STABILITY CRITERION FOR A SYSTEM OF DELAY-DIFFERENTIAL EQUATIONS (Qualitative Theory on ODEs and their applications to Mathematical Modeling)
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(2) 38 We introduce the necessary and sufficient condition for the absolute stability of the system × nn square = system (1.1). (1.1). To To this this end, end, we we prepare prepare some some notations. notations. For For nn\cross square matrix matrix XX=. \hat{X} and X \overline{X} as (x jk )1≤j,k≤n (x_{jk})_{1\leq j,k\leq n} , we define the matrices X ⎞ ⎛ ⎞ ⎛ −|x12 | · · · −|x1n | x11 |x11 | · · · |x1n | ⎜ −|x21 | x22 · · · −|x2n | ⎟ .. ⎠ . .... := ⎝ ... := ⎜ ⎟, X X .. .. .. ... . ⎝ ⎠ . . . |xn1 | · · · |xnn | −|xn1 | −|xn2 | · · · xnn. \hat{X}:=(\begin{ar y}{l x_{l1} -|x_{12}| \cdots -|x_{ln}| -x_{21}| x_{2} \cdots -|x_{2n}| \vdots \vdots \dots \vdots -|x_{n1}| -x_{n2}| \cdots x_{n} \end{ar y}),. \overline{X}:=(\begin{ar y}{l |x_{1 }| \cdots |x_{1n}| \vdots \d ots |x_{n1}| \cdots |x_{n }| \end{ar y}). Furthermore, Furthermore, to to mention mention Stability Stability Condition, Condition, we we define define principal principal minors minors (cf. (cf. Leslie Leslie [4]). [4]).. M be n\cross n square matrix. Let µ Definition 1.2. Let M be aa n×n be aa nonempty set of of row indices \mu be \nu M and ν a nonempty set of column indices. A submatrix of M is a matrix M [µ, ν] obtained and a of column of a obtained M[\mu, \nu] M \nu. by choosing the entries of M , which lie in rows µ and columns ν. A principal submatrix by choosing entries of \mu and columns M is a of [µ,\mu]. µ]. A principal minor is the determinant of M a submatrix of of the form M determinant of of aa M[\mu, principal submatrix.. A and B B as Now, we define the constant matrices A ⎛ ⎞ ⎛ a11 · · · a1n .. ⎠ , ... A = ⎝ ... B=⎝ . an1 · · · ann. ⎞ b11 · · · b1n .. ⎠ , .. . . . . . bn1 · · · bnn. A=(\begin{ar y}{l a_{1 } \cdots a_{1n} \vdots \d ots a_{n1} \cdots a_{n } \end{ar y}),B=(\begin{ar y}{l b_{1 } \cdots b_{1n} \vdots \d ots b_{n1} \cdots b_{n } \end{ar y}),. and and introduce introduce Stability Stability Condition Condition (SC) (SC) as as follows. follows.. Stability Stability Condition Condition (SC): (SC): The The coefficient coefficient matrices matrices of of (1.1) (1.1) satisfy satisfy the the following following conditions. (i) + B) = 0, (i) det(A \det(A+B)\neq 0, (ii) ≤ jj\leq ≤ n, n, jj − |bjj | > 0 or jj = bjj > 0 for (ii) aa_{jj}-|b_{jj}|>0 or aa_{jj}=b_{jj}>0 for all all jj with with 11\leq. (iii) − B are (iii) all all principal principal minors minors of of A —B‐ are nonnegative nonnegative definite. definite. Since Since Stability Stability Condition Condition (SC), (SC), we we derive derive the the following following theorem. theorem.. Theorem Theorem 1.3. 1.3. If If the the system system (1.1) (1.1) satisfies satisfies Stability Stability Condition Condition (SC), (SC), then then the the equilibequilib‐ rium point is absolutely absolutely stable. Furthermore, Furthermore, under under the the AA=diag = diag(a , · · · 11 \cdots,, a nn ),, we (a_{11}, a_{nn}). A in condition condition that that the the matrix matrix A in (1.1) (1.1) is is diagonal, diagonal, that that is, is, also obtain the following theorem.. Theorem 1.4. Suppose AA=diag = diag(a · · ,, aa_{nn}) If the equilibrium equilibrium point of of the system 11 , ·\cdots nn ).. If (a_{11}, (1.1) (1.1) is is absolutely absolutely stable, stable, then then the the system system (1.1) (1.1) satisfies satisfies Stability Stability Condition Condition (SC). (SC). Consequently, the simple combination of Theorem 1.3 and Theorem 1.4 gives the following corollary. Corollary = diag(a · · ,, aa_{nn}) 11 , ·\cdots nn ).. The Corollary 1.5. 1.5. Suppose Suppose AA=diag (a_{11}, The system system (1.1) (1.1) satisfies satisfies Stability Stability Condition (SC) if and only if the equilibrium point is absolutely stable. Condition (SC) if and only if the equilibrium point is absolutely stable. 2. Sufficient SUFFICIENT condition CONDITION 2.. To To prove prove Theorem Theorem 1.3, 1.3, we we start start from from the the definition definition of of the the irreducible irreducible matrix matrix (cf. (cf. Lancaster Lancaster and and Tismenetsky Tismenetsky [6]), [6]), 2.
(3) 39 M be M is said to be Definition 2.1. Let M × nn square matrix. The be aa nn\cross The matrix M be reducible P n if there is an permutation matrix P of order n such that if an of order
(4). M11 O −1 (2.1) P MP = , (2.1) M21 M22 ,. where M M_{11} 11 no such PP. P^{-1}MP=(\begin{ar ay}{l } M_{1 } O M_{21} M_{2 } \end{ar ay}). O is a and M_{22} and M are square matrices of of order order less than nn and and O a zero matrix. If If 22 are M exists, then M is irreducible. exists,. We introduce the following two lemmas to show Theorem 1.3. M be Lemma × nn real Lemma 2.2. 2.2. (cf. (cf. Fiedler Fiedler [5]) [5]) Let Let M be aa nn\cross real matrix matrix whose whose off-diagonal off‐liagonal entries entries M is irreducible, then there are are nonpositive and and all all principal minors are are nonnegative. If If M is aa vector vv>0 > 0 such that M v ≥ 0. Mv\geq 0.. Here, vv>0 > 0 or vv\geq ≥ 0 means that all components of the vector vv are positive or nonnegative, respectively. Lemma 2.3. Let Q = (αjk + βjk )1≤j,k≤n × nn square matrix, where α be aa nn\cross and β jkk} and Q=(\alpha_{jk}+\beta_{jk})_{1\leq j,k\leq n} be \alpha_{j \betjk a_{jk} are for 11\leq ≤ j,j, kk\leq ≤ n. n . Then Q lies in at are complex complex numbers for Then every every eigenvalue eigenvalue of of Q at least one one of of the disks disks n n. (2.2) z ∈ C ; |z − αjj | ≤ |αjk | + |βjk | (2.2). \{z\in \mathb {C} ; |z-\alpha_{j }|\leq\sum_{k=1,k\neq j}^{n}|\alpha_{jk}|+ \sum_{k=1}^{n}|\beta_{jk}|\} k=1,k=j. k=1. z ‐plane. for 11\leq ≤ jj\leq ≤ nn in the complex complex z-plane.. w with w w= Proof. Let λ\lambda be an eigenvalue of Q = (w · · · ,, w_{n})^{T}. w n )T . Q with the associated eigenvector w 1 , \cdots (w_{1}, Since Qw = λw, Qw=\lambda w , we have n (αjk + βjk )wk = λwj. \sum_{k=1}^{n}(\alpha_{jk}+\beta_{jk})w_{k}=\lambda w_{j}. k=1. for 11\leq ≤ jj\leq ≤ n. n . This means. n . n . ( \lambda-\alpha_{j })w_{j}=\sum_{k=1,k\neq j}^{n}\alpha_{\dot{J}^{k} w_{k}+ \sum_{k=1}^{n}\beta_{jk}w_{k}.. (λ − αjj )wj =. αjk wk +. k=1,k=j. βjk wk .. k=1. Let pp be a natural number which satisfies |w maxj |wj |.. Then the p‐th p-th equation p | =\max_{j}|w_{j}| |w_{p}|= gives n n αpk wk + βpk wk |λ − αpp ||wp | = . | \lambda-\alpha_{p }|w_{p}|=|\sum_{k=1,k\neq p}^{n}\alpha_{pk}w_{k}+\sum_{k= 1}^{n}\beta_{pk}w_{k}| \leq\sum_{k=1,k\neq p}^{n}|\alpha_{pk}|w_{k}|+\sum_{k=1}^{n}|\beta_{pk} |w_{k}| \leq(\sum_{k=1,k\neq p}^{n}|\alpha_{pk}|+\sum_{k=1}^{n}|\beta_{pk}|)w_{p}|. ≤. ≤. k=1,k=p n . k=1 n . |αpk ||wk | +. k=1,k=p n . |αpk | +. k=1,k=p. |βpk ||wk |. k=1. n k=1. |βpk | |wp |.. Because of w = 0, w\neq 0 , we have |w p | \= |w_{p}| neq0. 0 . Consequently, we obtain n n |λ − αpp | ≤ |αpk | + |βpk |,. | \lambda-\alpha_{p }|\leq\sum_{k=1,k\neq p}^{n}|\alpha_{pk}|+\sum_{k=1}^{n} |\beta_{pk}|, k=1,k=p. k=1. and this estimate gives the conclusion of Lemma 2.3. 3. \square .
(5) 40 Remark 1. If ≤ j,j, kk\leq ≤ nn in Lemma 2.3, this lemma If we suppose that β\beta_{jk}=0 jk = 0 for 11\leq becomes sgorin’s theorem becomes Ger˘ Geršgorin’s theorem (cf. (cf. Lancaster Lancaster and and Tismenetsky Tismenetsky [6]). [6]). −B. is Proof Proof of of Theorem Theorem 1.3. 1.3. We We suppose suppose that that A —B‐ is irreducible. irreducible. By By employing employing Theorem Theorem. 2.2, − B whose 2.2, for for an an irreducible irreducible matrix matrix A —B‐ whose off-diagonal off‐diagonal entries entries are are nonpositive nonpositive and and all all. v>0 principal minors are nonnegative, there is a vector v > 0 such that ( A − B)v ≥ \geq principal minors are nonnegative, there is a vector such that (—B‐)v 0.0. Namely, there exists vv_{j}>0 j > 0 such that n . n . -a_{j }v_{j}+ \sum_{k=1,k\neq j}^{n}|a_{\dot{J}^{k} |v_{k}+\sum_{k=1}^{n} |b_{jk}|v_{k}\leq 0. (2.3) (2.3). −ajj vj +. |ajk |vk +. k=1,k=j. |bjk |vk ≤ 0. k=1. for 11\leq ≤ jj\leq ≤ n. n. We suppose {\rm Re}\lambda_{0}\geq 0 . Then, 0 ≥ 0. We suppose that that there there exists exists aa root root λ\lambd0a_{0} of of (1.2) (1.2) satisfying satisfying Reλ Then, we we −λ0 τjk introduce the square matrix EE := −(a + b e ) . We remark that λ\lambd0a_{0} is jk jk 1≤j,k≤n :=-(a_{jk}+b_{jk}e^{-\lambda_{0}\tau_{jk}})_{1\leq j,k\leq n} an −(vj−1 (ajk + an eigenvalue eigenvalue of of EE because because of of (1.2). (1.2). On On the the other other hand, hand, we we define define FF := :=-(v_{j}^{-1}(a_{jk}+ −λ τ bb_{jk jk}e^{e-\lambda_{0 0jk }\tau_{)vjk} )kv_{)k}1≤j,k≤n )_{1\leq j,k\leq n} . Then, every eigenvalue of EE is equivalent to every eigenvalue of FF . Indeed, since FF=V^{-1}EV = V −1 EV , where VV := diag(v · · ,, v_{n}) vn ),, we obtain :=diag 1 , ·\cdots (v_{1}, −1 EV ) = det(V −1 ) det(λI − I-E)\det E) detV= \det(\lambda V = det(λI − I-E) E). det(λI − F )I-F)=\det(\lambda = det(λI − VI-V^{-1}EV)=\det(V^{-1})\det(\lambda \det(\lambda .. Namely, λ\lambd0a_{0} is an eigenvalue of FF.. We apply Lemma 2.3 to the matrix FF and derive the following. For every eigenvalue of FF, there exists pp such that the eigenvalue lies within the disk n n. vk vk Dp := z ∈ C ; |z + app | ≤ |bpk e−λ0 τpk | . |apk | + vp k=1 vp k=1,k=p. D_{p}:= \{z\in \mathb {C};|z+a_{p }|\leq\sum_{k=1,k\neq p}^{n}|a_{pk} |\frac{v_{k} {v_{p} +\sum_{k=1}^{n}|b_{pk}e^{-\lambda_{0}\tau_{pk} |\frac{v_{k} {v_{p} \}.. From {\rm Re}\lambda_{0}\geq 0 ≥ 00 and From Reλ and (2.3), (2.3), we we compute compute n . n n n vk vk vk −λ0 τpk vk |apk | + |bpk e | = |ajk | + |bpk |e−Reλ0 τpk v v v vp p p p k=1,k=p k=1 k=1,k=p k=1. \sum^{n} |a_{pk}|\frac{v_{k} {v_{p} +\sum^{n}|b_{pk}e^{-\lambda_{0}\tau_{pk} | \frac{v_{k} {v_{p} = \sum^{n} |a_{jk}|\frac{v_{k} {v_{p} +\sum^{n}|b_{pk}|e^{- {\rm Re}\lambda_{0}\tau_{pk} \frac{v_{k} {v_{p} \leq \sum^{n} |a_{jk}|\frac{v_{k} {v_{p} +\sum^{n}|b_{pk}|\frac{v_{k} {v_{p} \leq a_{p }.. k=1,k\neq p k=1 k=1,k\neq p k=1. ≤. n . n. |ajk |. vk vk + |bpk | ≤ app . vp k=1 vp. k=1,k =p k=1,k\neq p k=1. This estimate gives that D C; |z +z+a_{pp}| app | \≤ app }.. Therefore, we conclude p ⊆ {z ∈ \mathbb{C};| D_{p}\subseteq\{z\in leq a_{pp}\} that, for every eigenvalue of FF, there is pp such that the eigenvalue lies within the disk {z C; |z +z+a_{pp}| app | \≤ app }. \{z\in∈\mathbb{C};| leq a_{pp}\}. Consequently, there exists the natural number pp such that λ\lambda_{0}\in\{z\in |z + z+a_{pp}| app | \≤ 0 ∈ {z ∈ C; \mathbb{C};| leq aa_{pp}\} = {\rm Re}\lambda_{0}\geq \lambda_{0}=0 pp }.. Furthermore, the assumption Reλ 0 ≥ 00 gives λ 0 = 0.. However, since det(A+B) \det(A+B)\neq 0, \lambd0a_{0} must satisfy λ 0, λ \lambda_{0}\neq 0 . Eventually, this fact is a contradiction and we conclude that 0 = 0. the the real real part part of of the the eigenvalues eigenvalues of of (1.2) (1.2) must must be be negative negative under under Stability Stability Condition Condition (SC). (SC). − B,. we −B. into In In the the case case of of aa reducible reducible matrix matrix A —B‐, we can can rewrite rewrite A —B‐ into aa lower lower block block triangular triangular matrix matrix with with irreducible irreducible blocks blocks along along the the diagonal diagonal similar similar to to (2.1) (2.1) by by using using aa suitable permutation matrix. Furthermore, this translation does not affect the principal − B.. Thus, minors minors of of A —B‐. Thus, the the result result follows follows by by applying applying the the previous previous argument argument to to each each \square irreducible diagonal block. 4.
(6) 41 41 3. Necessary NECESSARY condition CONDITION 3.. Next, we show Theorem 1.4. To prove this theorem, we derive three lemmas. For this this purpose, purpose, we we prepare prepare the the following following theorem theorem (cf. (cf. Bellman Bellman and and Cooke Cooke [1]). [1]). Theorem e’s theorem) (z) and Theorem 3.1. 3.1. (Rouch´ (Rouché’s theorem) If If ff(z) and g(z) g(z) are are analytic analytic inside inside and and on on aa closed closed C , and C , then ff(z) contour (z)| > |g(z)| for for each (z) and (z) + g(z) have the contour C, and |f|f(z)|>|g(z)| each point on on C, and ff(z)+g(z) C. same number of of zeros inside C. Then we will obtain the following lemmas. Lemma + B) = 0.. Then, Lemma 3.2. 3.2. Suppose Suppose det(A \det(A+B)=0 Then, (1.2) (1.2) has has aa zero zero eigenvalue. eigenvalue. Lemma 3.3. Suppose aa_{pp}<|b_{pp}| ≤ pp\leq ≤ n, or aa_{pp}=b_{pp}=0 p with 11\leq n , Then, Then, pp < |bpp | or pp = bpp = 0 for some p {\rm Re}\lambda>0. there ≤ n) > 0. jk (1 ≤ j, j,kk\leq there exist exist τ\tau_{jk}(1\leq such that that (1.2) (1.2) has has aa root root λ\lambda with with Reλ n) such Lemma 3.4. Suppose that aa_{jj}>0 ≤ jj\leq ≤ nn and any jj with 11\leq and some principal minors jj > 0 for any. A-\overline{B} of A − B is negative. Then, there exist τ (1 ≤ j, k ≤ n) such jk of is negative. Then, there exist \tau_{jk}(1\leq that (1.2) (1.2) has has aa root root λ\lambda j, k\leq n) such that {\rm Re}\lambda>0. with Reλ > 0. From these lemmas, it is easy to give a proof of Lemma 3.2, 3.3 and. prove Theorem 1.4. At the rest of this section, we 3.4.. Proof + B) = 0, λ\lambda=0 = 0 satisfies Proof of of Lemma Lemma 3.2. 3.2. Because Because of of det(A \det(A+B)=0, satisfies (1.2) (1.2) and and we we complete complete \square the proof. Proof Proof of of Lemma 3.3. We introduce the ⎛ λ + dll −λτl+1l ⎜ b l+1l e Gl11 (λ) := ⎜ .. ⎝ .. matrix bll+1 e−λτll+1 λ + dl+1l+1 .. .. ⎞ ··· bln e−λτln −λτl+1n ⎟ · · · bl+1n e ⎟ .. ... ⎠ .. G_{1}^l(\ambd):=(\begin{ary}l \ambd+_{l}b l+1}e^{-\lambd\tauio l+1} b_{ln}e^- \lambd\tau_{ln} b_{l+1}e^{-\lambd\tau_{l+\iota} \lmbda+_{l1+}\cdotsb_{l+1n}e^{- \lambd\tauio+ln} \vdots\vdots. b_{nl}e^-\ambd\tau_{nl} b_{nl+1}e^{-\lambd\tau_{nl+1} \lambd+ _{n} \end{ary}) bnl e−λτnl. bnl+1 e−λτnl+1 · · ·. λ + dnn. −λτjj for 11\leq ≤ jj\leq ≤ n. for 11\leq ≤ l l\leq ≤ n− n-11,, where dd_{jj}=a_{jj}+b_{jj}e^{-\lambda\tau_{jj}} n . Then, because of jj = ajj + bjj e 1 AA=diag = diag(a , ann ),, we have G(λ) = G11 (λ).. Applying the cofactor expansion to 11 , · ·. ·, a_{nn}) G(\lambda)=G_{11}^{1}(\lambda) (a_{11} det G(λ),, we obtain \det G(\lambda). (3.1) (3.1). n . \det G(\lambda)=(\lambda+d_{1 })\det G_{1 }^{2}(\lambda)+\sum_{j=2}^{n}(-1)^{j -1}b_{j1}e^{-\lambda\tau_{j1} \det G_{j1}^{2}(\lambda). det G(λ) = (λ + d11 ) det G211 (λ) +. (−1)j−1 bj1 e−λτj1 det G2j1 (λ),,. j=2. 2 where G j ‐th row and k-th k‐th column G(\lambda) obtained by removeing j-th G_{\dotjk {J}^{k} ^(λ) {2}(\lambda) is a submatrix of G(λ) h from G(λ). G11 (λ), G(\lambda) . Similarly, we apply the cofactor expansion to det \det G_{11}^{h}(\l ambda) , and get. (3.2) (3.2). n . \det G_{1 }^{l}(\lambda)=(\lambda+d_{l })\det G_{1 }^{l+1}(\lambda)+\sum_{j=l+ 1}^{n}(-1)^{j-l}b_{jl}e^{-\lambda\tau_{J}\prime\iota}\det G_{j-l+1 }^{l+1} (\lambda). det Gl11 (λ) = (λ + dll ) det Gl+1 11 (λ) +. (−1)j−l bjl e−λτjl det Gl+1 j−l+11 (λ). j=l+1. l+1 l for 11\leq ≤l≤ n− l\leq n-11,, where G G_{11}11^{l}((λ) \lambda) obtained by striking out G_{jkjk}^{l+1}(\(λ) lambda) isl also a submatrix of G j-th row j‐th row and and k-th k‐th column column from from G G_{11}11^{l}(\(λ). lambda) . Therefore, Therefore, using using (3.1) (3.1) and and (3.2), (3.2), we we obtain obtain the the. 5.
(7) 42 expansion for det G(λ) that \det G(\lambda) n . \det G(\lambda)=(\lambda+d_{1 })(\lambda+d_{2 })\det G_{1 }^{3}(\lambda)+\sum_ {j=2}^{n}(-1)^{j-1}b_{j1}e^{-\lambda\tau_{j1} \det G_{j1}^{2}(\lambda) +( \lambda+d_{2 })\sum_{j=3}^{n}(-1)^{j-2}b_{j2}e^{-\lambda\tau_{j2} \det G_{j- 1 }^{3}(\lambda) = \prod_{j=1}^{n}(\lambda+d_{j })+\sum_{j=2}^{n}(-1)^{j-1}b_{j1}e^{- \lambda\tau_{j1} \det G_{j1}^{2}(\lambda). det G(λ) = (λ + d11 )(λ + d22 ) det G311 (λ) +. (−1)j−1 bj1 e−λτj1 det G2j1 (λ). j=2. + (λ + d22 ). (3.3) (3.3). =. n . n . (−1)j−2 bj2 e−λτj2 det G3j−11 (λ). j=3 n . (λ + djj ) +. j=1. (−1)j−1 bj1 e−λτj1 det G2j1 (λ). j=2. k n−1 . n . + \sum_{k=2}^{n-1}\prod_{h=2}^{k}(\lambda+d_{\dot{j} )\sum_{j=h+1}^{n}(-1)^{j- h}b_{jh}e^{-\lambda\tau_{jh} \det G_{j-h+1 }^{h+1}(\lambda). +. (λ + djj ). k=2 h=2. (−1)j−h bjh e−λτjh det Gh+1 j−h+11 (λ). .. j=h+1. Here, = 2.. Summarizing Here, the the last last term term in in (3.5) (3.5) is is neglected neglected if if nn=2 Summarizing the the above, above, we we define define. f( \lambda):=\prod_{j=1}^{n}(\lambda+a_{j }+b_{j }e^{-\lambda\tau_{j } ) g( \lambda) :=\sum_{j=2}^{n}e^{-\lambda\tau\prime}\det G_{j1}^{2}(\lambda). f (λ) :=. (3.4) (3.4). n . g(λ) :=. (λ + ajj + bjj e−λτjj ),. j=1 n . ,. (−1)j−1 bj1 e−λτj1 det G2j1 (λ). j=2. n−1 k . n . + \sum_{k=2}^{n-1}\prod_{h=2}^{k}(\lambda+a_{j }+b_{j }e^{-\lambda\tau_{\dot{0} } j)\sum_{j=h+1}^{n}(-1)^{j-h}b_{jh}e^{-\lambda\tau_{jh} \det G_{j-h+1 }^{h+1} (\lambda). +. (λ + ajj + bjj e−λτjj ). k=2 h=2. (−1)j−h bjh e−λτjh det Gh+1 j−h+11 (λ), ,. j=h+1. and obtain det G(λ) = f (λ) + g(λ).. \det G(\lambda)=f(\lambda)+g(\lambda) Next, we consider ff(\lambda)=0 (λ) = 0 to indicate ff(\l(λ) zero‐solution in the right half of ambda) has the zero-solution the complex plane. Because of ff(\lambda)=0 (λ) = 0,, we find the eigenvalue λ\lambda which satisfies (3.5) (3.5). λ\lambda+a_{pp}+b_{pp}e^{-\lambda\tau}=0, + app + bpp e−λτ = 0,. where = τpp .. Now, = iω1 with where τ\tau=\tau_{pp} Now, we we show show that that (3.5) (3.5) has has aa purely purely imaginary imaginary root root λ\lambda=i\omega_{1} with some some \lambda=i\omega delay term τ = τ . Substituting λ = iω with ω ≥ 0 into (3.5) yields \omega\geq 0 \tau=\tau_{1} 1 delay term . Substituting with into (3.5) yields iω + app + bpp e−iωτ = 0. i\omega+a_{pp}+b_{pp}e^{-i\omega\tau}=0. Then, we put ff_{1}(\omega) iω + app , and obtain |f|f_{1}(0)|=|a_{pp}| →arrow\infty. ∞. 1 (ω) := 1 (0)| = |app | and lim ω→∞ |f1 (ω)| \lim_{\omegaarrow\infty}| f_{1}(\omega)| :=i\omega+a_{pp} Therefore, under the assumption |a|a_{pp}|<|b_{pp}| | < |b |, there is a positive number ω such that \omega_{11} pp pp , |f 1 (ω1 )| = =||b pp | by the intermediate value theorem. This tells us that there exists a |f_{1}(\omega_{1})| b_{pp}| −iθ positive number θ\theta such that ff_{1}(\omega_{1})=-b_{pp}e^{-i\theta} , and we get 1 (ω1 ) = −bpp e −iθ iω = 0. 1 + app + bpp e i\omega_{1}+a_{pp}+b_{pp}e^{-i\theta}=0.. Thus, choosing τ\tau_{11} such that ω\omega_{1}\tau_{1}=\theta+2\pi ∈ N0 , the pair (ω m with m m\in \mathbb{N}_{0} 1 τ1 = θ + 2πm (\omega_{1 ,1}, τ\tau_{11)}) satisfies (3.5). (3.5). We We note note that that τ\tau_{11} can can be be taken taken suitably suitably large. large. := {\rm Re}\lambda>0 The > 0.. We The next next purpose purpose is is to to show show that that (3.5) (3.5) has has aa root root λ\lambda with with Reλ We put put h(λ, h(\lambda, τ\tau)) := λ\lambda+a_{pp}+b_{pp}e^{+ app + bpp e−λτ .. Then, Then, using using (3.5), (3.5), we we have have \lambda\tau} hh_{\lambda}(\lambda, − bpp τ e−λτ = 1 + τ (λ + app ), λ (λ, τ ) = 1 \tau)=1-b_{pp}\tau e^{-\lambda\tau}=1+\tau(\lambda+a_{pp}) ,. −λτ. τ )\tau)== b−b = λ(λ app ). hh_{\tau}(\l τ (λ,ambda, _{pp}\lpp ambdaλee^{-\lambda\tau}=\l ambda(\l+ ambda+a_{pp}) .. 6.
(8) 43 By By the the implicit implicit function function theorem, theorem, we we obtain obtain aa solution solution λ(τ \lambda(\ta)u) of of (3.5) (3.5) around around τ\tau_{11} .. FurFur‐ thermore the equality. gives us that. iω1 (iω1 + app ) hτ (iω1 , τ1 ) =− hλ (iω1 , τ1 ) 1 + τ1 (iω1 + app ). \lambda'(\tau_{1})=-\frac{h_{\tau}(i\omega_{1},\tau_{1}) {h_{\lambda}(i\omega_ {1},\tau_{1}) =-\frac{i\omega_{1}(i\omega_{1}+a_{p }) {1+\tau_{1}(i\omega_{1}+a_ {p }). λ′ (τ1 ) = −. ω12 > 0. (1 + τ1 app )2 + (τ1 ω1 )2 Therefore there exists τ\tau_{2}>\tau_{1} 2 > τ1 such that Reλ(τ 2 ) > 0. {\rm Re}\lambda(\tau_{2})>0. \lambda := :=x\inx\mathbb{R} On the other hand, under the assumption aa_{pp}\leq-|b_{pp}| ∈ R and find pp ≤ −|bpp |,, we put λ the solution of. {\rm Re} \lambda'(\tau_{1})=\frac{\omega_{1}^{2} {(1+\tau_{1}a_{p })^{2}+(\tau_ {1}\omega_{1})^{2} >0.. Reλ′ (τ1 ) =. (3.6) (3.6). xx+a_{pp}+b_{pp}e^{-x\tau}=0. + app + bpp e−xτ = 0.. We put that ff_{1}(x) x + app and ff_{2}(x) −bpp e−xτ . We consider graphs of yy=f_{1}(x) = f1 (x) 1 (x) := 2 (x) := :=x+a_{pp} :=-b_{pp}e^{-x\tau} and yy=f_{2}(x) = f2 (x).. Then there exists an intersection at xx>0 > 0 of these two graphs for suitably suitably large large τ\tau .. Hence, Hence, (3.6) (3.6) has has positive positive solution solution for for large large τ\tau.. Finally, e’s theorem G(λ) = f (λ) + g(λ).. Finally, we we go go back back to to (3.4) (3.4) and and apply apply Rouch´ Rouché’s theorem to to det \det G(\lambda)=f(\lambda)+g(\lambda) We put CC := {λ ∈\mathbb{C};{\rm C; Reλ > 0} and C {λn \mathbb{C}; ∈ C;|\|λ λ0 |<\varepsi < ε}. C_{\varepsεilon} := :=\{\lambda\in Re}\lambda>0\} :=\{\lambda\i lambda-− \lambda_{0}| lon\} . Then, there exists ε\varepsilon_{0}>0 \ove¯rline{C}_{\varεepsilon} is a closure of C C_{\varepsεilon} . Since all terms of g(λ) 0 > 0 such that C \o¯verliεne{C0}_{\vare⊂ psilon_{0} \sC ubset C holds, where C g(\lambda) −λτ jk e^{ \ l a mbda\t a u_{ j k } contain e which τ\tau_{jk} (λ)| > |g(λ)| on ∂C \tau_{00} such that |f jk is not τ\tau_{pp} pp , there exists τ |f(\lambda)| >|g(\lambda)| \partial C_{\varepsεilon_0{0} provided τ\tau_{jk}>\tau_{0} \partial C_{\varepsεilon} denotes a boundary of C C_{\varepsεilon} . Therefore, jk > τ0 except for τ\tau_{jk}=\tau_{pp} jk = τpp . Here, ∂C we can apply Rouch´ e’s theorem and conclude that ff(\lambda)+g(\lambda)=0 (λ) + g(λ) = 0 has at least one Rouché’s solution in C = 0 has a root which real part is positive. \ove¯rline{C}_{\vεarepsilo0n_{0} . This means det \det G(λ) G(\lambda)=0 Furthermore, in the case that aa_{pp}=b_{pp}=0 = bpp = 0 for some pp with 11\leq ≤ pp\leq ≤ n, n , we consider pp \square the approximation and derive the desired result. Thus we complete the proof. Proof Proof of of Lemma 3.4. We modify the proof of Lemma 3.3. Since some principal minors. is negative, there exists rr with 11\leq of AA-B −B ≤ rr\leq ≤ nn such that ⎞ ⎛ −|brn | arr − |brr | · · · .. .. ... ⎠ <<0. \det ⎝ 0. (3.7) det (3.7) . . −|bnr | · · · ann − |bnn |. (\begin{ar ay}{l} a_{r }-|b_{r }| -|b_{rn}| \vdots -|b_{nr}| a_{n }-|b_{n }| \end{ar ay}). {\rm Re}\lambda>0 To > 0,, we To show show that that (1.2) (1.2) has has aa root root λ\lambda with with Reλ we consider consider the the following following function function ⎞ ⎛ −zηrr −zηrn ··· brn e κz + arr + brr e .. ... ⎠, ⎝ ... γ\gamma_{\kappa}(z):=\det κ (z) := det . −zηnr −zηnn bnr e · · · κz + ann + bnn e. where. (\begin{ar y}{l } \kap az+a_{r}+b_{r}e^{-z\eta_{r} \cdots b_{rn}e^{-z\eta_{rn} \vdots \d ots b_{nr}e^{-z\eta_{nr} \cdots \kap az+a_{n }+b_{n }e^{-z\eta_{n } \end{ar y}) . \eta_{jk}:=\{ begin{ar ay}{l} 1/2 (b_{jk}\geq0), 1 (b_{jk}<0), \end{ar ay}. ηjk :=. ,. 1/2 (bjk ≥ 0), 1 (bjk < 0),. for rr\leq ≤ j,j, kk\leq ≤ n. n . We show that γ\gamma_{\kappa}(z)=0 zero‐solution in the right half of the κ (z) = 0 has the zero-solution complex plane. For zz=x+2\pi = x + 2πii with xx\in \mathbb{R} ∈ R,, we obtain δ(x) γ0 (x + 2πi)i) and \delta(x) := :=\gamma_{0}(x+2\pi ⎞ ⎛ −xηrr −xηrn ··· −|brn |e arr − |brr |e .. .. ... ⎠. δ(x) = det ⎝ \delta(x)=\det . . −xηnr −xηnn −|bnr |e · · · an − |bnn |e. (\begin{ar y}{l a_{r}-|b_{r}|e^{-x\eta_{r} \cdots -|b_{rn}|e^{-x\eta_{rn} \d ots -|b_{nr}|e^{-x\eta_{nr} \cdots a_{n}-|b_{n }|e^{-x\eta_{n } \end{ar y}) 7.
(9) 44 Because < 0.. On jj > 0 Because of of (3.7), (3.7), we we have have δ(0) \delta(0)<0 On the the other other hand, hand, under under the the assumption assumption aa_{jj}>0 for all jj with 11\leq ≤ jj\leq ≤ n, > 0.. Therefore, by n , we derive lim x→∞ δ(x) = arr · · · anna_{nn}>0 \lim_{xarrow\infty}\delta(x)=a_{rr}\cdots the intermediate value theorem, there exists xx_{0}>0 0 > 0 such that δ(x 0 ) = 0.. Namely, \delta(x_{0})=0 . zz_{0}=x_{0}+2\pi = x + 2πi is a solution of γ (z). i 0 0 0 0}(z) \gamma_{ Because of this fact and similar argument as in Lemma 3.3, we prove Lemma 3.4 and \square complete the proof. Acknowledgments: ACKNOWLEDGMENTS: The author is partially supported by Grant-in-Aid Grant‐in‐Aid for Scientific 18K03369 from Research Research (C) (C) No. No. 18K03369 from Japan Japan Society Society for for the the Promotion Promotion of of Science. Science. REFERENCES References. [1] [1] Bellman, Bellman, R., R., Cooke, Cooke, K.L.: K.L.: Differential-difference Differential‐difference equations. equations. Academic Academic Press, Press, New New York-London York‐London (1963) (1963) [2] [2] Hale, J.K.: Theory of Functional Differential Differential Equations. Applied Mathematics Sciences, 3, 3, Springer-Verlag, Springer‐Verlag, New New York-Heidelberg York‐Heidelberg (1977) (1977) [3] [3] Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Differential Equations. Applied MatheMathe‐ matical Sciences, 99, 99, Springer-Verlag, Springer‐Verlag, New York (1993) (1993) [4] [4] Leslie Leslie Hogben.: Hogben.: Handbook Handbook of of Linear Linear Algebra Algebra 1st 1st Edition. Edition. Discrete Discrete Mathematics Mathematics and and Its Its ApplicaApplica‐ tions, Chapman and Hall/CRC Hal1/CRC (2006) [5] [5] M. Fiedler.: Special matrices and their applications in numerical mathematics, Martinus Nijhoff Nijhoff Publ. Publ. (Kluwer), (Kluwer), Dordrecht, Dordrecht, (1986) (1986) [6] [6] P. Lancaster and M. Tismenetsky, The theory of matrices (2nd (2nd ed.), Academic Press, New York(1985) York(1985) [7] [7] Ruan, Ruan, S.: S.: Absolute Absolute stability, stability, conditional conditional stability stability and and bifurcation bifurcation in in Kolmogorov-type Kolmogorov‐type predatorpredator‐ prey systems with discrete delays. Q. Appl. Math. 59(1), 159-173 159‐173 (2001) [8] ep´ an, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Pitman Re[8] St´ Stépán, Re‐ search Notes in Mathematics Series, 210, 210, Longman Scientific Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1989) (1989) Faculty FACULTY of OF Maritime MARITIME Sciences, SCIENCES, Kobe KOBE University, UNIVERSITY, Kobe KOBE 658-0022, 658‐0022, Japan JAPAN Email Email address: address: [email protected] [email protected]‐u.ac.jp. 8.
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