Hölder Regularity of Grobman–Hartman Theorem for Dynamic Equations on Measure Chains

Abstract

It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Hölder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Hölder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Hölder continuous.

1 Introduction

In 1973, Palmer [27] generalized the classical Hartman–Grobman theorem [13, 14] to nonautonomous cases under the assumption that the linear system possesses an exponential dichotomy. By means of the (h, k) dichotomy proposed by Pinto [28–30], Fenner and Pinto [12] presented a generalized Grobman–Hartman theorem for a class of impulsive differential equation. Recently, Xia et al. [38] gave a generalization of the linearization theorem of Fenner and Pinto. Other generalizations of the linearization theorem for dynamic systems can be found in [3–5, 18, 19, 25, 26, 32, 35, 36].

An interesting field of research is that of dynamic systems on measure chains introduced by Hilger [16]. It would be worthy to mention that the dynamic equations in measure chains allow to unify difference and differential equations in only one study, and it is indeed more inclusive than that. We refer the reader to the monographs [6, 7] and the papers [1, 2, 8–11, 15, 20–24, 33, 34, 39–41]. Hilger [17] generalized Palmer’s linearization theorem to dynamic equations on measure chains. In order to obtain more easily verifiable results, a new analytic method to study the topological equivalence problem is presented in Xia et al. [37]. Those authors constructed a topologically equivalent function H(t, x) transforming the (c, d)-quasibounded solutions of the nonlinear system $x^{\Delta }=A(t)x+f(t,x)$ onto those of the linear system $x^{\Delta }=A(t)x$. Moreover, the authors studied the periodicity of the equivalent function H(t, x) when the parameters and measure chains are all periodic. Pötzche [31] extended the main results in [37] to a more general form. He investigated the behavior of the topological conjugacy under parameter variation.

Xia et al. [37], however, showed that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system for dynamic systems on measure chains. To the best of our knowledge, there has been no previous work studying Hölder regularity of the transformation H(t, x). Therefore, it is worthwhile and interesting to prove the Hölder regularity of the transformation H(t, x). We present a rigorous proof to show that the conjugating function H(t, x) in the Hartman–Grobman theorem is always Hölder continuous (and has Hölder continuous inverse). Also, the Hölder exponent is estimated.

The paper is organized as follows. In the next section, we introduce some notations and definitions on measure chains. In Sect. 3, we state our main result. In Sect. 4, we present the proof of our main result.

2 Notations, Definitions, and Lemmas

In this section, we introduce some notations and basic terminology from the calculus on measure chains. For further details, see Hilger [16] and Bohner [6]. Suppose for the following that $({\mathbb {T}},\preceq ,\mu )$ is an arbitrary measure chain, i.e., a conditionally complete totally ordered set $({\mathbb {T}},\preceq )$ (see [2, Axiom 2]) with bounded growth calibration $\mu :\mathbb {T}\rightarrow \mathbb {R}$ (see [2, Axiom 3]). Let $\chi$ be a $\mathbb {K}$-Banach space with the norm $\Vert \cdot \Vert$, where $\mathbb {K}=\mathbb {R}$ or $\mathbb {K}=\mathbb {C}$. ${\mathcal L}(\chi _1,\chi _2)$ stands for the linear space of continuous homomorphisms with the norm $\Vert T\Vert :=\sup _{\Vert x\Vert =1}\Vert Tx\Vert$ for any $T\in {{\mathcal {L}}}(\chi _1,\chi _2)$, and ${\mathcal {GL}}(\chi _1,\chi _2)$ for the set of toplinear isomorphisms between two linear subspaces $\chi _1$ and $\chi _2$ of $\chi$; $I_{\chi }$ is the identity mapping on $\chi$. In addition, write ${{\mathcal {L}}}(\chi ):={{\mathcal {L}}}(\chi ,\chi )$. We also introduce some notions, which are specific for the calculus on measure chains. In particular, ${\mathbb {T}}^+_{ {t}_{0}}$ and ${\mathbb {T}}^-_{ {t}_{0}}$ are the ${\mathbb {T}}$-intervals $\{t\in {\mathbb {T}}: {t}_{0}\preceq t\}$ and $\{t\in {\mathbb {T}}:t\preceq {t}_{0}\}$, respectively, for any ${t}_{0}\in {\mathbb {T}}$; ${\mathbb {T}}^{\kappa }:=\{t\in {\mathbb {T}}: t$ nonmaximal or left-dense$\}$. A subset $J\subseteq {\mathbb {T}}$ is said to the unbounded above (resp. below), if the set $\{\mu (t, {t}_{0})\in {\mathbb {R}}:t\in J\}$ is unbounded above (resp. below) for one and hence (by the properties of the growth calibration $\mu$) every ${t}_{0}\in {\mathbb {T}}$. The partial derivative of a mapping $\varPhi :{\mathbb {T}}\times {\mathbb {T}}\rightarrow \chi$ with respect to the first variable is denoted by $\Delta _1\varPhi$. ${{\mathcal {C}}}_{rd}({\mathbb {T}}^{\kappa },\chi )$ are the rd-continuous mappings from ${\mathbb {T}}^{\kappa }$ into $\chi$ and ${{\mathcal {C}}}_{rd}^+{\mathcal R}({\mathbb {T}}^{\kappa },{\mathbb {R}}): =\{a\in {\mathcal {C}}_{rd}({\mathbb {T}}^{\kappa },{\mathbb {R}}):1+\mu ^*(t)a(t)>0$ for $t\in {\mathbb {T}}^{\kappa }\}$ is the linear space of positively regressive functions with the algebraic operations

$$\begin{aligned} (a\oplus b)(t):= & {} a(t)+b(t)+\mu ^*(t)a(t)b(t), \\ (\alpha \odot a)(t):= & {} \lim \limits _{h\searrow \mu ^*(t)}\frac{(1+ha(t))^{\alpha }-1}{h} \quad \mathrm{for} \ t\in {\mathbb {T}}^{\kappa } \end{aligned}$$

for $a,b\in {\mathcal {C}}_{rd}^+{{\mathcal {R}}}({\mathbb {T}}^{\kappa },{\mathbb {R}})$ and reals $\alpha \in {\mathbb {R}}$.

The most intuitive and relevant examples of measure chains are time scales, where $\mathbb {T}$ is a canonically ordered closed subset of the real numbers, and $\mu$ measures the oriented distance by $\mu (t, s)=t-s$. Furthermore, the function $\sigma :{\mathbb {T}}\rightarrow {\mathbb {T}}$, $\sigma (t):=\inf \{s \in \mathbb {T}: t\prec s\}$ defines the forward jump operator and $\mu ^*:\mathbb {T}\rightarrow \mathbb {R}, \mu ^*(t):=\mu (\sigma (t), t)= \sigma (t)-t$ the graininess.

Since we deal with the linearization problem, the following standard assumption is legitimate:

Hypothesis

$\mu (\mathbb {T}, {t}_{0})\subset \mathbb {R}$, ${t}_{0}\in \mathbb {T}$, is unbounded above, and the graininess $\mu ^*$ is bounded with $0\le \mu ^*<{\overline{\mu }}^*$. Growth rates are functions $a,\,b \in {{\mathcal {C}}}_{rd}^{+}{\mathcal R}({\mathbb {T}},{\mathbb {R}})$ with $\sup b(t)\le \infty$.

With fixed ${t}_{0} \in {\mathbb {T}}$ and $c,d\in {{\mathcal {C}}}_{rd}^+{{\mathcal {R}}}({\mathbb {T}}^{\kappa },{\mathbb {R}})$, we define the three linear spaces:

$$\begin{aligned}&{{\mathcal {B}}}^+_{ {t}_{0},c}(\chi ):=\Big \{\lambda \in {\mathcal {C}}_{rd}({\mathbb {T}}^+_{ {t}_{0}},\chi ):\ \sup \limits _{ {t}_{0}\preceq t}\Vert \lambda (t)\Vert e_{\ominus c}(t, {t}_{0})<\infty \Big \},\\&{{\mathcal {B}}}^-_{ {t}_{0},d}(\chi ):=\Big \{\lambda \in {\mathcal {C}}_{rd}({\mathbb {T}}^-_{ {t}_{0}},\chi ):\ \sup \limits _{t\preceq {t}_{0}}\Vert \lambda (t)\Vert e_{\ominus d}(t, {t}_{0})<\infty \Big \},\\&{{\mathcal {B}}}^{\pm }_{ {t}_{0},c,d}(\chi )=\left\{ \lambda \in {\mathcal {C}}_{rd}({\mathbb {T}}^{\kappa },\chi )\Big | \left. \begin{array}{l} \sup \limits _{ {t}_{0}\preceq t}\Vert \lambda (t)\Vert e_{\ominus c}(t, {t}_{0})<\infty \\ \sup \limits _{t\preceq {t}_{0}}\Vert \lambda (t)\Vert e_{\ominus d}(t, {t}_{0})<\infty \end{array}\right. \right\} \end{aligned}$$

of the so-called $c^+$-quasibounded and $d^-$-quasibounded mappings, which are immediately seen to be Banach spaces with norms

$$\begin{aligned}&\displaystyle \Vert \lambda \Vert ^+_{ {t}_{0},c} := \sup \limits _{ {t}_{0}\preceq t}\Vert \lambda (t)\Vert e_{\ominus c}(t, {t}_{0}), \quad \Vert \lambda \Vert ^-_{ {t}_{0},d}:=\sup \limits _{t\preceq {t}_{0}}\Vert \lambda (t)\Vert e_{\ominus d}(t, {t}_{0}),\\&\displaystyle \Vert \lambda \Vert ^{\pm }_{ {t}_{0},c,d} = \max \left\{ \big \Vert \lambda |_{{\mathbb {T}}^+_{ {t}_{0}}}\big \Vert ^+_{ {t}_{0},c},\ \big \Vert \lambda |_{{\mathbb {T}}^-_{ {t}_{0}}}\big \Vert ^-_{ {t}_{0},d}\right\} \end{aligned}$$

respectively, where $e_c(t, {t}_{0})$ is the real exponential function on ${\mathbb {T}}$. It is easy to derive that

$$\begin{aligned}&\Vert \lambda (t)\Vert \le \Vert \lambda \Vert ^+_{ {t}_{0},c}e_c(t, {t}_{0}) \quad \hbox { for all } t\in {\mathbb {T}}^+_{ {t}_{0}},\quad \Vert \lambda (t)\Vert \le \Vert \lambda \Vert ^-_{ {t}_{0},d}e_d(t, {t}_{0}) \quad \mathrm{for\ all}\ t\in {\mathbb {T}}^-_{ {t}_{0}}. \end{aligned}$$

Throughout this paper, we use the abbreviation $\lfloor b-a\rfloor :=\inf \nolimits _{t\in {\mathbb {T}}^{\kappa }}(b(t)-a(t))$ and introduce the notations $a\lhd b:\Leftrightarrow 0<\lfloor b-a\rfloor$, $a\unlhd b:\Leftrightarrow 0\le \lfloor b-a\rfloor$, where two positively regressive functions $a,b\in {\mathcal {C}}_{rd}^+{{\mathcal {R}}}({\mathbb {T}}^{\kappa },{\mathbb {R}})$ are denoted as growth rates, if $\sup \nolimits _{t\in {\mathbb {T}}^{\kappa }}\mu ^*(t)a(t)<\infty$ and $\sup \nolimits _{t\in {\mathbb {T}}^{\kappa }}\mu ^*(t)b(t)<\infty$, respectively. Then we obtain the limits

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }e_{a\ominus b}(t, {t}_{0})=0, \ \ \ \lim \limits _{t\rightarrow -\infty }e_{b\ominus a}(t, {t}_{0})=0, \end{aligned}$$

for growth rates $a\lhd b$ and on a measure chain, which is unbounded above (resp. below).

Definition 2.1

A mapping $\phi :{\mathbb {T}}\rightarrow \chi$ is said to be differentiable (in a point $t_0\in {\mathbb {T}}$), if there exists a unique derivative $\phi ^{\Delta }(t_0)\in \chi$, such that for any $\varepsilon >0$ the estimate

$$\begin{aligned} \Vert \phi (\sigma (t_0))-\phi (t)-\mu (\sigma (t_0),t)\phi ^{\Delta }(t_0)\Vert \le \varepsilon |\mu (\sigma (t_0),t)|\ \quad \mathrm{for }\ t\in U \end{aligned}$$

holds in a ${\mathbb {T}}$-neighborhood U of $t_0$. The Cauchy integral of $\phi$ is denoted as $\int _{ {t}_{0}}^t\phi (s)\Delta s$ for ${t}_{0},t\in {\mathbb {T}}$, provided it exists.

Now consider the following systems:

$$\begin{aligned} x^{\Delta }=A(t)x, \end{aligned}$$

(2.1)

where $A\in {{\mathcal {C}}}_{rd}({\mathbb {T}},\mathcal {L}(\chi ))$, ($\mathcal {L}(\chi )$ is the Banach space of linear bounded endomorphisms) $N\in {\mathbb {N}}$. Let $\varPhi _A(t,t_{0})\in \mathcal {L}(\chi )$ denote the transition operator of (2.1), i.e., the solution of the corresponding operator-valued initial value problem $X^{\Delta }=A(t)X,\,X(t_{0})=I$ for $t_{0}, t\in {\mathbb {T}}$, $t\succeq t_{0}$.

Definition 2.2

(Exponential dichotomy on measure chains) [31, 37] Let $P:{\mathbb {T}}\rightarrow \mathcal {L}(\chi )$ be an invariant projector of (2.1) such that the regularity condition $P(t) {\varPhi }_A(t,s)= {\varPhi }_A(t,s)P(s)$ is fulfilled for $s\preceq t$. Then Eq. (2.1) is said to possess an exponential dichotomy, if the estimates

$$\begin{aligned} \left. \begin{array}{l} \Vert \varPhi _A(t,s)P(s)\Vert \le K_1e_a(t,s) \quad \mathrm{for }\ s\preceq t, \quad s,t\in {\mathbb {T}},\\ \Vert {\varPhi }_A(t,s)[I-P(s)]\Vert \le K_2e_b(t,s) \quad \mathrm{for } \ t\preceq s, \quad s,t\in {\mathbb {T}} \end{array}\right. \end{aligned}$$

(2.2)

hold for real constants $K_1,K_2\ge 1$ and growth rates $a,b\in {{\mathcal {C}}}_{rd}^{+}{{\mathcal {R}}}({\mathbb {T}},{\mathbb {R}})$, $a\lhd b$.

Lemma 2.1

(Bellman inequality) Let $-p\in \mathcal {R}^{+}(\mathbb {T},\mathbb {R})$, $y\in {\mathcal {C}}_{rd}\mathcal {R}(\mathbb {T},\mathbb {R})$. Suppose that $p\ge 0$, $y\ge 0$, and $\alpha >0$. Then

$$\begin{aligned} y(t)\le \alpha + |\int _{t_0}^ty(s)p(s)\Delta s|,\quad \mathrm{for\, all}\quad t\in \mathbb {T} \end{aligned}$$

implies

$$\begin{aligned} y(t)\le \left\{ \begin{array}{lll} \alpha e_{p}(t,t_{0}), &{}\quad \mathrm{for }\quad t\in [t_0,+\infty )_{\mathbb {T}},\\ \alpha e_{-p}(t,t_{0}), &{}\quad \mathrm{for }\quad t\in (-\infty ,t_0]_{\mathbb {T}}. \end{array} \right. \end{aligned}$$

3 Statement of the Main Result

In what follows, for convenience, we denote a n-dimension $\mathbb {K}$-Banach space $\chi$ by $\chi ^n$.

Considering the following equations of the form:

$$\begin{aligned} x^{\Delta }= & {} f(t,x),\end{aligned}$$

(3.1)

$$\begin{aligned} y^{\Delta }= & {} g(t,y), \end{aligned}$$

(3.2)

where $f,g\in {\mathcal {C}}_{rd}({\mathbb {T}}\times \chi \rightarrow \chi$), the Banach Space $\chi$ are all ${\mathbb {R}}^{N}$ or ${\mathbb {C}}^{N}$ with the norm $\Vert \cdot \Vert$.

Definition 3.1

Suppose that a continuous function $H:\mathbb {T}\times \chi \rightarrow \chi$ satisfies the followings:

• (i) for every $t\in \mathbb {T}$, the mapping $H_{t}:\mathbf \chi \rightarrow \chi$, $H_{t}(x)=H(t,x)$ is a homeomorphism;

• (ii) the inverse $G:\mathbb {T}\times \chi \rightarrow \chi$, $G(t,x)=H^{-1}(t,x)$ is continuous also;

• (iii) $\lim \nolimits _{x\rightarrow 0}H(t,x)=0$ and $\lim \nolimits _{x\rightarrow 0}G(t,x)=0$, uniformly with respect to $t\in \mathbb {T}$;

• (iv) for every solution x(t) of (3.1), then $y(t)=H(t,x(t))$ solves (3.2); for every solution y(t) of (3.2), then $x(t)=G(t,y(t))$ solves (3.1).

If such a mapping H exists, then (3.1) and (3.2) are called topologically conjugated.

Consider the nonautonomous nonlinear system:

$$\begin{aligned} \left\{ \begin{array}{rcl} x_1^{\Delta }&{}=&{}A_1(t)x_1+f(t,x_1,x_2),\\ x_2^{\Delta } &{}=&{}A_2(t)x_2, \end{array}\right. \end{aligned}$$

(3.3)

and its linear system

$$\begin{aligned} \left\{ \begin{array}{rcl} x_1^{\Delta } &{} = &{} A_1(t)x_1,\\ x_2^{\Delta } &{} = &{} A_2(t)x_2, \end{array}\right. \end{aligned}$$

(3.4)

where $x_1\in \chi ^n$, $x_2\in \chi ^m$, $A_1\in {\mathcal {C}}_{rd}({\mathbb {T}},{{\mathcal {L}}}(\chi ^n))$, $A_2\in {\mathcal {C}}_{rd}({\mathbb {T}},{{\mathcal {L}}}(\chi ^m))$.

Now we are ready to state our main result.

Theorem 3.1

(Hölder conjugacy) Let ${\mathbb {T}}$ be unbounded above and below. For any ${t}_{0}\in {\mathbb {T}}$, $c,d\in {\mathcal {C}}^+_{rd}({\mathbb {T}},\mathbb {R})$, $a\lhd c\lhd b$, $a\lhd d\lhd b$, suppose that the following conditions hold:

$(\mathrm{H}_1)$ :

The linear subsystem $x_1^{\Delta }=A_1(t)x_1$ has exponential dichotomy on ${\mathbb {T}}$, that is, $x_1^{\Delta }=A_1(t)x_1$ has a fundamental matrix $\varPhi _{A_1}(t)$ satisfying (2.2);

$(\mathrm{H}_2)$ :

$f:{\mathbb {T}}\times \chi ^n\times \chi ^m\rightarrow \chi ^{n+m}$ is rd-continuous and satisfies the following:

$$\begin{aligned} \Vert f(t,x_1,x_2)\Vert ^{\pm }_{ {t}_{0},c,d}\le & {} \mu , \ \ \Vert f(t,x_1,x_2)-f(t,y_1,y_2)\Vert \nonumber \\\le & {} \gamma [\Vert x_1-y_1\Vert +\Vert x_2-y_2\Vert ] \ \ \mathrm{for} \ t\in {\mathbb {T}}; \end{aligned}$$

$(\mathrm{H}_3)$ :

$\gamma C_2(c,d)<1$, where $C_2(c,d)=\max \Big \{C_1(c)+\frac{K_1}{\lfloor {d-a}\rfloor },\ C_1(d)+\frac{K_2}{\lfloor {b-c}\rfloor }\Big \}>0$, $C_1(c)=\frac{K_1}{\lfloor {c-a}\rfloor }+\frac{K_2}{\lfloor {b-c}\rfloor }>0$, $C_1(d)=\frac{K_1}{\lfloor {d-a}\rfloor }+\frac{K_2}{\lfloor {b-d}\rfloor }>0$.

Then we have the following conclusions:

(I):

system (3.3) is topologically conjugated to its linear system (3.4);

(II):

the equivalent function $H(t,x)\,(x=(x_1,x_2)^T)$ satisfies:

when $\Vert x-{\bar{x}}\Vert <1$ for all t, we have $\Vert H(t,x)-H(t,{\bar{x}})\Vert ^{\pm }_{ {t}_{0},c,d}\le p\Vert x-{\bar{x}}\Vert ^q$ for any constants $p,q>0$;

(III):

letting $G(t,\cdot )=H^{-1}(t,\cdot )$, then $G(t,\cdot )$ satisfies:

when $\Vert y-{\bar{y}}\Vert <1$ for all t, we have $\Vert H(t,y)-H(t,{\bar{y}})\Vert ^{\pm }_{ {t}_{0},c,d}\le {\tilde{p}}\Vert y-{\bar{y}}\Vert ^{{\tilde{q}}}$ for any constants ${\tilde{p}},{\tilde{q}}>0$.

4 Proof of Main Result

Throughout this section, we always assume that the assumptions in Theorem 3.1 are satisfied.

Assume that $\varPhi _{A_1}(t,t_0)$ denotes a fundamental matrix of $x_1^{\Delta }=A_1(t)x$, $\left[ \begin{array}{cc} X_1(t,t_0,x_{10},x_{20}) \\ X_2(t,t_0,x_{10},x_{20})\end{array}\right]$ is a solution of (3.3) satisfying the initial condition $\left[ \begin{array}{cc} X_1(t_0) \\ X_2(t_0)\end{array}\right] =\left[ \begin{array}{cc} x_{10} \\ x_{20}\end{array}\right]$, and $\left[ \begin{array}{cc} Y_1(t,t_0,y_{10},y_{20}) \\ Y_2(t,t_0,y_{10},y_{20})\end{array}\right]$ is a solution of (3.4) satisfying the initial condition $\left[ \begin{array}{cc} Y_1(t_0) \\ Y_2(t_0)\end{array}\right] =\left[ \begin{array}{cc} y_{10} \\ y_{20}\end{array}\right]$.

Proposition 1

For any fixed $(\tau ,\xi ,\eta )$, it follows that system

$$\begin{aligned} z^{\Delta }=A_1(t)z-f\big (t,X_1(t,\tau ,\xi ,\eta ),X_2(t,\tau ,\xi ,\eta )\big ) \end{aligned}$$

(4.1)

has a unique (c, d)-quasibounded solution $h(t,(\tau ,\xi ,\eta ))$ which can be represented as

$$\begin{aligned}&h(t,(\tau ,\xi ,\eta ))\nonumber \\&\quad =\,-\int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s)) f\big (s,X_1(s,\tau ,\xi ,\eta ),X_2(s,\tau ,\xi ,\eta )\big )\Delta s\nonumber \\&\qquad +\,\int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] f\big (s,X_1(s,\tau ,\xi ,\eta ),X_2(s,\tau ,\xi ,\eta )\big )\Delta s. \nonumber \\ \end{aligned}$$

(4.2)

and it satisfies

$$\begin{aligned} \Vert h(\cdot ,(\tau ,\xi ,\eta ))\Vert ^{\pm }_{ {t}_{0},c,d}\le \mu C_2(c,d), \end{aligned}$$

where $C_2(c,d)$ is defined in Theorem 3.1.

Now we introduce two functions as follows:

$$\begin{aligned}&\left[ \begin{array}{cl} H_1(t,x_{1},x_{2}) \\ H_2(t,x_{1},x_{2})\end{array}\right] = \left[ \begin{array}{ll} x_1+h(t,(t,x_{1},x_{2})) \\ x_{2}\end{array}\right] \end{aligned}$$

(4.3)

$$\begin{aligned}&\left[ \begin{array}{cc} G_1(t,y_{1},y_{2}) \\ G_2(t,y_{1},y_{2})\end{array}\right] = \left[ \begin{array}{ll} y_1+g(t,(t,y_{1},y_{2})) \\ y_{2}\end{array}\right] . \end{aligned}$$

(4.4)

Proposition 2

For any fixed $(\tau ,\xi ,\eta )$, it follows that system

$$\begin{aligned} z^{\Delta }=A_1(t)z+f\big (t,Y_1(t,\tau ,\xi ,\eta )+z,Y_2(t,\tau ,\xi ,\eta )\big ) \end{aligned}$$

(4.5)

has a unique (c, d)-quasibounded solution $g(t,(\tau ,\xi ,\eta ))$ which can be represented as

$$\begin{aligned}&g(t,(\tau ,\xi ,\eta ))\nonumber \\&\quad =\int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s)) f\big (s,Y_1(s,\tau ,\xi ,\eta )+g(s,\tau ,\xi ,\eta ),Y_2(s,\tau ,\xi ,\eta )\big )\Delta s\nonumber \\&\qquad -\int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))]\nonumber \\&\qquad \times \,f\big (s,Y_1(s,\tau ,\xi ,\eta )+g(s,\tau ,\xi ,\eta ),Y_2(s,\tau ,\xi ,\eta )\big )\Delta s. \end{aligned}$$

(4.6)

and it satisfies

$$\begin{aligned} \Vert g(\cdot ,(\tau ,\xi ,\eta ))\Vert ^{\pm }_{ {t}_{0},c,d}\le \mu C_2(c,d), \end{aligned}$$

where $C_2(c,d)$ is defined in Theorem 3.1.

Proof of assertion (I)

From Propositions 1, 2, and a similar procedure in the proof of Theorem 2.1 in [37], it is not difficult to prove the first assertion (I) of Theorem 3.1 (so we omit the proof). Next we focus on the proof of assertions (II) and (III). To prove these two assertions, we need the following proposition. $\square$

Proposition 3

Set $\sup \limits _{t\in \mathbb {R}}\Vert A_1(t)\Vert +\sup \limits _{t\in \mathbb {R}}\Vert A_2(t)\Vert =M$. Then

$$\begin{aligned}&\Vert X_1(t, {t}_{0},x_{10},x_{20})-X_1(t, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\qquad +\,\Vert X_2(t, {t}_{0},x_{10},x_{20})-X_2(t, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\quad \le \,\left\{ \begin{array}{l} {[}\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert ]e_{p_1}(t, {t}_{0}),\quad t\in [ {t}_{0},+\infty )_{\mathbb {T}},\\ {[}\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert ] e_{-p_1}(t, {t}_{0}),\quad t\in (-\infty , {t}_{0}]_{\mathbb {T}}, \end{array} \right. \end{aligned}$$

(4.7)

$$\begin{aligned}&\Vert Y_1(t, {t}_{0},y_{10},y_{20})-Y_1(t, {t}_{0},{\widetilde{y}}_{10},{\widetilde{y}}_{20})\Vert \nonumber \\&\qquad +\,\Vert Y_2(t, {t}_{0},y_{10},y_{20})-Y_2(t, {t}_{0},{\widetilde{y}}_{10},{\widetilde{y}}_{20})\Vert \nonumber \\&\quad \le \, \left\{ \begin{array}{l} {[}\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert ] e_{p_1}(t, {t}_{0}),\quad t\in [ {t}_{0},+\infty )_{\mathbb {T}},\\ {[}\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert ] e_{-p_1}(t, {t}_{0}),\quad t\in (-\infty , {t}_{0}]_{\mathbb {T}}, \end{array} \right. \end{aligned}$$

(4.8)

where $p_1(t)\equiv M+\gamma ,\,p_2(t)\equiv M.$ $\square$

Proof

Integrating (3.3) over $[ {t}_{0},t]$, we have

$$\begin{aligned}&\left[ \begin{array}{c} X_1(t, {t}_{0},x_{10},x_{20}) \\ X_2(t, {t}_{0},x_{10},x_{20})\end{array}\right] = \left[ \begin{array}{c} x_{10} \\ x_{20} \end{array}\right] \nonumber \\&\quad +\, \left[ \begin{array}{l} \int _{ {t}_{0}}^t \big [A_1(s)X_1(s, {t}_{0},x_{10},x_{20})+f\big (s,X_1(s, {t}_{0},x_{10},x_{20}),X_2(s, {t}_{0},x_{10},x_{20})\big )\big ]\Delta s\\ \int _{ {t}_{0}}^t \big [A_2(s)X_2(s, {t}_{0},x_{10},x_{20}) \big ]\Delta s\end{array}\right] . \end{aligned}$$

A simple computation leads to

$$\begin{aligned}&\Vert X_1(t, {t}_{0},x_{10},x_{20})-X_1(t, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\qquad +\,\Vert X_2(t, {t}_{0},x_{10},x_{20})-X_2(t, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\quad \le \,\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\displaystyle \Big |\int _{ {t}_{0}}^t \Big [\Vert A_1(s)\Vert \Vert X_1(s, {t}_{0},x_{10},x_{20})-X_1(s, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\qquad +\,\gamma \Big (\Vert X_1(s, {t}_{0},x_{10},x_{20})-X_1(s, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\qquad + \,\Vert X_2(s, {t}_{0},x_{10},x_{20})-X_2(s, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \Big )\Big ]\Delta s\Big |\nonumber \\&\qquad +\, \Vert x_{20}-{\widetilde{x}}_{20}\Vert +\displaystyle \Big |\int _{{t}_{0}}^t \Big [\Vert A_2(s)\Vert \Vert X_2(s, {t}_{0},x_{10},x_{20})-X_2(s,{t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \Delta s\Big |\nonumber \\&\quad \le \, \big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \big ]\nonumber \\&\qquad +\,(M+\gamma )\displaystyle \Big |\int _{ {t}_{0}}^t \Big (\Vert X_1(s,{t}_{0},x_{10},x_{20})-X_1(s,{t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&\qquad + \,\Vert X_2(s, {t}_{0},x_{10},x_{20})-X_2(s, {t}_{0},{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \Big )\Big ]\Delta s. \end{aligned}$$

(4.9)

From Bellman inequality, (4.7) follows immediately. The proof of the second inequality is similar, so we omit it. $\square$

Remark 4.1

Since $M=\sup _{t\in \mathbb {T}}\Vert A_1(t)\Vert +\sup _{t\in \mathbb {T}}\Vert A_2(t)\Vert$, it is clear that $\lfloor a+p_{1}\rfloor >0$ and $\lfloor p_{1}-b \rfloor > 0$.

Proof of assertion (II)

To prove (II), It is sufficient to prove that there exist two positive constants $M_0>0$ and $q>0$ such that if $\Vert x_{10}-{\widetilde{x}}_{10}\Vert + \Vert x_{20}-{\widetilde{x}}_{20}\Vert <1$, then

$$\begin{aligned} \Vert H(t,x_1,x_2)-H(t,{\widetilde{x}}_1,{\widetilde{x}}_2)\Vert \le M_0[\Vert x_{10}-{\widetilde{x}}_{10}\Vert + \Vert x_{20}-{\widetilde{x}}_{20}\Vert ]^{q}. \end{aligned}$$

From Proposition 1 and (4.2), we know

$$\begin{aligned}&h(t,(t,\xi ,\eta ))\nonumber \\&\quad =\displaystyle -\int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s))f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big )\Delta s\nonumber \\&\qquad +\displaystyle \int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big )\Delta s. \nonumber \\ \end{aligned}$$

(4.10)

Then,

$$\begin{aligned}&h(t,(t,\xi ,\eta ))-h(t,(t,{\widetilde{\xi }},{\widetilde{\eta }}))\nonumber \\&\quad =\,-\int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s)) \Big [f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big ) \nonumber \\&\qquad \,\,\, -\,f\big (s,X_1(s,t,{\widetilde{\xi }},{\widetilde{\eta }}),X_2(s,t,{\widetilde{\xi }},{\widetilde{\eta }})\big )\Big ] \Delta s\nonumber \\&\qquad \,\,\, +\,\int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] \Big [f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big )\nonumber \\&\qquad \,\,\, -\,f\big (s,X_1(s,t,{\widetilde{\xi }},{\widetilde{\eta }}),X_2(s,t,{\widetilde{\xi }},{\widetilde{\eta }})\big )\Big ] \Delta s\nonumber \\&\quad {:=} \, I_1+I_2 . \end{aligned}$$

(4.11)

If $1+k\mu >0$, $y=\displaystyle \frac{\ln (1+k\mu )}{\mu },(k<0)$ is nonincreasing with respect to $\mu$. Thus, by using $0\le \mu ^{*}(t)\le {\overline{\mu }}^*$, if $z\lhd 0$, then

$$\begin{aligned} \xi _{\mu ^*}(z)=\left\{ \begin{array}{ll} z(t) \le -\lfloor -z\rfloor ,&{}\quad \mu ^*(t)=0,\\ \frac{\ln [1+\mu ^*(t)z(t) ]}{\mu ^*(t)} \le \lim _{\mu ^*(t)\rightarrow 0}\frac{\ln (1-\mu ^*(t) \lfloor -z\rfloor )}{\mu ^*(t)}=-\lfloor -z\rfloor ,&{}\quad \mu ^*(t)>0, \end{array} \right. \end{aligned}$$

which implies

$$\begin{aligned} \xi _{\mu ^*}(z) \le -\lfloor -z\rfloor ,\quad \text{ if }\quad z\lhd 0. \end{aligned}$$

Consequently,

$$\begin{aligned} \xi _{\mu ^*}(a\ominus c)\le & {} \lfloor -(a\ominus c)\rfloor := q_1,\nonumber \\ \xi _{\mu ^*}(a\ominus d)\le & {} \lfloor -(a\ominus d)\rfloor := q_2,\nonumber \\ \xi _{\mu ^*}(c\ominus b)\le & {} \lfloor -(c\ominus b)\rfloor := q_3, \nonumber \\ \xi _{\mu ^*}(d\ominus b)\le & {} \lfloor -(d\ominus b)\rfloor := q_4. \end{aligned}$$

(4.12)

Now $y=\displaystyle \frac{\ln (1+k\mu )}{\mu },(k>0)$ is nondecreasing with respect to $\mu$. Thus, by means of $0\le \mu ^{*}(t)\le {\overline{\mu }}^*$, if $0\lhd z$, then

$$\begin{aligned} \xi _{\mu ^*}(a\ominus c)=\left\{ \begin{array}{ll} z(t) \le \sup z(t),&{}\quad \mu ^*(t)=0,\\ \frac{\ln [1+\mu ^*(t)z(t) ]}{\mu ^*(t)} \le \frac{\ln [1+{\overline{\mu }}^*\sup z(t) ]}{{\overline{\mu }}^*},&{}\quad \mu ^*(t)>0, \end{array} \right. \end{aligned}$$

Since $(\mathcal {R}^+,\oplus )$ is a subgroup of the regressive group, noting that $p_1=M+\gamma$, $a\lhd c,d\lhd b$ and a, b are growth rates with $\sup b(t)<\infty$, it is not difficult to see that

$$\begin{aligned} \xi _{\mu ^*}[c\ominus (a\ominus p_1)]\le & {} \max \left\{ \sup c\ominus (a\ominus p_1),\right. \nonumber \\&\left. \frac{\ln [1+{\overline{\mu }}^*\sup \sup c\ominus (a\ominus p_1) ]}{{\overline{\mu }}^*}\right\} :=q_5, \nonumber \\ \xi _{\mu ^*}[d\ominus (a\ominus p_1)]\le & {} \max \left\{ \sup d\ominus (a\ominus p_1),\right. \nonumber \\&\left. \frac{\ln [1+{\overline{\mu }}^*\sup \sup d\ominus (a\ominus p_1) ]}{{\overline{\mu }}^*}\right\} :=q_6,\nonumber \\ \xi _{\mu ^*}[c\ominus (b\ominus p_1)]\le & {} \max \left\{ \sup c\ominus (b\ominus p_1),\right. \nonumber \\&\left. \frac{\ln [1+{\overline{\mu }}^*\sup \sup c\ominus (b\ominus p_1)]}{{\overline{\mu }}^*}\right\} :=q_7, \nonumber \\ \xi _{\mu ^*}[d\ominus (b\ominus p_1)]\le & {} \max \left\{ \sup d\ominus (b\ominus p_1),\right. \nonumber \\&\left. \frac{\ln [1+{\overline{\mu }}^*\sup \sup d\ominus (b\ominus p_1) ]}{{\overline{\mu }}^*}\right\} :=q_8. \end{aligned}$$

(4.13)

Choose $\xi ,\eta ,{\widetilde{\xi }},{\widetilde{\eta }}$ suitably such that $0<\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert <1$ and set

$$\begin{aligned} T=\frac{1}{q_0}\ln \frac{1}{\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert },\quad \text{ where }\,\,\,\,q_0=\max \{q_5,q_6,q_7,q_8\}. \end{aligned}$$

Now divide $I_1$ and $I_2$ into two parts, respectively, as follows:

$$\begin{aligned} I_1= & {} \int _{-\infty }^{t-T}+\int _{t-T}^t:=I_{11}+I_{12},\\ I_2= & {} \int _{t }^{t+T}+\int _{t+T}^{+\infty }:=I_{21}+I_{22}. \end{aligned}$$

By using the assumptions, it is not difficult to obtain

$$\begin{aligned} \Vert I_{11}\Vert= & {} \displaystyle \Big |\int _{-\infty }^{t-T}\varPhi _{A_1}(t,\sigma (s))P(\sigma (s))\Big [f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big )\nonumber \\&-\,f\big (s,X_1(s,t,{\widetilde{\xi }},{\widetilde{\eta }}),X_2(s,t,{\widetilde{\xi }},{\widetilde{\eta }})\big )\Big ]\Delta s\Big |\nonumber \\\le & {} \displaystyle \int _{-\infty }^{t-T}K_1e_{a}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s. \end{aligned}$$

(4.14)

For fixed ${t}_{0}\in {\mathbb {T}}$, without loss of generality, first we consider (4.14) on ${\mathbb {T}}^+_{ {t}_{0}}$, and there are two cases here: $t-T\prec {t}_{0}$ and $t-T\preceq {t}_{0}$. If $t-T\succeq {t}_{0}$, then

$$\begin{aligned} \Vert I_{11}\Vert\le & {} \displaystyle \int _{-\infty }^{ {t}_{0}}K_1e_{a}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s+ \int _{ {t}_{0}}^{t-T}K_1e_{a}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s \\\le & {} \displaystyle \int _{-\infty }^{ {t}_{0}}K_1e_{a}(t,\sigma (s))e_{d}(s, {t}_{0})\cdot 2\Vert f\Vert _{ {t}_{0},d}^{-}\Delta s \\&+\, \displaystyle \int _{ {t}_{0}}^{t-T}K_1e_{a}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot 2\Vert f\Vert _{ {t}_{0},c}^{+}\Delta s\\\le & {} \displaystyle \left[ \int _{-\infty }^{ {t}_{0}}K_1e_{a}(t,\sigma (s))e_{d}(s, {t}_{0}) \Delta s \right. \\&\left. +\, \displaystyle \int _{ {t}_{0}}^{t-T}K_1e_{a}(t,\sigma (s))e_{c}(s, {t}_{0}) \Delta s\right] \cdot 2\Vert f\Vert _{ {t}_{0},c,d}^{\pm } \\= & {} \displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor }[e_{d\ominus a}( {t}_{0}, {t}_{0})-e_{d\ominus a}(-\infty , {t}_{0})]2\mu \\&+\, \displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }[e_{c\ominus a}(t-T, {t}_{0})-e_{c\ominus a}( {t}_{0}, {t}_{0})]2\mu \\= & {} \displaystyle \left[ \frac{K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor } +\displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }e_{c\ominus a}(t-T, {t}_{0})\!-\!\frac{K_1e_{a}(t, {t}_{0})}{\lfloor c\!-\!a\rfloor }\!\right] 2\mu \\= & {} \displaystyle \left[ \!\frac{K_1e_{a}(t, {t}_{0})}{\lfloor d\!-\!a\rfloor } \!+\! \displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor c\!-\!a\rfloor } \!\cdot \!\frac{e_{c\ominus a}(t,t\!-\!T)}{e_{c\ominus a}(t,t\!-\!T)}e_{c\ominus a}(t-T, {t}_{0})-\frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }\!\right] 2\mu \\= & {} \displaystyle \left[ \frac{K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor } +\displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }\cdot \frac{e_{c\ominus a}(t, {t}_{0})}{e_{c\ominus a}(t,t-T)} -\frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }\right] 2\mu \nonumber \\= & {} \displaystyle \left[ \frac{K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor } +\displaystyle \frac{K_1e_{c}(t, {t}_{0})}{\lfloor c-a\rfloor }e_{a\ominus c}(t,t-T) -\frac{K_1e_{a}(t, {t}_{0})}{\lfloor c-a\rfloor }\right] 2\mu \nonumber \\\le & {} \displaystyle \left[ \frac{K_1}{\lfloor d-a\rfloor }e_{a}(t, {t}_{0}) +\displaystyle \frac{K_1}{\lfloor c-a\rfloor }e_{a\ominus c}(t,t-T) e_{c}(t, {t}_{0})\right] 2\mu ,\quad \text{ for } \text{ all }\quad t\succeq {t}_{0}, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus c}(t, {t}_{0})\le & {} \displaystyle \Big (\frac{K_1}{\lfloor d-a\rfloor }e_{a\ominus c}(t, {t}_{0}) +\displaystyle \frac{K_1}{\lfloor c-a\rfloor }e_{a\ominus c}(t,t-T)\Big )2\mu \nonumber \\= & {} \displaystyle \left( \frac{K_1}{\lfloor d-a\rfloor }e_{a\ominus c}(t,t-T)e_{a\ominus c}(t-T, {t}_{0}) \right. \nonumber \\&\left. +\,\displaystyle \frac{K_1}{\lfloor c-a\rfloor }e_{a\ominus c}(t,t-T)\right) 2\mu \nonumber \\\le & {} 2\mu \Big (\displaystyle \frac{K_1}{\lfloor d-a\rfloor } +\displaystyle \frac{K_1}{\lfloor c-a\rfloor }\Big )e_{a\ominus c}(t,t-T),\quad t-T\succeq {t}_{0}. \nonumber \\ \end{aligned}$$

(4.15)

In view of (4.12),

$$\begin{aligned} e_{a\ominus c}(t,t-T)=\displaystyle \exp \left( \int ^t_{t-T}\xi _{\mu ^*}(a\ominus c)\Delta s\right) \le \displaystyle \exp \left( \int ^t_{t-T}-q_1\Delta s\right) =e^{-q_1T}. \nonumber \\ \end{aligned}$$

(4.16)

It follows from (4.15) and (4.16) that

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus c}(t, {t}_{0}) \le 2\mu \left( \displaystyle \frac{K_1}{\lfloor d-a\rfloor } +\displaystyle \frac{K_1}{\lfloor c-a\rfloor }\right) e^{-q_1T},\quad t-T\succeq {t}_{0}. \end{aligned}$$

(4.17)

If $t-T\prec {t}_{0}$, then there exists $0<\theta _1<1$ such that $t-T\prec t-\theta _1 T\preceq {t}_{0}$. Using the fact that $e_{d\ominus a}(t-T, {t}_{0})\le e_{d\ominus a}(t-T,t-\theta _1 T)$, we see that

$$\begin{aligned} \Vert I_{11}\Vert\le & {} \displaystyle \int _{-\infty }^{t-T}K_1e_{a}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s \\\le & {} \displaystyle \int _{-\infty }^{t-T}K_1e_{a}(t,\sigma (s))e_{d}(s, {t}_{0})\cdot 2\Vert f\Vert _{ {t}_{0},d}^{-}\Delta s \\\le & {} \displaystyle \int _{-\infty }^{t-T}K_1e_{a}(t,\sigma (s))e_{d}(s, {t}_{0})\Delta s\cdot 2\Vert f\Vert _{ {t}_{0},c,d}^{\pm }\cdot 2\mu \\= & {} \displaystyle \frac{K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor }[e_{d\ominus a}(t-T, {t}_{0})-e_{d\ominus a}(-\infty , {t}_{0})]\cdot 2\mu \\\le & {} \displaystyle \frac{2\mu K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor }\cdot e_{d\ominus a}(t-T,t-\theta _1 T) \\= & {} \displaystyle \frac{2\mu K_1e_{a}(t, {t}_{0})}{\lfloor d-a\rfloor }\cdot e_{a\ominus d}(t-\theta _1 T,t- T), \end{aligned}$$

which implies that

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus c}(t, {t}_{0})\le \frac{2 K_1 \mu }{\lfloor d-a\rfloor } e_{a\ominus d}(t-\theta _1 T,t- T),\quad t-T\prec {t}_{0}\preceq t. \end{aligned}$$

(4.18)

In view of (4.12),

$$\begin{aligned} e_{a\ominus d}(t-\theta _1 T,t-T)=\displaystyle \exp \left( \int ^{t-\theta _1 T}_{t-T}\xi _{\mu ^*}(a\ominus d)\Delta s\right) \le e^{-(1-\theta _1)q_2T}. \end{aligned}$$

(4.19)

It follows from (4.18) and (4.19) that

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus c}(t, {t}_{0}) \le 2\mu \frac{K_1}{\lfloor d-a\rfloor } e^{-(1-\theta _1)q_2T},\quad t-T\prec {t}_{0}\preceq t. \end{aligned}$$

(4.20)

Thus, from (4.15) and (4.18), we obtain that for any $t\succeq {t}_{0}$,

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus c}(t, {t}_{0})\le & {} \displaystyle \max \left\{ \frac{2 K_1 \mu }{\lfloor d-a\rfloor }e^{-(1-\theta _1)q_2T},\,\,\,\Big [\frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor d-a\rfloor }\Big ]2\mu e^{-q_1T}\right\} \nonumber \\= & {} 2\mu \left[ \displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }\right] e^{-\min \{q_1,(1-\theta _1)q_2\}T},\quad t\succeq {t}_{0}. \end{aligned}$$

(4.21)

Now we consider (4.14) on ${\mathbb {T}}^-_{ {t}_{0}}$, and there is only one case: $t-T\preceq t\preceq {t}_{0}$. Then similar arguments lead to

$$\begin{aligned} \Vert I_{11}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle \frac{2 K_1 \mu e_{a\ominus d}(t, {t}_{0})}{\lfloor d-a\rfloor }[e_{d\ominus a}(t-T, {t}_{0})-e_{d\ominus a}(-\infty , {t}_{0})]\nonumber \\\le & {} \displaystyle \frac{2 K_1 \mu e_{a\ominus d}(t, {t}_{0})}{\lfloor d-a\rfloor }e_{a\ominus d}( {t}_{0},t-T)\nonumber \\= & {} \displaystyle \frac{2 K_1 \mu }{\lfloor d-a\rfloor }e_{a\ominus d}(t,t-T)\le \frac{2 K_1 \mu }{\lfloor d-a\rfloor }e^{-q_2T} ,\quad t\preceq {t}_{0}. \end{aligned}$$

(4.22)

Taking the supremum, it follows from (4.21) and (4.22) that

$$\begin{aligned} \Vert I_{11}\Vert ^{\pm }_{ {t}_{0},c,d}\le & {} \displaystyle \max \left\{ 2\mu \left[ \displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }\right] e^{-\min \{q_1,(1-\theta _1)q_2\}T},\,\,\,\frac{2 K_1 \mu }{\lfloor d-a\rfloor }e^{-q_2T}\right\} \nonumber \\= & {} 2\mu \left[ \displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }\right] e^{-\min \{q_1,(1-\theta _1)q_2\}T}\nonumber \\= & {} 2\mu \left[ \displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }\right] e^{-q_0T\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}}\nonumber \\= & {} 2\mu \left[ \displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }\right] \Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}}\nonumber \\:= & {} M_1(K_1)\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}} ,\,\,\,\,\,\,\text{ for } \text{ all }\,\,\, t\in \mathbb {T}, \end{aligned}$$

(4.23)

where $M_1(K_1)=2\mu [\frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }]$. Now we estimate $\Vert I_{22}\Vert ^{\pm }_{ {t}_{0},c,d}$. In view of (4.11),

$$\begin{aligned} \Vert I_{22}\Vert= & {} \displaystyle \Big |\int _{t+T}^{+\infty } {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] \Big [f\big (s,X_1(s,t,\xi ,\eta ),X_2(s,t,\xi ,\eta )\big )\nonumber \\&-\,f\big (s,X_1(s,t,{\widetilde{\xi }},{\widetilde{\eta }}),X_2(s,t,{\widetilde{\xi }},{\widetilde{\eta }})\big )\Big ] \Delta s\Big |\nonumber \\\le & {} \displaystyle \int _{t+T}^{+\infty }K_2e_{b}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s. \end{aligned}$$

(4.24)

For fixed ${t}_{0}\in {\mathbb {T}}$, without loss of generality, first we consider (4.24) on ${\mathbb {T}}^+_{ {t}_{0}}$, and there is only one case: $t+T\succeq t\succeq {t}_{0}$. Now

$$\begin{aligned} \Vert I_{22}\Vert= & {} \displaystyle \int _{t+T}^{+\infty }K_2e_{b}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s \\\le & {} \displaystyle \int _{t+T}^{+\infty }K_2e_{b}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot 2\Vert f\Vert _{ {t}_{0},c}^{+}\Delta s \\\le & {} \displaystyle \int _{t+T}^{+\infty }K_2e_{b}(t,\sigma (s))e_{c}(s, {t}_{0})\Delta s\cdot 2\Vert f\Vert _{ {t}_{0},c,d}^{\pm }\cdot 2\mu \\= & {} \displaystyle \frac{K_2e_{b}(t, {t}_{0})}{\lfloor b-c\rfloor }[e_{c\ominus b}(t+T, {t}_{0})-e_{c\ominus b}(+\infty , {t}_{0})]\cdot 2\mu \\= & {} \displaystyle \frac{2\mu K_2e_{b}(t, {t}_{0})}{\lfloor b-c\rfloor }\cdot e_{c\ominus b}(t+T, {t}_{0}) \\= & {} \displaystyle \frac{2\mu K_2e_{b}(t, {t}_{0})}{\lfloor b-c\rfloor }\cdot e_{c\ominus b}(t+T,t)e_{c\ominus b}(t, {t}_{0}), \end{aligned}$$

which, when combined with (4.12) leads to

$$\begin{aligned} \Vert I_{22}\Vert e_{\ominus c}(t, {t}_{0})\le \frac{2 K_2 \mu }{\lfloor b-c\rfloor } e_{c\ominus b}(t+ T,t)\le \frac{2 K_2 \mu }{\lfloor b-c\rfloor } e^{-q_3T},\quad t\succeq {t}_{0}. \end{aligned}$$

(4.25)

Now we consider (4.24) on ${\mathbb {T}}^-_{ {t}_{0}}$, and there are only two cases: $t+T\preceq {t}_{0}$ and $t\preceq {t}_{0}\prec t+T$. If $t+T\preceq {t}_{0}$, then

$$\begin{aligned} \Vert I_{22}\Vert= & {} \displaystyle \int _{t+T}^{ {t}_{0}}K_2e_{b}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s+ \int _{ {t}_{0}}^{+\infty }K_2e_{b}(t,\sigma (s))\cdot 2\Vert f\Vert \Delta s \\\le & {} \displaystyle \left[ \int _{t+T}^{ {t}_{0}}K_2e_{b}(t,\sigma (s))e_{d}(s, {t}_{0}) \Delta s \right. \\&\left. +\, \displaystyle \int _{ {t}_{0}}^{+\infty }K_2e_{b}(t,\sigma (s))e_{c}(s, {t}_{0}) \Delta s\right] \cdot 2\Vert f\Vert _{ {t}_{0},c,d}^{\pm } \\= & {} \displaystyle \frac{K_2e_{b}(t, {t}_{0})}{\lfloor b-d\rfloor }[e_{d\ominus b}(t+T, {t}_{0})-e_{d\ominus b}( {t}_{0}, {t}_{0})]2\mu \\&+\,\displaystyle \frac{K_2e_{b}(t, {t}_{0})}{\lfloor b-c\rfloor }[e_{c\ominus b}( {t}_{0}, {t}_{0})-e_{c\ominus b}(+\infty , {t}_{0})]2\mu \\= & {} \displaystyle \left[ \frac{K_2e_{b}(t, {t}_{0})}{\lfloor b-d\rfloor }e_{d\ominus b}(t+T, {t}_{0})+\frac{K_2e_{b}(t, {t}_{0})}{\lfloor b-c\rfloor }\right] 2\mu \quad \text{ for } \,\,\, t\preceq t+T\preceq {t}_{0}, \end{aligned}$$

which, when combined with (4.12) leads to

$$\begin{aligned} \Vert I_{22}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle 2\mu \left[ \frac{K_2e_{b \ominus d}(t, {t}_{0})}{\lfloor b-d\rfloor }e_{d\ominus b}(t+T, {t}_{0})+\frac{K_2e_{b \ominus d}(t, {t}_{0})}{\lfloor b-c\rfloor }\right] \nonumber \\= & {} \displaystyle 2\mu \left. [\frac{K_2}{\lfloor b-d\rfloor }e_{d\ominus b}(t+T, {t}_{0})e_{d \ominus b}( {t}_{0},t) \right. \nonumber \\&\left. \displaystyle +\,\frac{K_2}{\lfloor b-c\rfloor }e_{d \ominus b}(t+T,t)e_{d \ominus b}( {t}_{0},t+T)\right] \nonumber \\\le & {} 2\mu \left[ \displaystyle \frac{K_2}{\lfloor b-d\rfloor } +\displaystyle \frac{K_2}{\lfloor b-c\rfloor }\right] e_{d\ominus b}(t+T,t)\nonumber \\\le & {} 2\mu \left[ \displaystyle \frac{K_2}{\lfloor b-d\rfloor } +\displaystyle \frac{K_2}{\lfloor b-c\rfloor }\right] e^{-q_4T} ,\quad t\preceq t+T\preceq {t}_{0}. \end{aligned}$$

(4.26)

If $t\preceq {t}_{0}\prec t+T$, then there exists $0<\theta _2<1$ such that ${t}_{0}\preceq t+\theta _2 T\prec t+T$. Using the fact that $e_{c\ominus b}(t+T, {t}_{0})\le e_{c\ominus b}(t+T,t+\theta _2 T)$, we obtain that

$$\begin{aligned} \Vert I_{22}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle \frac{2 K_2 \mu }{\lfloor b-c\rfloor } e_{b\ominus d}(t, {t}_{0})e_{c\ominus b}(t+T, {t}_{0})\nonumber \\\le & {} \displaystyle \frac{2 K_2 \mu }{\lfloor b-c\rfloor } e_{c\ominus b}(t+T,t+\theta _2 T) \nonumber \\\le & {} \displaystyle \frac{2 K_2 \mu }{\lfloor b-c\rfloor }e^{-(1-\theta _2)q_3 T} ,\quad t\preceq {t}_{0}\prec t+T. \end{aligned}$$

(4.27)

Thus, from (4.26) and (4.27), we obtain that for any $t\preceq {t}_{0}$,

$$\begin{aligned} \Vert I_{22}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle \max \left\{ \frac{2 K_2 \mu }{\lfloor b-c\rfloor }e^{-(1-\theta _2)q_3T},\,\,\,\left[ \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }\right] 2\mu e^{-q_4T}\right\} \nonumber \\= & {} 2\mu [\displaystyle \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }]e^{-\min \{q_4,(1-\theta _2)q_3\}T},\quad t\preceq {t}_{0}. \end{aligned}$$

(4.28)

Taking the supremum, it follows from (4.25) and (4.28) that

$$\begin{aligned}&\Vert I_{22}\Vert ^{\pm }_{ {t}_{0},c,d} \nonumber \\&\quad \le \displaystyle \max \left\{ 2\mu [\displaystyle \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }]e^{-\min \{q_4,(1-\theta _2)q_3\}T},\,\,\,\frac{2 K_2 \mu }{\lfloor b-c\rfloor }e^{-q_3T}\right\} \nonumber \\&\quad = 2\mu \left[ \displaystyle \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }\right] e^{-\min \{q_4,(1-\theta _2)q_3\}T} \nonumber \\&\quad = 2\mu \left[ \displaystyle \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }\right] e^{-q_0T\frac{\min \{q_4,(1-\theta _2)q_3\}}{q_0}} \nonumber \\&\quad := M_1(K_2) \Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_4,(1-\theta _2)q_3\}}{q_0}} ,\quad \text{ for } \text{ all }\quad t\in \mathbb {T}. \end{aligned}$$

(4.29)

Now we estimate $I_{21}$. For convenience, denote $P(s)=\frac{1}{1+\mu ^{*}(s)p_1}$. Now

$$\begin{aligned} \Vert I_{21}\Vert\le & {} \displaystyle \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [ \Vert X_1(s,t,x_{10},x_{20})-X_1(s,t,{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \nonumber \\&+\,\Vert X_2(s,t,x_{10},x_{20})-X_2(s,t,{\widetilde{x}}_{10},{\widetilde{x}}_{20})\Vert \big ]\Delta s \nonumber \\\le & {} \displaystyle \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] e_{p_{1}}(s,t) \Delta s \nonumber \\= & {} \displaystyle \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \frac{e_{p_{1}}(\sigma (s),t)}{1+\mu ^{*}(s)p_1} \Delta s \nonumber \\= & {} K_2\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t} \frac{e_{b}(t,\sigma (s)) e_{\ominus p_{1}}(t,\sigma (s))}{1+\mu ^{*}(s)p_1} \Delta s \nonumber \\\le & {} K_2\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t}e_{b\ominus p_{1}}(t,\sigma (s))\cdot \Vert P\Vert \Delta s. \nonumber \\ \end{aligned}$$

(4.30)

For fixed ${t}_{0}\in {\mathbb {T}}$, without loss of generality, first we consider (4.30) on ${\mathbb {T}}^+_{ {t}_{0}}$, and there is only one case: $t+T\succeq t\succeq {t}_{0}$. Thus,

$$\begin{aligned} \Vert I_{21}\Vert\le & {} K_2\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t}e_{b\ominus p_{1}}(t,\sigma (s))e_{c}(s, {t}_{0})\Vert P\Vert ^{+}_{ {t}_{0},c} \Delta s\\= & {} K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\\&\cdot \, \displaystyle \frac{e_{b\ominus p_{1}}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \Big [e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})-e_{c\ominus (b\ominus p_{1})}(t, {t}_{0})\Big ]\\\le & {} K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}\!-\!{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \frac{e_{b\ominus p_{1}}(t, {t}_{0})e_{c\ominus (b\ominus p_{1})}(t\!+\!T, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor }, \end{aligned}$$

which, when combined with (4.13) leads to

$$\begin{aligned}&\Vert I_{21}\Vert e_{\ominus c}(t, {t}_{0})\nonumber \\&\quad \le K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert \!+\!\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \frac{e_{(b\ominus p_{1})\ominus c}(t, {t}_{0})e_{c\ominus (b\ominus p_{1})}(t\!+\!T, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \nonumber \\&\quad = K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \displaystyle \frac{e_{c\ominus (b\ominus p_{1})}(t+T,t)}{\lfloor c-(b\ominus p_1)\rfloor }\nonumber \\&\quad \le \displaystyle \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] e^{q_7T},\,\,\,\,\,t\succeq {t}_{0}. \end{aligned}$$

(4.31)

Now we consider (4.30) on ${\mathbb {T}}^-_{ {t}_{0}}$, and there are only two cases: $t+T\preceq {t}_{0}$ and $t\preceq {t}_{0}\prec t+T$. If $t+T\preceq {t}_{0}$, a similar argument to (4.31) gives

$$\begin{aligned} \Vert I_{21}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor d-(b\ominus p_1)\rfloor } \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] e^{q_8T},\,\,\,\,\,t+T\preceq {t}_{0}. \nonumber \\ \end{aligned}$$

(4.32)

If $t\preceq {t}_{0}\prec t+T$, then

$$\begin{aligned} \Vert I_{21}\Vert\le & {} K_2\gamma \Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \\&\cdot \, \Big [ \displaystyle \int ^{ {t}_{0}}_{t}e_{b\ominus p_{1}}(t,\sigma (s))e_{d}(s, {t}_{0})\Vert P\Vert ^{-}_{ {t}_{0},d} \Delta s \\&+\,\displaystyle \int ^{t+T}_{ {t}_{0}}e_{b\ominus p_{1}}(t,\sigma (s))e_{c}(s, {t}_{0})\Vert P\Vert ^{+}_{ {t}_{0},c} \Delta s \Big ] \\= & {} K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \\&\cdot \, \left\{ \displaystyle \frac{e_{b\ominus p_{1}}(t, {t}_{0})}{\lfloor d-(b\ominus p_1)\rfloor } \Big [e_{d\ominus (b\ominus p_{1})}( {t}_{0}, {t}_{0})-e_{d\ominus (b\ominus p_{1})}(t, {t}_{0})\right] \\&+\,\displaystyle \frac{e_{b\ominus p_{1}}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \Big [e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})-e_{c\ominus (b\ominus p_{1})}( {t}_{0}, {t}_{0})\Big ] \Big \} \\\le & {} K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ] \\&\cdot \, \displaystyle \left[ \frac{e_{b\ominus p_{1}}(t, {t}_{0})}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{e_{b\ominus p_{1}}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})\right] , \end{aligned}$$

which, when combined with (4.13) leads to

$$\begin{aligned}&\Vert I_{21}\Vert e_{\ominus d}(t, {t}_{0})\nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\nonumber \\&\qquad \cdot \, \displaystyle \left[ \frac{e_{(b\ominus p_{1})\ominus d}(t, {t}_{0})}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{e_{(b\ominus p_{1})\ominus d}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})\right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\nonumber \\&\qquad \cdot \,\displaystyle \left[ \frac{e_{d\ominus (b\ominus p_{1})}( {t}_{0},t)}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\nonumber \\&\qquad \cdot \, \displaystyle \left[ \frac{e_{d\ominus (b\ominus p_{1})}(t+T,t)}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{e_{c\ominus (b\ominus p_{1})}(t+T,t)}{\lfloor c-(b\ominus p_1)\rfloor } \right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{{t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\nonumber \\&\qquad \cdot \, \displaystyle \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{q_8,q_7\}T} ,\quad t\preceq {t}_{0}\prec t+T. \nonumber \\ \end{aligned}$$

(4.33)

It follows from (4.32) and (4.33) that for any $t\preceq {t}_{0}$,

$$\begin{aligned} \Vert I_{21}\Vert e_{\ominus d}(t, {t}_{0})\le & {} \displaystyle K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}\Big [\Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \Big ]\nonumber \\&\cdot \,\displaystyle \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{q_8,q_7\}T},\quad t\preceq {t}_{0}. \nonumber \\ \end{aligned}$$

(4.34)

Taking the supremum, it follows from (4.31) and (4.34) that for any $t\in \mathbb {T}$,

$$\begin{aligned} \Vert I_{21}\Vert ^{\pm }_{ {t}_{0},c,d} \le \,&\displaystyle K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \displaystyle \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] \nonumber \\&\cdot \,\left[ \Vert x_{10}-{\widetilde{x}}_{10}\Vert +\Vert x_{20}-{\widetilde{x}}_{20}\Vert \right] e^{\max \{q_7,q_8\}T}\nonumber \\ =&\,\displaystyle K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \displaystyle \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{-q_0T}e^{\max \{q_7,q_8\}T}\nonumber \\ =&\,\displaystyle K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \displaystyle \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor }+ \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{-q_0T\cdot \frac{q_0-\max \{q_7,q_8\}}{q_0}} \nonumber \\ :=&\, M_2(K_2)\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{q_0-\max \{q_7,q_8\}}{q_0}}, \end{aligned}$$

(4.35)

where $M_2(K_2)= \Big [\frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } \Big ]$. A similar estimation for $I_{12}$ gives for any $t\in \mathbb {T}$

$$\begin{aligned} \Vert I_{12}\Vert ^{\pm }_{ {t}_{0},c,d} \le M_2(K_1)\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{q_0-\max \{q_5,q_6\}}{q_0}}. \end{aligned}$$

(4.36)

Therefore, it follows from (4.23), (4.29), (4.35) and (4.36) that

$$\begin{aligned}&\Vert h(t,(t,\xi ,\eta ))-h(t,(t,{\widetilde{\xi }},{\widetilde{\eta }}))\\&\quad \le [M_1(K_1)+M_1(K_2)+M_2(K_1)+M_2(K_2)]\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{{\widetilde{q}}}, \end{aligned}$$

where ${\widetilde{q}}=\min \Big \{\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}, \frac{\min \{q_3,(1-\theta _2)q_4\}}{q_0} \frac{q_0-\max \{q_5,q_6\}}{q_0},\frac{q_0-\max \{q_7,q_8\}}{q_0}\Big \}$, and clearly, ${\widetilde{q}}<1$. Consequently,

$$\begin{aligned}&\Vert H(t,x_1,x_2)-H(t,\widetilde{x_1},\widetilde{x_2})\\&\quad =\,[\Vert x_1-\widetilde{x_1}\Vert +\Vert x_2-\widetilde{x_2}\Vert ]+\Vert h(t,(t,x_1,x_2))-h(t,\widetilde{x_1},\widetilde{x_2})\Vert \\&\quad \le \, [1+M_1(K_1)+M_1(K_2)+M_2(K_1)+M_2(K_2)]\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{{\widetilde{q}}}. \end{aligned}$$

This completes the proof of the assertion (II). $\square$

Proof of assertion (III)

To prove (III), it suffices to prove that there exist two positive constants ${{\overline{M}}}_0>0$ and ${{\overline{q}}}>0$ such that if $\Vert y_{10}-{\widetilde{y}}_{10}\Vert + \Vert y_{20}-{\widetilde{y}}_{20}\Vert <1$, then

$$\begin{aligned} \Vert G(t,y_1,y_2)-G(t,{\widetilde{y}}_1,{\widetilde{y}}_2)\Vert \le {{\overline{M}}}_0[\Vert y_{10}-{\widetilde{y}}_{10}\Vert + \Vert y_{20}-{\widetilde{y}}_{20}\Vert ]^{{{\overline{q}}}}. \end{aligned}$$

From Proposition 2 and (4.6), we know $g(t,( {t}_{0},\xi ,\eta ))$ is the fixed point of the map $\mathcal {T}$ defined by

$$\begin{aligned} \mathcal {T}Z(t)= & {} \displaystyle \int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s)) f\big (s,Y_1(s, {t}_{0},\xi ,\eta )+Z(s),Y_2(s, {t}_{0},\xi ,\eta )\big )\Delta s\\&-\displaystyle \int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s)) [I_{\chi }-P(\sigma (s))] \nonumber \\&\times \, f\big (s,Y_1(s, {t}_{0},\xi ,\eta )+Z(s),Y_2(s,{t}_{0},\xi ,\eta )\big )\Delta s. \end{aligned}$$

For all $t, {t}_{0} \in \mathbb {T}$, we define $g_{0}(t,( {t}_{0},\xi ,\eta ))\equiv 0$, and recursively define

$$\begin{aligned}&g_{m+1}(t,( {t}_{0},\xi ,\eta ))\nonumber \\&\quad = \displaystyle \int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s))\nonumber \\&\qquad \times f\big (s,Y_1(s, {t}_{0},\xi ,\eta )+g_{m}(s,( {t}_{0},\xi ,\eta )),Y_2(s, {t}_{0},\xi ,\eta )\big )\Delta s\nonumber \\&\qquad -\displaystyle \int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] \nonumber \\&\qquad \times f\big (s,Y_1(s, {t}_{0},\xi ,\eta )+g_{m}(s,( {t}_{0},\xi ,\eta )),Y_2(s,{t}_{0},\xi ,\eta )\big )\Delta s. \end{aligned}$$

(4.37)

It is easy to conclude that

$$\begin{aligned} g_{m}(t,( {t}_{0},\xi ,\eta ))\rightarrow g(t,( {t}_{0},\xi ,\eta )) \end{aligned}$$

(4.38)

as $n\rightarrow \infty$, uniformly with respect to $t, {t}_{0}, \xi , \eta$. Since

$$\begin{aligned} Y_1(t, {t}_{0},\xi ,\eta )= & {} Y_1(t,t,Y_1(t, {t}_{0},\xi ,\eta ),Y_2(t, {t}_{0},\xi ,\eta ))\\ Y_2(t, {t}_{0},\xi ,\eta )= & {} Y_2(t,t,Y_1(t, {t}_{0},\xi ,\eta ),Y_2(t, {t}_{0},\xi ,\eta ))\\ g_{0}(t,( {t}_{0},\xi ,\eta ))= & {} g_{0}(t,(t,Y_1(t, {t}_{0},\xi ,\eta ),Y_2(t, {t}_{0},\xi ,\eta )))\equiv 0, \end{aligned}$$

it is easy to prove recursively that for all $m>0$, we have

$$\begin{aligned} g_{m}(t,( {t}_{0},\xi ,\eta ))=g_{m}(t,(t,Y_1(t, {t}_{0},\xi ,\eta ),Y_2(t, {t}_{0},\xi ,\eta ))). \end{aligned}$$

(4.39)

For any natural number m, there exists a constant $\lambda > 0$ and sufficient small constant ${\overline{q}} > 0$ such that

$$\begin{aligned} \lambda\ge & {} 3[M_{1}(k_{1})+M_{1}(k_{2})+M_{2}(k_{1})+M_{2}(k_{2})],\nonumber \\ {\overline{q}}< & {} \min \left\{ \frac{q_{0}-\max \{q_{5},q_{6}\} }{q_{0}}, \frac{q_{0}-\max \{q_{7},q_{8}\} }{q_{0}}, \frac{\min \{q_{1},(1-\theta _{1})q_{2}\}}{q_{0}}, \right. \nonumber \\&\left. \frac{\min \{q_{4},(1-\theta _{2})q_{3}\}}{q_{0}}\right\} ,\nonumber \\ 0< & {} M_{3}(K_{1}) +M_{3}(K_{2}) <\frac{2}{3}. \end{aligned}$$

(4.40)

We claim that when $0<\Vert \xi +\widetilde{\xi }\Vert +\Vert \eta +\widetilde{\eta }\Vert <1$, for all m, we have

$$\begin{aligned} \Vert g_{m}(t,(t,\xi ,\eta ))-g_{m}(t,(t,\widetilde{\xi },\widetilde{\eta }))\Vert < \lambda [\Vert \xi +\widetilde{\xi }\Vert +\Vert \eta +\widetilde{\eta }\Vert ]^{{\overline{q}}}. \end{aligned}$$

(4.41)

Obviously, (4.41) holds when $m=0$. Assume that (4.41) holds for some natural number m, and by (4.37) we get

$$\begin{aligned}&\Vert g_{m+1}(t,(t,\xi ,\eta ))-g_{m+1}(t,(t,\widetilde{\xi },\widetilde{\eta }))\Vert \nonumber \\&\quad =\, \displaystyle \int _{-\infty }^t\varPhi _{A_1}(t,\sigma (s))P(\sigma (s)) \Big [ f\big (s,Y_1(s, t,\xi ,\eta )+g_{m}(s,( t,\xi ,\eta )),Y_2(s, t,\xi ,\eta )\big )\nonumber \\&\qquad -\,f\big (s,Y_1(s, t,\widetilde{\xi },\widetilde{\eta })+g_{m}(s,( t,\widetilde{\xi },\widetilde{\eta })), Y_2(s, t,\widetilde{\xi },\widetilde{\eta })\big ) \Big ]\Delta s\nonumber \\&\qquad -\, \displaystyle \int ^{+\infty }_t {\varPhi }_{A_1}(t,\sigma (s))[I_{\chi }-P(\sigma (s))] \nonumber \\&\qquad \times \,\Big [f\big (s,Y_1(s, t,\xi ,\eta )+g_{m}(s,( t,\xi ,\eta )),Y_2(s,t,\xi ,\eta )\big )\nonumber \\&\qquad -\,f\big (s,Y_1(s, t,\widetilde{\xi },\widetilde{\eta })+g_{m}(s,( t,\widetilde{\xi },\widetilde{\eta })), Y_2(s,t,\widetilde{\xi },\widetilde{\eta })\big ) \Big ]\Delta s\nonumber \\&\quad :=\, J_{1}-J_{2}. \end{aligned}$$

Divide the integrals $J_{1}$ and $J_{2}$ into two parts:

$$\begin{aligned} J_{1}= & {} \displaystyle \int _{-\infty }^{t-T}+\displaystyle \int _{t-T}^t := J_{11}+J_{12},\\ J_{2}= & {} \displaystyle \int _{t}^{t+T}+\displaystyle \int _{t+T}^{+\infty }:=J_{21}+J_{22}. \end{aligned}$$

By computation, we obtain

$$\begin{aligned} \Vert J_{11}\Vert ^{\pm }_{ {t}_{0},c,d} \le \,&2\mu [\displaystyle \frac{K_1}{\lfloor d-a\rfloor } + \frac{K_1}{\lfloor c-a\rfloor }]\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}}\nonumber \\ :=\,&M_1(K_1)\Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_1,(1-\theta _1)q_2\}}{q_0}} ,\quad \text{ for } \text{ all }\,\,\, t\in \mathbb {T}, \end{aligned}$$

(4.42)

and

$$\begin{aligned} \Vert J_{22}\Vert ^{\pm }_{ {t}_{0},c,d}\le & {} 2\mu [\displaystyle \frac{K_2}{\lfloor b-d\rfloor } + \frac{K_2}{\lfloor b-c\rfloor }]e^{-q_0T\frac{\min \{q_4,(1-\theta _2)q_3\}}{q_0}}\nonumber \\:= & {} M_1(K_2) \Big [\Vert \xi -{\widetilde{\xi }}\Vert +\Vert \eta -{\widetilde{\eta }}\Vert \Big ]^{\frac{\min \{q_4,(1-\theta _2)q_3\}}{q_0}} ,\quad \text{ for } \text{ all }\quad t\in \mathbb {T}. \nonumber \\ \end{aligned}$$

(4.43)

From (4.8) and (4.41), we get

$$\begin{aligned}&\Vert g_{m}(s,(t,\xi ,\eta ))-g_{m}(s,(t,\xi ^{\prime },\eta ^{\prime }))\Vert \nonumber \\&\quad =\Vert g_{m}(s,(s,Y_1(s,t,\xi ,\eta ),Y_2(s,t,\xi ,\eta )))\nonumber \\&\qquad -\,g_{m}(s,(s,Y_1(s,t,\widetilde{\xi },\widetilde{\eta }),Y_2(s,t,\widetilde{\xi },\widetilde{\eta }))) \Vert \nonumber \\&\quad \le \lambda [\Vert Y_1(s,t,\xi ,\eta )-Y_1(s,t,\widetilde{\xi },\widetilde{\eta })\Vert +\Vert Y_2(s,t,\xi ,\eta )-Y_2(s,t,\widetilde{\xi },\widetilde{\eta })\Vert ]^{{\overline{q}}} \nonumber \\&\quad \le \left\{ \begin{array}{l} \lambda [\Vert \xi _{10}-\widetilde{\xi }_{10}\Vert +\Vert \eta _{20}-\widetilde{\eta }_{20}\Vert ]^{{\overline{q}}} e_{{\overline{q}}p_1}(s, t),\quad s\in [ t,+\infty )_{\mathbb {T}}, \\ \lambda [\Vert \xi _{10}-\widetilde{\xi }_{10}\Vert +\Vert \eta _{20}-\widetilde{\eta }_{20}\Vert ]^{{\overline{q}}}e_{-{\overline{q}}p_1}(s, t),\quad s\in (-\infty ,t]_{\mathbb {T}}, \end{array} \right. \end{aligned}$$

(4.44)

Now, we are in a position to estimate $J_{21}$. For convenience, denote ${\overline{P}}(s)=\frac{1}{1+\mu ^{*}(s)({\overline{q}}p_1)(s)}$, and

$$\begin{aligned}&\xi _{\mu ^*}[c\ominus (a\ominus ({\overline{q}}p_1))]\nonumber \\&\quad \le \max \left\{ \sup c\ominus (a\ominus ({\overline{q}}p_1)),\frac{\ln [1+{\overline{\mu }}^*\sup \sup c\ominus (a\ominus ({\overline{q}}p_1)) ]}{{\overline{\mu }}^*}\right\} :=\overline{q_5},\nonumber \\&\xi _{\mu ^*}[d\ominus (a\ominus ({\overline{q}}p_1))]\nonumber \\&\quad \le \max \left\{ \sup d\ominus (a\ominus ({\overline{q}}p_1)),\frac{\ln [1+{\overline{\mu }}^*\sup \sup d\ominus (a\ominus ({\overline{q}}p_1)) ]}{{\overline{\mu }}^*}\right\} := \overline{q_6},\nonumber \\&\xi _{\mu ^*}[c\ominus (b\ominus ({\overline{q}}p_1))]\nonumber \\&\quad \le \max \left\{ \sup c\ominus (b\ominus ({\overline{q}}p_1)),\frac{\ln [1+{\overline{\mu }}^*\sup \sup c\ominus (b\ominus ({\overline{q}}p_1)) ]}{{\overline{\mu }}^*}\right\} :=\overline{q_7}, \nonumber \\&\xi _{\mu ^*}[d\ominus (b\ominus ({\overline{q}}p_1))]\nonumber \\&\quad \le \max \left\{ \sup d\ominus (b\ominus ({\overline{q}}p_1)),\frac{\ln [1+{\overline{\mu }}^*\sup \sup d\ominus (b\ominus ({\overline{q}}p_1)) ]}{{\overline{\mu }}^*}\right\} :=\overline{q_8}. \nonumber \\ \end{aligned}$$

(4.45)

Now

$$\begin{aligned} \Vert J_{21}\Vert\le & {} \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [ \Vert Y_1(s,t,y_{10},y_{20})-Y_1(s,t,{\widetilde{y}}_{10},{\widetilde{y}}_{20})\Vert \nonumber \\&+\,\Vert Y_2(s,t,y_{10},y_{20})-Y_2(s,t,{\widetilde{y}}_{10},{\widetilde{y}}_{20})\Vert \nonumber \\&+\,\Vert g_m(s,t,y_{10},y_{20})-g_m(s,t,{\widetilde{y}}_{10},{\widetilde{y}}_{20})\Vert \Big ]\Delta s \nonumber \\\le & {} \,\displaystyle \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert e_{p_{1}}(s,t) \nonumber \\&+\,\lambda \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert ^{{\overline{q}}}e_{{\overline{q}}p_{1}}(s,t)\Big ]\Delta s\nonumber \\= & {} \, \displaystyle \int ^{t+T}_{t}K_2e_{b}(t,\sigma (s))\gamma \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \frac{e_{p_{1}}(\sigma (s),t)}{1+\mu ^{*}(s)p_1} \nonumber \\&+\,\lambda \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert ^{{\overline{q}}} \frac{e_{{\overline{q}}p_{1}}(\sigma (s),t)}{1+\mu ^{*}(s){\overline{q}}p_1}\Big ]\Delta s \nonumber \\= & {} \, K_2\gamma \Big [\Vert y_{10}{-}{\widetilde{y}}_{10}\Vert +\Vert y_{20}{-}{\widetilde{y}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t} \frac{e_{b}(t,\sigma (s)) e_{\ominus p_{1}}(t,\sigma (s))}{1{+}\mu ^{*}(s)p_1} \Delta s \nonumber \\&+ \,K_2\gamma \lambda \Big [\Vert y_{10}{-}{\widetilde{y}}_{10}\Vert {+}\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \displaystyle \int ^{t+T}_{t} \frac{e_{b}(t,\sigma (s)) e_{\ominus ({\overline{q}}p_{1})}(t,\sigma (s))}{1{+}\mu ^{*}(s){\overline{q}}p_1} \Delta s\ \nonumber \\\le & {} \, K_2\gamma \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t}e_{b\ominus p_{1}}(t,\sigma (s))\cdot \Vert P\Vert \Delta s \nonumber \\&+\, K_2\gamma \lambda \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \displaystyle \int ^{t+T}_{t}e_{b\ominus ({\overline{q}}p_{1})}(t,\sigma (s))\cdot \Vert {\overline{P}}\Vert \Delta s. \nonumber \\ \end{aligned}$$

(4.46)

For fixed ${t}_{0}\in {\mathbb {T}}$, without loss of generality, first we consider (4.46) on ${\mathbb {T}}^+_{ {t}_{0}}$, and there is only one case: $t+T\succeq t\succeq {t}_{0}$. Thus,

$$\begin{aligned} \Vert J_{21}\Vert\le & {} K_2\gamma \Big [\Vert y_{10}{-}{\widetilde{y}}_{10}\Vert +\Vert y_{20}{-}{\widetilde{y}}_{20}\Vert \Big ] \displaystyle \int ^{t+T}_{t}e_{b\ominus p_{1}}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot \Vert P\Vert ^{+}_{ {t}_{0},c}\Delta s\nonumber \\&+\, K_2\gamma \lambda \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\times \,\displaystyle \int ^{t+T}_{t}e_{b\ominus ({\overline{q}}p_{1})}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot \Vert {\overline{P}}\Vert ^{+}_{ {t}_{0},c} \Delta s \nonumber \\\le & {} \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\times \, \displaystyle \frac{e_{b\ominus p_{1}}(t, {t}_{0})e_{c\ominus (b\ominus p_{1})}(t+T, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \nonumber \\&+\, K_2\gamma \lambda |{\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\times \,\displaystyle \frac{e_{b\ominus ({\overline{q}}p_{1})}(t, {t}_{0})e_{c\ominus (b\ominus ({\overline{q}}p_{1}))}(t+T, {t}_{0})}{\lfloor c-(b\ominus ({\overline{q}}p_1)\rfloor },\end{aligned}$$

(4.47)

$$\begin{aligned} \Vert J_{21}\Vert e_{\ominus c}\le & {} K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ] \displaystyle \frac{e_{c\ominus (b\ominus p_{1})}(t+T,t)}{\lfloor c-(b\ominus p_1)\rfloor }\nonumber \\&+\, K_2\gamma \lambda |{\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \displaystyle \frac{e_{c\ominus (b\ominus ({\overline{q}}p_{1}))}(t+T,t)}{\lfloor c-(b\ominus ({\overline{q}}p_1)\rfloor } \nonumber \\= & {} \, \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\cdot \, e^{q_{7}T} +\frac{K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}e^{{\overline{q}}_{7}T}}{\lfloor c-(b\ominus ({\overline{q}}p_1))\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\\le & {} \, \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\times \, \frac{1}{\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{\frac{q_{0}-_{q_{7}}}{q_{0}}} \nonumber \\&+\,\frac{K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}e^{{\overline{q}}_{7}T}}{\lfloor c-(b\ominus ({\overline{q}}p_1))\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}. \end{aligned}$$

(4.48)

From (4.40), it is easy to see that ${\overline{q}}<\frac{q_{0}-{q_{7}}}{q_{0}}$ and we have

$$\begin{aligned} \Vert J_{21}\Vert e_{\ominus c}\le & {} \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&+\, \frac{K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}e^{{\overline{q}}_{7}T}}{\lfloor c-(b\ominus ({\overline{q}}p_1))\rfloor } \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\= & {} \left[ \frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor c-(b\ominus p_1)\rfloor } +\frac{K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}e^{{\overline{q}}_{7}T}}{\lfloor c-(b\ominus ({\overline{q}}p_1))\rfloor } \right] \nonumber \\&\times \,\Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} . \end{aligned}$$

(4.49)

Now we consider (4.46) on ${\mathbb {T}}^-_{ {t}_{0}}$, and there are only two cases: $t+T\preceq {t}_{0}$ and $t\preceq {t}_{0}\prec t+T$. If $t+T\preceq {t}_{0}$, a similar argument to (4.49) gives

$$\begin{aligned} \Vert J_{21}\Vert e_{\ominus d}\le & {} \Big [\frac{K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor d-(b\ominus p_1)\rfloor } +\frac{K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}}{\lfloor d-(b\ominus ({\overline{q}}p_1))\rfloor } \cdot e^{{\overline{q}}_{8}T} \Big ]\nonumber \\&\times \,\Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ] ^{{\overline{q}}}. \end{aligned}$$

(4.50)

If $t\preceq {t}_{0}\prec t+T$, then

$$\begin{aligned} \Vert J_{21}\Vert&\le \, K_2\gamma \Big [\Vert y_{10}{-}{\widetilde{y}}_{10}\Vert +\Vert y_{20}{-}{\widetilde{y}}_{20}\Vert \Big ] \left[ \displaystyle \int ^{ {t}_{0}}_{t}e_{b\ominus p_{1}}(t,\sigma (s))e_{d}(s, {t}_{0})\cdot \Vert P\Vert ^{-}_{ {t}_{0},d}\Delta s\right. \nonumber \\&\left. \quad +\,\displaystyle \int ^{t+T}_{ {t}_{0}}e_{b\ominus p_{1}}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot \Vert P\Vert ^{+}_{ {t}_{0},c}\Delta s \right] \nonumber \\&\quad +\, K_2\gamma \lambda \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\quad \times \,\left[ \displaystyle \int ^{ {t}_{0}}_{t}e_{b\ominus ({\overline{q}}p_{1})}(t,\sigma (s))e_{d}(s, {t}_{0})\cdot \Vert {\overline{P}}\Vert ^{-}_{ {t}_{0},d}\Delta s\right. \nonumber \\&\left. \quad +\,\displaystyle \int ^{t+T}_{ {t}_{0}}e_{b\ominus ({\overline{q}}p_{1})}(t,\sigma (s))e_{c}(s, {t}_{0})\cdot \Vert {\overline{P}}\Vert ^{+}_{ {t}_{0},c}\Delta s \right] \Delta s\nonumber \\&\le K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\quad \times \, \left[ \frac{e_{b\ominus p_1}(t, {t}_{0})}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{e_{b\ominus p_1}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } e_{c\ominus (b\ominus p_1)}(t+T, {t}_{0}) \right] \nonumber \\&\quad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&\quad \times \, \left[ \frac{e_{b\ominus ({\overline{q}} p_1)}(t, {t}_{0})}{\lfloor d-(b\ominus ({\overline{q}} p_1))\rfloor } + \frac{e_{b\ominus ({\overline{q}}p_1)}(t, {t}_{0})}{\lfloor c-(b\ominus ({\overline{q}} p_1))\rfloor } e_{c\ominus (b\ominus ({\overline{q}}p_1))}(t+T, {t}_{0}) \right] ,\nonumber \\ \end{aligned}$$

(4.51)

which, when combined with (4.45) leads to

$$\begin{aligned}&\Vert J_{21}\Vert e_{\ominus d}(t, {t}_{0})\le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ] \nonumber \\&\qquad \times \,\left[ \frac{e_{(b\ominus p_1)\ominus d}(t, {t}_{0})}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{e_{(b\ominus p_1)\ominus d}(t, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } e_{c\ominus (b\ominus p_1)}(t+T, {t}_{0}) \right] \nonumber \\&\qquad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&\qquad \times \, \left[ \frac{e_{(b\ominus ({\overline{q}} p_1))\ominus d}(t, {t}_{0})}{\lfloor d-(b\ominus ({\overline{q}} p_1))\rfloor } + \frac{e_{(b\ominus ({\overline{q}}p_1))\ominus d}(t, {t}_{0})}{\lfloor c-(b\ominus ({\overline{q}} p_1))\rfloor } e_{c\ominus (b\ominus ({\overline{q}}p_1))}(t+T, {t}_{0}) \right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\qquad \times \, \left[ \frac{e_{d\ominus (b\ominus p_1)}( {t}_{0},t)}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{e_{c\ominus (b\ominus p_1)}(t+T, {t}_{0})}{\lfloor c-(b\ominus p_1)\rfloor } \right] \nonumber \\&\qquad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\qquad \times \, \left[ \frac{e_{d\ominus (b\ominus ({\overline{q}} p_1))}( {t}_{0},t)}{\lfloor d-(b\ominus ({\overline{q}} p_1))\rfloor } + \frac{e_{c\ominus (b\ominus ({\overline{q}}p_1))\ominus d}(t+T, {t}_{0})}{\lfloor c-(b\ominus ({\overline{q}} p_1))\rfloor } \right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\qquad \times \, \left[ \frac{e_{d\ominus (b\ominus p_1)}(t+T,t)}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{e_{c\ominus (b\ominus p_1)}(t+T,t)}{\lfloor c-(b\ominus p_1)\rfloor } \right] \nonumber \\&\qquad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&\qquad \times \, \left[ \frac{e_{d\ominus (b\ominus ({\overline{q}} p_1))}(t+T,t)}{\lfloor d-(b\ominus ({\overline{q}} p_1))\rfloor } + \frac{e_{c\ominus (b\ominus ({\overline{q}}p_1))\ominus d}(t+T,t)}{\lfloor c-(b\ominus ({\overline{q}} p_1))\rfloor } \right] \nonumber \\&\quad \le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ] \nonumber \\&\qquad \times \, \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{q_{8},q_{7}\}T} \nonumber \\&\qquad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&\qquad \times \, \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{{\overline{q}}_{8},{\overline{q}}_{7}\}T}. \end{aligned}$$

(4.52)

It follows from (4.50) and (4.52) that for any $t\preceq {t}_{0}$,

$$\begin{aligned} \Vert J_{21}\Vert e_{\ominus d}(t, {t}_{0})&\le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\quad \times \, \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{q_{8},q_{7}\}T} \nonumber \\&\quad + \, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}} \nonumber \\&\quad \times \, \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{{\overline{q}}_{8},{\overline{q}}_{7}\}T}, \quad t\preceq {t}_{0}. \nonumber \\ \end{aligned}$$

(4.53)

Taking the supremum, it follows from (4.49) and (4.53) that for any $t\in \mathbb {T}$,

$$\begin{aligned} \Vert J_{21}\Vert ^{\pm }_{ {t}_{0},c,d}&\le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]\nonumber \\&\quad \times \, \left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{q_{8},q_{7}\}T}\nonumber \\&\quad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\quad \times \,\left[ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \right] e^{\max \{{\overline{q}}_{8},{\overline{q}}_{7}\}T} \nonumber \\&\le \, K_2\gamma \Vert P\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \Big ]\nonumber \\&\quad \times \, \left[ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \right] ^{\frac{q_{0}-\max \{q_{7},q_{8}\}}{q_{0}}}\nonumber \\&\quad +\, K_2\gamma \lambda \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \Big ]e^{\max \{{\overline{q}}_{8},{\overline{q}}_{7}\}T} \nonumber \\&\quad \times \, \left[ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \right] ^{{\overline{q}}} \nonumber \\&:= \, M_{2}(K_{2}) \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{\frac{q_{0}-\max \{q_{7},q_{8}\}}{q_{0}}} +M_{3}(K_{2})\lambda \nonumber \\&\quad \times \,\left[ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \right] ^{{\overline{q}}}, \end{aligned}$$

(4.54)

where $M_{3}(K_{2}) = K_2\gamma \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d} \Big [ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \Big ]e^{\max \{{\overline{q}}_{8},{\overline{q}}_{7}\}T}.$ A similar estimation for $J_{12}$, gives for any $t\in \mathbb {T}$,

$$\begin{aligned} \Vert J_{12}\Vert ^{\pm }_{ {t}_{0},c,d}\le & {} M_{2}(K_{1}) \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{\frac{q_{0}-\max \{q_{5},q_{6}\}}{q_{0}}} \nonumber \\&+\, M_{3}(K_{1})\lambda \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}, \end{aligned}$$

(4.55)

where $M_{3}(K_{1})=K_1\gamma \Vert {\overline{P}}\Vert ^{\pm }_{ {t}_{0},c,d}\Big [ \frac{1}{\lfloor d-(b\ominus p_1)\rfloor } + \frac{1}{\lfloor c-(b\ominus p_1)\rfloor } \Big ]e^{\max \{{\overline{q}}_{6},{\overline{q}}_{5}\}T}$.

Therefore, it follows from (4.42), (4.43), (4.54), and (4.55) that

$$\begin{aligned}&\Vert g_{m+1}(t,(t,\xi ,\eta ))-g_{m+1}(t,(t,\widetilde{\xi },\widetilde{\eta }))\Vert \\&\quad \le \, \big [ M_{1}(K_{1})+M_{1}(K_{2})+M_{2}(K_{1})+M_{2}(K_{2})\\&\qquad +\,\big (M_{3}(K_{1})+M_{3}(K_{2})\big )\lambda \big ]\nonumber \Big [\Vert \xi -\widetilde{\xi }\Vert +\Vert \eta -\widetilde{\eta }\Vert \Big ]^{{\overline{q}}}\\&\quad \le \, \lambda \Big [\Vert \xi -\widetilde{\xi }\Vert +\Vert \eta -\widetilde{\eta }\Vert \Big ]^{{\overline{q}}}, \end{aligned}$$

where ${\overline{q}} < \min \left\{ \frac{q_{0}-\max \{q_{5},q_{6}\} }{q_{0}}, \frac{q_{0}-\max \{q_{7},q_{8}\} }{q_{0}}, \frac{\min \{q_{1},(1-\theta _{1})q_{2}\}}{q_{0}}, \frac{\min \{q_{4},(1-\theta _{2})q_{3}\}}{q_{0}}\right\}$. Clearly, ${\overline{q}}<1$, for any m, that

$$\begin{aligned} \Vert g(t,(t,\xi ,\eta ))-g(t,(t,\widetilde{\xi },\widetilde{\eta }))\Vert \le \lambda \Big [\Vert \xi -\widetilde{\xi }\Vert +\Vert \eta -\widetilde{\eta }\Vert \Big ]^{{\overline{q}}}. \end{aligned}$$

Consequently,

$$\begin{aligned}&\Vert G(t,y_1,y_2)-G(t,{\widetilde{y}}_1,{\widetilde{y}}_2)\Vert \\&\quad \le \, (1+ \lambda ) \Big [ \Vert y_{10}-{\widetilde{y}}_{10}\Vert +\Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{\overline{q}}}\nonumber \\&\quad := \,{{\overline{M}}}_0\Big [\Vert y_{10}-{\widetilde{y}}_{10}\Vert + \Vert y_{20}-{\widetilde{y}}_{20}\Vert \Big ]^{{{\overline{q}}}} . \end{aligned}$$

This ends the proof of assertion (III). $\square$