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RETRACTED ARTICLE: Fuzzy fractional Ostrowski inequality with Caputo differentiability

  • 1,
  • 2Email author,
  • 3,
  • 4, 5, 6 and
  • 7
Journal of Inequalities and Applications20132013:50

https://doi.org/10.1186/1029-242X-2013-50

  • Received: 25 November 2012
  • Accepted: 24 January 2013
  • Published:

The Retraction Note to this article has been published in Journal of Inequalities and Applications 2013 2013:417

Abstract

Abstract

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 ( 0 < ν < 1 ). 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.

Keywords

  • fuzzy fractional Ostrowski inequality
  • fuzzy Caputo differentiability
  • Hukuhara difference
  • fuzzy-valued function

1 Introduction

In 1938, Ostrowski proved the following important inequality [1]:

Theorem 1.1 Let f : [ a , b ] R be continuous on [ a , b ] and differentiable on ( a , b ) whose derivative f : ( a , b ) R is bounded on ( a , b ) , i.e., f = sup t ( a , b ) | f ( t ) | < . Then
| 1 b a a b f ( t ) d t f ( x ) | [ 1 4 + ( x a + b 2 ) 2 ( b a ) 2 ] ( b a ) f

for any x [ a , b ] .

The constant 1 4 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 1 b a a b f ( t ) d t by the value f at a point x ( a , b ) . We notice that the first generalization of Ostrowski’s inequality was given by Milovanovic and Pecaric in [2].

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. [310] 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 [11], and it was generalized and improved by Matic, Pecaric and Ujevic in [12]. Also, Cheng gave a sharp version of the mentioned inequality in [13].

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. [1518] 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. [19] and the references therein). In [20] the authors have applied this inequality for the error bounds of general Riemman’s quadrature formulae in terms of f 1 .

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. [2124] 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 [ ( i ) ν ] -differentiability and [ ( i i ) ν ] -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 , and the set of all fuzzy numbers on is indicated by R F . A fuzzy number is a mapping u : R [ 0 , 1 ] with the following properties:
  1. (a)

    u is upper semi-continuous,

     
  2. (b)

    u is fuzzy convex, i.e., u ( λ x + ( 1 λ ) y ) min { u ( x ) , u ( y ) } for all x , y R , λ [ 0 , 1 ] ,

     
  3. (c)

    u is normal, i.e., x 0 R for which u ( x 0 ) = 1 ,

     
  4. (d)

    supp u = { x R | u ( x ) > 0 } is the support of the u, and its closure cl ( supp u ) is compact.

     
For u , v R F and λ R , we define uniquely the sum u v and the product λ u by
[ u v ] r = [ u ] r + [ v ] r , [ λ u ] r = λ [ u ] r , r [ 0 , 1 ] ,

where [ u ] r + [ v ] r means the usual addition of two integrals (as subsets of ) and λ [ u ] r means the usual product between a scalar and a subset of .

Define D : R F × R F R + by
D ( u , v ) : = sup r [ 0 , 1 ] max { | u ( r ) v ( r ) | , | u + ( r ) v + ( r ) | } ,
(1)
where
[ v ] r = [ v ( r ) , v + ( r ) ] , u , v R F .
(2)

We have that D is a metric on R F . Then ( R F , D ) is a complete metric space with the following properties:

D ( u w , v w ) = D ( u , v ) , u , v , w R F ,

D ( k u , k v ) = | k | D ( u , v ) , u , v R F , k R ,

D ( u v , w e ) = D ( u , w ) + D ( v , e ) , u , v , w , e R F .

Let us consider x , y R F . If there exists z R F : x = y z , then we call z the H-difference on x and y, denoted by x y .

Let us consider f : [ a , b ] R F . We say that f is fuzzy Riemann integrable to I R F if for any ϵ > 0 , there exists δ > 0 such that for any division P = { [ u , v ] ; ξ } of [ a , b ] with the norms Δ ( P ) < δ , we have
D ( P ( v u ) f ( ξ ) , I ) < ϵ ,
(3)
and we write
I : = ( F R ) a b f ( x ) d x .
(4)

Let f : [ a , b ] R F be fuzzy continuous. Then ( F R ) a b f ( x ) d x exists and belongs to R F ; furthermore, it holds [ ( F R ) a b f ( x ) d x ] r = [ a b ( f ) ( r ) ( x ) d x , ( f ) + ( r ) ( x ) d x ] , r [ 0 , 1 ] .

We consider f C F ( [ a , b ] ) and c [ a , b ] . Then ( F R ) a b f ( x ) d x = ( F R ) a c f ( x ) d x + ( F R ) c b f ( x ) d x .

Let f , g C F ( [ a , b ] ) and c 1 , c 2 [ a , b ] . Then ( F R ) a b ( c 1 f ( x ) + c 2 g ( x ) ) d x = c 1 ( F R ) a b f ( x ) d x + c 2 ( F R ) a b g ( x ) d x . Also, if f , g : [ a , b ] R R F are fuzzy continuous functions, then the function F : [ a , b ] R + defined by F ( x ) : = D ( f ( x ) , g ( x ) ) is continuous on [ a , b ] , and
D ( ( F R ) a b f ( x ) d x , ( F R ) a b g ( x ) d x )
(5)
a b D ( f ( x ) , g ( x ) ) d x .
(6)
Let f C F ( [ a , b ] ) , 0 < ν < 1 . Then we define the fuzzy fractional left Riemann-Liouville operator as
I a + ν f ( x ) = 1 Γ ( ν ) a x ( x t ) ν 1 f ( t ) d t , x [ a , b ] .
(7)
Also, we define the fuzzy fractional right Riemann-Liouville operator by
I b ν f ( x ) = 1 Γ ( ν ) x b ( t x ) ν 1 f ( t ) d t , x [ a , b ] .
(8)
Above, Γ denotes the gamma function
Γ ( ν ) = 0 e t t ν 1 d t .
(9)

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 f C F [ a , b ] L F [ a , b ] , f be integrable. Then the right fuzzy Caputo derivative of f for 0 < ν < 1 and x [ a , b ] is denoted by D b ν f ( x ) R F and defined by
D b ν f ( x ) = ( 1 ) Γ ( 1 ν ) x b ( t x ) ν f ( t ) d t .
(10)

Now, we state an efficient result.

Theorem 3.1 Let f C F [ a , b ] L F [ a , b ] , 0 < ν < 1 , a x b .
  1. (i)
    Let f be ( i i ) -differentiable, then we have [ ( i ) ν ] -differentiable right fuzzy Caputo derivative and
    ( D b ν f ) ( x , r ) = [ ( D b ν f ) ( x , r ) , ( D b ν f + ) ( x , r ) ] , 0 r 1 .
    (11)
     
  2. (ii)
    Let f be ( i ) -differentiable, then we have [ ( i i ) ν ] -differentiable right fuzzy Caputo derivative and
    ( D b ν f ) ( x , r ) = [ ( D b ν f + ) ( x , r ) , ( D b ν f ) ( x , r ) ] , 0 r 1 .
    (12)
     

Proof It is straightforward. □

Theorem 3.2 Let f C F [ a , b ] L F [ a , b ] , 0 < ν < 1 , a x b . If D b ν f ( x ) exists and it is Lebesgue integrable, then we state the equivalent integral form of the original fuzzy fractional differential equation D b ν f ( x ) = g ( x , f ( x ) ) with the initial condition f 0 = f ( b ) as follows:
  1. (i)
    if f is a [ ( i ) ν ] -differentiable fuzzy-valued function, then
    f ( x ) = f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t .
    (13)
     
  2. (ii)
    if f is a [ ( i i ) ν ] -differentiable fuzzy-valued function, then
    f ( x ) = f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t .
    (14)
     

Proof [25]. □

Now, we state the following result which will be used later.

Theorem 3.3 Let f C F [ a , b ] L F [ a , b ] and 0 < ν < 1 . Then
D ( 1 b a ( F R ) a b f ( x ) d x , f ( b ) ) ( b a ) ν Γ ( ν + 2 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .
Proof We observe that
D ( 1 b a ( F R ) a b f ( x ) d x , f ( b ) ) = D ( 1 b a ( F R ) a b f ( x ) d x , f ( b ) b a a b d x ) = D ( 1 b a ( F R ) a b f ( x ) d x , 1 b a ( F R ) a b f ( b ) d x ) = 1 b a D ( ( F R ) a b f ( x ) d x , ( F R ) a b f ( b ) d x ) 1 b a a b D ( f ( x ) , f ( b ) ) d x
for the case [ ( i ) ν ] -differentiable. We notice that f C F [ a , b ] L F [ a , b ] and 0 < ν < 1 ,
f ( x ) = f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t .
For a x b , we have
D ( f ( x ) , f ( b ) ) = D ( f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , f ( b ) ) = D ( 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , 0 ˜ ) 1 Γ ( ν ) D ( x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , 0 ˜ ) 1 Γ ( ν ) D ( x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , x b 0 ˜ d t ) 1 Γ ( ν ) x b ( t x ) ν 1 D ( ( D b ν f ) ( t ) , 0 ˜ ) d t 1 Γ ( ν ) ( b x ) ν ν sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .
As a result, we prove that
D ( f ( x ) , f ( b ) ) ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) , a x b .
Thus, we obtain
a c D ( f ( x ) , f ( b ) ) d x ( b a ) ν + 1 Γ ( ν + 2 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .
In the case [ ( i ) ν ] -differentiable, we have
D ( 1 b a ( F R ) a b f ( x ) d x , f ( b ) ) 1 b a a b D ( f ( x ) , f ( b ) ) d x ( b a ) ν Γ ( ν + 2 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ )
for the case [ ( i i ) ν ] -differentiable. We notice that f C F [ a , b ] L F [ a , b ] and 0 < ν < 1 , also
f ( x ) = f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t .
Thus, for all a x b , we have
D ( f ( x ) , f ( b ) ) = D ( f ( b ) 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , f ( b ) ) = D ( 1 Γ ( ν ) x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , 0 ˜ ) 1 Γ ( ν ) D ( x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , 0 ˜ ) 1 Γ ( ν ) D ( x b ( t x ) ν 1 ( D b ν f ) ( t ) d t , x b 0 ˜ d t ) 1 Γ ( ν ) x b ( t x ) ν 1 D ( ( D b ν f ) ( t ) , 0 ˜ ) d t 1 Γ ( ν ) ( b x ) ν ν sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .
So, we prove that
D ( f ( x ) , f ( b ) ) ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) , a x b .
Thus, we obtain
a c D ( f ( x ) , f ( b ) ) d x ( b a ) ν + 1 Γ ( ν + 2 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .
When [ ( i i ) ν ] -differentiable, we have
D ( 1 b a ( F R ) a b f ( x ) d x , f ( b ) ) 1 b a a b D ( f ( x ) , f ( b ) ) d x ( b a ) ν Γ ( ν + 2 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) .

This completes the proof. □

Now, we state the main result as given below.

Theorem 3.4 Let f , g , h C F [ a , b ] L F [ a , b ] and 0 < ν < 1 . Then
D ( 3 a b f ( x ) g ( x ) h ( x ) d x , a b [ f ( b ) g ( x ) h ( x ) + f ( x ) g ( b ) h ( x ) + f ( x ) g ( x ) h ( b ) ] d x ) I a + ν + 1 ( h ( b ) g ( b ) ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) + I a + ν + 1 ( f ( b ) h ( b ) ) sup t [ a , b ] D ( ( D b ν g ) ( t ) , 0 ˜ ) + I a + ν + 1 ( f ( b ) g ( b ) ) sup t [ a , b ] D ( ( D b ν h ) ( t ) , 0 ˜ ) .
Proof For f , g , h C F [ a , b ] L F [ a , b ] we have
D ( 3 a b f ( x ) g ( x ) h ( x ) d x , a b [ f ( b ) g ( x ) h ( x ) + f ( x ) g ( b ) h ( x ) + f ( x ) g ( x ) h ( b ) ] d x ) = a b ( D ( 3 f ( x ) g ( x ) h ( x ) , [ f ( b ) g ( x ) h ( x ) + f ( x ) g ( b ) h ( x ) + f ( x ) g ( x ) h ( b ) ] ) ) d x .
So we have the following:
a b ( D ( 3 f ( x ) g ( x ) h ( x ) , [ f ( b ) g ( x ) h ( x ) + f ( x ) g ( b ) h ( x ) + f ( x ) g ( x ) h ( b ) ] ) ) d x = a b [ D ( f ( x ) g ( x ) h ( x ) + f ( x ) g ( x ) h ( x ) + f ( x ) g ( x ) h ( x ) , [ f ( b ) g ( x ) h ( x ) + f ( x ) g ( b ) h ( x ) + f ( x ) g ( x ) h ( b ) ] ) ] d x = a b [ D ( f ( x ) g ( x ) h ( x ) , f ( b ) g ( x ) h ( x ) ) + D ( f ( x ) g ( x ) h ( x ) , f ( x ) g ( b ) h ( x ) ) + D ( f ( x ) g ( x ) h ( x ) , f ( x ) g ( x ) h ( b ) ) ] d x = a b [ g ( x ) h ( x ) D ( f ( x ) , f ( b ) ) + f ( x ) h ( x ) D ( g ( x ) , g ( b ) ) + f ( x ) g ( x ) D ( h ( x ) , h ( b ) ) ] d x = a b [ g ( x ) h ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) ) + f ( x ) h ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν g ) ( t ) , 0 ˜ ) ) + f ( x ) g ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν h ) ( t ) , 0 ˜ ) ) ] d x a b [ g ( x ) h ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) ) ] d x + a b [ f ( x ) h ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν g ) ( t ) , 0 ˜ ) ) ] d x + a b [ f ( x ) g ( x ) ( ( b x ) ν Γ ( ν + 1 ) sup t [ a , b ] D ( ( D b ν h ) ( t ) , 0 ˜ ) ) ] d x sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) ( 1 Γ ( ν + 1 ) a b [ g ( x ) h ( x ) ( b x ) ν ] d x ) + sup t [ a , b ] D ( ( D b ν g ) ( t ) , 0 ˜ ) ( 1 Γ ( ν + 1 ) a b [ f ( x ) h ( x ) ( b x ) ν ] d x ) + sup t [ a , b ] D ( ( D b ν h ) ( t ) , 0 ˜ ) ( 1 Γ ( ν + 1 ) a b [ f ( x ) g ( x ) ( b x ) ν ] d x ) = I a + ν + 1 ( h ( b ) g ( b ) ) sup t [ a , b ] D ( ( D b ν f ) ( t ) , 0 ˜ ) + I a + ν + 1 ( f ( b ) h ( b ) ) sup t [ a , b ] D ( ( D b ν g ) ( t ) , 0 ˜ ) + I a + ν + 1 ( f ( b ) g ( b ) ) sup t [ a , b ] D ( ( D b ν h ) ( t ) , 0 ˜ ) .
Finally, by using the following property:
I a + ν + 1 ( h ( b ) g ( b ) ) = 1 Γ ( ν + 1 ) a b [ g ( x ) h ( x ) ( b x ) ν ] d x ,

we complete the proof. □

4 Conclusion

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.

Notes

Declarations

Acknowledgements

The reviewers’ comments, which have improved the quality of this paper, are greatly appreciated.

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
(2)
Department of Mathematics, Yasooj Branch, Islamic Azad University, Yasooj, Iran
(3)
Faculty of Marine Science, Chabahar Maritime University, Chabahar, Iran
(4)
Department of Mathematics and Computer Sciences, Cankaya University, Ankara, 06530, Turkey
(5)
Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia
(6)
Institute of Space Sciences, Magurele, Bucharest, Romania
(7)
Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran

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