A note on the initial value problem of fractional evolution equations
 Pengyu Chen^{1}Email author,
 Xuping Zhang^{1, 2} and
 Yongxiang Li^{1}
https://doi.org/10.1186/s1366201504702
© Chen et al.; licensee Springer. 2015
Received: 16 November 2014
Accepted: 13 April 2015
Published: 14 May 2015
Abstract
In this paper, we discuss the continuous dependence of mild solutions on initial values and orders for the initial value problem of fractional evolution equations in infinite dimensional spaces. The results obtained in this paper improve and extend some related conclusions on this topic. This paper can be considered as a contribution to this emerging field.
Keywords
fractional evolution equation initial value problem continuous dependenceMSC
35R11 47J351 Introduction
Fractional differential equations have recently come to be considered to be a very powerful tool to help scientists explore the hidden properties of the dynamics of complex systems in various fields of sciences and engineering. In recent years, fractional differential equations played the key role of a fundamental, efficient, and convenient theoretical framework for more adequate modeling of complex dynamic processes. Indeed, we can find numerous applications in viscoelasticity, electromagnetism, diffusion, control, mechanics, physics, signal processing, chemistry, bioengineering, medicine, and in many other areas. For more details as regards fractional differential equations we refer to the monographs by Miller and Ross [1], Podlubny [2] and Kilbas et al. [3], the papers by Eidelman and Kochubei [4] and Lakshmikantham and Vatsala [5], and the survey by Agarwal et al. [6].
In recent years, there has been a significant development in the theory of fractional evolution equations. Due to fractional semilinear evolution equations being abstract formulations for many problems arising in engineering and physics, fractional evolution equations have attracted much attention in recent years; see [7–17] and the references cited therein. In addition, some numerical methods have been used to solve spacetime or space fractional evolution equations; see [18–23].
However, we observed that all of the existing articles are only devoted to the study of the existence, uniqueness, and controllability of mild solutions for fractional evolution equations; up to now the continuous dependence of mild solutions on parameters for fractional evolution equations has not been considered in the literature. In order to fill this gap, we are concerned with the continuous dependence of mild solutions on the initial values and orders for IVP (1.1).
2 Preliminaries
In this section, we introduce some notations, definitions, and preliminary facts which are used in the sequel.
Let E be a Banach space with the norm \(\\cdot\\) and let \(a>0\) be a constant. We denote by \(C([0,a],E)\) the Banach space of all continuous Evalue functions on interval \([0,a]\) with the supnorm \(\ u\_{C}=\sup_{t\in[0,a]}\ u(t)\\). Throughout this paper, we assume that \(A:D(A)\subset E\to E\) is a closed linear operator and −A generates a uniformly bounded \(C_{0}\)semigroup \(T(t)\) (\(t\geq0\)) on E. Let \(M=\sup_{t\in [0,+\infty)}\ T(t)\_{\mathcal{L}(E)}\), where \(\mathcal{L}(E)\) stands for the Banach space of all linear and bounded operators in E. For more details of the theory of operator semigroups, see [24].
If u is an abstract function with values in E, then the integrals which appear in (2.1) and (2.2) are taken in Bochner’s sense. For the details as regards the definitions of fractional derivative and integral, please see [1–3].
Definition 2.1
The following lemma will be of fundamental importance in what follows (see [7, 9]).
Lemma 2.2
 (1)For every fixed \(t\geq0\), \(\mathcal{T}_{\alpha}(t)\) and \(\mathcal{S}_{\alpha}(t)\) are linear bounded operators, i.e.,$$\bigl\Vert \mathcal{T}_{\alpha}(t)u\bigr\Vert \leq M\ u\,\qquad \bigl\Vert \mathcal{S}_{\alpha}(t)u\bigr\Vert \leq\frac{M}{\Gamma(\alpha)}\ u \, \quad u\in E. $$
 (2)
The operators \(\mathcal{T}_{\alpha}(t)\) (\(t\geq0\)) and \(\mathcal{S}_{\alpha}(t)\) (\(t\geq0\)) are strongly continuous on \([0,\infty)\).
In what follows, we recall the following GronwallBellman type inequalities, which can be used in fractional differential equations and integral equations with singular kernel.
Lemma 2.3
([25])
Lemma 2.4
([26])
3 Main results
 (H_{ f }):

For any \(\overline{t}>0\) and constant \(r>0\), there exists a positive constant \(L=L(r,\overline{t})\) such thatfor every \(t\in[0,\overline{t}]\) and all \(u, v\in E\) satisfying \(\ u(t)\\leq r\), \(\v(t)\\leq r\).$$\bigl\Vert f\bigl(t,u(t)\bigr) f\bigl(t,v(t)\bigr)\bigr\Vert \leq L\bigl\Vert u(t)v(t)\bigr\Vert $$
Theorem 3.1
Proof
Next, we discuss the continuous dependence of the mild solutions for IVP (1.1) on the orders.
Theorem 3.2
Proof
Remark 3.3
The results obtained in this paper can be applied to all kinds of timefractional partial differential equations. Using Theorems 3.1 and 3.2 one can obtain the continuous dependence results for the concrete fractional partial differential equations. Based on Theorem 3.2, obtained in this paper, we can approximate the solutions of timefractional differential equations by the solutions of integer order differential equations, which are easier to obtain. Therefore, Theorem 3.2 gives an approach to obtain an approximate solution of fractional differential equations.
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by NWNULKQN143 and NNSF of China (11261053).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
 Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
 Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. NorthHolland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
 Eidelman, SD, Kochubei, AN: Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199, 211255 (2004) View ArticleMATHMathSciNetGoogle Scholar
 Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69, 16771682 (2008) MathSciNetGoogle Scholar
 Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 9731033 (2010) View ArticleMATHMathSciNetGoogle Scholar
 ElBorai, MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433440 (2002) View ArticleMATHMathSciNetGoogle Scholar
 ElBorai, MM, ElNadi, KS, ElAkabawy, EG: On some fractional evolution equations. Comput. Math. Appl. 59, 13521355 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262272 (2011) View ArticleMATHMathSciNetGoogle Scholar
 Wang, J, Zhou, Y, Fečkan, M: Abstract Cauchy problem for fractional differential equations. Nonlinear Dyn. 74, 685700 (2013) View ArticleGoogle Scholar
 Wang, R, Xiao, TJ, Liang, J: A note on the fractional Cauchy problems with nonlocal conditions. Appl. Math. Lett. 24, 14351442 (2011) View ArticleMATHMathSciNetGoogle Scholar
 Chen, P, Li, Y: Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions. Z. Angew. Math. Phys. 65, 711728 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Chen, P, Li, Y, Chen, Q, Feng, B: On the initial value problem of fractional evolution equations with noncompact semigroup. Comput. Math. Appl. 67, 11081115 (2014) View ArticleMathSciNetGoogle Scholar
 Chen, P, Li, Y, Li, Q: Existence of mild solutions for fractional evolution equations with nonlocal initial conditions. Ann. Pol. Math. 110, 1324 (2014) View ArticleMATHGoogle Scholar
 Herzallah, MAE, Baleanu, D: On fractional order hybrid differential equations. Abstr. Appl. Anal. 2014, Article ID 389386 (2014) View ArticleMathSciNetGoogle Scholar
 Herzallah, MAE, ElShahed, M, Baleanu, D: Mild and strong solutions for a fractional nonlinear Neumann boundary value problem. J. Comput. Anal. Appl. 15, 341352 (2013) MATHMathSciNetGoogle Scholar
 Kerboua, M, Debbouche, A, Baleanu, D: Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. Abstr. Appl. Anal. 2013, Article ID 262191 (2013) View ArticleMathSciNetGoogle Scholar
 Bhrawy, AH, Zaky, MA, Baleanu, D: New numerical approximations for spacetime fractional Burgers’ equations via a Legendre spectralcollocation method. Rom. Rep. Phys. 67(2), 113 (2015) Google Scholar
 Bhrawy, AH, Baleanu, D: A spectral LegendreGaussLobatto collocation method for a spacefractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219233 (2013) View ArticleMATHMathSciNetGoogle Scholar
 Bhrawy, AH, AlZahrani, AA, Alhame, YA, Baleanu, D: A new generalized LaguerreGauss collocation scheme for numerical solution of generalized fractional pantograph equations. Rom. J. Phys. 59, 646657 (2014) Google Scholar
 Bhrawy, AH, Zaky, MK: A method based on the Jacobi tau approximation for solving multiterm timespace fractional partial differential equations. J. Comput. Phys. 281, 876895 (2015) View ArticleMathSciNetGoogle Scholar
 Bhrawy, AH, Doha, EH, Baleanu, D, EzzEldien, SS: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusionwave equations. J. Comput. Phys. (2014). doi:https://doi.org/10.1016/j.jcp.2014.03.039 Google Scholar
 Abdelkawy, MA, Zaky, MA, Bhrawy, AH, Baleanu, D: Numerical simulation of time variable fractional order mobileimmobile advectiondispersion model. Rom. Rep. Phys. 67(3), 119 (2015) Google Scholar
 Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) View ArticleMATHGoogle Scholar
 Henry, D: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., vol. 840. Springer, New York (1981) MATHGoogle Scholar
 Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 10751081 (2007) View ArticleMATHMathSciNetGoogle Scholar