# Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

- Fang Li
^{1}Email author

**2010**:287861

**DOI: **10.1155/2010/287861

© Fang Li. 2010

**Received: **8 January 2010

**Accepted: **21 January 2010

**Published: **26 January 2010

## Abstract

## 1. Introduction

The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensively (see, e.g., [1–8] and references therein).

In this paper, we discuss the existence and uniqueness of mild solution for

where , , and generates an analytic compact semigroup of uniformly bounded linear operators on a Banach space . The term which may be interpreted as a control on the system is defined by

where (the set of all positive function continuous on ) and

The functions and are continuous.

The nonlocal condition can be applied in physics with better effect than that of the classical initial condition . There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 9–14] and references cited there).

In this paper, motivated by [1–7, 9–15] (especially the estimating approach given by Xiao and Liang [14]), we study the semilinear fractional differential equations with nonlocal condition (1.1) in a Banach space , assuming that the nonlinear map is defined on and is defined on where , for , the domain of the fractional power of . New and general existence and uniqueness theorem, which extends many previous results, are given.

## 2. Preliminaries

In this paper, we set , a compact interval in . We denote by a Banach space with norm . Let be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators , that is, there exists such that ; and without loss of generality, we assume that . So we can define the fractional power for , as a closed linear operator on its domain with inverse , and one has the following known result.

Lemma 2.1 (see [15]).

is a Banach space with the norm for .

Definition 2.2.

for is called a mild solution of (1.1).

In this paper, we use to denote the norm of whenever for some with . We denote by the Banach space endowed with the sup norm given by

The following well-known theorem will be used later.

Theorem 2.3 (Krasnoselkii, see [16]).

Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that

## 3. Main Results

- (H1)
- (H2)

Theorem 3.1.

Let be the infinitesimal generator of an analytic compact semigroup with and . If the maps and satisfy (H1), (H2), respectively, and , then (1.1) has a mild solution for every .

Proof.

which means that is continuous.

we can see that is uniformly bounded on .

It follows from the assumption of that tends to 0 as . For , using the Hölder inequality, we can see that tends to 0 as and .

Using the compactness of in implies the continuity of for integrating with , we see that tends to , as . For , from the assumption of and the Hölder inequality, it is easy to see that tends to 0 as and .

Thus, , as , which does not depend on .

So, is relatively compact. By the Arzela-Ascoli Theorem, is compact.

We now conclude the result of the theorem by Krasnoselkii's theorem.

Now we assume the following.

(H3) There exists a positive function such that

Theorem 3.2.

Let be the infinitesimal generator of an analytic semigroup with and . If and (H2)–(H4) hold, then (1.1) has a unique mild solution .

Proof.

Obviously, is well defined on .

and the result follows from the contraction mapping principle.

## Declarations

### Acknowledgment

This work is supported by the NSF of Yunnan Province (2009ZC054M).

## 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 - El-Borai MM, Amar D:
**On some fractional integro-differential equations with analytic semigroups.***International Journal of Contemporary Mathematical Sciences*2009,**4**(25–28):1361–1371.MATHMathSciNetGoogle Scholar - Lakshmikantham V:
**Theory of fractional functional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3337–3343. 10.1016/j.na.2007.09.025MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Vatsala AS:
**Basic theory of fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(8):2677–2682. 10.1016/j.na.2007.08.042MATHMathSciNetView ArticleGoogle Scholar - Lv ZW, Liang J, Xiao TJ:
**Solutions to fractional differential equations with nonlocal initial condition in Banach spaces.**reprint, 2009Google 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.2008.04.009MATHMathSciNetView ArticleGoogle Scholar - N'Guérékata GM:
**A Cauchy problem for some fractional abstract differential equation with non local conditions.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):1873–1876. 10.1016/j.na.2008.02.087MATHMathSciNetView ArticleGoogle Scholar - Zhu X-X:
**A Cauchy problem for abstract fractional differential equations with infinite delay.***Communications in Mathematical Analysis*2009,**6**(1):94–100.MATHMathSciNetGoogle 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-XMATHMathSciNetView ArticleGoogle Scholar - Liang J, Liu J, Xiao T-J:
**Nonlocal Cauchy problems governed by compact operator families.***Nonlinear Analysis: Theory, Methods & Applications*2004,**57**(2):183–189. 10.1016/j.na.2004.02.007MATHMathSciNetView ArticleGoogle Scholar - Liang J, Liu JH, Xiao T-J:
**Nonlocal Cauchy problems for nonautonomous evolution equations.***Communications on Pure and Applied Analysis*2006,**5**(3):529–535. 10.3934/cpaa.2006.5.529MATHMathSciNetView ArticleGoogle Scholar - Liang J, Liu JH, Xiao T-J:
**Nonlocal impulsive problems for nonlinear differential equations in Banach spaces.***Mathematical and Computer Modelling*2009,**49**(3–4):798–804. 10.1016/j.mcm.2008.05.046MATHMathSciNetView ArticleGoogle 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)90071-5MATHMathSciNetView ArticleGoogle Scholar - Xiao T-J, Liang J:
**Existence of classical solutions to nonautonomous nonlocal parabolic problems.***Nonlinear Analysis: Theory, Methods & Applications*2005,**63**(5–7):e225-e232. 10.1016/j.na.2005.02.067MATHView ArticleGoogle 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 - Smart DR:
*Fixed Point Theorems*. Cambridge University Press; 1980.MATHGoogle Scholar

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