Application of Delay Equations

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1 Application of Delay Equations István Győri and Ferenc Hartung Department of Mathematics and Computing University of Pannonia Veszprém, Hungary Sarajevo 2006 István Győri and Ferenc Hartung ( Department of Application Mathematics of Delay and Computing Equations University of Pannonia Veszprém, Sarajevo Hungary) / 1

2 short history Stability of several classes of difference equations has been studied extensively in the recent literature. Without completeness, we refer to K. L. Cooke, I. Győri, Comp. Math. Appl. 28:1-3 (1994) K. Gopalsamy, Kluwer Aacademic Publishers (1992) I. Győri, M. Pituk, Proc. 2nd Int. Conf. on Difference Eqns (1997) V. L. Kocic, G. Ladas, Kluwer Aacademic Publishers (1993) I. Kovácsvölgyi, Appl. Math. Letters 13 (2000) G. Ladas, C. Qian, P. N. Vlahos, J. Yan, Appl. Anal. 41 (1991) R. Ogita, H. Matsunaga, T. Hara,JMAA 248 (2000)

3 Consider Stability of difference equations short history x(n + 1) x(n) = ax(n k), n = 0,1,... (83) where a R, k 0. Look for solutions of the form x(n) = λ n. Then or equivalently, λ n+1 λ n = aλ n k, λ k+1 λ k = a. (84) Theorem 34 (Levin and May, 1976). The following statements are equivalent (i) (83) is asymptotically stable; (ii) all roots of (84) satisfy λ < 1; (iii) 0 < ak < 2k cos kπ 2k+1.

4 Equivalence with EPCAs Consider the linear delay difference equation x(n + 1) x(n) = a i x(n k i (n)), n Z +, (85) where a i > 0 and k i : Z + Z +, (i = 1,...,m), and there exists r > 0 such that k i (n) r for n Z + and i = 1,...,m. Equation (85) has a unique solution, assuming that x(n) = ϕ(n), ϕ: [ r,0] R. (86) We compare the stability of the discrete equation (85) to that of a differential equation. We associate the linear delay differential equation with piecewise constant argument ( ) ẏ(t) = a i y [t] k i ([t]), t 0, (87) and the initial condition y(t) = ϕ(t), t [ r,0] (88)

5 Equivalence with EPCAs to (85)-(86), where [ ] is the greatest integer function. Integrating both sides of (87) from n to t [n,(n + 1)), we get y(t) y(n) = ( ) a i y n k i (n) (t n). Therefore IVP (87)-(88) has a unique solution, which is piecewise linear between nonnegative integers, and y(n + 1) y(n) = ( ) a i y n k i (n), n N 0. (89) We can observe that the solutions of (85) and (87) are related by y(n) = x(n). Therefore the trivial solution of (85) is asymptotically stable, if and only if, so is the trivial solution of (87).

6 Stability theorems x(n + 1) x(n) = ẏ(t) = a i x(n k i (n)), n Z +, (85) ( ) a i y [t] k i ([t]), t 0, (87) Rewrite (87) as where ẏ(t) = ( ) a i y t σ i (t), t 0, (90) σ i (t) k i ([t]) + t [t]. Define the function Φ(r) = v(t) dt, 0

7 Stability theorems where v is the fundamental solution of the equation v(t) = v(t r), t 0, { 1, t = 0, v(t) = 0, t < 0. ẏ(t) = ( ) a i y t τ + ( a i (y(t τ) y t σ i (t)) ), Theorem 19 yields that the trivial solution of (90) (i.e., that of (87)) is asymptotically stable, if for some τ [0,π/(2a)) it follows τa 1 m Φ(τa) < a i lim σ i (t) t a i lim t σ i (t) < τa + 1 Φ(τa), (91) where a m a i. Since ( ) lim σ i (t) = lim k i ([t]) + t [t] lim k i (n) t t

8 and Stability of difference equations Stability theorems ( ) lim i(t) = lim t t k i ([t]) + t [t] lim i(n) + 1, we get the following result.

9 Stability theorems x(n + 1) x(n) = a i x(n k i (n)), n Z +, (85) Theorem 35 (I. Győri, F. Hartung (2000)). Suppose a i > 0 (i = 1,...,m), a m a i, and for some τ [0,π/(2a)) τa 1 m Φ(τa) < a i lim k i (n) a i lim k i(n) < (τ 1)a + 1 Φ(τa) holds. Then the trivial solution of (85) is asymptotically stable. Let τ = 1 ae. Then Φ(aτ) = 1, and we get

10 Corollary 36. Stability of difference equations Suppose 0 < a i (i = 1,...,m), and Stability theorems a i lim k i(n) < m e a i. Then the trivial solution of (85) is asymptotically stable.

11 Stability theorems Consider the time-dependent scalar linear delay difference equation x(n + 1) x(n) = a(n)x(n k(n)), n Z +, (92) where a: Z + [0, ), k : Z + Z +. Theorem 37. Assume n=0 a(n) =, and there exists τ [0,π/2) such that τ 1 Φ(τ) < lim n 1 i=n k(n) a(i) lim n i=n k(n) Then the trivial solution of (92) is asymptotically stable. a(i) < τ + 1 Φ(τ). If τ = 1 e we get Φ(τ) = 1, and

12 Corollary 38. Stability of difference equations Assume n=0 a(n) =, and lim n i=n k(n) Stability theorems a(i) < e. Then the trivial solution of (92) is asymptotically stable.

13 Comparison of our results to known conditions x(n + 1) x(n) = a(n)x(n k), a(n) =, a(n) 0 n=0 Condition of Ladas, Qian, Vlahos, Yan (1991): lim n i=n k Condition of Győri and Pituk (1997): a(i) < 1 Our condition: lim n 1 i=n k lim n i=n k a(i) < 1 a(i) < e.

14 Comparison of our results to known conditions x(n + 1) x(n) = a i x(n k i ), a i > 0 Condition of Cooke, Győri (1994): a i k i < 1, Our condition a i k i < m e a i

15 Comparison of our results to known conditions x(n + 1) x(n) = a(n)x(n k), a(n) =, a(n) 0 n=0 Condition of Erbe, Xia and Yu (1995): lim n i=n k a(i) < (k + 1) Applying this for equation x(n + 1) x(n) = ax(n k) we get ak 3 k 2 k k 2(k + 1) 2

16 Comparison of our results to known conditions Our condition is τ 1 Φ(τ) < ak < k ( τ + 1 ) k + 1 Φ(τ) for some τ [0, π/2) Exact condition (Levin and May (1976)): ak < 2k cos kπ 2k + 1

17 Comparison of our results to known conditions τ k 1 ( k = τ k + 1 ) Φ(τ k ) k + 1 Φ(τ k ) τ k Φ(τ k ) = 2k + 1 τ k π 2 k exact Erbe our τ k

18 Comparison of our results to known conditions graphs of τ 1/Φ(τ) and (τ + 1/Φ(τ))

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,