.
Keywords
- Opial-type inequality
- The left operator
- The right operator
- The left integral
- The right integral
- Caputo–Fabrizio fractional derivative
MSC
- 26D10
- 26D15
- 26A33
- 26A40
- 26A42
- 26A51
1 Introduction and preliminaries
Since the discovery of Opial’s inequality, it has found interesting applications. Really, Opial’s inequality and its generalizations, extensions, and discretizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations besides difference equations [3, 4, 21, 39, 42].
In 1960, Opial [43] obtained the following integral inequality:
Theorem 1.1
From that time, Opial’s inequality [43] has been studied extensively by many mathematicians. This inequality has been extended, generalized in different ways, see [2, 5, 6, 22, 24–28, 39, 40, 44–47, 54]. Also, various mathematicians studied Opial-type integral inequalities for different types of fractional derivative and integral operators involving Caputo, Canavati, Riemann–Liouville, and so on, see [9, 14, 16–19, 29–31] and the references therein.
In 2000, Anastassiou [7] obtained Opial-type inequalities involving functions and their ordinary and fractional derivatives. In 2002, Anastassiou and Goldstein [12] presented the Opial-type inequalities involving fractional derivatives of different orders. The same year, Anastassiou et al. [13] studied a class of \(L_{p}\)-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author in 1998. In 2004, Anastassiou [8] established the Opial-type inequalities including fractional derivatives of two functions in different order and power. In 2008, he presented Opial-type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications, see [9]. Also, in 2009, he presented fractional Opial-type inequalities subject to high order boundary conditions in \(L_{p}\) for \(p > 1\), and in 2012, he extended Opial’s integral inequality using the right and left Caputo as well as Riemann–Liouville fractional derivatives, respectively, see [10, 11].
In 2013, Andrić et al. [17] obtained several Opial-type inequalities including Caputo, Canavati, and Riemann–Liouville fractional derivatives. The same year, they presented developments of composition identities for the Caputo fractional derivatives. They gave applications to Opial-type inequalities in [18]. Also, the same year, they studied some Opial-type inequalities for Riemann–Liouville fractional derivatives obtained by Fink in [34] and Pang and Agarwal in [48], see [19].
In 2014, Andrić et al. [15] gave expansions of the Opial-type integral inequalities. Also, they presented a generalization of an inequality obtained by Agarwal and Pang [4].
In 2015, Farid et al. [32] studied the Opial-type inequalities by using generalized fractional integral operator including the Mittag-Leffler function in the kernel. One year later, they presented Opial-type integral inequalities for Hilfer differential and fractional integral operators involving a generalized Mittag-Leffler function in the kernel, see [33].
In 2017, Tomovski et al. [53] gave the generalization of weighted Opial-type inequalities for fractional integral and differential operators involving generalized Mittag-Leffler functions by using Hölder’s integral inequality motivated by the work of Koliha and Pečarić [38].
In 2017, Sarıkaya and Budak [50] obtained new inequalities of Opial-type for conformable integrals.
Recently, researchers have proposed different fractional-time operators from the well-known Riemann–Liouville operator, see [20, 35–37, 52]. They are defined by nonsingular memory kernels. Also, they used these new operators to generalize the usual diffusion equation. In fact, these new operators can describe better the evolution of some dynamics of complex systems which cannot be done within the standard fractional calculus operators (for more details, see Refs. [35–37] and the references therein).
The purpose of this paper is to establish some Opial-type integral inequalities for the left and right operators with nonsingular kernel. The organization of this paper is given below. The introduction is given in Sect. 1. In Sect. 2, basic definitions and theorems are introduced. Motivated by [4] and [15], we establish several Opial-type inequalities in Sect. 3. Several examples are given for our results in Sect. 4.
2 Basic definitions and theorems
In this section, we present the following theorems, corollaries, and definitions which are useful in the proofs of our results.
Theorem 2.1
([15])
When \(\psi (x )=t^{p_{1}+q_{1}}\), the following corollary is obtained.
Corollary 2.1
([15])
Theorem 2.2
([15])
When \(\psi (x )=t^{p_{1}+q_{1}}\), the following corollary is obtained.
Corollary 2.2
([15])
Definition 2.1
([49])
Definition 2.2
([51])
3 Main results
In this section, we give the Opial-type integral inequalities for the left and right of the operator using the inequalities obtained by Andrić et al. [15], which is the generalization of an inequality of Agarwal and Pang [4].
The following result is obtained by using Theorem 2.1 and the left operator.
Theorem 3.1
Proof
When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.1, the following corollary is obtained.
Corollary 3.1
Theorem 3.2
When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.2, the following corollary is obtained.
Corollary 3.2
The next result is obtained by using Theorem 2.1 and the left integral operator, see for more details [41].
Theorem 3.3
Proof
When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.3, we obtain the following corollary.
Corollary 3.3
Theorem 3.4
When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.4, we obtain the following corollary.
Corollary 3.4
4 Examples
Below, we will show the application of our main results with two examples.
Example 4.1
Example 4.2
5 Conclusion
Caputo–Fabrizio operator has recently started to play an important role in modeling of a class of real world dissipative phenomena [35]. In fact some real data have confirmed that this operator is important for describing the dynamics of specific classes of real world problems. At the same time new mathematical generalizations of this operator were developed. In our manuscript, with the help of inequalities obtained by Andrić et al. [15], we proposed, within four theorems and their related corollaries, several Opial-type integral inequalities for Caputo–Fabrizio operators. Finally, we analyzed two illustrative examples carefully. The results reported in this manuscript can find applications within the evaluation of the existence and uniqueness of initial and boundary value problems related to diffusion process within the Caputo–Fabrizio operators.
Declarations
Acknowledgements
The authors would like to thank the referees for their useful comments and remarks.
Funding
No funding.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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