Mild solutions of fractional evolution equations on an unbounded interval
© Zhang et al.; licensee Springer. 2014
Received: 19 August 2013
Accepted: 16 December 2013
Published: 22 January 2014
This paper is concerned with the existence results for mild solutions of semilinear fractional evolution equations on an unbounded interval. The methods used in the paper are based on the concept of measure of noncompactness in Fréchet space and classical Tichonov fixed-point theorem. An example is also given to illustrate our main results.
where is the Caputo fractional derivative of order , A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operator in Banach space E, and is a given function.
Fractional differential equations have appeared in many branches of physics, economics, and technical sciences [1, 2]. There has been a considerable developments in fractional differential equations in the last decades. Recently, the definition for mild solutions of fractional evolution equations was successfully given in two ways: one was given by using some probability densities [3–6], the other was given by the so-called solution operator [7–10]. In the scheme of these definitions, many interesting existence results for mild solutions were established by various fixed-point theorems.
We notice that all the papers mentioned above were investigated mild solutions on a bounded interval. On the other hand, research on mild solutions on an unbounded interval of the integer order evolution equations could be found in papers [11, 12] and the references therein. Very recently, Banaś, O’Regan  studied existence and attractiveness of solutions of a nonlinear quadratic integral equations of fractional order on an unbounded interval. By means of Darbo’s fixed-point theorem, Su  considered the existence of solutions to boundary value problems of fractional differential equations on unbounded domains. So we think there is a real need to concern existence results for fractional evolution equation on an unbounded interval. But as far as we known, there are few works on this subject up to now. Motivated by this, we concern ourselves with existence results for mild solutions of problem (1.1) in the present paper by the Tichonov fixed-point theorem.
The rest of paper will be organized as follows. In Section 2 we will introduce some basic definitions and lemmas from the measure of noncompactness, fractional derivation, and integration. Section 3 is devoted to the existence results for problem (1.1). We shall present in Section 4 an example which illustrates our main theorems.
In this section, we collect some definitions and results which will be used in the rest of the paper. Let be a real Banach space. Define be the space of E-valued Bochner functions on with the norm , . Denote by the space of continuous functions from into E.
Definition 2.1 ()
provided the right-hand side is pointwise defined on , where Γ is the gamma function.
Definition 2.2 ()
According to , a sequence is convergent to x in if and only if is uniformly convergent to x on compact subsets of . Moreover, a subset is relatively compact if and only if the restrictions on of all functions from X form an equicontinous set for each and is relatively compact in E for each , where .
Next, we present some basic facts concerning the measure of noncompactness on . Let 0 be the zero element of E. Denote by the closed ball centered at x with radius and by the ball . If X is a subset of E, then the symbols and ConvX stand for the closure and convex closure of X, respectively. Further we assume to be the family of all nonempty and bounded subsets of E, represents its subfamily consisting of relatively compact sets.
Definition 2.3 ()
If is a sequence of nonempty, bounded, and closed subsets of E such that () and if , then the intersection is nonempty.
Denote by the family of all relatively compact members of .
Remark 2.4 ()
We observe that functions from the set are equicontinuous on compact intervals of if and only if for each .
where is a given function such that for .
Theorem 2.5 ()
The family .
If is a sequence of closed sets from such that () and if , then the intersection is nonempty.
For , let us denote .
Lemma 2.6 ()
Lemma 2.7 ()
for any bounded subset , where is the Hausdorff distance between X and Y.
Lemma 2.8 ()
Similar to Cauchy’s formula, we have the following lemma.
Proof By changing the integral order and some calculations, one can prove the lemma easily. We omit the proof here. □
Theorem 2.10 (, Tikhonov fixed-point theorem)
Let V be a locally convex topological vector space. For any nonempty compact convex X in V, if the function is continuous, then F has a fixed point in X.
3 Main results
In this section we will establish the existence results. Based on reference , we give the definition of the mild solutions of problem (1.1) as follows.
Remark 3.2 ()
To state and prove our main results for the existence of mild solutions of problem (1.1), we need the following hypotheses:
(H1) The -semigroup generated by A is compact and there exists a constant such that .
(H2) The function satisfies the Carathéodory type conditions, i.e. is continuous for a.e. and is strongly measurable for each .
(H3) There exists a locally -integrable () function such that for all and a.e. .
for a.e. and bounded subsets X of E, where μ is a regular measure of noncompactness on E.
Remark 3.3 If , , , then we get for each bounded and a.e. .
- (i)For any fixed , and defined in (3.1) are linear and bounded operators, i.e. for any ,
and are continuous in the uniform operator topology for .
Proof (i) was proved in  and (ii) can easily be proved by the compactness of the semigroup . We omit the proof here. □
Theorem 3.5 Under the assumptions (H1)-(H4) problem (1.1) has at least one mild solution x in for each .
Moreover, is nondecreasing.
where , .
In view of , we see that Q is nonempty. Moreover, Q is a closed and convex subset of . From (3.4) and Remark 2.4, we find that the set Q is the family consisting of functions equicontinuous on compact intervals of . By (3.2) we find that F maps Q into itself.
Hence in by the Lebesgue dominated convergence theorem and hypothesis (H2).
which together with (3.4) implies the continuity of on .
where , obviously, is nondecreasing.
In view of Theorem 2.5 we get . Since , we have , which implies is a compact subset in .
Consider . From the above arguments, we see that all the conditions of the Tichonov fixed-point theorem are satisfied. Therefore F has at least one fixed point x in , which is the mild solution of problem (1.1). The proof is completed. □
4 An example
In this section, we give an example to illustrate the applications of Theorem 3.5 established in the previous sections.
where is the Caputo fractional partial derivative of order , and f is a given function.
It is well known that A generates a strongly continuous semigroup which is compact, analytic, and self-adjoint.
where , that is, , .
Let us take , . Firstly, we see that (H1)-(H3) are satisfied. From and Remark 3.3 we find that (H4) is satisfied. According to Theorem 3.5, problem (4.1) has at least one mild solution in .
This work was partially supported by the Professor (Doctor) Scientific Research Foundation of Suzhou University (2013jb04), the Nature Science Foundation of Anhui Provincial Education (KJ2013A248, KJ2012Z403).
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