ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(2009) No.4,pp.412-418
Existence and Uniqueness of Neutral Functional Differential Equations with Random Impulses A.Anguraj ∗ , A.Vinodkumar∗∗ Department of Mathematics, PSG College of Arts and Science, Coimbatore-641 014, Tamil Nadu, INDIA (Received 6 December 2008, accepted 15 July 2009)
Abstract:In this paper, the existence and uniqueness of the solution of random impulsive neutral functional differential equation is investigated under sufficient condition. The results are obtained by using the Contraction principle. Keywords:neutral functional differential equation; random impulse; existence; uniqueness 2000 MR Subject Classification: 34K05, 34K45, 60H99.
1
Introduction
Many evolution processes from fields as diverse as physics, population dynamics, aeronautics, economics, telecommunications and engineering are characterized by the fact that they undergo abrupt change of state at certain moments of time between intervals of continuous evolution. The duration of these changes are often negligible compared to the total duration of process act instantaneously in the form of impulses. It is now being recognized that the theory of impulsive differential equations is not only richer than the corresponding theory of differential equations but also represents a more natural frame work for mathematical modeling of many real world phenomena see [3, 4, 7, 8, 10, 11] and reference therein. The impulses are exist at fixed time or at random time ie., they are deterministic or random. There are lot of papers which investigate the qualitative properties of fixed type impulses see [3, 4, 7, 8, 10, 11], but there are few papers which studies the random type impulses. Wu and Meng [12], first brought forward random impulsive ordinary differential equations and investigated boundedness of solutions to these models by Liapunov’s direct function. Wu et al. [13–16], have been studied some qualitative properties of differential equations with random impulses. In [3, 4, 7], the authors study the impulsive neutral type of differential equations at fixed points. In [9, 17], the authors studied the existence results under measure of noncompactness. Afrouzi1 et. al [1, 2], studied the existence of numerical methods for multiple solutions of Dirichlet boundary value problems. In this paper, we investigate a neutral type of differential equation which was first studied by Dhage [5, 6], in Banach algebra. The physical situations for the problem studied in [5] and in this paper occurs are not known so far. Therefore its importance in applications is yet to be investigated. So, the problem under study is new to the literature and so are the existence results to the theory of nonlinear problems of ordinary differential equation. The paper is organized as follows: Some preliminaries are presented in section 2. In section 3, we investigate the existence of solution of neutral differential equations with random impulses by using Contraction mapping principal and finally in section 4, we present an example for the system studied. ∗
Corresponding author. E-mail address:
[email protected] E-mail address:
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∗∗
c Copyright⃝World Academic Press, World Academic Union IJNS.2009.12.30/296
A.Anguraj, A.Vinodkumar: Existence and Uniqueness of Neutral Functional Differential Equations ⋅ ⋅ ⋅ 413
2
Preliminaries
Let ℜ𝑛 be the 𝑛−dimensional Euclidean space and Ω a nonempty set. Assume that 𝜏𝑘 is a random variable 𝑑𝑒𝑓.
defined from Ω to 𝐷𝑘 = (0, 𝑑𝑘 ) for all 𝑘 = 1, 2, ⋅ ⋅ ⋅ , where 0 < 𝑑𝑘 < +∞. Furthermore, assume that 𝜏𝑖 and 𝜏𝑗 are independent with each other as 𝑖 ∕= 𝑗 for 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ . Let 𝜏 ∈ ℜ be a constant. For the sake of simplicity, we denote ℜ𝜏 = [𝜏, 𝑇 ], ℜ+ = [0, +∞). We consider neutral functional differential equations with random impulses of the form ⎧ [ ]′ 𝑥(𝑡) = 𝑓 (𝑡, 𝑥𝑡 ), 𝑡 ∕= 𝜉𝑘 , 𝑡 ∈ [𝜏, 𝑇 ], ⎨ 𝑔(𝑡,𝑥𝑡 ) 𝑥(𝜉𝑘 ) ⎩ 𝑥𝑡0
= 𝑏𝑘 (𝜏𝑘 )𝑥(𝜉𝑘− ), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , = 𝜑,
(2.1)
where the functional 𝑔 : ℜ𝜏 × 𝐶 → ℜ𝑛 − {0}, 𝑓 : ℜ𝜏 × 𝐶 → ℜ𝑛 , for each 𝛾 > 0, we define 𝐶(𝛾) = {𝜍 ∈ 𝐶 : ∥𝜍∥2 ≤ 𝛾}, 𝐶 = 𝐶([−𝑟, 0], ℜ𝑛 ) is the set of piecewise continuous functions mapping [−𝑟, 0] into ℜ𝑛 with some given 𝑟 > 0; 𝑥𝑡 is a function when 𝑡 is fixed, defined by 𝑥𝑡 (𝑠) = 𝑥(𝑡 + 𝑠) for all 𝑠 ∈ [−𝑟, 0]; 𝜉0 = 𝑡0 and 𝜉𝑘 = 𝜉𝑘−1 + 𝜏𝑘 for 𝑘 = 1, 2, ⋅ ⋅ ⋅ , here 𝑡0 ∈ ℜ𝜏 is arbitrary given real number. Obviously, 𝑡0 = 𝜉0 < 𝜉1 < 𝜉2 < ⋅ ⋅ ⋅ < 𝜉𝑘 < ⋅ ⋅ ⋅ ; 𝑏𝑘 : 𝐷𝑘 → ℜ𝑛×𝑛 is a matrix-valued function for each 𝑘 = 1, 2, ⋅ ⋅ ⋅ ; 𝑥(𝜉𝑘− ) = lim 𝑥(𝑡) according to their paths with the norm ∥𝑥∥𝑡 = sup ∣𝑥(𝑠)∣ for each 𝑡 𝑡↑𝜉𝑘
𝑡−𝑟≤𝑠≤𝑡
satisfying 𝜏 ≤ 𝑡 ≤ 𝑇 and 𝑇 is a given number satisfying 𝜏 < 𝑇 ; 𝜑 is a function defined from [−𝑟, 0] to ℜ𝑛 . Denote {𝐵𝑡 , 𝑡 ≥ 0} the simple counting process generated by {𝜉𝑛 }, that is, {𝐵𝑡 ≥ 𝑛} = {𝜉𝑛 ≤ 𝑡}, and denote ℱ𝑡 the 𝜎-algebra generated by {𝐵𝑡 , 𝑡 ≥ 0}. Then (Ω, 𝑃, {ℱ𝑡 }) is a probability space. For the 𝑑𝑒𝑓.
simplification, denote Φ𝑇 = Φ𝑇 (𝜓) the Banach space with norm ∥𝜓∥Φ𝑇 = (𝐸∥𝜓∥2𝑡 )1/2 , where 𝜓 is any function defined from [𝜏 − 𝑟, 𝑇 ] to ℜ𝑛 . In the next section, we will discuss the existence and uniqueness of system (2.1) in (Ω, 𝑃, {ℱ𝑡 }). Definition 2.1 A stochastic process {𝑥(𝑡), 𝑡0 − 𝑟 ≤ 𝑡 ≤ 𝑇 } is called a solution to equation (2.1) in (Ω, 𝑃, {ℱ𝑡 }), if (i) 𝑥(𝑡) is ℱ𝑡 −adapted for 𝑡 ≥ 𝑡0 ; (ii) 𝑥(𝑡0 + 𝑠) = 𝜑(𝑠) when 𝑠 ∈ [−𝑟, 0], and ⎡ ∫ 𝜉𝑖 𝑘 𝑘 +∞ ∏ 𝑘 ∏ [ 𝜑(0) ] ∑ ∑ ⎣ 𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 ) 𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 ) 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝑥(𝑡) = + 𝑔(0, 𝜑(0)) 𝜉𝑖−1 𝑖=1 𝑗=𝑖 𝑘=0 𝑖=1 (2.2) ] ∫ 𝑡 +𝑔(𝑡, 𝑥𝑡 ) 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡), 𝑡 ∈ [𝑡0 , 𝑇 ], 𝜉𝑘
where
𝑘 ∏ 𝑗=𝑖
𝑏𝑗 (𝜏𝑗 ) = 𝑏𝑘 (𝜏𝑘 )𝑏𝑘−1 (𝜏𝑘−1 ) ⋅ ⋅ ⋅ 𝑏𝑖 (𝜏𝑖 ),
𝑛 ∏
{
𝐼𝐴 (𝑡) = (iii)
3
𝑥(𝑡) 𝑔(𝑡,𝑥𝑡 )
(⋅) = 1 as 𝑚 > 𝑛 and 𝐼𝐴 (⋅) is the index function, i.e.,
𝑗=𝑚
1, if 𝑡 ∈ 𝐴, 0, if 𝑡 ∈ / 𝐴.
is differentiable and satisfy the equation (2.1) for 𝑡 ∈ [𝜏, 𝑇 ].
Existence and uniqueness by contraction mapping
In this section, we discuss the existence and uniqueness of solutions of system (2.1). Before give the main results, we introduce the following hypotheses. (𝐻1 ) : The functions 𝑔 and 𝑓 satisfies the Lipschitz condition and there exists positive constants 𝐿1 , 𝐿2 > 0 for 𝜓, 𝜁 ∈ 𝐶 and 𝑡 ∈ [𝜏, 𝑇 ] such that ∥𝑔(𝑡, 𝜓𝑡 ) − 𝑔(𝑡, 𝜁𝑡 )∥2 ≤ 𝐿1 ∥𝜓 − 𝜁∥2𝑡 ∥𝑓 (𝑡, 𝜓𝑡 ) − 𝑓 (𝑡, 𝜁𝑡 )∥2 ≤ 𝐿2 ∥𝜓 − 𝜁∥2𝑡 . IJNS homepage:http://www.nonlinearscience.org.uk/
414
International Journal of Nonlinear Science,Vol.8(2009),No.4,pp. 412-418
(𝐻2 ) : The functions 𝑔 : ℜ𝜏 × 𝐶 → ℜ𝑛 − {0}, 𝑓 : ℜ𝜏 × 𝐶 → ℜ𝑛 are continuous and there exists a non-negative constant 𝜅 such that ∥𝑔(𝑡, 0)∥2 ≤ 𝜅 ∥𝑓 (𝑡, 0)∥2 ≤ 𝜅. (𝐻3 ) There exists a constants 0 ≤ 𝛾1 < 1 and 𝛾2 > 0 for 𝜑(0) ∈ 𝐶 such that ∥𝑔(0, 𝜑(0))∥2 ≤ 𝛾1 ∥𝜑(0)∥2 + 𝛾2 [ { }] 𝑘 ∏ 2 (𝐻4 ) : 𝐸 max ∥𝑏𝑗 (𝜏𝑗 )∥ < ∞. 𝑖,𝑘
𝑗=𝑖
Theorem 3.1 Let the hypotheses (𝐻1 )–(𝐻4 ) holds, then there exists a constant 𝐶 > 0 such that ⎡ 𝐸 ⎣max 𝑖,𝑘
⎧ 𝑘 ⎨∏ ⎩
𝑗=𝑖
⎫⎤ ⎬ ∥𝑏𝑗 (𝜏𝑗 )∥2 ⎦ ≤ 𝐶, ⎭
If the inequality 2𝐶𝐿1 𝐸∥𝜑(0)∥2 𝛾1 𝐸∥𝜑(0)∥2 +𝛾2
) ( ) ] [ ( + 2 max{1, 𝐶}(𝑇 − 𝑡0 )2 𝐿1 𝐿2 𝛾 + 𝜅 + 𝐿1 𝛾 + 𝜅 𝐿2 < 1
(3.1)
is satisfied, then the equation (2.1) has a unique solution in Φ𝑇 . Proof. Let 𝑇 be an arbitrary positive number 𝑡0 < 𝑇 < +∞. To apply the principle, we define the nonlinear operator 𝑃 : Φ𝑇 → Φ𝑇 as follows. (𝑃 𝑥)(𝑡) = 𝜑(𝑡 − 𝑡0 ),
𝑡 ∈ [𝑡0 − 𝑟, 𝑡0 ],
and ⎡ +∞ ∏ 𝑘 ∑ ⎣ 𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 ) (𝑃 𝑥)(𝑡) =
∫ 𝜉𝑖 𝑘 ∏ 𝑘 ∑ 𝜑(0) + 𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 ) 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝑔(0, 𝜑(0)) 𝜉 𝑖−1 𝑖=1 𝑗=𝑖 𝑘=0 𝑖=1 ] ∫ 𝑡 +𝑔(𝑡, 𝑥𝑡 ) 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡), 𝑡 ∈ [𝑡0 , 𝑇 ]. 𝜉𝑘
Now, we have to show that 𝑃 maps Φ𝑇 into itself. Clearly (𝑃 𝑥) : Φ𝑇 → Φ𝑇 is continuous with (𝑃 𝑥)𝑡0 = 𝜑 and 2 ∥(𝑃 𝑥)(𝑡)∥ [ 𝑘 +∞ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∑ ∏ ∑ 𝜑(0) 𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 ) 𝑔(0,𝜑(0)) =∥ 𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑖−1 + 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝑖=1 𝑗=𝑖 𝑘=0 𝑖=1 ] ∫𝑡 +𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡)∥2 [ 𝑘 ] +∞ ∑ ∏ 𝜑(0) 2 2 2 ∥𝑏𝑖 (𝜏𝑖 )∥ ∥𝑔(𝑡, 𝑥𝑡 )∥ ∥ 𝑔(0,𝜑(0)) ∥ 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) ≤2 𝑘=0 𝑖=1 [ [ +∞ ] ]2 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∫𝑡 ∑ ∑ ∥𝑏𝑗 (𝜏𝑗 )∥∥𝑔(𝑡, 𝑥𝑡 )∥ 𝜉𝑖−1 +2 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 +∥𝑔(𝑡, 𝑥𝑡 )∥ 𝜉𝑘 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) 𝑘=0
𝑖=1 𝑗=𝑖
IJNS email for contribution:
[email protected]
A.Anguraj, A.Vinodkumar: Existence and Uniqueness of Neutral Functional Differential Equations ⋅ ⋅ ⋅ 415 𝐸∥(𝑃 𝑥)(𝑡)∥2 ] [ 𝑘 +∞ ∑ ∏ 𝜑(0) 2 2 2 ≤ 2𝐸 ∥𝑏𝑖 (𝜏𝑖 )∥ ∥𝑔(𝑡, 𝑥𝑡 )∥ ∥ 𝑔(0,𝜑(0)) ∥ 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) 𝑘=0 𝑖=1 [ ] ]2 [ +∞ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∫𝑡 ∑ ∑ +2𝐸 ∥𝑏𝑗 (𝜏𝑗 )∥∥𝑔(𝑡, 𝑥𝑡 )∥ 𝜉𝑖−1 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 +∥𝑔(𝑡, 𝑥𝑡 )∥ 𝜉𝑘 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) 𝑖=1 𝑗=𝑖 [ 𝑘=0{∏ }] 𝜑(0) 𝑘 2 ≤ 2𝐸 max ∥𝑏 (𝜏 )∥ 𝐸∥𝑔(𝑡, 𝑥𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) ∥2 𝑗 𝑗 𝑗=𝑖 𝑘 [ ] [ +∞ ]2 ∫𝑡 ∏ ∑ +2𝐸 max{1, 𝑘𝑗=𝑖 ∥𝑏𝑗 (𝜏𝑗 )∥2 } 𝐸 ∥𝑔(𝑡, 𝑥𝑡 )∥ 𝑡0 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) 𝑖,𝑘 𝑘=0 [ ]2 ∫𝑡 𝜑(0) 2 2 ≤ 2𝐶𝐸∥𝑔(𝑡, 𝑥𝑡 )∥ 𝐸∥ 𝑔(0,𝜑(0)) ∥ + 2 max{1, 𝐶}𝐸 ∥𝑔(𝑡, 𝑥𝑡 )∥ 𝑡0 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 ∫𝑡 𝜑(0) ∥2 + 2 max{1, 𝐶}𝐸∥𝑔(𝑡, 𝑥𝑡 )∥2 (𝑇 − 𝑡0 ) 𝑡0 𝐸∥𝑓 (𝑠, 𝑥𝑠 )∥2 𝑑𝑠 ≤ 2𝐶𝐸∥𝑔(𝑡, 𝑥𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) ] [ ][ ∫𝑡 𝜑(0) ≤ 𝐸∥𝑔(𝑡, 𝑥𝑡 )∥2 2𝐶𝐸∥ 𝑔(0,𝜑(0)) ∥2 + 2 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝑡0 𝐸∥𝑓 (𝑠, 𝑥𝑠 )∥2 𝑑𝑠 [ ][ 𝜑(0) ≤ 𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 0)∥2 + 𝐸∥𝑔(𝑡, 0)∥2 2𝐶𝐸∥ 𝑔(0,𝜑(0)) ∥2 + 4 max{1, 𝐶}(𝑇 − 𝑡0 ) ] } ∫𝑡 { × 𝑡0 𝐸∥𝑓 (𝑠, 𝑥𝑠 ) − 𝑓 (𝑠, 0)∥2 + 𝐸∥𝑓 (𝑠, 0)∥2 𝑑𝑠 [ ][ } ] ∫𝑡 { 𝐸∥𝜑(0)∥2 2 + 𝜅 𝑑𝑠 𝐿 𝐸∥𝑥∥ ≤ 𝐿1 𝐸∥𝑥∥2𝑡 + 𝜅 2𝐶 𝛾1 𝐸∥𝜑(0)∥ 2 2 +𝛾 + 4 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝑡 𝑠 0 2 ] [ ][ ∫𝑡 𝐸∥𝜑(0)∥2 2 2 ≤ 𝐿1 𝐸∥𝑥∥𝑡 + 𝜅 2𝐶 𝛾1 𝐸∥𝜑(0)∥2 +𝛾2 + 4 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝜅 + 4 max{1, 𝐶}𝐿2 (𝑇 − 𝑡0 ) 𝑡0 𝐸∥𝑥∥2𝑠 𝑑𝑠 [ ][ ] ≤ 𝐿1 𝐸∥𝑥∥2𝑡 + 𝜅 𝛽1 + 𝛽2 𝐸∥𝑥∥2𝑡
2
𝐸∥𝜑(0)∥ 2 2 where 𝛽1 = 2𝐶 𝛾1 𝐸∥𝜑(0)∥ 2 +𝛾 + 4 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝜅 and 𝛽2 = 4 max{1, 𝐶}𝐿2 (𝑇 − 𝑡0 ) 2
sup 𝐸∥(𝑃 𝑥)∥2𝑡 ≤ [𝐿1 sup 𝐸∥𝑥∥2𝑡 + 𝜅][𝛽1 + 𝛽2 sup 𝐸∥𝑥∥2𝑡 ]
𝑡0 ≤𝑡≤𝑇
𝑡0 ≤𝑡≤𝑇
𝑡0 ≤𝑡≤𝑇
(3.2)
from (3.2), we deduce that the operator 𝑃 maps Φ𝑇 into itself. To see that 𝑃 is a contraction mapping, observe for 𝑡 ∈ [𝑡0 , 𝑇 ],
(𝑃 𝑥)(𝑡) − (𝑃 𝑦)(𝑡) [ 𝑘 [ { ] ∑ +∞ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∑ ∏ 𝜑(0) 𝑏𝑖 (𝜏𝑖 ) [𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )] 𝑔(0,𝜑(0)) = 𝑏𝑗 (𝜏𝑗 ) 𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑖−1 + 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝑖=1 𝑗=𝑖 𝑘=0 𝑖=1 } { }] ∫ 𝜉𝑖 ∫𝑡 ∫𝑡 −𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠 + 𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡)
∥(𝑃 𝑥)(𝑡) − (𝑃 𝑦)(𝑡)∥2 [ 𝑘 +∞ ∑ ∏ 𝜑(0) ∥𝑏𝑖 (𝜏𝑖 )∥2 ∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 ∥ 𝑔(0,𝜑(0)) ∥2 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) ≤2 𝑘=0 𝑖=1 [ +∞ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∫ 𝜉𝑖 ∑[∑ ∥𝑏𝑗 (𝜏𝑗 )∥∥𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑖−1 +2 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠∥ 𝑘=0 𝑖=1 𝑗=𝑖 ] ]2 ∫𝑡 ∫𝑡 +∥𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠∥ 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) IJNS homepage:http://www.nonlinearscience.org.uk/
416
International Journal of Nonlinear Science,Vol.8(2009),No.4,pp. 412-418 𝐸∥(𝑃[𝑥)(𝑡) − (𝑃 𝑦)(𝑡)∥2 {∏ }] 𝜑(0) 𝑘 2 ≤ 2𝐸 max ∥𝑏 (𝜏 )∥ 𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) ∥2 𝑗 𝑗 𝑗=𝑖 𝑘 [ +∞ [ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 ∫ 𝜉𝑖 ∑[∑ +2𝐸 ∥𝑏𝑗 (𝜏𝑗 )∥ ∥𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠∥ 𝑘=0 𝑖=1 𝑗=𝑖 ] [ ∫ 𝜉𝑖 ∫ 𝜉𝑖 ∫𝑡 +∥𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠∥ + ∥𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) ]] ]2 ∫𝑡 ∫𝑡 ∫𝑡 × 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠∥ + ∥𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 − 𝑔(𝑡, 𝑦𝑡 ) 𝜉𝑘 𝑓 (𝑠, 𝑦𝑠 )𝑑𝑠∥ 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) [ +∞ [ 𝑘 ∏ 𝑘 ∑[∑ 𝜑(0) ∥2 + 2𝐸 ≤ 2𝐶𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) ∥𝑏𝑗 (𝜏𝑗 )∥ ∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥ 𝑘=0 𝑖=1 𝑗=𝑖 ] ∫ 𝜉𝑖 ∫ 𝜉𝑖 × 𝜉𝑖−1 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 + ∥𝑔(𝑡, 𝑦𝑡 )∥ 𝜉𝑖−1 ∥𝑓 (𝑠, 𝑥𝑠 ) − 𝑓 (𝑠, 𝑦𝑠 )∥𝑑𝑠 [ ]] ]2 ∫𝑡 ∫𝑡 + ∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥ 𝜉𝑘 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 + ∥𝑔(𝑡, 𝑦𝑡 )∥ 𝜉𝑘 ∥𝑓 (𝑠, 𝑥𝑠 ) − 𝑓 (𝑠, 𝑦𝑠 )∥𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) [ +∞ ∑( 𝜑(0) ∥2 + 2 max{1, 𝐶}𝐸 ∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥ ≤ 2𝐶𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) 𝑘=0 ) ]2 ∫𝑡 ∫𝑡 × 𝑡0 ∥𝑓 (𝑠, 𝑥𝑠 )∥𝑑𝑠 + ∥𝑔(𝑡, 𝑦𝑡 )∥ 𝑡0 ∥𝑓 (𝑠, 𝑥𝑠 ) − 𝑓 (𝑠, 𝑦𝑠 )∥𝑑𝑠 𝐼[𝜉𝑘 ,𝜉𝑘+1 ) (𝑡) ( 𝜑(0) ∥2 + max{1, 𝐶} 𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 (𝑇 − 𝑡0 ) ≤ 2𝐶𝐸∥𝑔(𝑡, 𝑥𝑡 ) − 𝑔(𝑡, 𝑦𝑡 )∥2 𝐸∥ 𝑔(0,𝜑(0)) ) ∫𝑡 ∫𝑡 × 𝑡0 𝐸∥𝑓 (𝑠, 𝑥𝑠 )∥2 𝑑𝑠 + 𝐸∥𝑔(𝑡, 𝑦𝑡 )∥2 (𝑇 − 𝑡0 ) 𝑡0 𝐸∥𝑓 (𝑠, 𝑥𝑠 ) − 𝑓 (𝑠, 𝑦𝑠 )∥2 𝑑𝑠 𝜑(0) ∥2 𝐸∥𝑥 − 𝑦∥2𝑡 + max{1, 𝐶}(𝑇 − 𝑡0 ) ≤ 2𝐶𝐿1 𝐸∥ 𝑔(0,𝜑(0)) [ ] ) ( ) ∫𝑡 ∫𝑡 ( × 𝐿1 𝐸∥𝑥 − 𝑦∥2𝑡 𝑡0 𝐿2 𝐸∥𝑥∥2𝑠 + 𝜅 𝑑𝑠 + 𝐿1 𝐸∥𝑥∥2𝑡 + 𝜅 𝐿2 𝑡0 𝐸∥𝑥 − 𝑦∥2𝑠 𝑑𝑠 .
Then, sup 𝐸∥(𝑃 𝑥) − (𝑃 𝑦)∥2𝑡
𝑡0 ≤𝑡≤𝑇
[ 2 2 sup 𝐸∥𝑥 − 𝑦∥ ≤ 𝑡 + 2 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝐿1 sup 𝐸∥𝑥 − 𝑦∥𝑡 (𝑇 − 𝑡0 ) 𝑡0 ≤𝑡≤𝑇 𝑡0 ≤𝑡≤𝑇 ] ( ) ( ) × 𝐿2 sup 𝐸∥𝑥∥2𝑡 + 𝜅 + 𝐿1 sup 𝐸∥𝑥∥2𝑡 + 𝜅 𝐿2 (𝑇 − 𝑡0 ) sup 𝐸∥𝑥 − 𝑦∥2𝑡 𝑡0 ≤𝑡≤𝑇 𝑡0 ≤𝑡≤𝑇 [ 𝑡0 ≤𝑡≤𝑇 𝜑(0) 2 2 ≤ 2𝐶𝐿1 𝐸∥ 𝑔(0,𝜑(0)) ∥ sup 𝐸∥𝑥 − 𝑦∥𝑡 + 2 max{1, 𝐶}(𝑇 − 𝑡0 ) 𝐿1 sup 𝐸∥𝑥 − 𝑦∥2𝑡 (𝑇 − 𝑡0 ) 𝑡0 ≤𝑡≤𝑇 𝑡0 ≤𝑡≤𝑇 ] ( ) ( ) 2 × 𝐿2 𝛾 + 𝜅 + 𝐿1 𝛾 + 𝜅 𝐿2 (𝑇 − 𝑡0 ) sup 𝐸∥𝑥 − 𝑦∥𝑡 𝑡0 ≤𝑡≤𝑇 { ) ( ) ]} [ ( 𝐸∥𝜑(0)∥2 sup 𝐸∥𝑥 − 𝑦∥2𝑡 ≤ 2𝐶𝐿1 𝛾1 𝐸∥𝜑(0)∥2 +𝛾2 + 2 max{1, 𝐶}(𝑇 − 𝑡0 )2 𝐿1 𝐿2 𝛾 + 𝜅 + 𝐿1 𝛾 + 𝜅 𝐿2 𝜑(0) ∥2 2𝐶𝐿1 𝐸∥ 𝑔(0,𝜑(0))
𝑡0 ≤𝑡≤𝑇
It results that ∥(𝑃{𝑥) − (𝑃 𝑦)∥2Φ𝑇 [ ( ) ( ) ]} 𝐸∥𝜑(0)∥2 2 𝐿 𝐿 𝛾+𝜅 + 𝐿 𝛾+𝜅 𝐿 ≤ 2𝐶𝐿1 𝛾1 𝐸∥𝜑(0)∥ + 2 max{1, 𝐶}(𝑇 − 𝑡 ) ∥𝑥 − 𝑦∥2 0 1 2 1 2 2 +𝛾 2 Therefore by (3.1), 𝑃 is a contraction mapping. Thus the mapping 𝑃 has a unique fixed point 𝑥 in Φ𝑇 which is a solution of (2.1) with 𝜑 ∈ 𝐶(𝛾). For any 𝑡 ∈ [𝑡0 , 𝑇 ] and 𝑡 ∕= 𝜉𝑘 , there exists (𝜉𝑘 , 𝜉𝑘+1 ) such that 𝑡 ∈ (𝜉𝑘 , 𝜉𝑘+1 ), which yields 𝑥(𝑡) =
𝑘 ∏
[ 𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 )
𝑖=1
𝑘
𝑘
𝑖=1 𝑗=𝑖
[ and therefore,
𝜑(0) ] ∑ ∏ + 𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 ) 𝑔(0, 𝜑(0))
𝑥(𝑡) 𝑔(𝑡,𝑥𝑡 )
]′
𝑥𝜉𝑘 =
= 𝑓 (𝑡, 𝑥𝑡 ), ie., 𝑘 ∏ 𝑖=1
[
𝑥(𝑡) 𝑔(𝑡,𝑥𝑡 )
]′
∫
𝜉𝑖
𝜉𝑖−1
∫ 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 + 𝑔(𝑡, 𝑥𝑡 )
= 𝑓 (𝑡, 𝑥𝑡 ), as 𝑡 ∕= 𝜉𝑘 . Furthermore,
] ∑ [ 𝑘 ∏ 𝑘 ∫ 𝜉𝑖 𝜑(0) + 𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 ) 𝑔(0,𝜑(0)) 𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 ) 𝜉𝑖−1 𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 𝑖=1 𝑗=𝑖
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𝑡
𝜉𝑘
𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠
A.Anguraj, A.Vinodkumar: Existence and Uniqueness of Neutral Functional Differential Equations ⋅ ⋅ ⋅ 417 and 𝑥− 𝜉𝑘 =
𝑘−1 ∏ 𝑖=1
𝑏𝑖 (𝜏𝑖 )𝑔(𝑡, 𝑥𝑡 )
[
𝜑(0) 𝑔(0,𝜑(0))
]
+
𝑘−1 ∑ 𝑘−1 ∏ 𝑖=1 𝑗=𝑖
𝑏𝑗 (𝜏𝑗 )𝑔(𝑡, 𝑥𝑡 )
∫ 𝜉𝑖
𝜉𝑖−1
𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠 + 𝑔(𝑡, 𝑥𝑡 )
∫𝑡
𝜉𝑘−1
𝑓 (𝑠, 𝑥𝑠 )𝑑𝑠
imply that 𝑥𝜉𝑘 = 𝑏𝑘 (𝜏𝑘 )𝑥− 𝜉𝑘 . Thus, 𝑥(𝑡) is a solution to system (2.1).
4
An Application:
In this section, we present an example to show value of (2.1) and an application for the obtained result of Theorem 3.1. Example 1 Let 𝜏𝑘 be a random variable defined in 𝐷𝑘 ≡ (0, 𝑑𝑘 ) for all 𝑘 = 1, 2, ⋅ ⋅ ⋅ , where 0 < 𝑑𝑘 < +∞. Further more, assume that 𝜏𝑖 and 𝜏𝑗 be independent with each other as 𝑖 ∕= 𝑗 for 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ . ⎧ [ ⎨
]′
𝑥(𝑡)
5(1−𝑡)+
𝑥(𝜉𝑘 ) ⎩ 𝑥𝑡0
ln(2+𝑡) 2(1+𝑡)
sin 𝑥(𝑡/2)
= 𝑎𝑟𝑔𝑡𝑔
[
𝑡 𝑡2 +1
] exp(𝑡/2) +
𝑥 sin 𝑥 3+∣ sin 𝑥∣ ,
= 𝑝(𝑘)(𝜏𝑘 )𝑥(𝜉𝑘− ), = 𝜑,
𝑡 ∕= 𝜉𝑘 , 𝑡 ∈ [𝑡0 , 𝑇 ], 𝑘 = 1, 2, ⋅ ⋅ ⋅ ,
(4.1)
Observe that the problem (4.1) is a special case of the problem (2.1), where the functions 𝑓 and 𝑔 has the form 𝑔(𝑡, 𝑥𝑡 ) = 5(1 − 𝑡) + ln(2+𝑡) 2(1+𝑡) ]sin 𝑥(𝑡/2) [ 𝑡 𝑥 𝑓 (𝑡, 𝑥𝑡 ) = 𝑎𝑟𝑔𝑡𝑔 𝑡2 +1 exp(𝑡/2) + 3+∣𝑥 sin sin(𝑥)∣ 𝑏𝑘 (𝜏𝑘 )
=
𝑝(𝑘)(𝜏𝑘 ).
It is easily seen that the functions 𝑓 and 𝑔 satisfies the assumptions (𝐻1 ) with 𝐿1 = 1/2 and 𝐿2 = 1/3. On the other hand it satisfies (𝐻2 ) with ∥𝑔(𝑡, 0)∥2 ∥𝑓 (𝑡, 0)∥ and it also satisfies (𝐻3 ) that
2
≤ 5(1 − 𝑡)[ ≤ 𝜅1
≤ ∥𝑎𝑟𝑔𝑡𝑔
𝑡
𝑡2 +1
] exp(𝑡/2) ∥2 ≤
𝑡 𝑡2 +1
ln(2) 2 2 ∥𝜑(0)∥
+5
∥𝑔(0, 𝜑(0))∥2 ≤
exp(𝑡/2) ≤ 𝜅1
Moreover the assumption (𝐻4 ) is satisfied. Summing up all the above facts, in view of Theorem 3.1 we conclude that the problem (4.1) has a unique solution 𝑥 = 𝑥(𝑡) provided that 𝐶 𝐸∥𝜑(0)∥2
ln(2) 𝐸∥𝜑(0)∥2 +5 2
) ( ) [ ( ] + 2 max{1, 𝐶}(𝑇 − 𝑡0 )2 (1/2) (1/3)𝛾 + 𝜅1 + (1/2)𝛾 + 𝜅1 (1/3) < 1.
References [1] G.A. Afrouzi, S. Heidarkhani: On the Existence of Three Solutions for a Dirichlet Boundary Value Problem in N-dimensional Case. International Journal of Nonlinear Science. 2:111-116(2006) [2] G.A.Afrouzi, Z.Naghizadeh and S.Mahdavi: Numerical Methods for Finding Multiple Solutions of a Dirichlet Problem with Nonlinear Terms. International Journal of Nonlinear Science. 2:147-152(2006) [3] A.Anguraj, K.Karthikeyan: Existence of solutions for impulsive neutral functional differential equations with non-local conditions. Nonlinear Analysis Theory Methods and Applications. 70(7): 27172721(2009) [4] A. Anguraj, M. Mallika Arjunan, E. Hern´andez: Existence results for an impulsive partial neutral functional differential equations with state - dependent delay. Applicable Analysis. 86(7):861-872(2007) IJNS homepage:http://www.nonlinearscience.org.uk/
418
International Journal of Nonlinear Science,Vol.8(2009),No.4,pp. 412-418
[5] B.C. Dhage: Existence results for neutral functional differential inclusions in Banach algebras. Nonlinear Analysis. 64:1290-1306(2006) [6] B.C. Dhage, M. Kumpulainen: Nonlinear functional boundary value problems involving the product of two nonlinearities. Applied Mathematics letter. 21:537-544(2008) [7] E. Hern´andez, Marco Rabello, H.R.Henriquez: Existence of solutions for impulsive partial neutral functional differential equations. J.Math.Anal.Appl. 331:1135-1158(2007) [8] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov: Theory of Impulsive Differential Equations. World Scientific, Singapore. (1989) [9] Qixiang Dong , Zhenbin Fan, Gang Li: Existence of Solutions to Nonlocal Neutral Functional Differential and Integrodifferential Equations. International Journal of Nonlinear Science. 5:140-151(2008) [10] Yu.V. Rogovchenko: Impusive evolution systems: main results and new trends. Dynamics Contin. Diser. Impulsive Sys. 3:57-88(1997) [11] A.M. Samoilenko and N.A Perestyuk: Impulsive Differential Equations. World Scientific, Singapore. (1995) [12] S.J.Wu, X.Z. Meng: Boundedness of nonlinear differential systems with impulsive effect on random moments. Acta Mathematicae Applicatae Sinica. 20(1):147-154(2004) [13] S.J.Wu, Y.R. Duan: Oscillation, stability, and boundedness of second-order differential systems with random impulses. Computers and Mathematics with Applications. 49(9-10):1375-1386(2005) [14] S.J.Wu, X.L. Guo, S.Q. Lin: Existence and uniqueness of solutions to random impulsive differential systems. Acta Mathematicae Applicatae Sinica. 22(4):595-600(2006) [15] S.J.Wu, X.L. Guo, Y. Zhou: 𝑝−moment stability of functional differential equations with random impulses. Computers and Mathematics with Applications. 52:1683-1694(2006) [16] S.J.Wu, X.L. Guo, R.H. Zhai: Almost sure stability of functional differential equations with random impulses. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 15:403-415(2008) [17] Zhenbin Fan, Qixiang Dong, Gang Li: Semilinear Differential Equations with Nonlocal Conditions in Banach Spaces. International Journal of Nonlinear Science. 2:131-139(2006)
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