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Axioms 2017, 6(4), 30; doi:10.3390/axioms6040030
Mild Solutions to the Cauchy Problem for Some Fractional Differential Equations with Delay
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
Author to whom correspondence should be addressed.
Received: 5 October 2017 / Accepted: 14 November 2017 / Published: 20 November 2017
In this paper, we present new existence theorems of mild solutions to Cauchy problem for some fractional differential equations with delay. Our main tools to obtain our results are the theory of analytic semigroups and compact semigroups, the Kuratowski measure of non-compactness, and fixed point theorems, with the help of some estimations. Examples are also given to illustrate the applicability of our results.
Keywords:fractional differential equations; analytic semigroup; compact semigroup; fixed point; mild solution
In this paper, we consider the following Cauchy problem for fractional differential equations with delay in a Banach space X which could be an infinite dimensional space:where , is the Liouville-Caputo fractional derivative of order A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operator on X, f is a given function, is defined byand
As shown in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein, differential equations with delay or differential equations of fractional order have appeared in many branches of science and technology. They have received a lot of attention in all these years.
The paper is organized as follows. In Section 2, we first recall and give some basic facts or results about semigroup theory and related tools which will be used in our investigation. Then, we study the existence of mild solutions to the Cauchy Problem (1) and prove our main results. In Section 3, we give some examples to to illustrate our abstract results.
2. Results and Proofs
Beta function:Gamma function:It is well known thatThroughout this paper, is a Banach space, denotes the space of the continuous functions from to X with the normSetwith the norm
(cf., e.g., ) The Liouville-Caputo derivative of order q for a function can be written as
Since is the infinitesimal generator of an analytic semigroup of uniformly bounded operators, we know from  that, there exists such that for all . Moreover, is continuous in the uniform operator topology for all , i.e.,As in many papers on fractional differential equations, for , we define two operators and bywhereand is a probability density function defined on and satisfiesandClearly,
() and are strongly continuous on X for .
() and are norm-continuous on X for .
A function satisfying the equationis called a mild solution of the problem (1.1).
The following lemma is a generalization of Gronwall’s inequality.
() Suppose and is a nonnegative function locally integrable on (), and suppose is nonnegative and locally integrable on withon this interval, then we have that
Kuratowski measure of noncompactness:
On each bounded subset B in the Banach space X, defineThen, is called the Kuratowski measure of noncompactness on B.
Some basic properties of are given in the following Lemma.
() Let X a Banach space, be a completely continuous operator, if the setis bounded. Then Q has a fixed point.
() Let X be a Banach space and T an operator on X. If there exists a positive integer n such that is a contractive map, i.e., there exists a constant such thatthen has a unique fixed point on X and it is also the unique fixed point of T.
Before we give the main theorems, we need the following lemma.
Let . Suppose that is nonnegative continuous function on withon this interval. Then
WriteThen is a non-decreasing nonnegative continuous function on .
Given . Then for any ,Hence,By Lemma 3, we haveTherefore,The proof ends then. ☐
First we discuss the case f is not necessarily Lipschitz.
In this case, A needs to not only generate an analytic semigroup, but also needs to generate a compact semigroup.
Our first main result is as follows, where the space X could be an infinite dimensional space.
Let A be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operator, and is continuous. If there are almost everywhere nonnegative measurable functions on such thatfor wherethen for any , the Problem (1) has at least one mild solution on .
For every we defineBy Lemma 1, we see that
SetLetThen, it is obvious that u satisfies Equation (2) if and only if and for
We consider the operator as follows:Because f is continuous, by using the Lebesgue dominated convergence theorem, it is easy to prove that Set Next, we will show that P is a compact operator on .
Clearly, is compact.
For letThen, we obtainSince is compact, and the setis bounded, we see that the setis relatively compact in X. Lemma 4(1) tells us thatMoreover, it is clear thatThus, we getOn the other hand, it is easy to see that there exists a positive constant C such thatBy Lemma 4(6), we haveThis means that,Similarly, we can prove that
By Lemma 4(4), we obtainLetting we getConsequently, we see that is relatively compact in X for all .
Clearly, for ,Thus, for we obtainThis, together with Lemma 2, implies that Obviously is bounded in . By the Arzela-Ascoli theorem, we know that P is a compact operator. Hence,
Set . Take . Then for eachThusWriteThenBy Lemma 7, we haveTherefore, By virtue of Lemma 5, we see that P has a fixed point . Thus, is a mild solution of the Problem (1). ☐
If the semigroup (generated by A) satisfies that there exists a such that is compact for all , then we can see from the proof above that the theorem still holds.
The mild solution in this case is usually not unique.
Suppose that is not Lipschitz continuous, i.e., there does not exist a positive constant C such thatbut there exists a positive constant M such that (therefore g is bounded on X). SetLet be a fixed element, and be continuous functions on and Then f satisfies the condition of this theorem, but f is usually not Lipschitz continuous.
- Next we discuss the case when f is Lipschitz continuous.
- In this case, A needs only to generate an analytic semigroup.
- Our second main result is as follows.
Let A be the infinitesimal generator of an analytic semigroup of uniformly bounded linear operator, and be continuous. If f satisfies the Lipschitz condition, i.e., there exists a constant such thatthen for any , the problem (1) has a unique mild solution on .
As in the proof of last theorem, for every we define , and the operator . Then we know that u satisfies Equation (2) if and only if and forand is continuous.
For anyWrite Then we haveHenceWe can deduce by induction thatIn fact, suppose that this inequality holds for , that is, for any ,Then, by the similar argument as above, we obtainThus we have proved thatThereforeSo is a contractive map on for a positive integer . Thus by Lemma 6, we know that P has a unique fixed point on , that is, is the unique mild solution of the Problem (1).
A similar result holds for the following first-order differential equation in the case f is Lipschitz continuous
For details, please refer to , p. 183–185.
If we want to get the unique mild solution, we can do as follows. Set as in the proof of Theorem 2),Then converges uniformly to the unique mild solution of the equation.
It is known that there are many concrete fractional differential equations from anomalous diffusion on fractals (e.g., some amorphous semiconductors or strongly porous materials), which are concrete models of the abstract Cauchy Problem (1). We refer the reader to [2,16] and references therein. Moreover, from [2,16] and references therein, we see that the following Example 1 with the delay effect models some type of anomalous dynamical behaviors of anomalous transport processes.
Letand define its natural norm and inner product respectively, for by
Consider the following Cauchy problem for fractional partial differential equations with finite delay:where are constants.
Let the operator be define byIt is well known (cf., e.g., ) that—A has a discrete spectrum with eigenvalues of the form , and corresponding normalized eigenfunctions given byMoreover, A generates a compact analytic semigroup on X, andIt is not difficult to verify thatHence, we take Thus, when f satisfies the conditions in Remark 3 and ϕ is a continuous function, we see by Theorem 1, the Problem (5) has at least one mild solution.
For the special case A=0,where are constants, f satisfies the condition in Remark 3, and is a continuous function. Then the Problem (6) has at least one mild solution.
Consider the following problemwhere X is a Banach space, are constants, A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operator on a Banach space X,is a fixed element, are continuous functions on and
It is easy to verify that f satisfies the condition of Theorem 2. So the Problem (3) has a unique mild solution.
For the special case A=0,where are constants, a Banach space is a fixed element, are continuous functions on So the Problem (8) has a unique mild solution.
The work was supported partly by the National Natural Science Foundation of China (11571229). The authors would like to thank the referees very much for their helpful comments and suggestions.
Two authors contributed equally and significantly in writing this paper. Two authors read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest
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