Abstract
This paper is concerned with the existence of mild solutions to impulsive differential equations with nonlocal conditions. We firstly establish a property of the measure of noncompactness in the space of piecewise continuous functions. Then, by applying this property and DarboSadovskii’s fixed point theorem, we get the existence results of impulsive differential equations in a unified way under compactness conditions, Lipschitz conditions and mixedtype conditions, respectively.
MSC: 34K30, 34G20.
Keywords:
impulsive conditions; nonlocal conditions; Hausdorff measure of noncompactness; fixed point theorem1 Introduction
In this paper, we discuss the existence of mild solutions for the following impulsive differential equation with nonlocal conditions:
where
Impulsive differential equations are recognized as excellent models to study the evolution processes that are subject to sudden changes in their states; see the monographs of Lakshmikantham et al.[1], Benchohra et al.[2]. In recent years impulsive differential equations in Banach spaces have been investigated by many authors; see [38] and references therein. Liu [9] discussed the existence and uniqueness of mild solutions for a semilinear impulsive Cauchy problem with Lipschitz impulsive functions. NonLipschitzian impulsive equations are considered by Nieto et al.[10]. Cardinali and Rubbioni [11] proved the existence of mild solutions for the impulsive Cauchy problem controlled by a semilinear evolution differential inclusion. In [12], Abada et al. studied the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions.
On the other hand, the study of abstract nonlocal initial value problems was initiated
by Byszewski, and the importance of the problem consists in the fact that it is more
general and has better effect than the classical initial conditions
From the viewpoint of theory and practice, it is natural for mathematics to combine
impulsive conditions and nonlocal conditions. Recently, the nonlocal impulsive differential
problem of type (1.1) has been discussed in the papers of Liang et al.[23] and Fan et al.[24,25], where a semigroup
This paper is organized as follows. In Section 2, we present some concepts and facts about the strongly continuous semigroup and the measure of noncompactness. In Section 3, we give four existence theorems of the problem (1.1) by using a condensing operator and the measure of noncompactness. At last, an example of an impulsive partial differential system is given in Section 4.
2 Preliminaries
Let
The semigroup
(HA) The semigroup
For the sake of simplicity, we put
Definition 2.1 A function
for all
Now, we introduce the Hausdorff measure of noncompactness (in short MNC)
for each bounded subset B in a Banach space X. We recall the following properties of the Hausdorff measure of noncompactness β.
Lemma 2.2 ([28])
LetXbe a real Banach space and
(1) Bis relatively compact if and only if
(2)
(3)
(4)
(5)
(6)
(7) If the map
The map
Lemma 2.3 (See [28], DarboSadovskii)
If
In order to remove the strong restriction on the coefficient in DarboSadovskii’s
fixed point theorem, Sun and Zhang [29] generalized the definition of a βcondensing operator. At first, we give some notation. Let
where
Definition 2.4 Let
Obviously, if
Lemma 2.5 ([29])
If
Now, we give an important property of the Hausdorff MNC in
Lemma 2.6 ([28])
If
By applying Lemma 2.6, we shall extend the result to the space
Lemma 2.7If
(1) Wis equicontinuous on
(2) Wis equicontinuous at
Then
Proof For arbitrary
where
for
By the arbitrariness of ε, we get that
Next, if the conditions (1) and (2) are satisfied, it remains to prove that
and obviously
Moreover, we define the map
by
And from the fact that
Lemma 2.8 ([28])
If
for all
Lemma 2.9If the hypothesis (HA) is satisfied, i.e.,
Proof We let
If
Here, as
uniformly for u.
Then from (2.1), (2.2) and the absolute continuity of integrals, we get that
3 Main results
In this section we give the existence results for the problem (1.1) under different
conditions on g and
Let r be a finite positive constant, and set
with
for all
We list the following hypotheses:
(Hf)
(i)
for a.e.
(ii) there exists a constant
for a.e.
(Hg1)
(HI1)
Theorem 3.1Assume that the hypotheses (HA), (Hf), (Hg1), (HI1) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on
Proof We will prove that the solution map G has a fixed point by using the fixed point theorem about the βconvexpower condensing operator.
Firstly, we prove that the map G is continuous on
Then by the continuity of g,
Secondly, we claim that
for each
Now, we show that
Thus,
which implies the equicontinuity of
Set
for every nonprecompact bounded subset
From Lemma 2.2 and Lemma 2.8, noticing the compactness of g and
for
for
for
By the fact that
which implies that
Remark 3.2 By using the method of the measure of noncompactness, we require f to satisfy some proper conditions of MNC, but do not require the compactness of a
semigroup
Remark 3.3 When we apply DarboSadovskii’s fixed point theorem to get the fixed point of a map, a strong inequality is needed to guarantee its condensing property. By using the βconvexpower condensing operator developed by Sun et al.[29], we do not impose any restrictions on the coefficient L. This generalized condensing operator also can be seen in Liu et al.[30], where nonlinear Volterra integral equations are discussed.
In the following, by using Lemma 2.7 and DarboSadovskii’s fixed point theorem, we give the existence results of the problem (1.1) under Lipschitz conditions and mixedtype conditions, respectively.
We give the following hypotheses:
(Hg2)
(HI2)
for
Theorem 3.4Assume that the hypotheses (HA), (Hf), (Hg2), (HI2) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on
and (3.3) are satisfied.
Proof From the proof of Theorem 3.1, we have that the solution operator G is continuous and maps
By the conditions (Hg2) and (HI2), we get that
Thus, from Lemma 2.2(7), we obtain that
For the operator
Combining (3.5) and (3.6), we have
From the condition (3.4),
Among the previous works on nonlocal impulsive differential equations, few are concerned with the mixedtype conditions. Here, by using Lemma 2.7, we can also deal with the mixedtype conditions in a similar way.
Theorem 3.5Assume that the hypotheses (HA), (Hf), (Hg1), (HI2) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on
and (3.3) are satisfied.
Proof We will also use DarboSadovskii’s fixed point theorem to obtain a fixed point of
the solution operator G. From the proof of Theorem 3.1, we have that G is continuous and maps
Subsequently, we show that G is βcondensing in
On the other hand, for
Then by Lemma 2.2(7), we obtain that
Combining (3.6), (3.8) and (3.9), we get that
From the condition (3.7), the map G is βcondensing in
Theorem 3.6Assume that the hypotheses (HA), (Hf), (Hg2), (HI1) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on
and (3.3) are satisfied.
Proof From the proof of Theorem 3.1, we have that the solution operator G is continuous and maps
By the Lipschitz continuity of g, we have that for
which implies that
Similar to the discussion in Theorem 3.1, from the compactness of
Combining (3.6), (3.11) and (3.12), we have that
From condition (3.10), the map G is βcondensing in
Remark 3.7 With the assumption of compactness on the associated semigroup, the existence of mild solutions to functional differential equations has been discussed in [6,2325]. By using the method of the measure of noncompactness, we deal with the four cases of impulsive differential equations in a unified way and get the existence results when the semigroup in not compact.
4 An example
In the application to partial differential equations, such as a class of parabolic
equations, the semigroup corresponding to the differential equations is an analytic
semigroup. We know that an analytic semigroup or a compact semigroup must be equicontinuous;
see Pazy [31]. So, our results can be applied to these problems. If the operator
We consider the following partial differential system (based on [23]) to illustrate our abstract results:
Take
From Pazy [31], we know that A is the infinitesimal generator of an analytic semigroup
Let
(1)
(2)
(3)
(4)
(5)
Then we obtain that
Case 1. Under the conditions (1) + (3) + (5), the assumptions in Theorem 3.1 are satisfied
for large
Case 2. Under the conditions (1) + (2) + (4), the assumptions in Theorem 3.4 are satisfied
for large
Case 3. Under the conditions (1) + (3) + (4), the assumptions in Theorem 3.5 are satisfied
for large
Case 4. Under the conditions (1) + (2) + (5), the assumptions in Theorem 3.6 are satisfied
for large
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
Research is partially supported by the National Natural Science Foundation of China (11271316), the Postgraduate Innovation Project of Jiangsu Province (No. CXZZ120890), the NSF of China (11101353), the first author is also supported by the Youth Teachers Foundation of Huaiyin Institute of Technology (2012).
References

Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, Hindawi Publishing, New York (2006)

Guo, M, Xue, X, Li, R: Controllability of impulsive evolution inclusions with nonlocal conditions. J. Optim. Theory Appl.. 120, 355–374 (2004)

Hernández, E, Rabelo, M, Henríquez, HR: Existence of solutions for impulsive partial neutral functional differential equations. J. Math. Anal. Appl.. 331, 1135–1158 (2007). Publisher Full Text

Ji, S, Wen, S: Nonlocal Cauchy problem for impulsive differential equations in Banach spaces. Int. J. Nonlinear Sci.. 10(1), 88–95 (2010)

Ji, S, Li, G: Existence results for impulsive differential inclusions with nonlocal conditions. Comput. Math. Appl.. 62, 1908–1915 (2011). Publisher Full Text

Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of firstorder differential equations. J. Math. Anal. Appl.. 325, 226–236 (2007). Publisher Full Text

Benchohra, M, Henderson, J, Ntouyas, SK: An existence result for firstorder impulsive functional differential equations in Banach spaces. Comput. Math. Appl.. 42, 1303–1310 (2001). Publisher Full Text

Liu, JH: Nonlinear impulsive evolution equations. Dyn. Contin. Discrete Impuls. Syst.. 6, 77–85 (1999)

Nieto, JJ, RodriguezLopez, R: Periodic boundary value problem for nonLipschitzian impulsive functional differential equations. J. Math. Anal. Appl.. 318, 593–610 (2006). Publisher Full Text

Cardinali, T, Rubbioni, P: Impulsive semilinear differential inclusion: topological structure of the solution set and solutions on noncompact domains. Nonlinear Anal.. 14, 73–84 (2008)

Abada, N, Benchohra, M, Hammouche, H: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ.. 246, 3834–3863 (2009). Publisher Full Text

Byszewski, L, Lakshmikantham, V: Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space. Appl. Anal.. 40, 11–19 (1990)

Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl.. 162, 494–505 (1991). Publisher Full Text

Fu, X, Ezzinbi, K: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions. Nonlinear Anal.. 54, 215–227 (2003). Publisher Full Text

Aizicovici, S, McKibben, M: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal.. 39, 649–668 (2000). Publisher Full Text

Xue, X: Nonlinear differential equations with nonlocal conditions in Banach spaces. Nonlinear Anal.. 63, 575–586 (2005). Publisher Full Text

Xue, X: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal.. 70, 2593–2601 (2009). Publisher Full Text

Banas, J, Zajac, T: A new approach to the theory of functional integral equations of fractional order. J. Math. Anal. Appl.. 375, 375–387 (2011). Publisher Full Text

Cardinali, T, Rubbioni, P: On the existence of mild solutions of semilinear evolution differential inclusions. J. Math. Anal. Appl.. 308, 620–635 (2005). Publisher Full Text

Dong, Q, Li, G: Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces. Electron. J. Qual. Theory Differ. Equ.. 2009, (2009) Article ID 47

Agarwal, RP, Benchohra, M, Seba, D: On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Results Math.. 55, 221–230 (2009). Publisher Full Text

Liang, J, Liu, JH, Xiao, TJ: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model.. 49, 798–804 (2009). Publisher Full Text

Fan, Z: Impulsive problems for semilinear differential equations with nonlocal conditions. Nonlinear Anal.. 72, 1104–1109 (2010). Publisher Full Text

Fan, Z, Li, G: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal.. 258, 1709–1727 (2010). Publisher Full Text

Zhu, L, Dong, Q, Li, G: Impulsive differential equations with nonlocal conditions in general Banach spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 10

Barbu, V: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976)

Banas, J, Goebel, K: Measure of Noncompactness in Banach Spaces, Dekker, New York (1980)

Sun, J, Zhang, X: The fixed point theorem of convexpower condensing operator and applications to abstract semilinear evolution equations. Acta Math. Sin. Chin. Ser.. 48, 439–446 (2005)

Liu, LS, Guo, F, Wu, CX, Wu, YH: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl.. 309, 638–649 (2005). Publisher Full Text

Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983)