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Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations
Advances in Difference Equations volume 2010, Article number: 158789 (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.
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 wellknown 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).
However, among the previous researches on the fractional differential equations, few are concerned with mild solutions of the fractional integrodifferential equations as follows:
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.
A continuous function satisfying the equation
for is called a mild solution of (1.1), where
and is a probability density function defined on such that its Laplace transform is given by
Remark 2.2.
Noting that (cf., [23]), we can see that
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 supnorm given by for . Set .
The following wellknown theorem will be used later.
Theorem 2.3 (Krasnosel'skii).
Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that

(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.
(H1) The function is continuous, and there exists such that
(H2) The function , , satisfies
Theorem 3.1.
Let be the infinitesimal generator of a strongly continuous semigroup with , . If the maps and satisfy (H1), satisfies (H2), and
then (1.1) has a unique mild solution for every .
Proof.
Define the mapping by
Set , .
Choose such that
Let be the nonempty closed and convex set given by
Then for , we have
Noting that
we obtain
for . Hence .
Let and be two elements in . Then
So
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.
If the maps and satisfy (H1), (H3), respectively, and
then (1.1) has a mild solution for every .
Proof.
Define
Choose such that
where .
Let be the closed convex and nonempty subset of the space .
Letting , we have
Set .
According to the Hölder inequality, (H1), and (3.8), for , we have
Thus, .
For and , using (H1), we obtain
So, we know that is a contraction mapping.
Set .
Fix . For , set
Since is compact for each , the sets are relatively compact in for each , . Furthermore,
which implies that is relatively compact in .
Next, we prove that is equicontinuous.
For , we have
By (H3), we get
In view of the assumption of , we see that tends to 0 as , and one
Clearly, the last term tends to as . Hence as , and
The righthand 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 ArzelaAscoli theorem.
Next we show that is continuous.
Let be a sequence of such that in . By the continuity of on , we have
Noting the continuity of , we get
Thus, we have
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
(H4)
holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.
Actually, from what we have just proved, (1.1) has a mild solution and
Let be another mild solution of (1.1). Then
which implies by Gronwall's inequality that (1.1) has a unique mild solution .
Example 3.4.
Let . Define
Then generates a compact, analytic semigroup of uniformly bounded linear operators.
Let , , and let , be positive constants. We set
, and .
It is not hard to see that and satisfy (H1), (H3), respectively, and if
then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.
References
ElBorai MM, Debbouche A: On some fractional integrodifferential equations with analytic semigroups. International Journal of Contemporary Mathematical Sciences 2009,4(25–28):13611371.
Ding HS, Liang J, Xiao TJ: Positive almost automorphic solutions for a class of nonlinear delay integral equations. Applicable Analysis 2009,88(2):231242. 10.1080/00036810802713875
Ding HS, Xiao TJ, Liang J: Existence of positive almost automorphic solutions to nonlinear delay integral equations. Nonlinear Analysis: Theory, Methods & Applications 2009,70(6):22162231. 10.1016/j.na.2008.03.001
Liang J, Liu JH, Xiao TJ: Nonlocal problems for integrodifferential equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2008,15(6):815824.
Liang J, Nagel R, Xiao TJ: Approximation theorems for the propagators of higher order abstract Cauchy problems. Transactions of the American Mathematical Society 2008,360(4):17231739. 10.1090/S0002994707045515
Liang J, Xiao TJ: Semilinear integrodifferential equations with nonlocal initial conditions. Computers & Mathematics with Applications 2004,47(67):863875. 10.1016/S08981221(04)900715
Xiao TJ, Liang J: The Cauchy Problem for HigherOrder Abstract Differential Equations, Lecture Notes in Mathematics. Volume 1701. Springer, Berlin, Germany; 1998:xii+301.
Xiao TJ, Liang J: Approximations of Laplace transforms and integrated semigroups. Journal of Functional Analysis 2000,172(1):202220. 10.1006/jfan.1999.3545
Xiao TJ, Liang J: Second order differential operators with FellerWentzell type boundary conditions. Journal of Functional Analysis 2008,254(6):14671486. 10.1016/j.jfa.2007.12.012
Xiao TJ, Liang J: Blowup and global existence of solutions to integral equations with infinite delay in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1442e1447. 10.1016/j.na.2009.01.204
Xiao TJ, Liang J, van Casteren J: Time dependent DeschSchappacher type perturbations of Volterra integral equations. Integral Equations and Operator Theory 2002,44(4):494506. 10.1007/BF01193674
Hilfer R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.
Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):20912105. 10.1016/j.na.2008.02.111
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):33373343. 10.1016/j.na.2007.09.025
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):26772682. 10.1016/j.na.2007.08.042
Liu H, Chang JC: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):30763083. 10.1016/j.na.2008.04.009
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A WileyInterscience Publication. John Wiley & Sons, New York; 1993:xvi+366.
N'Guérékata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):18731876. 10.1016/j.na.2008.02.087
Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315322. 10.1007/s002330089117x
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.
Zhu XX: A Cauchy problem for abstract fractional differential equations with infinite delay. Communications in Mathematical Analysis 2009,6(1):94100.
ElBorai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons and Fractals 2002,14(3):433440. 10.1016/S09600779(01)002089
ElBorai MM: On some stochastic fractional integrodifferential equations. Advances in Dynamical Systems and Applications 2006,1(1):4957.
Liang J, van Casteren J, Xiao TJ: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2002,50(2):173189. 10.1016/S0362546X(01)00743X
Liang J, Xiao TJ: Solvability of the Cauchy problem for infinite delay equations. Nonlinear Analysis: Theory, Methods & Applications 2004,58(34):271297. 10.1016/j.na.2004.05.005
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.
Xiao TJ, Liang J: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):e225e232. 10.1016/j.na.2005.02.067
Acknowledgments
The authors are grateful to the referees for their valuable suggestions. The first author is supported by the NSF of Yunnan Province (2009ZC054M).
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Li, F., N'Guérékata, G. Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations. Adv Differ Equ 2010, 158789 (2010). https://doi.org/10.1155/2010/158789
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DOI: https://doi.org/10.1155/2010/158789