一类带时滞的分数阶非线性阻尼问题解的存在性与唯一性

Existence and uniqueness of solutions for a class of fractional-order nonlinear damped problems with time delay

摘要

研究了一类带时滞的分数阶非线性阻尼问题解的存在性和唯一性.利用Banach压缩映射原理、Krasnoselskii不动点定理以及一个广义Henry型Gronwall不等式，给出了几个保证解存在唯一性的充分条件.所得结果推广了已有文献中的相关结论.

Abstract

The existence and uniqueness of solutions to a class of fractional-order nonlinear damped problems with time delay is studied. Several sufficient conditions are established to ensure the existence and uniqueness of solutions. The proofs depend mainly on Banach’s contraction mapping principle， Krasnoselskii’s fixed-point theorem， and a generalized Henry-type Gronwall inequality. The obtained results extend those conclusions in the existing literature.

关键词

分数阶时滞方程 / 非线性阻尼方程 / 存在性 / 唯一性 / 不动点定理

Key words

fractional-order delayed equation / nonlinear damped equation / existence / uniqueness / fixed point theorem

Fractional-order calculus has garnered considerable attention due to its remarkable applicability in a wide range of fields^［1-4］. In contrast to integer-order calculus， fractional-order calculus inherently exhibits infinite memory and enhanced degrees of freedom. Therefore， it can more accurately characterize phenomena in many fields such as physics， economics， medicine， and biology.

Damping is a ubiquitous physical phenomenon in oscillatory or vibratory systems. Its effects cannot be ignored in the analysis of some structures that contain rubbers， polymers， or viscoelastic materials. Over the past several decades， various material damping models have been formulated using fractional-order differential or integral equations. For certain damping materials， fractional calculus offers a particularly efficient modeling tool， yielding simpler representations that involve only a small number of parameters. In recent years， many scholars have established different types of fractional-order damped equations and have reported a wealth of interesting results regarding the properties of their solutions， including existence^［5-7］， uniqueness^［5-6］，controllability^［8-12］， and stability^［13-14］.

It is well known that the existence and uniqueness of solutions are the prerequisite for analyzing the dynamics of systems. In ［6］， Boulares et al. studied the existence and uniqueness of positive solutions for the following fractional differential equation：

\begin{array}{c}\left\{\begin{array}{cc}{}_{\mathrm{0}}{}^C{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{2}}}h\left(t\right)=f\left(t,h\left(t\right)\right)+{}_{\mathrm{0}}{}^C\mathrm{𝒟}{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{2}}-\mathrm{1}}g\left(t,h\left(t\right)\right),& t\in \left[\mathrm{0},T\right],\\ h\left(\mathrm{0}\right)=\varphi \left(\mathrm{0}\right)>\mathrm{0},h\mathrm{\text{'}}\left(\mathrm{0}\right)=\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)>\mathrm{0},& \end{array}\right.\end{array}（1）

where . The proof depends mainly on the method of upper and lower solutions， the Schauder fixed point theorem， and the Banach fixed point theorem. In ［5］， Sheng and Jiang considered the existence and uniqueness of the solution to the following damped system：

\begin{array}{c}\left\{\begin{array}{ll}{}_{\mathrm{0}}{}^C{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{2}}}h\left(t\right)-Q{}_{\mathrm{0}}{}^C\mathrm{𝒟}{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{1}}}h\left(t\right)=f\left(t,h\left(t\right)\right),& t\in \left[\mathrm{0},T\right],\\ h\left(\mathrm{0}\right)=\varphi \left(\mathrm{0}\right),h\mathrm{\text{'}}\left(\mathrm{0}\right)=\varphi \mathrm{\text{'}}\left(\mathrm{0}\right),& \end{array}\right.\end{array}（2）

where . The proofs rely mainly on the Banach fixed point theorem， the Krasnoselskii fixed-point theorem， and the Schauder fixed point theorem. In ［15］， Wang et al. considered the following nonlinear fractional-order system with delay：

\begin{array}{c}\left\{\begin{array}{l}{}_{\mathrm{0}}{}^C{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{1}}}h\left(t\right)=Q_{\mathrm{0}}h\left(t\right)+Q_{\mathrm{1}}h\left(t-\eta \right)+f\left(t,h\left(t\right),h\left(t-\eta \right)\right),t\in \left[\mathrm{0},\mathrm{T}\right],\\ h\left(t\right)=\varphi \left(t\right),t\in \left[-\eta ,\mathrm{0}\right],\end{array}\right.\end{array}（3）

where . The existence and uniqueness of the solution were proved by using the supremum technique and a generalized Gronwall inequality.

Motivated by the above-mentioned works， we are concerned with the following fractional nonlinear damped problem with time delay：

\begin{array}{c}\left\{\begin{array}{ll}{}_{\mathrm{0}}{}^C{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{2}}}h\left(t\right)-Q{}_{\mathrm{0}}{}^C\mathrm{𝒟}{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{1}}}h\left(t\right)=f\left(t,h\left(t\right),h\left(t-\eta \right)\right),& t\in \left[\mathrm{0},T\right],\\ h\left(t\right)=\varphi \left(t\right),h\mathrm{\text{'}}\left(t\right)=\varphi \mathrm{\text{'}}\left(t\right),& t\in \left[-\eta ,\mathrm{0}\right],\end{array}\right.\end{array}（4）

where ； Q is a constant matrix in {\mathbb{R}}^{n\times n}， \eta >\mathrm{0} is the time delay； f:\left[\mathrm{0},T\right]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n is continuous， and \varphi \in {\mathrm{𝒞}}^{\mathrm{1}}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right) is the given initial function. We establish several sufficient conditions to ensure the existence and uniqueness of the solution to Problem （4）. The proofs depend mainly on Banach’s contraction mapping principle， Krasnoselskii’s fixed-point theorem， and a generalized Henry-type Gronwall inequality. It is worth pointing out that the assumption on f covers those in ［5，6，15］. Our proof methods differ from those in ［6，15］. In particular， Theorems 1-2 in this paper extend those in ［5］. In addition， we also establish an existence and uniqueness theorem （Theorem 3） using a generalized Henry-type Gronwall inequality with delay.

The rest of this paper is organized as follows. Section 1 recalls some necessary preliminaries. Section 2 establishes several theorems on the existence and uniqueness of solutions， along with a corollary. In Section 3， we provide an example to show the correctness of our results.

1 Preliminaries

This section presents some necessary notation and recalls several definitions as well as auxiliary lemmas.

Let {\mathbb{N}}_{\mathrm{0}} denote the set of nonnegative integers and {\mathbb{R}}_{+} the set of nonnegative real numbers. Let T>\mathrm{0} be a constant， and let \left|x\right|=\underset{\mathrm{1}\le i\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|x_i\right| be the infinite norm in {\mathbb{R}}^n. Denote by \mathrm{𝒞}\left(\left[a,b\right],{\mathbb{R}}^n\right) the Banach space of all continuous functions mapping \left[a,b\right] into {\mathbb{R}}^n， equipped with the norm {‖h‖}_c=\underset{t\in \left[a,b\right]}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|h\left(t\right)\right|. For an arbitrary matrix Q=\left(a_{ij}\right)\in {\mathbb{R}}^{n\times n}， let ‖Q‖=\underset{\mathrm{1}\le i\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}{\sum }_{j=\mathrm{1}}^n\left|a_{ij}\right|. For ， L^p\left(\left[a,b\right],{\mathbb{R}}^n\right) denotes the Lebesgue space of p-integrable functions， endowed with the norm {‖f‖}_p={\left({\int }_a^b{\left|f\left(t\right)\right|}^{p}dt\right)}^{\mathrm{1}/p}. \mathrm{𝒜}\mathrm{𝒞}\left(\left[a,b\right],{\mathbb{R}}^n\right) denotes the space of absolutely continuous functions on \left[a,b\right]， and \mathrm{𝒜}{\mathrm{𝒞}}^m\left(\left[a,b\right],{\mathbb{R}}^n\right) denotes the space of functions f\in {\mathrm{𝒞}}^{m-\mathrm{1}}\left(\left[a,b\right],{\mathbb{R}}^n\right) with f^{\left(m-\mathrm{1}\right)}\in \mathrm{𝒜}\mathrm{𝒞}\left(\left[a,b\right],{\mathbb{R}}^n\right)， where m\ge \mathrm{2} is an integer.

Definition 1^［16］. Let \alpha \in {\mathbb{R}}^{+} and f\in L^{\mathrm{1}}\left([\mathrm{0},\mathrm{\infty }\right),{\mathbb{R}}^n)， the Riemann-Liouville fractional integral of order \alpha is defined by

I_t^{\alpha }f\left(t\right)=\frac{\mathrm{1}}{\Gamma \left(\alpha \right)}{\int }_{\mathrm{0}}^t{\left(t-\tau \right)}^{\alpha -\mathrm{1}}f\left(\tau \right)d\tau , t>\mathrm{0},

where \Gamma \left(\cdot \right) denotes the Gamma function.

Definition 2^［16］. Let \left[\alpha \right] denote the integer part of \alpha \in {\mathbb{R}}^{+}. Let m=\left[\alpha \right]+\mathrm{1}. For f\in \mathrm{𝒜}{\mathrm{𝒞}}^m\left([\mathrm{0},\mathrm{\infty }\right),{\mathbb{R}}^n)， the Riemann-Liouville fractional derivative of order \alpha is defined by

{}_{\mathrm{0}}{}^L{}_t^{\alpha }f\left(t\right)=\frac{\mathrm{1}}{\Gamma \left(m-\alpha \right)}\frac{d^m}{d{t}^m}{\int }_{\mathrm{0}}^t{\left(t-\tau \right)}^{m-\alpha -\mathrm{1}}f\left(\tau \right)d\tau , t>\mathrm{0}.

Definition 3^［17］. For \alpha \in {\mathbb{R}}^{+}， let m=\left[\alpha \right]+\mathrm{1} when \alpha \notin {\mathbb{N}}_{\mathrm{0}}. If f\in \mathrm{𝒜}{\mathrm{𝒞}}^m\left([\mathrm{0},\mathrm{\infty }\right),{\mathbb{R}}^n)， then the Caputo fractional derivative of order \alpha is defined by

{}_{\mathrm{0}}{}^C{}_t^{\alpha }f\left(t\right)=\frac{\mathrm{1}}{\Gamma \left(m-\alpha \right)}{\int }_{\mathrm{0}}^t{\left(t-\tau \right)}^{m-\alpha -\mathrm{1}}{f}^{\left(m\right)}\left(\tau \right)d\tau , t>\mathrm{0}.

Based on Definitions 2-3， we have

{}_{\mathrm{0}}{}^C{}_t^{\alpha }f\left(t\right){=}^L{\mathrm{𝒟}}_t^{\alpha }\left[f\left(t\right)-{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}\frac{t^k}{k!}{f}^{\left(k\right)}\left(\mathrm{0}\right)\right].

In particular， when \alpha \in {\mathbb{N}}_{\mathrm{0}}， we have {}_{\mathrm{0}}{}^C{}_t^{\alpha }f\left(t\right)=f^{\left(\alpha \right)}\left(t\right).

Lemma 1 （Krasnoselskii’s Fixed-Point Theorem ^［18］）. Let \Omega be a nonempty， bounded， closed， and convex subset of a Banach space \left(\mathbb{X},‖\cdot ‖\right). Suppose that two operators {\mathrm{𝒫}}_{\mathrm{1}},{\mathrm{𝒫}}_{\mathrm{2}}:\Omega \to \mathbb{X} satisfy：

（i） {\mathrm{𝒫}}_{\mathrm{1}}x+{\mathrm{𝒫}}_{\mathrm{2}}y\in \Omega for every x,y\in \Omega；

（ii） {\mathrm{𝒫}}_{\mathrm{1}} is continuous and compact；

（iii） {\mathrm{𝒫}}_{\mathrm{2}} is a contraction with constant .

Then there exists a point \omega \in \Omega such that {\mathrm{𝒫}}_{\mathrm{1}}\omega +{\mathrm{𝒫}}_{\mathrm{2}}\omega =\omega.

Lemma 2 （Arzelà-Ascoli Theorem ^［19］）. Let K be a compact metric space and let \mathrm{ℱ}\subset \mathrm{𝒞}\left(K,{\mathbb{R}}^n\right). If \mathrm{ℱ} is uniformly bounded and equicontinuous on K， then \mathrm{ℱ} is relatively compact in the Banach space \left(\mathrm{𝒞}\left(K,{\mathbb{R}}^n\right),{‖\cdot ‖}_c\right).

Lemma 3 （Banach Fixed-Point Theorem ^［20］）. Let \left(X,d\right) be a complete metric space and let \mathrm{𝒫}:X\to X be a contraction， i.e.， there exists k\in [\mathrm{0,1}) such that

d\left(\mathrm{𝒫}x,\mathrm{𝒫}y\right)\le kd\left(x,y\right) \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in X.

Then \mathrm{𝒫} admits a unique fixed point in X.

Lemma 4 （Hölder’s Inequality^［21］）. Letp,q\ge \mathrm{1} be conjugate exponents satisfying \mathrm{1}/p+\mathrm{1}/q=\mathrm{1}. If \varphi \in L^p\left(\left[a,b\right]\right) and \phi \in L^q\left(\left[a,b\right]\right)， then the product \varphi \phi belongs toL^{\mathrm{1}}\left(\left[a,b\right]\right) and

{‖\varphi \phi ‖}_{L^{\mathrm{1}}\left(\left[a,b\right]\right)}\le {‖\varphi ‖}_{L^p\left(\left[a,b\right]\right)}{‖\phi ‖}_{L^q\left(\left[a,b\right]\right)}.

Lemma 5^［22］. Let \alpha ,\beta >\mathrm{0}，{\hslash },\mathrm{𝓁},\mathrm{\Im },{\mu }_{\mathrm{1}},{\mu }_{\mathrm{2}},{\kappa }_{\mathrm{1}},{\kappa }_{\mathrm{2}}\in \mathrm{𝒞}\left(\left[\mathrm{0},T\right],{\mathbb{R}}_{+}\right)， and \mathrm{\varpi }\left(t\right)\in \mathrm{𝒞}\left(\left[-\tau ,\mathrm{0}\right],{\mathbb{R}}_{+}\right). Let \mathrm{\hslash }\left(\mathrm{0}\right)=\mathrm{\varpi }\left(\mathrm{0}\right). Suppose that {\hslash },\mathrm{𝓁},\mathrm{\Im },{\mu }_{\mathrm{1}},{\mu }_{\mathrm{2}},{\kappa }_{\mathrm{1}},{\kappa }_{\mathrm{2}} are nondecreasing functions. If w\in \mathrm{𝒞}\left(\left[-\tau ,T\right],{\mathbb{R}}_{+}\right) satisfies

\left\{\begin{array}{c}w\left(t\right)\le \mathrm{\hslash }\left(t\right)+\mathrm{l}\left(t\right){\int }_{\mathrm{0}}^t\frac{{\left(t-s\right)}^{\alpha -\mathrm{1}}}{\Gamma \left(\alpha \right)}\left[{\mu }_{\mathrm{1}}\left(s\right)w\left(s\right)+{\kappa }_{\mathrm{1}}\left(s\right)w\left(s-\tau \right)\right]ds\\ +\mathrm{\Im }\left(t\right){\int }_{\mathrm{0}}^t\frac{{\left(t-s\right)}^{\beta -\mathrm{1}}}{\Gamma \left(\beta \right)}\left[{\mu }_{\mathrm{2}}\left(s\right)w\left(s\right)+{\kappa }_{\mathrm{2}}\left(s\right)w\left(s-\tau \right)\right]ds,t\in \left[\mathrm{0},T\right],\\ w\left(t\right)\le \mathrm{\varpi }\left(t\right),t\in \left[-\tau ,\mathrm{0}\right],\end{array}\right.（5）

then

w\left(t\right)\le P^{\mathrm{1}/q}\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\{G\left(t\right)\mathrm{2}t/q\}, t\in \left[\mathrm{0},T\right],

where

\begin{array}{l}Z\left(t\right)={\mathrm{3}}^{q-\mathrm{1}}{\left\{\frac{\mathrm{l}\left(t\right)}{\Gamma \left(\alpha \right)}\frac{t^{\alpha -\mathrm{1}/q}{\mu }_{\mathrm{1}}\left(t\right)}{{\left(\left(\alpha -\mathrm{1}\right)p+\mathrm{1}\right)}^{\mathrm{1}/p}}+\frac{\mathrm{\Im }\left(t\right)}{\Gamma \left(\beta \right)}\frac{t^{\beta -\mathrm{1}/q}{\mu }_{\mathrm{2}}\left(t\right)}{{\left(\left(\beta -\mathrm{1}\right)p+\mathrm{1}\right)}^{\mathrm{1}/p}}\right\}}^q,\\ R\left(t\right)={\mathrm{3}}^{q-\mathrm{1}}{\left\{\frac{\mathrm{l}\left(t\right)}{\Gamma \left(\alpha \right)}\frac{t^{\alpha -\mathrm{1}/q}{\kappa }_{\mathrm{1}}\left(t\right)}{{\left(\left(\alpha -\mathrm{1}\right)p+\mathrm{1}\right)}^{\mathrm{1}/p}}+\frac{\mathrm{\Im }\left(t\right)}{\Gamma \left(\beta \right)}\frac{t^{\beta -\mathrm{1}/q}{\kappa }_{\mathrm{2}}\left(t\right)}{{\left(\left(\beta -\mathrm{1}\right)p+\mathrm{1}\right)}^{\mathrm{1}/p}}\right\}}^q,\\ G\left(t\right)=\mathrm{m}\mathrm{a}\mathrm{x}\{Z\left(t\right),R\left(t\right)\}, P\left(t\right)={\mathrm{3}}^{q-\mathrm{1}}{\mathrm{\hslash }}^q\left(t\right),\end{array}

p,q>\mathrm{0}，\mathrm{1}/p+\mathrm{1}/q=\mathrm{1}，\alpha >\mathrm{1}/q， and \beta >\mathrm{1}/q.

2 Main results

This section focuses on the existence and uniqueness of solutions to Problem （4）. Let us first present some assumptions on the nonlinear function f.

（H1） The function f:\left[\mathrm{0},T\right]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n is continuous.

（H2） There exists a non-negative function \delta \in L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) such that

\begin{array}{l}\left|f\left(t,\alpha ,\beta \right)\right|\le \delta \left(t\right), \\ \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left[\mathrm{0},T\right]\mathrm{a}\mathrm{n}\mathrm{d}\alpha ,\beta \in {\mathbb{R}}^n,\end{array}

where {\rho }_{\mathrm{1}}^{\mathrm{*}}>\mathrm{1} is a given constant.

（H3） There exist two functions {\theta }_{\mathrm{1}}\in L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) and {\theta }_{\mathrm{2}}\in L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) such that

\left|f\left(t,{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}\right)-f\left(t,{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{2}}\right)\right|\le {\theta }_{\mathrm{1}}\left(t\right)\left|{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}\right|+{\theta }_{\mathrm{2}}\left(t\right)\left|{\beta }_{\mathrm{1}}-{\beta }_{\mathrm{2}}\right|,

for all t\in \left[\mathrm{0},T\right] and all {\alpha }_{\mathrm{1}},{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}},{\beta }_{\mathrm{2}}\in {\mathbb{R}}^n， where {\rho }_{\mathrm{2}}^{\mathrm{*}},{\rho }_{\mathrm{3}}^{\mathrm{*}}>\mathrm{1} are two given constants.

For convenience in the subsequent analysis， let {\rho }_{\mathrm{1}},{\rho }_{\mathrm{2}}， and {\rho }_{\mathrm{3}} be the conjugate exponents corresponding to {\rho }_{\mathrm{1}}^{\mathrm{*}},{\rho }_{\mathrm{2}}^{\mathrm{*}}， and {\rho }_{\mathrm{3}}^{\mathrm{*}} in （H2） and （H3）， respectively. That is，

{\rho }_k=\frac{{\rho }_k^{\mathrm{*}}}{{\rho }_k^{\mathrm{*}}-\mathrm{1}}, k=\mathrm{1,2},\mathrm{3}.

Define {\mathrm{𝒞}}_{\omega } by

{\mathrm{𝒞}}_{\omega }=\left\{h\in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right):h\left(t\right)=\varphi \left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\in \left[-\eta ,\mathrm{0}\right],\hspace{0.25em}{‖h‖}_c\le \omega \right\},

where \omega is a constant such that \omega \ge {‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right)}.

By Lemma 3.1 in ［7］， a function h\in \mathit{𝒞}\left(\left[-\eta ,T\right],{\mathit{ℝ}}^n\right) is a solution of Problem （4） if and only if it satisfies the following Volterra-type integral equation

\begin{array}{c}\begin{array}{c}h\left(t\right)=\left\{\begin{array}{cc}\varphi \left(t\right),& t\in \left[-\eta ,\mathrm{0}\right],\\ \varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}& \\ +\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds& \\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h\left(s\right),h\left(s-\eta \right)\right)ds,& t\in (\mathrm{0},T].\end{array}\right.\end{array}\end{array}（6）

Lemma 6. The set {\mathrm{𝒞}}_{\omega } is a non-empty， closed， and convex subset of\mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right).

Proof. It is clear that {\mathrm{𝒞}}_{\omega } is not empty. Indeed， consider the following function z defined by

z\left(t\right)=\left\{\begin{array}{ll}\varphi \left(t\right),& t\in \left[-\eta ,\mathrm{0}\right],\\ \varphi \left(\mathrm{0}\right),& t\in (\mathrm{0},T].\end{array}\right.

Obviously， it is continuous on \left[-\eta ,T\right] since \varphi is continuous on \left[-\eta ,\mathrm{0}\right]. Moreover，

{‖z‖}_c=\mathrm{m}\mathrm{a}\mathrm{x}\{{‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right]\right)},\left|\varphi \left(\mathrm{0}\right)\right|\}\le \omega .

Thus， z\in {\mathrm{𝒞}}_{\omega }.

Next， we prove that {\mathrm{𝒞}}_{\omega } is convex. For z_{\mathrm{1}},z_{\mathrm{2}}\in {\mathrm{𝒞}}_{\omega } and \theta \in \left[\mathrm{0,1}\right]， put w=\theta z_{\mathrm{1}}+\left(\mathrm{1}-\theta \right)z_{\mathrm{2}}. For any t\in \left[-\eta ,\mathrm{0}\right]， we have

w\left(t\right)=\theta \varphi \left(t\right)+\left(\mathrm{1}-\theta \right)\varphi \left(t\right)=\varphi \left(t\right).

Moreover， the triangle inequality allows us to obtain

{‖w‖}_c\le \theta {‖z_{\mathrm{1}}‖}_c+\left(\mathrm{1}-\theta \right){‖z_{\mathrm{2}}‖}_c\le \theta \omega +\left(\mathrm{1}-\theta \right)\omega =\omega .

Consequently， w\in {\mathrm{𝒞}}_{\omega }. This indicates that {\mathrm{𝒞}}_{\omega } is convex.

It remains to show that {\mathrm{𝒞}}_{\omega } is closed. Let \{z^{\left(k\right)}\}\subset {\mathrm{𝒞}}_{\omega } be a sequence such that z^{\left(k\right)}\to z in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right). It is easy to see that

z\left(t\right)=\underset{k\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{z}^{\left(k\right)}\left(t\right)=\varphi \left(t\right),\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\forall t\in \left[-\eta ,\mathrm{0}\right].

Additionally， the continuity of the norm implies

{‖z‖}_c=\underset{k\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{‖z^{\left(k\right)}‖}_c\le \omega .

Therefore， z\in {\mathrm{𝒞}}_{\omega }. We conclude that {\mathrm{𝒞}}_{\omega } is a closed subset of \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right).

Theorem 1. Suppose that （H1）-（H3） hold. If

then Problem （4） has a unique solution on \left[-\eta ,T\right].

Proof. Notice that the condition （7） evidently implies . This allows us to choose a constant \mathrm{\varrho } such that

\begin{array}{l}\mathrm{\varrho }\ge \frac{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)-‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}\\ \left({‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right)}+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|\varphi \left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{1}}^{\mathrm{*}}}{‖\delta ‖}_{L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{1}}}}\right).\end{array}

Let

{\mathrm{𝒞}}_{\mathrm{\varrho }}=\{h\in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right):h\left(t\right)=\varphi \left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\in \left[-\eta ,\mathrm{0}\right],\hspace{0.25em}{‖h‖}_c\le \mathrm{\varrho }\}.

By Lemma 6， {\mathrm{𝒞}}_{\mathrm{\varrho }} equipped with the supremum norm {‖\cdot ‖}_c is a complete metric space.

The operator F:{\mathrm{𝒞}}_{\mathrm{\varrho }}\to \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right) is defined as

\begin{array}{c}\left(Fh\right)\left(t\right)=\left\{\begin{array}{ll}\varphi \left(t\right),& t\in \left[-\eta ,\mathrm{0}\right],\\ \varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}& \\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em} +\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds& \\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em} +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h\left(s\right),h\left(s-\eta \right)\right)ds,& t\in (\mathrm{0},T].\end{array}\right.\end{array}

First， let us prove F\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right)\subseteq {\mathrm{𝒞}}_{\mathrm{\varrho }}. Let h\in {\mathrm{𝒞}}_{\mathrm{\varrho }}. Evidently， Fh\in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right). For t\in \left[-\eta ,\mathrm{0}\right]， it is obvious that \left|\left(Fh\right)\left(t\right)\right|=\left|\varphi \left(t\right)\right|\le {‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right)}\le \mathrm{\varrho }. For t\in (\mathrm{0},T]， by （H2）， Hölder’s inequality and {‖h‖}_c\le \mathrm{\varrho }， we derive

\begin{array}{l}\left|\left(Fh\right)\left(t\right)\right|=|\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h\left(s\right),h\left(s-\eta \right)\right)ds|\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|\varphi \left(\mathrm{0}\right)\right|\\ +\frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h\left(s\right)\right|ds+\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\delta \left(s\right)ds\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|\varphi \left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\mathrm{\varrho }\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{1}}}{‖\delta ‖}_{L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}}\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left(\left|\varphi \left(\mathrm{0}\right)\right|+\mathrm{\varrho }\right)+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{1}}^{\mathrm{*}}}{‖\delta ‖}_{L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{1}}}}\\ \le \mathrm{\varrho }.\end{array}

This indicates {‖Fh‖}_c\le \mathrm{\varrho }. Hence， F maps {\mathrm{𝒞}}_{\mathrm{\varrho }} into {\mathrm{𝒞}}_{\mathrm{\varrho }}.

Now， let us show that F is a contraction mapping. Suppose that h_{\mathrm{1}},h_{\mathrm{2}}\in {\mathrm{𝒞}}_{\mathrm{\varrho }}. When t\in \left[-\eta ,\mathrm{0}\right]， it is evident that \left|\left(F{h}_{\mathrm{1}}\right)\left(t\right)-\left(F{h}_{\mathrm{2}}\right)\left(t\right)\right|=\mathrm{0}. For t\in (\mathrm{0},T]， we apply （H3） to get

\begin{array}{l}\left|\left(F{h}_{\mathrm{1}}\right)\left(t\right)-\left(F{h}_{\mathrm{2}}\right)\left(t\right)\right|\\ \le \frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right|ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\left|f\left(s,h_{\mathrm{1}}\left(s\right),h_{\mathrm{1}}\left(s-\eta \right)\right)-f\left(s,h_{\mathrm{2}}\left(s\right),h_{\mathrm{2}}\left(s-\eta \right)\right)\right|ds\\ \le \frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right|ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{1}}\left(s\right)\left|h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right|ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{2}}\left(s\right)\left|h_{\mathrm{1}}\left(s-\eta \right)-h_{\mathrm{2}}\left(s-\eta \right)\right|ds.\end{array}

Applying Hölder’s inequality to the integrals involving {\theta }_{\mathrm{1}} and {\theta }_{\mathrm{2}}， we obtain

\begin{array}{l}\left|\left(F{h}_{\mathrm{1}}\right)\left(t\right)-\left(F{h}_{\mathrm{2}}\right)\left(t\right)\right|\\ \le \frac{‖Q‖{‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}ds\\ +\frac{{‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{1}}\left(s\right)ds+{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{2}}\left(s\right)ds\right)\\ \le {‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c\left[\frac{‖Q‖t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\right.\\ \left.+\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}\left({\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{2}}}{‖{\theta }_{\mathrm{1}}‖}_{L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}}+{\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{3}}}{‖{\theta }_{\mathrm{2}}‖}_{L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}}\right)\right]\\ \le {‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c\left[\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{2}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{1}}‖}_{L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{2}}}}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{3}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{2}}‖}_{L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{3}}}}\right].\end{array}

Therefore，

\begin{array}{rr}& \\ & \end{array}

\begin{array}{l}{‖\left(F{h}_{\mathrm{1}}\right)-\left(F{h}_{\mathrm{2}}\right)‖}_c\\ \le {‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c\left[\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{2}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{1}}‖}_{L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{2}}}}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{3}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{2}}‖}_{L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{3}}}}\right].\end{array}

Combining this with the condition （7）， we conclude that F is a contraction mapping on the complete metric space {\mathrm{𝒞}}_{\mathrm{\varrho }}. According to Banach’s contraction mapping principle， F has a unique fixed point h^{\mathrm{*}}\in {\mathrm{𝒞}}_{\mathrm{\varrho }}， which is the unique solution to Problem （4） on \left[-\eta ,T\right]. This completes the proof.

Remark 1. When the time delay \eta is absent in Problem （4）， Theorem 1 reduces to Theorem 4.1 in ［5］.

Theorem 2. Suppose that （H1）-（H3） hold. If

（8）

and

（9）

then Problem （4） admits at least one solution on \left[-\eta ,T\right].

Proof. By （8）， we can choose a constant \mathrm{\varrho } such that

\begin{array}{l}\mathrm{\varrho }\ge \frac{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)-‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}\\ \left({‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right)}+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|\varphi \left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{1}}^{\mathrm{*}}}{‖\delta ‖}_{L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right){\left(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}+\mathrm{1}\right)}^{\mathrm{1}/{\rho }_{\mathrm{1}}}}\right).\end{array}

Let

{\mathrm{𝒞}}_{\mathrm{\varrho }}=\left\{h\in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right):h\left(t\right)=\varphi \left(t\right)\mathrm{f}\mathrm{o}\mathrm{r}t\in \left[-\eta ,\mathrm{0}\right],\hspace{0.25em}{‖h‖}_c\le \mathrm{\varrho }\right\}.

By Lemma 6， {\mathrm{𝒞}}_{\mathrm{\varrho }} is a nonempty， bounded， closed， and convex subset of \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right). Define the operators F_{\mathrm{1}},F_{\mathrm{2}}:{\mathrm{𝒞}}_{\mathrm{\varrho }}\to \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right) by

\begin{array}{c}\left(F_{\mathrm{1}}h\right)\left(t\right)=\left\{\begin{array}{ll}\varphi \left(t\right),& t\in \left[-\eta ,\mathrm{0}\right],\\ \varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds,& t\in (\mathrm{0},T],\end{array}\right.\end{array}

and

\begin{array}{c}\left(F_{\mathrm{2}}h\right)\left(t\right)=\left\{\begin{array}{ll}\mathrm{0},& t\in \left[-\eta ,\mathrm{0}\right],\\ \frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h\left(s\right),h\left(s-\eta \right)\right)ds,& t\in (\mathrm{0},T].\end{array}\right.\end{array}

We first verify F_{\mathrm{1}}x+F_{\mathrm{2}}h\in {\mathrm{𝒞}}_{\mathrm{\varrho }} for all x,h\in {\mathrm{𝒞}}_{\mathrm{\varrho }}. Clearly， F_{\mathrm{1}}x+F_{\mathrm{2}}h is continuous. For any t\in \left[-\eta ,\mathrm{0}\right],\left(F_{\mathrm{1}}x+F_{\mathrm{2}}h\right)\left(t\right)=\varphi \left(t\right). Moreover， we get that \left|\left(F_{\mathrm{1}}x+F_{\mathrm{2}}h\right)\left(t\right)\right|=\left|\varphi \left(t\right)\right|\le {‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right],{\mathbb{R}}^n\right)}\le \mathrm{\varrho }. For t\in (\mathrm{0},T]， the use of （H2） and Hölder’s inequality yield

\begin{array}{l}\left|\left(F_{\mathrm{1}}x\right)\left(t\right)+\left(F_{\mathrm{2}}h\right)\left(t\right)\right|\\ =|\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}x\left(s\right)ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h\left(s\right),h\left(s-\eta \right)\right)ds|\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|\varphi \left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\mathrm{\varrho }\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\delta \left(s\right)ds\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}‖Q‖\left(\left|\varphi \left(\mathrm{0}\right)\right|+\mathrm{\varrho }\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{1}}}{\left({\int }_{\mathrm{0}}^t\delta {\left(s\right)}^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{1}}^{\mathrm{*}}}\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}‖Q‖\left(\left|\varphi \left(\mathrm{0}\right)\right|+\mathrm{\varrho }\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{1}}^{\mathrm{*}}}{‖\delta ‖}_{L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{1}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{1}}}}\\ \le \mathrm{\varrho }.\end{array}

This implies {‖F_{\mathrm{1}}x+F_{\mathrm{2}}h‖}_c\le \mathrm{\varrho }. Hence， F_{\mathrm{1}}x+F_{\mathrm{2}}h\in {\mathrm{𝒞}}_{\mathrm{\varrho }} for all x,h\in {\mathrm{𝒞}}_{\mathrm{\varrho }}.

Next， we demonstrate that F_{\mathrm{2}} is a contraction mapping. Let h_{\mathrm{1}},h_{\mathrm{2}}\in {\mathrm{𝒞}}_{\mathrm{\varrho }}. For t\in \left[-\eta ,\mathrm{0}\right]， \left(F_{\mathrm{2}}{h}_{\mathrm{1}}\right)\left(t\right)-\left(F_{\mathrm{2}}{h}_{\mathrm{2}}\right)\left(t\right)=\mathrm{0}. For t\in (\mathrm{0},T]， using （H3）， we obtain

\begin{array}{l}\left|\left(F_{\mathrm{2}}{h}_{\mathrm{1}}\right)\left(t\right)-\left(F_{\mathrm{2}}{h}_{\mathrm{2}}\right)\left(t\right)\right|\\ =\left|\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h_{\mathrm{1}}\left(s\right),h_{\mathrm{1}}\left(s-\eta \right)\right)ds-\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h_{\mathrm{2}}\left(s\right),h_{\mathrm{2}}\left(s-\eta \right)\right)ds\right|\\ \le \frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\left[{\theta }_{\mathrm{1}}\left(s\right)\left|h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right|+{\theta }_{\mathrm{2}}\left(s\right)\left|h_{\mathrm{1}}\left(s-\eta \right)-h_{\mathrm{2}}\left(s-\eta \right)\right|\right]ds\end{array}

Applying Hölder’s inequality， we get

\begin{array}{l}\left|\left(F_{\mathrm{2}}{h}_{\mathrm{1}}\right)\left(t\right)-\left(F_{\mathrm{2}}{h}_{\mathrm{2}}\right)\left(t\right)\right|\le \frac{{‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}\left[{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{1}}\left(s\right)ds+{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}{\theta }_{\mathrm{2}}\left(s\right)ds\right]\\ \le \frac{{‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}\left[{\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{2}}}{\left({\int }_{\mathrm{0}}^t{\theta }_{\mathrm{1}}{\left(s\right)}^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{2}}^{\mathrm{*}}}\right.\\ \left.+{\left({\int }_{\mathrm{0}}^t{\left(t-s\right)}^{\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{3}}}{\left({\int }_{\mathrm{0}}^t{\theta }_{\mathrm{2}}{\left(s\right)}^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}ds\right)}^{\mathrm{1}/{\rho }_{\mathrm{3}}^{\mathrm{*}}}\right]\\ \le {‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c\left[\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{2}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{1}}‖}_{L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{2}}}}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{3}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{2}}‖}_{L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{3}}}}\right].\end{array}

Therefore，

{‖\left(F_{\mathrm{2}}{h}_{\mathrm{1}}\right)-\left(F_{\mathrm{2}}{h}_{\mathrm{2}}\right)‖}_c\le \left[\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{2}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{1}}‖}_{L^{{\rho }_{\mathrm{2}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{2}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{2}}}}+\frac{T^{{\gamma }_{\mathrm{2}}-\mathrm{1}/{\rho }_{\mathrm{3}}^{\mathrm{*}}}{‖{\theta }_{\mathrm{2}}‖}_{L^{{\rho }_{\mathrm{3}}^{\mathrm{*}}}}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)(\left({\gamma }_{\mathrm{2}}-\mathrm{1}\right){\rho }_{\mathrm{3}}{+\mathrm{1})}^{\mathrm{1}/{\rho }_{\mathrm{3}}}}\right]{‖h_{\mathrm{1}}-h_{\mathrm{2}}‖}_c.

With the condition （9）， F_{\mathrm{2}} is a contraction mapping.

In what follows， we will prove that the operator F_{\mathrm{1}} is continuous and compact.

Let us first discuss the continuity of F_{\mathrm{1}}. Suppose that \{h_n{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}\subset {\mathrm{𝒞}}_{\mathrm{\varrho }} is a sequence such that h_n\to h uniformly on \left[-\eta ,T\right]. For any t\in \left[-\eta ,\mathrm{0}\right]， \left(F_{\mathrm{1}}{h}_n-F_{\mathrm{1}}h\right)\left(t\right)=\mathrm{0}. For any t\in \left[\mathrm{0},T\right]， we obtain

\begin{array}{l}\left|\left(F_{\mathrm{1}}{h}_n\right)\left(t\right)-\left(F_{\mathrm{1}}h\right)\left(t\right)\right|\\ =\left|\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}{h}_n\left(s\right)ds\right.\\ \left.-\left(\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds\right)\right|\\ =\left|\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left(h_n\left(s\right)-h\left(s\right)\right)ds\right|\\ \le \frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h_n\left(s\right)-h\left(s\right)\right|ds\\ \le \frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}{‖h_n-h‖}_c.\end{array}

Hence，

{‖F_{\mathrm{1}}{h}_n-F_{\mathrm{1}}h‖}_c\le \frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}{‖h_n-h‖}_c.

Therefore， we obtain that {‖F_{\mathrm{1}}{h}_n-F_{\mathrm{1}}h‖}_c converges to 0 in \mathrm{𝒞}\left(\left[-\eta ,T\right],{\mathbb{R}}^n\right). This implies the continuity of F_{\mathrm{1}}.

In order to prove the compactness of F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right)， we first consider the uniform boundedness of F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right). Let h\in {\mathrm{𝒞}}_{\mathrm{\varrho }}. For t\in \left[-\eta ,\mathrm{0}\right]， \left|\left(F_{\mathrm{1}}h\right)\left(t\right)\right|\le {‖\varphi ‖}_{\mathrm{𝒞}\left(\left[-\eta ,\mathrm{0}\right]\right)}\le \mathrm{\varrho }. For t\in (\mathrm{0},T]， we obtain

\begin{array}{l}\left|\left(F_{\mathrm{1}}h\right)\left(t\right)\right|=\left|\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}+\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds\right|\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}‖Q‖\left|\varphi \left(\mathrm{0}\right)\right|+\frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h\left(s\right)\right|ds\\ \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}‖Q‖\left|\varphi \left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\mathrm{\varrho }.\end{array}

Thus，

\begin{array}{rr}{‖F_{\mathrm{1}}h‖}_c& \le \left|\varphi \left(\mathrm{0}\right)\right|+T\left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|+\frac{T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}‖Q‖\left|\varphi \left(\mathrm{0}\right)\right|+\frac{‖Q‖T^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\mathrm{\varrho }.\end{array}

This gives that， F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right) is uniformly bounded.

Now we prove that F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right) is equicontinuous on \left[-\eta ,T\right]. On one hand， for ， we have

\begin{array}{l}\left|\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{2}}\right)-\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{1}}\right)\right|\\ \le \left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|\left(t_{\mathrm{2}}-t_{\mathrm{1}}\right)+\frac{\left|\varphi \left(\mathrm{0}\right)\right|‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left(t_{\mathrm{2}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}-t_{\mathrm{1}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}\right)\\ +\frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}\left|{\int }_{\mathrm{0}}^{t_{\mathrm{2}}}{\left(t_{\mathrm{2}}-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds-{\int }_{\mathrm{0}}^{t_{\mathrm{1}}}{\left(t_{\mathrm{1}}-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}h\left(s\right)ds\right|\\ \le \left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|\left(t_{\mathrm{2}}-t_{\mathrm{1}}\right)+\frac{\left|\varphi \left(\mathrm{0}\right)\right|‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left(t_{\mathrm{2}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}-t_{\mathrm{1}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}\right)\\ +\frac{\mathrm{\varrho }‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}\left[{\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}{\left(t_{\mathrm{2}}-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}ds+{\int }_{\mathrm{0}}^{t_{\mathrm{1}}}\left|{\left(t_{\mathrm{2}}-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}-{\left(t_{\mathrm{1}}-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\right|ds\right].\end{array}

Consequently，

\begin{array}{l}\left|\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{2}}\right)-\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{1}}\right)\right|\\ \le \left|\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)\right|\left(t_{\mathrm{2}}-t_{\mathrm{1}}\right)+\frac{\left|\varphi \left(\mathrm{0}\right)\right|‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left|t_{\mathrm{2}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}-t_{\mathrm{1}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}\right|\\ +\frac{\mathrm{\varrho }‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\left[\mathrm{2}{\left(t_{\mathrm{2}}-t_{\mathrm{1}}\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}+\left|t_{\mathrm{2}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}-t_{\mathrm{1}}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}\right|\right].\end{array}

Obviously， the term on the right hand side of this inequality is irrelevant to h. As t_{\mathrm{2}} tends to t_{\mathrm{1}}， we have \left|\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{2}}\right)-\left(F_{\mathrm{1}}h\right)\left(t_{\mathrm{1}}\right)\right|\to \mathrm{0}. As a result， F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right) is equicontinuous on \left[\mathrm{0},T\right].

On the other hand， for t\in \left[-\eta ,\mathrm{0}\right]， \left(F_{\mathrm{1}}h\right)\left(t\right)=\varphi \left(t\right). It is evident that F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right) is equicontinuous on \left[-\eta ,\mathrm{0}\right]. Moreover， F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right) is equicontinuous on \left[-\eta ,T\right]. Combining this with the uniform boundedness of F_{\mathrm{1}}\left({\mathrm{𝒞}}_{\mathrm{\varrho }}\right)， we use the Arzelà-Ascoli theorem to show that F_{\mathrm{1}} is a compact operator.

With the help of Krasnoselskii’s fixed point theorem， we conclude that there exists h^{\mathrm{*}}\in {\mathrm{𝒞}}_{\mathrm{\varrho }} such that F_{\mathrm{1}}{h}^{\mathrm{*}}+F_{\mathrm{2}}{h}^{\mathrm{*}}=h^{\mathrm{*}}. Consequently， h^{\mathrm{*}} is a solution to Problem （4） on \left[-\eta ,T\right]. The proof is complete.

Remark 2. When the time delay \eta is absent in Problem （4）， Theorem 2 reduces to Theorem 4.2 in ［5］.

To further investigate the uniqueness of the solution to Problem （4）， we introduce the following assumption， which is stronger than （H3）.

（H4） There exist two nondecreasing functions {\theta }_{\mathrm{1}},{\theta }_{\mathrm{2}}\in \mathrm{𝒞}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) such that

\begin{array}{l}\left|f\left(t,{\alpha }_{\mathrm{1}},{\beta }_{\mathrm{1}}\right)-f\left(t,{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{2}}\right)\right|\le {\theta }_{\mathrm{1}}\left(t\right)\left|{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}\right|\\ +{\theta }_{\mathrm{2}}\left(t\right)\left|{\beta }_{\mathrm{1}}-{\beta }_{\mathrm{2}}\right|,\end{array}

for all t\in \left[\mathrm{0},T\right] and all {\alpha }_{\mathrm{1}},{\alpha }_{\mathrm{2}},{\beta }_{\mathrm{1}},{\beta }_{\mathrm{2}}\in {\mathbb{R}}^n.

Theorem 3. Suppose that （H1）， （H2） and （H4） hold. If

and

then Problem （4） has a unique solution on \left[-\eta ,T\right].

Proof. According to Theorem 2， there exists at least one solution to Problem （4）. Let h_{\mathrm{1}} and h_{\mathrm{2}} be two solutions to Problem （4）， and let \xi \left(t\right)=\left|h_{\mathrm{1}}\left(t\right)-h_{\mathrm{2}}\left(t\right)\right|.

For t\in \left[-\eta ,\mathrm{0}\right]， h_{\mathrm{1}}=h_{\mathrm{2}}=\varphi \left(t\right). For t\in \left[\mathrm{0},T\right]， it is obvious that h_{\mathrm{1}} and h_{\mathrm{2}} satisfy the equivalent integral equation （6）. Namely，

\begin{array}{l}{h}_i\left(t\right)=\varphi \left(\mathrm{0}\right)+t\varphi \mathrm{\text{'}}\left(\mathrm{0}\right)-\frac{t^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}}Q\varphi \left(\mathrm{0}\right)}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}+\mathrm{1}\right)}\\ +\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}{h}_i\left(s\right)ds+\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}f\left(s,h_i\left(s\right),h_i\left(s-\eta \right)\right)ds,\end{array}

where \mathrm{i}=\mathrm{1,2}.

Clearly， \xi \left(t\right)=\mathrm{0}， t\in \left[-\eta ,\mathrm{0}\right]. For t\in \left[\mathrm{0},T\right]，

\begin{array}{l}\left|\mathrm{\xi }\left(t\right)\right|=\left|\frac{Q}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left[h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right]ds\right.\\ \left.+\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\left[f\left(s,h_{\mathrm{1}}\left(s\right),h_{\mathrm{1}}\left(s-\eta \right)\right)-f\left(s,h_{\mathrm{2}}\left(s\right),h_{\mathrm{2}}\left(s-\eta \right)\right)\right]ds\right|\\ \le \frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|h_{\mathrm{1}}\left(s\right)-h_{\mathrm{2}}\left(s\right)\right|ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\left|f\left(s,h_{\mathrm{1}}\left(s\right),h_{\mathrm{1}}\left(s-\eta \right)\right)-f\left(s,h_{\mathrm{2}}\left(s\right),h_{\mathrm{2}}\left(s-\eta \right)\right)\right|ds\\ \le \frac{‖Q‖}{\Gamma \left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}-\mathrm{1}}\left|\mathrm{\xi }\left(s\right)\right|ds\\ +\frac{\mathrm{1}}{\Gamma \left({\gamma }_{\mathrm{2}}\right)}{\int }_{\mathrm{0}}^t{\left(t-s\right)}^{{\gamma }_{\mathrm{2}}-\mathrm{1}}\left[{\theta }_{\mathrm{1}}\left(s\right)\left|\mathrm{\xi }\left(s\right)\right|+{\theta }_{\mathrm{2}}\left(s\right)\left|\mathrm{\xi }\left(s-\eta \right)\right|\right]ds.\end{array}

The condition implies {\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}>\mathrm{0} and {\gamma }_{\mathrm{2}}>\mathrm{0}. Applying Lemma 5， we get

\left|\mathrm{\xi }\left(t\right)\right|\le \mathrm{0}, t\in \left[\mathrm{0},T\right].

Hence， \left|\mathrm{\xi }\left(t\right)\right|=\mathrm{0} for any t\in \left[-\eta ,T\right]. Moreover， h_{\mathrm{1}}\left(t\right)\equiv h_{\mathrm{2}}\left(t\right) for t\in \left[-\eta ,T\right]. This completes the proof of Theorem 3. □

In the absence of time delay， Problem （4） reduces to the following form

\begin{array}{c}\left\{\begin{array}{cc}{}_{\mathrm{0}}{}^C{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{2}}}h\left(t\right)-Q{}_{\mathrm{0}}{}^C\mathrm{𝒟}{\mathrm{𝒟}}_t^{{\gamma }_{\mathrm{1}}}h\left(t\right)=f\left(t,h\left(t\right)\right),& t\in \left[\mathrm{0},T\right],\\ h\left(\mathrm{0}\right)=\varphi \left(\mathrm{0}\right),h\mathrm{\text{'}}\left(\mathrm{0}\right)=\varphi \mathrm{\text{'}}\left(\mathrm{0}\right),& \end{array}\right.\end{array}（10）

where .

In this case， （H1）， （H2） and （H4） reduce to the following assumptions.

（H1）' The function f:\left[\mathrm{0},T\right]\times {\mathbb{R}}^n\to {\mathbb{R}}^n is continuous.

（H2）' There exists a non-negative function \delta \in L^{{\rho }_{\mathrm{1}}^{\mathrm{*}}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) such that

\left|f\left(t,\alpha \right)\right|\le \delta \left(t\right), \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left[\mathrm{0},T\right]\mathrm{a}\mathrm{n}\mathrm{d}\alpha \in {\mathbb{R}}^n,

where {\rho }_{\mathrm{1}}^{\mathrm{*}}>\mathrm{1} is a given constant.

（H4）' There exists \theta \in \mathrm{𝒞}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right) such that

\left|f\left(t,{\alpha }_{\mathrm{1}}\right)-f\left(t,{\alpha }_{\mathrm{2}}\right)\right|\le \theta \left(t\right)\left|{\alpha }_{\mathrm{1}}-{\alpha }_{\mathrm{2}}\right|

for all t\in \left[\mathrm{0},T\right] and all {\alpha }_{\mathrm{1}},{\alpha }_{\mathrm{2}}\in {\mathbb{R}}^n.

Corollary 1. Suppose that （H1）'， （H2）' and （H4）' hold. If

（11）

then Problem （10） has a unique solution on \left[\mathrm{0},T\right].

3 Application

Example 1. For Problem （4）， the parameters are given as follows： {\gamma }_{\mathrm{1}}=\mathrm{1}/\mathrm{2}， {\gamma }_{\mathrm{2}}=\mathrm{3}/\mathrm{2}， Q=\mathrm{0.2},T=\mathrm{1}， \eta =\mathrm{0.2}， f(t,x,y)=\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{50}}sinx+\frac{\mathrm{1}}{\mathrm{50}}siny， \varphi (t)=\mathrm{1}+\frac{t}{\mathrm{2}},{\varphi }^{\mathrm{\text{'}}}(t)=\frac{\mathrm{1}}{\mathrm{2}}. The corresponding form is written as：

（12）

Obviously， f is continuous on [\mathrm{0,1}]\times \mathbb{R}\times \mathbb{R} and satisfies the following conditions：

|f(t,x,y)|\le \frac{\mathrm{7}}{\mathrm{50}},

and

|f(t,x_{\mathrm{1}},y_{\mathrm{1}})-f(t,x_{\mathrm{2}},y_{\mathrm{2}})|\le \frac{\mathrm{1}}{\mathrm{50}}|x_{\mathrm{1}}-x_{\mathrm{2}}|+\frac{\mathrm{1}}{\mathrm{50}}|y_{\mathrm{1}}-y_{\mathrm{2}}|.

Let \delta (t)=\mathrm{7}/\mathrm{50}and {\theta }_{\mathrm{1}}(t)={\theta }_{\mathrm{2}}(t)=\mathrm{1}/\mathrm{50}， and let {\rho }_{\mathrm{1}}^{\mathrm{*}}={\rho }_{\mathrm{2}}^{\mathrm{*}}={\rho }_{\mathrm{3}}^{\mathrm{*}}={\rho }_{\mathrm{2}}={\rho }_{\mathrm{3}}=\mathrm{2}. Obviously， \delta \in L^{\mathrm{2}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right)， {\theta }_{\mathrm{1}}\in L^{\mathrm{2}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right)， and {\theta }_{\mathrm{2}}\in L^{\mathrm{2}}\left(\left[\mathrm{0},T\right],{\mathbb{R}}^{+}\right).

（1） With the above parameters， we derive

and

.

According to Theorem 2， Problem （12） has at least one solution on [-\mathrm{0.2,1}].

（2） On the basis of the above parameters， we obtain

By Theorem 1， Problem （12） has a unique solution on [-\mathrm{0.2,1}].

（3） It is easy to see that {\theta }_{\mathrm{1}}(t) and {\theta }_{\mathrm{2}}(t) are continuous and nondecreasing on [\mathrm{0,1}]. Based on the above data， the conditions of Theorem 3 are satisfied. Therefore， by Theorem 3， there exists a unique solution to Problem （12） on [-\mathrm{0.2,1}].

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基金资助

国家自然科学基金资助项目(62476292)

湖北省自然科学基金资助项目(2020CFB637)