Qualitative Analysis of Solutions of Nonlinear Delay Dynamic Equations

Abstract

We use the fixed point theory to investigate the qualitative analysis of a nonlinear delay dynamic equation on an arbitrary time scales. We illustrate our results by applying them to various kind of time scales.

1. Introduction

In this paper, we investigate the qualitative analysis of solutions of nonlinear delay dynamic equation of the form 𝑥^Δ(𝑡)=−𝑎(𝑡)𝑔(𝑥(𝛿(𝑡)))𝛿^Δ(𝑡),𝑡∈𝑡₀,∞_𝕋 (1) on an arbitrary time scale 𝕋 which is unbounded above, where the functions 𝑎 and 𝑔 are rd-continuous, the delay function 𝛿∶[𝑡₀,∞)_𝕋→[𝛿(𝑡₀),∞)_𝕋 is strictly increasing, invertible, and delta differentiable such that 𝛿(𝑡)<𝑡, |𝛿^Δ(𝑡)|<∞ for 𝑡∈𝕋, and 𝛿(𝑡₀)∈𝕋.

Although it is assumed that the reader is already familiar with the time scale calculus, for completeness, we will provide some essential information about time scale calculus in the Section 1.1. We should only mention here that this theory was introduced in order to unify continuous and discrete analysis; however it is not only unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases to cases “in between,” for example, to so-called 𝑞-difference equations. Also note that, when 𝕋=ℝ, (1) is reduced to the nonlinear delay differential equation 𝑥^′(𝑡)=−𝑎(𝑡)𝑔(𝑥(𝑡−𝑟)) (2) and when 𝕋=ℤ, it becomes a nonlinear delay difference equation Δ𝑥(𝑡)=−𝑎(𝑡)𝑔(𝑥(𝑡−𝑟)). (3) In the case of quantum calculus which defined as 𝕋=𝑞^ℕ∶={𝑞^𝑚∶𝑚∈ℕ}, 𝑞>1 is a real number, (1) leads to the nonlinear delay 𝑞-difference equation Δ_𝑞𝑥(𝑡)=−𝑎(𝑡)𝑔(𝑥(𝛿(𝑡)))Δ_𝑞𝛿(𝑡), (4) where Δ_𝑞𝑓(𝑡)=(𝑓(𝑞𝑡)−𝑓(𝑡))/(𝑞−1)𝑡.

Motivated by the papers [1, 2], in this paper we study the qualitative properties of solution of nonlinear delay dynamic equation (1) by means of fixed point theory. The results of this paper unify the results given by [1] for (2) and by [2] for (3). Moreover, we obtain new results for the 𝑞-difference equation (4) and explicitly provide an example in which we show how our conditions can be applied. Our technique in proving the results naturally has some common features with the ones employed in both [1] and [2] but it is actually quite different due to difficulties that are peculiar to the time scale calculus. Also, our results may be considered as generalization of the ones obtained in [3, 4] and [5] in which the authors studied the stability of the delay dynamic equation 𝑥^Δ(𝑡)=−𝑎(𝑡)𝑥(𝛿(𝑡))𝛿^Δ(𝑡). (5) In [6], the authors establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation 𝑥^Δ(𝑡)=𝑝(𝑡)𝑥(𝑡−𝜏(𝑡)). (6) One can easily see that the results of (6) cannot be applied to 𝑞-difference equations. Moreover, it requires that 𝑡−𝜏(𝑡) be in the time scale. For resent results regarding existence, uniqueness and continuous dependence of the solution for nonlinear delay dynamic equations, we refer to [7].

1.1. Preliminaries on Time Scales

In this subsection, we recall some of the notations, definitions, and theorems on time scale calculus that we use throughout the paper. An excellent comprehensive treatment of calculus on time scales can be found in [8, 9]. Most of the material in this subsection can be found in [8, Chapter 1]. We should start mentioning that this theory was introduced in order to unify continuous and discrete analysis; however, it does not only unify the theories of differential equations and difference equations, but also it enable us to extend these classical cases to cases “in between,” for example, to so-called 𝑞-difference equations.

A time scale is an arbitrary nonempty closed subset of the real numbers ℝ and is denoted by the symbol 𝕋. The two most popular examples are 𝕋=ℝ and 𝕋=ℤ. Several other interesting time scales exist, and they give rise to plenty of applications such as the study of population dynamic models (see [8, pages 15 and 71]). Define the time scale interval [𝑎,𝑏]_𝕋 by [𝑎,𝑏]_𝕋∶=[𝑎,𝑏]∩𝕋. Other time scale intervals are defined similarly. The forward jump operator 𝜎(𝑡) at 𝑡∈𝕋 for 𝑡<sup𝕋 is defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡} with 𝜎(sup𝕋)=sup𝕋. The graininess function 𝜇∶𝕋→ℝ is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. The backward jump operator 𝜌(𝑡) at 𝑡∈𝕋 for −∞<inf𝕋 is defined by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡} with 𝜌(inf𝕋)=inf𝕋. The point 𝑡∈𝕋 is called right scattered if 𝜎(𝑡)>𝑡, left scattered if 𝜌(𝑡)<𝑡, and dense if 𝜎(𝑡)=𝑡.

Fix 𝑡∈𝕋 and let 𝑓∶𝕋→ℝ. Define 𝑓^Δ(𝑡) (called the delta derivatives of 𝑓 at 𝑡) to be the number (if exists) with the property that given 𝜖>0 there is a neighborhood 𝑈 of 𝑡 such that, for all 𝑠∈𝑈, 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜖|𝜎(𝑡)−𝑠|. (7)

Some elementary facts concerning the delta derivative are as follows:(1)if 𝑓 is differentiable at 𝑡, then 𝑓^𝜎(𝑡)=𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡); (8)(2)if 𝑓 and 𝑔 are differentiable at 𝑡, then 𝑓𝑔 is differentiable at 𝑡 with 𝑓𝑔^Δ(𝑡)=𝑓^𝜎(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔^𝜎(𝑡); (9)(3)if 𝑓 and 𝑔 are differentiable at 𝑡 and 𝑔(𝑡)𝑔(𝜎(𝑡))≠0, then 𝑓/𝑔 is differentiable at 𝑡 with 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔^𝜎(𝑡). (10)

We say 𝑓∶𝕋→ℝ is right-dense continuous (𝑓∈𝐶_rd(𝕋,ℝ)) provided 𝑓 is continuous at right-dense points in 𝕋 and its left-sided limit exists (finite) at left-dense points in 𝕋. The importance of rd-continuous functions is that every rd-continuous function possesses an antiderivative. A function 𝐹∶𝕋^𝜅→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝕋^𝜅.

Some elementary facts concerning the delta integral are as follows:

if 𝑎, 𝑏, 𝑐∈𝕋, 𝛼∈ℝ, and 𝑓, 𝑔∈𝐶_rd, then(a)∫^𝑏_𝑎𝑓(𝜎(𝑡))𝑔^Δ(𝑡)Δ𝑡=(𝑓𝑔)(𝑏)−(𝑓𝑔)(𝑎)−∫^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝑡)Δ𝑡;(b)∫^𝑏_𝑎𝑓(𝑡)𝑔^Δ(𝑡)Δ𝑡=(𝑓𝑔)(𝑏)−(𝑓𝑔)(𝑎)−∫^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝜎(𝑡))Δ𝑡;(c)If |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)_𝕋, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡;(d)If 𝑓(𝑡)≥0 for all 𝑡∈[𝑎,𝑏)_𝕋, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0.

Now, we present a chain rule: Assume that 𝜈∶𝕋→ℝ is a strictly increasing and 𝕋∶=𝜐(𝕋) is a time scale. Let 𝜔∶𝕋→ℝ. If 𝜈^Δ(𝑡) and 𝜔Δ(𝜈(𝑡)) exists for 𝑡∈𝕋^𝜅, then (𝑤∘𝜈)^Δ=(𝜔Δ∘𝜈)𝜈^Δ.

A function 𝑝∶𝕋→ℝ is called regressive if it is rd-continuous and satisfies 1+𝜇(𝑡)𝑝(𝑡)≠0∀𝑡∈𝕋. (11) The set of all regressive functions will be denoted by ℛ. Also 𝑝∈ℛ⁺ (positively regressive) if and only if 𝑝∈ℛ and 1+𝜇(𝑡)𝑝(𝑡)>0, for all 𝑡∈𝕋.

For ℎ>0, define the cylinder transformation 𝜉_ℎ(𝑧)=ℂ_ℎ→ℤ_ℎ by 𝜉_ℎ(𝑧)=1ℎLog(1+𝑧ℎ), (12) where Log is the principal logarithm function. For ℎ=0, we define 𝜉_ℎ(𝑧)=𝑧 for all 𝑧∈ℂ.

If 𝑝∈ℛ, then we define the generalized exponential function 𝑒_𝑝𝑡,𝑡₀=exp^𝑡_𝑡0𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏 (13) for all 𝑡, 𝑡₀∈𝕋. Note that one can also define the generalized exponential function 𝑒_𝑝(𝑡,𝑡₀) to be the unique solution of the initial value problem 𝑥^Δ(𝑡)=𝑝(𝑡)𝑥(𝑡),𝑥𝑡₀=1. (14) Also, it is well know that, if 𝑝∈ℛ⁺, then 𝑒_𝑝(𝑡,𝑠)>0 for all 𝑡∈𝕋. We will use many of the following properties of the generalized exponential function 𝑒_𝑝(𝑡,𝑡₀) in our calculations.

If 𝑝, 𝑞∈ℛ and 𝑡, 𝑠, 𝑟, 𝑐∈𝕋, then the properties of generalized exponential function are:(1)𝑒₀(𝑡,𝑠)≡1 and 𝑒_𝑝(𝑡,𝑡)≡1;(2)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(3)1/𝑒_𝑝(𝑡,𝑠)=𝑒_⊖𝑝(𝑡,𝑠), where ⊖𝑝∶=−𝑝/(1+𝜇(𝑡)𝑝(𝑡));(4)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡);(5)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);(6)𝑒_𝑝(𝑡,𝑠)𝑒_𝑞(𝑡,𝑠)=𝑒_𝑝⊕𝑞(𝑡,𝑠), where 𝑝⊕𝑞∶=𝑝+𝑞+𝜇𝑝𝑞;(7)𝑒_𝑝(𝑡,𝑠)/𝑒_𝑝(𝑡,𝑠)=𝑒_𝑝⊖𝑞(𝑡,𝑠);(8)[𝑒_𝑝(𝑐,⋅)]^Δ=−𝑝[𝑒_𝑝(𝑐,⋅)]^𝜎.

Another useful tool is a variation of parameters formula for first order linear nonhomogeneous dynamic equations which now we state next. Suppose that 𝑝∈ℛ and 𝑓 is rd-continuous function. Let 𝑡₀∈𝕋 and 𝑥₀∈ℝ. Then the unique solution of 𝑥^Δ=𝑝(𝑡)𝑥+𝑓(𝑡),𝑥𝑡₀=𝑥₀ (15) is given by 𝑥(𝑡)=𝑒_𝑝𝑡,𝑡₀𝑥₀+^𝑡_𝑡0𝑒_𝑝(𝑡,𝜎(𝜏))𝑓(𝜏)Δ𝜏. (16)

1.2. Solution

For each 𝑡₀∈𝕋 and for a given rd-continuous initial function 𝜓∶=[𝛿(𝑡₀),𝑡₀]_𝕋→ℝ, we say that 𝑥(𝑡)∶=𝑥(𝑡;𝑡₀,𝜓) is the solution of (1) if 𝑥(𝑡)=𝜓(𝑡) on [𝛿(𝑡₀),𝑡₀]_𝕋 and satisfies (1) for all 𝑡≥𝑡₀. The zero solution of (1) is called stable if for any 𝜖>0 and 𝑡₀∈𝕋, there exists a 𝜂(𝑡₀,𝜖)>0 such that |𝜓|<𝜂 implies |𝑥(𝑡;𝑡₀,𝜓)|<𝜖 for all 𝑡≥𝑡₀.

We need the following lemmas in proving our main theorem.

Lemma 1 (see [10, Lemma 2]). For nonnegative 𝑝 with −𝑝∈ℛ⁺ one has the inequalities 1−^𝑡_𝑠𝑝(𝑢)Δ𝑢≤𝑒_−𝑝(𝑡,𝑠)≤exp−^𝑡_𝑠𝑝(𝑢)Δ𝑢∀𝑡≥𝑠. (17)

Lemma 2 (see [3, Lemma 3], [4]). Suppose that 𝕋 is a time scale having a strictly increasing, and invertible delay function 𝛿∶[𝑡₀,∞]_𝕋→[𝛿(𝑡₀),∞]_𝕋, 𝑡₀∈𝕋 such that 𝛿(𝑡)<𝑡 and |𝛿^Δ(𝑡)|<∞. Then for a given rd-continuous function 𝑓∶𝕋→ℝ, one has ^𝑡_𝛿(𝑡)𝑓(𝑠)Δ𝑠^Δ=𝑓(𝑡)−𝑓(𝛿(𝑡))𝛿^Δ(𝑡). (18)

Lemma 3. Assume 𝕋 is a time scale having a strictly increasing and invertible delay function 𝛿∶[𝑡₀,∞]_𝕋→[𝛿(𝑡₀),∞]_𝕋, 𝑡₀∈𝕋 such that 𝛿(𝑡)<𝑡 and |𝛿^Δ(𝑡)|<∞. Then, the nonlinear delay equation (1) is equivalent to 𝑥^Δ(𝑡)=−𝑎𝛿⁻¹(𝑡)𝑔(𝑥(𝑡+))_{𝑡𝛿(𝑡)}𝑎𝛿⁻¹(𝑠)𝑔(𝑥(𝑠))Δ𝑠^Δ. (19)

Proof. Assume 𝑥 is a solution of (1). Then the proof immediately follows from Lemma 2 since ^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑔(𝑥(𝑠))Δ𝑠^Δ=𝑎𝛿⁻¹(𝑡)𝑔(𝑥(𝑡))−𝑎(𝑡)𝑔(𝑥(𝛿(𝑡)))𝛿^Δ(𝑡). (20)

Lemma 4. Suppose that −𝑎∈ℛ. If 𝑥 is a solution of (1) with initial function 𝜓, then 𝑥(𝑡)=𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀𝜓𝑡₀+^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑔(𝑥(𝑠))Δ𝑠−𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀×^𝑡0_𝛿𝑡0𝑎𝛿⁻¹(𝑠)𝑔𝜓(𝑠)−Δ𝑠_𝑡𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑔(𝑥(𝑢))Δ𝑢Δ𝑠+^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑥(𝑠)−𝑔(𝑥(𝑠))Δ𝑠. (21)

Proof. We know from Lemma 3 that (1) is equivalent to (19). To create a linear term, we add and subtract 𝑎(𝛿⁻¹(𝑡))𝑥(𝑡) in (19) to obtain 𝑥^Δ(𝑡)=−𝑎𝛿⁻¹(𝑡)𝑥(𝑡)+𝑎𝛿⁻¹(𝑡)𝑥(𝑡)(𝑥(𝑡+−𝑔))_{𝑡𝛿(𝑡)}𝑎𝛿⁻¹(𝑠)𝑔(𝑥(𝑠))Δ𝑠^Δ. (22) Using the variation of constants formula page 77 [8] for (22) yields 𝑥(𝑡)=𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀𝑥𝑡₀+^𝑡_𝑡0𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑎𝛿⁻¹(𝑠)𝑥(𝑠)(𝑥(𝑠+−𝑔))Δ𝑠_𝑡𝑡0𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑔(𝑥(𝑢))Δ𝑢^Δ𝑠Δ𝑠, (23) where Δ_𝑠 denotes the delta derivative with respect to 𝑠. The proof follows from using integration by parts formula on the last term of the right hand side of (23).

2. Main Results

Let 𝕋 be an arbitrary time scale which is unbounded above and consider the nonlinear delay dynamic equation 𝑥^Δ(𝑡)=−𝑎(𝑡)𝑔(𝑥(𝛿(𝑡)))𝛿^Δ(𝑡),𝑡∈𝑡₀,∞_𝕋, (24) with rd-continuous initial function 𝜓∶=[𝛿(𝑡₀),𝑡₀]_𝕋→ℝ.

In the sequel we assume the following:(a)sup𝕋=∞;(b)𝑎∶[𝑡₀,∞)_𝕋→ℝ  rd-continuous and −𝑎∈ℛ⁺;(c)the delay function 𝛿∶[𝑡₀,∞)_𝕋→[𝛿(𝑡₀),∞)_𝕋 is strictly increasing, invertible, and delta differentiable such that 𝛿(𝑡)<𝑡, |𝛿^Δ(𝑡)|<∞ for 𝑡∈𝕋, and 𝛿(𝑡₀)∈𝕋;(d)𝑔∶ℝ→ℝ is continuous, locally Lipschitz and odd while 𝑔(𝑥) is rd-continuous, 𝑥−𝑔(𝑥) is nondecreasing and 𝑔(𝑥) is increasing on an interval [0,𝐿] with 𝐿>0, where 𝑥∶𝕋→ℝ is rd-continuous function.

We should remark here that condition (d) in our hypotheses ensures that the function 𝑔(𝑥) and 𝑥−𝑔(𝑥) are locally Lipschitz with the same Lipschitz constant 𝐾>0. Also it is clear that if 0<𝐿₁<𝐿, then the condition on 𝑔 given by (d) hold on [−𝐿₁,𝐿₁]. Moreover, for any given rd-continuous function 𝜙∶[𝑡₀,∞)_𝕋→ℝ with 𝜙₀=𝜓 is bounded with |𝜙(𝑡)|≤𝐿, then for 𝑡≥𝑡₀ we have 𝜙(𝑡)−𝑔𝜙(𝑡)≤𝐿−𝑔(𝐿), (25) since 𝑥−𝑔(𝑥) is odd and nondecreasing on (0,𝐿).

We need to construct a mapping which is suitable for fixed point theory. Instead of using the supremum norm, we will use nonconventional metric to define a new norm in order to overcome the difficulties that arise from the contraction constant which, in turn rely on the Lipschitz constant.

Lemma 5. Let (a)–(d) of our hypotheses hold. Let 𝐿>0 and 𝜓(𝑡₀) be a fixed number for 𝑡₀∈𝕋, 𝑆=𝜙∶𝑡₀,∞_𝕋⟶ℝ∣𝜙∈𝐶_𝑟𝑑,𝜙𝑡₀=𝜓𝑡₀,𝜙(𝑡)≤𝐿, (26) and 𝑓∶[−𝐿,𝐿]→ℝ satisfy a Lipschitz condition with constant 𝐾>0. Suppose that ℎ∶[𝑡₀,∞)_𝕋→ℝ is rd-continuous and for 𝜙∈𝑆 define 𝑃𝜙(𝑡)=ℎ(𝑡)+^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑓𝜙(𝑠)Δ𝑠. (27) If 𝑃∶𝑆→𝑆, then for each 𝑑>1 there is a metric 𝜈 on 𝑆 such that 𝑃 is a contraction with constant 1/𝑑 and (𝑆,𝜈) is a complete metric space.

Proof. Let (ℬ,|⋅|_𝐾) be the Banach space of rd-continuous function 𝜙∶[𝑡₀,∞)_𝕋→ℝ for which 𝜙_𝐾∶=sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀𝜙(𝑡) (28) exists. If 𝜙, 𝜑∈𝑀, then 𝑃𝜙−𝑃𝜑_𝐾≤sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑓𝜙(𝑠)𝜑(𝑠)𝑡,𝜎))−𝑓Δ𝑠≤sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝐾𝜙(𝑠)−𝜑(𝑠)Δ𝑠=sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠𝜙(𝑠)(𝑠𝑒𝑡,𝜎))×𝐾−𝜑)_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀Δ𝑠. (29) Since 𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀=𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|(𝑡,𝑠)𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑠,𝑡₀,𝜙−𝜑_𝐾=sup_𝑠≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑠,𝑡₀𝜙(𝑠)−𝜑(𝑠), (30) we obtain 𝑃𝜙−𝑃𝜑_𝐾≤𝐾𝜙−𝜑_𝐾×sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))×𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|(𝑡,𝑠)Δ𝑠. (31) On the one hand, using Lemma 1 in the integral on the right hand side of (31), we have ^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|()≤𝑡,𝑠Δ𝑠_𝑡𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑒𝑡,𝜎))_{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}Δ𝑠. (32) In addition, by setting 𝑓(𝑠)=𝑒_{−|𝑎(𝛿}⁻¹_)|(),𝑔(𝑠)𝑡,𝑠=𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}, (33) and using the fact that 𝑔^Δ𝑠(𝑠)=(𝑑𝐾+1)𝑎𝛿⁻¹(𝑠)𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢} (34) yield ^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}=1Δ𝑠(𝑑𝐾+1)^𝑡_𝑡0𝑓(𝜎(𝑠))𝑔^Δ𝑠(𝑠)Δ𝑠, (35) where Δ_𝑠 denotes the delta derivative with respect to variable 𝑠. And hence, by fixing 𝑡 and using the integration by part formula by taking into account that 𝑓^Δ𝑠(𝑠)=𝑒_{−|𝑎(𝛿}⁻¹_)|(𝑡,𝑠)^Δ𝑠=𝑎𝛿⁻¹𝑒_{−|𝑎(𝛿}⁻¹_)|(𝑡,𝜎(𝑠)), (36) we obtain ^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}=1Δ𝑠()−1𝑑𝐾+1()𝑒𝑑𝐾+1^{−|𝑎(𝛿−1)|}×𝑡,𝑡₀𝑒^{−(𝑑𝐾+1)∫𝑡𝑡0|𝑎(𝛿−1(𝑢))|Δ𝑢}−1(𝑑𝐾+1)^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑒𝑡,𝜎))_{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}Δ𝑠. (37) Thus, we get ^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}=1Δ𝑠(𝑑𝐾+2)1−𝑒_{−|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀𝑒^{−(𝑑𝐾+1)∫𝑡𝑡0|𝑎(𝛿−1(𝑢))|Δ𝑢}≤1().𝑑𝐾+2 (38) Finally, substituting (38) into (31) we get 𝑃𝜙−𝑃𝜑_𝐾≤𝐾𝑑𝐾+2𝜙−𝜑_𝐾≤1𝑑𝜙−𝜑_𝐾. (39) Since 𝑆 is a subset of the Banach space ℬ and 𝑆 is closed, hence 𝑆 is complete. Thus, 𝑃∶𝑆→𝑆 has a unique fixed point.

For any given rd-continuous initial function 𝜓 defined on [𝛿(𝑡₀),𝑡₀]_𝕋with |𝜓(𝑡)|<𝐿, we let 𝑆=𝜙∶𝛿𝑡₀,∞_𝕋⟶ℝ∣𝜙∈𝐶_rd,𝜙₀=𝜓,𝜙(𝑡)≤𝐿, (40) where 𝜙₀ denotes the segment of 𝜙 on [𝛿(𝑡₀),𝑡₀].

Theorem 6. Let (a)–(d) of our hypotheses hold, and suppose that for each 𝐿₁∈(0,𝐿] the inequality 𝐿₁−𝑔𝐿₁sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠𝐿𝑡,𝜎))Δ𝑠+𝑔₁sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠𝐿)Δ𝑠+𝑔₁sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)Δ𝑢Δ𝑠<𝐿₁ (41) and for 𝐽>0 the inequality 𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀≤𝐽,𝑡≥𝑡₀ (42) hold. Then every solution of (24) is bounded. In addition if 𝑔(0)=0, then the zero solution of (24) is stable.

Proof. Define a mapping 𝑃 on 𝑆 using (21) in such a way that for 𝜙∈𝑆 we have 𝑃𝜙(𝑡)=𝜓(𝑡),𝛿𝑡₀≤𝑡≤𝑡₀,𝑃𝜙(𝑡)=𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀𝜓𝑡₀+^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑔𝜙(𝑠)Δ𝑠−𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀^𝑡0_𝛿(𝑡0)𝑎𝛿⁻¹(𝑠)𝑔𝜓(𝑠)−Δ𝑠_𝑡𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑔𝜙(𝑢)+Δ𝑢Δ𝑠_𝑡𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝜙(𝑠)𝜙(𝑠)𝑡,𝜎))−𝑔Δ𝑠,𝑡≥𝑡₀. (43) By (41) there exists an 𝛼∈(0,1) such that if 𝜙∈𝑆 and for 𝑡≥𝑡₀ we have 𝑃𝜙(𝑡)≤𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀𝜓+𝑔(𝐿)sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)Δ𝑠+𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀𝑔𝜓^𝑡0_𝛿(𝑡0)𝑎𝛿⁻¹(𝑠(𝐿))Δ𝑠+𝑔sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)+(𝐿Δ𝑢Δ𝑠𝐿−𝑔)sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎×(𝑡,𝜎(𝑠))Δ𝑠. (44) Since 𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝑡₀)≤𝐽 we obtain 𝑃𝜙(𝑡)≤𝐽𝜓+𝑔𝜓^𝑡0_𝛿(𝑡0)𝑎𝛿⁻¹(𝑠)Δ𝑠+𝛼𝐿. (45) By choosing the initial function 𝜓 small enough we have 𝐽𝜓+𝐾𝜓^𝑡0_𝛿(𝑡0)𝑎𝛿⁻¹(𝑠)Δ𝑠≤(1−𝛼)𝐿, (46) where 𝐾 is the Lipschitz constant of 𝑔 on [0,𝐿]. Hence, we obtain 𝑃𝜙(𝑡)≤(1−𝛼)𝐿+𝛼𝐿=𝐿. (47) Thus, we have showed that 𝑃∶𝑆→𝑆 and any solution of (24) that is in 𝑆 is bounded.
Next we need to show that 𝑃 is a contraction. To do this, we proceed as in the proof of Lemma 5. First note that for 𝜙, 𝜑∈𝑆 we have (𝑡)−(𝑡≤𝑃𝜙𝑃𝜑)_{𝑡𝛿(𝑡)}𝑎𝛿⁻¹(𝑠)𝑔𝜙(𝑠)−𝑔𝜑(𝑠)Δ𝑠+^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑔𝜙(𝑢)𝜑(𝑢)+−𝑔Δ𝑢Δ𝑠_𝑡𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝜙(𝑠)𝜙(𝑠)(𝑠)𝜑(𝑠)𝑡,𝜎))−𝑔−𝜑+𝑔Δ𝑠, (48) and hence we take the metric on 𝑆 which is induced by the norm 𝜙_𝐾∶=sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀𝜙(𝑡). (49) Our aim is to simplify (48). It follows from Lemma 5 that the last term on the right hand side of (48) has a contraction constant 1/𝑑 since 𝑔(𝑥) and 𝑥−𝑔(𝑥) both satisfy a Lipschitz condition with the same constant 𝐾>0 that is; sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝜙(𝑠)𝜙(𝑠)(𝑠)𝜑(𝑠)≤1𝑡,𝜎))−𝑔−𝜑+𝑔Δ𝑠𝑑𝜙−𝜑_𝐾. (50)
The first term satisfies sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑔𝜙(𝑠)−𝑔𝜑(𝑠)Δ𝑠≤sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝐾𝜙(𝑠)−𝜑(𝑠)×𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀Δ𝑠≤sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝐾𝜙(𝑠)−𝜑(𝑠)×𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|(𝑡,𝑠)𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑠,𝑡₀Δ𝑠≤𝐾𝜙−𝜑_𝐾sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|(𝑡,𝑠)Δ𝑠≤𝐾𝜙−𝜑_𝐾sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}Δ𝑠. (51) Setting 𝑓(𝑠)=𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢} (52) and using the fact that 𝑓^Δ𝑠(𝑠)=(𝑑𝐾+1)𝑎𝛿⁻¹(𝑠)𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢} (53) we obtain sup_𝑡≥𝑡0^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝑢))|Δ𝑢}Δ𝑠=sup_𝑡≥𝑡01𝑑𝐾+11−𝑒^{−(𝑑𝐾+1)∫𝑡𝛿(𝑡)|𝑎(𝛿−1(𝑢))|Δ𝑢}≤1.𝑑𝐾+1 (54) Thus, sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑔𝜙(𝑠)𝜑(𝑠)≤𝐾−𝑔Δ𝑠𝑑𝐾+1𝜙−𝜑_𝐾≤1𝑑𝜙−𝜑_𝐾. (55) Now, we turn our attention to the second term of (48). Multiply this term by 𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀ (56) we get sup_𝑡≥𝑡0𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑔𝜙(𝑢)−𝑔𝜑(𝑢)Δ𝑢Δ𝑠≤sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝐾𝜙(𝑢)−𝜑(𝑢)×𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|𝑡,𝑡₀Δ𝑢Δ𝑠≤𝐾𝜙−𝜑_𝐾^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)×𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|(𝑡,𝑢)Δ𝑢Δ𝑠≤𝐾𝜙−𝜑_𝐾^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎×(𝑡,𝜎(𝑠))𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|()×𝑡,𝑠_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)×𝑒^{−(𝑑𝐾+1)∫𝑠𝑢|𝑎(𝛿−1(𝜏))|Δ𝜏}≤𝐾Δ𝑢Δ𝑠𝑑𝐾+1𝜙−𝜑_𝐾×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑒_{−(𝑑𝐾+1)|𝑎(𝛿}⁻¹_)|×()≤𝐾𝑡,𝑠Δ𝑠𝑑𝐾+1𝜙−𝜑_𝐾×^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))×𝑒^{−(𝑑𝐾+1)∫𝑡𝑠|𝑎(𝛿−1(𝜏))|Δ𝜏}≤𝐾Δ𝑠𝑑𝐾+1𝜙−𝜑_𝐾1()≤1𝑑𝐾+2𝑑𝜙−𝜑_𝐾. (57) A substitution of (50), (55), and (57) into (48) gives 𝑃𝜙(𝑡)−𝑃𝜑(𝑡)_𝐾≤1𝑑+1𝑑+1𝑑𝜙−𝜑_𝐾. (58) Hence, we have showed that 𝑃 is a contraction for 𝑑>3. Thus, taking into account Lemma 5, 𝑃∶𝑆→𝑆 has a unique fixed point. This completes the proof.

When 𝕋=ℝ and 𝕋=ℤ, we have the following two corollaries which are immediate consequences of Theorem 6 and thus the proofs are omitted.

Corollary 7. Let (a)–(d) of our hypotheses hold in case of  𝕋=ℝ and suppose that for each 𝐿₁∈(0,𝐿] the inequality 𝐿₁−𝑔𝐿₁sup_𝑡≥𝑡0^𝑡_𝑡0|𝑎(𝑠+𝑟)|𝑒^{−∫𝑡𝑠𝑎(𝑢+𝑟)𝑑𝑢}𝐿𝑑𝑠+𝑔₁sup_𝑡≥𝑡0^𝑡_𝑡−𝑟|𝑎(𝐿𝑠+𝑟)|𝑑𝑠+𝑔₁sup_𝑡≥𝑡0^𝑡_𝑡0|𝑎(𝑠+𝑟)|𝑒^{−∫𝑡𝑠𝑎(𝑢+𝑟)𝑑𝑢}×^𝑠_𝑠−𝑟|𝑎(𝑢+𝑟)|𝑑𝑢𝑑𝑠<𝐿₁ (59) and for 𝐽>0 the inequality 𝑒^{−∫𝑡𝑡0𝑎(𝑠+𝑟)𝑑𝑠}≤𝐽,𝑡≥𝑡₀ (60) hold. Then every solution of (2) is bounded. In addition, if 𝑔(0)=0, then the zero solution of (2) is stable.

Corollary 8. Let (a)–(d) of our hypotheses hold in case of 𝕋=ℤ, and suppose that for each 𝐿₁∈(0,𝐿] the inequality 𝐿₁−𝑔𝐿₁max_𝑡≥0^𝑡−1_𝑠=0|𝑎(𝑠+𝑟)|^𝑡−1_𝑢=𝑠(𝐿|(1−𝑎𝑢+𝑟))|+𝑔₁max_𝑡≥0^𝑡−1_{𝑠=𝑡−𝑟}|𝑎(𝐿𝑠+𝑟)|+𝑔₁max_𝑡≥0^𝑡−1_𝑠=0|𝑎(𝑠+𝑟)|^𝑡−1_𝑢=𝑠(×|(1−𝑎𝑢+𝑟))|_{𝑠−1𝑘=𝑠−𝑟}|𝑎(𝑘+𝑟)|<𝐿₁ (61) and for 𝐽>0 the inequality |(1−𝑎(^𝑡−1_𝑠=0𝑠+𝑟))|≤𝐽,𝑡≥0 (62) hold. Then every solution of (3) is bounded. In addition if 𝑔(0)=0, then the zero solution of (3) is stable.

Remark 9. We may deduce [11, Theorem 4.1] and [2, Theorem 2.2] as Corollaries 7 and 8, respectively. Thus, we have unified these results and moreover, we have extended them the general time scales. In particular, the next results concerning the 𝑞-difference equation (4) are new.

Example 10. For any 𝑞>1 and fixed positive integer 𝑚, define 𝕋∶=𝑞^−𝑚,𝑞^−𝑚+1,…,𝑞⁻¹,1,𝑞,𝑞²,…. (63) For any initial function 𝜓(𝑡), 𝑡∈[𝑞^−𝑚,1]_𝕋, Consider the following nonlinear delay initial value problem 𝐷_𝑞𝑥(𝑡)𝑐=−𝑡𝑥³𝑞^−𝑚𝑡𝑞^−𝑚,𝑡∈[1,∞)_𝕋𝑥(𝑡)=𝜓(𝑡),𝑡∈𝑞^−𝑚,1_𝕋, (64) where 0<𝑐<𝑞^𝑚/2𝑚(𝑞−1). We show that the solution of (64) is bounded and zero solution is stable.

A simple comparison of (64) to (24) yield that 𝑎(𝑡)=𝑐/𝑡, 𝛿(𝑡)=𝑞^−𝑚𝑡, 𝛿⁻¹(𝑡)=𝑞^𝑚𝑡, 𝐷_𝑞𝛿(𝑡)=𝑞^−𝑚, and 𝑎(𝛿⁻¹(𝑡))=𝑐/𝑞^𝑚𝑡=𝑞^−𝑚(𝑐/𝑡).

Notice that 𝑔(𝑥(𝑡))=𝑥³(𝑡) is increasing, continuous and odd and that 𝑥−𝑔(𝑥) and 𝑔(𝑥) locally Lipschitz on [0,1/√3𝑞]. Thus, every condition of our hypotheses is satisfied.

In order to make use of Theorem 6, we must perform some calculations so that conditions (41) and (42) are satisfied.

For 𝑡₀=1, 𝑡=𝑞^𝑛 we first calculate 𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝑡₀): 𝑒_−𝑎(𝛿⁻¹₎𝑡,𝑡₀=𝑒_−𝑎(𝛿⁻¹₎(𝑡,1)=𝑒^{(1/(𝑞−1))∫𝑞𝑛1(1/𝑠)log[1−(𝑞−1)𝑠(𝑐/(𝑞𝑚𝑠))]𝑑𝑞𝑠}=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐]∫𝑞𝑛1(1/𝑠)𝑑𝑞𝑠}, (65) and since ^𝑞𝑛₁1𝑠𝑑_𝑞𝑠=^𝑞_𝑞01𝑠𝑑_𝑞𝑠+_𝑞^𝑞21𝑠𝑑_𝑞𝑠+^𝑞3_𝑞21𝑠𝑑_𝑞𝑠+⋯+^𝑞𝑛_𝑞𝑛−11𝑠𝑑_𝑞𝑠=^𝑛−1_𝑘=0^{𝜎(𝑞𝑘)}_𝑞𝑘1𝑠𝑑_𝑞𝑠=^𝑛−1_𝑘=0𝜇𝑞^𝑘1𝑞^𝑘=^𝑛−1_𝑘=0𝑞−1𝑞^𝑘1𝑞^𝑘=𝑛𝑞−1, (66) we obtain 𝑒_−𝑎(𝛿⁻¹₎(𝑡,1)=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐]∫𝑞𝑛1(1/𝑠)𝑑𝑞𝑠}=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐]𝑛(𝑞−1)}=𝑒^{𝑛log[1−(𝑞−1)𝑞−𝑚𝑐]}=1−𝑞−1𝑞^−𝑚𝑐^𝑛. (67)

Next, we calculate ∫^𝑡₁|𝑎(𝛿⁻¹(𝑠))|𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠. For 1≤𝑠≤𝑡=𝑞^𝑛, we have 𝑠=𝑞^𝑘−1 where 1≤𝑘−1≤𝑛⇒𝜎(𝑠)=𝑞^𝑘. This implies that 𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))=𝑒^{∫𝑡𝜎(𝑠)(1/(𝑞−1)𝑢)log[1−(𝑞−1)𝑢𝑎(𝛿−1(𝑢))]𝑑𝑞𝑢}=𝑒^{∫𝑞𝑛𝑞𝑘(1/(𝑞−1)𝑢)log[1−(𝑞−1)𝑢𝑎(𝛿−1(𝑢))]𝑑𝑞𝑢}=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐]∫𝑞𝑛𝑞𝑘(1/𝑢)𝑑𝑞𝑢}. (68)

Since ^𝑞𝑛_𝑞𝑘1𝑢𝑑_𝑞𝑢=^𝑞𝑘+1_𝑞𝑘1𝑢𝑑_𝑞𝑢+^𝑞𝑘+2_𝑞𝑘+11𝑢𝑑_𝑞𝑢+^𝑞𝑘+3_𝑞𝑘+21𝑢𝑑_𝑞𝑢+⋯+^𝑞𝑛_𝑞𝑛−11𝑢𝑑_𝑞𝑢=^{𝜎(𝑞𝑘)}_𝑞𝑘1𝑢𝑑_𝑞𝑢+^{𝜎(𝑞𝑘+1)}_𝑞𝑘+11𝑢𝑑_𝑞𝑢+^{𝜎(𝑞𝑘+2)}_𝑞𝑘+21𝑢𝑑_𝑞𝑢+⋯+^{𝜎(𝑞𝑛−1)}_𝑞𝑛−11𝑢𝑑_𝑞𝑢=𝜇𝑞^𝑘1𝑞^𝑘+𝜇𝑞^𝑘+11𝑞^𝑘+1+𝜇𝑞^𝑘+21𝑞^𝑘+2+⋯+𝜇𝑞^𝑛−11𝑞^𝑛−1=^𝑛−1_𝑖=𝑘𝜇𝑞^𝑖1𝑞^𝑖=^𝑛−1_𝑖=𝑘𝑞−1𝑞^𝑖1𝑞^𝑖=𝑞−1(𝑛−𝑘), (69) we have 𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐]∫𝑞𝑛𝑞𝑘(1/𝑢)𝑑𝑞𝑢}=𝑒^{(1/(𝑞−1))log[1−(𝑞−1)𝑞−𝑚𝑐](𝑞−1)(𝑛−𝑘)}=𝑒^{(𝑛−𝑘)log[1−(𝑞−1)𝑞−𝑚𝑐]}=1−𝑞−1𝑞^−𝑚𝑐^𝑛−𝑘. (70) Since |𝑎(𝛿⁻¹(𝑠))|=𝑐/𝑞^𝑚𝑠, we have that ^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠=^𝑛−1_𝑘=0^{𝜎(𝑞𝑘)}_𝑞𝑘𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠=^𝜎(𝑞0)_𝑞0+^𝜎(𝑞1)_𝑞1+⋯+^{𝜎(𝑞𝑛)}_𝑞𝑛×𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠=^𝑛−1_𝑘=0𝜇𝑞^𝑘𝑎𝛿⁻¹𝑞^𝑘𝑒_−𝑎(𝛿⁻¹₎𝑡,𝜎𝑞^𝑘=^𝑛−1_𝑘=0𝑞−1𝑞^𝑘𝑐𝑞^𝑚+𝑘1−𝑞−1𝑞^−𝑚𝑐^{𝑛−(𝑘+1)}=^𝑛−1_𝑘=0𝑐𝑞−1𝑞^𝑚1−𝑞−1𝑞^−𝑚𝑐^{𝑛−(𝑘+1)}=𝑐𝑞−1𝑞^𝑚1−𝑞−1𝑞^−𝑚𝑐^𝑛−1×^𝑛−1_𝑘=01−𝑞−1𝑞^−𝑚𝑐^−𝑘, (71) where we have used the fact that 𝑒_−𝑎(𝛿⁻¹₎𝑡,𝜎𝑞^𝑘=1−𝑞−1𝑞^−𝑚𝑐^{𝑛−(𝑘+1)}. (72)

We know that ∑^𝑛−1_𝑖=0𝑎^𝑖=(1−𝑎^𝑛)/(1−𝑎), and hence, by taking 𝑎=1/(1−(𝑞−1)𝑞^−𝑚𝑐), we obtain 1−𝑎^𝑛=𝑞1−𝑎1−1/(1−𝑞−1^−𝑚𝑐)^𝑛1−1/1−𝑞−1𝑞^−𝑚𝑐=1−1/1−𝑞−1𝑞^−𝑚𝑐^𝑛1−1/1−𝑞−1𝑞^−𝑚𝑐=1−𝑞−1𝑞^−𝑚𝑐^𝑛−1/1−𝑞−1𝑞^−𝑚𝑐^𝑛1−𝑞−1𝑞^−𝑚𝑐−1/1−𝑞−1𝑞^−𝑚𝑐=1−𝑞−1𝑞^−𝑚𝑐^𝑛−11−𝑞−1𝑞^−𝑚𝑐^𝑛1−𝑞−1𝑞^−𝑚𝑐−𝑞−1𝑞^−𝑚𝑐=1−𝑞−1𝑞^−𝑚𝑐𝑞−1𝑞^−𝑚𝑐1−1−𝑞−1𝑞^−𝑚𝑐^𝑛1−𝑞−1𝑞^−𝑚𝑐^𝑛. (73) Thus 𝑐𝑞−1𝑞^𝑚1−𝑞−1𝑞^−𝑚𝑐^{𝑛−1𝑛−1}_𝑘=01−𝑞−1𝑞^−𝑚𝑐^−𝑘=𝑐𝑞−1𝑞^𝑚1−𝑞−1𝑞^−𝑚𝑐^𝑛−1×1−𝑞−1𝑞^−𝑚𝑐𝑞−1𝑞^−𝑚𝑐1−1−𝑞−1𝑞^−𝑚𝑐^𝑛1−𝑞−1𝑞^−𝑚𝑐^𝑛=1−𝑞−1𝑞^−𝑚𝑐^𝑛−11−𝑞−1𝑞^−𝑚𝑐×1−1−𝑞−1𝑞^−𝑚𝑐^𝑛1−𝑞−1𝑞^−𝑚𝑐^𝑛=1−1−𝑞−1𝑞^−𝑚𝑐^𝑛. (74) As a consequence, we obtain ^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠=1−1−𝑞−1𝑞^−𝑚𝑐^𝑛. (75)

Left to calculate ∫^𝑡_𝛿(𝑡)|𝑎(𝛿⁻¹(𝑠))|𝑑_𝑞𝑠. Consider ^𝑡_𝛿(𝑡)𝑎𝛿⁻¹(𝑠)𝑑_𝑞𝑠=^𝑡_{𝑞−𝑚𝑡}𝑐𝑞^𝑚𝑠𝑑_𝑞𝑠=^𝑡_{𝑞−𝑚𝑡}𝑐𝑞^𝑚𝑠𝑑_𝑞𝑠=𝑐𝑞^{𝑚𝑞−𝑚+1𝑡}_𝑞^−𝑚_𝑡1𝑠𝑑_𝑞𝑠+^{𝑞−𝑚+2𝑡}_{𝑞−𝑚+1𝑡}1𝑠𝑑_𝑞𝑠+⋯+^𝑡_𝑞−1𝑡1𝑠𝑑_𝑞𝑠=𝑐𝑞^𝑚𝑚_𝑘=1^{𝜎(𝑞−𝑘}_𝑞^𝑡)−𝑘_𝑡1𝑠𝑑_𝑞𝑐𝑠=𝑞^𝑚𝑚_𝑘=1𝜇𝑞^−𝑘𝑡1𝑞^−𝑘𝑡=𝑐𝑞^𝑚𝑚_𝑘=1𝑞−1𝑞^−𝑘𝑡1𝑞^−𝑘𝑡=𝑐𝑞^𝑚𝑞−1(𝑚). (76) Thus, the above integral is independent of 𝑡.

Left to compute ^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))^𝑠_𝛿(𝑠)𝑎𝛿⁻¹(𝑢)𝑑_𝑞𝑢𝑑_𝑞𝑠. (77)

We have already shown ∫^𝑠_𝛿(𝑠)|𝑎(𝛿⁻¹(𝑢))|𝑑_𝑞𝑢=(𝑐/𝑞^𝑚)(𝑞−1)(𝑚) which is independent of integration variable. This leads us to ^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))^𝑠_𝛿(𝑠)𝑎𝛿⁻¹(𝑢)𝑑_𝑞𝑢𝑑_𝑞𝑠=𝑚𝑐𝑞^𝑚𝑞−1^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠=𝑚𝑐𝑞^𝑚𝑞−11−1−𝑞−1𝑞^−𝑚𝑐^𝑛. (78)

As consequence of all of the above calculations, we arrive at, for 𝐿₁∈[0,1/√3𝑞] that 𝐿₁−𝑔𝐿₁sup_𝑡≥𝑡0^𝑡_𝑡0𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎(𝑡,𝜎(𝑠))𝑑_𝑞𝑠+𝑔𝐿₁sup_𝑡≥𝑡0^𝑡_𝛿𝑡0𝑎𝛿⁻¹(𝑠)𝑑_𝑞𝑠+𝑔𝐿₁sup_𝑡≥𝑡0^𝑡₁𝑎𝛿⁻¹(𝑠)𝑒_−𝑎(𝛿⁻¹₎((𝑠×𝑡,𝜎))_{𝑠𝛿(𝑠)}𝑎𝛿⁻¹(𝑢)𝑑_𝑞𝑢𝑑_𝑞𝑠≤𝐿₁−𝑔𝐿₁sup_𝑡≥11−1−𝑞−1𝑞^−𝑚𝑐^𝑛+𝑔𝐿₁sup_𝑡≥1𝑚𝑐𝑞^𝑚𝐿𝑞−1+𝑔₁sup_𝑡≥1𝑚𝑐𝑞^𝑚𝑞−11−1−𝑞−1𝑞^−𝑚𝑐^𝑛≤𝐿₁. (79) Since 𝑐<𝑞^𝑚/2𝑚(𝑞−1), we obtain sup_𝑡≥11−1−𝑞−1𝑞^−𝑚𝑐^𝑛=1,sup_𝑡≥1𝑚𝑐𝑞^𝑚=1𝑞−12,sup_𝑡≥1𝑚𝑐𝑞^𝑚𝑞−11−1−𝑞−1𝑞^−𝑚𝑐^𝑛=12. (80) Also since 𝑒_−𝑎(𝛿⁻¹₎(𝑡,1)=1−𝑞−1𝑞^−𝑚𝑐^𝑛, (81) we can find sufficiently large 𝑁 such that for 𝑛≥𝑁𝑒_−𝑎(𝛿⁻¹₎(𝑡,1)=1−𝑞−1𝑞^−𝑚𝑐^𝑛≤11−2𝑚^𝑁=𝐽 (82) since 𝑐𝑞^−𝑚(𝑞−1)<1/2𝑚<1. Hence all the conditions of Theorem 6 are satisfied and a result solution of (64) are bounded and its zero solution is stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for their kindly suggestions, comments, and remarks that helped improve the presentation of this paper.