Stability conditions for scalar delay differential equations with a nondelay term
aa r X i v : . [ m a t h . D S ] J un Stability conditions for scalar delay differential equations with anon-delay term
Leonid Berezansky a , Elena Braverman b a Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel b Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary,AB T2N 1N4, Canada
Abstract
The problem considered in the paper is exponential stability of linear equations and globalattractivity of nonlinear non-autonomous equations which include a non-delay term and oneor more delayed terms. First, we demonstrate that introducing a non-delay term with a non-negative coefficient can destroy stability of the delay equation. Next, sufficient exponentialstability conditions for linear equations with concentrated or distributed delays and globalattractivity conditions for nonlinear equations are obtained. The nonlinear results are appliedto the Mackey-Glass model of respiratory dynamics.
Keywords:
Linear and nonlinear delay differential equations, global asymptotic stability,Mackey-Glass equation of respiratory dynamics
AMS Subject Classification:
1. Introduction
Stability of the autonomous delay differential equation˙ x ( t ) + bx ( t − τ ) = 0 (1.1)(the sharp asymptotic stability condition for τ > < bτ < π/
2) and of the equationwith a non-delay term ˙ x ( t ) + ax ( t ) + bx ( t − τ ) = 0 (1.2)was investigated in detail, and stability of (1.1) implies stability of (1.2) for any a ≥ x ( t ) + ax ( t ) + b ( t ) x ( h ( t )) = 0 , t ≥ , (1.3)where a > b is a locally essentially bounded nonnegative function, h ( t ) ≤ t isa delay function, is a generalization of (1.2) and also is a special case of the non-autonomousequation with two variable coefficients˙ x ( t ) + a ( t ) x ( t ) + b ( t ) x ( h ( t )) = 0 , t ≥ , a ( t ) ≥ . (1.4) Preprint submitted to Applied Mathematics and Computation March 27, 2018 et us note that, generally, asymptotic stability of the equation without the non-delay term˙ x ( t ) + b ( t ) x ( h ( t )) = 0 , t ≥ Example 1.
Consider equations (1.4) and (1.5) for b ( t ) ≡ b > and h ( t ) = [ t ] , where [ t ] isthe maximal integer not exceeding t . The equation ˙ x ( t ) + bx ([ t ]) = 0 , t ≥ is asymptotically stable for any b satisfying < b < , since the solution on [ n, n + 1] is x ( t ) = x ( n )[1 − b ( t − n )] which is a linear function on any [ n, n +1] . Thus x ( n ) = (1 − b ) n x (0) and | x ( n ) | ≤ δ n | x (0) | , where < δ = | − b | < .Let us choose . < b < . and consider the equation ˙ x ( t ) + a ( t ) x ( t ) + bx ([ t ]) = 0 , t ≥ with a periodic piecewise constant nonnegative function a ( t ) with the period T = 1 . If a ( t ) ≡ α on [0 , ε ] for < ε < then x ( t ) = (cid:18) bα + 1 (cid:19) x (0) e − αt − bα x (0) , t ∈ [0 , ε ] . Let us choose α = 3 b and ε in such a way that x ( ε ) = 0 , i.e. ε = b ln 4 , and a ( t ) = (cid:26) b, n ≤ t ≤ n + ε, , n + ε < t < n + 1 , (1.8) where n ≥ is an integer. For . < b < . we have . < ε < . , thus | x (1) | = b | x (0) | (1 − ε ) > . | x (0) | . Further, | x ( n ) | > . n | x (0) | , which means that (1.7) isunstable, while (1.6) is asymptotically stable. Fig. 1, left, illustrates the solutions of (1.6)and (1.7) with b = 1 . , x (0) = 1 , here | x ( n + 1) | ≈ . | x ( n ) | for (1.7), so (1.7) is unstablewhile (1.6) is stable.It is also possible to construct an example of asymptotically stable equation (1.6) with a ( t ) satisfying inf t> a ( t ) > such that (1.7) is unstable. For example, consider a ( t ) = (cid:26) b, n ≤ t ≤ n + ε, . , n + ε < t < n + 1 , (1.9) where b = 1 . , x (0) = 1 . As previously, x ( t ) = x ( n ) e − α ( t − n ) − x ( n ) on [ n, n + ε ] ; thesolution on [ n + ε, n + 1] is x ( t ) = 2 bx ( n )( e − . t − n − ε ) − and | x ( n + 1) | ≈ . | x ( n ) | for(1.7). In this case a ( t ) ≥ . for any t , and the solution is unstable and unbounded (seeFig. 1, right), though the divergence is slower than in the case when a is defined by (1.8). For scalar differential equation (1.3), where a > b is a locally essentiallybounded nonnegative function, h ( t ) ≤ t is a delay function, the following result is a corollaryof [1, Theorem 2.9]. 2 Figure 1: Solutions of equations (1.6) and (1.7) with b = 1 . x (0) = 1, ε ≈ . a is defined by (1.8) and can vanish (left) and a is described by (1.9) and satisfies a ( t ) ≥ . Theorem 1.
Suppose ≤ b ( t ) ≤ b , ≤ t − h ( t ) ≤ h and the inequality ab e − ah > ln b + abb + a (1.10) holds. Then equation (1.3) is exponentially stable. The aim of this paper is to extend Theorem 1 to other classes of equations, including(1.4), models with variable coefficients and several delays, as well as with distributed delays.In Section 3 we consider nonlinear delay differential equations and apply the results obtainedto the Mackey-Glass model of respiratory dynamics in Section 4.For other recent stability results, different from the results in the present paper, for linearscalar delay differential equations see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20] and in[19, 21, 22, 23, 24, 25, 26] for nonlinear equations.
2. Linear Equations
Consider the equation ˙ x ( t ) + a ( t ) x ( t ) + b ( t ) x ( h ( t )) = 0 , t ≥ , (2.1)under the following assumptions:(a1) a, b are essentially bounded on [0 , ∞ ) Lebesgue measurable nonnegative functions;(a2) h is a Lebesgue measurable function, h ( t ) ≤ t, lim t →∞ h ( t ) = ∞ .Together with (2.1) consider the initial condition x ( t ) = ϕ ( t ) , t ≤ . (2.2)3e assume that(a3) ϕ is a Borel measurable bounded function.The solution of problem (2.1)-(2.2) is an absolutely continuous on [0 , ∞ ) function satis-fying (2.1) almost everywhere for t ≥ t ≤ . Instead of the initialpoint t = 0 we can consider any t > Theorem 2.
Suppose a ( t ) ≥ a > , b ( t ) ≥ , h := lim sup t →∞ Z th ( t ) a ( s ) ds < ∞ , (2.3) and the inequality β e − h > ln β + ββ + 1 (2.4) holds, where β := lim sup t →∞ b ( t ) a ( t ) . (2.5) Then equation (2.1) is exponentially stable.Proof.
By (2.4), with the notation introduced in (2.3) and (2.5), there exists t ≥ β e − H > ln B + BB + 1 , holds, where H = sup t ≥ t Z th ( t ) a ( s ) ds, B = sup t ≥ t b ( t ) a ( t ) . Without loss of generality we can assume t = 0. After the substitution s = p ( t ) = Z t a ( τ ) dτ, y ( s ) = x ( t )(the function p ( t ) is one-to-one since a ( t ) ≥ a > y ′ ( s ) + y ( s ) + b ( p − ( s )) a ( p − ( s )) y ( l ( s )) = 0 , (2.6)where l ( s ) = R h ( p − ( s ))0 a ( τ ) dτ . Moreover, the function p ( t ) is monotone increasing and abso-lutely continuous, therefore p − ( t ) is also a continuous increasing function. Thus h ( p − ( · )), a ( p − ( · )) > b ( p − ( · )) are Lebesgue measurable functions as compositions of a contin-uous and a Lebesgue measurable function. Therefore the coefficients and the arguments inequation (2.6) are Lebesgue measurable. We have b ( p − ( s )) a ( p − ( s )) = b ( t ) a ( t ) ≤ B, s − l ( s ) = Z p − ( s ) h ( p − ( s )) a ( τ ) dτ ≤ H, s ≥ p ( t ) .
4y Theorem 1 equation (2.6) is exponentially stable. It means that there exist
M > α > y of equation (2.6) with the initial function ϕ the inequality | y ( s ) | ≤ M k ϕ k e − αs holds, where k · k is the sup-norm. Thus for the solution x ( t ) = y ( s ) ofproblem (2.1),(2.2) we have | x ( t ) | ≤ M k ϕ k e − α R t a ( τ ) dτ ≤ M k ϕ k e − αa t . Hence equation (2.1) is exponentially stable, which concludes the proof.Consider the equation with several delays˙ x ( t ) + a ( t ) x ( t ) + m X k =1 b k ( t ) x ( h k ( t )) = 0 , t ≥ , (2.7)where for the functions a, b k , h k conditions (a1)-(a2) hold. Theorem 3.
Suppose a ( t ) ≥ a > , b k ( t ) ≥ , h := lim sup t →∞ Z t min k h k ( t ) a ( s ) ds < ∞ , (2.8) and inequality (2.4) holds, where b ( t ) = P mk =1 b k ( t ) , β is defined in (2.5).Then equation (2.7) is exponentially stable.Proof. Suppose x is a solution of equation (2.7). The functions defined as h ( t ) := min ≤ k ≤ m h k ( t ) , u ( t ) := m X k =1 b k ( t ) x ( h k ( t )) , m X k =1 b k ( t ) (2.9)are both Lebesgue measurable. Define h ( t ) = inf s ∈ [ h ( t ) ,t ] { s | x ( s ) = u ( t ) } , (2.10)the fact that the set { s ∈ [ h ( t ) , t ] | x ( s ) = u ( t ) } is non-empty was justified in [2, Lemma 5].Further, let us notice that for any C > u , h defined in (2.9) and (2.10), respectively,the set { t | h ( t ) ≤ C } has the form { t | h ( t ) ≤ C } = (cid:26) t (cid:12)(cid:12)(cid:12)(cid:12) max s ∈ [ h ( t ) ,C ] x ( s ) ≥ u ( t ) or t ≤ C (cid:27) = (cid:26) t (cid:12)(cid:12)(cid:12)(cid:12) max s ∈ [ h ( t ) ,C ] x ( s ) ≥ u ( t ) (cid:27) ∪ [0 , C ] . Since x : [0 , ∞ ) → R is continuous and h ( t ) is measurable, the function max s ∈ [ h ( t ) ,C ] x ( s ) is aLebesgue measurable function of t . Therefore the set (cid:8) t (cid:12)(cid:12) max s ∈ [ h ( t ) ,C ] x ( s ) ≥ u ( t ) (cid:9) is measur-able for any C , which by definition implies that h is measurable. Since u ( t ) = x ( h ( t )) then x is a solution of equation (2.1) with nonnegative measurable coefficients and a measurabledelay which is exponentially stable by Theorem 2. Thus equation (2.7) is also exponentiallystable. 5onsider now the equation with a distributed delay˙ x ( t ) + a ( t ) x ( t ) + m X k =1 b k ( t ) Z th k ( t ) x ( s ) d s R k ( t, s ) = 0 , (2.11)where for a, b k , h k conditions (a1)-(a2) hold, ϕ in (2.2) is continuous and(a4) R k ( t, s ) are nondecreasing in s for almost all t and R t d s R k ( t, s ) ≡ k = 1 , . . . , m . Theorem 4.
Suppose a ( t ) ≥ a > , b k ( t ) ≥ , conditions (2.8) and (2.4) hold, where b ( t ) = P mk =1 b k ( t ) , β is defined in (2.5). Then equation (2.11) is exponentially stable.Proof. Suppose x is a solution of equation (2.11). By [3, Theorem 9], there exists a function g ( t ) ≤ t such that min ≤ k ≤ m h k ( t ) ≤ g ( t ) ≤ t and any solution of (2.11) is also a solution ofthe equation ˙ y ( t ) + a ( t ) y ( t ) + m X k =1 b k ( t ) ! y ( g ( t )) = 0 . (2.12)The fact that g ( t ) can be chosen as a Lebesgue measurable function is verified similarlyto the proof of Theorem 3. By Theorem 2 equation (2.12) and thus equation (2.11) areexponentially stable.Consider now the integro-differential equation˙ x ( t ) + a ( t ) x ( t ) + m X l =1 b l ( t ) Z th l ( t ) K l ( t, s ) x ( s ) ds = 0 , (2.13)where for a, b l , h l conditions (a1)-(a2) hold and(a5) K l ( t, s ) ≥ R th l ( t ) K l ( t, s ) ds ≡ l = 1 , . . . , m . Corollary 1.
Suppose a ( t ) ≥ a > , b l ( t ) ≥ , conditions (2.8) and (2.4) hold, where b ( t ) = P ml =1 b l ( t ) , β is defined in (2.5). Then equation (2.13) is exponentially stable.
3. Nonlinear Equations
Consider now the nonlinear equation˙ x ( t ) + f ( t, x ( t )) + m X k =1 g k ( t, x ( h k ( t ))) = 0 (3.1)with initial condition (2.2), where everywhere in this section we assume that the functions h k , k = 1 , . . . , m , satisfy (a2), (a3) and the following conditions hold:(a6) f ( t, u ), g k ( t, u ) are continuous, f ( t,
0) = g k ( t,
0) = 0, f ( t, u ) u > g k ( t, u ) u > u = 0 and k = 1 , . . . , m ;(a7) there exist x , x , x , x , where −∞ ≤ x ≤ ≤ x ≤ ∞ and −∞ < x ≤ ≤ x < ∞ such that for any x ≤ ϕ ≤ x there exists the unique global solution x of problem (3.1),(2.2), and it satisfies x ≤ x ( t ) ≤ x . 6 heorem 5. Suppose that there exist positive numbers a , A , b k , k = 1 , . . . , m such that forany x ≤ u ≤ x , u = 0 we have a ≤ f ( t, u ) u ≤ A, ≤ g k ( t, u ) u ≤ b k . Assume also that t − h k ( t ) ≤ h , b = P mk =1 b k and a b e − Ah > ln b + a b b + a . (3.2) Then all solutions of problem (3.1), (2.2) with x ≤ ϕ ≤ x converge to zero.Proof. Suppose x is a solution of problem (3.1), (2.2) with x ≤ ϕ ≤ x . Denote a ( t ) = ( f ( t,x ( t )) x ( t ) , x ( t ) = 0 , , x ( t ) = 0 ,b k ( t ) = ( g k ( t,x ( h k ( t ))) x ( h k ( t )) , x ( h k ( t )) = 0 , , x ( h k ( t )) = 0 , then equation (3.1) has form (2.7). All conditions of Theorem 3 are satisfied with β = b a and Ah instead of h in (2.8), hence for any solution y of equation (2.7) we have lim t →∞ y ( t ) = 0.Then lim t →∞ x ( t ) = 0.Consider now the nonlinear equation with a distributed delay˙ x ( t ) + f ( t, x ( t )) + m X k =1 Z th k ( t ) g k ( t, x ( s )) d s R k ( t, s ) = 0 , (3.3)where conditions (a2),(a4),(a6) and (a7) hold, the initial function ϕ is continuous. Theorem 6.
Assume that for any x ≤ u ≤ x , u = 0 a ≤ f ( t, u ) u ≤ A, ≤ g k ( t, u ) u ≤ b k . Assume also that t − h k ( t ) ≤ h , b = P mk =1 b k and inequality (3.2) holds. Then the zero solu-tion is an attractor of all solutions of problem (3.3), (2.2) with the initial function satisfying x ≤ ϕ ≤ x . The proof applies Theorem 4 and is similar to the proof of Theorem 5.
Remark 1.
Nonlinear integro-differential equations, mixed differential equations with con-centrated delay and integral terms are partial cases of equation (3.3). . Mackey-Glass Model of Respiratory Dynamics As an application we consider the Mackey-Glass model of respiratory dynamics (for reviewand recent results see [4]) ˙ x ( t ) = r ( t ) (cid:20) α − βx ( t ) x n ( h ( t ))1 + x n ( h ( t )) (cid:21) , (4.1)where α > , β > n > R ≥ r ( t ) ≥ r > h ( t ) ≤ t is a measurable delay function, t − h ( t ) ≤ h . Equation (4.1)has a nontrivial equilibrium K , where K is a unique positive solution determined by theequation βK n +1 = α (1 + K n ) . (4.2) Lemma 1. [4, Lemma 3.1] For any ϕ ( t ) ≥ , ϕ (0) > , problem (4.1), (2.2) has a uniqueglobal positive solution.For any ε > there exists sufficiently large t such that for t ≥ t the solution satisfies µ ε ≤ x ( t ) ≤ M ε , where µ ε = αβ − ε, M ε = αβ (cid:20) (cid:18) βα (cid:19) n (cid:21) + ε. (4.3)After the substitution y ( t ) = ln x ( t ) K equation (4.1) has the form˙ y ( t ) + r ( t ) αK (cid:0) − e − y ( t ) (cid:1) + βK n r ( t ) (cid:18)
11 + K n e − ny ( h ( t )) −
11 + K n (cid:19) = 0 . (4.4) Lemma 2. [4, Theorem 3.3] For any ε > and sufficiently large t , for any solution y ofproblem (4.4),(2.2), the inequality c ε ≤ y ( t ) ≤ C ε is satisfied, where c ε = ln µ ε K , C ε = ln M ε K , (4.5) and µ ε , M ε are denoted by (4.3). Denote µ = αβ , M = αβ (cid:20) (cid:18) βα (cid:19) n (cid:21) , c = ln µK , C = ln MK .
Theorem 7.
Suppose t − h ( t ) ≤ h , < r ≤ r ( t ) < R and inequality (3.2) holds, where a = αK − e − C C r , A = αK − e − c c R, b = βnR . Then K is a global attractor for all solutions of problem (4.1), (2.2) with ϕ ( t ) ≥ , ϕ (0) > . roof. It is sufficient to prove that y ( t ) = 0 is a global attractor for all solutions of problem(4.4),(2.2). By Lemma 2, there exist ε > t ≥ c ε ≤ y ( t ) ≤ C ε for t ≥ t , and inequality (3.2) holds if a , A are changedby a ε = αK − e − C ε C ε r , A ε = αK − e − c ε c ε , respectively, where c ε , C ε are denoted by (4.5).Equation (4.4) has form (3.1) for m = 1 with f ( t, x ) = r ( t ) αK (1 − e − x ) , g ( t, x ) = βK n r ( t ) (cid:18)
11 + K n e − nx −
11 + K n (cid:19) . In [4, the proof of Theorem 5.4] for these functions the following inequalities were justified: a ε ≤ f ( t, u ) u ≤ A ε , ≤ g ( t, u ) u ≤ b . By Theorem 6, the zero solution is a global attractor for all solutions of problem (4.4),(2.2).
Example 2.
Consider equation (4.1) with K = 1 . , α = 1 , β = 0 . , n = 4 , r ( t ) =2 . . t , t − h ( t ) ≤ h .To apply Theorem 7, we compute R = 3 , r = 2 . , µ = 2 , M = 2 . , c ≈ . , C ≈ . , a ≈ . , b = 1 . , A ≈ . and obtain that K is a global attractor if h < . .For comparison, [4, Theorem 5.4] gives the condition βh nR < e for the global at-tractivity of K which leads to the estimate h < . . The results of [5, Corollary 4] cannotbe applied since the coefficients are variable.
5. Discussion
Everywhere above for linear equations˙ x ( t ) + a ( t ) x ( t ) + b ( t ) x ( h ( t )) = 0 (5.1)we assumed a positive lower bound a ( t ) ≥ a > a , theywould also yield that the equation is stable for any greater lower bound. However, Example 1demonstrated that in a stable equation with a single delay term (1.5) which has a positivevariable coefficient, the introduction of a non-delay term with a nonnegative (or even positive)coefficient as in (1.4) may destroy its stability.Let us note that the condition Z th ( t ) b ( s ) ds < e (5.2)9uarantees that (1.5) is stable and also that (5.1) is stable for any a ( t ) ≥ z ( t ) = x ( t ) exp (cid:26)Z t a ( s ) ds (cid:27) ,we can rewrite (5.1) as˙ x ( t ) + r ( t ) z ( h ( t )) = 0 , r ( t ) = b ( t ) e − R th ( t ) a ( s ) ds , (5.3)where nonoscillation of z is equivalent to nonoscillation of x . For any a ( t ) ≥
0, equation(5.3) is nonoscillatory as (5.2) implies Z th ( t ) r ( s ) ds < e , (5.4)thus (5.1) is stable (and even nonoscillatory). The possibility to destabilize oscillatory so-lutions was illustrated in Example 1. However, it is still an open problem whether someother conditions which would guarantee that stability of (1.5) implies stability of (1.4) canbe established, where (5.2) does not hold, and the inequality 0 < b ( t ) < λa ( t ) is not satisfiedfor any 0 < λ < f ( t,
0) = g k ( t,
0) = 0, f ( t, u ) u >
0, and g k ( t, u ) u > u = 0. Such equations are obtained from a given mathematical model after the substitution x = y + K , where K is a positive equilibrium or a positive periodic/almost periodic solution.However, in (3.1) every term in the sum contains only one delay. It would be interestingto extend global stability results obtained here to more general equations, for example, ofthe form ˙ x ( t ) + f ( t, x ( t )) + m X k =1 g k ( t, x ( h ( t )) , . . . , x ( h l ( t ))) = 0 . Acknowledgment
The first author was partially supported by Israeli Ministry of Absorption. The secondauthor was partially supported by the NSERC Discovery Grant.
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