- Research Article
- Open Access

# Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations

- Fei Xiao
^{1}Email author

**Received:**28 November 2010**Accepted:**29 January 2011**Published:**24 February 2011

## Abstract

## Keywords

- Probability Density Function
- Fractional Order
- Fixed Point Theorem
- Mild Solution
- Fractional Differential Equation

## 1. Introduction

During the past decades, the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in engineering and physics; they attracted many researchers (cf., e.g., [1–9]). On the other hand, the autonomous and nonautonomous evolution equations and related topics were studied in, for example, [6, 7, 10–20], and the nonlocal Cauchy problem was considered in, for example, [2, 5, 18, 21–26].

In this paper, we denote that is a positive constant and assume that a family of closed linear satisfying

(A1) the domain of is dense in the Banach space and independent of ,

(A2) the operator exists in for any with and

(A3) There exists constant and such that

where , , ([11]).

We study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness. An example is given to show an application of the abstract results.

## 2. Preliminaries

Throughout this work, we set . We denote by a Banach space, the space of all linear and bounded operators on , and the space of all -valued continuous functions on .

- (1)
- (2)

Definition 2.2.

From the definition, we can get some properties of -measure immediately, see ([27]).

Lemma 2.3 (see [27]).

The following lemma will be needed.

Lemma 2.4 (see [27]).

Lemma 2.5 (see [28]).

We need to use the following Sadovskii's fixed point theorem.

Definition 2.6 (see [29]).

Let be an operator in Banach space . If is continuous and takes bounded, sets into bounded sets, and for every bounded set of with , then is said to be a condensing operator on .

Lemma 2.7 (Sadovskii's fixed point theorem [29]).

Let be a condensing operator on Banach space . If for a convex, closed, and bounded set of , then has a fixed point in .

According to [4], a mild solution of (1.1) can be defined as follows.

Definition 2.8.

To our purpose, the following conclusions will be needed. For the proofs refer to [4].

Lemma 2.9 (see [4]).

Remark 2.10.

- (1)
- (2)

## 3. Existence of Solution

Assume that

where is a nonnegative function, and ,

such that

Theorem 3.1.

Suppose that (B1)–(B4) are satisfied, and if , then (1.1) has a mild solution on .

Proof.

Then we proceed in five steps.

Step 1.

Step 2.

We show that maps bounded sets of into bounded sets in .

Step 3.

We show that there exists such that .

which contradicts (B4).

Step 4.

We show that is equicontinuous.

It follows from Lemma 2.9, (B1), and (3.20) that .

Step 5.

In summary, we have proven that has a fixed point . Consequently, (1.1) has at least one mild solution.

Our next result is based on the Banach's fixed point theorem.

(G1) There exists a positive function and a constant such that

(G2) There exists a constant such that the function defined by

Theorem 3.2.

Assume that (G1), (G2) are satisfied, then (1.1) has a unique mild solution.

Proof.

By the Banach contraction mapping principle, has a unique fixed point, which is a mild solution of (1.1).

## 4. An Example

Then generates an analytic semigroup .

Then (4.1) has a mild solution by Theorem 3.1.

## Authors’ Affiliations

## References

- Agarwal RP, Belmekki M, Benchohra M:
**A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative.***Advances in Difference Equations*2009,**2009:**-47.Google Scholar - Anguraj A, Karthikeyan P, N'Guérékata GM:
**Nonlocal Cauchy problem for some fractional abstract integro-differential equations in Banach spaces.***Communications in Mathematical Analysis*2009,**6**(1):31-35.MathSciNetMATHGoogle Scholar - El-Borai MM:
**Some probability densities and fundamental solutions of fractional evolution equations.***Chaos, Solitons and Fractals*2002,**14**(3):433-440. 10.1016/S0960-0779(01)00208-9MathSciNetView ArticleMATHGoogle Scholar - El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis 2004, (3):197-211.Google Scholar
- Li F:
**Mild solutions for fractional differential equations with nonlocal conditions.***Advances in Difference Equations*2010,**2010:**-9.Google Scholar - Li F:
**Solvability of nonautonomous fractional integrodifferential equations with infinite delay.***Advances in Difference Equations*2011,**2011:**-18.Google Scholar - Liang J, Xiao T-J:
**Solutions to nonautonomous abstract functional equations with infinite delay.***Taiwanese Journal of Mathematics*2006,**10**(1):163-172.MathSciNetMATHGoogle Scholar - Mophou GM:
**Existence and uniqueness of mild solutions to impulsive fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(3-4):1604-1615. 10.1016/j.na.2009.08.046MathSciNetView ArticleMATHGoogle Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach Science, New York, NY, USA; 1993:xxxvi+976.MATHGoogle Scholar - Diagana T:
**Pseudo-almost automorphic solutions to some classes of nonautonomous partial evolution equations.***Differential Equations & Applications*2009,**1**(4):561-582.MathSciNetView ArticleMATHGoogle Scholar - Hille E, Phillips RS:
*Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications*.*Volume 31*. American Mathematical Society, Providence, RI, USA; 1957:xii+808.Google Scholar - Josić K, Rosenbaum R:
**Unstable solutions of nonautonomous linear differential equations.***SIAM Review*2008,**50**(3):570-584. 10.1137/060677057MathSciNetView ArticleMATHGoogle Scholar - Kunze M, Lorenzi L, Lunardi A:
**Nonautonomous Kolmogorov parabolic equations with unbounded coefficients.***Transactions of the American Mathematical Society*2010,**362**(1):169-198.MathSciNetView ArticleMATHGoogle Scholar - Liang J, Nagel R, Xiao T-J:
**Approximation theorems for the propagators of higher order abstract Cauchy problems.***Transactions of the American Mathematical Society*2008,**360**(4):1723-1739.MathSciNetView ArticleMATHGoogle Scholar - Pazy A:
*Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences*.*Volume 44*. Springer, New York, NY, USA; 1983:viii+279.View ArticleGoogle Scholar - Xiao T-J, Liang J:
*The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics*.*Volume 1701*. Springer, Berlin, Germany; 1998:xii+301.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Approximations of Laplace transforms and integrated semigroups.***Journal of Functional Analysis*2000,**172**(1):202-220. 10.1006/jfan.1999.3545MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Existence of classical solutions to nonautonomous nonlocal parabolic problems.***Nonlinear Analysis: Theory, Methods and Applications*2005,**63**(5–7):e225-e232.View ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Second order differential operators with Feller-Wentzell type boundary conditions.***Journal of Functional Analysis*2008,**254**(6):1467-1486. 10.1016/j.jfa.2007.12.012MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J, van Casteren J:
**Time dependent Desch-Schappacher type perturbations of Volterra integral equations.***Integral Equations and Operator Theory*2002,**44**(4):494-506. 10.1007/BF01193674MathSciNetView ArticleMATHGoogle Scholar - Byszewski L, Lakshmikantham V:
**Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space.***Applicable Analysis*1991,**40**(1):11-19. 10.1080/00036819008839989MathSciNetView ArticleMATHGoogle Scholar - Fan Z:
**Impulsive problems for semilinear differential equations with nonlocal conditions.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(2):1104-1109. 10.1016/j.na.2009.07.049MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J:
**Semilinear integrodifferential equations with nonlocal initial conditions.***Computers & Mathematics with Applications*2004,**47**(6-7):863-875. 10.1016/S0898-1221(04)90071-5MathSciNetView ArticleMATHGoogle Scholar - Liang J, Liu JH, Xiao T-J:
**Nonlocal problems for integrodifferential equations.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2008,**15**(6):815-824.MathSciNetMATHGoogle Scholar - Liang J, van Casteren J, Xiao T-J:
**Nonlocal Cauchy problems for semilinear evolution equations.***Nonlinear Analysis: Theory, Methods & Applications*2002,**50**(2):173-189. 10.1016/S0362-546X(01)00743-XMathSciNetView ArticleMATHGoogle Scholar - Liu H, Chang J-C:
**Existence for a class of partial differential equations with nonlocal conditions.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(9):3076-3083. 10.1016/j.na.2008.04.009MathSciNetView ArticleMATHGoogle Scholar - Banaś J, Goebel K:
*Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics*.*Volume 60*. Marcel Dekker, New York, NY, USA; 1980:vi+97.Google Scholar - Heinz H-P:
**On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions.***Nonlinear Analysis: Theory, Methods & Applications*1983,**7**(12):1351-1371. 10.1016/0362-546X(83)90006-8MathSciNetView ArticleMATHGoogle Scholar - Sadovskii B:
**On a fixed point principle.***Functional Analysis and Its Applications*1967,**2:**151-153.MathSciNetMATHGoogle Scholar - Bothe D:
**Multivalued perturbations of m-accretive differential inclusions.***Israel Journal of Mathematics*1998,**108:**109-138. 10.1007/BF02783044MathSciNetView ArticleMATHGoogle Scholar

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