- Research Article
- Open Access
Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay
© Fang Li. 2011
- Received: 4 September 2010
- Accepted: 29 October 2010
- Published: 31 October 2010
We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with infinite delay in a Banach space . The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii's fixed point theorem. An application of the abstract results is also given.
- Banach Space
- Probability Density Function
- Mild Solution
- Fractional Differential Equation
- Dominate Convergence Theorem
, defined by for , belongs to the phase space , and . The fractional derivative is understood here in the Riemann-Liouville sense.
In recent years, the fractional differential equations have been proved to be good tools in the investigation of many phenomena in engineering, physics, economy, chemistry, aerodynamics, electrodynamics of complex medium and they have been studied by many researchers (cf., e.g., [13, 14, 16, 17] and references therein). Moreover, many phenomena cannot be described through classical differential equations but the integral and integrodifferential equations in abstract spaces in fields like electronic, fluid dynamics, biological models, and chemical kinetics. So many significant works on this topic have been appeared (cf., e.g., [10, 15, 18–25] and references therein).
In this paper, we study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup generated by is compact (see Theorem 3.1). An example is given to show an application of the abstract results.
Next, we recall the definition of the Riemann-Liouville integral.
Definition 2.1 (see ).
where is the Gamma function. Moreover, , for all .
where is a beta function.
See the following definition about phase space according to Hale and Kato .
A linear space consisting of functions from into , with seminorm , is called an admissible phase space if has the following properties.
for and as in (1).
The space is complete.
Equation (2.4) in (1) is equivalent to , for all .
Next, we consider the properties of Kuratowski's measure of noncompactness.
From the definition we can get some properties of -measure immediately, see .
Lemma 2.6 (see ).
Let and be bounded subsets of , then
, if ;
, where denotes the closure of ;
if and only if is precompact;
, where ;
, for any .
The following lemma will be needed.
If is a bounded, equicontinuous set, then
(ii) , for .
For a proof refer to .
Lemma 2.8 (see ).
We need to use the following Sadovskii's fixed point theorem here, see .
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 2.10 (Sadovskii's fixed point theorem ).
Let be a condensing operator on Banach space . If for a convex, closed, and bounded set of , then has a fixed point in .
In this paper, we denote that is a positive constant, and assume that a family of closed linear operators satisfying the following.
(A1)The domain of is dense in the Banach space and independent of .
where , , (see ).
According to , a mild solution of (1.1) can be defined as follows.
To our purpose the following conclusions will be needed. For the proofs refer to .
Lemma 2.12 (see ).
We need the hypotheses as follows:
and set ,
where , and ,
where , and .
Suppose that (H1)–(H3) are satisfied, and if , then for (1.1) there exists a mild solution on .
It is easy to see that is well-defined.
Let , .
and , .
Thus is a Banach space.
Obviously, the operator has a fixed point if and only if has a fixed point. So it turns out to prove that has a fixed point.
This means that is continuous.
This means .
This contradicts (3.3). Hence for some positive number , .
It follows from Lemma 2.12, (H1) and (3.23) that , , as .
So, the set is equicontinuous.
In view of the Sadovskii's fixed point theorem, we conclude that has at least one fixed point in . Let , , then is a fixed point of the operator which is a mild solution of (1.1).
Now we assume that
Assume that (H1') and (H2') are satisfied, then (1.1) has a unique mild solution.
and the result follows from the contraction mapping principle.
, are continuous functions, and .
Then generates an analytic semigroup satisfying assumptions (A1)–(A3) (see ).
then we can see that in (2.5).
Then the above equation (3.37) can be written in the abstract form as (1.1).
where , satisfy (H1).
Suppose further that
there exists such that ,
then (3.37) has a mild solution by Theorem 3.1.
The author is grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (no. 2009ZC054M).
- Fan Z: Existence and continuous dependence results for nonlinear differential inclusions with infinite delay. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2379-2392. 10.1016/j.na.2007.08.011MathSciNetView ArticleMATHGoogle Scholar
- Favini A, Vlasenko L: Degenerate non-stationary differential equations with delay in Banach spaces. Journal of Differential Equations 2003,192(1):93-110. 10.1016/S0022-0396(03)00090-1MathSciNetView ArticleMATHGoogle Scholar
- Guo Faming, Tang B, Huang F: Robustness with respect to small delays for exponential stability of abstract differential equations in Banach spaces. The ANZIAM Journal 2006,47(4):555-568. 10.1017/S1446181100010130MathSciNetView ArticleMATHGoogle Scholar
- Liu JH: Periodic solutions of infinite delay evolution equations. Journal of Mathematical Analysis and Applications 2000,247(2):627-644. 10.1006/jmaa.2000.6896MathSciNetView ArticleMATHGoogle Scholar
- Liu J, Naito T, Van Minh N: Bounded and periodic solutions of infinite delay evolution equations. Journal of Mathematical Analysis and Applications 2003,286(2):705-712. 10.1016/S0022-247X(03)00512-2MathSciNetView ArticleMATHGoogle Scholar
- Liang J, Xiao TJ: Functional-differential equations with infinite delay in Fréchet space. Sichuan Daxue Xuebao 1989,26(4):382-390.MathSciNetMATHGoogle Scholar
- Liang J, Xiao TJ: Functional-differential equations with infinite delay in Banach spaces. International Journal of Mathematics and Mathematical Sciences 1991,14(3):497-508. 10.1155/S0161171291000686MathSciNetView ArticleMATHGoogle Scholar
- Liang J, Xiao TJ: Solutions to abstract functional-differential equations with infinite delay. Acta Mathematica Sinica 1991,34(5):631-644.MathSciNetMATHGoogle Scholar
- Liang J, Xiao T-J: The Cauchy problem for nonlinear abstract functional differential equations with infinite delay. Computers & Mathematics with Applications 2000,40(6-7):693-703. 10.1016/S0898-1221(00)00188-7MathSciNetView ArticleMATHGoogle Scholar
- Liang J, Xiao T-J, van Casteren J: A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Applied Mathematics Letters 2004,17(4):473-477. 10.1016/S0893-9659(04)90092-4MathSciNetView ArticleMATHGoogle 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.2004.05.005MathSciNetView ArticleMATHGoogle 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, 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
- Mophou GM, N'Guérékata GM: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Applied Mathematics and Computation 2010,216(1):61-69. 10.1016/j.amc.2009.12.062MathSciNetView ArticleMATHGoogle 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.2009.01.204MathSciNetView 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.MathSciNetView ArticleMATHGoogle 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-xMathSciNetView ArticleMATHGoogle Scholar
- Chen J-H, Xiao T-J, Liang J: Uniform exponential stability of solutions to abstract Volterra equations. Journal of Evolution Equations 2009,9(4):661-674. 10.1007/s00028-009-0028-4MathSciNetView 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
- Naito T, Van Minh N, Liu JH: On the bounded solutions of Volterra equations. Applicable Analysis 2004,83(5):433-446. 10.1080/00036810310001632781MathSciNetView ArticleMATHGoogle Scholar
- Ngoc PHA, Murakami S, Naito T, Shin JS, Nagabuchi Y: On positive linear Volterra-Stieltjes differential systems. Integral Equations and Operator Theory 2009,64(3):325-335. 10.1007/s00020-009-1692-zMathSciNetView ArticleMATHGoogle 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, 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
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11-41.MathSciNetMATHGoogle 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 1967,1(2):151-153.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
- Bothe D: Multivalued perturbations of -accretive differential inclusions. Israel Journal of Mathematics 1998, 108: 109-138. 10.1007/BF02783044MathSciNetView ArticleMATHGoogle Scholar
- Friedman A: Partial Differential Equations. Holt, Rinehat and Winston, New York, NY, USA; 1969:vi+262.Google Scholar
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