See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/283038283 On the Convergence of Solutions to Dynamic Equations on Time Scales Article in Qualitative Theory of Dynamical Systems · October 2015 DOI: 10.1007/s12346-015-0166-8 CITATION READS 92 4 authors, including: Nguyen Huu Du Vietnam National University, Hanoi 65 PUBLICATIONS 444 CITATIONS SEE PROFILE All content following this page was uploaded by Nguyen Huu Du on 29 December 2015 The user has requested enhancement of the downloaded file All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately On the Convergence of Solutions to Dynamic Equations on Time Scales Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi & Do Duc Thuan Qualitative Theory of Dynamical Systems ISSN 1575-5460 Qual Theory Dyn Syst DOI 10.1007/s12346-015-0166-8 23 Your article is protected by copyright and all rights are held exclusively by Springer Basel This e-offprint is for personal use only and shall not be self-archived in electronic repositories If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website The link must be accompanied by the following text: "The final publication is available at link.springer.com” 23 Author's personal copy Qual Theory Dyn Syst DOI 10.1007/s12346-015-0166-8 Qualitative Theory of Dynamical Systems On the Convergence of Solutions to Dynamic Equations on Time Scales Nguyen Thu Ha1 · Nguyen Huu Du2 · Le Cong Loi2 · Do Duc Thuan3 Received: 23 February 2015 / Accepted: 15 September 2015 © Springer Basel 2015 Abstract In this paper, we study the convergence of solutions to dynamic equations x = f (t, x) on time scales {Tn }∞ n=1 when this sequence converges to the time scale T The convergent rate of solutions is estimated when f satisfies the Lipschitz condition in both variables By a general view, we derive a new approach to the approximation of dynamic equations on time scales, especially the Euler method for differential equations Some examples are given to illustrate our results Keywords Dynamic equations · Time scales · Grownall–Bellman inequality · Convergence of solutions Mathematics Subject Classification B 06B99 · 34D99 · 47A10 · 47A99 · 65P99 Nguyen Huu Du nhdu@viasm.edu.vn; dunh@vnu.edu.vn Nguyen Thu Ha ntha2009@yahoo.com Le Cong Loi loilc@vnu.edu.vn Do Duc Thuan ducthuank7@gmail.com Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet Str., Hanoi, Vietnam Author's personal copy N T Ha et al Introduction Ordinary differential equations (ODEs) occur in many scientific disciplines, for instance in physics, chemistry, biology, and economics Therefore, finding solutions of ODEs is important both in theory and practice Unfortunately, almost ODEs can not be solved analytically, which causes in science and engineering, to find a numerical approximation to the solutions The Euler method is well-known because it is simple and useful, see [4,8,9,17] For solving the stiff initial value problem x(t) ˙ = f (t, x(t)), x(t0 ) = x0 , (1.1) at each step [tm , tm+1 ], the explicit Euler approximation of (1.1) is xm+1 = xm + h f (tm , xm ), (1.2) where tm+1 = tm + h and xm is the approximative value of x(t) at t = tm The quantity en := |x(tm ) − xm | is called the error of this method after n time steps which characterizes the difference between the approximative solution and the exact solution The interesting problem is how the error en can be estimated when the mesh step h tends to zero With some added assumptions on f , we have known that en tends to zero as h tends to zero, see [8,9,13] In recent years, in order to unify the continuous and discrete analysis, the theory of the analysis on time scales was introduced by Stefan Hilger in his PhD thesis (supervised by Bernd Aulbach) [14] and has received a lot of attention, see [1,2,6,10, 11,15] By using the notation of the analysis on time scales, Eqs (1.1) and (1.2) can be rewritten by the form x (t) = f (t, x(t)), (1.3) x(t0 ) = x0 , with the time t belongs to the time scales T = R or Th = hZ Thus, in view of analysis on time scale, using the Euler method means we consider Eq (1.1) on the time Th , which is “close” to T = R in some sense Then the problem of the error estimation above can be restated as follows: How the solutions of (1.3) on Th converge to the solution of (1.3) on T = R as the mesh step h tends to zero? Following this idea in a more general context, in this paper, we will consider the behavior of solutions of Eq (1.3) on time scales {Tn }∞ n=1 when Tn tends to T by the Hausdorff distance Assume that on time scales {Tn }∞ n=1 Eq (1.3) has the solutions and on time scale T Eq (1.3) has the solution x(t) Then, we will prove {xn (t)}∞ n=1 that (1.4) xn (t) → x(t) as Tn → T, under the assumption that f (t, x) satisfies the Lipschitz condition in the variable x Moreover, if f satisfies the Lipschitz condition in both variables t and x then the convergent rate of solutions is estimated as the same degree as the Hausdorff distance between Tn and T, i.e., Author's personal copy On the Convergence of Solutions to Dynamic Equations xn (t) − x(t) C2 d H (T, Tn ), for all t ∈ T ∩ Tn : t0 t T (1.5) Using these results, we obtain the convergence of the Euler method as a consequence It can be considered as a new and general approach to the convergence problems of the approximative solutions This paper is organized as follows Section summarizes some preliminary results on time scales In Sect 3, we study the convergence of solutions of dynamic equations on time scales The main results of the paper are derived here In Sect 4, we give some examples and show the convergence of the Euler method The last section deals with some conclusions and open problems Preliminaries Let T be a closed nonempty subset of R, endowed with the topology inherited from the standard topology on R 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) (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 called r d-continuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left dense points f is ld-continuous if it is continuous at every left-dense point and if the right-sided limit exists in every right-dense point It is easy to see that a function is continuous if and only if it is both r d-continuous and ld-continuous Definition 2.1 (Delta derivative) A function f : T → Rd is called delta differentiable at t if there exists a vector f (t) such that for all > f (σ (t)) − f (s) − f (t)(σ (t) − s) |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 = R then the delta derivative is f (t) from continuous calculus; if T = Z then the delta derivative is the forward difference, f (t) = f (t + 1) − f (t), from discrete calculus For any r d-continuous function p(·) from T to R, the solution of the dynamic equation x = p(t)x, with the initial condition x(s) = 1, defines a so-called exponential function We denote this exponential function by e p (T; t, s) For the properties of the exponential function e p (T; t, s) the interested reader can see [2,6] To simplify notation, we write e p (T; t, s) for e p (t, s) if there is no confusion Let f be a rd-continuous function and a, b ∈ T Then, the Riemann integral b a f (s) s exists (see [12]) In case b ∈ / T, writing where b = max{t < b : t ∈ T} b a f (s) s means b a f (s) s, Author's personal copy N T Ha et al Consider the dynamic equation on the time scale T x (t) = f (t, x), x(t0 ) = x0 , (2.1) where f : T × Rd → Rd For the existence, uniqueness and extensibility of the solution of Eq (2.1) we refer to [5] In particular, for any positive regressive number α, the Cauchy problem x (t) = αx(t); x(t0 ) = has a unique solution, namely eα (t, t0 ) satisfying the estimate < eα (t, t0 ) eα(t−t0 ) , (2.2) (see [1,2]) It is easy to see that if f (t, x) is a continuous function in (t, x) then x(t) is a solution to (2.1) if and only if x(t) = x0 + t f (s, x(s)) s t0 Lemma 2.2 (Gronwall–Bellman lemma, see [5]) Let x(t) be a continuous function and k > 0, x0 ∈ R Assume that x(t) satisfies the inequality x(t) x0 + k t x(s) s, for all t ∈ T, t t0 (2.3) t0 Then, the relation x(t) x0 ek (t, t0 ) for all t ∈ T, t t0 (2.4) holds Fix t0 ∈ R Let T = T(t0 ) 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 defined by d H (T1 , T2 ) := max sup d(t1 , T2 ), sup d(t2 , T1 ) , t1 ∈T1 t2 ∈T2 where d(t1 , T2 ) = inf |t1 − t2 | and d(t2 , T1 ) = inf |t2 − t1 | t2 ∈T2 t1 ∈T1 (2.5) Author's personal copy On the Convergence of Solutions to Dynamic Equations For properties of the Hausdorff distance, we refer the interested readers to [3,7,16] Throughout this paper, all considered time scales belong to T Convergence of Solutions Let {Tn }n∈N ⊂ T be a sequence of time scales satisfying: lim Tn = T, n→∞ by the Hausdorff distance We define the time scale T = ∪n∈N Tn ∪ T (3.1) Assume that f is continuous on T and satisfies the Lipschitz condition in the variable x, that is there exists a constant k > such that f (t, x) − f (t, y) k x−y , t ∈ T : t0 for all t T and x, y ∈ Rd (3.2) By these assumptions, the initial value problems (IVPs) xn (t) = f (t, xn (t)), t ∈ Tn , xn (t0 ) = x0 , n = 1, 2, , (3.3) x (t) = f (t, x(t)), t ∈ T, x(t0 ) = x0 , (3.4) and have a unique solution xn (t) defined on Tn (respectively solution x(t) defined on T) It is clear that the solutions of the IVPs (3.3) and (3.4) are given by xn (t) = x0 + t f (s, xn (s)) ns (3.5) t0 and t x(t) = x0 + f (s, x(s)) s, (3.6) t0 respectively, where f n s denotes the integral on the time scale Tn The following lemma gives the uniformly bounded property of solutions of the IVPs (3.3) and (3.4) on different time scales Lemma 3.1 Let xS (t) be the solution to the dynamic equation x (t) = f (t, x(t)), t ∈ S, x(t0 ) = x0 Then, for any T > t0 one has sup S∈T;S⊂T sup t∈S:t0 t T xS (t) < ∞ (3.7) Author's personal copy N T Ha et al Proof Let S ∈ T; S ⊂ T For any t ∈ S, we have t xS (t) = x0 + f (s, xS (s)) s t0 t x0 + x0 + t0 t t f (s, 0) s + f (s, 0) s+ t0 t0 t t0 f (s, 0) s C(t − t0 ) f (s, 0) s t0 f (s, xS (s)) − f (s, 0) By virtue of continuity of f on T, one has C = sups∈T;t0 Hence, t t f (s, xS (s)) s − s T s f (s, 0) (3.8) ∞ C(T − t0 ) t0 Moreover, since f satisfies the Lipschitz condition (3.2), t t0 f (s, xS (s)) − f (s, 0) t s k t0 xS (s) s Therefore, xS (t) x0 + C(T − t0 ) + k t t0 xS (s) s By using the Gronwall–Bellman lemma, we get xS (t) x0 + C(T − t0 ) ek (S; t, t0 ), where ek (S; t, t0 ) is the exponential function defined on S Thus, by (2.2), we obtain sup S∈T sup t∈S:t0 t T xS (t) ( x0 + C(T − t0 )) ek(T −t0 ) < ∞ The proof is complete Let n ∈ N, we denote by σn the forward jump operator on the time scale Tn For any t ∈ T, there exists a unique s ∈ Tn , say s = γ T,Tn (t), such that either s = t or t ∈ (s, σn (s)) It is easy to check that the function γ T,Tn (t) is r d-continuous on T Also, there exists tn∗ = tn∗ (t) ∈ Tn satisfying |t − tn∗ | = d(t, Tn ) = inf{|t − s| : s ∈ Tn } (3.9) We choose tn∗ = γ T,Tn (t) if |t − γ T,Tn (t)| = d(t, Tn ), otherwise tn∗ = σn (γ T,Tn (t)) Define f n (t, x) = f (γ T,Tn (t), x), t ∈ T; x ∈ Rd , (3.10) Author's personal copy On the Convergence of Solutions to Dynamic Equations xn (t) = x(γ T,Tn (t)), t ∈ T (3.11) Assume that Tn ⊂ T Then, by the definition of Riemann integral on time scales, we have t f (s, x(s)) ns t = t0 f n (s, xn (s)) s, t0 for any t ∈ Tn (see, e.g [12]) Since d H (T, Tn ) → as n → ∞, we can assume that tn∗ (t) < T + when t By Lemma 3.1, A = supS∈T supt∈S:t0 t T +1 xS (t) < ∞, and hence let M = sup{ f (t, x) : t0 t T T + 1, x < A} Now, we need the following lemmas for proving the convergence of the solution sequence {xn (t)} of the IVPs (3.3) when Tn tends to T Lemma 3.2 Let xn (t), n = 1, 2, be solutions to the IVPs (3.3) and x(t) be the solution to the IVP (3.4) Assume that Tn ⊂ T Then, (n) x(t) − xn (t) δT ek (Tn ; t, t0 ), t ∈ Tn : t0 t T, (3.12) t T, (3.13) and x(t) − xn (tn∗ ) (n) δT +1 ek (Tn ; tn∗ , t0 ) + Md H (T, Tn ), t ∈ T : t0 where tn∗ is defined by (3.9) and (n) δt Proof For any t ∈ Tn , t t = f (s, x(s)) − f n (s, xn (s)) s (3.14) t0 T we have x(t) − xn (t) = = t t0 t f (s, x(s)) s − f (s, x(s)) s − t0 = t0 t f (s, xn (s)) f (s, x(s)) ns ns t0 t + t [ f (s, x(s)) − f (s, xn (s))] ns t0 t [ f (s, x(s)) − f n (s, xn (s))] s t0 + t t0 [ f (s, x(s)) − f (s, xn (s))] n s Author's personal copy N T Ha et al By virtue of the Lipschitz condition f (s, x(s)) − f (s, xn (s)) k x(s) − xn (s) , it follows that t x(t) − xn (t) f (s, x(s)) − f n (s, xn (s)) s+k t0 t x(s) − xn (s) ns t0 t (n) δT + k x(s) − xn (s) n s t0 By using Gronwall–Bellman lemma, we obtain (3.12) If t ∈ T, t T then x(t) − xn (tn∗ ) Since tn∗ x(t) − x(tn∗ ) + x(tn∗ ) − xn (tn∗ ) T + and supt0 t T +1 x(t)) A, (n) x(tn∗ ) − xn (tn∗ ) δT +1 ek (Tn ; tn∗ , t0 ), t ∈ T : t0 t T Further, x(t) − x(tn∗ ) = tn∗ M|t − tn∗ | f (s, x(s)) s t Md H (T, Tn ) Summing up, (3.13) holds The proof is complete Lemma 3.3 Assume that Tn ⊂ T For each such that if d H (T, Tn ) < θ then (n) δT (T − t0 ) + and T ∈ T, there exists θ = θ ( , T ) 2M(T − t0 ) d H (T, Tn ), θ (3.15) (n) where δT is defined by (3.14) Proof Since f is continuous, f is uniformly continuous on [t0 , T ] × B(0, A) where B(0, A) is the ball with the center and radius A Therefore, for each , there exists δ = δ( ) such that if |t1 − t2 | + x1 − x2 < δ then f (t1 , x1 ) − f (t2 , x2 ) We choose θ = θ ( ) = x(t) − xn (t) = on [t0 , T ] × B(0, A) δ( ) If t − γ T,Tn (t) < θ then M +1 t γ T,Tn (t) f (s, x(s)) s M(t − γ T,Tn (t)) < Mθ, Author's personal copy On the Convergence of Solutions to Dynamic Equations and |t − γ T,Tn (t)| + x(t) − xn (t) < (M + 1)θ = δ This implies that if t − γ T,Tn (t) < θ then f (t, x(t)) − f n (t, xn (t)) < We see that the number of values s ∈ Tn satisfying t0 {t ∈ T : s < t < σn (s), t − s s T and θ } = ∅, is less than or equal to [ T −t θ ] Assume that these values are s1 < s2 < · · · < sr T −t0 with r [ θ ] In case d H (T, Tn ) < θ , we see that if t ∈ T such that si < t < σn (si ), t − si θ then σn (si ) − t = d(t, Tn ) d H (T, Tn ) Let τi = min{t ∈ T : si < t < σn (si ), t − si It is clear σn (si ) − τi θ }; i = 1, r d H (T, Tn ) Further, (n) δT = = T t0 τ1 t0 r −1 + i=1 r + i=1 f (s, x(s)) − f n (s, xn (s)) s f (s, x(s)) − f n (s, xn (s)) s T ∧τi+1 σn (si ) T ∧σn (si ) T ∧τi f (s, x(s)) − f n (s, xn (s)) s f (s, x(s)) − f n (s, xn (s)) s, where a ∧ b = min{a, b} Since f (s, x(s)) − f n (s, xn (s)) < for all s ∈ [t0 , τ1 ) ∪ [σn (si ), τi+1 ), r − 1, τ1 f (s, x(s)) − f n (s, xn (s)) s i (τ1 − t0 ), t0 T ∧τi+1 σn (si ) f (s, x(s)) − f n (s, xn (s)) s (T ∧ τi+1 − σn (si )) (T ∧ τi+1 −τi ) Author's personal copy N T Ha et al On the other hand, for i = 1, 2, , r we have T ∧σn (si ) T ∧τi f (s, x(s)) − f n (s, xn (s)) 2M(T ∧ σn (si ) − T ∧ τi ) s 2Md H (T, Tn ) Thus, we obtain (n) δT r −1 (τ1 − t0 ) + r (T ∧ τi+1 − τi ) + i=1 2Md H (T, Tn ) i=1 = (T − t0 ) + 2r Md H (T, Tn ) (T − t0 ) + 2M(T − t0 ) d H (T, Tn ) θ The proof is complete We are now derive the convergence theorem for the IVPs (3.3) and (3.4) Theorem 3.4 Let the sequence of time scales {Tn }∞ n=1 satisfy lim n→∞ Tn = T Let xn (t), n = 1, 2, be the solutions to the IVPs (3.3) and x(t) be the solution to the IVP (3.4) Then, for any T > t0 we have lim sup n→∞ t∈T;t t T x(t) − xn (tn∗ ) = 0, (3.16) where tn∗ is defined by (3.9) Proof First, we assume that Tn ⊂ T for all n ∈ N From Lemma 3.2, it follows that x(t) − xn (tn∗ ) δT(n)+1 ek (Tn ; tn∗ , t0 ) + Md H (Tn , T) (n) ∗ δT(n)+1 ek(tn −t0 ) + Md H (Tn , T) δT +1 ek(T +1−t0 ) + Md H (Tn , T), (n) t T By Lemma 3.3, we get limn→∞ δT +1 = Therefore, (3.16) for any t0 holds In the general case, we put Tn = Tn ∪ T Then, it is easy to see that d H (T, Tn ) = max{d H (Tn , T), d H (Tn , Tn )} (3.17) Let xn (t) be the solution to equation (2.1) on the time scale Tn For t ∈ T, we have x(t) − xn (tn∗ ) xn (t) − x(t) + xn (t) − xn (tn∗ ) Author's personal copy On the Convergence of Solutions to Dynamic Equations Since T ⊂ Tn and Tn ⊂ Tn , we can apply Lemma 3.2 to obtain xn (t) − x(t) xn (t) − xn (tn∗ ) (n1) k(T −t0 ) δT e , k(T +1−t0 ) δT(n2) +1 e + Md H (Tn , Tn ), where (n1) δT T = (n2) δT +1 = f (s, xn (s)) − f (γ Tn ,T (s), xn (γ Tn ,T (s))) t0 T +1 Tn s, f (s, xn (s)) − f (γ Tn ,Tn (s), xn (γ Tn ,Tn (s))) Tn s t0 (n1) (3.18) (n2) → 0, δT +1 → as By Lemmas 3.1, 3.3 and equality (3.17), we imply that δT n → ∞ Thus, (3.16) holds The proof is complete For estimating the convergent rate, we need the following lemma Lemma 3.5 Assume that Tn ⊂ T Then, we have T (s − γ T,Tn (s)) s 2(T − t0 )d H (T, Tn ) t0 Proof With the value θ = d H (T, Tn ), we have a similar way as in the proof of Lemma 3.3 to construct the sequence s1 , s2 , , sr and the sequence τ1 , τ2 , , τr satisfying s1 < τ1 < σn (s1 ) < · · · < sr < τr < σn (sr ) σn (si ) − si = μn (si ) for all s ∈ [τi , σn (si )] and s − Note that s − γ T,Tn (s) d H (T, Tn ) for all s ∈ [σn (si ), τi+1 ) Therefore, γ T,Tn (s) T ∧σn (si ) T ∧τi (s − γ T,Tn (s)) s μn (si )(T ∧ σn (si ) − T ∧ τi ) μn (si )d H (T, Tn ) Thus, we get T (s − γ T,Tn (s)) s r (T − t0 )d H (T, Tn ) + t0 μn (si )d H (T, Tn ) i=1 2(T − t0 )d H (T, Tn ) The proof is complete Assume further that f (t, x) satisfies the Lipschitz condition in both variables t and x, that is f (t, x) − f (s, y) k(|t − s| + x − y ), for all s, t ∈ T and x, y ∈ Rd (3.19) Author's personal copy N T Ha et al We now estimate the convergent rate of approximation Theorem 3.6 Assume that assumption (3.19) is satisfied Let xn (t), n = 1, 2, be solutions of the IVPs (3.3) and x(t) be the solution of the IVP (3.4) If t ∈ T : t0 t < T then C1 d H (T, Tn ), (3.20) x(t) − xn (tn∗ ) where C1 = 2k(2T + − 2t0 )(M + 1)ek(T +1−t0 ) + M and tn∗ is defined by (3.9) Moreover, if t ∈ T ∩ Tn : t0 t < T then x(t) − xn (t) C2 d H (T, Tn ), where C2 = 4k(T − t0 )(M + 1)ek(T −t0 ) Proof Let Tn = Tn ∪ T, and xn (t) be the solution of Eq (2.1) on the time scale Tn It is showed in Theorem 3.4, for t ∈ T : t0 t T , we have x(t) − xn (tn∗ ) xn (t) − x(t) + xn (t) − xn (tn∗ ) , k(T +1−t0 ) + Md H (Tn , Tn ), (δT(n1) + δT(n2) +1 )e (n1) (n2) where δt , δt are given by (3.18) Note that if t ∈ T ∩ Tn : t0 t T then x(t) − xn (t) Since f (t, x) satisfies the Lipschitz condition (3.19), (n1) δT T = (n1) (δ T f (s, xn (s)) − f (γ Tn ,T (s), xn (γ Tn ,T (s))) t0 T k t0 (n2) + δT )ek(T −t0 ) Tn s (|s − γ Tn ,T (s)| + xn (s) − xn (γ Tn ,T (s)) ) Tn s We have xn (s) − xn (γ Tn ,T (s)) = s γ Tn ,T (s) f (u, xn (γ Tn ,T (u))) Tn u M|s − γ Tn ,T (s)| Therefore, by Lemma 3.5, we get δT(n1) k(M + 1) T t0 |s − γ Tn ,T (s)| Tn s 2k(M + 1)(T − t0 )d H (Tn , T) Author's personal copy On the Convergence of Solutions to Dynamic Equations Similarly, we get that (n2) δT +1 2k(M + 1)(T + − t0 )d H (Tn , Tn ) Thus, we obtain x(t) − xn (tn∗ ) k(T −t0 ) (δT(n1) + δT(n2) +1 )e 2k(M + 1)ek(T +1−t0 ) (T − t0 )d H (Tn , T) +(T + − t0 )d H (Tn , Tn ) + Md H (Tn , Tn ) 2k(2T + − 2t0 )(M + 1)ek(T +1−t0 ) d H (T, Tn ) + Md H (T, Tn ) = C1 d H (T, Tn ), where C1 = 2k(2T + − 2t0 )(M + 1)ek(T +1−t0 ) + M Similarly, if t ∈ T ∩ Tn : t0 t < T then x(t) − xn (t) C2 d H (T, Tn ), where C2 = 4k(T − t0 )(M + 1)ek(T −t0 ) The proof is complete Examples Example 4.1 Consider the IVP x = f (t, x), t0 t T, x(t0 ) = x0 (4.1) In numerical analysis, approximations to the solution x(t) of (4.1) will be generated at various values, called mesh points, in the interval [t0 , T ] For a positive integer n, we select a subdivision of the interval [t0 , T ] (n) t0 = t0 (n) < t1 (n) (n) < · · · < tkn −1 < tkn := T, kn ∈ N (4.2) Associating with (4.2), we study a difference equation, called the Euler method [8,9,13], as follows (n) (n) x0(n) = x0 , xi+1 = xi(n) + ti+1 − ti(n) f ti(n) , xi(n) , i = 0, 1, , kn − (4.3) (n) (n) (n) (n) Let T := [t0 , T ] and Tn := {t0 , t1 , , tkn −1 , tkn } Then T and Tn are time scales Therefore, we can rewrite (4.1) and (4.3) as follows x (t) = f (t, x(t)), t ∈ T, x(t0 ) = x0 , and xn (τ ) = f (τ, xn (τ )), τ ∈ Tn , xn (t0 ) = x0 , Author's personal copy N T Ha et al respectively In this case, it is easy to see that 2d H (Tn , T) = h n := sup (n) i kn −1 (n) ti+1 − ti for all n ∈ N (4.4) Suppose that f (t, x) is continuous and satisfies Lipschitz condition f (t, x1 ) − f (t, x2 ) k x1 − x2 , for all t ∈ [t0 , T ] By Theorem 3.4 and (4.4), we see that lim n→∞ xn (γnTn ,T (t)) = limn→∞ xn (tn∗ (t)) = x(t) uniformly in [t0 , T ] Hence, we obtain the well-known result for the convergence of the Euler method in numerical analysis [13] Assume further that f satisfies the Lipschitz condition in both variables with constant k That is, f (t, x1 ) − f (s, x2 ) k(|t − s| + x1 − x2 ), for all t, s ∈ [t0 , T ], x1 , x2 ∈ Rd By Theorem 3.6, we get an estimate for the convergent rate as well as an error bound for the Euler method as follows sup i kn (n) x ti (n) − xi C3 h n , for all n ∈ N, where C3 = 2k(T − t0 )(M + 1)ek(T −t0 ) Now, we consider some numerical examples Example 4.2 (Approximation of solutions to logistic equations on R+ ) Let T = [0, ∞) We now consider plant population models Let x(t) be the number of plants of one particular kind at time t ∈ T in a certain area By experiments we know that x(t) grows according to the logistic equation x (t) = x(t)[1 − 4x(t)], t ∈ T and x(0) = > (4.5) Suppose that we are unable to observe the values of x(t) but xn (t) with xn (t) to be the number of plants of one particular kind at time t ∈ Tn in a certain area, subject to the equation xn (t) = xn (t)[1 − 4xn (t)], t ∈ Tn and xn (0) = for all where Tn is a time scale given by ∞ Tn = {0} ∪ k=1 2k − 2k , n n for all n ∈ N n ∈ N, (4.6) Author's personal copy On the Convergence of Solutions to Dynamic Equations This means that we lack the observation, for some reasons, at the times in the 2k+1 It is easy to see that d H (Tn , T) = 2n Hence, intervals 2k n , n lim Tn = T n→∞ We have x(t) = et ; t ∈ T and + 4(et − 1) 2k + n xn (0) = 1; xn xn (t) = t− xn ( 2k−1 n )e + 4xn ( 2k−1 n ) 2k n = xn e 1+ 2k−1 n t− 2k−1 n n − , ∀ t∈ −1 2k n x n n ; 2k − 2k , , n n for all k = 1, 2, ; n ∈ N Note that if f satisfies the local Lipschitz condition and the solution sequence {xn (t)} is bounded then Theorem 3.4 also holds Therefore, by this theorem, we get xn (t) → x(t) as n → ∞ The discrete graph of solutions xn (t) and x(t) on the interval [0, 1] is shown in Fig Example 4.3 (Approximation of solutions to a logistic equation on Cantor set) Let K be be the Cantor set in [0, 1] Following the construction of this Cantor set, we define K = [0, 1] We obtain K by removing the “middle third” of K , i.e., the open interval 13 , 23 from K K is obtained by removing two “middle thirds of K , i.e., the two open intervals 19 , 29 and 79 , 89 from K Proceeding in this manner we obtain a sequence of time scales (K n )n∈N The Cantor set is defined ∞ K = Kn n=0 1 the graph of soluions to x(t) on [0,1] values of Euler method xn((2i+1)/n) 0.9 the graph of soluions to x(t) on [0,1] values of Euler method xn((2i+1)/n) 0.9 the graph of soluions to xn(t) when n=20 0.8 0.7 0.7 x(t) x(t) the graph of soluions to xn(t) when n=10 0.8 0.6 0.6 0.5 0.5 0.4 0.4 0.2 0.4 0.6 0.8 0.2 0.4 0.6 t t (a) (b) 0.8 Fig The graph of the solution xn (t) on the time scale Tn a xn (t) and x(t) with n = 10 on the interval [0, 1] in Example 4.2 b xn (t) and x(t) with n = 20 on the interval [0, 1] in Example 4.2 Author's personal copy N T Ha et al the graph of soluions to x(t) on [0,2] the graph of soluions to x4(t) on T4 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Fig The graph of the solution x4 (t) on the time scale T4 Let (Tn ) be a sequence of time scales, where Tn = K n ∪ (K n + 1) and T = K ∪ (K + 1) Consider the dynamic equation (4.5) with x(0) = It is known that we are unable to give an explicit formula for solutions as well as a numerical solution to Eq (4.5) However, we can use Theorem 3.4 to approximate these solutions We illustrate this approximation by Fig It is seen that the graph on T4 of the Eq (4.5) (the green line) is rather different to one on T0 = [0, 2] (the red line) (colour figure online) Conclusion In this paper, we have proved the convergence of solutions of dynamic equations x (t) = f (t, x) on time scales {Tn }∞ n=1 when this sequence converges to the time scale T The convergent rate of solutions is estimated when f satisfies the Lipschitz condition in both variables A similar problem for nabla dynamic equations x (t) = f (t, x) which relates to the implicit Euler method is still an open problem Acknowledgments This work was supported financially by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 101.03-2014.58 References Agarwal, R., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey J Comput Appl Math 141, 1–26 (2002) Akin-Bohner, E., Bohner, M., Akin, F.: Pachpatte inequalities 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radius of implicit dynamic equations with constant coefficients on time scales Syst Control Lett 60, 596–603 (2011) 11 Gard, T., Hoffacker, J.: Asymptotic behavior of natural growth on time scales Dyn Syst Appl 12(1–2), 131–148 (2003) 12 Guseinov, GS.h.: Integration on time scales J Math Anal Appl 285, 107–127 (2003) 13 Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I nonstiff problems, 2nd revised edn Springer, Berlin (1993) 14 Hilger, S.: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten Ph.D thesis, Universität Würzburg (1988) 15 Hilger, S.: Analysis on measure chains–a unified approach to continuous and discrete calculus Res Math 18, 18–56 (1990) 16 Memoli, F.: Some properties of Gromov–Hausdorff distance Discret Comput Geom 48, 416–440 (2012) 17 Salinetti, G., Wets, R.J.-B.: On the convergence of sequence of convex sets in finite dimensions SIAM Rev 21, 16–33 (1979) View publication stats ... results on time scales In Sect 3, we study the convergence of solutions of dynamic equations on time scales The main results of the paper are derived here In Sect 4, we give some examples and show the. .. 23 Author's personal copy Qual Theory Dyn Syst DOI 10.1007/s12346-015-0166-8 Qualitative Theory of Dynamical Systems On the Convergence of Solutions to Dynamic Equations on Time Scales Nguyen Thu.. .On the Convergence of Solutions to Dynamic Equations on Time Scales Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi & Do Duc Thuan Qualitative Theory of Dynamical Systems ISSN 1575-5460 Qual Theory