- Open Access
Fractional evolution equations with infinite delay under Carathéodory conditions
© Zhou and Yin; licensee Springer. 2014
Received: 28 April 2014
Accepted: 10 July 2014
Published: 4 August 2014
This paper studies fractional evolution equations with infinite delay. We use the means of the successive approximation to establish the existence and uniqueness of mild solutions for this class of equations under global and local Carathéodory conditions. An example is given to illustrate our results.
where is a Caputo fractional derivative of order . takes the value in the Banach space X; A is the infinitesimal generator of an analytic semigroup ; , , , belongs to an abstract phase space ℬ (specified later); . Throughout this paper, we employ the norm denoted by for X. The initial data is a ℬ-valued function.
Fractional differential equations are well known to describe many sophisticated dynamical systems in physics, fluid dynamics, praxiology, viscoelasticity and engineering. The greatest merit of systems including fractional derivative is their nonlocal property and history memory . For more details on the basic theory of fractional differential equations, one can see the monographs [2, 3]. At present, the existence of solutions for fractional equations were discussed, for example, in [4–6], but these equations are usually assumed to satisfy the Lipschitz condition. Wang and Zhou in  addressed the existence of solutions for a class of fractional evolution equations with delay with locally Lipschitz condition. The existence of mild solutions for fractional neutral evolution equations with nonlocal initial condition was obtained by the assumption of Lipschitz condition by Zhou and Jiao in . Besides, Agarwal et al. in  examined the existence of fractional neutral functional differential equations with Lipschitz condition. At present, some important results of impulsive fractional equations have been obtained. Wang et al. in [10, 11] addressed the existence of solutions for impulsive fractional equations. Further, Dabas et al. in  investigated the existence of mild solutions for impulsive fractional equations with infinite delay which possess the Lipschitz condition. The existence of solutions for fractional evolution equations with Lipschitz condition was obtained by means of the monotone iterative technique by Mu in . However, as far as we know, there are few works to research the existence of solutions for fractional evolution equations without Lipschitz condition. To fill this gap, this paper studies system (1.1) which has no assumption of Lipschitz condition.
the conditions on f are nonlinear case and more general, and they do not need any Lipschitz one and take values in X;
the key condition that is compact is not required.
The rest of this paper is organized as follows. In Section 2, we introduce some notations, concepts and basic results. In Section 3, the main results are presented. In Section 4, we give an example to illustrate our results.
First, we introduce some definitions and lemmas on fractional derivation and fractional evolution equation.
Obviously, Caputo’s derivative of any constant is zero.
if is continuous on J and , then for every , we have and ;
for the function in (i), is a ℬ-valued continuous function on J;
the space ℬ is complete.
and is continuous on J;
- (ii)define the norm in(2.1)
Then with norm (2.1) is a Banach space. In the sequel, if there is no ambiguity, we will use for this norm.
- (i)the following integral equation is satisfied:(2.2)
is the function of Wright type defined on .
Lemma 2.5 
and are strongly continuous operators on X;
- (ii)for any , and are linear and bounded operators on X, i.e., there exists a positive constant M such that
In this paper, we will work under the following assumption:
(1a) there exists a function such that for and ;
(1b) is locally integrable in t for each fixed and is continuous and monotone nondecreasing in u for each fixed ;
has a global solution for any initial value ;
for all and ;
where D is a positive constant, then for ;
(H3) the local condition
for with and ,
where is a positive constant, then for .
3 Main results
, , .
The first result is the following theorem.
Theorem 3.1 Let the assumptions of (H1)-(H2) hold. Then there exists a unique mild solution of (1.1) in the sense of the space .
Proof In order to prove this theorem, we divide the proof into the following steps.
which shows the boundedness of the sequence .
By assumption (H2), we can obtain . As a result, it is known that is a Cauchy sequence.
Step 3. The existence and uniqueness of the solution for (1.1). Let , it follows that holds uniformly for . So, taking limits on both sides of (3.1) for , we have that is a solution for (1.1). This shows the existence of solution for (1.1). The uniqueness of the solution could be gotten following the same procedure as in Step 2. By Step 1, we can know that . □
Next, we prove the existence and uniqueness of mild solutions for (1.1) under the local Carathéodory conditions.
Theorem 3.2 Let the assumptions of (H1)-(H3) hold. Then there exists a unique mild solution of (1.1) in the sense of the space .
so this proof is finished. □
possesses a unique solution of 0. Thus, according to Theorems 3.1 and 3.2, system (4.1) has a unique mild solution.
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