Stability of Analytic and Numerical Solutions for Differential Equations with Piecewise Continuous Arguments

and Applied Analysis 3 Theorem 2.3. Suppose that f : R ×Rd×Rd → ×Rd is continuous, a, b : R → R are continuous, strictly increasing functions satisfying a 0 b 0 0. Let constants M > 0 and q ∈ 0, 1 exist such that for n ∈ Z, a |x| ≤ V t, x ≤ b |x| , t ∈ R, x ∈ R, 2.3 V n 1, x n 1 ≤ qV n, x n , 2.4 V t, x t ≤ MV n, x n , for t ∈ n, n 1 . 2.5 Then the trivial solution of 1.1 is asymptotically stable. Proof. Let ε > 0 be given, then there exist ε1 > 0 and δ δ ε such that ε1 ≤ ε, a ε1 ≤ a ε /M, and b δ < a ε1 . Let |x0| < δ, and Let x t be a solution of 1.1 , then it follows from 2.4 that the function V n, x n is decreasing with respect to n. Making use of 2.3 and 2.4 , we obtain successively the inequalities for any integer n, a |x n | ≤ V n, x n ≤ V 0, x0 ≤ b δ < a ε1 , 2.6 so |x n | < ε1 ≤ ε. From 2.3 , 2.5 , and 2.6 , we have for t ∈ n, n 1 , a |x t | ≤ V t, x t ≤ MV n, x n < Ma ε1 ≤ a ε . 2.7 Hence, |x t | < ε for all t > 0, which implies that the trivial solution is stable. From 2.3 , 2.4 , and 2.5 , we have for t ∈ n, n 1 , a |x t | ≤ V t, x t ≤ MV n, x n ≤ MqV n − 1, x n − 1 ≤ · · · ≤ MqV 0, x 0 . 2.8 Hence, limt→∞x t 0. Example 2.4. The trivial solution of the following system: ẋ1 t − tan x1 t 2 x2 t , ẋ2 t − sinx1 t − x2 2 t x2 t − x2 t 2 t > 0, 2.9 is asymptotically stable. 4 Abstract and Applied Analysis Proof. Let h > 0 be a constant such that |x| ≤ h, V t, x t 1 − cosx1 t x2 2 t /2 , a s s/h mins≤|x|≤hV x , b s max|x|≤sV x s,M 1, and q e−1, then V̇ t, x t sinx1 t ẋ1 t x2 t ẋ2 t − sinx1 t tan x1 t 2 x2 t sinx1 t − x2 t sinx1 t − x 2 2 t x 2 2 t − x2 2 t 2 ≤ −2sin21 t 2 − x 2 2 t 2 cosx1 t − 1 − x2 2 t 2 −V t, x t . 2.10 Hence, for t ∈ n, n 1 , we have V t, x t ≤ e− t−n V n, x n ≤ MV n, x n , V n 1, x n 1 ≤ qV n, x n . 2.11 Therefore, the trivial solution is asymptotically stable. In the following, we consider the following equation: x′ t a t x t b t x t , x 0 x0, 2.12 where a t and b t are continuous. Theorem 2.5. The trivial solution of 2.12 is asymptotically stable if there exist constants M > 0 and q ∈ 0, 1 such that for n 1, 2, . . ., max n≤t 0 , then for t ∈ n, n 1 , ∣ ∣ ∣ ∣ ∣ e ∫ t n a s ds e ∫ t n a s ds ∫ t n e− ∫s n a u b s ds ∣ ∣ ∣ ∣ ∣ ≤ e ∫ t n a s ds e ∫ t n a s α ∫ t n e− ∫s n a u du −a s ds e ∫ t n a s ds αe ∫ t n a s dse− ∫s n a u du ∣ ∣ ∣ t n e ∫ t n a s ds αe ∫ t n a s ds [ e− ∫ t n a s ds − e− ∫n n a s ds ] e ∫ t n a s ds α − αe ∫ t n a s ds α 1 − α e ∫ t n a s . 2.21 Therefore, we have the following corollary. 6 Abstract and Applied Analysis Corollary 2.7. Assume that a t ≤ −β, |b t | ≤ −αa t , then the trivial solution of 2.12 is asymptotically stable if 0 ≤ α < 1, β > 0. 2.22 3. The Stability of the Discrete System In this section, we will consider the discrete system with the form xkm l 1 φ km l, xkm l, xkm l−1, . . . , xkm , 3.1 where k ∈ Z, l 0, 1, . . . , m − 1. We assume that φ km l, 0, 0, . . . , 0 0 k ∈ Z, l 0, 1, . . . , m − 1 and 3.1 has a unique solution. The solution x n ≡ 0 is the trivial solution of 3.1 . Like 2.1 , we can define the stability and asymptotical stability. Theorem 3.1. Suppose φ : R × R × R × · · · × R → R are continuous, a, b : R → R are continuous, strictly increasing functions satisfying a 0 b 0 0. Let constants M > 0 and q ∈ 0, 1 exist such that for k ∈ Z a |x| ≤ V t, x ≤ b |x| , 3.2 V ( k 1 m,x k 1 m ) ≤ qV km, xkm , 3.3 V km l, xkm l ≤ MV km, xkm , l 0, 1, . . . , m − 1. 3.4 Then the trivial solution of 3.1 is asymptotically stable. Proof. Firstly, we will prove the stability. Let ε > 0 be given, then there exists a ε1 > 0 and δ δ ε such that ε1 ≤ ε, a ε1 ≤ a ε /M, and b δ < a ε1 . Let |x0| < δ, and Let xkm l be a solution of 3.1 , then it follows from 3.3 that the function V km, xkm is nonincreasing with respect to k. Making use of 3.2 and 3.3 , we obtain successively the inequalities a |xkm| ≤ V km, xkm ≤ V 0, x0 ≤ b δ < a ε1 , 3.5 so |xkm| < ε1 ≤ ε. From 3.2 , 3.4 , and 3.5 a |xkm l| ≤ V km l, xkm l ≤ MV km, xkm < Ma ε1 ≤ a ε . 3.6 Therefore, for all k ∈ Z, l 0, 1, . . . , m − 1, |xkm l| < ε. Abstract and Applied Analysis 7 Nextly, we will prove the asymptotic stability. We have, from 3.2 and 3.4 ,and Applied Analysis 7 Nextly, we will prove the asymptotic stability. We have, from 3.2 and 3.4 , a |xkm l| ≤ V km l, xkm l ≤ MV km, xkm ≤ MqV ( k − 1 m,x k−1 m ) ≤ · · · ≤ MqV 0, x 0 , 3.7 so limk→∞V km, xkm 0. The proof is complete. In the rest of the section, we consider the following scalar system: xkm l 1 f akm l 1, akm l, . . . , akm, bkm l 1, bkm l, . . . , bkm xkm l g akm l 1, akm l, . . . , akm, bkm l 1, bkm l, . . . , bkm xkm. 3.8 Let V t, x |x t | a |x| b |x| . The following corollary is easy to prove. Corollary 3.2. If there exists a α ∈ 0, 1 , such that for k ∈ Z, l 0, 1, . . . , m, Sk,l ∣ ∣f akm l 1, . . . , akm, bkm l 1, . . . , bkm ∣ ∣ ∣ ∣g akm l 1, . . . , akm, bkm l 1, . . . , bkm ∣ ∣ ≤ α, 3.9 then the trivial solution of 3.1 is asymptotically stable. 4. The Stability of the Numerical Solution In this section, we will investigate the numerical asymptotic stability of θ-methods. 4.1. θ-Methods Let h 1/m be a given stepsize with integer m ≥ 1 and the gridpoints tn nh n 0, 1, . . . . The linear θ-method applied to 1.1 can be represented as follows: xn 1 xn h { θf ( n 1 h, xn 1, x n 1 h ) 1 − θ f(nh, xn, x nh )} , 4.1 and the one-leg θ-method xn 1 xn h { f ( n θ h, x n θ h , x n θ h )} . 4.2 8 Abstract and Applied Analysis Table 1: Linear θ-methods for problem 5.1 . m θ 0 θ 1/2 θ 1 AE RE AE RE AE RE 3 1.8930E − 3 6.5172E − 1 1.9237E − 4 6.6228E − 2 3.0508E − 3 1.0503E − 0 5 1.2740E − 3 4.3861E − 1 6.9633E − 5 2.3973E − 2 1.6936E − 3 5.8308E − 1 10 6.8950E − 4 2.3738E − 1 1.7448E − 5 6.0071E − 3 7.9471E − 4 2.7360E − 1 20 3.5792E − 4 1.2322E − 1 4.3646E − 6 1.5026E − 3 3.8424E − 4 1.3229E − 1 50 1.4633E − 4 5.0378E − 2 6.9845E − 7 2.4046E − 4 1.5054E − 4 5.1828E − 2 100 7.3691E − 5 2.5370E − 2 1.7462E − 7 6.0117E − 5 7.4744E − 5 2.5733E − 2 Ratio 1.9857 1.9857 3.9998 3.9999 2.0141 2.0141 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Linear θ-method with θ 0 for 5.1 . Here, n 0, 1, . . . , θ is a parameter with 0 ≤ θ ≤ 1, specifying the method, x t denotes an approximation to x t , and x t is an approximation to x t defined by x t t − nh h xn 1 n 1 h − t h xn, for nh < t ≤ n 1 h, n 0, 1, . . . . 4.3 4.2. Numerical Stability Applying 4.1 and 4.2 to 2.12 , we arrive at the following recurrence relations, respectively: xn 1 xn h { θ ( a tn 1 xn 1 b tn 1 x n 1 h ) 1 − θ ( a tn xn b tn x nh )} , xn 1 xn h { a tn θ θxn 1 1 − θ xn b tn θ x n θ h } . 4.4 Abstract and Applied Analysis 9 Table 2: One-leg θ-methods for problem 5.1 . m θ 0 θ 1/2 θ 1 AE RE AE RE AE RE 3 1.8930E − 3 6.5172E − 1 1.4260E − 4 4.9094E − 2 3.0508E − 3 1.0503E 000 5 1.2740E − 3 4.3861E − 1 5.1570E − 5 1.7754E − 2 1.6936E − 3 5.8308E − 1 10 6.8950E − 4 2.3738E − 1 1.2917E − 5 4.4470E − 3 7.9471E − 4 2.7360E − 1 20 3.5792E − 4 1.2322E − 1 3.2307E − 6 1.1123E − 3 3.8424E − 4 1.3229E − 1 50 1.4633E − 4 5.0378E − 2 5.1699E − 7 1.7799E − 4 1.5054E − 4 5.1828E − 2 100 7.3691E − 5 2.5370E − 2 1.2925E − 7 4.4498E − 5 7.4744E − 5 2.5733E − 2 Ratio 1.9857 1.9857 3.9999 4.0000 2.0141 2.0141and Applied Analysis 9 Table 2: One-leg θ-methods for problem 5.1 . m θ 0 θ 1/2 θ 1 AE RE AE RE AE RE 3 1.8930E − 3 6.5172E − 1 1.4260E − 4 4.9094E − 2 3.0508E − 3 1.0503E 000 5 1.2740E − 3 4.3861E − 1 5.1570E − 5 1.7754E − 2 1.6936E − 3 5.8308E − 1 10 6.8950E − 4 2.3738E − 1 1.2917E − 5 4.4470E − 3 7.9471E − 4 2.7360E − 1 20 3.5792E − 4 1.2322E − 1 3.2307E − 6 1.1123E − 3 3.8424E − 4 1.3229E − 1 50 1.4633E − 4 5.0378E − 2 5.1699E − 7 1.7799E − 4 1.5054E − 4 5.1828E − 2 100 7.3691E − 5 2.5370E − 2 1.2925E − 7 4.4498E − 5 7.4744E − 5 2.5733E − 2 Ratio 1.9857 1.9857 3.9999 4.0000 2.0141 2.0141 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 2: Linear θ-method with θ 1/2 for 5.1 . Let n km l l 0, 1, . . . , m − 1 , then we define x tn δh , 0 ≤ δ ≤ 1, as xkm according to Definition 1.1. As a result, 4.4 reduce to xkm l 1 1 h 1 − θ a tkm l 1 − hθa tkm l 1 xkm l hθb tkm l 1 h 1 − θ b tkm l 1 − hθa tkm l 1 xkm, xkm l 1 [ 1 ha tkm l θ 1 − hθa tkm l θ ] xkm l hb tkm l θ 1 − hθa tkm l θ xkm. 4.5 In fact, in each interval n, n 1 , 2.12 can be seen as ordinary differential equation. Hence, the θ-methods are convergent of order 1 if θ / 1/2 and order 2 if θ 1/2. Definition 4.1. 1 The numerical methods are called asymptotically stable if there exists an h0 > 0, such that xn → 0 as n → ∞ for any given x0 and any stepsize h < h0. 2 The numerical methods are called general asymptotically stable if xn → 0 as n → ∞ for any given x0 and any stepsize. 10 Abstract and Applied Analysis 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: One-leg θ-method with θ 1/2 for 5.2 . 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4: One-leg θ-method with θ 1 for 5.2 . Theorem 4.2. Assume that a t ≤ −β, |b t | ≤ −αa t β > 0, 0 ≤ α < 1 , and there exists a r > 0 such that −r ≤ a t , then 1 the linear θ-method and the one-leg θ-method are asymptotically stable if h < 1/ 1 − θ r, 2 the one-leg θ-method is general asymptotically stable if 1 a/2 ≤ θ ≤ 1, and the linear θ-method is general asymptotically stable if 1 a/2 ≤ θ ≤ 1 and a t is nonincreasing. Abstract and Applied Analysis 11and Applied Analysis 11 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: One-leg θ-method with θ 0 for 5.3 . Proof. Denote akm l a tkm l , akm l θ a tkm l θ , bkm l b tkm l , and bkm l θ b tkm l θ . 1 For any integer k, and l 0, 1, . . . , m − 1, we have, from h < 1/ 1 − θ r, 1 h 1 − θ akm l > 0, 1 h 1 − θ akm l θ > 0. 4.6 For the linear θ-method, Sk,l ∣ ∣ ∣ ∣ 1 h 1 − θ akm l 1 − θhakm l 1 θhbkm l 1 1 − θhakm l 1 1 − θ hbkm l 1 − θhakm l 1 ∣ ∣ ∣ ∣ ≤ 1 h 1 − θ akm l − αθhakm l 1 − α 1 − θ hakm l 1 − θhakm l 1 1 1 − θ 1 − α hakm l θ 1 − α hakm l 1 1 − θhakm l 1 ≤ 1 − 1 − θ 1 − α hβ − θ 1 − α hβ 1 θhr 1 − 1 − α hβ 1 θhr < 1. 4.7 12 Abstract and Applied Analysis 50 100 150 200 250 300 350 400 450 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6: Linear θ-method with θ 1 for 5.3 . For the one-leg θ-method, Sk,l ∣ ∣ ∣ ∣ 1 hakm l θ 1 − θhakm l θ hbkm l θ 1 − θhakm l θ ∣ ∣ ∣ ∣ ≤ 1 1 − α hakm l θ 1 − θhakm l θ ≤ 1 − 1 − α hβ 1 θhβ < 1. 4.8 2 For the linear θ-method, if 1 1 − θ hakm l ≥ 0, then we have, from 1 , Sk,l ≤ 1 1 − θ 1 − α hakm l θ 1 − α hakm l 1 1 − θhakm l 1 ≤ 1 − 1 − α hβ 1 θhr < 1. 4.9 If 1 1 − θ hakm l < 0, then since a t is nonincreasing, we have Sk,l ≤ −1 − 1 − θ 1 α hakm l − θ 1 α hakm l 1 1 − θhakm l 1 ≤ −1 − 1 α hakm l 1 1 − θakm l 1 ≤ −1 1 α hr 1 θhr < 1. 4.10 For one-leg θ-method, we have the following two cases. Abstract and Applied Analysis 13 If 1 1 − θ hakm l θ ≥ 0, then Sk,l ≤ 1 1 α hakm l θ / 1 − θhakm l θ ≤ 1 − 1 − α hβ / 1 θhβ < 1. If 1 1−θ hakm l θ < 0, then Sk,l ≤ −1 − 1 α hakm l θ / 1−θhakm l θ ≤ −1 1 α hr/ 1 θhr < 1. 5. Numerical Experiments In this section, we will give some examples to illustrate the conclusions in the paper. We consider the following three problems: ẋ t (−e−t − 1)x t e −t 1 3 x t , t > 0, x 0 1, 5.1and Applied Analysis 13 If 1 1 − θ hakm l θ ≥ 0, then Sk,l ≤ 1 1 α hakm l θ / 1 − θhakm l θ ≤ 1 − 1 − α hβ / 1 θhβ < 1. If 1 1−θ hakm l θ < 0, then Sk,l ≤ −1 − 1 α hakm l θ / 1−θhakm l θ ≤ −1 1 α hr/ 1 θhr < 1. 5. Numerical Experiments In this section, we will give some examples to illustrate the conclusions in the paper. We consider the following three problems: ẋ t (−e−t − 1)x t e −t 1 3 x t , t > 0, x 0 1, 5.1 ẋ t sin t − 2 x t e −t 2 x t , t > 0, x 0 1, 5.2 ẋ t (−e−t − 1)x t − 13x t , t > 0,x 0 1.5.3 It is easy to verify that the above examples satisfy the conditions of Theorem 4.2.Hence, the solutions of three equations are asymptotically stable according to Corollary 2.7.In Tables 1 and 2, we list the absolute errors AEs and the relative errors REs at t 10of the θ-methods for the first problem.We can see from these tables that the methods preservetheir orders of convergence.In Figures 1, 2, 3, 4, 5, and 6, we draw the numerical solutions of the θ-methods withm 50. It is easy to see that the numerical solutions are asymptotically stable.AcknowledgmentThis work is supported by the NSF of P.R. China no. 10671047References1 J. Wiener, “Differential equations with piecewise constant delays,” in Trends in the Theory and Practiceof Nonlinear Differential Equations, V. 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