Multi-term fractional differential equations in a nonreflexive Banach space
© Agarwal et al.; licensee Springer. 2013
Received: 7 August 2013
Accepted: 1 October 2013
Published: 8 November 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.
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].
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 .
- (b)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.
The function is called a pseudo-derivative of and it will be denoted by or by .
is scalarly integrable; that is, for every , the real function is Lebesgue integrable on T;
- (b)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 .
- (b)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
Proposition 2.3 
If is Pettis integrable on T, then the indefinite Pettis integral (4) is AC on T and is a pseudo-derivative of .
- (b)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 .
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].
is wAC on T;
is a pseudo-derivative of ;
If E is a weakly sequentially complete space, then wAC is replaced by AC.
is called the fractional weak derivative of on T.
Remark 2.3 
If is a pseudo-differentiable function with a pseudo-derivative , then (a) on T; (b) on T.
Remark 2.4 
3 Fractional differential equations
where the integral is in the sense of Pettis.
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. □
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 ;
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 
the function is continuous on T,
where denotes the weak measure of noncompactness in and , .
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 ;
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 ).
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 .
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 
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