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Multi-term fractional differential equations in a nonreflexive Banach space
Advances in Difference Equations volume 2013, Article number: 302 (2013)
In this paper we establish an existence result for the multi-term fractional differential equation
where and are fractional pseudo-derivatives of a weakly absolutely continuous and pseudo-differentiable function of order and , , respectively, the function is weakly-weakly sequentially continuous for every and is Pettis integrable for every weakly absolutely continuous function , T is a bounded interval of real numbers and E is a nonreflexive Banach space, and are real numbers such that .
The mathematical field that deals with derivatives of any real order is called fractional calculus. Fractional calculus has been successfully applied in various applied areas like computational biology, computational fluid dynamics and economics etc. .
In certain situations, we need to solve fractional differential equations containing more than one differential operator, and this type of fractional differential equation is called a multi-term fractional differential equation. Multi-term fractional differential equations have numerous applications in physical sciences and other branches of science . The existence of solutions of multi-term fractional differential equations was studied by many authors [3–8]. The main tool used in [3–5] is the Krasnoselskii’s fixed point theorem on a cone, while the main tool used in  is the technique associated with the measure of noncompactness and fixed point theorem. In , the author established the existence of a monotonic solution for a multi-term fractional differential equation in Banach spaces, using the Riemann-Liouville fractional derivative and in that paper no compactness condition is assumed on the nonlinearity of the function f.
When and , the existence of weak solutions to multi-term fractional differential equation (1) was discussed in [9–13]. In , the author studied the existence of a weak solution of Cauchy problem (1) in reflexive Banach spaces equipped with the weak topology, and the author assumed a weak-weak continuity assumption on f. In , the author established the existence of a global monotonic solution for Cauchy problem (1), and the author assumed f is Carathéodory with linear growth.
In this present article, we prove the existence of a solution of Cauchy problem (1) in nonreflexive Banach spaces equipped with the weak topology. In comparison to other results in the literature, we use more general assumptions so that the function f is assumed to be weakly-weakly sequentially continuous and is Pettis integrable for each weakly absolutely continuous function .
For convenience here we present some notations and the main properties for Pettis integrable, weakly-weakly continuous functions, and we state some properties of the measure of noncompactness. Also, we present definitions and preliminary facts of fractional calculus in abstract spaces. Let E be a Banach space with the norm , and let be the topological dual of E. If , then its value on an element will be denoted by . The space E endowed with the weak topology will be denoted by . Consider an interval of ℝ endowed with the Lebesgue σ-algebra and the Lebesgue measure λ. We will denote by the space of all measurable and Lebesgue integrable real functions defined on T, and by the space of all measurable and essentially bounded real functions defined on T.
Definition 2.1 (a) A function is said to be strongly measurable on T if there exists a sequence of simple functions such that for a.e. .
A function is said to be weakly measurable (or scalarly measurable) on T if for every , the real-valued function is Lebesgue measurable on T. It is well known that a weakly measurable and almost separable valued function is strongly measurable [, Theorem 1.1].
Definition 2.2 (a) A function is said to be absolutely continuous on T (AC, for short) if for every , there exists such that
for every finite disjoint family of subintervals of T such that . If the last inequality is replaced by , then we say that is a strongly absolutely continuous (sAC) function.
A function is said to be weakly absolutely continuous (wAC) on T if for every , the real-valued function is AC on T.
Remark 2.1 Each sAC function is an AC function, and each AC function is a wAC function. If E is a weakly sequentially complete space, then every wAC function is an AC function .
Definition 2.3 (a) A function is said to be strongly differentiable at if there exists an element such that
The element will also be denoted by and it is called the strong derivative of at .
A function is said to be weakly differentiable at if there exists an element such that
for every . The element will be also denoted by and it is called the weak derivative of at .
Proposition 2.1 [, Theorem 7.3.3]
If E is a weakly sequentially complete space and is a function such that for every , the real function is differentiable T, then is weakly differentiable on T.
Proposition 2.2 [, Theorem 1.2]
If is an a.e. weakly differentiable on T, then its weak derivative is strongly measurable on T.
Definition 2.4 A function is said to be pseudo-differentiable on T to a function if for every , there exists a null set such that the real function is differentiable on and
The function is called a pseudo-derivative of and it will be denoted by or by .
Definition 2.5 A weakly measurable function is said to be Pettis integrable on T if
is scalarly integrable; that is, for every , the real function is Lebesgue integrable on T;
for every set , there exists an element such that(3)
for every . The element is called the Pettis integral on A and it will be denoted by .
Remark 2.2 (a) It is known that if is Bochner integrable on T, then the function , given by
is AC and a.e. differentiable on T, and for a.e. . Also, if a function is AC and a.e. strongly differentiable on T, then is Bochner integrable on T and
In the case of the Pettis integral, in  it was shown that if is an AC and a.e. weakly differentiable on T, then is Pettis integrable on T and
In 1994 Kadets  proved that there exists a strongly measurable and Pettis integrable function such that the indefinite Pettis integral
is not weakly differentiable on a set of positive Lebesgue measures (see also [18, 19]).
Proposition 2.3 
Let be a weakly measurable function.
If is Pettis integrable on T, then the indefinite Pettis integral (4) is AC on T and is a pseudo-derivative of .
If is an AC function on T and it has a pseudo-derivative on T, then is Pettis integrable on T and
It is known that the Pettis integrals of two strongly measurable functions and coincide over every Lebesgue measurable set in T if and only if a.e. on T [, Theorem 5.2]. Since a pseudo-derivative of the pseudo-differentiable function is not unique and two pseudo-derivatives of need not be a.e. equal, then the concept of weak equivalence plays an important role in the following.
Definition 2.6 Two weak measurable functions and are said to be weakly equivalent on T if for every , we have that for a.e. .
Proposition 2.4 A weakly measurable function is Pettis integrable on T and for every if and only if the function is Pettis integrable on T for every .
Let us denote by the space of all weakly measurable and Pettis integrable functions with the property that for every . Since for each the real-valued function is Lebesgue integrable on for every then, by Proposition 2.4, the fractional Pettis integral
exists for every function as a function from T into E. Moreover, we have that
for every , and the real function is continuous (in fact, bounded and uniformly continuous on T if ) on T for every [, Proposition 1.3.2].
Proposition 2.5 If is a function such that for every wAC function , then the function given by
has the following properties:
is wAC on T;
is a pseudo-derivative of ;
If E is a weakly sequentially complete space, then wAC is replaced by AC.
In the following, consider . If is a pseudo-differentiable function with a pseudo-derivative on T, then the following fractional Pettis integral
exists on T. The fractional Pettis integral is called a fractional pseudo-derivative of on T and it will be denoted by ; that is,
If is an a.e. weakly differentiable function with the weak derivative on T, then
is called the fractional weak derivative of on T.
The following results will be useful (see  and ).
Remark 2.3 
If is a pseudo-differentiable function with a pseudo-derivative , then (a) on T; (b) on T.
Remark 2.4 
The fractional Pettis integral is a linear operator from into . Moreover, if , then for , , we have
3 Fractional differential equations
In this section we establish an existence result for the fractional differential equation
where and are fractional pseudo-derivatives of the function of order and , , respectively and is a given function, and are real numbers such that . Along with Cauchy problem (9), consider the integral equation
where the integral is in the sense of Pettis.
Definition 3.1 A wAC function (or an AC function, if E is a weakly sequentially complete space) is said to be a solution of (9) if
has pseudo-derivative of order , ,
the pseudo-derivative of of order , , belongs to ,
for all ,
Lemma 3.1 Let be a function such that for every wAC function . Then a wAC function is a solution of (9) if and only if it satisfies the integral equation (10).
Proof Indeed, if a wAC function is a solution of (9), then from Remark 2.3(a) and Remark 2.4 it follows that on T; that is, satisfies the integral equation (10). Conversely, suppose that a wAC function satisfies the integral equation (10). Since , then from Proposition 2.5 it follows that the function has a pseudo-derivative belonging to .Thus, using Remark 2.3(b), (10) and (6), we obtain that on T. □
Let us denote by the set of all weakly compact subsets of E. The weak measure of noncompactness is the set function defined by
where is the closed unit ball in E. The properties of weak noncompactness measure are analogous to the properties of measure of noncompactness. If ,
() implies that ;
() , where denotes the weak closure of A;
() if and only if is weakly compact;
() , for all ;
() , for all ;
Let and let denote the space of all strong continuous functions , endowed with the supremum norm . Also, denotes the space of all weakly continuous functions from T into endowed with the topology of weak uniform convergence. It is known that (see [23, 24])
where is the space of all bounded regular vector measures from into which are of bounded variation. Here, denotes the σ-algebra of Borel measurable subsets of T.
Lemma 3.2 
Let be bounded and equicontinuous. Then
the function is continuous on T,
where denotes the weak measure of noncompactness in and , .
By a Kamke function we mean a function such that is continuous, nondecreasing with and is the only solution of
We recall that a function is said to be sequentially continuous from into (or weakly-weakly sequentially continuous) if for every weakly convergent sequence , the sequence is weakly convergent in E.
Theorem 3.1 Let . Let be a function such that:
(h1) is weakly-weakly sequentially continuous for every ;
(h2) is Pettis integrable for every wAC function ;
(h3) for all , where ;
(h4) for all , we have
where is a Kamke function. Then (9) admits a solution on an interval with
Proof Let denote the set of all wAC functions , and we consider the nonlinear operator defined by
for all . Since the function for every , then by Proposition 2.5 it follows that the real function is wAC on T for every . Now, since
we obtain that
Thus Q maps into . Next, we show that Q is weakly-weakly sequentially continuous. For this let be a sequence in such that as (that is, converges weakly to in ). Since , it follows that
for all . Let and . If we take , where is the Dirac measure concentrated in t, then and from (12) it follows that
Therefore, for each , converges weakly to in E. Further, by (h1) it follows that converges weakly to in E for each . Hence, using Lemma 2.2 from , it follows that converges weakly to in E for all , and also we have that converges to u weakly. Then, by the Lebesgue dominated convergence theorem for Pettis integral (see ), we obtain
for all . Therefore, Q is weakly-weakly continuous. Next, for any , we define the sequence as follows:
Obviously, for all . Further, for all , if , then
If , then
Thus we obtain that for all ,
Let and . If and , then
If and , then
If and , then
It follows from the above three cases that W is equicontinuous, and by (13) we obtain that V is also equicontinuous. By virtue of Lemma 3.2 and (13), we have that
Since for all we have that , it follows that
On the other hand, for all , we have , and so
From (14) and (15), we infer that
Further, fix , and choose such that . Then
Hence we conclude that
Using the property () of noncompactness measure, we infer
Since by Lemma 3.2 the function is continuous on , it follows that is continuous on . Hence, there exists such that
If and with , then it follows that
for all with and . Consider the following partition of the interval into n parts such that (). By Lemma 3.2, for each i, there exists such that , . Then we have (see , Theorem 2.2)
Using (18) we have that
This implies that
By using (17) we claim that
If we let
then for , which implies that . From relations (19) and (20), we obtain
then by virtue of (17) and (21), we have
As the last inequality is true for every , we infer
and thus by using (16) it follows that
Since is a Kamke function, then for . Using Lemma 3.2, we infer
Thus V is relatively compact in . Therefore, taking a subsequence if necessary, we can assume that converges weakly in to a function . Since Q is weakly-weakly sequentially continuous, then converges weakly to in . Recalling that the norm is weakly lower semicontinuous , we obtain that
Then from (13) it follows that , and so
Using Lemma 3.1, we conclude that is a solution of (9). □
Remark 3.1 If and , then we obtain Theorem 5.1 from .
Let E be a weakly sequentially complete space. It is known that if is a continuous function from into , then the function is Pettis integrable for every AC function (see [, Lemma 15]). Therefore, in the case of weakly sequentially complete spaces, we obtain the following result (see also ).
Corollary 3.1 Let E be a weakly sequentially complete space. If is a continuous function from into and conditions (h3)-(h4) in Theorem 3.1 hold, then (9) admits a solution on an interval with
Remark 3.2 If and , then we obtain some known results. In this case, Corollary 3.1 is a generalization of a result from  and . Also, for any Banach space, the following result is a generalization of Theorem 2.1 in  (see also [13, 32–34]) for and .
Corollary 3.2 If is a continuous function from into such that , then (9) admits a solution on with
If E is a reflexive Banach space, it is not necessary to assume any compactness conditions since in this case a subset of E is weakly compact if and only if it is weakly closed and norm bounded. Thus, arguing similarly as in the proof of Theorem 3.1, we obtain the following result.
Corollary 3.3 
Let E be a reflexive Banach space. If is a continuous function from into such that for all , then (9) admits a solution on with
Remark 3.3 If and , then we obtain Theorem 8 from , Theorem 3.1 from , and using the conditions of Corollary 3.3, we obtain some known results from .
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All authors declare that they have no competing interests.
All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Agarwal, R.P., Lupulescu, V., O’Regan, D. et al. Multi-term fractional differential equations in a nonreflexive Banach space. Adv Differ Equ 2013, 302 (2013). https://doi.org/10.1186/1687-1847-2013-302
- weak measure of noncompactness
- nonreflexive Banach spaces
- Pettis integral
- multi-term fractional differential equation
- fractional pseudo-derivative