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Existence and uniqueness of mild solutions for a class of nonlinear fractional evolution equation
Advances in Difference Equations volume 2014, Article number: 150 (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.
The nonlinear fractional evolution equation is a general form for fractional ordinary differential equations, fractional partial differential equations, and fractional functional differential equations related to the time variable. We can widely find the applications in several fields of sciences and technology. Many real phenomena in those fields can be described very successfully by models using mathematical tools of fractional calculus, such as dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, modeling of earthquakes, fluid dynamics, traffic models with fractional derivative, measurements of viscoelastic material properties, modeling of viscoplasticity, control theory, and economy (see [1–4]). There has been a great deal of interest in the solutions of fractional evolution equations in infinite dimensional space. One is referred to the monographs of N’Guérékata et al. , Mophou et al. [6, 7], Liu et al. [8, 9], EI-Borai , Zhou et al. [11, 12], and the references therein. But to the best of the author’s knowledge, most of them have researched the fractional evolution equation with Caputo derivative and the initial conditions such as and so on; there are few papers on the Riemann-Liouville fractional derivative. We note that on a series of examples from the field of viscoelasticity, Heymans and Podlubny  have demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations. In this paper, we discuss the existence and uniqueness of the following fractional evolution equation with Riemann-Liouville fractional derivative:
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
In this section, we introduce preliminary facts which are used throughout this paper. Let us denote by the space of all X-valued continuous functions defined on , which turns out to be a Banach space with the norm
We define similarly another Banach space , which X-valued function is continuous on and is continuous on with the norm
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 )
The fractional primitive of order of function is given by
From  we know exists for all , when and is bounded; notice also that when then , and, moreover, .
Definition 2.2 (see )
The fractional derivative of order of a function is given by
We have for all .
Lemma 2.3 (see )
Let . If we assume , then the fractional differential equation
has , , as solutions.
From this lemma we can obtain the following law of composition.
Lemma 2.4 (see )
Assume that with a fractional derivative of order that belongs to . Then
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 )
Let A be a closed and linear operator with domain defined on Banach space X and . Let be the resolvent set of A. We call A the generator of a q-resolvent family, if there exist and a strongly continuous function satisfying such that and
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 .
Then using the fact that
we deduce for the Laplace transform of (1.1)
if A generates the q-resolvent family .
Throughout this work f will be a continuous function .
Definition 2.8 A function is said to be a mild solution of (1.1) if x satisfies
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 ;
(H1) There exist a constant and such that
where and .
Then (1.1) has a unique mild solution on .
Consider the operator defined by
Let , which is a bounded and closed subset of . For any , we have
The right-hand side will be positive if
Therefore, N maps the ball of of radius R into itself, when T satisfies (3.1).
Next we show that N is a contraction on . For this, let us take , then we get
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
(H2) There exist a constant and a function such 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.
Now we prove that . For any , we have
Notice that is continuous on , therefore .
In view of the continuity of f, it is easy to show that the operator N is continuous. Now we show that N is a completely continuous operator. For each , let with . Then
Actually, and tend to 0 as independently of . Indeed, since is equicontinuous, we have , . Hence .
In view of (H0), (H2), and , we have
so . According to the Hölder inequality we can obtain , then
In view of the Lebesgue’s dominated convergence theorem, we can deduce that as ,
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
(H4) There exist a constant and a function such 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.
Example 4.1 Let , , . Consider the following fractional evolution equation:
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 .
Obviously, is continuous on since ,
We know , ,
So (4.1) has a unique mild solution on by Theorem 3.1.
Example 4.2 Let , , . Consider the following fractional evolution equation:
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.
We choose , . Obviously, is continuous on since and
We know , ,
So (4.2) has at least one mild solution on by Theorem 3.2 and Corollary 3.3.
Example 4.3 To illustrate our results, we give another more concrete example of application. We consider the following fractional anomalous diffusion equation:
To study this system in the abstract form (1.1), we choose the space and the operator A defined by , with domain .
Then A generates a uniformly bounded analytic semigroup which satisfies the condition (H0), (H3). Furthermore, A has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors . Then the following properties hold.
If , then
For each ,
In particular, .
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.
Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Machens W, Voss H, Werther J. Springer, Heidelberg; 1999:217–224.
Caputo M: Linear models of dissipation whose Q is almost independent. Geophys. J. R. Astron. Soc. 1967, 13: 529–539. 10.1111/j.1365-246X.1967.tb02303.x
Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5: 81–88. 10.1016/0888-3270(91)90016-X
Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.
N’Guérékata GM: A Cauchy problem for some abstract fractional differential equation with nonlocal conditions. Nonlinear Anal. 2009, 70: 1873–1876. 10.1016/j.na.2008.02.087
Mophou GM, N’Guérékata GM: Optimal control of a fractional diffusion equation with state constraints. Comput. Math. Appl. 2011, 62: 1413–1426. 10.1016/j.camwa.2011.04.044
Mophou GM, N’Guérékata GM: Existence of mild solutions for some semilinear neutral fractional functional evolution equation with infinite delay. Appl. Math. Comput. 2010, 216: 61–69. 10.1016/j.amc.2009.12.062
Liu ZH, Li XW: Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 1362–1373. 10.1016/j.cnsns.2012.10.010
Liu ZH, Sun JH: Nonlinear boundary value problems of fractional functional integro-differential equation. Comput. Math. Appl. 2012, 64: 3228–3234. 10.1016/j.camwa.2012.02.026
EI-Borai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 2002, 14: 433–440. 10.1016/S0960-0779(01)00208-9
Zhou Y, Jiao F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 2010, 11: 4465–4475. 10.1016/j.nonrwa.2010.05.029
Zhou Y, Jiao F, Li J: Existence and uniqueness for p -type fractional neutral differential equations. Nonlinear Anal., Theory Methods Appl. 2009, 71(7–8):2724–2733. 10.1016/j.na.2009.01.105
Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 2006, 45(5):765–771. 10.1007/s00397-005-0043-5
Miller KS, Ross B: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609–625. 10.1006/jmaa.1996.0456
Belmekki M, Nieto JJ, Rodrguez-Lopez R: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009., 2009: Article ID 324561
Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1980.
Agarwal RP, de Andrade B, Cuevas C: On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differ. Equ. 2010., 2010: Article ID 179750
Lizama C: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 2000, 243: 278–292. 10.1006/jmaa.1999.6668
Lizama C: On approximation and representation of K-regularized resolvent families. Integral Equ. Oper. Theory 2001, 41(2):223–229. 10.1007/BF01295306
Lizama C, Sánchez J: On perturbation of k -regularized resolvent families. Taiwan. J. Math. 2003, 7(2):217–227.
Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. TMA 2009, 69(11):3692–3705.
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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Wang, F., Wang, P. Existence and uniqueness of mild solutions for a class of nonlinear fractional evolution equation. Adv Differ Equ 2014, 150 (2014). https://doi.org/10.1186/1687-1847-2014-150
- fractional derivative
- evolution equation
- Riemann-Liouville integral