DSpace at VNU: On The Convergence of Stability Domain on Time Scales

VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 On The Convergence of Stability Domain on Time Scales Nguyen Thu Ha1, Khong Chi Nguyen2,*, Le Hong Lan3 Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam Tan Trao University, Trung Mon Affairs, Yen Son, Tuyen Quang, Vietnam Faculty of Basic Sciences, Hanoi University Transportation, Hanoi, Vietnam Received 18 August 2015 Revised 28 September 2015; Accepted 20 November 2015 Abstract: This paper studies convergence of the stability domains for a sequence of time scales It is proved that if the sequence of time scales (Tn ) converges to a time scale T in Hausdorff topology then their stability domains UTn will converge to the stability domain UT of T Keywords: Implicit dynamic equations, time scales, convergence, stability domain, stability radius Introduction∗ It is known that the linear system x(t ) = Ax(t ); t ∈ » + = [0, ∞) is exponentially stable if and only if the spectrum σ ( A) lies within the half plan ⊂ » − of the complex numbers » Also, if we consider the different system xn +1 = xn + hAxn , n ∈ h» + , then this system is exponentially stable ⇔ σ ( A) lies in the 1 disk UTh := {z :| z + |< } Where h > and h» +h = {0, h, 2h, } h h In view of theory of time scales, the sets » + , » +h are time scales and » − and UTh are respectively their stability domains Further, when h → , the sequence of time scales (» +h ) is “close” to » + and the set of the disks UTh will “enlarge” to the set » − in some sense Indeed, the Hausdorff distance d (» + , » +h ) = sup{d ( x, » +h ) : x ∈ » + } of » + and » +h is h and ∪ UT h = »+ h >0 The question rises here if we can generalize this idea to an arbitrary set of time scales (Tn ) ? That is if the sequence (Tn ) tends to the time scale T in Hausdorff topology, can we conclude that their respective stability domain converges to one of T _ ∗ Corresponding author Tel.: 84-984732576 Email: nguyenkc69@gmail.com N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 This paper concerns with such problem Firstly, we consider the continuous dependence of the exponential functions to time scales Then, we show that the stability domain UTn corresponding to the time scale Tn converges when Tn tends to T The paper is organized as follows Section summarizes some preliminary results on time scales, property of exponential functions on time scales and characterizes the stability domain of a time scale The main results of the paper are derived in Section We study the convergence of the stability domains here The last section deals with some conclusions and open problems Preliminaries 2.1 Time scales Let T be a closed subset of » , enclosed with the topology inherited from the standard topology » Let σ (t ) = inf{s ∈ T : s > t}, µ (t ) = σ (t ) − t and ρ (t ) = sup{s ∈ T : s < t},ν (t ) = t − ρ (t ) on (supplemented by sup ∅ = inf T,inf ∅ = sup T ) A point t ∈ T is said to be right-dense if σ (t ) = t , right-scattered if σ (t ) > t , left-dense if ρ (t ) = t , left-scattered if ρ (t ) < t and isolated if t is simultaneously right-scattered and left-scattered A function f defined on T is regulated if there exist the left-sided limit at every left-dense point and right-sided limit at every right-dense point A regulated function is called rd-continuous if it is continuous at every right-dense point, and ldcontinuous if it is continuous at every left-dense point It is easy to see that a function is continuous if and only if it is both rd-continuous and ld-continuous A function f from T to » is positively regressive if + µ (t ) f (t ) > for every t ∈ T We denote by R + the set of positively regressive functions from T to » Definition 2.1 (Delta Derivative) A function f : T → » d is called delta differentiable at t if there exists a vector f ∆ (t ) such that for all ε > ‖ f (σ (t )) − f (s) − f ∆ ‖ (t )(σ (t ) − s ) ≤ ε | σ (t ) − s | for all s ∈ (t − δ , t + δ ) ∩ T and for some δ > The vector f ∆ (t ) is called the delta derivative of f at t If T = » then the delta derivative is f ′(t ) from continuous calculus; if T = » then the delta derivative is the forward difference, ∆f (t ) = f (t + 1) − f (t ) , from discrete calculus Let f be a rd-continuous function and a, b ∈ T Then, the Riemann integral [1]) In a, b ∈/ T , case writing ∫ b a f ( s )∆ T s means ∫ b a ∫ b a f ( s )∆ T s exists (see f ( s) ∆ T s , a = min{t > a : t ∈ T}; b = max{t < b : t ∈ T} If there is no confusion, we write simply (resp ∫ b a f ( s )∆ n s ) for ∫ b a f ( s )∆ T s (resp ∫ b a f ( s )∆ Tn s ) where ∫ b a f ( s )∆s N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 Fix t0 ∈ » Let T be the set of all time scales with bounded graininess such that t0 ∈ T for all T ∈ T We endow T with the Hausdorff distance, that is Hausdorff distance between two time scales T1 and T2 is defined by d H (T1 , T2 ) := max{sup d (t1 , T2 ), sup d (t2 , T1 )}, t1 ∈T1 (2.1) t2 ∈T2 where d (t1 , T2 ) = inf | t1 − t2 | and d (t2 , T1 ) = inf | t2 − t1 | t2 ∈T2 t1∈T1 For properties of the Hausdorff distance, we refer the interested readers to [2, 3] 2.2 Exponential Function Let T be an unbounded above time scale, that is sup T = ∞ Definition 2.2 (Exponential stability) Let p : T → » is regressive, we define the exponential function by t e p (t , t0 ) = exp{∫ lim t0 h µ (s) Ln(1+hp(s)) ∆s}, h where Lna is the principal logarithm of the number a Theorem 2.3 (see [4]) If p is regressive and t0 ∈ T , then e p (., t0 ) is a unique solution of the initial value problem y ∆ (t ) = p (t ) y (t ), y (t0 ) = When p(t ) = λ , where λ is a constant in » , we write eλ (t , s ) for e p (·) (t , s ) Theorem 2.4 (Properties of the Exponential Function) If continuous functions and t , r , s ∈ T then the following hold: p, q : T → » are regressive, rd- e0 (t , s ) = 1, and e p (t , t ) = ; e p (σ (t ), s ) = (1 + µ (t ) p (t ))e p (t , s ) ; = e− p (t , s ) = e p ( s, t ) e p (t , s ) e p (t , s )eq (t , s ) = e p + q (t , s ) e p (t , s)e p ( s, r ) = e p (t , r ) e p (t , s ) eq (t , s ) = e p − q (t , s ) Lemma 2.5 (Gronwall-Bellman lemma, see [4]) Let f (t ) be a positive continuous function and k > 0, f (t0 ) ∈ » Assume that f (t ) satisfies the inequality f (t ) ≤ f + k ∫ t t0 f ( s ) ∆s , for all t ∈ T, t ≥ t0 (2.2) N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 Then, the following relation holds, f (t ) ≤ f ek (t , t0 ), for all t ∈ T,t ≥ t0 (2.3) We see that t | eλ (t , t0 ) |= exp{∫ lim µ (s) t0 h ln |1 + hλ | ∆s} h By using the notation ζ λ ( s ) = lim h ln | + hλ | s h if s = ℜλ  =  ln | + sλ | if s ≠ 0,  s we can rewrite t | eλ (t , t0 ) |= exp{∫ ζ λ ( µ ( s ))∆s} t0 (2.4) We note that | eλ (t , s ) |=| eλ (t , s ) | for any λ ∈ » Further, it is easy to see that ζ λ ( x) ≤ | λ | for all x ≥ For the properties of exponential function eλ (t , s ) the interested readers can refer to [5] 2.3 Exponential stability Let Tt0 = {t ∈ T : t ≥ t0 } Consider a dynamic equation x ∆ = f (t , x), t ≥ t0 (2.5) We assume that the function f : Tt0 × » m → » m satisfies conditions such that Equation (2.5) has a unique solution x(t , s, x0 ), t ≥ s with the initial condition x( s, s, x0 ) = x0 for any s ∈ Tt0 and x0 ∈ » m Definition 2.1 (Exponential stability) The dynamic equation (2.5) is called uniformly exponentially stable if there exist constants α > with −α ∈ R + and K > such that for every s < t , s, t ∈ Tt0 , the inequality ‖x(t, s, x )‖≤ K‖x ‖e 0 −α (t , s) (2.6) holds for any x0 ∈ » m In the linear homogeneous case, i.e., f (t , x) = Ax we have the equation x ∆ (t ) = Ax(t ), t ≥ t0 (2.7) It is known that Equation (2.7) is uniformly exponentially stable if and only if so is for the scalar equation x ∆ (t ) = λ x(t ), t ≥ t0 , (2.8) for any λ ∈ σ ( A) (see [6, 7]) Denote by UT the set of complex values λ such that (2.8) is uniformly exponentially stable We call UT the domain of uniformly exponential stability (or stability domain for short) of the time scale T Denote N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 L(λ ) := lim sup t − s →∞ t ζ λ ( µ (τ ))∆τ < t − s ∫s (2.9) Propotion 2.7 Let λ ∈ » , then λ ∈ UT if and only if L(λ ) := lim sup t − s →∞ t ζ λ ( µ ( s ))∆s < t − s ∫s (2.10) Proof Assume that (2.7) is uniformly exponentially stable It implies that there exist constants α > 0, K > such that ln |1 + hλ | t | eλ (t , s ) |= exp{∫ lim s h ∫ t s h ln |1 + hλ | lim h µ (τ ) µ (τ ) h ∆τ } ≤ Ke −α ( t − s ) , forall t ≥ s ∆τ ≤ −α (t − s ) + lnK , forall t ≥ s Therefore, we have t ∫ ζ λ (µ ( s))∆s < −α < t − s →∞ t − s s Assume that L(λ ) := −α < Then there is an integer number N large enough such that lim sup t ∫ζ t−s s λ ( µ ( s )) ∆s < Thus | eλ (t , s ) | < | e −α −α , forall t − s > N (t − s ) |, ∀t − s > N By virtue of the inequality ζ λ ( x) ≤| λ | for any x ≥ , we have t | eλ (t , s ) |= exp{∫ ζ λ ( µ ( s ))∆s} ≤ e|λ | N , ∀t − s < N s Hence | eλ (t , s ) |≤ Ke with K := e |λ | N + −α ( t − s ) , ∀t ≥ s αN > The proposition is proved □ Further, it is easy to verify that if | b1 | ≥ | b2 | then ζ a + ib1 ( x) ≥ ζ a + ib2 ( x) for any a ∈ » and x ≥ So UT is symmetric with respect to the real axis of the complex plan » This means that λ ∈ UT implies the segment [λ , λ ] ⊂ UT Propotion 2.8 Let T be a time scale with bounded graininess, then stability domain UT is an open set in » Proof Let λ ∈ UT ,then there are K > and λ ∈ R + such that | eλ (t , s ) |≤ Ke −α ( t − s ) , forall t ≥ s (2.11) We now proof that there exists ε > such that the equation x∆ = β x (2.12) N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 is also uniformly exponentially stable for all β ∈ » with| β − λ |≤ ε We rewrite Equation (2.12) under the form x ∆ = λ x + ( β − λ ) x Using variation of constants formula yields t eβ (t , s ) = eλ (t , s ) + ∫ eλ (t , σ (u ))( β − λ )eβ (u , s )∆u (2.13) s This implies that t | eβ (t , s ) |≤ Ke −α (t − s ) + K | β − λ | ∫ |eλ (t ,σ (u ))eβ (u , s ) | ∆u s Hence, t eα ( t − s ) | eβ (t , s ) |≤ K + K | β − λ | ∫ eαµ (u ) eα ( u − s ) | eβ (u , s ) | ∆u s Let f (t ) = eα ( t − s ) | eβ (t , s ) | , we have t f (t ) ≤ K + K | β − λ | eα H ∫ f (u )∆u s Using Gronwall's inequality (with f ( s ) = ) obtains f (t ) ≤ Ke M (t , s ) |, forall t ≥ s , where M = K | β − λ | eα H Thus | eβ (t , s ) |≤ Ke ( −α + M )( t − s ) , forall t ≥ s By choosing ε = α αH Ke proof is complete we get that | eβ (t , s ) |≤ Ke − α (t − s ) , forall t ≥ s , for any β such that | β − λ |≤ ε The □ Main results In this section, we consider a sequence {Tn }n∈» ⊂ T of time scales satisfying: lim Tn = T n→∞ Denote by µn (t ) (resp µ (t ) ) the graniness of Tn (resp T ) at time t Since T ∈ T , sup{µ (t ) : t ∈ T} < ∞ Therefore, it is easy to prove that if lim Tn = T then n→∞ sup{µn (t ) : t ∈ Tn , n ∈ »} < ∞, {sup{µ (t ) : t ∈ T, n ∈ »} < ∞ Denote µ * = max{sup{µ (t ) : t ∈ T},sup{µ n (t ) : t ∈ Tn , n ∈ »}} First, we need the following lemmas to derive some characteristics of stability domains UTn when Tn tends to T Lemma 3.1 For any λ ∈ » » we have L(λ ) ≤ if and only if λ ∈ UT N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 Proof Denote µ* = sup{µ (t ) : t ∈ T} and let λ ∈ UT » Then, there is a sequence {λn } ⊂ UT such that lim λn = λ Let λ = a + ib with b ≠ and λn = an + ibn Using Lagrange finite increments n →∞ formula, for all x > we have ζ λ ( x) − ζ λ ( x) = n x(| λn |2 − | λ |2 ) + 2(an − a ) , θ ∈ (0,1) 2(1 + x(a + θ (an − a)) + x (| λ |2 +θ (| λn |2 − | λ |2 ) Since < + xa + x | λ |2 for all x ≥ , we can choose a n0 ∈ » and a constant c1 > such that c1 < + x(a + θ (an − a )) + x (| λ |2 +θ (| λn |2 − | λ |2 ) for all ≤ x ≤ µ* and n > n0 Thus, for any ε > , there exists n1 > n0 satisfying ζ λ ( x) − ζ λ ( x) < ε, ∀ ≤ x ≤ µ* , ∀ n > n1 n This implies that t t s s ∫ ζ λ (µ (τ ))∆τ < ∫ ζ λ (µ (τ ))∆τ + ε(t − s) < ε(t − s), n ∀ t0 ≤ s ≤ t , ∀ n > n1 Hence lim sup t − s →∞ t ζ λ ( µ (τ ))∆τ < ε, ∀ε > t − s ∫s Thus t ∫ ζ λ (µ (τ ))∆τ ≤ t − s →∞ t − s s Conversely, let λ = a + ib ∈ » » such that lim sup t ∫ ζ λ (µ (τ ))∆τ ≤ t − s →∞ t − s s For any ε > , let λε = aε + ibε be chosen such that | λ | −ε Since lim sup < + xa + x | λ |2 , we can choose aε and bε such that < 2(1 + x(a + θ (aε − a )) + x (| λ |2 +θ (| λε |2 − | λ |2 )) < c2 for all ≤ x ≤ µ* Thus, ζ λ ( x) − ζ λ ( x) < ε aε − a , ∀0 ≤ x ≤ µ* c2 This implies that t t ∫ ζ λ ( µ (τ ))∆τ < ∫ ζ λ ( µ (τ ))∆τ + s ε s aε − a (t − s), ∀t0 ≤ s ≤ t c2 Hence, lim sup t − s →∞ a −a t ζ λε ( µ (τ ))∆τ ≤ ε < ∫ s t−s c2 which follows that λε ∈ UT This means that λ ∈ UT The proof is completed □ N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 Lemma 3.2 Let K ⊂ » » be a compact set Suppose that lim Tn = T, then for any ε > , there n→∞ are δ > 0, n0 ∈ » such that | ∫ ζ λ (µ (h))∆ h − ∫ ζ λ (µ (h))∆h |< 2ε(t − s) + 8M t − s d t s t n n δ s for n > n0 , λ ∈ K and M = sup λ∈K , x∈[0, µ * ] Proof Since K ⊂ » H (T, Tn ), ∀s < t , (3.1) | ζ λ ( x) | » is a compact set, M < ∞ First, assume that Tn ⊂ T We see that the function  ℜλ + x | λ |2 ln(1 + xℜλ + x | λ |2 ) − dζ λ ( x)   (1 + xℜλ + x | λ |2 ) x x2 = dx  | λ | − (ℜ λ )  if x > 0, if x = 0, is continuous in ( x, λ ), provided ℑλ ≠ Therefore, the family of functions (ζ λ (u ))λ∈K is equicontinuous in u on [0, µ * ], i.e., for any ε > , there exists δ = δ (ε ) > such that if | u − v |< δ then | ζ λ (u ) − ζ λ (v) |< ε for any λ ∈ K Since lim Tn = T , we can choose n0 such that d H (T, Tn ) < n→∞ δ when n > n0 Fix t0 ≤ s < t; s, t ∈ [0, ∞) and n > n0 Denote A1 = {h ∈ Tn ∩ [ s, t ]: µn (h) ≥ δ }, A2 = {h ∈ Tn ∩ [ s, t ] : µ n (h) < δ } The assumption Tn ⊂ T implies that ≤ µ (h) ≤ µ n (h) for all h ∈ Tn If h ∈ A2 then µ (h) ≤ µn (h) < δ , which implies | ζ λ ( µ (h)) − ζ λ ( µ n (h)) |< ε On the other hand, the cardinal of A1 , t − s  say r, is finite and r ≤  Thus, we can write A1 = {s1 < s2 < … < sr }  δ  Denote sequence τ i by si + σ n ( si ) };and τ i = σ (τ i ), i = 1, 2,…, r Since d H (T, Tn ) ≥ max{| τ i − si |,| σ n ( si ) − τ i |} , it follows that τ i = max{h ∈ T : h ≤ | τ i − si |< δ ,| σ n ( si ) − τ i |< δ Therefore, | µ (τ i ) − µn ( si ) |= µn ( si )− | τ i − τ i |=| τ i − si | + | σ n ( si ) − τ i |< δ , which implies | ζ λ ( µ (τ i )) − ζ λ ( µn ( si )) |< ε N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 For any h ∈ T , there exists a unique u ∈ Tn , say u = γ T,Tn (h) , h ∈ (u , σ n (u )) It is easy to check that the function γ T , Tn such that either h = u or (h) is rd-continuous on T By definition of integral on time scales we have t t ∫ ζ λ (µ (h))∆ h = ∫ ζ λ (µ (γ n s n n s T , Tn (h)))∆h Therefore, t t t s s | ∫ ζ λ (µ (h))∆ h − ∫ ζ λ (µ (h))∆h |≤ ∫ |ζ λ ( µ (γ n s n r = ∫ |ζ λ ( µ (h)) − ζ λ ( µn (γ T,Tn (h))) | ∆h + ∑ (∫ s1 s T , Tn n t ∧τ i i =1 (h))) − ζ λ ( µ (h)) | ∆h |ζ λ ( µ (h)) − ζ λ ( µ n (γ T,Tn (h))) | ∆h si r s1 = ∫ |ζ λ ( µ (h)) − ζ λ ( µn (γ T , Tn s +∫ t ∧τ i t ∧σ n ( si ) +∑ ∫ si +1 σ n ( si ) i =1 t ∧τ i si i =1 |ζ λ ( µ (h)) − ζ λ ( µn (γ T,Tn (h))) | ∆h + ∫t ∧τ t ∧τ i r −1 (h))) | ∆h + ∑ ( ∫ |ζ λ ( µ ( h)) − ζ λ ( µ n (γ |ζ λ ( µ ( h)) − ζ λ ( µ n (γ T , Tn T , Tn ( h))) | ∆h ( h))) | ∆h) i |ζ λ ( µ (h)) − ζ λ ( µ n (γ T,Tn (h))) | ∆h , where a ∧ b = min{a, b} Since | µ (h) − µ (γ T,Tn (h)) |< δ for all h ∈ [t0 , s1 ) ∪ [σ ( si ), si +1 ∧ t ),1 ≤ i ≤ r − 1, ∫ s1 ∫ si +1 s |ζ λ ( µ (h)) − ζ λ ( µn (γ T,Tn (h))) | ∆h < ε (τ − s ) , σ n ( si ) |ζ λ ( µ ( h)) − ζ λ ( µ n (γ T , Tn ( h))) | ∆h < ε ( si +1 − σ n ( si )) On the other hand, for i = 1,2, , r we have ∫ t ∧τ i t ∧τ i |ζ λ ( µ (h)) − ζ λ ( µn (γ T,Tn (h))) | ∆h = (t ∧ τ i − t ∧ τ i ) | ζ λ ( µ (τ i )) − ζ λ ( µn (γ T,Tn (τ i ))) | = (t ∧ τ i − t ∧ τ i ) | ζ λ ( µ (τ i )) − ζ λ ( µn ( si )) |< ε (t ∧ τ i − t ∧ τ i ), and ∫ t ∧τ i ∫ t ∧σ n ( si ) si |ζ λ ( µ (h)) − ζ λ ( µn (γ T,Tn (h))) | ∆h ≤ 2M (t ∧ τ i − si ) ≤ Md H (T, Tn ) t ∧τ i |ζ λ ( µ ( h)) − ζ λ ( µn (γ T, Tn (h))) | ∆h ≤ M (t ∧ σ n ( si ) − t ∧ τ i ) ≤ 2Md H (T, Tn ) Thus, we obtain t ∫ |ζ s λ ( µ ( h)) − ζ λ ( µ n (γ T , Tn ( ( h))) | ∆h < ε (τ − s ) + r −1 r ∑ (t ∧ τ i − t ∧ τ i ) + ∑ ( si +1 − σ n ( si ))) i =1 i =1 r t−s i =1 δ +4 M ∑ d H (T, Tn ) < ε (t − s ) +4 Mrd H (T, Tn ) < ε (t − s) + M Therefore, d H (T, Tn ) N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 10 t | ∫ ζ λ (µ s n t t−s s δ (h))∆ n h − ∫ ζ λ ( µ (h))∆h |< ε (t − s ) + 4M d H (T, Tn ) If Tn ⊂/ T , we put T n = Tn ∪ T It is easy to see that d H (T, Tn ) = max{d H (T n , T), d H (T n , Tn )} (3.2) By the above proof, we have t | ∫ ζ λ (µ s n t t−s s δ ( h))∆ n h − ∫ ζ λ ( µ (h))∆ Tn h |< ε (t − s) + M | ∫ ζ λ (µ (h))∆h − ∫ ζ λ ( µ (h))∆ h |< ε(t − s) + 4M t − s d t t s s δ Tn d H (T, Tn ), H (T, Tn ) This implies that t | ∫ ζ λ (µ s n t t−s s δ ( h))∆ n h − ∫ ζ λ ( µ (h))∆h |< 2ε (t − s) + 8M The proof is complete d H (T, Tn ) □ Denote by UTn (resp UT ) the domain of stability of the time scale Tn (resp T ) Proposition 3.3 Suppose that lim Tn = T Then, for any λ ∈ UT we can find a neighborhood n →∞ B (λ , δ ) of λ and nλ > such that B (λ , δ ) ⊂ UT ∩ U Tn n > nλ Proof Firstly, we prove the proposition with λ ∈ UT » Following the proof of Lemma 3.1 and by Proposition 2.8, there exists a δ1 > satisfying B (λ , δ1 ) ⊂ UT and − L(λ ) ζ λ ( x) − ζ λ ( x) < ; ∀ ≤ x ≤ µ* , ∀ λ ∈ B (λ , δ1 ) Hence, by (2.10) L (λ ) ≤ L (λ ) + − L (λ ) = L (λ ) for any λ ∈ B (λ , δ1 ) (3.3) | ℑλ | } > we see that B (λ , δ λ ) ⊂ UT » Using Lemma 3.2 with − L (λ ) K = B (λ , δ λ ) and ε = we can find a δ > and n0 such that t t − L(λ ) t−s (3.4) ∫s ζ λ ( µn (h))∆ n h < ∫s ζ λ ( µ (h))∆h + (t − s) + 8M δ d H (T, Tn ), ∀ s < t , By choosing δ λ := min{δ1 , for n > n0 and λ ∈ B(λ , δ λ ) We choose nλ > n0 such that d H (T, Tn ) < From (3.4) we get lim sup t − s →∞ t ∫ζ t−s s λ ( µ n ( h))∆ n h ≤ lim sup t − s →∞ t ∫ζ t−s s λ ( µ (h))∆h − L(λ ) − L(λ ) −δ L(λ ) for any n > nλ 32 M N.T Ha et al / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 1-12 ≤ 3L(λ ) − L( λ ) − L (λ ) = L(λ ) 11 < 0, ∀ n > n0 , λ ∈ B (λ , δ λ ) This means that B (λ , δ λ ) ⊂ UTn for all n > nλ We now consider the case λ ∈ UT ∩ » Since UT is open set, there exists δ > such that δ δ B (λ , δ ) ⊂ UT Let λ1 = λ + i Following above argument, there exist nλ1 > and < δ λ < such 2 that B (λ1 , δ λ1 ) ⊂ UT ∩ UTn for all n > nλ1 Since UTn is symmetric with respect to the real axis, the segment [λ ′, λ ′] ⊂ UTn , for all λ ′ ∈ B (λ1 , δ λ1 ) Thus B (λ , δ λ1 ) ⊂ UT ∩ UTn for all n > nλ1 The proposition is proved □ Theorem 3.4 If lim Tn = T then n →∞ ∞ UT ⊂ ∪ ∩ U Tm and n =1 m ≥ n ∞ ∩ ∪ UT » ⊂ UT m » (3.5) n =1 m ≥ n Proof The first relation follows immediately from Proposition 3.3 ∞ To prove the second one, let λ ∈ ∩ ∪ U Tm » By definition, there is a sequence {nk } → ∞ n =1 m ≥ n » for all k Using again inequality (3.1), for any ε > , yields such that λ ∈ UTnk lim sup t − s →∞ t ∫ζ t−s s λ ( µ (τ ))∆τ ≤ lim sup t − s →∞ t ∫ζ t−s s λ ( µ n (τ )) ∆ n τ + 2ε + k k 8M δ d H (Tn , T) k (3.6) Taking limit as nk → ∞ we obtain lim sup t − s →∞ t ζ λ ( µ (τ ))∆τ < 2ε, ∀ ε > t − s ∫s This implies that lim sup t − s →∞ t ζ λ ( µ (τ ))∆τ ≤ t − s ∫s Thus, λ ∈ U T » by Lemma 3.1 The proof is complete □ Conclusion In this paper, we study the convergence of the stability domains on the time scales in the sense of Hausdorff topology We prove that if lim Tn = T in Hausdorff distance then n →∞ UT ⊂ lim inf UTn and lim sup( UTn n →∞ » ) ⊂ UT » n →∞ So far the question whenever lim UTn = UT is still an open problem n →∞ 12 N.T Ha et al / 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Lett (1986) 105-113  ... of exponential functions on time scales and characterizes the stability domain of a time scale The main results of the paper are derived in Section We study the convergence of the stability domains... Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkhäuser, Boston, 2001 [5] M Bohner and A Peterson, Advances in dynamic equations on time scales, Birkhäuser, Boston,... that lim sup t − s →∞ t ζ λ ( µ (τ ))∆τ ≤ t − s ∫s Thus, λ ∈ U T » by Lemma 3.1 The proof is complete □ Conclusion In this paper, we study the convergence of the stability domains on the time scales