Research Article | Open Access
An Existence Result for Nonlocal Impulsive Second-Order Cauchy Problems with Finite Delay
We deal with the existence of mild solutions of a class of nonlocal impulsive second-order functional differential equations with finite delay in a real Banach space . An existence result on the mild solution is obtained by using the theory of the measures of noncompactness. An example is presented.
On the other hand, the impulsive conditions have advantages over traditional initial value problems because they can be used to model phenomena that cannot be modeled by traditional initial value problems, such as the dynamics of populations subject to abrupt changes (harvesting, diseases, etc.) (see [16–27] and references therein). For this reason, the theory of impulsive differential equations has become an important area of investigation in recent years. Partial differential equations of first and second order with impulses have been studied by Rogovchenko , Liu , Cardinali and Rubbioni , Liang et al. , Henríquez and Vásquez , Hernández et al., [21–23], Arthi and Balachandran , and so forth.
Moreover, we consider the nonlocal condition , where is a mapping from some space of functions so that it constitutes a nonlocal condition (see [24, 28–30] and the references therein), where it is demonstrated that nonlocal conditions have better effects in applications than traditional initial value problems.
In this paper, we pay our attention to the investigation of the existence of mild solutions to the following impulsive second-order functional differential equations with finite delay in a real Banach space : where is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on . , are given functions to be specified later. , where denotes the space of all continuous functions from to .
The impulsive moments are given such that , are appropriate functions, represents the jump of a function at , which is defined by , where and are, respectively, the right and the left limits of at .
For any continuous function defined on the interval and any , we denote by the element of defined by for .
In this paper, motivated by above works, we study (1)–(4) in and obtain the existence theorem based on theory on measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the cosine family of bounded linear operators generated by is compact.
Throughout this paper, we set , a compact interval in . We denote by a Banach space with norm , by the Banach space of all linear and bounded operators on . We abbreviate with , for any .
It is easy to check that is a Banach space with the norm
We let , .
For , we denote by the set .
A family in is called a cosine function on if (i) is the identity operator in ; (ii) for all ; (iii)The map is strongly continuous for each . The associated sine function is the family of operators defined by
One can define the infinitesimal generator of by
Next, we recall that the Hausdorff measure of noncompactness on each bounded subset of Banach space is defined by
This measure of noncompactness satisfies some basic properties as follows.
Lemma 1 (see ). Let be a real Banach space, and let be bounded. Then (1) if and only if is precompact; (2), where and mean the closure and convex hull of , respectively; (3) if ; (4); (5), where ; (6), for any ; (7)let be a Banach space and Lipschitz continuous with constant . Then for all being bounded.
Proposition 2 (see , Page 125). Let be a bounded set for a real Banach space . Then, for every there exists a sequence in such that
Proposition 3. Let be a sequence in such that there exist with the properties:(i); (ii). Then, for every , we have where is from (10).
A continuous map is said to be a -contraction if there exists a positive constant such that for any bounded closed subset .
Theorem 4 (see  (Darbo-Sadovskii)). If is bounded closed and convex, the continuous map is a -contraction, then the map has at least one fixed point in .
3. Existence Result and Proof
Let stand for the space endowed with norm
We will require the following assumptions.
(H1) satisfies is measurable for all and is continuous for a.e. , and there exists a function such that for almost all ;
there exists a function such that for any bounded sets , (H2) are compact operators and there exist positive constants such that (H3) is a compact operator and there exists a constant such that where . (H4) There exists such that .
Proof. Define the operator in the following way:
It is clear that the operator is well defined, and the fixed point of is the mild solution of problems (1)–(4).
The operator can be written in the form , where the operators are defined as follows:
Obviously, under the assumptions of , is continuous. For , we can prove that is continuous.
Indeed, let be a sequence such that in as . Since satisfies (H1)(i), for almost every , we get
Noting that in , we can see that there exists such that for sufficiently large. Therefore, we have It follows from the Lebesgue’s dominated convergence theorem that Moreover, noting that (H2), we obtain that This shows that is continuous. Therefore, is continuous.
Let us introduce in the space the equivalent norm defined as where is a constant chosen so that Noting that for any , we have so, we can take the appropriate to satisfy (29).
Consider the set where is a constant chosen so that where and .
Now, if , , then For , , we have then therefore, It results that
Let , we obtain Hence for some positive number , .
Using the strong continuity of and the compactness condition on the operators , for , there exists such that when . If and such that , then
For , by the hypothesis (H1)(i) and (40), we get As and , the right-hand side of the inequality above tends to zero independent of , so maps bounded sets into equicontinuous sets.
For bounded set , we consider the map where is the Hausdorff measure of noncompactness on the Banach space and is defined in (7).
Furthermore, we define the Hausdorff measure of noncompactness on as follows:
For every bounded subset , by applying Proposition 2, for any there exists a sequence such that noting that the definition of , we have Then, noting the equicontinuity of , , we can apply Lemmas 2.1 and 2.2 of  to obtain Then from (45) and (46), we have
For every bounded subset , we have moreover, by applying Proposition 2, for any there exists a sequence such that Combining with (48) and (49), we have Using the induction of (45)–(47) above, we can see Thus, from (51), (H2) and Proposition 3 and (3) in Lemma 1, we can see where .
Noting that Thus, by (52), we see
Since is arbitrary, we can obtain
Combining with (H3), we have hence is a -contraction on . According to Theorem 4, the operator has at least one fixed point in . This completes the proof.
Next, we establish a condition that guarantee that a mild solution satisfies (3).
Proof. Clearly, as . Moreover, noting that and (11), we have is of class . Therefore, we can see that which shows the assertion.
In this section, we consider an application of the theory developed in Section 3 to the study of an impulsive partial differential equation with unbounded delay.
Example 9. , is the map defined by with domain .
We consider the following integrodifferential model: where , , , , and . and satisfy the following assumptions. (1)The function is a continuous function and . (2)The function is measurable and there exists a constant such that . (3)For every , the function , is measurable and there exists a constant such that
To treat the above problem, we define is the infinitesimal generator of a strongly continuous cosine function on . Moreover, has a discrete spectrum, the eigenvalues are , with the corresponding normalized eigenvectors ; the set is an orthonormal basis of and the following properties hold. (a)If , then . (b)For each , and . Consequently, for , and is compact for every .
For and , we set Then the above equation (59) can be reformulated as the abstract (1)–(4).
For , we can see where .
For any , ,
Therefore, for any bounded sets , , we have where .
Suppose further that there exists a constant such that and , then (59) has at least a mild solution by Theorem 7.
This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M), the Educational Commission of Yunnan Province (2012Z010) and the Foundation of Key Program of Yunnan Normal University.
- H. R. Henríquez and C. H. Vásquez, “Differentiability of solutions of second-order functional differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 284–312, 2003.
- F. Li, “Solvability of nonautonomous fractional integrodifferential equations with infinite delay,” Advances in Difference Equations, vol. 2011, Article ID 806729, 18 pages, 2011.
- F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,” Electronic Journal of Qualitative Theory of Differential Equations, no. 52, pp. 1–15, 2011.
- J. Liang and T. J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004.
- T. J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442–e1447, 2009.
- H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.
- J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1985.
- J. Kisyński, “On cosine operator functions and one-parameter groups of operators,” Studia Mathematica, vol. 44, no. 1, pp. 93–105, 1972.
- J. Liang and T. J. Xiao, “A characterization of norm continuity of propagators for second order abstract differential equations,” Computers & Mathematics with Applications, vol. 36, no. 2, pp. 87–94, 1998.
- J. Liang, R. Nagel, and T. J. Xiao, “Approximation theorems for the propagators of higher order abstract Cauchy problems,” Transactions of the American Mathematical Society, vol. 360, no. 4, pp. 1723–1739, 2008.
- C. C. Travis and G. F. Webb, “Second order differential equations in Banach space,” in Nonlinear Equations in Abstract Spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977), pp. 331–361, Academic Press, New York, NY, USA, 1978.
- T. J. Xiao and J. Liang, “Second order linear differential equations with almost periodic solutions,” Acta Mathematica Sinica (New Series), vol. 7, no. 4, pp. 354–359, 1991.
- T. J. Xiao and J. Liang, “Differential operators and -wellposedness of complete second order abstract Cauchy problems,” Pacific Journal of Mathematics, vol. 186, no. 1, pp. 167–200, 1998.
- T. J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
- T. J. Xiao and J. Liang, “Higher order abstract Cauchy problems: their existence and uniqueness families,” Journal of the London Mathematical Society, vol. 67, no. 1, pp. 149–164, 2003.
- N. U. Ahmed, “Optimal feedback control for impulsive systems on the space of finitely additive measures,” Publicationes Mathematicae Debrecen, vol. 70, no. 3-4, pp. 371–393, 2007.
- G. Arthi and K. Balachandran, “Controllability of second-order impulsive functional differential equations with state-dependent delay,” Bulletin of the Korean Mathematical Society, vol. 48, no. 6, pp. 1271–1290, 2011.
- M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006.
- T. Cardinali and P. Rubbioni, “Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 1, pp. 73–84, 2008.
- T. Cardinali and P. Rubbioni, “Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 2, pp. 871–879, 2012.
- E. Hernández, K. Balachandran, and N. Annapoorani, “Existence results for a damped second order abstract functional differential equation with impulses,” Mathematical and Computer Modelling, vol. 50, no. 11-12, pp. 1583–1594, 2009.
- E. Hernández M., H. R. Henríquez, and M. A. McKibben, “Existence results for abstract impulsive second-order neutral functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2736–2751, 2009.
- E. Hernández M. and S. M. T. Aki, “Global solutions for abstract impulsive differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1280–1290, 2010.
- J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.
- J. H. Liu, “Nonlinear impulsive evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 77–85, 1999.
- Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 1997.
- S. T. Zavalishchin, “Impulse dynamic systems and applications to mathematical economics,” Dynamic Systems and Applications, vol. 3, no. 3, pp. 443–449, 1994.
- J. Liang, J. Liu, and T. J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 2, pp. 183–189, 2004.
- J. Liang and T. J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,” Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
- T. J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e225–e232, 2005.
- J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
- D. Bothe, “Multivalued perturbations of -accretive differential inclusions,” Israel Journal of Mathematics, vol. 108, pp. 109–138, 1998.
- M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001.
- T. Cardinali and P. Rubbioni, “On the existence of mild solutions of semilinear evolution differential inclusions,” Journal of Mathematical Analysis and Applications, vol. 308, no. 2, pp. 620–635, 2005.
- A. Ambrosetti, “Un teorema di esistenza per le equazioni differenziali negli spazi di Banach,” Rendiconti del Seminario Matematico della Università di Padova, vol. 39, pp. 349–361, 1967.
Copyright © 2013 Fang Li and Huiwen Wang. 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.