- Research Article
- Open Access
Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions
© R.-N. Wang and J. Xia. 2011
- Received: 23 November 2010
- Accepted: 7 March 2011
- Published: 15 March 2011
We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of the paper, we construct an example to illustrate the feasibility of our results.
- Banach Space
- Cauchy Problem
- Mild Solution
- Fractional Brownian Motion
- Sectorial Operator
Let denote a complex Banach space and denote by the space of all bounded linear operators from into with the usual operator norm . Let us recall the following definitions.
Definition 1.1 (see ).
is called the Riemann-Liouville integral of order .
Definition 1.2 (see ).
where is a constant and stands for the resolvent of . In this case, we also say that is a solution operator generated by .
It is to be noted that in the border case , the family corresponds to a classical strongly continuous semigroup, whereas in the case a solution operator corresponds to the concept of a cosine family. Moreover, according to , one can find that solution operators are a particular case of -regularized families and a solution operator corresponds to a -regularized family.
Note that solution operator does not satisfy the semigroup property.
Starting from some speculations of Leibniz and Euler, the fractional calculus (such as the Riemann-Liouville fractional integral) which allows us to consider integration and differentiation of any order, not necessarily integer, have been the object of extensive study for analyzing not only stochastic processes driven by fractional Brownian motion, but also nonrandom fractional phenomena in physics and optimal control (cf. e.g., [1, 12, 13]). One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional integration or fractional differentiation (see, e.g., [1, 14–17]). Let us point out that many phenomena in engineering, physics, economy, chemistry, aerodynamics, and electrodynamics of complex medium can be modeled by this class of equations.
where , is a generator of a solution operator , , and stand for the right and left limits of at , respectively, and , are appropriate functions to be specified later. As can be seen, the convolution integral in (1.3) is the Riemann-Liouville fractional integral, and the function constitutes a nonlocal condition.
As usual, the solution with the points of discontinuity at the moments follows that , that is, at which it is continuous from the left.
We mention that 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., [2, 18–23] and references therein).
where , is a -almost sectorial operator (not necessarily densely defined).
In this work, motivated by the above contributions, we shall combine these earlier work and extend the study to the Cauchy problem (1.3). New existence and uniqueness results in the case when is a generator of a solution operator, under various criterions, are proved. In the last part of paper, we construct an example to illustrate the feasibility of our results.
with , , and let be the restriction of a function to .
It is easy to see is a Banach space.
holds for all .
Motivated by the above consideration, we give the following definition.
where is the solution operator generated by .
We list the following basic assumptions of this paper.
is continuous in on and there exists a constant such that
for all .
is continuous and there exists a function such that
for all .
is completely continuous and there exists a continuous nondecreasing function such that for each ,
is Lipschitz continuous with Lipschitz constant .
For , is Lipschitz continuous with Lipschitz constant .
For , is completely continuous and there exists a continuous nondecreasing function such that for each ,
The following fixed-point theorem plays a key role in the proof of our main results.
Lemma 2.2 (see ).
Let be a convex, bounded, and closed subset of a Banach space and let be a condensing map. Then, has a fixed point in .
To set the framework for our main existence results, we will make use of the following lemma.
Assume that is a mild solution of (2.14) in the sense of Definition 2.1. Obviously, if , then one sees from Definition 2.1, that the assertion of theorem remains true. Thus, the rest proof of the theorem is done under .
This proves, for the case , that the conclusion of theorem holds.
here and are given by (3.2) and (3.3) with , respectively. A continuation of the same process shows that for any , the assertion of theorem holds.
In this work, we adopt the following concept of mild solution for the problem (1.3).
here , is called a mild solution of the Cauchy problem (1.3).
Note that if there is no discontinuity, that is, if , , then Definition 2.1 is equivalent to Definition 3.2.
Now we present and prove our main results.
Then it is clear that is well defined.
To prove the theorem, it is sufficient to prove that has a fixed point in .
for as selected below.
which contradicts (3.12).
maps into , here is a positive number yet to be determined, as the cases for other subintervals are the same.
This is a contradiction to (3.12). Thus, we prove that there exists an integer such that .
which means that is a contraction due to (3.12).
Thus, is a condensing map on . Then, it follows from Lemma 2.2 that the Cauchy problem (1.3) admits at least one mild solution. This completes the proof.
This is a contradiction to (3.25).
Next, we will verify that for each , is a completely continuous operator, while, is a contraction. Obviously, by assumptions , it easily seen that is a completely continuous operator. Moreover, by a similar proof with that in Theorem 3.4, we can prove that is a contraction.
As a consequence of the above discussion and Lemma 2.2, we can conclude that the problem (1.3) admits at least one mild solution. The proof is completed.
which implies is a contractive mapping on due to (3.29). Thus has a unique fixed point , this means that is a mild solution of (1.3). This completes the proof of the theorem.
In this section, we present an example to illustrate the abstract results of this paper, which do not aim at generality but indicate how our theorems can be applied to concrete problems.
Clearly is densely defined in and is sectorial of type . Hence is a generator of a solution operator satisfying the estimate (2.7) on . Here, without lost of generality, we take .
which implies that one can choose large enough such that the first inequality of (3.29) is satisfied. Hence, according to Theorem 3.6, the Cauchy problem (4.1) has a unique mild solution.
This research was supported in part by the NSF of JiangXi Province of China (2009GQS0018) and the Youth Foundation of JiangXi Provincial Education Department of China (GJJ10051).
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Cuevas C, César de Souza J:Existence of -asymptotically -periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Analysis. Theory, Methods & Applications 2010,72(3-4):1683-1689. 10.1016/j.na.2009.09.007View ArticleMathSciNetMATHGoogle Scholar
- Lizama C: Regularized solutions for abstract Volterra equations. Journal of Mathematical Analysis and Applications 2000,243(2):278-292. 10.1006/jmaa.1999.6668MathSciNetView ArticleMATHGoogle Scholar
- Batty CJK, Liang J, Xiao T-J: On the spectral and growth bound of semigroups associated with hyperbolic equations. Advances in Mathematics 2005,191(1):1-10. 10.1016/j.aim.2004.01.005MathSciNetView ArticleMATHGoogle Scholar
- Engel K-J, Nagel R: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Volume 194. Springer, New York, NY, USA; 2000:xxii+586.MATHGoogle Scholar
- Fattorini HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985:xiii+314.Google 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
- Müller C: Solving abstract Cauchy problems with closable operators in reflexive spaces via resolvent-free approximation. Forum Mathematicum 2007,19(1):1-18. 10.1515/FORUM.2007.001MathSciNetView 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: A solution to an open problem for wave equations with generalized Wentzell boundary conditions. Mathematische Annalen 2003,327(2):351-363. 10.1007/s00208-003-0457-2MathSciNetView ArticleMATHGoogle Scholar
- Xiao T-J, Liang J: Complete second order differential equations in Banach spaces with dynamic boundary conditions. Journal of Differential Equations 2004,200(1):105-136. 10.1016/j.jde.2004.01.011MathSciNetView ArticleMATHGoogle Scholar
- Anh VV, Leonenko NN: Spectral analysis of fractional kinetic equations with random data. Journal of Statistical Physics 2001,104(5-6):1349-1387.MathSciNetView ArticleMATHGoogle Scholar
- Anh VV, Mcvinish R: Fractional differential equations driven by Lévy noise. Journal of Applied Mathematics and Stochastic Analysis 2003,16(2):97-119. 10.1155/S1048953303000078MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis. Theory, Methods & Applications 2008,60(10):3337-3343.MathSciNetView ArticleMATHGoogle 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.2007.08.042MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Jiao F: Existence of mild solutions for fractional neutral evolution equations. Computers & Mathematics with Applications 2010,59(3):1063-1077.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Jiao F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Analysis. Real World Applications 2010,11(5):4465-4475. 10.1016/j.nonrwa.2010.05.029MathSciNetView ArticleMATHGoogle Scholar
- Cuesta E: Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete and Continuous Dynamical Systems. Series A 2007, 277-285.Google 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. 10.1080/00036810802713875MathSciNetView ArticleMATHGoogle Scholar
- Guo DJ, Liu X: Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1993,177(2):538-552. 10.1006/jmaa.1993.1276MathSciNetView 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, 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
- 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: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Journal of Mathematical Analysis and Applications 1991,162(2):494-505. 10.1016/0022-247X(91)90164-UMathSciNetView 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
- Deng K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. Journal of Mathematical Analysis and Applications 1993,179(2):630-637. 10.1006/jmaa.1993.1373MathSciNetView ArticleMATHGoogle 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
- Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations. Volume 14. World Scientific Publishing, River Edge, NJ, USA; 1995:x+462.MATHGoogle Scholar
- Benchohra M, Gatsori EP, Henderson J, Ntouyas SK: Nondensely defined evolution impulsive differential inclusions with nonlocal conditions. Journal of Mathematical Analysis and Applications 2003,286(1):307-325. 10.1016/S0022-247X(03)00490-6MathSciNetView ArticleMATHGoogle Scholar
- Fan ZB: 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, 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.2008.05.046MathSciNetView ArticleMATHGoogle Scholar
- Wang R-N, Li Z-Q, Ding X-H: Nonlocal Cauchy problems for semilinear evolution equations involving almost sectorial operators. Indian Journal of Pure and Applied Mathematics 2008,39(4):333-346.MathSciNetMATHGoogle Scholar
- Bajlekova EG: Fractional evolution equations in Banach spaces, Ph.D. thesis. Eindhoven University of Technology; 2001.MATHGoogle Scholar
- Sadovskiĭ BN: On a fixed point principle. Akademija Nauk SSSR 1967,1(2):74-76.MathSciNetMATHGoogle Scholar
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.