Stability of abstract dynamic equations on time scales

  • Alaa E Hamza1Email author and

    Affiliated with

    • Karima M Oraby2

      Affiliated with

      Advances in Difference Equations20122012:143

      DOI: 10.1186/1687-1847-2012-143

      Received: 28 February 2012

      Accepted: 25 July 2012

      Published: 12 August 2012

      Abstract

      In this paper, we investigate many types of stability, like uniform stability, asymptotic stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, uniform exponential stability, of the homogeneous first-order linear dynamic equations of the form

      x Δ ( t ) = A x ( t ) , t > t 0 , t , t 0 T x ( t 0 ) = x 0 D ( A ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equa_HTML.gif

      where A is the generator of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup { T ( t ) : t T } L ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq2_HTML.gif, the space of all bounded linear operators from a Banach space X into itself. Here, T R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq3_HTML.gif is a time scale which is an additive semigroup with the property that a b T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq4_HTML.gif for any a , b T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq5_HTML.gif such that a > b http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq6_HTML.gif. Finally, we give an illustrative example for a nonregressive homogeneous first-order linear dynamic equation and we investigate its stability.

      1 Introduction and preliminaries

      The history of asymptotic stability of dynamic equations on a time scale goes back to Aulbach and Hilger [3]. For a real scalar dynamic equation, stability and instability results were obtained by Gard and Hoffacker [12]. Pötzche [20] provides sufficient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time-varying systems on time scales. Doan, Kalauch, and Siegmund [10] established a necessary and sufficient condition for the existence of uniform exponential stability and characterized the uniform exponential stability of a system by the spectrum of its matrix. Properties of exponential stability of a time varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [7], DaCunha [9], Du and Tien [11], Hoffacker and Tisdell [16], Martynyuk [17], and Peterson and Raffoul [19].

      The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 [14], in order to unify continuous and discrete calculus [4, 15]. A time scale T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif is a nonempty closed subset of R http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq8_HTML.gif. The forward jump operator σ : T T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq9_HTML.gif is defined by σ ( t ) = inf { s T : s > t } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq10_HTML.gif (supplemented by inf = sup T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq11_HTML.gif) and the backward jump operator ρ : T T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq12_HTML.gif is defined by ρ ( t ) = sup { s T : s < t } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq13_HTML.gif (supplemented by sup = inf T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq14_HTML.gif). The graininess function μ : T R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq15_HTML.gif is given by μ ( t ) = σ ( t ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq16_HTML.gif. A point t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif is said to be right-dense if σ ( t ) = t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq18_HTML.gif, right-scattered if σ ( t ) > t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq19_HTML.gif, left-dense if ρ ( t ) = t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq20_HTML.gif, left-scattered if ρ ( t ) < t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq21_HTML.gif, isolated if ρ ( t ) < t < σ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq22_HTML.gif, and dense if ρ ( t ) = t = σ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq23_HTML.gif. A time scale T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif is said to be discrete if t is left-scattered and right-scattered for all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif, and it is called continuous if t is right-dense and left-dense at the same time for all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif. Suppose that T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif has the topology inherited from the standard topology on R http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq8_HTML.gif. We define the time scale interval [ a , b ] : = [ a , b ] T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq24_HTML.gif. Open intervals and open neighborhoods are defined similarly. A set we need to consider is T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq25_HTML.gif which is defined as T k = T { M } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq26_HTML.gif if T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif has a left-scattered maximum M, and T k = T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq27_HTML.gif otherwise. A function f : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq28_HTML.gif is called right dense continuous, or just rd-continuous, if
      1. (i)

        f is continuous at every right-dense point t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif;

         
      2. (ii)

        lim s t f ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq29_HTML.gif exists (finite) for every left-dense point t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

         

      The set of rd-continuous functions f : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq28_HTML.gif will be denoted by C rd = C rd ( T ) = C rd ( T , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq30_HTML.gif.

      A function f : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq28_HTML.gif is called delta differentiable ( or simply differentiable ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq31_HTML.gif at t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif provided there exists an α such that for every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq33_HTML.gif there is a neighborhood U of t with
      f ( σ ( t ) ) f ( s ) α ( σ ( t ) s ) ϵ | σ ( t ) s | for all  s U . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equb_HTML.gif
      In this case, we denote the α by f Δ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq34_HTML.gif; and if f is differentiable for every t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif, then f is said to be differentiable on T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif. If f is differentiable at t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif, then it is easy to see that
      f Δ ( t ) = { f ( σ ( t ) ) f ( t ) μ ( t ) , if  μ ( t ) > 0 ; lim s t f ( t ) f ( s ) t s , if  μ ( t ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equc_HTML.gif
      A function F : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq35_HTML.gif is called an antiderivative of f : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq28_HTML.gif if F Δ ( t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq36_HTML.gif, t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif. The Cauchy integral is defined by
      s t f ( τ ) Δ τ = F ( t ) F ( s ) , s , t T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equd_HTML.gif

      where F is an antiderivative of f. Every rd-continuous function f : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq28_HTML.gif has an antiderivative and F ( t ) = s t f ( τ ) Δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq37_HTML.gif is an antiderivative of f, i.e., F Δ ( t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq36_HTML.gif, t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif. Equations which include Δ-derivatives are called dynamic equations. We refer the reader to the very interesting monographs of Bohner and Peterson [5, 6].

      Definition 1.1 A mapping A : T L ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq38_HTML.gif is called regressive if I + μ ( t ) A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq39_HTML.gif is invertible for every t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif, and we say that
      x Δ ( t ) = A ( t ) x ( t ) , t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Eque_HTML.gif

      is regressive if A is regressive. We say that a real valued function p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq40_HTML.gif on T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif is regressive (resp. positively regressive) if 1 + μ ( t ) p ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq41_HTML.gif (resp. 1 + μ ( t ) p ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq42_HTML.gif), t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq43_HTML.gif. The family of all regressive functions (resp. positively regressive functions) is denoted by R http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq44_HTML.gif (resp. R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq45_HTML.gif).

      It is well known that if A B C rd R ( T , L ( X ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq46_HTML.gif, the space of all right dense continuous and regressive bounded functions from T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif to L ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq47_HTML.gif, then the initial value problem (IVP)
      x Δ ( t ) = A ( t ) x ( t ) , t T , x ( s ) = x s X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ1_HTML.gif
      (1.1)
      has the unique solution
      x ( t ) = e A ( t , s ) x s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equf_HTML.gif
      Here, e A ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq48_HTML.gif is the exponential operator function. For more details, see [2]. When X = R http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq49_HTML.gif and A ( t ) = p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq50_HTML.gif is a real valued function, Eq. (1.1) yields
      x Δ ( t ) = p ( t ) x ( t ) , t T , x ( s ) = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ2_HTML.gif
      (1.2)
      whose solution has the closed form
      x ( t ) = e p ( t , s ) = exp s t ξ μ ( τ ) ( p ( τ ) ) Δ τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ3_HTML.gif
      (1.3)
      where
      ξ μ ( z ) = { 1 μ Log ( 1 + μ z ) , if  μ > 0 ; z , if  μ = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ4_HTML.gif
      (1.4)
      and Log z = log | z | + i arg z http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq51_HTML.gif, π < arg z π http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq52_HTML.gif is the principal logarithm function. It is evident that when p ( τ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq53_HTML.gif, τ T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq54_HTML.gif, then
      ξ μ ( τ ) ( p ( τ ) ) = { 1 μ ( τ ) log ( 1 + μ ( τ ) p ( τ ) ) , if  μ ( τ ) > 0 ; p ( τ ) , if  μ ( τ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ5_HTML.gif
      (1.5)
      = lim u μ ( τ ) + log ( 1 + u p ( τ ) ) u . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ6_HTML.gif
      (1.6)
      It can be seen that for λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq55_HTML.gif with λ R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq56_HTML.gif, the following claim is true
      e λ ( t , τ ) e λ ( t τ ) , t τ , t , τ T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ7_HTML.gif
      (1.7)
      Indeed, by taking p ( τ ) = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq57_HTML.gif in Eqs. (1.5) and (1.6), we have
      ξ μ ( s ) ( λ ) = { 1 μ ( s ) ln ( 1 λ μ ( s ) ) , for  μ ( s ) > 0 ; λ , for  μ ( s ) = 0 , s T = lim u μ ( s ) + log ( 1 λ u ) u http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ8_HTML.gif
      (1.8)
      λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ9_HTML.gif
      (1.9)

      This implies that the claim is true.

      In the sequel, we denote by T R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq3_HTML.gif for a time scale which is an additive semigroup with the property that a b T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq4_HTML.gif for any a , b T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq5_HTML.gif such that a > b http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq6_HTML.gif. In this case, T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif is called a semigroup time scale. We assume X is a Banach space. Finally, we assume that T = { T ( t ) : t T } L ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq58_HTML.gif is a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup on T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif, that is, it satisfies
      1. (i)

        T ( t + s ) = T ( t ) T ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq59_HTML.gif for every t , s T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq60_HTML.gif (the semigroup property).

         
      2. (ii)

        T ( 0 ) = I http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq61_HTML.gif (I is the identity operator on X).

         
      3. (iii)

        lim t 0 + T ( t ) x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq62_HTML.gif (i.e., T ( ) x : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq63_HTML.gif is continuous at 0) for each x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif.

         
      If in addition lim t 0 + T ( t ) I = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq65_HTML.gif, then T is called a uniformly continuous semigroup. A linear operator A is called the generator [1] of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup T if
      A x = lim s 0 + T ( μ ( t ) ) x T ( s ) x μ ( t ) s , x D ( A ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ10_HTML.gif
      (1.10)

      where the domain D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif of A is the set of all x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif for which the above limit exists uniformly in t. Clearly, when T = R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq67_HTML.gif, the concept of the generator defined by relation (1.10) coincides with the classical definition by Hille. See [13].

      In Section 2 of this paper we present some results from [1] that we need in our study. One of them is that an abstract Cauchy problem
      x Δ ( t ) = A x ( t ) , t > τ , t , τ T , x ( τ ) = x τ D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ11_HTML.gif
      (1.11)
      has the unique solution
      x ( t ) = T ( t τ ) x τ , t T , t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equg_HTML.gif

      when A is the generator of the C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup T. When T = R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq68_HTML.gif, we get the classical existence and uniqueness theorem of the abstract Cauchy problem (1.11); see [21]. The other results include some properties of T and its generator A, which we use in the subsequent sections. The solution x ( t ) = x ( t , τ , x τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq69_HTML.gif is a function of the variables t, τ and the initial value x τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq70_HTML.gif. Generally, we consider τ and x τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq70_HTML.gif as parameters. Therefore, when we investigate the asymptotic behavior of x ( t , τ , x τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq71_HTML.gif with respect to T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif, we must investigate whether or not the asymptotic behavior uniformly depends on τ or x τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq70_HTML.gif. Accordingly, there are many types of stability which we give in Section 3.

      S. K. Choi, D. M. Im, and N. Koo in [8], Theorem 3.5] proved that the stability of the time variant abstract Cauchy problem
      x Δ ( t ) = A ( t ) x ( t ) , x ( t 0 ) = x 0 , t 0 T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ12_HTML.gif
      (1.12)

      where A C rd R ( T , M n ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq72_HTML.gif, n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq73_HTML.gif and M n ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq74_HTML.gif is the family of all n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq75_HTML.gif real matrices is equivalent to the boundedness of all its solutions. DaCunha in [9] defined the concepts of uniform stability and uniform exponential stability. These two concepts involve the boundedness of the solutions of the regressive time varying linear dynamic Eq. (1.12). He established a characterization of uniform stability and uniform exponential stability in terms of the transition matrix for system (1.12). Also, he illustrated the relationship between the uniform asymptotic stability and the uniform exponential stability.

      In Section 4, we extend these results for the case where A is the generator of T and we prove that the concepts of stability and uniform stability are same.

      Sections 5 and 6 are devoted to establishing characterizations for many other types of stability, like asymptotic stability, uniform asymptotic stability, global asymptotic stability, exponential stability, and uniform exponential stability for the abstract Cauchy problem (1.11).

      We end this paper with a new illustrative example including non-regressive dynamic equation and we investigate its stability.

      2 The existence and uniqueness of solutions of dynamic equations

      Our aim in this section is to prove that the first order initial value problem
      x Δ ( t ) = A x ( t ) , t T , x ( 0 ) = x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ13_HTML.gif
      (2.1)
      has the unique solution
      x ( t ) = T ( t ) x , t T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equh_HTML.gif

      when A is the generator of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup T = { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq76_HTML.gif.

      At first, we establish some properties of T and its generator A which we use to arrive at our aim.

      Theorem 2.1 For x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif, the following statements are true:
      1. 1.
        For t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif,
        lim s t 1 s σ ( t ) σ ( t ) s T ( τ ) x Δ τ = lim h 0 1 h μ ( t ) μ ( t ) + t h + t T ( τ ) x Δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ14_HTML.gif
        (2.2)
         
      = T ( t ) x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ15_HTML.gif
      (2.3)
      and
      lim h 0 1 h μ ( t ) μ ( t ) h T ( τ ) x Δ τ = x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ16_HTML.gif
      (2.4)
      1. 2.
        For t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif,
        0 t T ( τ ) x Δ τ D ( A ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equi_HTML.gif
         
      and
      A ( 0 t T ( τ ) x Δ τ ) = T ( t ) x x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ17_HTML.gif
      (2.5)
      Proof 1. Set f ( t ) = 0 t T ( τ ) x Δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq77_HTML.gif. Then
      lim s t 1 s σ ( t ) σ ( t ) s T ( τ ) x Δ τ = lim s t 1 s σ ( t ) [ 0 s T ( τ ) x Δ τ 0 σ ( t ) T ( τ ) x Δ τ ] = lim s t f ( s ) f ( σ ( t ) s σ ( t ) = f Δ ( t ) = T ( t ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equj_HTML.gif
      Also, we have
      lim h 0 1 h μ ( t ) μ ( t ) + t h + t T ( s ) x Δ s = lim h 0 f ( h + t ) f ( μ ( t ) + t ) h μ ( t ) = f Δ ( t ) = T ( t ) x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equk_HTML.gif
      and
      lim h 0 1 h μ ( t ) μ ( t ) h T ( s ) x Δ s = lim u t + 1 u σ ( t ) σ ( t ) t u t T ( s ) x Δ s = lim u t + 1 u σ ( t ) σ ( t ) u T ( s t ) x Δ s = ϕ Δ ( t ) = x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equl_HTML.gif
      where ϕ ( u ) = t u T ( s t ) x Δ s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq78_HTML.gif.
      1. 2.
        Let h > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq79_HTML.gif be a number in T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif. We have
        A 0 t T ( s ) x Δ s = lim h 0 + 1 μ ( t ) h [ 0 t T ( μ ( t ) + s ) x Δ s 0 t T ( h + s ) x Δ s ] = lim h 0 + 1 μ ( t ) h [ t + h μ ( t ) + t T ( s ) x Δ s + μ ( t ) h T ( s ) x Δ s ] = T ( t ) x x by Eqs. (2.3), (2.4). http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equm_HTML.gif
         

       □

      Theorem 2.2 For x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq80_HTML.gif, the following statements are true:
      1. 1.
        For t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif, x ( t ) = T ( t ) x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq81_HTML.gif and
        x Δ ( t ) = A T ( t ) x = T ( t ) A x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ18_HTML.gif
        (2.6)
         
      2. 2.
        For t , s T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq82_HTML.gif, we have
        T ( t ) x T ( s ) x = s t T ( τ ) A x Δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ19_HTML.gif
        (2.7)
         
      = s t A T ( τ ) x Δ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ20_HTML.gif
      (2.8)
      and
      A 0 t T ( τ ) x Δ τ = 0 t T ( τ ) A x Δ τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ21_HTML.gif
      (2.9)

      Proof 1. Let x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq80_HTML.gif. It is evident that T ( t ) x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq83_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq84_HTML.gif.

      Now, we show that x ( t ) = T ( t ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq85_HTML.gif solves the initial value problem
      x Δ ( t ) = A x ( t ) , x ( 0 ) = x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equn_HTML.gif
      We have either μ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq86_HTML.gif or μ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq87_HTML.gif. The case μ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq86_HTML.gif implies
      lim u t x ( σ ( t ) ) x ( u ) σ ( t ) u = lim u t T ( t ) x T ( u ) x t u = lim s 0 + T ( t s ) T ( s ) x x s = T ( t ) A x = A T ( t ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equo_HTML.gif
      On the other hand,
      lim u t + x ( σ ( t ) ) x ( u ) σ ( t ) u = lim u t + T ( t ) x T ( u ) x t u = lim s 0 + T ( t ) x T ( s ) x s = T ( t ) A x = A T ( t ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equp_HTML.gif
      When μ ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq87_HTML.gif, we obtain
      lim u t T ( σ ( t ) ) x T ( u ) x σ ( t ) u = T ( μ ( t ) + t ) x T ( t ) x μ ( t ) = T ( t ) T ( μ ( t ) ) x x μ ( t ) = T ( t ) A x = A T ( t ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equq_HTML.gif

      2. Relations (2.7) and (2.8) can be obtained by integrating both sides of Eq. (2.6) from s to t. Relation (2.9) follows from Eqs. (2.5) and (2.7). □

      Corollary 2.3 If A is the generator of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup T on T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif, then D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif is dense in X and A is a closed linear operator.

      Proof For every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif and fixed t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif, set
      x h = 1 h μ ( t ) μ ( t ) h T ( s ) x Δ s , h T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equr_HTML.gif
      Theorem 2.1 implies that
      μ ( t ) h T ( s ) x Δ s = 0 h T ( s ) x Δ s 0 μ ( t ) T ( s ) x Δ s D ( A ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equs_HTML.gif

      By the same theorem, x h x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq88_HTML.gif as h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq89_HTML.gif. So D ( A ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq90_HTML.gif, the closure of D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif, is equal to X. The linearity of A is evident.

      To prove its closeness, let x n D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq91_HTML.gif, x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq92_HTML.gif and A x n y http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq93_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq94_HTML.gif. In view of equality (2.7), we obtain
      T ( h ) x n T ( μ ( t ) ) x n = μ ( t ) h T ( s ) A x n Δ s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ22_HTML.gif
      (2.10)
      The integrand on the right-hand side of (2.10) converges to T ( s ) y http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq95_HTML.gif uniformly on bounded intervals. Consequently, letting n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq94_HTML.gif in (2.10), we get
      T ( h ) x T ( μ ( t ) ) x = μ ( t ) h T ( s ) y Δ s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ23_HTML.gif
      (2.11)

      Dividing Eq. (2.11) by h μ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq96_HTML.gif, h > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq79_HTML.gif and letting h 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq89_HTML.gif, we see, using identity (2.4), that x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq80_HTML.gif and A x = y http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq97_HTML.gif. □

      Theorem 2.4 Equation (2.1) has the unique solution
      x ( t ) = T ( t ) x , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equt_HTML.gif
      Proof The existence of the solution x ( t ) = T ( t ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq98_HTML.gif follows by Theorem 2.2. To prove the uniqueness, assume that V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq99_HTML.gif is another solution. Consider the function
      G ( s ) : = H t ( s ) V ( s ) , s [ 0 , t ] , s , t T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equu_HTML.gif
      where H t ( s ) = T ( t s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq100_HTML.gif. We have
      G Δ ( s ) = H t ( σ ( s ) ) V Δ ( s ) + H t Δ ( s ) V ( s ) = T ( t σ ( s ) ) A V ( s ) + H t Δ ( s ) V ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equv_HTML.gif
      On the other hand, we have
      H t Δ ( s ) x = lim u s T ( t σ ( s ) ) T ( t u ) σ ( s ) u x = lim r 0 T ( t σ ( s ) ) I T ( μ ( s ) r ) μ ( s ) r x = T ( t σ ( s ) ) A x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equw_HTML.gif

      from which we obtain that G Δ ( s ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq101_HTML.gif on [ 0 , t [ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq102_HTML.gif. Then G ( t ) = G ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq103_HTML.gif, i.e. V ( t ) = T ( t ) V ( 0 ) = T ( t ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq104_HTML.gif. □

      3 Types of stability

      In this section, the definitions of the various types of stability for dynamic equations of the form
      x Δ ( t ) = F ( t , x ) , x ( t 0 ) = x 0 X , t , t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ24_HTML.gif
      (3.1)

      are presented, where F C rd ( T × X , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq105_HTML.gif and x Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq106_HTML.gif is the delta derivative of x : T X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq107_HTML.gif with respect to t T k http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq32_HTML.gif. See [8, 18].

      Definition 3.1 Equation (3.1) is said to be stable if, for every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif and for every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq33_HTML.gif, there exists a δ = δ ( ϵ , t 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq109_HTML.gif such that, for any two solutions x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif and x ¯ ( t ) = x ( t , t 0 , x ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq111_HTML.gif of Eq. (3.1), the inequality x 0 x ¯ 0 < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq112_HTML.gif implies x ( t ) x ¯ ( t ) < ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq113_HTML.gif, for all t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq114_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

      Definition 3.2 Equation (3.1) is said to be uniformly stable if, for each ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq115_HTML.gif, there exists a δ = δ ( ϵ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq116_HTML.gif independent on any initial point t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq117_HTML.gif such that, for any two solutions x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif and x ¯ ( t ) = x ( t , t 0 , x ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq111_HTML.gif of Eq. (3.1), the inequality x 0 x ¯ 0 < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq112_HTML.gif implies x ( t ) x ¯ ( t ) < ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq113_HTML.gif, for all t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq114_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

      Definition 3.3 Equation (3.1) is said to be asymptotically stable if it is stable and for every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there exists a δ = δ ( t 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq118_HTML.gif such that, the inequality x 0 < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq119_HTML.gif implies lim t x ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq120_HTML.gif.

      Definition 3.4 Equation (3.1) is said to be uniformly asymptotically stable if it is uniformly stable and there exists a δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq121_HTML.gif such that for every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif the inequality x 0 < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq119_HTML.gif implies lim t x ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq120_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

      Definition 3.5 Equation (3.1) is said to be globally asymptotically stable if it is stable and for any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of Eq. (3.1), we have lim t x ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq120_HTML.gif.

      Definition 3.6 Equation (3.1) is said to be exponentially stable if there exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif such that for every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there is γ = γ ( t 0 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq124_HTML.gif such that, for any two solutions x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif and x ¯ ( t ) = x ( t , t 0 , x ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq125_HTML.gif of Eq. (3.1), we have x ( t ) x ¯ ( t ) γ x 0 x ¯ 0 e α ( t , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq126_HTML.gif, for all t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq114_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

      Definition 3.7 Equation (3.1) is said to be uniformly exponentially stable if there exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif and there is γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq127_HTML.gif independent on any initial point t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq117_HTML.gif such that, for any two solutions x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif and x ¯ ( t ) = x ( t , t 0 , x ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq125_HTML.gif of Eq. (3.1), we have x ( t ) x ¯ ( t ) γ x 0 x ¯ 0 e α ( t , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq126_HTML.gif, for all t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq114_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif.

      4 Characterization of stability and uniformly stability

      In this section, we obtain some results concerning characterizations of stability and uniform stability of linear dynamic equations of the form
      C P ( 0 ) : x Δ ( t ) = A x ( t ) , x ( t 0 ) = x 0 D ( A ) , t t 0 , t , t 0 T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equx_HTML.gif
      where A is the generator of T. The initial value problem C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif has the unique solution
      x ( t ) = T ( t t 0 ) x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ25_HTML.gif
      (4.1)

      In the following two lemmas, by linearity of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif, we get an equivalent definition of stability and uniform stability of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif.

      Lemma 4.1 The following statements are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is stable;

         
      2. (ii)
        For every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif and for every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq33_HTML.gif, there exists δ = δ ( ϵ , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq129_HTML.gif such that for any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif, we have
        x 0 < δ x ( t ) < ϵ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equy_HTML.gif
         
      Lemma 4.2 The following statements are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly stable;

         
      2. (ii)
        For every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq33_HTML.gif there exists δ = δ ( ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq130_HTML.gif such that for any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif, we have
        x 0 < δ x ( t ) < ϵ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equz_HTML.gif
         

      S. K. Choi, D. M. Im, and N. Koo in [8], Theorem 3.5] proved that the stability of (1.12) is equivalent to the boundedness of all its solutions when A C rd R ( T , M n ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq72_HTML.gif, n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq73_HTML.gif where M n ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq74_HTML.gif is the family of all n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq75_HTML.gif real matrices. Also, DaCunha in [9] proved that the uniform stability of (1.12) is equivalent to the uniform boundedness of all its solutions with respect to the initial point t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq117_HTML.gif, when A C rd R ( T , M n ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq72_HTML.gif.

      In the following theorem, we extend these results for the case where A is the generator of a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup T and we prove that the concepts of stability and uniform stability are the same.

      Theorem 4.3 The following statements are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is stable.

         
      2. (ii)

        { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq131_HTML.gif is bounded.

         
      3. (iii)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly stable.

         
      Proof (i) ⟹ (ii) Assume C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is stable. Let t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif. Fix ϵ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq132_HTML.gif. There exists δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq121_HTML.gif such that for any solution x ( t ) = T ( t t 0 ) x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq133_HTML.gif, where x 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq134_HTML.gif, we have
      x 0 < δ T ( t t 0 ) x 0 < 1 , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equaa_HTML.gif
      Let 0 y 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq135_HTML.gif. Take x 0 = δ y 0 2 y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq136_HTML.gif. Since x 0 < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq119_HTML.gif, then
      T ( t t 0 ) δ y 0 2 y 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equab_HTML.gif
      i.e.
      T ( t t 0 ) y 0 < 2 δ y 0 , y 0 D ( A ) t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equac_HTML.gif
      The density of D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif in X, by Corollary 2.3, implies that
      T ( t t 0 ) x < 2 δ x , x X t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equad_HTML.gif
      Thus, for every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif, { T ( t t 0 ) x : t T , t t 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq137_HTML.gif is bounded. By the uniform boundedness theorem [22], { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq138_HTML.gif is bounded.
      1. (ii)

        ⟹ (iii) Assume that there is M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq139_HTML.gif such that T ( t ) M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq140_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif. Clearly, condition (ii) of Lemma 4.2 holds, because for ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq33_HTML.gif, choose δ = ϵ / M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq141_HTML.gif. □

         

      5 A characterization of global asymptotic stability

      In the following result, we establish necessary and sufficient conditions for C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif to be globally asymptotically stable.

      Theorem 5.1 The following statement are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is asymptotically stable;

         
      2. (ii)

        lim t T ( t ) x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq142_HTML.gif, for every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif;

         
      3. (iii)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is globally asymptotically stable;

         
      4. (iv)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly asymptotically stable.

         
      Proof (i) ⟹ (ii) Suppose that C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is asymptotically stable. Let t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif. There exists γ = γ ( t 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq143_HTML.gif such that any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif with initial value x 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq134_HTML.gif, vanishes at ∞ whenever x 0 < γ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq144_HTML.gif. Fix 0 x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq145_HTML.gif. Then
      lim t T ( t t 0 ) γ x / ( 2 x ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equae_HTML.gif
      Hence,
      lim t T ( t t 0 ) x = 0 , x D ( A ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equaf_HTML.gif
      Consequently, we obtain
      lim t T ( t ) x = 0 , x D ( A ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equag_HTML.gif
      By the boundedness of { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq131_HTML.gif and the density of D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif in X, we deduce that
      lim t T ( t ) x = 0 , x X . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equah_HTML.gif
      1. (ii)

        ⟹ (iii) Condition (ii) implies that { T ( t ) x : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq146_HTML.gif is bounded for every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif. The uniform boundedness theorem insures the boundedness of { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq138_HTML.gif. Consequently, C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is stable, and by our assumption, C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is globally asymptotically stable.

         
      2. (iii)

        ⟹ (iv) Condition (iii) implies that { T ( t ) x : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq147_HTML.gif is bounded for every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq64_HTML.gif. Again the uniform boundedness theorem guarantees the boundedness of { T ( t ) : t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq148_HTML.gif. Consequently, C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly stable by Theorem 4.3, and by our assumption, C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly asymptotically stable. □

         

      6 A characterization of exponential stability and uniform exponential stability

      We need the following lemmas to establish a characterization of the exponential stability of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif. Their proofs are straightforward and will be omitted.

      Lemma 6.1 C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is exponentially stable if and only if there exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif such that for any t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there exists γ = γ ( t 0 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq124_HTML.gif such that for any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif with initial value x 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq134_HTML.gif we have
      x ( t ) γ x 0 e α ( t , t 0 ) , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equai_HTML.gif
      Lemma 6.2 C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly exponentially stable if and only if there exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif and there exists γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq149_HTML.gif such that for any t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, and any solution x ( t ) = x ( t , t 0 , x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq110_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif with initial value x 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq134_HTML.gif we have
      x ( t ) γ x 0 e α ( t , t 0 ) , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equaj_HTML.gif

      In the following two theorems, we extend the results of DaCunha [9], Theorem 2.2] when A C rd R ( T , M n ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq72_HTML.gif to the case where A is the generator of T.

      Theorem 6.3 The following statements are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is exponentially stable;

         
      2. (ii)
        There exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif such that for any t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there exists γ = γ ( t 0 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq150_HTML.gif such that
        T ( t ) γ e α ( t + t 0 , t 0 ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equak_HTML.gif
         
      Proof (i) ⟹ (ii) Let C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif be exponentially stable. Then there is α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif such that for any t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there exists γ = γ ( t 0 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq124_HTML.gif such that for any solution x ( t ) = T ( t t 0 ) x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq133_HTML.gif of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif with initial value x 0 D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq134_HTML.gif, we have
      T ( t t 0 ) x 0 γ x 0 e α ( t , t 0 ) , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equal_HTML.gif
      Fix t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, and let 0 x D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq145_HTML.gif. Then
      T ( t t 0 ) x γ x e α ( t , t 0 ) , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equam_HTML.gif
      Using D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq66_HTML.gif is dense in X and Corollary 2.3, we obtain
      T ( t t 0 ) x γ x e α ( t , t 0 ) , x X , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equan_HTML.gif
      This implies that
      T ( t ) γ e α ( t + t 0 , t 0 ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equao_HTML.gif
      (ii) ⟹ (i) Assume there exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif such that for every t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif, there exists γ = γ ( t 0 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq124_HTML.gif such that
      T ( t ) γ e α ( t + t 0 , t 0 ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equap_HTML.gif
      Let x ( t , t 0 , x 0 ) = T ( t t 0 ) x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq151_HTML.gif be any solution of C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif with initial value x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq152_HTML.gif. Then
      x ( t ) = T ( t t 0 ) x 0 T ( t t 0 ) x 0 γ e α ( t , t 0 ) x 0 , t t 0 , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equaq_HTML.gif

       □

      By same way as in the proof of Theorem 6.3, we can obtain the following result.

      Theorem 6.4 The following statements are equivalent:
      1. (i)

        C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is uniformly exponentially stable;

         
      2. (ii)
        There exists α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq122_HTML.gif with α R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq123_HTML.gif and there exists γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq149_HTML.gif such that for any t 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq108_HTML.gif,
        T ( t ) γ e α ( t + t 0 , t 0 ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equar_HTML.gif
         

      From Theorem 5.1 (Theorem 6.4), Lemma 6.1 (Lemma 6.2), and relation (1.7), we get the following result.

      Corollary 6.5 If C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is (uniformly) exponentially stable, then C P ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq128_HTML.gif is (uniformly) asymptotically stable.

      7 Example

      Choi in [8] gave an example to illustrate many types of stability. He considered the linear dynamic system
      x Δ ( t ) = A x , x ( 0 ) = x 0 , t > 0 , t T , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ26_HTML.gif
      (7.1)
      where T R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq153_HTML.gif is a time scale and A = ( 0 0 0 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq154_HTML.gif and investigated some types of stability of Eq. (7.1) when A is regressive, i.e., μ ( t ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq155_HTML.gif for all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif. In this case the equation has the unique solution x ( t ) = e A ( t , 0 ) x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq156_HTML.gif, where e A ( t , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq157_HTML.gif is the matrix exponential function. It is given by
      e A ( t , 0 ) = ( 1 0 0 e 2 ( t , 0 ) ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ27_HTML.gif
      (7.2)
      We see that the generalized exponential function e 2 ( t , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq158_HTML.gif is given by
      e 2 ( t , 0 ) = e 2 t , if  t T = R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ28_HTML.gif
      (7.3)
      and
      e 2 ( t , 0 ) = ( 1 2 h ) t h , if  t T = h Z 0 , h 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equ29_HTML.gif
      (7.4)
      The following stability results [8] for (7.1) were obtained in different cases of T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq7_HTML.gif.
      1. (1)

        If T = R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq68_HTML.gif, then (7.1) is uniformly stable, exponentially stable and asymptotically stable, since e A ( t , 0 ) = e 2 t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq159_HTML.gif as t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq160_HTML.gif.

         
      2. (2)

        If T = Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq161_HTML.gif, then (7.1) is uniformly stable but not asymptotically stable, since e A ( t , 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq162_HTML.gif.

         
      3. (3)

        If T = h Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq163_HTML.gif with 0 < h < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq164_HTML.gif and h 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq165_HTML.gif, then (7.1) is not asymptotically stable. However, e 2 ( t , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq158_HTML.gif goes to zero as t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq160_HTML.gif.

         
      4. (4)

        If T = h Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq163_HTML.gif with h > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq166_HTML.gif, then (7.1) is not asymptotically stable.

         
      Now we consider the time scale T = { n 2 : n Z 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq167_HTML.gif with the graininess function μ ( t ) = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq168_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif. So A is nonregressive and the matrix exponential function e A ( t , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq157_HTML.gif does not exist. On the other hand, A is the generator of the C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq1_HTML.gif-semigroup
      T ( t ) = ( I + 1 2 A ) 2 t , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equas_HTML.gif
      Indeed, for x R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq169_HTML.gif, we have
      lim s 0 + T ( μ ( t ) ) x T ( s ) x μ ( t ) s = lim s 0 + T ( 1 / 2 ) x T ( s ) x 1 2 s = 2 [ ( I + 1 2 A ) x x ] = A x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equat_HTML.gif
      Then
      T ( t ) = ( 1 0 0 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_Equau_HTML.gif

      Consequently, T ( t ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq170_HTML.gif, t T http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-143/MediaObjects/13662_2012_Article_290_IEq17_HTML.gif which implies that Eq. (7.1) is uniformly stable but is not asymptotically stable.

      Declarations

      Authors’ Affiliations

      (1)
      Department of Mathematics, Faculty of Science, Cairo University
      (2)
      Department of Mathematics, Faculty of Science, Suez Canal University

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