- Open Access
RETRACTED ARTICLE: Fuzzy fractional Ostrowski inequality with Caputo differentiability
© Allahviranloo et al.; licensee Springer 2013
- Received: 25 November 2012
- Accepted: 24 January 2013
- Published: 14 February 2013
The Retraction Note to this article has been published in Journal of Inequalities and Applications 2013 2013:417
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.
- fuzzy fractional Ostrowski inequality
- fuzzy Caputo differentiability
- Hukuhara difference
- fuzzy-valued function
In 1938, Ostrowski proved the following important inequality :
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.
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.
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 be fuzzy continuous. Then exists and belongs to ; furthermore, it holds , .
We consider and . Then .
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.
Now, we state an efficient result.
- (i)Let f be -differentiable, then we have -differentiable right fuzzy Caputo derivative and(11)
- (ii)Let f be -differentiable, then we have -differentiable right fuzzy Caputo derivative and(12)
Proof It is straightforward. □
- (i)if f is a -differentiable fuzzy-valued function, then(13)
- (ii)if f is a -differentiable fuzzy-valued function, then(14)
Proof . □
Now, we state the following result which will be used later.
This completes the proof. □
Now, we state the main result as given below.
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.
The reviewers’ comments, which have improved the quality of this paper, are greatly appreciated.
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