Abstract
Using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral secondorder functional integrodifferential equations with infinite delay in Banach spaces. The compactness condition of the impulsive term, some restrictive conditions on a priori estimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As applications, some examples are provided to illustrate the obtained results.
MSC: 34K30, 34K40, 35R10, 47D09.
Keywords:
impulsive partial neutral functional integrodifferential equations; mild solutions; fixed point; Banach spaces1 Introduction
Consider the following impulsive partial neutral secondorder functional integrodifferential systems with infinite delay in a Banach space X:
where A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, , on X. In both cases, the history , and belongs to some abstract phase space ß defined axiomatically; g, f, , , () are appropriate functions; are fixed numbers and the symbol represents the jump of the function x at , which is defined by , .
The study of impulsive functional differential equations is linked to their utility in simulating processes and phenomena subject to shorttime perturbations during their evolution. The perturbations are performed discretely and their duration is negligible in comparison with the total duration of the processes and phenomena. Now impulsive partial neutral functional differential equations have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. With regard to this matter, we refer the reader to [112] and references therein. However, in order to obtain the existence of solutions in these study papers, the compactness condition on the associated family of operators and the impulsive term, some similar restrictive conditions on a priori estimation,
are used. In [1316], authors used a strict set contraction mapping fixed point theorem without the compactness assumption on the associated family of operators to obtain the existence results of system (1) when is not an integral operator and the following system:
improved and generalized some results in [1,7]. However, the compactness condition of the impulsive terms , some similar restrictive conditions on a priori estimation (3), (4) and the restrictive condition on measure of noncompactness estimation
are used in [1316]. So far we have not seen the existence results of system (2).
In this paper, using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions of impulsive partial neutral secondorder functional integrodifferential systems (1) and (2). The compactness condition of impulsive terms , some restrictive conditions on a priori estimation and measure of noncompactness estimation (3), (4) and (6) have been deleted. Our conditions are simple and our results essentially improve and extend some corresponding results in papers [1,2,13,14]. As applications, some examples are provided to illustrate the obtained results.
2 Preliminaries
In this paper, X is a Banach space with the norm and A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, , on X and is the sine function associated with , which is defined by , , . We designate by N, certain constants such that and for every . We refer the reader to [17] for the necessary concepts about cosine functions. Next, we only mention a few results and notations needed to establish our results. As usual we denote by the domain of A endowed with the graph norm , . Moreover, the notation E stands for the space formed by the vector , for which the function is of class . It was proved by Kisyński [18] that the space E endowed with the norm
is a Banach space. The operatorvalued function is a strongly continuous group of linear operators on the space generated by the operator defined on . It follows from this that is a bounded linear operator and that () for each . Furthermore, if is a locally integrable function, then defines an Evalued continuous function. This is a consequence of the fact that
defines an valued continuous function. Next, we denote , in which stands for the Banach space of bounded linear operators from E into X, and we abbreviate this notation to when .
To describe appropriately our system (1), we say that the function is a normalized piecewise continuous function on if u is piecewise continuous and left continuous on . We denote by the space formed by the normalized piecewise continuous functions from into X. In particular, we introduce the space PC formed by all functions such that u is continuous at , and exists for all . It is clear that PC endowed with the norm is a Banach space.
Then . Moreover, for and , we use the notation for . From Lemma 1.1 in [1], we know that a set is relatively compact if and only if each set is relatively compact in ().
For system (2), we give the precise meaning of the derivative in (2). We say that is piecewise smooth if x is continuously differentiable at , , and for , , there are the right derivative and the left derivative . Furthermore, we denote the space by . Then endowed with the norm is a Banach space.
In this work we employ an axiomatic definition of the phase space ß introduced by Hale and Kato [19] which appropriated to treat retarded impulsive differential equations. For other abstract phase spaces, we can refer to [20,21].
Definition 2.1[19]
The phase space ß is a linear space of functions mapping into X endowed with a seminorm . We assume that ß satisfies the following axioms.
(A) If () is such that and , then for every the following conditions hold:
(iii) , where is a constant; , K is continuous, M is locally bounded and H, K, M are independent of .
(B) The space ß is complete.
In this paper we denote by the Kuratowski measure of noncompactness of X, by the Kuratowski measure of noncompactness of and by the Kuratowski measure of noncompactness of PC.
The following lemma is easy to get.
Lemma 2.2If the cosine function family, , is equicontinuous and, then the set
(1) Ifis bounded, thenfor any, where.
(2) IfWis piecewise equicontinuous on, thenis piecewise continuous forand.
(3) Ifis bounded and piecewise equicontinuous, thenis piecewise continuous forand
(4) Ifis bounded and the elements ofare equicontinuous on each (), then
wheredenotes the Kuratowski measure of noncompactness in the space.
Lemma 2.4[23]
Letbe an integrable function such that. Then the functionbelongs to, the functionis integrable onforand
Lemma 2.5[24]
Let. If there is () such thatforand a.e. , thenand
Lemma 2.6[25] (Mónch)
LetXbe a Banach space, Ω be a bounded open subset inXand. Assume that the operatoris continuous and satisfies the following conditions:
3 Main results
Firstly, we discuss the existence of mild solutions for the impulsive secondorder system (1).
Definition 3.1 A function is said to be a mild solution of system (1) if , and
For system (1), we make the following hypotheses.
(H_{1}) The functions () satisfy the following conditions:
(1) For every , are strongly measurable and are continuous for every ;
(2) There are integrable functions () such that
(3) For any bounded set , there are integrable functions () such that
(H_{2}) (, ) satisfies the following conditions:
(1) For every , are strongly measurable and are continuous for every ;
(2) There are continuous functions () such that
(3) For any bounded set , there are continuous functions () such that
(H_{3}) The functions (, ) are continuous and there are constants , such that
Let the function be defined by and
Theorem 3.2Suppose that the cosine function family, , is equicontinuous, , satisfy the condition (H_{1}), (H_{2}) and (H_{3}) are satisfied. Then the impulsive secondorder system (1) has at least one mild solution.
Proof Let be the space endowed with the supremum norm . The map is defined by
Clearly, , where , , . Thus F is well defined with values in . In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (H_{1}), (H_{2}) and (H_{3}), we can show that F is continuous (see [5]). It is easy to see that if x is a fixed point of F, then is a mild solution of system (1).
Firstly, we show that the set
is bounded. In fact, if , then there exists a such that .
When , notice that and is continuous nondecreasing on . We have, by (8) and (H_{1}),
Consequently,
By wellknown Gronwall’s lemma and (10), there are constants independent of x and such that and , . It follows from this and the condition (H_{3}) that
Then . Similar to (10), we get
where . Equation (11) implies that
where . Using Gronwall’s lemma once again and (12), there is a constant independent of v and such that , . Thence and for .
It is similar to the proof above, there is a constant independent of x and such that , (). Let , then , , i.e., is bounded.
Lastly, we verify that all the conditions of Lemma 2.6 are satisfied. Let and
Then is a bounded open set and . Since , we know that for any and .
Nextly, let be a countable set and . Then
It follows from (H_{1})(H_{3}) and Lemma 2.2 that is equicontinuous on every interval (), which together with (13) implies that V is equicontinuous on every ().
When , by the property of noncompactness measure, (H_{1})(3), (H_{2})(3) and Lemma 2.5, we have
where . Let , . Lemma 2.3 implies that and
From this and Gronwall’s lemma, we know that and , . Therefore V is a relative compact set in . Since
When , similar to (14), it is easy to get
where . Let , . Equation (15) implies that
Therefore , and V is a relative compact set in .
Similarly, we can show that V is a relative compact set in (), so V is a relative compact set in . Lemma 2.6 concludes that F has a fixed point in . Let x be a fixed point of F on . Then is a mild solution of system (1). □
Nextly, we discuss the existence of mild solutions for the impulsive system (2).
Definition 3.3 A function is said to be a mild solution of system (2) if , , and
Differentiate (16) to get
Let functions be defined by , and
Clearly,
Let be the space endowed with the supremum norm .
We make the following hypotheses for convenience.
(H_{f}) satisfies the following conditions:
(1) For every , the function is strongly measurable and is continuous for every ;
(2) There is an integrable function such that
(3) For any bounded set , there is an integrable function such that
(H_{g}) satisfies the following conditions:
(1) The function is continuous, there are constants , such that and
(2) For every bounded set , the set of functions is uniformly equicontinuous on for every , where ;
(H_{I}) The functions , () are continuous and there are constants , such that
Theorem 3.4Let the conditions (H_{f}), (H_{g}) and (H_{I}) be satisfied, the cosine function family, , be equicontinuous and. Then system (2) has at least one mild solution.
Proof Let the function be defined by , , , the map be defined by
The product space is endowed with the norm . Then , are well defined and with values in . In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (H_{f}), (H_{g}) and (H_{I}), we can show that is continuous. It is easy to see that if is a fixed point of Γ, then is a mild solution of system (2).
Firstly, we show that the set
is bounded. If , there exists a such that and .
When , it follows from (18), (19) and (H_{f})(2), (H_{g})(1), (H_{I}) that
Equations (20) and (21) imply that
Since , by Gronwall’s lemma and (22), there is a constant such that , . Therefore , and , , . It follows from this and the condition (H_{I}) that
Then . Similar to (20) and (21), we get
We have, by (23) and (24),
where
Using Gronwall’s lemma once again and (25), there is a constant such that , , and so , .
It is similar to the proof above, there are constants such that , (). Let , then and is bounded.
Then is a bounded open set and . Since , we know that for any and .
Suppose that is a countable set and . Let
Then we have
It follows from (18), (19) and (H_{g})(2) that () are equicontinuous on every interval (), which together with (26) implies that () are equicontinuous on every interval .
In the following, we verify that the set , is relatively compact in PC. Without loss of generality, we do not distinguish and , where () is the restriction of on .
When , by the condition (H_{f})(3), (H_{g})(3) and Lemma 2.5, we have
Since () are continuous nondecreasing on , (27) and (28) imply that
By Gronwall’s lemma and (29), we have (), . Lemma 2.3 implies that () is relatively compact in . Note that and () is continuous, we have
When , it is similar to (27) and (28), we get
Equations (30) and (31) imply that
Consequently, (), . Lemma 2.3 implies that () are relatively compact in .
Similarly, we can show that () are relatively compact in (). So () are relatively compact in . In view of Lemma 2.6, we conclude that Γ has a fixed point in . Let be a fixed point of Γ on . Then is a mild solution of system (2). □
Theorem 3.5Let the conditions (H_{f}), (H_{g})(2) and (H_{I}) be satisfied, the cosine function family, , be equicontinuous and. Furthermore, suppose that the following condition is satisfied:
(H_{g})(1′) The functionis continuous andsatisfies the Lipschitz condition, that is, there is a constantsuch that
and. Then system (2) has at least one mild solution.
Proof We have, by the condition (H_{g})(1′),
The rest of the proof is similar to the proof of Theorem 3.4, we omit it. □
Remark 3.6 The similar restrictive conditions (3), (4) and (6) were used in [1,2,13,14] even if .
4 Examples
Let and let A be the operator given by with the domain
It is well known that A is the infinitesimal generator of a strongly continuous cosine family , , on X. Moreover, A has discrete spectrum, the eigenvalues are , with corresponding normalized eigenvectors , and the following properties hold:
(a) is an orthonormal basis of X.
(b) For , . Moreover, it follows from this expression that , that is compact for and that and for every . Additionally, we observe that the operators , , are not compact.
(c) If Φ denotes the group of translations on X defined by , where is the extension of x with period 2π, then ; , where B is the infinitesimal generator of the group Φ and ; see [1] for details. In particular, we observe that the inclusion is compact.
In the next application, ß should be the phase space in [1], where is a positive Lebesgue integrable function. We can take , and for .
Example 4.1 Consider the partial neutral functional integrodifferential system:
(i) The function is continuous and there is a continuous function such that , .
(iv) The function φ defined by belongs to ß.
Assuming that the conditions (i)(iv) are satisfied, then system (32) can be modeled as the abstract impulsive Cauchy problem (1) by defining
where
Moreover, , (, ) and are bounded linear operators, (, ),
where . If the cosine function family , , is equicontinuous, all the conditions of Theorem 3.2 are satisfied (see [1] for details), so system (32) has at least one mild solution. However, if we select , in (iii), we have . But
the restrictive conditions (3), (4) and (6) do not hold. Thus, our results are different from the corresponding known results.
Example 4.2 Consider the partial neutral functional integrodifferential system:
(v) The functions , are measurable, and
(vi) The function is continuous and .
Assuming that the conditions (ii)(vi) are satisfied, system (33) can be modeled as the abstract Cauchy problem (2) by defining
where , (, ), are bounded linear operators and
Moreover, for every bounded set , it follows from (34) and the proof in [26] that the set of functions is uniformly equicontinuous on for every . If the cosine function family , , is equicontinuous, all the conditions of Theorem 3.4 are satisfied, so system (33) has at least one mild solution.
Remark 4.3 From the results of this paper, we know that the compactness condition of the impulsive term, the restrictive conditions on a priori estimation and noncompactness measure estimation can be deleted for the existence results of abstract impulsive functional Volterra integrodifferential equations Cauchy problems.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author is grateful to the referees for their careful reading and helpful suggestions that have led to the present improved version of the original paper. The work was supported by the Natural Science Foundation of Anhui Province (11040606M01), Anhui Educational Committee (KJ2011A061, KJ2011B052), China.
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