Existence and uniqueness of mild solutions for a class of nonlinear fractional evolution equation
© Wang and Wang; licensee Springer. 2014
Received: 30 December 2013
Accepted: 6 May 2014
Published: 20 May 2014
In this paper, we discuss a class of fractional evolution equations with the Riemann-Liouville fractional derivative and obtain the existence and uniqueness of mild solutions by using some classical fixed point theorem. Then we give some examples to demonstrate the main results.
Keywordsfractional derivative evolution equation Riemann-Liouville integral existence uniqueness
where is the Riemann-Liouville fractional derivative of order , A is the infinitesimal generator of a q-resolvent family defined on a Banach space X and . The function is given and satisfies some conditions which are weak compared to the existing results and the conclusion is generalized.
2 Definitions and preliminary results
is the space of all linear and bounded operators on X. The definitions and results of the fractional calculus reported below are not exhaustive but rather oriented to the subject of this paper. For the proofs, which are omitted, we refer the reader to [14, 15] or other texts on basic fractional calculus. As x is an abstract function with values in X, the integrals which appear in Definition 2.1 and Definition 2.2 are taken in Bochner’s sense.
Definition 2.1 (see )
From  we know exists for all , when and is bounded; notice also that when then , and, moreover, .
Definition 2.2 (see )
We have for all .
Lemma 2.3 (see )
has , , as solutions.
From this lemma we can obtain the following law of composition.
Lemma 2.4 (see )
for some . When the function x is in , then .
Recall that the Laplace transform of a function is defined by , , if the integral is absolutely convergent for .
Theorem 2.5 (see )
Let E be a closed, convex and bounded and nonempty subset of a Banach space X and be a completely continuous operator. Then N has at least one fixed point in E.
Definition 2.6 (see )
In this case, is called the q-resolvent family generated by A.
Remark 2.7 Note that if A is the generator of an q-resolvent family then the Laplace transform of is .
if A generates the q-resolvent family .
Throughout this work f will be a continuous function .
Since we have the uniqueness of the Laplace transform, a 1-resolvent family is the same as a -semigroup whereas a 2-resolvent family corresponds to the concept of sine family; see . We note that q-resolvent families are a particular case of -regularized families introduced in . These have been studied in a series of several papers in recent years (see [20, 21] and so on). According to  a q-resolvent family corresponds to a regularized family. For more details on the q-resolvent family, we refer to  and the references therein. We also refer to  for more information as regards the resolvent or solution operator. As in the situation of -semigroups we have diverse relations of a q-resolvent family and its generator. So we can assume the following condition to present the first result in this paper.
We present now our first result.
Theorem 3.1 Assume that
(H0) There exist and such that ;
where and .
Then (1.1) has a unique mild solution on .
Therefore, N maps the ball of of radius R into itself, when T satisfies (3.1).
From the condition (3.1), we conclude that N is a contraction. Therefore, N has a unique fixed point in . So (2.1) is the unique mild solution of (1.1) on . □
Now we assume that
(H3) The q-resolvent family is equicontinuous.
Theorem 3.2 Assume that (H0), (H2), and (H3) hold. Then (1.1) has at least one mild solution on .
Observe that is a closed, bounded, and convex subset of Banach space X.
Notice that is continuous on , therefore .
Actually, and tend to 0 as independently of . Indeed, since is equicontinuous, we have , . Hence .
as . Hence N maps bounded sets of X into equicontinuous sets of X.
Next we will prove the operator N maps into a relatively compact set in X. Indeed from the equicontinuity of and , according to the Arzela-Ascoli theorem the set is relatively compact in X, for every . Therefore, we can obtain N is a completely continuous operator. Thus the conclusion of Theorem 2.5 implies that (1.1) has at least one mild solution on . □
Now we assume that
Corollary 3.3 Assume that (H0), (H3), and (H4) hold. Then (1.1) has at least one mild solution on .
The proof of the Corollary 3.3 is similar to Theorem 3.2.
Assume that , , defined by , , where .
It is well know that A is an infinitesimal generator of a semigroup in and given by , for , is a strongly continuous semigroup on and . We choose .
So (4.1) has a unique mild solution on by Theorem 3.1.
Assume that , , defined by , , where .
It is well known that A is an infinitesimal generator of a semigroup in and given by , for , is a strongly continuous semigroup on , , and we assume that is equicontinuous.
So (4.2) has at least one mild solution on by Theorem 3.2 and Corollary 3.3.
To study this system in the abstract form (1.1), we choose the space and the operator A defined by , with domain .
- (i)If , then
- (ii)For each ,
- (iii)The operator is given by
on the space .
If the nonlinear term satisfies the condition (H1), then (4.3) has a unique mild solution on by Theorem 3.1. If the nonlinear term satisfies the conditions (H2) and (H4), then (4.3) has at least one mild solution on by Theorem 3.2 and Corollary 3.3.
The authors are highly grateful for the referee’s careful reading and comments on this note. The present project is supported by Science and Technology Department of Hunan Province granted 2014FJ3071, Hunan Provincial Educational Department Science Foundation no. 13C1035 and NNSF of China Grant no. 11301039, no. 11301040.
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