On the Theory of Fractional Calculus in the Pettis-Function Spaces

Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem.


Introduction and Preliminaries
The issue of fractional calculus for the functions that take values in Banach space where the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives has been studied for the first time by the authors of [1,2].Following the appearance of [1], there has been a significant interest in the study of this topic (see, e.g., [3][4][5][6]; see also [7][8][9]).This paper is devoted to presenting general results and examples for the existence of the fractional integral (and corresponding fractional differential) operators in arbitrary Banach space where it is endowed with its weak topology.In our investigations, we show that the well-known properties of the fractional calculus for functions taking values in finite dimensional spaces also hold in infinite dimensional spaces.Our results extend all previous contributions of the same type in the Bochner integrability setting and in the Pettis integrability one.
For the readers convenience, here we present some notations and the main properties for the Pettis integrals.For further background, unexplained terminology and details pertaining to this paper can be found in Diestel et al. [10,11] and Pettis [12].
Throughout this paper, we consider the measure space (, Ω, ), where  = [0, 1], 0 ≤  <  < ∞ denote a fixed interval of the real line, Ω denotes the Lebesgue -algebra L(), and  stands for the Lebesgue measure. denotes a real Banach space with a norm ‖ ⋅ ‖ and  * is its dual.By   we denotes the space  when endowed with the weak topology generated by the continuous linear functionals on .We will let [,   ] denote the Banach space of weakly continuous functions  → , with the topology of weak uniform convergence.And [, ] denotes the space of valued Pettis integrable functions in the interval  (see [10,12] for the definition).Recall that (see, e.g., [10,[13][14][15][16][17][18][19]) the weakly measurable function  :  →  is said to be Dunford (or Gelfand) integrable on  if and only if  is Lebesgue integrable on  for each  ∈  * .Definition 1.Let  ∈ [1, ∞].Define H  () to be the class of all weakly measurable functions  :  →  having  ∈   () for every  ∈  * .
In the following proposition, we summarize some important facts which are the main tool in carrying out our investigations (see [10,12,16,17]).
A fundamental property of Pettis integral is contained in the following.

Fractional Integrals of Vector-Valued Functions
In this section, we define and study the Riemann-Liouville fractional integral operators and the corresponding fractional derivatives in Banach spaces.Devoted by the definition of the Riemann-Liouville fractional integral of real-valued function, we introduce the following.
Definition 5. Let  :  → .The Riemann-Liouville fractional Pettis integral (shortly RFPI) of  of order  > 0 is defined by In the preceding definition, "∫ " stands for the Pettis integral.
When  = R, it is well known (see, e.g., [20,21]) that the operator This seems to be a good place to put the following.
As cited in ( [13], Corollary 4),  is Pettis integrable functions on [0, 1] whose indefinite integral is nowhere weakly differentiable on [0, 1].Here we will show that  has RFPI of all order  ≥ 3/4 and Arguing similarly as in ( [13], page 368), we have in view of Also, for any  ∈ [0, 1] and fixed  ∈ N, we have for some Therefore, for any measurable  ⊂ [0, 1], we arrive at Obviously, the latter series converges whenever  ≥ 3/4,  = 0.9 which allows us to interchange the integral and summation below to see that which approaches zero as ,  → ∞ as needed for (9).
Remark 7. Observe Example 6.We remark the following: (1) There is a reflexive Banach space for which the indefinite Pettis integral of the function  defined by ( 7) is nowhere weakly differentiable on [0, 1] (see [13], the remark below Corollary 4).Meanwhile,  has a RFPI of all order  ≥ 3/4.(2) The function I  (⋅) is weakly continuous on [0, 1]: this follows easily from the definition of the Pettis integral.In fact, we have in view of (9) that holds for every  ∈  * .Since then the infinite series of continuous functions in the left hand side of (15) converges uniformly in [0, 1].Hence, the function (I  (⋅)) is continuous on [0, 1] (this yields the weak continuity of I  (⋅) on [0, 1]).
In the following lemma, we gather together some simple particular sufficient conditions that ensure the existence of the Riemann-Liouville fractional integral of the functions from H  ().Lemma 8. Let  :  →  be weakly measurable function.The RFPI of the function  of order  > 0 makes sense a.e. on  if at least one of the following cases holds: In all cases, (I  ) = I   holds for every  ∈  * .
In the assertions ((a) and (b)), we find sufficient conditions needed for the existence of I   in the situation in which no restriction is placed on .In the third assertion, the properties of  allow us to characterize a function  ∈ H  () for which I   exists.
Similarly, when  is reflexive, the result follows from part (3) of Proposition 2.
However, in all cases, the function   → ( − ) −1 () is Pettis integrable on [0, ] for almost every  ∈ .That is, for almost every  ∈ , there exists an element in  denoted by holds for every  ∈  * .This completes the proof.
Remark 9.If  ∈ H  0 () such that I  () does not exist for some  ∈ , then it does not exist even when we "enlarge" the space  into .To see this, let  :  →  such that () ⊂ .If I  () exists for some  ∈ , then ( − ⋅) −1 (⋅) ∈ ([0, ], ).Since  assumes only values in , it follows by the mean value theorem for Pettis integral (Theorem 4) that the RFPI of  should lie in .
Before we come to a deep study of the mathematical properties of the RFPI operator, let us take a look at the following miscellaneous examples.
This function is weakly measurable, Pettis integrable on [0, 1], and  is a function of bounded variation (see, e.g., [18]).That is,  ∈ H ∞ 0 ( ∞ ).Hence, in view of Lemma 8 with  = ∞, the RFPI of  exists on [0, 1].Further, calculations (cf.[4]) show that Example 11.Let  > 0. Define the function  from the interval [0, 1] into the Hilbert space ℓ 2 as We note that Thus, the function  is well defined.We claim that  is Dunford integrable on [0, 1].Once our claim is established, Lemma 8 guarantees the existence of I   on [0, 1].It remains to prove this claim and to calculate I  .
In what follows, we will show that the RFPI of  does not exist on a subinterval of positive measure on [0, 1].
The following theorem provides a useful characterization of the space H  (), for which the statements reveal how much the fractional integral I  is better than the function  ∈ H  ().Indeed, based on Lemma 8 using an inequality of Young, we can easily prove the following.Lemma 14.For any  > 0, the following holds.In particular, if 0 <  < 1, it can be easily seen that ∫  (())   < ∞ for every  ∈ [1, 1/(1 − )).That is,  ∈  1/(1−)− [],  > 0. By Young's inequality, it follows that (I  ) =  *  ∈  1/(1−)− [] for every  ∈  * however small  > 0 is.Now, the assertion (a) follows because of the reflexivity of .
To show that With no loss of generality, we may assume that ,  ∈   0 , for some  0 ∈ N. Since the nonzero terms of the sequence in (50) are nonincreasing, then, in view of we have Thus, the RFPI of  is norm continuous on [0, 1].Precisely, since 0 <  − 1/2 <  − 1/ < 1, for any  ≤ 2, then This is precisely what one would expect from Lemma 14 (part (c)).
Analogously, an explicit calculation reveals that We now consider additional mapping properties of the operator I  .Precisely, we will show that the RFPI enjoys the following commutative property which is folklore in case  = R.However, the proof is completely similar to that of [8], Lemma 3.5.
When  is reflexive, the result follows as a direct application of Lemma 14.

Fractional Derivatives of Vector-Valued Functions
After the notation of the fractional integrals of vectorvalued functions, the fractional derivatives become a natural requirement.Before we come to the definitions and a detailed study of the mathematical properties of fractional differential operators, we recall the following.
Definition 18.Consider the vector-valued function  :  → : (1) Let  be differentiable on  for every  ∈  * .The function  is said to be weakly differentiable on  if there exists  :  →  such that for every  ∈  * we have  ()  =  () , for every  ∈ .
The function  is called the weak derivative of the function .
(2) Let  be differentiable a.e. on  for every  ∈  * (the null set may vary with  ∈  * ).The function  is said to be pseudo differentiable on  if there exists a function  :  →  such that for every  ∈  * there exists a null set () ⊂  such that In this case, the function  is called the pseudo derivative of .
If  is pseudo differentiable on  and the null set invariant for every  ∈  * , then  is a.e.weakly differentiable on .
Clearly, if  is a.e.weakly differentiable on , then () is a.e.differentiable on .The converse holds in a weakly sequentially complete space (see [25], Theorem 7.3.3).
For more details of the derivatives of vector-valued functions we refer to [10,12,26].
The following results play a major role in our analysis Proposition 19 (see [27], Theorem 5.1).The function  :  →  is an indefinite Pettis integrable, if and only if  is weakly absolutely continuous on  and have a pseudo derivative on .
In this case,  is an indefinite Pettis integral of any of its pseudo derivatives.
Now we are in the position to define the fractional-type derivatives of vector-valued functions.Definition 20.Let  :  → .For the positive integer  such that  ∈ ( − 1, ),  ∈ N 0 fl {0, 1, 2, . ..} we define the Caputo fractional-pseudo (weak) derivative "shortly CFPD (CFWD)" of  of order  by where the sign "" denotes the pseudo (or weak) differential operator.We use the notation    /  and    /  to characterize the Caputo fractional-pseudo derivatives and Caputo fractional weak derivatives, respectively.
It is well known that, although the weak derivative of a weakly differentiable function is uniquely determined, the pseudo derivative of the pseudo differentiable function is not unique.Also, although any two pseudo derivatives ,  of a function  :  →  need not be a.e.equal (see [12,Example 9.1] and [26, p. 2]), the functions ,  are weakly equivalent on  (that is,  =  holds a.e. for every  ∈  * ).The next lemma provides a useful characterization property of the CFPD.Really, it can be easily seen that the CFPD of a Caputo fractional-pseudo differentiable function does not depend on the choice of a pseudo derivative of the function.
Lemma 21.Let  :  →  be pseudo differentiable function where pseudo derivatives lie in H  0 ().If (   /  ) exists on , then the CFPD of  depends on the choice of the pseudo derivatives of .
Remark 24.As shown in Example 23, there is an infinite dimension Banach space  and weakly absolutely continuous function  :  →  which is nowhere weakly differentiable (hence the CFWD of  does not exist).Also, even when  is separable, and  is Lipschitz function, the pseudo derivatives (hence the CFPD) of  need not to exist [28].
However, Definition 20 of the CFPD (CFWD) has the disadvantage that it completely loses its meaning if the function  fails to be pseudo (weakly) differentiable.Precisely, the CFPD (in particular the CFWD) of a function  loses its meaning if  is not weakly absolutely continuous.
The next lemma gives sufficient conditions that ensure the existence of the Caputo fractional derivatives of a function  ∈ H  ().
Lemma 25.Let  ∈ (0, 1).For the function  :  → , the following hold: where  is defined as in Definition 20.We use the notation D   (D   ) to characterize the Riemann-Liouville fractionalpseudo (weak) derivatives.
Clearly, in infinite dimension Banach spaces, the weak absolute continuity of I − , is necessarily (but not sufficient) condition for the existence of RFPD (in particular RFWD) of .Lemma 27.Let 0 <  < 1.For any  ∈ H  0 () with  > max{1/, 1/(1 − )}, we have If  is reflexive, this is also true for every  ≥ 1.
The claim now follows immediately, since the indefinite integral of weakly contentious function is weakly absolutely continuous and it is weakly differentiable with respect to the right endpoint of the integration interval and its weak derivative equals the integrand at that point.
The following lemma is folklore in case  = R, but to see that it also holds in the vector-valued case, we provide a proof.
In particular, if  passes a weak derivative in [,   ], then If  is reflexive, this is also true for every  ≥ 1.
Proof.We observe that, under the assumption imposed on    together with Proposition 3, the weakly absolutely continuity of  is equivalent to Hence, owing to Lemma 27, it follows that which is what we wished to show.The proof of ( 69) is very similar to that in (68); therefore, we omit the details.
Proof.We omit the proof since it is almost identical to that in the proof in ([6], Theorem 3.3) with (small) necessary changes.
In the following example we assume that  ∈ [,   ] solves (77) and we will show that, not only do we have that  no longer necessarily solves (74) (when the Caputo fractional weak derivative is taken in the sense of Definition 20), but even worse, it could happen that the problem (74) is "meaningless" on .
Proposition 19)the indefinite integral of Pettis integrable function is weakly absolutely continuous and it is pseudo differentiable with respect to the right endpoint of the integration interval and its pseudo derivative equals the integrand at that point.Remark 28.When we replace D  by D   , then Lemma 27 is no longer necessarily true for arbitrary  ∈ H 1 0 () even when  is reflexive: evidently, in [13, remark below Corollary 4] the existence of a reflexive Banach space  and a strong measurable Pettis integrable function  :  →  was proved such that  has nowhere weakly differentiable integral.In this case,   I 1  lost its meaning.This gives a reason to believe that (65) (hence (64)) with D    could not happen.
However, we have the following result.Lemma 29.Let 0 <  < 1.For any  ∈ [,   ], we have Remark 31.As we remark above, the definition of the CFPD of a function  loses completely its meaning if  is not weakly absolutely continuous.For this reason, we are able to use Lemma 30 to define the Caputo fractional derivative in general; that is, we put