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Existence results for third-order impulsive neutral differential equations with deviating arguments
Advances in Difference Equations volume 2014, Article number: 38 (2014)
Abstract
By using the fixed-point theorem of Leray-Schauder or Banach, we discuss the existence of solutions for third-order impulsive neutral differential equations with deviating arguments. Two examples are given to demonstrate our main results.
MSC:34A37, 34k05.
1 Introduction
Impulsive differential equations are a class of important models which describe many evolution processes that abruptly change their state at a certain moment. The theory of impulsive differential equations has become an active area of investigation in recent years due to their numerous applications for problems arising in mechanics, electrical engineering, medicine biology, ecology, and other areas of science (see, for example, [1–3] and the references therein). With regard to ordinary impulsive differential equations, we have refer to some work [1, 4–6]. Partial neutral differential equations with infinite delay have been used for modeling the evolution of physical systems in which the response of the system depends on the current state as well as the past history of the system (see, for instance, [7, 8] on the description of heat conduction in materials with fading memory). First-order abstract neutral functional differential equations with finite delay have been studied in [9–11] among others. The work on first-order abstract neutral functional differential equations with unbounded delay was initiated in [12, 13]. Relative to second-order abstract neutral differential equations, we must mention [14]. Recently, weighted pseudo almost periodic solutions to some partial neutral functional differential equations have been considered in [15]; Baghli et al. [16] have investigated sufficient conditions for the existence of mild solutions, on the positive half-line, for two classes of first-order functional and neutral functional perturbed differential equations with infinite delay. Hernández et al. [17, 18] have studied the existence of mild solutions for a class of autonomous impulsive partial neutral functional differential equations with infinite delay of first and second order. The authors in [19–30] have further investigated the existence results for many kinds of impulsive neutral differential equations.
In [19–30], the neutral terms of those equations are
respectively.
The histories , , , belong to an abstract phase space .
Motivated by [19–30], we will consider the case that the neutral term of the equation is . Obviously, the neutral term is different from the previous literature. Besides, the authors in [17, 19–30] have studied the impulsive neutral differential equations of the first or the second order, but, in this paper, we focus on the third-order impulsive neutral differential equations. Here, we introduce a real Banach space, which has been adopted by us the first time.
Our results are based on the fixed-point theorem of Leray-Schauder or Banach.
Consider the following impulsive neutral differential equations:
where , , (). Throughout this paper, we assume , θ is monotone increasing with respect to t, (), , and . Let with , . Also let be continuous everywhere except for some at which and exist and . with (). Denote by , where , , the set of all functions which are piecewise continuous in X with points of discontinuity of the first kind at the points , where the limits and .
2 Preliminaries
Note that , , , and . It is clear that and .
Let . Evidently, is a real Banach space with norm
where
Further, let . We can check that E is also a real Banach space with norm
where
Define the operator , where and . It is evident that B is topological linear isomorphic, which implies that E is a real Banach space.
Since (), i.e., , we get , next we have
Similarly, we have
So
Lemma 1 Let .
Then u is a solution of (1.1) if and only if is a solution of the following integral equation:
Proof (i) Necessity.
For (), by (1.1), we get
Adding these together, we obtain
Similarly, we obtain, respectively,
Substituting (2.5) into (2.4), it is easy to get
Substituting (2.6) into (2.3), it is easy to get (2.2).
(ii) Sufficiency.
According to (2.2), it is clear that
Differentiating both sides of (2.2), we get
It follows that
Differentiating both sides of (2.8), we get
It follows that
Differentiating both sides of (2.10), we get
From (2.7), (2.9), (2.11), and (2.12), we see that is a solution of (1.1). □
Lemma 2 (Leray-Schauder [31])
Let the operator be completely continuous, where X is a real Banach space. If the set
is bounded, then the operator A has at least one fixed point in the closed ball
where .
Lemma 3 (Compactness criterion [32])
is a relatively compact set if and only if is uniformly bounded and equicontinuous on every (), where , ().
3 Main result
Let us introduce the following conditions for later use:
(H1) There exist nonnegative constants , (in the condition, the first three formulas have , the last formulas , and also ), , , and such that
where , , , , .
(H2) There exist positive constant M such that
Theorem 1 If conditions (H1), (H2) are satisfied, and
(H3) , where
then (1.1) has at least one solution in the closed ball
where , in which
Proof (i) For any , define the operator A by
It is easy to see that . According to the properties of θ, for any , we have
Let . Next, it is clear that . It follows that A maps E into E. Thus with
A is a completely continuous operator, as will be verified by the following three steps.
Step 1. A is continuous.
Let any (), with as .
By (3.1) and (H1), we have
Then, from (3.4) and (2.1), we have
Thus
Similarly, from (3.2) and (2.1), we get
Thus
Similarly, from (3.3) and (2.1), we get
by (3.7) and the properties of θ, it follows that
so
Thus
By (3.6), (3.9), and (3.11), it is easy to see that as , that is to say, A is continuous.
Step 2. A maps any bounded subset of E into one bounded subset of E.
Let T be any bounded subset of E. Then there exists such that for all .
By (3.1), (H1), (H2), and (2.1), we have
so
Similarly, from (3.2), (H1), (H2), and (2.1), we get
so
Similarly, from (3.3), (H1), (H2), (3.13), (2.1), and the properties of θ, we get
so
According to (3.12), (3.14), and (3.15), we obtain
Therefore, is uniformly bounded.
Step 3. is equicontinuous on every (), where , ().
For any and any , take
Then if and with , from (3.1), (H1), (H2), and (2.1), we have
Thus, is equicontinuous on every ().
As a consequence of Steps 1-3, A is completely continuous.
-
(ii)
For any , similar to getting (3.12), (3.14), and (3.15), we have, respectively,
Then
where
and .
It follows that , i.e., G is bounded.
From (i) and (ii), all conditions of Lemma 2 are satisfied. Therefore, the proof is complete. □
Theorem 2 If condition (H1) ( are not needed) is satisfied, and
(H4) , where
then (1.1) has a unique solution.
Proof Similar with getting (3.5), (3.8) and (3.10), for any , we obtain, respectively,
Then
Thus all conditions of the fixed-point theorem of Banach are satisfied. The proof is complete. □
Remark 1 By comparing Theorem 1 with Theorem 2, it is easy to see that each of them has its own strong and weak points. The condition (H3) of Theorem 1 is satisfied more easily than the condition (H4) of Theorem 2. The condition (H2) of Theorem 1 is also satisfied easily, but it does not need in Theorem 2.
4 Examples
Example 1 Consider the equation
Firstly, it is easy to verify that
all satisfy the requisitions of (1.1). From and , we get that
Next, since
and
where is located between and , we have
where , , , , , . From , , , we have
where , , . Further, we have
where .
Finally, we get
Thus, (4.1) satisfies all conditions of Theorem 1. It follows that (4.1) has at least one solution in the closed ball .
Example 2 Consider the equation
Firstly, it is easy to verify that
all satisfy the requisitions of (1.1). From and , we get
Next, since
and
where is located between and , is located between and , we have
where , , , , . From , , , we have
where , , .
Finally, we get
Thus, (4.2) satisfies all conditions of Theorem 2. It follows that (4.2) has a unique solution.
References
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow; 1993.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.
He ZM, Yu JS: Periodic boundary value problem for first-order impulsive functional differential equations. J. Comput. Appl. Math. 2002, 138: 205-217. 10.1016/S0377-0427(01)00381-8
Luo ZG, Shen JH: Stability results for impulsive functional differential equations with infinite delays. J. Comput. Appl. Math. 2001, 131: 55-64. 10.1016/S0377-0427(00)00323-X
Sun JT, Zhang YP: Impulsive control of a nuclear spin generator. J. Comput. Appl. Math. 2003, 157: 235-242. 10.1016/S0377-0427(03)00454-0
Gurtin ME, Pipkin AC: A general theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 1968, 31: 113-126. 10.1007/BF00281373
Nunziato JW: On heat conduction in materials with memory. Q. Appl. Math. 1971, 29: 187-204.
Adimy M, Ezzimbi K: A class of linear partial neutral functional-differential equations with nondense domain. J. Differ. Equ. 1998, 147: 285-332. 10.1006/jdeq.1998.3446
Dako R: Linear autonomous neutral differential equations in Banach space. J. Differ. Equ. 1997, 25: 258-274.
Fu XL, Ezzimbi K: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 2003, 54: 215-227. 10.1016/S0362-546X(03)00047-6
Hernández E, HenrÃquez HR: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. J. Comput. Appl. Math. 1998, 221: 499-522.
Hernández E, HenrÃquez HR: Existence results for partial neutral functional differential equations with unbounded delay. J. Comput. Appl. Math. 1998, 221: 452-475.
Hernández E, Mckibben MA: Some comments on: existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations. Comput. Math. Appl. 2005, 50: 655-669. 10.1016/j.camwa.2005.08.001
Agarwal R, Diagana T, Hernández E: Weighted pseudo almost periodic solutions to some partial neutral functional differential equations. J. Nonlinear Convex Anal. 2007, 8: 397-415.
Baghli S, Benchohra M: Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces. Electron. J. Differ. Equ. 2008., 2008: Article ID 69
Hernández E, Rabelo M, HenrÃquez HR: Existence of solutions for impulsive partial neutral functional differential equations. J. Comput. Appl. Math. 2007, 331: 1135-1158.
Hernández E, Mckibben MA, HenrÃquez HR: Existence results for abstract impulsive second order neutral functional differential equations. Nonlinear Anal., Theory Methods Appl. 2008. 10.1016/j.na.2008.03.062
Benchohra M, Ouahab A: Impulsive neutral functional differential equations with variable times. Nonlinear Anal., Theory Methods Appl. 2003, 55: 679-693. 10.1016/j.na.2003.08.011
Hernández E, HenrÃquez HR: Impulsive partial neutral differential equations. Appl. Math. Lett. 2006, 19: 215-222. 10.1016/j.aml.2005.04.005
Chang YK, Anguraj A, Arjunan MM: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Anal. Hybrid Syst. 2008, 2: 209-218. 10.1016/j.nahs.2007.10.001
Li M, Han M: Existence for neutral impulsive functional differential equations with nonlocal conditions. Indag. Math. 2009, 20: 435-451. 10.1016/S0019-3577(09)80017-7
Balachandran K, Annapoorani N: Existence results for impulsive neutral evolution integrodifferential equations with infinite delay. Nonlinear Anal. Hybrid Syst. 2009, 3: 674-684. 10.1016/j.nahs.2009.06.004
Cuevas C, Hernández E, Rabelo M: The existence of solutions for impulsive neutral functional differential equations. Comput. Math. Appl. 2009, 58: 744-757. 10.1016/j.camwa.2009.04.008
Anguraj A, Karthikeyan K: Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 2009, 70: 2717-2721. 10.1016/j.na.2008.03.059
Park JY, Balachandran K, Annapoorani N: Existence results for impulsive neutral functional integrodifferential equations with infinite delay. Nonlinear Anal., Theory Methods Appl. 2009, 71: 3152-3162. 10.1016/j.na.2009.01.192
Chang YK, Nieto JJ, Zhao ZH: Existence results for a nondensely-defined impulsive neutral differential equation with state-dependent delay. Nonlinear Anal. Hybrid Syst. 2010, 4: 593-599. 10.1016/j.nahs.2010.03.006
Chang YK, Kavitha V, Arjunan MM: Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators. Nonlinear Anal. Hybrid Syst. 2010, 4: 32-43. 10.1016/j.nahs.2009.07.004
Ye RP: Existence of solutions for impulsive partial neutral functional differential equation with infinite delay. Nonlinear Anal., Theory Methods Appl. 2010, 73: 155-162. 10.1016/j.na.2010.03.008
Henrández E, HenrÃquez HR, Mckibben MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal., Theory Methods Appl. 2009, 70: 2736-2751. 10.1016/j.na.2008.03.062
Guo DJ: Nonlinear Functional Analysis. Shangdong Science and Technology Press, Jinan; 2002.
Fu XL, Yan BQ, Liu YS: Theory of Impulsive Differential System. Science Press, Beijing; 2005.
Acknowledgements
The authors are very grateful to the referees for the very useful comments and for detailed remarks that improved the presentation and the contents of the manuscript. This research is supported by the NNSF of China (No. 11171085), the SRF of Hunan Provincial Science and Technology Department (No. 2013FJ3096), the SRF of Hunan Provincial (No. 13JJ3106), and the SRF of Hunan Provincial Education Department (No. 12C0101).
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Ye, G., Zhao, Y. & Huang, L. Existence results for third-order impulsive neutral differential equations with deviating arguments. Adv Differ Equ 2014, 38 (2014). https://doi.org/10.1186/1687-1847-2014-38
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DOI: https://doi.org/10.1186/1687-1847-2014-38