ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS. 1. Introduction

Transcription

1 t m Mathematical Publications DOI: /tmmp Tatra Mt. Math. Publ ), ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS Jiří Jánsý ABSTRACT. The paper discusses some nown asymptotic stability conditions for the difference equation x n = αx n 1 + βx n, n =0, 1, 2... with complex numbers α, β. As the main result, we show that the system of necessary and sufficient stability conditions, derived in a previous wor, can be significantly simplified. 1. Introduction The problem of necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations is one of the most frequent topic of qualitative analysis of these equations. A considerable attention was paid especially to the three-term linear difference equation x n = αx n 1 + βx n, n =0, 1, 2,... 1) with either real or complex α, β and a positive integer 2. If α, β are real constants, then the system of necessary and sufficient conditions for the asymptotic stability of 1) was first derived by K u r u l i s [10]. We recall here this well-nown result. Theorem 1.1. Let α 0, β be arbitrary reals and 2 be an integer. The equation 1) is asymptotically stable if and only if α </ 1), and α 2 +1 cos φ) 1/2 > β > α 1 for even, 2) α+β < 1 and β < α 2 +1 cos φ) 1/2 for odd, where φ 0,π/) is the solution of sin 1)x ) / sinx) =1/ α. c 2015 Mathematical Institute, Slova Academy of Sciences M a t h e m a t i c s Subject Classification: 39A30, 39A06, 26C10. K e y w o r d s: delay difference equation, asymptotic stability, characteristic polynomial. 153

2 JIŘÍ JÁNSKÝ If α, β are complex numbers, then the corresponding generalization of the previous result was provided by K i p n i s [8]. The author considered complex coefficients in the goniometric form α = α expiθ α ), β = β expiθ β ), where θ α,θ β 0, 2π) are appropriate arguments, and formulated the following criterion. Theorem 1.2. Let α 0, β be arbitrary complex numbers and 2 be an integer. Then we have : i) If α < 1 β, then 1) is asymptotically stable. ii) If 1 β α < min { 1+ β,/ 1) }, then for the asymptotic stability of 1) is necessary and sufficient to fulfill simultaneously the following conditions: β < α 2 +1 cosφ) ) 1/2, 3) where φ 0,π/) is the root of the equation sinx) α = 4) sin 1)x) and 1) arccos 1+ α 2 β 2 + arccos 1 α 2 β 2 < β π π arg exp iθ β θ α ) )). 5) iii) If α min 1+ β,/ 1) ), then 1) is not asymptotically stable. The conditions of Theorem 1.1 and Theorem 1.2 were derived by thorough analysis of roots of the corresponding characteristic polynomial pλ) =λ αλ 1 β. 6) More precisely, these conditions equivalently formulate optimal criteria guaranteeing that all the roots of 6) are located inside the unit circle in the complex plane. Recently, an alternative system of conditions appeared in [1] the case α, β are real numbers) and in [2] the case α, β are complex numbers). Although both these systems are different from those of Theorem 1.1 and Theorem 1.2, some comparisons of the relevant results lead us to the hypothesis that condition 3), 4) of Theorem 1.2 is superfluous and can be in the corresponding system replaced simply by β < 1. In the next section, we present the result confirming this hypothesis, i.e., we show that the condition 3), 4) can be omitted. Also, some consequences and examples illustrating this simplified version of Theorem 1.2 are presented, which is a topic of the last section. 154

3 ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS 2. The main result and its proof Before we formulate a simplification of Theorem 1.2, we state the next auxiliary assertion. Let S) be the set of complex couples α, β) such that α </ 1), β < α 2 +1 cosφ) ) 1/2, 7) where φ 0,π/) is the root of the equation 4). Then we have: Proposition 2.1. Let α, β be nonzero complex numbers and 2 be an integer. Further, let 1 β α < 1 + β. Thenα, β) S) if and only if β < 1, 8) 1) arccos 1+ α 2 β 2 + arccos 1 α 2 β 2 β Proof. The case1 β = α is trivial. Let 1 β < α. At first, we rewrite 7) as α </ 1), cosφ) < 1+ α 2 β 2, where the last condition implies β < 1+ α. Thus 7) is equivalent to <π. 9) β < 1+ α, α </ 1), cosφ) < 1+ α 2 β 2. 10) We put ω = arccos 1+ α 2 β 2. 11) Further, we define the function sinx) fx) = 0, sin 1)x) α, x π ). This function is continuous and decreasing in 0,π / ) due to f x) cosx)sin 1)x) 1) sinx) = sin 2 < 0 for all x 0, π ). 1)x) Moreover, the function fx) has a unique root in this interval due to lim fx) = x α > 0, and f π ) = α < 0. Using this, we can reformulate 10) as β < 1+ α, α </ 1), cosφ) < cosω), where φ 0,π/) is such that fφ) =0. 155

4 JIŘÍ JÁNSKÝ Thus β < 1+ α, α </ 1), ω < φ. Equivalently, β < 1+ α, fω) > 0, ω 0,π/), 12) which is an equivalent expression of 10). Note that the condition fω) > 0 implies α </ 1). Now we reformulate the condition fω) > 0bythe following way: sin 1)ω + ω) fω) = α sin 1)ω) =cosω)+cot 1)ω ) sinω) α, hence considering 11) we get fω) =cot 1)ω ) sinω)+ 1 α 2 β 2 =sinω) cot 1)ω ) + 1 α 2 β 2 ) 1 sinω) =sinω) cot 1)ω ) cotarccos α 2 + β 2 1 ) β Since sinω) > 0, the condition fω) > 0 becomes cot 1)ω ) > cot arccos α 2 + β 2 ) 1, β i.e., 1+ α 2 β 2 ) 1) arccos < arccos α 2 + β 2 1. β Thus we get 9). Let 9) be valid at least for = 2. We obtain the condition 1+ α 2 β 2 α 2 + β 2 1 > 0, β i.e., 1 β )1 + α + β )1 α + β ) > 0, β which implies β < 1. Furthermore, using 9) and 11) we get 1)ω <arccos α 2 + β 2 1 β < arccos 1+ α 2 β 2 ) = π ω, hence ω<π/. To summarize this, we get the series of equivalencies 7) 10), 12), which proves Proposition ).

5 ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS Remar 2.2. Note that the arguments of goniometric functions involved in the proof of Proposition 2.1 are well defined due to β < 1+ α and 1 β α < 1+ β. Now we can formulate the following simplification of Theorem 1.2. Theorem 2.3. Let α 0, β be arbitrary complex numbers and 2 be an integer. Then the following holds : i) If α < 1 β, then 1) is asymptotically stable. ii) If 1 β α < 1+ β, β < 1, then for the asymptotic stability of 1) is necessary and sufficient to fulfill the condition 5). iii) If α 1 + β, then 1) is not asymptotically stable. P r o o f. The conditions 1 β α < min { 1+ β, / 1) } and 3), 4) can be rewritten as 1 β α < 1+ β, β < 1 and 9), by use of Proposition 2.1. Obviously, the condition 5) is more restrictive than 9). Hence 3), 4) can be omitted. 3. Consequences, examples and concluding remars As the first consequence of Theorem 2.3, we mention a reformulation and simplification of Theorem 1.1. Considering α, β as nonzero real numbers, the right-hand side of 5) has the form π π arg exp iθ β θ α ) )) { π for α β<0, = 0 for α β>0. Furthermore, in the case α =1 β and α β<0, the condition 5) is satisfied trivially. Thus Theorem 2.3 implies the following assertion see also [1]). Corollary 3.1. Let α 0, β be arbitrary reals and 2 be an integer. The equation 1) is asymptotically stable if and only if one of the following holds : i) β < 1 α, ii) β =1 α, α β<0, iii) β > 1 α, β < 1, α β<0 and 9) holds. 157

6 JIŘÍ JÁNSKÝ The equivalence of Theorem 1.1 and Corollary 3.1 is not obvious. We show it only for the case even, the case odd is analogous. Let be even. We have to distinguish two cases. Case I: Let β < 1 α. Then the case i) of Corollary 3.1 holds if and only if α < 1 <, 1 α > β > α 1, 1 and the inequality 2) holds due to α 2 +1 cos φ ) 1/2 > 1 α. Case II: Let the cases ii) and iii) of Corollary 3.1 hold simultaneously, i.e., β 1 α, 1 > β > α 1, α β<0 and 9) hold. Then β<0and, using Proposition 2.1, we obtain the equivalence of ii) and iii) with α < 1, α 2 +1 cos φ ) 1/2 > β 1 α, β> α 1. Thus the inequality 2) holds. A connection between Theorem 2.3 and Corollary 3.1 is illustrated in the following example. Example 1. We consider the linear difference equation x n = 0.8 x n x n, n =0, 1, 2,... 13) The case iii) of Corollary 3.1 implies that 13) is asymptotically stable if and only if <9.65 and is even. Now we apply Theorem 2.3 to 13). It occurs the case ii) of Theorem 2.3, where the left-hand side of 5) represents the linear function of variable and the right-hand side of 5) represents the piecewise linear function of variable. While the left-hand side of 5) increases, the right-hand side of 5) is equal to π the case even), or zero the case odd). Both the functions are depicted in Figure 1. Figure 1. Dependence of the stability domain on the delay. 158

7 ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS We can see that the inequality 5) is fulfilled and 13) is asymptotically stable) if and only if {2, 4, 6, 8}. Obviously, both the conditions on coincide. Now we consider 1) with purely imaginary coefficients, i.e., x n =iax n 1 ib x n, n=0, 1, 2,..., 14) where a, b are nonzero reals. We apply Theorem 2.3 to 14). It is not difficult to see that the right hand side of 5) has the form π π π arg exp iθ β θ α ) )) 2 for even, = π for ab 1) 1 2 < 0, and odd, 0 for ab 1) 1 2 > 0, and odd. Furthermore, if b =1 a, then the left-hand side of 5) is zero. Thus we obtain Corollary 3.2. Let a, b be nonzero reals and 2 be an integer. The equation 14) is asymptotically stable if and only if one of the following holds : i) b < 1 a, ii) b =1 a, even, iii) b =1 a, ab 1) 1 2 < 0, odd, iv) b > 1 a, b < 1and9) holds, or 1) arccos 1+a2 b 2 2 a ab 1) 1 2 < 0, odd, < arcsin 1 a2 b 2 2 ab, even. As an illustration of the previous result, we state the following example. Example 2. We consider the linear difference equation x n =i0.7x n 1 +i0.3x n, n=0, 1, 2,... 15) By Corollary 3.2, 15) is asymptotic stable if and only if 2isanyinteger such that 4l +1, l Z +. Remar 3.3. This conclusion can be obtained also from Theorem 2.3 via the following procedure. The left-hand side of 5) is equal to zero and the right-hand side of 5) is depicted in Figure

8 JIŘÍ JÁNSKÝ Figure 2. Dependence of the stability domain on the delay. It is easy to see that the case ii) of Theorem 2.3 implies the same conclusion as Example 2. Remar 3.4. We note that many other relevant results on the asymptotic stability of higher order linear difference equation utilizes the conditions of the type 3), 4), see [8] [10], [12], [13]; for other related results we refer to [3] [7], [11], [14], [15]). Our opinion is that also these conditions can be reformulated via conditions of the type 5). Acnowledgements. The research was supported by the Project for development of Department of Mathematics and Physics K-215) Support of the mathematical and physical research of University of Defence. Projet pro rozvoj pracoviště K-215 Podpora matematicého a fyziálního výzumu Univerzity Obrany.) REFERENCES [1] ČERMÁK, J. JÁNSKÝ, J. KUNDRÁT, P.: On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Difference Equ. Appl ), [2] ČERMÁK, J. JÁNSKÝ, J.: Stability switches in linear delay difference equations, Appl. Math. Comput ), [3] ČERMÁK, J. TOMÁŠEK, P.: On delay-dependent stability conditions for a three-term linear difference equation, Funcial. Evac ), [4] CHENG, S. S. HUANG, S. Y.: Alternate derivations of the stability region of a difference equation with two delays, Appl. Math. E-Notes , [5] FREEDMAN, H. I. KUANG, Y.: Stability switches in linear scalar neutral delay equations, Funcial.Evac ), [6] IVANOV, S. KIPNIS, M. M. MALYGINA, V. V.: The stability cone for a difference matrix equation with two delays, ISRN Appl. Math ), Article ID , 19 p. [7] KASLIK, E.: Stability results for a class of difference systems with delay, Adv. Difference Equ ), Article ID , 13 p. [8] KIPNIS, M. M. MALYGINA, V. V.: The stability cone for a matrix delay difference equation, Int.J.Math.Math.Sci ), Article ID , 15 p. 160

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