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Existence results for Riemann-Liouville fractional neutral evolution equations
Advances in Difference Equations volume 2014, Article number: 83 (2014)
In this paper, by using the fractional power of operators and the theory of measure of noncompactness, we discuss a class of fractional neutral evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the existence of mild solutions for fractional neutral evolution equations in the cases semigroup is compact or noncompact. We give an example to illustrate the applications of the abstract results.
A strong motivation for studying fractional differential equations comes from the fact that fractional order derivatives and integrals have extensive applications in viscoelasticity, analytical chemistry, electromagnetic, neuron modeling, and biological sciences, and the theory of fractional calculus has attracted great interest from the mathematical science research community. For more details of the theory and applications in this field, see the monographs of Samako et al. , Kilbas et al. , Miller and Ross , Podlubny  and the references therein.
Practical problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions. Initial conditions for Caputo fractional derivatives are expressed in terms of initial integer order derivatives. Heymans and Podlubny  demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions.
Recently, fractional evolution equations with the Caputo fractional derivative with difference conditions were studied by many authors (see, e.g., [6–15]), but much less is known about the fractional evolution equations with Riemann-Liouville fractional derivative; see [16–18].
In , Zhou, Zhang and Shen researched the existence of solutions for a fractional evolution equation with the Riemann-Liouville fractional derivative of the form
where is the Riemann-Liouville fractional derivative of order q, is the Riemann-Liouville integral of order , A is the infinitesimal generator of a -semigroup on a Banach space X. and are given functions satisfying some assumptions.
However, fractional neutral evolution equations with a Riemann-Liouville fractional derivative have not been considered in the literature. Motivated by the above work, the aim of this paper is to study the existence of solutions for fractional neutral evolution equations with a Riemann-Liouville fractional derivative of the form
where is the Riemann-Liouville fractional derivative of order q with the lower limit zero, is the Riemann-Liouville integral of order . X is a Banach space, is the infinitesimal generator of an analytic semigroup on a Banach space X. h and f are given functions satisfying some assumptions.
Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last few decades. The literature related to fractional neutral differential equations is very extensive; see for instance [12–14, 19]. The results we obtained in this paper are a generalization and continuation of the recent results on this issue.
The rest of this paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3, by using the fractional power of operators and the technique of using a measure of noncompactness, we give existence results for problem (1) under both compactness and noncompactness conditions on the semigroup. We present an example to demonstrate our main results in Section 4. Finally, the manuscript ends with our conclusions.
In this section, we introduce the notations, definitions, and preliminary facts that will be used in the remainder of this paper.
In this paper, we assume that X is a Banach space. Let denote the Banach space of all X-value continuous functions from into X with the norm . For measurable functions , we define the norm , . () is the Banach space of all Lebesgue measurable functions from J into ℝ with . Let denote the Banach space of functions which are Bochner integrable normed by . Let , to define the mild solution of (1), we also consider the Banach space with the norm . It is easy to see is a Banach space.
Definition 2.1 The fractional integral of order q with the lower limit zero for a function f is defined as
provided the right side is point-wise defined on , where is the gamma function.
Definition 2.2 The Riemann-Liouville derivative of order q with the lower limit zero for a function can be written as
Definition 2.3 The Caputo derivative of order q for a function can be written as
If , then
The Caputo derivative of a constant is equal to zero.
If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.
Throughout this paper, let −A be the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators on X; this means that there exists , such that . Let , where is the resolvent set of A. Then for , it is possible to define the fractional power as a closed linear operator on its domain . Moreover, the subspace is dense in X and the expression
defines a norm on . We denote by the Banach space normed with .
For analytic semigroup , the following property will be used:
for any , there exists a positive constant such that
Definition 2.5 Let X be a Banach space, denote the collection of all nonempty subsets of X, and a partially ordered set. A map is called a measure of noncompactness on X, MNC for short, if
for every , where is the closure of convex hull of Ω.
Definition 2.6 A measure of noncompactness β is called
monotone if , implies ;
nonsingular if for every , ;
invariant with respect to the union with compact sets if for every relatively compact set and ;
semi-additive if for every ;
regular if the condition is equivalent to the relative compactness of Ω.
One of the most important examples of the measure of noncompactness is the Kuratowski measure of noncompactness α defined on each bounded subset B of X by
, where ;
for any .
For any , we define
Proposition 2.7 If is bounded and equicontinuous, then is also bounded and equicontinuous.
Proposition 2.8 
If is bounded and equicontinuous, then is continuous on J and
Proposition 2.9 
X is a Banach space, if is bounded, then for each , there is a sequence , such that
Proposition 2.10 
Let be a sequence of Bochner integrable functions from J into X with for almost all and every , where , then the function belongs to and satisfies
Lemma 2.11 Let B be a closed and convex subset of a Banach space X. Suppose is a continuous operator and is bounded. For each bounded subset , set
If there exist a constant and a positive integer such that, for any bounded subset ,
then there exists a nonempty, compact, and convex subset such that and .
According to Lemma 2.4 in , we can obtain the results immediately.
The map is said to be an α-contraction if there exists a positive constant such that for any bounded closed subset .
Lemma 2.12 (Darbo-Sadovskii’s fixed point theorem) 
If B is a bounded closed convex subset of a Banach space X, the continuous map is an α-contraction, then the map F has at least one fixed point in B.
Definition 2.13 By a mild solution of system (1), we mean the function which satisfies
and is a probability density function which is defined by
Lemma 2.14 
The operator has the following properties.
For any fixed , is linear and bounded operators, i.e., for any ,
() is strongly continuous, which means that, for and , we have
for every , is a compact operator if is also compact.
Lemma 2.15 
For any , and , we have
3 Existence of mild solutions
Before stating and proving the main results, we introduce the following assumptions.
(H1) is a compact operator for every .
(H2) For almost all , the function is continuous and for each , the function is strongly measurable.
(H3) There exist a function , , and a constant such that
and the operator is continuous, where
(H4) is a continuous function and there exist a constant and such that and for any , , the function is strongly measurable and satisfies
and the inequality
For brevity, let
Define an operator F on by
For any , let
Assume that , and let
Lemma 3.1 If the assumptions (H1)-(H3) are satisfied and , then is relatively compact set in .
Proof Using (H1)-(H3) and Lemma 2.14, one can easily prove that for any . Then is well defined on . Let
For the sake of convenience, we divide the proof into several steps.
Step 1: For any , .
For any and , taking into account the imposed assumptions, applying Lemma 2.14 and the Hölder inequality, we obtain
Hence, for any , .
Step 2: is equicontinuous.
For any , let . Taking (H3) into account, we get
For , selecting sufficiently small. Taking (H3) and (H4) into account, we have
According to Lemma 2.14(ii), it is easy to see that as . Applying the absolute continuity of the Lebesgue integral, we see that and tend to zero independently of as , . Noting that
and that exists, by the Lebesgue dominated convergence theorem, we see that tends to zero independently of as . Since (H1) and Lemma 2.14 imply the continuity of () in t in the uniform operator topology, it is easy to see that tends to zero independently of as . Thus, tends to zero independently of as , which means that Ω is equicontinuous.
Step 3: For any , is relatively compact in X.
This is trivial for , since . So it is only necessary to consider . Let be fixed. For , , define
Then from the compactness of (), we find that the set is relatively compact in X for and .
Moreover, we have
Therefore, there are relatively compact sets arbitrarily close to the set , . Hence the set , is also relatively compact in X.
As a consequence of Step 1-Step 3, with the Arzola-Ascoli theorem, we can conclude that is a relatively compact set. Keeping in mind the relationship of Ω and , one can easily prove that is a relatively compact set in . The proof is complete. □
3.1 The case that is compact
Theorem 3.2 Assume that hypotheses (H1)-(H4) hold and , then system (1) has at least one mild solution.
Proof Using (H1)-(H4), Lemma 2.14 and Lemma 2.15, one can easily prove that for any . Then F is well defined on . We will show that F satisfies all conditions of Lemma 2.12. The proof will be given in several steps.
For any and , taking into account the imposed assumptions, applying Lemma 2.14 and the Hölder inequality, we obtain
Thus, F maps into .
Next, we will show that F is continuous in .
Let with in , we have
This means that F is continuous in .
According to Lemma 3.1, is relatively compact in , then .
For any , we have
which implies that
Noting that , we find that the operator F is an α-contraction in . It follows from Lemma 2.12 that F has at least one fixed point in . Then problem (1) has at least one mild solution in . The proof is complete. □
3.2 The case that is not compact
For any , set , define as follows:
If is noncompact, we will need the following assumptions.
(H1)′ () is continuous in the uniform operator topology for .
(H5) There exists a constant such that for any bounded ,
Theorem 3.3 If assumptions (H1)′, (H2)-(H5) are satisfied and , then problem (1) has at least one mild solution.
For any , , it is easy to see that . Let
According to (H1)′, Step 1 and Step 2 of Lemma 3.1, we find that and is equicontinuous. Let then is a bounded, closed, and convex set. According to Proposition 2.7, B is also equicontinuous. Since
we have . Using a similar method as in the proof of Theorem 3.2, we can easily see that is continuous. For any bounded subset , set
We know from Propositions 2.8-2.10, for any , that there is a sequence , such that
Since is arbitrary, we have
For any , there is a sequence , such that
Since is arbitrary, we have
It can be shown by mathematical induction that
By using the well-known Stirling formula, we get
Then there exists a such that
It can be shown that . Since B is bounded and equicontinuous, is also bounded and equicontinuous. It follows from Proposition 2.8 that
Then, by Lemma 2.11, there exists a such that
For any , we have , which means that . Since is relatively compact, we can easily prove that is relatively compact, that is
On the other hand, for any , we have
which implies that
Noting that , thus the operator F is an α-contraction in Q. It follows from Lemma 2.12 that F has at least one fixed point in Q. Then problem (1) has at least one mild solution in Q. The proof is complete. □
4 An example
Let . As an application of our results, consider the following fractional partial differential equations:
where is the Riemann-Liouville fractional derivative of order , , is a Riemann-Liouville integral of order . Define the operator A by , with the domain
Then −A generates a strongly continuous semigroup which is compact, analytic. Moreover, A can be written as
where , , is an orthonormal basis of X. We have
For each ,
in particular, .
The operator is given by
on the space
Choose , define
then system (5) can be written in the abstract form given by (1). Assume that (H2)-(H4) are satisfied and ; then by Theorem 3.2, system (5) has at least one mild solution.
In this manuscript, by using the fractional power of operators and the theory of a measure of noncompactness, the existence of fractional neutral evolution equations with a Riemann-Liouville fractional derivative were investigated. We give an example to illustrate the applications of the abstract results.
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The authors would like to express their gratitude to the editor and referees for their careful reading of the manuscript and a number of excellent criticisms and suggestions. The work is financially supported by the NNSF of China Grant Nos. 11271087, 61263006 and Innovation Project of Guangxi University for Nationalities (gxun-chx2012101, gxun-chx2013082).
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Liu, YL., Lv, JY. Existence results for Riemann-Liouville fractional neutral evolution equations. Adv Differ Equ 2014, 83 (2014). https://doi.org/10.1186/1687-1847-2014-83
- fractional neutral evolution equations
- Riemann-Liouville fractional derivative
- mild solutions
- analytic semigroup
- measure of noncompactness