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Fractional evolution equations with infinite delay under Carathéodory conditions
Advances in Difference Equations volume 2014, Article number: 216 (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.
In this paper, we investigate the existence of solutions for a class of fractional differential equations with infinite delay of the form
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.
In this paper, we show the existence and uniqueness results for (1.1) by means of the successive approximation. Compared with the earlier related existence results that appeared in [8, 13, 14], there are at least two essential differences:
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.
Definition 2.1 Caputo’s derivative of order q with the lower limit 0 for the function can be written as
Obviously, Caputo’s derivative of any constant is zero.
Axiom 2.2 ℬ is a linear space that denotes the family of functions from into X, endowed with the norm , which satisfies the following axioms:
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.
Definition 2.3 Denote by the space of all X-valued continuous mappings , such that
and is continuous on J;
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.
Definition 2.4 , is called a mild solution of (1.1) if
the following integral equation is satisfied:(2.2)
is the function of Wright type defined on .
Lemma 2.5 
The following properties are valid:
and are strongly continuous operators on X;
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 ;
(1c) for any constant , the fractional differential equation
has a global solution for any initial value ;
(H2) (2a) there exists a function such that
for all and ;
(2b) is locally integrable in t for each fixed and is continuous and nondecreasing in u for each fixed . In addition, and if a non-negative continuous function , satisfies
where D is a positive constant, then for ;
(H3) the local condition
(3a) for any integer , there exists a function such that
for with and ,
(3b) is locally integrable in t for each fixed and is continuous and nondecreasing in u for . Moreover, and if a nonnegative continuous function , satisfies
where is a positive constant, then for .
3 Main results
In this section, we establish the existence and uniqueness of mild solutions for (1.1). We construct the sequence of successive approximations defined as follows:
, , .
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.
Step 1. The boundedness of the sequence . From (2.2), we use the Hölder inequality and Lemma 2.5, and we obtain
Thus, by assumption (H1) we have
In view of (3.2), for any , we have
This indicates that
By assumption (H1), it implies that there is a solution satisfying
Since , we have
Further, since k is arbitrary, we have
So, we obtain
which shows the boundedness of the sequence .
Step 2. The sequence is a Cauchy sequence. From (3.1) and assumption (H2), for all and , we can get that
In view of (3.4), assumption (H2) and the Fatou lemma, we have
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 .
Proof Let N be a natural integral and . Define the sequence of the function for as follows:
Then the function satisfies (H3) and the following inequality holds:
for , . Therefore, by Theorem 3.1, there exist solutions and to the following equations, respectively:
From (3.5a) and (3.5b), we have
Define the stopping times
Thus, from (3.6), we obtain
Noting that for , we have
Therefore, this yields
By (H3), we get
By (H3) we have
Since , it holds that
Letting for , we have
so this proof is finished. □
In this section, we provide an example to illustrate our result. Consider the following nonlinear fractional partial differential equations with infinite delay:
with the initial data , for . Let and and they satisfy conditions (H1)-(H3). And we have known that
has a unique solution, i.e., , where , . In addition,
possesses a unique solution of 0. Thus, according to Theorems 3.1 and 3.2, system (4.1) has a unique mild solution.
Kilbas AA, Srivastava HM, Trujilio JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Morel JM, Takens F, Teissier B: The Analysis of Fractional Differential Equations. Springer, Berlin; 2004.
Klages R, Radons G, Sokolov IM: Anomalous Transport: Foundations and Applications. Wiley, New York; 2008.
Wang J, Zhou Y, Fěckan M: Abstract Cauchy problem for fractional differential equations. Nonlinear Dyn. 2013, 71: 685-700. 10.1007/s11071-012-0452-9
Wang J, Fěckan M, Zhou Y: Controllability of Sobolev type fractional evolution systems. Dyn. Partial Differ. Equ. 2014, 11: 71-87. 10.4310/DPDE.2014.v11.n1.a4
Wang J, Fěckan M, Zhou Y: On the nonlocal Cauchy problem for semilinear fractional order evolution equations. Cent. Eur. J. Math. 2014, 12: 911-922. 10.2478/s11533-013-0381-y
Wang J, Zhou Y: Existence of mild solutions for fractional delay evolution systems. Appl. Math. Comput. 2011, 218: 357-367. 10.1016/j.amc.2011.05.071
Zhou Y, Jiao F: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 2010, 59: 1063-1077. 10.1016/j.camwa.2009.06.026
Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010
Wang J, Fěckan M, Zhou Y: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 2011, 8: 345-361. 10.4310/DPDE.2011.v8.n4.a3
Wang J, Fěckan M, Zhou Y: Relaxed controls for nonlinear fractional impulsive evolution equations. J. Optim. Theory Appl. 2013, 156: 13-32. 10.1007/s10957-012-0170-y
Dabas J, Chauhan A, Kumar M: Existence of the mild solutions for impulsive fractional equations with infinite delay. Int. J. Differ. Equ. 2011., 2011: Article ID 793023
Mu J: Monotone iterative technique for fractional evolution equations in Banach spaces. J. Appl. Math. 2011., 2011: Article ID 767186
Hernández E, O’Regan D, Balachandran K: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal., Theory Methods Appl. 2010, 73: 3462-3471. 10.1016/j.na.2010.07.035
The authors declare that they have no competing interests.
JZ carried out the main parts of the draft. HY provided the main idea of this paper.
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Zhou, J., Yin, H. Fractional evolution equations with infinite delay under Carathéodory conditions. Adv Differ Equ 2014, 216 (2014). https://doi.org/10.1186/1687-1847-2014-216
- fractional differential equations
- infinite delay
- Carathéodory condition