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RETRACTED ARTICLE: Fuzzy fractional Ostrowski inequality with Caputo differentiability
Journal of Inequalities and Applications volume 2013, Article number: 50 (2013)
The use of fractional inequalities in mathematical models is increasingly widespread in recent years. In this manuscript, we firstly propose the right Caputo derivative of fuzzy-valued functions about fractional order ν (). To this end, we consider two types of differentiability (similar to the non-fractional case). Then we derive the equivalent integral forms of original fuzzy fractional differential equations. Finally, we prove the fuzzy Ostrowski inequality involving three functions under Caputo’s differentiability. In this regard, we state some new results.
In 1938, Ostrowski proved the following important inequality :
Theorem 1.1 Let be continuous on and differentiable on whose derivative is bounded on , i.e., . Then
for any .
The constant is best possible in the sense that it cannot be replaced by a smaller one. This inequality gives an upper bound for the approximation of the integral average by the value f at a point . We notice that the first generalization of Ostrowski’s inequality was given by Milovanovic and Pecaric in .
In recent years, these inequalities have been studied by many researchers, and numerous generalizations, extensions and variants of them have been considered in a number of papers (see Refs. [3–10] and the references therein). In this way, some new types of inequalities such as inequalities of Ostrowski-Gruss type, inequalities of Ostrowski-Chebyshev type, etc. were formed. The first inequality of Ostrowski-Gruss type was given by Dragomir and Wang in , and it was generalized and improved by Matic, Pecaric and Ujevic in . Also, Cheng gave a sharp version of the mentioned inequality in .
In [3, 14] Pachpatte has proved the Ostrowski inequality in three independent variables. In the past few years, many authors have obtained various generalizations of this type of inequality and many researchers worked on a fractional form of it as well as on time scale calculus (see, for example, Refs. [15–18] and the references therein).
We mention that these inequalities were applied for Euler’s beta mapping and special means such as the arithmetic mean, the geometric mean, the harmonic mean and so on (for more details, see Ref.  and the references therein). In  the authors have applied this inequality for the error bounds of general Riemman’s quadrature formulae in terms of .
The main purpose of this manuscript is to establish Ostrowski-type inequality involving Caputo differentiability. So, we propose the right fuzzy Caputo derivative and the fuzzy right fractional Taylor formula in order to prove Ostrowski’s inequality. Then we use these concepts to prove this inequality involving three functions. Some more details about fuzzy differential equations and their applications can be found in Refs. [21–24] and the references therein.
This manuscript is organized as follows. In Section 2, we recall some basic concepts. In Section 3, we firstly propose the right Caputo derivative in the sense of -differentiability and -differentiability, then the equivalent integral form of the original fuzzy fractional differential equation is obtained. After that, we prove the fuzzy Ostrowski inequality involving three functions.
2 Basic concepts
We denote the set of all real numbers by ℝ, and the set of all fuzzy numbers on ℝ is indicated by . A fuzzy number is a mapping with the following properties:
u is upper semi-continuous,
u is fuzzy convex, i.e., for all , ,
u is normal, i.e., for which ,
is the support of the u, and its closure is compact.
For and , we define uniquely the sum and the product by
where means the usual addition of two integrals (as subsets of ℝ) and means the usual product between a scalar and a subset of ℝ.
We have that D is a metric on . Then is a complete metric space with the following properties:
, , ,
Let us consider . If there exists , then we call z the H-difference on x and y, denoted by .
Let us consider . We say that f is fuzzy Riemann integrable to if for any , there exists such that for any division of with the norms , we have
and we write
Let be fuzzy continuous. Then exists and belongs to ; furthermore, it holds , .
We consider and . Then .
Let and . Then . Also, if are fuzzy continuous functions, then the function defined by is continuous on , and
Let , . Then we define the fuzzy fractional left Riemann-Liouville operator as
Also, we define the fuzzy fractional right Riemann-Liouville operator by
Above, Γ denotes the gamma function
3 Main results
In this section, we state some definitions and results about the right fuzzy Caputo derivative and the fuzzy right fractional Taylor formula in order to prove the Ostrowski inequality involving three functions.
Definition 3.1 Let , be integrable. Then the right fuzzy Caputo derivative of f for and is denoted by and defined by
Now, we state an efficient result.
Theorem 3.1 Let , , .
Let f be -differentiable, then we have -differentiable right fuzzy Caputo derivative and(11)
Let f be -differentiable, then we have -differentiable right fuzzy Caputo derivative and(12)
Proof It is straightforward. □
Theorem 3.2 Let , , . If exists and it is Lebesgue integrable, then we state the equivalent integral form of the original fuzzy fractional differential equation with the initial condition as follows:
if f is a -differentiable fuzzy-valued function, then(13)
if f is a -differentiable fuzzy-valued function, then(14)
Proof . □
Now, we state the following result which will be used later.
Theorem 3.3 Let and . Then
Proof We observe that
for the case -differentiable. We notice that and ,
For , we have
As a result, we prove that
Thus, we obtain
In the case -differentiable, we have
for the case -differentiable. We notice that and , also
Thus, for all , we have
So, we prove that
Thus, we obtain
When -differentiable, we have
This completes the proof. □
Now, we state the main result as given below.
Theorem 3.4 Let and . Then
Proof For we have
So we have the following:
Finally, by using the following property:
we complete the proof. □
Recently, the application of fractional differential equations under uncertainty received a considerable interest both in mathematics and in applications. In this manuscript, the fuzzy Caputo differentiability has been stated; then we made inquiries about the fuzzy fractional Ostrowski inequality involving three functions in the right Caputo fractional derivative. In this way, we have obtained some basic results in the fuzzy framework. To the best of our knowledge, this is the first time in the literature that such inequality has been considered under uncertainty. For future work, we will consider the mentioned fuzzy inequality possessing higher fractional order with different types of differentiability.
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The reviewers’ comments, which have improved the quality of this paper, are greatly appreciated.
The authors declare that they have no competing interests.
All authors have equal contributions.
An erratum to this article can be found at http://dx.doi.org/10.1186/1029-242X-2013-417.
The manuscript has been retracted by the editor as a request of one of the authors (Dumitru Baleanu). He realized that the version submitted by the corresponding author was not in the final form. The authors were advised to resubmit their paper after making necessary corrections.
A retraction note to this article can be found online at http://dx.doi.org/10.1186/1029-242X-2013-417.
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Allahviranloo, T., Avazpour, L., Ebadi, M.J. et al. RETRACTED ARTICLE: Fuzzy fractional Ostrowski inequality with Caputo differentiability. J Inequal Appl 2013, 50 (2013). https://doi.org/10.1186/1029-242X-2013-50
- fuzzy fractional Ostrowski inequality
- fuzzy Caputo differentiability
- Hukuhara difference
- fuzzy-valued function