- Research Article
- Open Access

# Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

- Fang Li
^{1}Email author and - GastonM N'Guérékata
^{2}

**2010**:158789

https://doi.org/10.1155/2010/158789

© Fang Li and Gaston M. N'Guérékata. 2010

**Received:**1 April 2010**Accepted:**17 June 2010**Published:**13 July 2010

## Abstract

We study the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations , , , in a Banach space , where . New results are obtained by fixed point theorem. An application of the abstract results is also given.

## Keywords

- Banach Space
- Probability Density Function
- Electric Circuit
- Nonempty Subset
- Complex Medium

## 1. Introduction

An integrodifferential equation is an equation which involves both integrals and derivatives of an unknown function. It arises in many fields like electronic, fluid dynamics, biological models, and chemical kinetics. A well-known example is the equations of basic electric circuit analysis. In recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established (see, e.g., [1–11] and references therein).

On the other hand, many phenomena in Engineering, Physics, Economy, Chemistry, Aerodynamics, and Electrodynamics of complex medium can be modeled by fractional differential equations. During the past decades, such problem attracted many researchers (see [1, 12–21] and references therein).

where , and the fractional derivative is understood in the Caputo sense.

In this paper, motivated by [1–27] (especially the estimating approaches given in [4, 6, 10, 24, 27]), we investigate the existence and uniqueness of mild solution of (1.1) in a Banach space : generates a compact semigroup of uniformly bounded linear operators on a Banach space . The function is real valued and locally integrable on , and the nonlinear maps and are defined on into . New existence and uniqueness results are given. An example is given to show an application of the abstract results.

## 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 semigroup of uniformly bounded linear operators. Then there exists such that for .

According to [22, 23], a mild solution of (1.1) can be defined as follows.

Definition 2.1.

Remark 2.2.

In this paper, we use to denote the norm of whenever for some with . denotes the Banach space of all continuous functions endowed with the sup-norm given by for . Set .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnosel'skii).

- (i)
whenever ,

- (ii)
is compact and continuous,

- (iii)
is a contraction mapping.

Then there exists such that .

## 3. Main Results

We will require the following assumptions.

Theorem 3.1.

then (1.1) has a unique mild solution for every .

Proof.

Set , .

for . Hence .

The conclusion follows by the contraction mapping principle.

We assume the following.

(H3) The function is continuous, and there exists a positive function ( ) such that

and set

Let be the infinitesimal generator of a compact semigroup of uniformly bounded linear operators. Then there exists a constant such that for .

Theorem 3.2.

then (1.1) has a mild solution for every .

Proof.

where .

Let be the closed convex and nonempty subset of the space .

Set .

Thus, .

So, we know that is a contraction mapping.

Set .

which implies that is relatively compact in .

Next, we prove that is equicontinuous.

The right-hand side of (3.24) tends to as as a consequence of the continuity of in the uniform operator topology for by the compactness of . So as . Thus, , as , which is independent of . Therefore is compact by the Arzela-Ascoli theorem.

Next we show that is continuous.

So is continuous.

By Krasnosel'skii's theorem, we have the conclusion of the theorem.

Remark 3.3.

In Theorem 3.2, if we furthermore suppose that the hypothesis

holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.

which implies by Gronwall's inequality that (1.1) has a unique mild solution .

Example 3.4.

Then generates a compact, analytic semigroup of uniformly bounded linear operators.

, and .

then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.

## Declarations

### Acknowledgments

The authors are grateful to the referees for their valuable suggestions. The first author is supported by the NSF of Yunnan Province (2009ZC054M).

## Authors’ Affiliations

## References

- El-Borai MM, Debbouche A:
**On some fractional integro-differential equations with analytic semigroups.***International Journal of Contemporary Mathematical Sciences*2009,**4**(25–28):1361-1371.MATHMathSciNetGoogle Scholar - Ding H-S, Liang J, Xiao T-J:
**Positive almost automorphic solutions for a class of non-linear delay integral equations.***Applicable Analysis*2009,**88**(2):231-242. 10.1080/00036810802713875MATHMathSciNetView ArticleGoogle Scholar - Ding H-S, Xiao T-J, Liang J:
**Existence of positive almost automorphic solutions to nonlinear delay integral equations.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(6):2216-2231. 10.1016/j.na.2008.03.001MATHMathSciNetView ArticleGoogle 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.MATHMathSciNetGoogle 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. 10.1090/S0002-9947-07-04551-5MATHMathSciNetView 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:
*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.1999.3545MATHMathSciNetView ArticleGoogle 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.2007.12.012MATHMathSciNetView ArticleGoogle Scholar - Xiao T-J, Liang J:
**Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(12):e1442-e1447. 10.1016/j.na.2009.01.204MATHMathSciNetView ArticleGoogle 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/BF01193674MATHMathSciNetView ArticleGoogle Scholar - Hilfer R (Ed):
*Applications of Fractional Calculus in Physics*. World Scientific, River Edge, NJ, USA; 2000:viii+463.MATHGoogle Scholar - Henderson J, Ouahab A:
**Fractional functional differential inclusions with finite delay.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):2091-2105. 10.1016/j.na.2008.02.111MATHMathSciNetView ArticleGoogle 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 - 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 - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication*. John Wiley & Sons, New York; 1993:xvi+366.Google 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 - Mophou GM, N'Guérékata GM:
**Existence of the mild solution for some fractional differential equations with nonlocal conditions.***Semigroup Forum*2009,**79**(2):315-322. 10.1007/s00233-008-9117-xMATHMathSciNetView ArticleGoogle Scholar - Mophou GM, N'Guérékata GM:
**A note on a semilinear fractional differential equation of neutral type with infinite delay.***Advances in Difference Equations*2010,**2010:**-8.Google 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 - 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)00208-9MATHMathSciNetView ArticleGoogle Scholar - El-Borai MM:
**On some stochastic fractional integro-differential equations.***Advances in Dynamical Systems and Applications*2006,**1**(1):49-57.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, Xiao T-J:
**Solvability of the Cauchy problem for infinite delay equations.***Nonlinear Analysis: Theory, Methods & Applications*2004,**58**(3-4):271-297. 10.1016/j.na.2004.05.005MATHMathSciNetView 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 - 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

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.