Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

Abstract

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] for 𝑖=0,1,…,𝐽 of the solution. Two examples are provided.

1. Introduction

This paper investigates the errors of the solutions of nonlinear differential equations 𝑦(̇𝑡)=𝑓(𝑦(𝑡),𝑡), where 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛), provided they exist and are unique for each given admissible initial condition 𝑦(𝑡₀)=𝑦₀, with respect to the solutions of its approximate differential equations 𝑥̇(𝑡)=∑_𝑘=0(𝑓^(𝑘)(𝑥(𝑡_𝑖),𝑡_𝑖)/𝑘!)(𝑥(𝑡)−𝑥(𝑡_𝑖))^𝑘; 𝑥(𝑡₀)=𝑥₀, for any given nonnegative integer ≤𝑛, obtained from truncated Taylor expansions of the solutions about certain sampling points 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] for 𝑖=0,1,…,𝐽. It is assumed that if a unique solution exists on some interval [𝑡₀,𝑡_𝐽] and that the choice of the sampling points is such that the intersample intervals [1–4] are subject to a certain maximum allowable upper-bound then the error of the solution in the whole interval [𝑡₀,𝑡_𝐽] satisfies a prescribed norm bound. Using the obtained results, the shadowing property [5–10] of the true solution with respect to the approximate one is investigated in the sense that “shadowing” initial conditions of the true solution exist, for each initial condition of the approximate differential equation, such that any approximated solution trajectory on the interval of interest is arbitrarily close to the true one under prescribed allowable maximum norms of the error between both the true solution and the approximate solutions. The problem is extended to the case when the approximated solution is perturbed either by a sequence of a certain allowable size at the sampling points or with perturbation functions of a certain size in norm about the whole considered interval. The main tool involved in the analysis is an “ad hoc” use of a known preparatory theorem to the celebrated Bernstein’s theorem, [11], which gives an upper-bound for the maximum norm of the error in between both the true and the approximate solutions. The results are potentially extendable to functional equations involving nonlinearities and the presence of delays subject to mixed types of uncertainties [12–18]. On the other hand, different characterizations of oscillatory solutions and limit oscillatory solutions (limit cycles) have received important interest in the literature concerning different types of nonlinear dynamic continuous-time, discrete and hybrid systems, and differential equations [19–28]. The shadowing property is naturally relevant for the characterization of limit oscillations. Therefore, the formulation is applied in the second example to the characterization of limit cycles generated as solutions to Van der Pol’s equation.

2. Calculation of the Exact Solution from Taylor Series Expansion

Lemma 1. Assume that 𝑓∈𝐶^(𝑛+1)((𝑎,𝑏);𝐑^𝑛) and divide the real interval (𝑎,𝑏) into 𝐽 subintervals with points 𝑦_𝑛∈[𝑎,𝑏] such that 𝑎≡𝑦₀<𝑦₁<𝑦₂<⋯<𝑦_𝐽−1<𝑦_𝐽≡𝑏. (1) Then 𝑓𝑦_𝑖+1=𝑓𝑦₀+^ℎ𝑖₀𝑓𝑦+𝑦₀𝑦𝑑𝑦=𝑓₀+^𝑖_𝑗=−1^ℎ𝑗+1_ℎ𝑗𝑓𝑦+𝑦₀𝑦𝑑𝑦=𝑓₀+^𝑖_𝑗=−1^ℎ𝑗₀𝑓𝑦+𝑦₀+ℎ_𝑗𝑦𝑑𝑦=𝑓₀+^𝑖𝑛_{𝑗=−1𝑘=0}𝑓^(𝑘)𝑐_𝑗𝑘!^ℎ𝑗₀𝑦+𝑦₀+ℎ_𝑗−𝑐_𝑗+1^𝑘+1𝑑𝑦𝑛!^𝑖_𝑗=−1^ℎ𝑗₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑗−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡𝑑𝑦, (2) where ℎ_𝑛=𝑦_𝑛+1−𝑦_𝑛, ℎ_𝑛=𝑦_𝑛+1−𝑦₀=∑^𝑛_𝑖=0ℎ_𝑖 for 𝑛=0,1,…,𝐽−1 with ℎ₋₁=0, and 𝑓𝑦_𝑖+1ℎ_𝑖+1=𝑓𝑦₀+^𝑖𝑛_{𝑗=−1𝑘=0}𝑓^(𝑘)𝑐_𝑗𝑘!^ℎ𝑗₀𝑦+𝑦₀+ℎ_𝑗−𝑐_𝑗+1^𝑘+1𝑑𝑦𝑛!^𝑖_𝑗=−1^ℎ𝑗₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑗−𝑡^𝑛𝑓^(𝑛+1)(𝑡)+𝑑𝑡𝑑𝑦_𝑛𝑘=0𝑓^(𝑘)𝑐_𝑖+1𝑘!^ℎ𝑖+1₀𝑦+𝑦₀+ℎ_𝑖+ℎ_𝑖+1−𝑐_𝑖+2^𝑘+1𝑑𝑦𝑛!^ℎ𝑖+1₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑖+ℎ_𝑖+1−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡𝑑𝑦; (3)∀𝑦∈[𝑦_𝑖,𝑦_𝑖+1] and any real 𝑐_𝑖∈[𝑦_𝑖,𝑦_𝑖+1] for 𝑖=0,1,…,𝐽−1; ∀ℎ_𝑖∈[0,ℎ_𝑖] for 𝑖=0,1,…,𝐽−2.

Proof. It follows from a well-known preparatory theorem to Bernstein’s theorem [5] that 𝑓𝑦=^𝑛_𝑘=0𝑓^(𝑘)𝑐_𝑖𝑘!𝑦−𝑐_𝑖^𝑘+1𝑛!^𝑦_𝑐𝑖𝑦−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡. (4)

Now, consider the nonlinear ordinary differential equation 𝑦̇(𝑡)=𝑓𝑦(𝑡),𝑡;𝑦𝑡₀=𝑦₀ (5) in the real interval 𝐑^𝑛×[𝑡₀,𝑡_𝐽] such that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽]. The following result follows from Lemma 1.

Theorem 2. The unique solution of (5) in [𝑡₀,𝑡_𝐽] is given by 𝑦(𝑡)=𝑦𝑡₀+^𝑖−2_𝑗=0^ℎ𝑗_0𝑘=0𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑡𝑡𝑖−1𝑘=0}𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (6)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽} and ∀(∈𝐙₀₊)≤𝑛, where 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] are any arbitrary strictly ordered points such that 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1.

Proof. Note that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽] so that the solution 𝑦(𝑡) on [𝑡₀,𝑡_𝐽] is unique, provided that 𝑡_𝐽=𝑡_𝐽(𝜃₀,𝑡₀) for the given 𝑡₀∈𝐑 and some 𝜃₀∈𝐑^𝑛 is such that 𝑦(𝑡)∈[𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]; ∀𝑡∈[𝑡₀,𝑡_𝐽] and 𝑡_𝐽∈(𝑡₀,𝑡_𝐽] since 𝑓∶[𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽]→𝐑^𝑛 is local Lipschitz-continuous as a result. Such a unique solution is given by 𝑦(𝑡)=𝑦_𝑎+^𝑡_𝑎𝑓𝑦(𝜏),𝜏𝑑𝜏;∀𝑡∈𝑡₀,𝑡_𝐽. (7) Take any set of 𝐽 strictly ordered points 𝑡_𝑛∈[𝑡₀,𝑡_𝐽] satisfying 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1, so that 𝑦(𝑡)=𝑦𝑡_𝑖−1+^𝑡_𝑡𝑖−1𝑓𝑦(𝜏)𝑡,𝜏𝑑𝜏=𝑦₀+^𝑖−2_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑓𝑦(𝜏),𝜏𝑑𝜏+^𝑡_𝑡𝑖−1𝑓𝑦(𝜏),𝜏𝑑𝜏; (8)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}, with 𝑦(𝑡₀), so that, by choosing the real 𝑐_𝑖∈[𝑡_𝑖−1,𝑡_𝑖] as 𝑐_𝑖=𝑡_𝑖 for 𝑖=0,1,…,𝐽−1, one gets from (3) in the proof of Lemma 1 into (4): 𝑦(𝑡)=𝑦𝑡₀+^𝑖−2_𝑗=0^ℎ𝑗₀^𝑛_𝑘=0𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘+1𝑛!^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗^𝑛×𝑓^(𝑛+1)𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑡−𝑡𝑖−10𝑛𝑘=0}𝑓^(𝑘)𝑦𝑡_𝑖−1,𝑡_𝑖−1×𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘+1𝑛!^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑛×𝑓^(𝑛+1)𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (9)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}. Note that, since 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛), then 𝑓∈𝐶⁽⁺¹⁾(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) for any nonnegative integer ≤𝑛. Thus, we obtain the result from a similar expression of (9) by replacing 𝑛 by (≤𝑛) while truncating the Taylor series expansion by its (+1)th term.

A consequence of Theorem 2 by using the same technique of the solution construction is as follows.

Corollary 3. Consider the nonlinear ordinary differential equation (5) with initial condition 𝑦(𝑡₀) on the real interval 𝐑^𝑛×𝐑₀₊, with initial conditions 𝑦^(𝑗)(𝑡₀) for 𝑗=0,1,…,𝑛−1, such that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽] for some 𝜃₀∈𝐑^𝑛, and consider also its th order truncation 𝑥̇(𝑡)=_𝑘=0𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥(𝑡)𝑡𝑘!−𝑥_𝑖^𝑘;𝑥𝑡₀=𝑥₀ (10) such that 𝑓^(𝑘)(𝑦(𝑡),𝑡) are bounded in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽] for 𝑘=0,1,…,+1 for some nonnegative integer ≤𝑛 and some 𝜃∈𝐵(𝜃₀,𝑅), where 𝐵(𝜃,𝑅)={𝑧∈𝐑^𝑛∶‖𝑧−𝜃₀‖≤𝑅} for some positive real 𝑅 with 𝑥^(𝑗)(𝑡₀)=𝑦^(𝑗)(𝑡₀) for 𝑗=0,1,…,+1.

Since 𝑓^(𝑘)(𝑦(𝑡),𝑡) are bounded in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽] for 𝑘=0,1,…,−1, then the right-hand-side of (10) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽]⊆[𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽]. Therefore, the unique solution of the truncated differential equation (10) in [𝑎,𝑏] is 𝑥(𝑡)=𝑥𝑡₀+_𝑘=0^𝑖−2_𝑗=0^ℎ𝑗₀𝑓^(𝑘)𝑥𝑡_𝑗,𝑡_𝑗𝑥𝑘!𝜏+𝑡_𝑗−𝑥𝑡_𝑗^𝑘+𝑑𝜏_{𝑡−𝑡𝑖−10}𝑓^(𝑘)𝑥𝑡_𝑖−1,𝑡_𝑖−1×𝑥𝑘!𝜎+𝑡_𝑖−1−𝑥𝑡_𝑖−1^𝑘𝑑𝜏; (11)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}, ∀(∈𝐙₀₊)≤𝑛, where 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] are arbitrary strictly ordered points such that 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1. The error in between the exact solution of (10) and that of its truncated form (5) is 𝑒(𝑡)(𝑡)(𝑡)==𝑦−𝑥_{𝑖−2𝑗=0𝑘=0ℎ𝑗0}𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘−𝑓^(𝑘)𝑥𝑡_𝑗,𝑡_𝑗×𝑥𝑘!𝜏+𝑡_𝑗−𝑥𝑡_𝑗^𝑘+1𝑑𝜏!^𝑖−2_𝑗=0^ℎ𝑗₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑘=0𝑡−𝑡𝑖−10}𝑓^(𝑘)𝑦𝑡_𝑖−1,𝑡_𝑖−1×𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘−𝑓^(𝑘)𝑥𝑡_𝑖−1,𝑡_𝑖−1×𝑥𝑘!𝜏+𝑡_𝑖−1−𝑥𝑡_𝑖−1^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖−1}₀^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (12)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽} and ∀(∈𝐙₀₊)≤𝑛.

Proof. Property (i) follows directly Theorem 2 applied to the truncated differential equation (10) leading to the solution (11) in [𝑡₀,𝑡_𝐽]. Property (ii) follows from (6) and (11).

Now, a preparatory result follows to be then used to guarantee sufficiency-type errors results in between the true and the approximate solutions in the interval [𝑎,𝑏].

Lemma 4. Assume that the following hypothesis holds.
(A1)  𝑓(𝑦(𝑡),𝑡) and its first (+1) derivatives are uniformly bounded from above on a bounded subset of their existence domain with the specific boundedness constraint:sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑦(𝑡)+𝐾₁,sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁, (13a)(13b)for 𝑗=0,1,…,+1 and some 𝐾,𝐾₁∈𝐑₀₊ with 𝐾<1 if 𝐾₁∈𝐑₊. Then, the following properties hold.
(i) Assume that the intersample intervals ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 fulfill the constraint ℎ_𝑖𝑎≤ℎ≤min_𝑖,1−𝜌_𝑥/2Λ_𝑥1−𝜌_𝑥/2,𝜌_𝑥2Λ_𝑥1−𝜌_𝑥/2⁺¹𝐽−1+𝜌_𝑥/2 (14) for 𝑖=0,1,…,𝐽−1 and any given real constant 𝜌_𝑥∈(0,2), where Λ_𝑥=_𝑘=0𝐾^𝑘𝑘!sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡)+𝐾,𝑡₁1−𝐾⁺¹1−𝐾𝑎_𝑖∶=minarg𝑡∈𝐑₊>𝑡_𝑖∶𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌_𝑥2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1. (15)(16) Then, the approximated solution fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2 provided that 𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶𝑥(𝑡)−𝑥𝑡₀≤𝜌_𝑥2. (17)
(ii) Assume that the intersample intervals ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 fulfill the constraint ℎ_𝑖𝑏≤ℎ≤min_𝑖,1−𝜌/2Λ1−𝜌/2,𝜌2Λ1−𝜌/2⁺¹𝐽−1+𝜌/2 (18) for 𝑖=0,1,…,𝐽−1 and any given real constant 𝜌∈(0,1), where Λ=⁺¹_𝑘=0𝐾^𝑘𝑘!sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡)+𝐾,𝑡₁1−𝐾^𝑘1−𝐾𝑏_𝑖∶=minarg𝑡>𝑡_𝑖∶𝑦(𝑡)−𝑦𝑡_𝑖≤𝜌2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1. (19)(20) Then, the true solution fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑦(𝑡)−𝑦(𝑡₀)‖≤𝜌/2 provided that 𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶𝑦(𝑡)−𝑦𝑡₀≤𝜌2. (21)
(iii) If 𝜌_𝑥=𝜌∈(0,1) and, furthermore, ℎ_𝑖𝑐≤ℎ≤min_𝑖,1−𝜌/2Λ1−𝜌/2,𝜌2Λ1−𝜌/2⁺¹;𝐽−1+𝜌/2𝑖=0,1,…,𝐽−1,𝑐_𝑖∶=minarg𝑡>𝑡_𝑖∶max𝑦(𝑡)−𝑦𝑡_𝑖,𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1,𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶max𝑦(𝑡)−𝑦𝑡₀,𝑥(𝑡)−𝑥𝑡₀≤𝜌2 (22)(23)(24) then the true, the approximated and the error solution fulfill sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑦(𝑡)−𝑦𝑡₀≤𝜌2,sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑥(𝑡)−𝑥𝑡₀≤𝜌2, (25) and the error in between them, 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡), fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑒(𝑡)−𝑒𝑡₀≤𝜌. (26)

Proof. Proceeding recursively, one gets from Assumption A1 that sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁≤𝐾²sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁(1+𝐾)≤𝐾^𝑘𝐹₀+𝐾₁^𝑘−1_𝑖=0𝐾^𝑖≤𝐾^𝑘𝐹₀+𝐾₁1−𝐾^𝑘1−𝐾≤𝐹₀+𝐾₁1−𝐾 (27) if 𝐾<1 and 𝐾₁≠0, and sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑘)𝑦(𝑡),𝑡≤𝐾^𝑘𝐹₀, (28) if 𝐾₁=0, where 𝐹₀=sup_{𝑦(𝑡)∈𝑦𝑡0𝑡−𝜃,𝑦𝐽+𝜃,𝑡∈[𝑡0,𝑡𝐽]}𝑓⁽⁰⁾𝑦(𝑡),𝑡=sup_{𝑦(𝑡)∈𝑦𝑡0𝑡−𝜃,𝑦𝐽+𝜃,𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡),𝑡<+∞. (29)
Case (a). If 𝐾<1 and 𝐾₁≠0 proceed by complete induction by assuming that sup_{𝑡∈[𝑡0,𝑡𝑖]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2 since the condition (𝑡(∈𝐑₊)>𝑡_𝑖∶‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2) guarantees that sup_{𝑡∈[𝑡0,𝑡1]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2. Thus, one gets from (11) that 𝑥(𝑡)−𝑥𝑡_𝑖≤^𝑖_𝑘=0𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘=_𝑘=0^𝑖−1_𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘+_𝑘=0ℎ_𝑖𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝑥(𝑡)𝑡1−𝐾−𝑥_𝑖^𝑘≤_𝑘=0^𝑖−1_𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘+_𝑘=0ℎ_𝑖𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝑥(𝑡)𝑡1−𝐾−𝑥_𝑖^𝑘=Λ_𝑥𝑘=0𝜌_𝑥2^𝑘𝑖−1_𝑗=0ℎ_𝑗+ℎ_𝑖Λ_𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘≤𝑖ℎΛ_𝑥𝑘=0𝜌_𝑥2^𝑘+ℎ_𝑖Λ_𝑥𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1𝑥(𝑡)−𝑥𝑡_𝑖;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1, (30)(31) where Λ_𝑥=∑_𝑘=0(1/𝑘!)(𝐾^𝑘𝐹₀+(𝐾₁(1−𝐾^𝑘)/(1−𝐾))) and ℎ≥max_{0≤𝑖≤𝐽−1}ℎ_𝑖, with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖, for 𝑖=0,1,…,𝐽−1, so that 1−ℎ_𝑖Λ_𝑥𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1×𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥𝑘=0𝜌_𝑥2^𝑘=𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−𝜌_𝑥,/2 (32) or 𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−ℎ_𝑖Λ_𝑥∑_𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−11−𝜌_𝑥≤/2𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−ℎ_𝑖Λ_𝑥1−𝜌_𝑥/2/1−𝜌_𝑥/21−𝜌_𝑥≤𝜌/2_𝑥2 (33) provided that 0<𝜌_𝑥<2, and 1>ℎ_𝑖Λ_𝑥(∑_𝑘=0(‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖^𝑘−1)) which is guaranteed if ℎ_𝑖<min(𝑎_𝑖,(1−𝜌_𝑥/2)/(Λ_𝑥(1−𝜌_𝑥/2))) holds with 𝑎_𝑖 for 𝑖=0,1,…,𝐽−1, defined in (16), provided that ‖𝑥(𝑡)−𝑥(𝑡_𝑗)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡_𝑗,𝑡_𝑗+1) for 𝑗(≤𝑖)=0,1,…,𝑖−1, and then (33) and ℎ_𝑗<((1−𝜌_𝑥/2)/(Λ_𝑥(1−𝜌_𝑥/2))) for 𝑗=0,1,…,𝑖−1 are jointly guaranteed for the given 𝑖=0,1,…,𝐽−1 if ℎ_𝑖<min1−𝜌_𝑥/2Λ_𝑥1−𝜌_𝑥/2,𝜌_𝑥2Λ_𝑥1−𝜌_𝑥/2⁺¹𝐽−1+𝜌_𝑥/2 (34) provided that ‖𝑥(𝑡)−𝑥(𝑡_𝑗)‖≤𝜌_𝑥/2 for 𝑡∈[𝑡_𝑗,𝑡_𝑗+1) for 𝑗=0,1,…,𝑖−1, the last condition being identical to 𝑡_𝑖+1≤𝑎_𝑖∶=minarg𝑡>𝑡_𝑖∶𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌_𝑥2. (35) The above two conditions (34)-(35) reduce to (14). Then, one gets from complete induction from (31), if (14) holds, the following.
sup_{𝑡∈[𝑡0,𝑡𝑖]}‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2⇒sup_{𝑡∈[𝑡0,𝑡𝑖+1)}‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2 and one gets also by continuity extension,
sup_{𝑡∈[𝑡0,𝑡]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡₀,𝑡_𝐽]. Hence, we got the result for Case (a).
Case (b). If 𝐾₁=0 then 𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥0𝑘=0𝜌_𝑥2^𝑘+ℎ_𝑖Λ_𝑥0𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1𝑥(𝑡)−𝑥𝑡_𝑖; (36)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1), where Λ_𝑥0=∑_𝑘=0((𝐾^𝑘𝐹₀)/(𝑘!))≤Λ_𝑥, so that (1−ℎΛ_𝑥0)‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖ ≤𝑖ℎΛ_𝑥0𝐸 for 𝑖=0,1,…,𝐽 and, thus, one gets the following.
‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤((𝑖ℎΛ_𝑥0𝐸)/(1−ℎΛ_𝑥0))≤𝜌_𝑥/2; ∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1) for 𝑖=0,1,…,𝐽 and one gets from complete induction the same conclusion sup_{𝑡∈[𝑡0,𝑡]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡₀,𝑡_𝐽) as in Case (a) provided that (14) holds. Then, (14) guarantees Property (i) for both Cases (a) and (b). Then, Property (i) has been proven.
Property (ii) is proven “mutatis-mutandis” by noting that Λ≥Λ_𝑥 from (15) and (19) and noting also that 𝑎_𝑖 in (16) is replaced with 𝑏_𝑖 in (20) so that the admissible intersample interval satisfying the constraint (14) is replaced by such an interval satisfying the constraint (18). Finally, Property (iii) follows from Properties (i)-(ii) by equalizing 𝜌_𝑥 and 𝜌 to take a maximum value less than 1/2.

The following comments address the fact that it is not necessary to deal with the solution of the true differential equation (5) to calculate upper-bounds of the solution and error in Lemma 4.

Remark 5. Note that one gets by direct integration from (5) that 𝑦(𝑡)≤sup_{𝑡0≤𝜏≤𝑡𝐽}𝑦(𝜏≤𝑦𝑡₀)+𝑡_𝐽−𝑡₀𝐾sup_{𝑡0≤𝜏≤𝑡𝐽}𝑦(𝜏) (37) leading to sup_{𝑡0≤𝑡≤𝑡𝐽}𝑦(𝑡≤𝐾𝑡)^𝐽−𝑡₀1−𝐾𝑡_𝐽−𝑡₀𝑦𝑡₀if^𝐽−1_𝑖=0ℎ_𝑖<1𝐾. (38) Thus, (25)-(26) might be guaranteed with sup_{𝑡0≤𝑡≤𝑡𝐽}‖𝑦(𝑡)−𝑦(𝑡₀)‖ ≤(1/(1−𝐾(𝑡_𝐽−𝑡₀)))‖𝑦(𝑡₀)‖≤(𝜌/2) if ‖𝑦(𝑡₀)‖≤(𝜌/2)(1−𝐾(𝑡_𝐽−𝑡₀))<(𝜌/2). Thus, there is no need to compute the solution of the true differential equation (5) and sup_{𝑡𝑖≤𝑡<𝑡𝑖+1}‖𝑦(𝑡)−𝑦(𝑡₀)‖≤(𝜌/2) for 𝑖=0,1,…,𝐽−1 in (20) and (23) if ‖𝑦(𝑡₀)‖≤(𝜌/2)(1−𝐾(𝑡_𝐽−𝑡₀)).

One gets directly from Lemma 4 the subsequent result.

Theorem 6. Assume the conditions (13a) and (13b) and (22)–(24) of Lemma 4(iii). Then maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,𝑒(𝑡)−𝑒𝑡₀≤𝜌<1;max_{𝑡𝑡∈0,𝑡𝐽}‖𝑒(𝑡)‖≤𝑒𝑡₀+𝜌; (39)∀𝑡∈[𝑡₀,𝑡_𝐽], and max_{0≤𝑖≤𝐽−1}𝑒(𝑡)−𝑒𝑡_𝑖𝑡≤𝜌;∀𝑡∈_𝑖,𝑡_𝑖+1𝑓𝑜𝑟𝑖=0,1,…,𝐽−1 (40) for 𝑖=0,1,…,𝐽−1.

3. Orbits of the Exact Solution, Pseudo-Orbits of the Approximated Solution, and the Shadowing Property

Now, consider a perturbed solution (11) of the approximated differential equation (10) associated with a perturbation 𝑥(𝑡_𝑖)=𝑥(𝑡⁻_𝑖)+𝑔(𝑡_𝑖) with {𝑔(𝑡_𝑖)}⊂𝐑^𝑛 at 𝑡=𝑡_𝑖 fulfilling ‖𝑔(𝑡_𝑖)‖≤𝑔_𝑖≤𝑔 for some given 𝑔∈𝐑, ∀𝑖∈𝐽. Note that a perturbation at the initial 𝑡=𝑡₀ is considered by choosing 𝑥(𝑡₀)=𝑦(𝑡₀)+𝑔₀ for some nonzero 𝑔₀=𝑔(𝑡₀)∈𝐑. The perturbed solution can be generated, in particular, from impulsive controls of amplitudes 𝑔(𝑡_𝑖) at the sequence {𝑡_𝑖∶𝑖∈𝐽}. The exact and approximate solutions (6) and (11) in [𝑡₀,𝑡_𝐽], provided that they exist, are 𝑦(𝑡)=𝑦𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖}₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑡𝑑𝜎𝑑𝜏;∀𝑡∈_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1,𝑥(𝑡)=𝑥𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘𝑑𝜏+𝑈𝑡−𝑡_𝑖+1𝑔𝑡_𝑖+1;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1, (41)(42) where 𝑈(𝑡) is the Heaviside function. The error between the exact and perturbed approximated solutions becomes 𝑒(𝑡)=𝑒𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘−𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖×𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖}₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑑𝜎𝑑𝜏−𝑈𝑡−𝑡_𝑖+1𝑔𝑡_𝑖+1; (43)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1]; 𝑖=0,1,…,𝐽−1. Now, one gets from (25)-(26) of Lemma 4: 𝑒(𝑡)−𝑒𝑡_𝑖≤_𝑘=02^𝑘+1𝑘!𝑡−𝑡_𝑖𝑀_𝑖𝑘𝜌2^𝑘+1!𝑡−𝑡_𝑖²2𝑀_𝑖,+1𝜌2+𝑔_𝑖;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1, (44) where 𝑀_𝑖𝑘=sup_{𝑦(𝑡)𝑡∈[𝑦𝑖𝑡−𝜃,𝑦𝑖+𝜃],𝑡∈[𝑡𝑖,𝑡𝑖+1]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾^𝑘𝐹₀+𝐾₁1−𝐾^𝑘≤𝐾1−𝐾_𝑘+1𝜌2+𝐾₁1−𝐾^𝑘≤𝜌1−𝐾2+𝐾₁1−𝐾 (45) from (27) and one gets after using the triangle inequality, 𝑒(𝑡)−𝑒𝑡_𝑖≤^𝑚_{𝑗=𝑖𝑘=0}2^𝑘+1𝑘!𝑡−𝑡_𝑗𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗²2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑗; (46)∀𝑡∈[𝑡_𝑖+𝑚,𝑡_𝑖+𝑚+1] for 𝑚=0,1,…,𝐽−𝑖−1; 𝑖=0,1,…,𝐽−1. One obtains easily from (46), either by complete induction or via recursive calculation, that𝑒(𝑡)−𝑒𝑡_𝑖≤^𝐽−1_𝑗=0𝑘=02^𝑘+1𝑘!𝑡−𝑡_𝑗𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗²2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑗≤𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+𝐽ℎ2!×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔,‖𝑒(𝑡)‖≤𝑒𝑡₀+^𝐽−1_𝑗=0𝑡−𝑡_𝑗𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹×𝜌1−𝐾2+𝑔_𝑗≤𝑒𝑡₀+𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+𝐽ℎ2!𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔; (47a)(47b)(47c)(47d)∀𝑡∈[𝑡₀,𝑡₀+∑^𝐽−1_𝑖=0ℎ_𝑖](⊆[𝑡₀,𝑡₀+𝐽ℎ]) with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 and any given nonnegative integer ≤𝑛. The following result follows directly from the above equations and Theorem 6.

Theorem 7. Consider an approximated perturbed solution (42) under a forcing perturbation sequence {𝑔(𝑡_𝑖)}⊂𝐑^𝑛 at 𝑡=𝑡_𝑖 fulfilling ‖𝑔(𝑡_𝑖)‖≤𝑔_𝑖≤𝑔≥‖𝑒(𝑡₀)‖ for 𝑖=1,2,… and some 𝑔∈𝐑₊. Then, there are numbers ℎ∈𝐑₊, 𝐽=𝐽(ℎ)∈𝐙₊, 𝜀₁=𝜀₁(ℎ,𝑔)∈𝐑₊, and 𝜀=𝜀(𝜀₁,‖𝑒(𝑡₀)‖) such that maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,max_{𝑡∈𝐑0+}𝑒(𝑡)−𝑒𝑡₀≤𝜀₁;max_{𝑡∈𝐑0+}‖𝑒(𝑡)‖≤𝜀 (48) on [𝑡₀,𝑡_𝐽] for any strictly ordered sequence of (𝐽+1) nonnegative real numbers {𝑡_𝑖∶𝑖=0,1,…,𝐽}, subject to the constraints 𝑡_𝐽=𝑡₀+^𝐽−1_𝑖=0ℎ_𝑖,ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ,𝑖=0,1,…,𝐽−1 (49) satisfying the constraints (22)–(24) of Lemma 4 subject to (18).

Proof. Note that fixing ∑^𝐽−1_𝑖=0ℎ_𝑖=𝑡_𝐽−𝑡₀≤𝐽ℎ, with ℎ=max_{0≤𝑖≤𝐽−1}(𝑡_𝑖+1−𝑡_𝑖), and the use of (46), (47a), and (47b) leads to𝑒(𝑡)−𝑒𝑡_𝑖≤^𝐽−1_𝑖=0ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖≤𝜀₁=𝜌+^𝐽−1_𝑖=0𝑔_𝑖,𝑒(𝑡)−𝑒𝑡_𝑖≤𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔≤𝜀₁=𝜌+𝐽𝑔; (50a)(50b)∀𝑡∈[𝑡₀,𝑡_𝐽]; 𝑖=0,1,…,𝐽−1 by using Lemma 4 and Theorem 6 for any given prefixed 𝜌∈𝐑₊. The result then follows since 𝑔₀=‖𝑒(𝑡₀)‖≤𝑔 and either 𝜀=𝜀₁+𝑒𝑡₀,𝜀₁=𝜌+^𝐽−1_𝑖=0𝑔_𝑖,_{𝐽−1𝑖=0}ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+ℎ2!𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2≤𝜌 (51) or 𝜀=𝜀₁+𝑒𝑡₀,𝜀₁=𝜌+𝐽ℎ𝑔,_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2≤𝜌; (52) and the result has been proven.

The following result extends Theorem 7 with results of Theorem 6 for the case when both the exact and approximated differential equations are subject to a piecewise-continuous bounded disturbance which might be dependent on the solution and also can have finite step discontinuities in the sequence {𝑡_𝑖∶𝑖=0,1,…,𝐽}.

Theorem 8. Consider the forced versions of the differential equations (5) and (10): 𝑦̇(𝑡)=𝑓𝑦(𝑡),𝑡+𝑔𝜏,𝑦(𝜏),𝑦𝑡₀=𝑦₀,𝑥̇(𝑡)=_𝑘=0𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥(𝑡)𝑡𝑘!−𝑥_𝑖^𝑘((𝜏,𝑥𝑡+𝑔𝜏,𝑥))₀=𝑥₀ (53)(54) under the additive forcing perturbation 𝑔∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) satisfying Assumption (A2) of Lemma 4 fulfilling 𝑔(𝑦(𝑡),𝑡)=𝜆(𝑡)𝑦(𝑡) and 𝑔(𝑥(𝑡),𝑡)=𝜆(𝑡)𝑥(𝑡)+𝑈(𝑡−𝑡_𝑖+1)𝑔_𝑖+1; ∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1] with ‖𝑔_𝑖+1‖≤𝑔 for 𝑖=0,1,…,𝐽−1 and some 𝑔∈𝐑₊ and 𝜆∶[𝑡₀,𝑡_𝐽]→𝐑^𝑛 being a bounded piecewise-continuous function. Then, there are numbers ℎ∈𝐑₊, 𝐽=𝐽(ℎ)∈𝐙₊, 𝜀₁=𝜀₁(ℎ,𝑔)∈𝐑₊ and 𝜀=𝜀(𝜀₁,‖𝑒(𝑡₀)‖) such that maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,max_{𝑡∈𝐑0+}𝑒(𝑡)−𝑒𝑡₀≤𝜀₁,max_{𝑡∈𝐑0+}‖𝑒(𝑡)‖≤𝜌 (55) on [𝑡₀,𝑡_𝐽] for a strictly ordered finite set of (𝐽+1) nonnegative real numbers {𝑡_𝑖∶𝑖=0,1,…,𝐽}, subject to the constraints 𝑡_𝐽=𝑡₀+∑^𝐽−1_𝑖=0ℎ_𝑖, ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ; 𝑖=0,1,…,𝐽−1, the constraints (22)–(24) subject to (18), and either^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖<1,_{𝐽−1𝑖=0}𝑔_𝑖<∞,∑𝑔≥^{𝐽−1𝑖=0}ℎ_𝑖𝜆_𝑖1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖𝑒𝑡₀, (56a)(56b)or𝐽ℎ𝜆<1,𝐽𝑔<∞,𝑔≥𝐽ℎ𝜆𝑒𝑡1−𝐽ℎ𝜆⁰. (57a)(57b)

Proof. Fix ∑^𝐽−1_𝑖=0ℎ_𝑖=𝑡_𝐽−𝑡₀≤𝐽ℎ, with ℎ=max_{0≤𝑖≤𝐽−1}(𝑡_𝑖+1−𝑡_𝑖). Equations (53)-(54) have the following solutions: 𝑦(𝑡)=𝑦𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑗