1 Introduction

In this paper, we are concerned with the neutral fractional differential equation of the form

$$\begin{aligned}& {}^{C} D^{q}\bigl(x(t)-g(t,x_{t}) \bigr)=Ax(t)+f(t,x_{t}), \quad t\in(0,T], \end{aligned}$$
(1.1)
$$\begin{aligned}& x_{0}(\vartheta)+\bigl(h(x_{t_{1}},x_{t_{2}}, \ldots,x_{t_{m}})\bigr) (\vartheta )=\phi(\vartheta)\in \mathcal{B}_{v}, \quad \vartheta\leq0, \end{aligned}$$
(1.2)

where \({}^{C}D^{q}\) is the Caputo fractional derivative of the order \(0< q\leq1\). \(0< t_{i}<T\), \(i=1,2,\ldots,m\). A is the infinitesimal generator of a \(C_{0}\)-semigroup of bounded linear operators, and densely defined on a Banach space \((X,\|\cdot\|)\). f, g, h, and ϕ are the given functions and satisfy some assumptions. The history \(x_{t}:(-\infty,0]\rightarrow X\), \(x_{t}(\theta)=x(t+\theta)\), \(\theta\in (-\infty,0]\), belongs to an abstract phase space \(\mathcal{B}_{v}\) which will be specified in Section 2. Moreover, the integral equation

$$ x(t)=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac{Ax(s)}{(t-s)^{1-q}}\,ds,\quad t\geq0, $$
(1.3)

has an associated resolvent operator \((\mathcal{S}(t))_{t\geq0}\) on X. By a resolvent operator, we mean the definition as follows.

Definition 1.1

[1, Definition 1.3]

A one parameter family of bounded linear operators \((\mathcal {S}(t))_{t\geq0}\) on X is called a resolvent operator for (1.3) if the following conditions hold:

  1. (a)

    \(\mathcal{S}(\cdot)x\in C([0,\infty),X)\) and \(\mathcal{S}(0)x=x\) for all \(x\in X\),

  2. (b)

    \(\mathcal{S}(t)D(A)\subset D(A)\) and \(A\mathcal{S}(t)x=\mathcal {S}(t)Ax\) for all \(x\in D(A)\) (the domain of A) and every \(t\geq0\),

  3. (c)

    for every \(x\in D(A)\) and \(t\geq0\),

    $$ \mathcal{S}(t)x=x+\frac{1}{\Gamma(q)}\int_{0}^{t} \frac{A\mathcal {S}(s)x}{(t-s)^{1-q}}\,ds. $$

To illustrate that our hypothesis on (1.3) is possible, we give the following example.

Example 1.1

We take \(X=L^{2}([0,\pi]; \mathbb{R})\) and let A be the operator given by \(Ax=x''\) with domain \(D(A):=\{x\in X:x''\in X, x(0)=x(\pi)=0\} \). It is well known that A is the infinitesimal generator of an analytic semigroup \((T(t))_{t\geq0}\) on X. Furthermore, A has discrete spectrum with eigenvalues of the form \(-n^{2}\), \(n\in\mathbb{N}\), and corresponding normalized eigenfunctions given by \(z_{n}(\xi):=\frac{2}{\pi}\sin(n\xi)\). In addition, \(\{z_{n}:n\in\mathbb{N}\}\) is an orthogonal basis for X; \(T(t)x=\sum_{n=1}^{\infty} e^{-n^{2}t}\langle x,z_{n}\rangle z_{n}\) for all \(x\in X\) and every \(t > 0\). From these expressions it follows that \((T(t))_{t\geq0}\) is a uniformly bounded compact semigroup, so that \(R(\lambda,A)=(\lambda-A)^{-1}\) is a compact operator for all \(\lambda\in\rho(A)\).

From [1, Example 2.2.1] we know that the integral equation

$$ x(t)=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac{Ax(s)}{(t-s)^{1-q}}\,ds, \quad t\geq0, $$

has an associated analytic resolvent operator \((\mathcal{S}(t))_{t\geq 0}\) on X. It is given by

$$\begin{aligned} \mathcal{S}(t)= \left \{ \begin{array}{@{}l@{\quad}l} \frac{1}{2\pi i}\int_{\Gamma_{r,\theta }}e^{\lambda t}(\lambda^{q}-A)^{-1}\,d\lambda, & t>0;\\ I, & t=0, \end{array} \right . \end{aligned}$$

where \(\Gamma_{r,\theta}\) denotes a contour consisting of the rays \(\{ re^{i\theta}:r\geq0\}\) and \(\{re^{-i\theta}:r\geq0\}\) for some \(\theta \in(\frac{\pi}{2},\pi)\). Obviously, there exists a constant \(\mathbb {S}_{A}\) such that \(\| [\mathcal{S}'(t)-\mathcal{S}'(s) ]x\|\leq \mathbb{S}_{A}|t-s|\|x\|_{[D(A)]}\) for all \(t,s\geq0\), where \(\|\cdot\| _{[D(A)]}\) denote the graph norm.

Recently, fractional differential equations have been investigated extensively. The motivation for those works arises from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, chemistry, aerodynamics, electrodynamics of the complex medium, and so on. For more details and examples see [1–5].

Especially, there are many papers treating the problem of the existence of a mild solution for abstract semilinear fractional differential equations; see [6–13]. But, as pointed out by Hernández et al. in [14], some concepts of mild solution are not realistic; see, for examples, [6, 9, 10]. Furthermore, in [14], the authors utilized an approach based on the well-developed theory of resolvent operators for integral equations to deal with the following abstract fractional equations

$$\begin{aligned}& D^{\alpha}\bigl(x(t)+g\bigl(t,x(t)\bigr)\bigr)=Ax(t)+f \bigl(t,x(t)\bigr), \quad t\in(0,a], \\& x(0)=x_{0}, \end{aligned}$$

where \(D^{\alpha}\) is the Riemann-Liouville fractional derivative of the order \(0<\alpha<1\). For the resolvent operators, we refer the interested reader to [1, 14–19] and the references therein for more details.

Nonlocal conditions were initiated by Byszewski [20] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski and Lakshmikantham [21], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena. There are many papers concerned with the nonlocal conditions; see [9, 12, 13, 20, 21] and the references therein, for examples.

In addition, Zhou and Jiao in [13] discuss a class of fractional neutral evolution equations with nonlocal conditions

$$\begin{aligned}& {}^{C}D^{q}\bigl[x(t)-h(t,x_{t}) \bigr]+Ax(t)=f(t,x_{t}), \quad t\in(0,a], \\& x_{0}(\vartheta)+\bigl(g(x_{t_{1}},x_{t_{2}}, \ldots,x_{t_{m}})\bigr) (\vartheta )=\varphi(\vartheta), \quad\vartheta \in[-r,0], \end{aligned}$$

where \(^{C} D^{q}\) is the Caputo fractional derivative of the order \(0< q\leq1\). \(0< t_{i}<a\), \(i=1,2,\ldots,m\). −A is the infinitesimal generator of an analytic semigroup on a Banach space E. By considering an integral equation which is given in terms of probability density and semigroup, they establish criteria on the existence and uniqueness of mild solutions.

In this paper, we used the theory of resolvent operators coupled with fixed point theorem, to obtain some new results about the global uniqueness and existence of the problem (1.1)-(1.2) in Section 3. Moreover, in Section 4, the continuous dependence of the mild solution is also investigated.

2 Preliminaries

In this section, we shall introduce some basic definitions, notations, and lemmas which are used throughout this paper.

Definition 2.1

The fractional (arbitrary) order integral of the function \(v(t)\in L^{1}([0,\infty],\mathbb{R})\) of \(\mu\in\mathbb{R}^{+}\) is defined by

$$I^{\mu}v(t)=\frac{1}{\Gamma(\mu)}\int_{0}^{t}(t-s)^{\mu-1}v(s)\,ds, \quad t>0, $$

where Γ is the Euler gamma function.

Definition 2.2

The Caputo fractional derivative of order \(\mu>0\) for a function \(v(t)\) given in the interval \([0,\infty)\) is defined by

$$^{C} D^{\mu}v(t)=\frac{1}{\Gamma(n-\mu)}\int_{a}^{t}(t-s)^{n-\mu-1}v^{(n)}(s)\,ds $$

provided that the right-hand side is point-wise defined. Here \(n = [\mu ] + 1\) and \([\mu]\) means the integral part of the number μ, and Γ is the Euler gamma function.

More details on fractional derivatives and their properties can be found in [2, 3, 5].

Now, let \((Z;\|\cdot\|_{Z})\) and \((W;\|\cdot\|_{W})\) be Banach spaces. We denote by \(\mathcal{L}(Z;W)\) the space of bounded linear operators from Z into W endowed with the operator norm denoted by \(\|\cdot\| _{\mathcal{L}(Z;W)}\), and we can write simply \(\mathcal{L}(Z)\) and \(\| \cdot\|_{\mathcal{L}(Z)}\) when \(Z=W\). The notation \([D(A)]\) stands for the domain of A endowed with the graph norm \(\|x\|_{[D(A)]}=\|x\|+\| Ax\|\). In addition, \(B_{r}(x; Z)\) represents the closed ball with center at x and radius r in Z.

In the sequel, we always assume that the resolvent operator \((\mathcal {S}(t))_{t\geq0}\) of (1.3) is analytic and compact; see [1, Chapter 2] for details. In addition, \(\|\mathcal {S}'(t)x\|\leq\varphi_{A}(t)\|x\|_{[D(A)]}\) for all \(t\geq0\) and \(\| [\mathcal{S}'(t)-\mathcal{S}'(s) ]x\|\leq\mathbb{S}_{A}|t-s|\|x\| _{[D(A)]}\) for all \(t,s\geq0\), where \(\varphi_{A}\) is a function in \(L^{1}_{\mathrm{loc}}([0,\infty); \mathbb{R}^{+})\) and \(\mathbb{S}_{A}\) is a constant.

Consider the abstract integral equation

$$\begin{aligned} x(t)=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac {Ax(s)}{(t-s)^{1-q}}\,ds+w(t), \quad t\in[0,T], \end{aligned}$$
(2.1)

where \(w\in C([0,T]; X)\). From [1, Definition 1.1.1], we note the following concept of a mild solution.

Definition 2.3

A function \(x\in C([0,T]; X)\) is called a mild solution of the integral equation (2.1) on \([0,T]\) provided that \(\int_{0}^{t}(t-s)^{q-1}x(s)\,ds\in D(A)\) for all \(t\in [0,T]\) and

$$\begin{aligned} x(t)=\frac{1}{\Gamma(q)}A\int_{0}^{t} \frac {x(s)}{(t-s)^{1-q}}\,ds+w(t), \quad t\in[0,T]. \end{aligned}$$
(2.2)

The next result plays an important part in our development, which follows from [1, Proposition I.1.2, Theorem II.2.4, Corollary II.2.6, Proposition I.1.3], and also can be found in [14, Lemma 1.1].

Lemma 2.1

Assume that the resolvent operator \((\mathcal{S}(t))_{t\geq0}\) of (1.3) is analytic and compact, and \(w\in C([0,T];D(A))\), then the function \(x:[0,T]\rightarrow X\) defined by

$$\begin{aligned} x(t)=\int_{0}^{t}\mathcal{S}'(t-s)w(s)\,ds+w(t), \quad t\in[0,T], \end{aligned}$$

is a mild solution of (2.1).

Lemma 2.2

[14, Lemma 2.2]

Assume that \(\mathcal{S}(t)\) is compact for all \(t>0\). Then \(\mathcal {S}'(t)\) is compact for all \(t>0\) and the inclusion map \(i_{c}:[D(A)]\rightarrow X\) is compact.

Next, we present the abstract phase space \(\mathcal{B}_{v}\), which has been used in [22]. Assume that \(v:(-\infty,0]\rightarrow (0,+\infty)\) is a continuous function with \(l=\int_{-\infty }^{0}v(s)\,ds<+\infty\). For any \(a>0\), we define

$$\mathcal{B}=\bigl\{ \xi:[-a,0]\rightarrow X \mbox{ such that } \xi(t) \mbox{ is bounded and measurable}\bigr\} , $$

and provide the space ℬ with the norm \(\|\xi\|_{[-a,0]}=\sup_{t\in[-a,0]}|\xi(t)|\). Then we define

$$\begin{aligned} \mathcal{B}_{v}={}&\biggl\{ \xi:[-\infty,0]\rightarrow X \mbox{ such that, for any } c>0,\xi|_{[-c,0]}\in\mathcal{B} \\ &{}\mbox{and } \int_{-\infty}^{0}v(s)\|\xi \|_{[s,0]}\,ds<+\infty\biggr\} . \end{aligned}$$

Obviously, the phase space \(\mathcal{B}_{v}\) endowed with the norm \(\| \xi\|_{\mathcal{B}_{v}}=\int_{-\infty}^{0}v(s)\|\xi\|_{[s,0]}\,ds\) is a Banach space and \(|\xi(0)|\leq l^{-1}\|\xi\|_{\mathcal{B}_{v}}\). Now we consider the space

$$\begin{aligned} \mathcal{B}_{vT}=\bigl\{ \xi:[-\infty,T]\rightarrow X \mbox{ such that}, \xi |_{[0,T]}\in C\bigl([0,T],X\bigr)\cap D(A) \mbox{ and } \xi_{0}\in\mathcal{B}_{v}\bigr\} , \end{aligned}$$

with the seminorm \(\|\cdot\|_{\ast}\) defined by

$$\begin{aligned} \|x\|_{\ast}=&\|x_{0}\|_{\mathcal{B}_{v}}+l\|x\|_{[0,T]},\quad x\in\mathcal{B}_{vT}. \end{aligned}$$

Lemma 2.3

[22]

Assume \(x\in\mathcal{B}_{vT}\), then for \(0\leq t\leq T\), \(x_{t}\in \mathcal{B}_{v}\). Moreover,

$$\begin{aligned} l\bigl|x(t)\bigr|\leq\|x_{t}\|_{\mathcal{B}_{v}}\leq\|x_{0} \|_{\mathcal {B}_{v}}+l\|x\|_{[0,T]}. \end{aligned}$$

We will now introduce the concept of a mild solution of (1.1)-(1.2). To this end, note that if \(x\in\mathcal{B}_{vT}\) is a solution of (1.1)-(1.2), we can expect that

$$ \left \{ \begin{array}{@{}l} x(t)=\phi(0)-(h(x_{t_{1}},x_{t_{2}},\ldots ,x_{t_{m}}))(0)-g(0,x_{0})+g(t,x_{t})\\ \hphantom{x(t)=}{}+\frac{1}{\Gamma(q)}\int_{0}^{t}\frac{f(s,x_{s})}{(t-s)^{1-q}}\,ds+\frac {1}{\Gamma(q)}\int_{0}^{t}\frac{Ax(s)}{(t-s)^{1-q}}\,ds,\quad 0< t\leq T;\\ x_{0}(\vartheta)+(h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}}))(\vartheta )=\phi(\vartheta), \quad \vartheta\leq0. \end{array} \right . $$

Furthermore, in view of \(x_{0}(t)=x(t)\), \(t\leq0\), we can give the following definition.

Definition 2.4

A function \(x\in\mathcal{B}_{vT}\) is called a mild solution of (1.1)-(1.2) provided that \(\int_{0}^{t}(t-s)^{q-1}x(s)\,ds\in D(A)\) for all \(t\in [0,T]\) and

$$\begin{aligned} x(t)=\left \{ \begin{array}{@{}l@{\quad}l} \phi(0)-(h(x_{t_{1}},x_{t_{2}},\ldots ,x_{t_{m}}))(0)-g(0,x_{0})+g(t,x_{t})\\ \quad{}+\frac{1}{\Gamma(q)}\int_{0}^{t}\frac{f(s,x_{s})}{(t-s)^{1-q}}\,ds+\frac {1}{\Gamma(q)}A\int_{0}^{t}\frac{x(s)}{(t-s)^{1-q}}\,ds,& 0< t\leq T;\\ \phi(t)-(h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}}))(t),& t\leq0. \end{array} \right . \end{aligned}$$
(2.3)

3 Existence and uniqueness of mild solutions

To simplify our development, in the rest of this work, for a function \(x\in\mathcal{B}_{vT}\) we use the notations \(G_{x}, F_{x}:[0,T] \rightarrow X\) given by

$$\begin{aligned} G_{x}(t)=\phi(0)-\bigl(h(x_{t_{1}},x_{t_{2}},\ldots ,x_{t_{m}})\bigr) (0)-g(0,x_{0})+g(t,x_{t}) \end{aligned}$$

and

$$\begin{aligned} F_{x}(t)=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac{f(s,x_{s})}{(t-s)^{1-q}}\,ds. \end{aligned}$$

By Lemma 2.1, one sees that

$$\begin{aligned} x(t)=\left \{ \begin{array}{@{}l@{\quad}l} \int_{0}^{t}\mathcal {S}'(t-s)[G_{x}(s)+F_{x}(s)]\,ds+G_{x}(t)+F_{x}(t),& 0< t\leq T;\\ \phi(t)-(h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}}))(t),& t\leq0, \end{array} \right . \end{aligned}$$
(3.1)

is a mild solution of (1.1)-(1.2).

From now on, we denote the set \(\{x:x\in\mathcal{B}_{vT},\|x\|_{\ast }\leq r\}\) by \(B_{r}\). Then \(B_{r}\), for each \(r>0\), is a bounded, closed, nonempty, and convex subset in X.

Now, we introduce the map \(\Psi:\mathcal{B}_{vT}\rightarrow\mathcal {B}_{vT}\) by

$$\begin{aligned} \Psi x(t)=\left \{ \begin{array}{@{}l@{\quad}l} \int_{0}^{t}\mathcal {S}'(t-s)[G_{x}(s)+F_{x}(s)]\,ds+G_{x}(t)+F_{x}(t), & 0< t\leq T;\\ \phi(t)-(h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}}))(t), & t\leq0. \end{array} \right . \end{aligned}$$

By Lemma 2.1 and the argument above, it is easy to see that the operator Ψ having a fixed point \(x_{*}\in\mathcal {B}_{vT}\) is equivalent to the problem (1.1)-(1.2) having a mild solution \(x_{*}(t)\), \(-\infty< t\leq T\).

To prove our results, we introduce the following conditions.

(H1) \(f,g\in C([0,T]\times\mathcal{B}_{v},X)\).

(H2) There exists a constant \(L_{f}\), such that, for all \(x,y\in\mathcal {B}_{v}\),

$$ \bigl\| f(t,x)-f(t,y)\bigr\| _{[D(A)]}\leq L_{f}\|x-y\|_{\mathcal{B}_{v}}, \quad t\in [0,T]. $$
(3.2)

(H3) There exist constants \(L_{f}\), \(L_{g}\), \(L_{h}\), such that, for all \(x,y\in\mathcal{B}_{v}\),

$$ \begin{aligned} &\bigl\| g(t,x)-g(t,y)\bigr\| _{[D(A)]}\leq L_{g}\|x-y\|_{\mathcal{B}_{v}}, \\ &\bigl\| h(y_{t_{1}},y_{t_{2}},\ldots,y_{t_{m}})-h(x_{t_{1}},x_{t_{2}}, \ldots ,x_{t_{m}})\bigr\| _{[D(A)]}\leq L_{h} \|x_{t}-y_{t}\|_{\mathcal{B}_{v}}. \end{aligned} $$
(3.3)

(H4) \(l(L_{h}+2L_{g})(\|\varphi_{A}\|_{L^{1}}+1)+l L_{h}<1\).

(H5)

$$\begin{aligned}& \lim_{\|y_{t}\|_{\mathcal{B}_{v}}\rightarrow\infty}\sup_{t\in[0,T]}\frac{\|f(t,y_{t})\|_{[D(A)]}}{\|y_{t}\|_{\mathcal {B}_{v}}}= \lambda_{3}, \qquad \lim_{\|y_{t}\|_{\mathcal{B}_{v}}\rightarrow\infty}\sup_{t\in[0,T]}\frac{\|g(t,y_{t})\|_{[D(A)]}}{\|y_{t}\|_{\mathcal {B}_{v}}}= \lambda_{4}. \end{aligned}$$

First, we present a uniqueness result.

Theorem 3.1

Suppose that (H1)-(H3) hold, then system (1.1)-(1.2) has a unique mild solution, provided that

$$ \wedge=l \biggl[L_{h}+2L_{g}+\frac{T^{q}L_{f}}{q\Gamma(q)} \biggr]\bigl(\| \varphi _{A}\|_{L^{1}}+1\bigr)+lL_{h}<1. $$

Proof

For \(y\in\mathcal{B}_{vT}\), From the assumption on f, g we see that

$$\begin{aligned} &\int_{0}^{t}\bigl\| \mathcal{S}'(t-s) \bigl[G_{y}(s)+F_{y}(s)\bigr]\bigr\| \,ds \\ &\quad\leq \int_{0}^{t}\varphi_{A}(t-s) \bigl\| G_{y}(s)+F_{y}(s)\bigr\| _{[D(A)]}\,ds \\ &\quad\leq \bigl\| G_{y}(s)+F_{y}(s)\bigr\| _{C([0,T],[D(A)])}\int _{0}^{t}\varphi _{A}(t-s)\,ds \\ &\quad\leq \bigl(\bigl\| G_{y}(s)\bigr\| _{C([0,T],[D(A)])}+\bigl\| F_{y}(s)\bigr\| _{C([0,T],[D(A)])}\bigr)\int_{0}^{t} \varphi_{A}(t-s)\,ds \\ &\quad\leq \biggl(\bigl\| G_{y}(s)\bigr\| _{C([0,T],[D(A)])}+\|f\| _{C([0,T],[D(A)])} \frac{T^{q}}{q\Gamma(q)}\biggr)\|\varphi_{A}\|_{L^{1}}, \end{aligned}$$

which implies that the function \(s\rightarrow\mathcal {S}'(t-s)[G_{y}(s)+F_{y}(s)]\) is integrable on \([0,t]\) for all \(t\in [0,T]\), since \(f,g\in C([0,T]\times\mathcal{B}_{v},X)\). Then Ψ is well defined.

From the definition of the \(G_{y}\), \(F_{y}\), we can see that, for \(x,y\in \mathcal{B}_{vT}\),

$$\begin{aligned} \|G_{x}-G_{y}\|_{[D(A)]} \leq& \bigl\| \bigl(h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}})\bigr) (0) \\ &{} -\bigl(h(y_{t_{1}},y_{t_{2}},\ldots,y_{t_{m}})\bigr) (0)\bigr\| _{[D(A)]} \\ &{} +\bigl\| g(0,x_{0})-g(0,y_{0})\bigr\| _{[D(A)]} \\ &{} +\bigl\| g(t,x_{t})-g(t,y_{t})\bigr\| _{[D(A)]} \\ \leq& L_{h}\|x_{t}-y_{t}\|_{\mathcal{B}_{v}}+2L_{g} \| x_{t}-y_{t}\|_{\mathcal{B}_{v}} \\ \leq& (L_{h}+2L_{g})\|x_{t}-y_{t} \|_{\mathcal{B}_{v}} \end{aligned}$$
(3.4)

and

$$\begin{aligned} \|F_{x}-F_{y}\|_{[D(A)]} \leq& \frac{T^{q}\| f(s,x_{s})-f(s,y_{s})\|_{[D(A)]}}{q\Gamma(q)} \\ \leq& \frac{T^{q}L_{f}}{q\Gamma(q)}\|x_{t}-y_{t}\|_{\mathcal{B}_{v}}. \end{aligned}$$
(3.5)

Then, for \(x,y\in\mathcal{B}_{vT}\), and \(t\in[0,T]\), we have

$$\begin{aligned} l\bigl\| \Psi x(t)-\Psi y(t)\bigr\| \leq{}&l\biggl\{ \int_{0}^{t} \bigl\| \mathcal {S}'(t-s)\bigl[F_{x}(s)-F_{y}(s) \bigr]\bigr\| \,ds \\ &{}+\int_{0}^{t}\bigl\| \mathcal{S}'(t-s) \bigl[G_{x}(s)-G_{y}(s)\bigr]\bigr\| \,ds \\ &{}+\|G_{x}-G_{y}\|_{[D(A)]}+\|F_{x}-F_{y} \|_{[D(A)]}\biggr\} \\ \leq{}&l\biggl\{ \int_{0}^{t}\varphi_{A}(t-s) \bigl\| F_{x}(s)-F_{y}(s)\bigr\| _{[D(A)]}\,ds \\ &{}+\int_{0}^{t}\varphi_{A}(t-s) \bigl\| G_{x}(s)-G_{y}(s)\bigr\| _{[D(A)]}\,ds \\ &{}+\|G_{x}-G_{y}\|_{[D(A)]}+\|F_{x}-F_{y} \|_{[D(A)]}\biggr\} \\ \leq{}&l\biggl\{ \biggl[L_{h}+2L_{g}+\frac{T^{q}L_{f}}{q\Gamma(q)} \biggr]\bigl(\|\varphi _{A}\|_{L^{1}}+1\bigr)\biggr\} \|x_{t}-y_{t} \|_{\mathcal{B}_{v}}. \end{aligned}$$

On the other hand, for \(x,y\in\mathcal{B}_{vT}\), and \(t<0\), we have

$$\begin{aligned} \bigl\| (\Psi x)_{0}-(\Psi y)_{0} \bigr\| _{\mathcal{B}_{v}} \leq& \int_{-\infty}^{0}v(s) \bigl\| h(y_{t_{1}},y_{t_{2}},\ldots,y_{t_{m}}) -h(x_{t_{1}},x_{t_{2}},\ldots,x_{t_{m}})\bigr\| _{[D(A)]}\,ds \\ \leq& l\bigl\| h(y_{t_{1}},y_{t_{2}},\ldots ,y_{t_{m}})-h(x_{t_{1}},x_{t_{2}}, \ldots,x_{t_{m}})\bigr\| _{[D(A)]} \\ \leq& lL_{h}\|x_{t}-y_{t}\|_{\mathcal{B}_{v}}. \end{aligned}$$
(3.6)

Hence,

$$\begin{aligned} \|\Psi x-\Psi y\|_{\ast}\leq&\wedge\|x-y\|_{\ast}, \end{aligned}$$

which implies that Ψ is a contraction and there exists a unique fixed point \(x_{*}\in\mathcal{B}_{vT}\) of Ψ. \(x_{*}(t)\), \(-\infty < t\leq T\) is a mild solution of system (1.1)-(1.2). The proof is complete. □

Next, we give some generally existence results. For this purpose, we present a fixed point theorem due to Krasnoselskii which can be found in [23, p.370] or Smart [24, p.31] and also can be found in [25, Theorem K].

Theorem 3.2

Let M be a closed, convex, and nonempty subset of a Banach space \((X, \|\cdot\|)\). Suppose that A and B map M into X such that

  1. (i)

    \(x,y\in M\Rightarrow Ax+By\in M\),

  2. (ii)

    A is compact and continuous,

  3. (iii)

    B is a contraction mapping.

Then \(\exists y\in M\) with \(y=Ay+By\).

In order to apply the fixed point theorem above, we introduce the decomposition \(\Psi=\Psi_{1}+\Psi_{2}+\Psi_{3}\), where

$$\begin{aligned}& \Psi_{1} y(t)=\left \{ \begin{array}{@{}l@{\quad}l} F_{y}(t),& 0\leq t\leq T;\\ 0, & t\leq0, \end{array} \right . \\& \Psi_{2} y(t)= \left \{ \begin{array}{@{}l@{\quad}l} \int_{0}^{t}\mathcal {S}'(t-s)F_{y}(s)\,ds,& 0\leq t\leq T;\\ 0, & t\leq0, \end{array} \right . \end{aligned}$$

and

$$\begin{aligned} \Psi_{3} y(t)=\left \{ \begin{array}{@{}l@{\quad}l} \int_{0}^{t}\mathcal {S}'(t-s)G_{y}(s)\,ds+G_{y}(t),& 0\leq t\leq T;\\ \phi(t)-(h(y_{t_{1}},y_{t_{2}},\ldots,y_{t_{m}}))(t), & t\leq0. \end{array} \right . \end{aligned}$$

Then we prove the following lemma.

Lemma 3.1

Suppose that (H1), (H3), and (H4) hold, then the operator \(\Psi_{1}\), \(\Psi_{2}\) are completely continuous and \(\Psi_{3}\) is a contraction mapping on \(B_{r}\), \(r>0\).

Proof

We divide this proof into three parts.

Part 1. \(\Psi_{1}\) is completely continuous.

Step 1: \(\Psi_{1}\) is continuous.

Let \(\{y^{n}\}\) be a sequence in \(\mathcal{B}_{vT}\) such that \(y^{n}\rightarrow y\) as \(n\rightarrow\infty\). Then for \(0\leq t\leq T\) we have

$$\begin{aligned} \bigl\| \Psi_{1} y^{n}(t)-\Psi_{1} y(t) \bigr\| \leq &\biggl\| \frac {1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1} \bigl[f\bigl(s,y^{n}_{s}\bigr)-f(s,y_{s}) \bigr]\,ds\biggr\| \\ \leq &\frac{1}{\Gamma(q)}\int_{0}^{t}(t-s)^{q-1} \bigl\| f\bigl(s,y^{n}_{s}\bigr)-f(s,y_{s})\bigr\| \,ds \\ \leq &\frac{T^{q}}{q\Gamma(q)}\bigl\| f\bigl(t,y^{n}_{t} \bigr)-f(t,y_{t})\bigr\| _{[0,T]}, \end{aligned}$$
(3.7)

which proves that \(\Psi_{1} y^{n}\rightarrow\Psi_{1} y\) as \(n\rightarrow \infty\) and \(\Psi_{1}\) is continuous, by the continuity of f.

Step 2: \(\Psi_{1}\) is a compact operator.

Let \(0<\epsilon<t<T\). From the mean value theorem for the Bochner integral (see [26, Lemma 2.1.3]), for \(y\in\mathcal{B}_{vT}\) we see that

$$\begin{aligned} \Psi_{1} y(t) =&\frac{1}{\Gamma(q)}\int _{0}^{\epsilon }\frac{f(s,y_{s})}{(t-s)^{1-q}}\,ds+\frac{1}{\Gamma(q)}\int _{\epsilon }^{t}\frac{f(s,y_{s})}{(t-s)^{1-q}}\,ds \\ \in & B_{\frac{M_{f}\epsilon^{q}}{q\Gamma(q)}}(\theta;X) \\ &{}+(t-\epsilon)\overline{co\biggl(\biggl\{ \frac {f(s,y_{s})}{(t-s)^{1-q}}:s\in[ \epsilon,t],y\in B_{r}\biggr\} \biggr)}, \end{aligned}$$
(3.8)

where \(M_{f}=\sup\{f(t,y):t\in[0,T],y\in B_{r}\}\) and \(co(S)\) denotes the convex hull of a set S. Since the map \(i_{c}\) is compact and f is continuous, we know that Diam of the first part of the right-hand side of (3.8) tends to 0 as \(\epsilon\rightarrow0\) and the second part is compact. This proves that the set \(\Psi_{1}B_{r}(t)\) is relatively compact in X for all \(t\in[0,T]\).

On the other hand, for \(0\leq t_{1}< t_{2}\leq T\), we have

$$\begin{aligned} &\bigl\| \Psi_{1} y(t_{2})-\Psi_{1} y(t_{1})\bigr\| \\ &\quad\leq \frac{1}{\Gamma(q)}\biggl\| \int_{0}^{t_{2}}(t_{2}-s)^{q-1}f(s,y_{s})\,ds -\int_{0}^{t_{1}}(t_{1}-s)^{q-1}f(s,y_{s})\,ds \biggr\| \\ &\quad\leq\frac{1}{\Gamma(q)}\biggl\{ \biggl\| \int_{t_{1}}^{t_{2}}(t_{2}-s)^{q-1}f(s,y_{s})\,ds \biggr\| \\ &\qquad{}+\int_{0}^{t_{1}}\bigl[(t_{2}-s)^{q-1}-(t_{1}-s)^{q-1} \bigr]\,ds\bigl\| f(t,y_{t})\bigr\| _{[0,T]}\biggr\} \\ &\quad\leq\frac{1}{q\Gamma(q)}\bigl(t_{2}^{q}-t_{1}^{q} \bigr)\bigl\| f(t,y_{t})\bigr\| _{[0,T]} \end{aligned}$$
(3.9)

and \(\|f(t,y_{t})\|_{[0,T]}<+\infty\) when \(y\in B_{r}\) since \(f\in C([0,T]\times\mathcal{B}_{v},X)\). Hence, (3.9) implies the set of functions \(\Psi_{1}B_{r}\) is equicontinuous. This proves that \(\Psi_{1}\) is a compact operator.

Part 2. \(\Psi_{2}\) is completely continuous.

It is easy to show that \(\Psi_{2}\) is continuous. We will prove that the set \(\Psi_{2}B_{r}(t)\) is relatively compact in X for all \(t\in [0,T]\) and the set of functions \(\Psi_{2}B_{r}\) is equicontinuous on \([0,T]\).

Firstly, let \(0< t\leq T\) and \(\epsilon<\min\{t,1\}\). Since \(\mathcal {S}'(t)\in C((0,T];\mathcal{L}(X))\), there are numbers \(0=s_{0}< s_{1}<s_{2}<\cdots<s_{n}<s_{n+1}=t\) such that \(|s_{i}-s_{i+1}|<\epsilon\) for all \(i=0,1,2,\ldots,n\), and \(\|\mathcal {S}'(s)-\mathcal{S}'(s_{j})\|_{\mathcal{L}(X)}<\epsilon\) for each \(s\in [s_{j},s_{j+1}]\) and every \(j=0,1,2,\ldots,n\). Under these conditions, for \(y\in B_{r}\) we get

$$\begin{aligned} \Psi_{2} y(t) = &\int_{0}^{s_{1}} \mathcal {S}'(s)F_{y}(t-s)\,ds+\sum _{i=1}^{n}\int_{s_{i}}^{s_{i+1}} \bigl[\mathcal {S}'(s)-\mathcal{S}'(s_{i}) \bigr]F_{y}(t-s)\,ds \\ &{}+\sum_{i=1}^{n}\mathcal{S}'(s_{i}) \int_{s_{i}}^{s_{i+1}}F_{y}(t-s)\,ds. \end{aligned}$$
(3.10)

Note now that

$$\begin{aligned}& \biggl\| \int_{0}^{s_{1}}\mathcal{S}'(s)F_{y}(t-s)\,ds \biggr\| \leq\|\varphi_{A}\| _{L^{1}([0,\epsilon],\mathbb{R}^{+})}\frac{M_{f}\epsilon^{q}}{q\Gamma (q)}, \\& \Biggl\| \sum_{i=1}^{n}\int_{s_{i}}^{s_{i+1}} \bigl[\mathcal{S}'(s)-\mathcal {S}'(s_{i}) \bigr]F_{y}(t-s)\,ds\Biggr\| \leq\epsilon\frac{M_{f}T^{q+1}}{q(q+1)\Gamma (q)}, \end{aligned}$$

where \(M_{f}\) is defined as in Part 1.

From Lemma 2.2 and similarly to the proof of Step 1 in Part 1, we infer that the set \(\Psi_{2}B_{r}(t)\) is relatively compact in X for all \(t\in[0,T]\).

Secondly, let \(0\leq\tau_{1}<\tau_{2}\leq T\), we have

$$\begin{aligned} &\bigl\| \Psi_{2} y(\tau_{2})-\Psi_{2}y( \tau_{1})\bigr\| \\ &\quad\leq\biggl\| \int_{0}^{\tau_{2}}\mathcal{S}'( \tau _{2}-s)F_{y}(s)\,ds-\int_{0}^{\tau_{1}} \mathcal{S}'(\tau _{1}-s)F_{y}(s)\,ds\biggr\| \\ &\quad\leq \biggl\| \int_{\tau_{1}}^{\tau_{2}}\mathcal{S}'( \tau _{2}-s)F_{y}(s)\,ds\biggr\| +\biggl\| \int_{0}^{\tau_{1}}\mathcal{S}'( \tau_{2}-s)F_{y}(s)-\mathcal {S}'( \tau_{1}-s)F_{y}(s)\,ds\biggr\| \\ &\quad\leq\biggl\| \int_{\tau_{1}}^{\tau_{2}}\mathcal{S}'( \tau _{2}-s)F_{y}(s)\,ds\biggr\| +\mathbb{S}_{A}| \tau_{2}-\tau_{1}|^{2}\|F_{y}\| _{[D(A)]} \\ &\quad\leq|\tau_{2}-\tau_{1}|\|\varphi_{A} \|_{L^{1}}\|F_{y}\| _{[D(A)]}+\mathbb{S}_{A}| \tau_{2}-\tau_{1}|^{2}\|F_{y} \|_{[D(A)]}. \end{aligned}$$
(3.11)

The equicontinuity of \(\Psi_{2}B_{r}\) is proved. Hence, \(\Psi_{2}\) is completely continuous.

Part 3. \(\Psi_{3}\) is a contraction on \(B_{r}\).

Proceeding as in the proof of Theorem 3.1, we can show that, for \(x,y\in\mathcal{B}_{vT}\),

$$\begin{aligned} &\|\Psi_{3} x-\Psi_{3} y\|_{\ast} \\ &\quad\leq l \bigl[(L_{h}+2L_{g}) \bigl(\|\varphi_{A} \|_{L^{1}}+1\bigr)+L_{h}\bigr]\| x-y\|_{\ast}. \end{aligned}$$

This proves \(\Psi_{3}\) is a contraction on \(B_{r}\), because of (H4). □

Theorem 3.3

Suppose that (H1), (H3)-(H5) hold, then system (1.1)-(1.2) has at least a mild solution, provided that

$$l\biggl(L_{h}+2\lambda_{4}+\frac{T^{q}\lambda_{3}}{q\Gamma(q)}\biggr) \bigl(1+\| \varphi _{A}\|_{L^{1}}\bigr)+lL_{h}<1. $$

Proof

Obviously, \(x(t)\) is a mild solution of (1.1)-(1.2) if and only if the operator equation \(\Psi x=\Psi_{1}x+\Psi_{2}x+\Psi _{3}x\) has a solution \(x\in B_{r}\).

First, we prove that there exists a sufficiently large constant R, such that \(\Psi_{1}x+\Psi_{2}x+\Psi_{3}y\in B_{R}\), for \(\forall x,y\in B_{R}\).

From the condition (H5), we know that, for a sufficiently small ϵ, such that

$$M_{\epsilon}=l\biggl[\biggl(L_{h}+2\epsilon+2 \lambda_{4}+\frac{T^{q}(\epsilon +\lambda_{3})}{q\Gamma(q)}\biggr) \bigl(1+\|\varphi_{A} \|_{L^{1}}\bigr)+L_{h}\biggr]<1, $$

there exists a constant \(R_{0}>0\), for \(y_{t}\in\mathcal{B}_{v}\), \(\|y_{t}\|_{\mathcal{B}_{v}}>R_{0}\), and we have

$$\begin{aligned} &\bigl\| f(t,y_{t})\bigr\| _{[D(A)]}\leq(\epsilon+\lambda_{3}) \|y_{t}\|_{\mathcal {B}_{v}}, \\ &\bigl\| g(t,y_{t})\bigr\| _{[D(A)]}\leq(\epsilon+\lambda_{4}) \|y_{t}\|_{\mathcal {B}_{v}}. \end{aligned}$$

On the other hand, by the continuity of f, g, we know

$$\begin{aligned} &\bigl\| f(t,y_{t})\bigr\| _{[D(A)]}\leq M_{f}, \\ &\bigl\| g(t,y_{t})\bigr\| _{[D(A)]}\leq M_{g},\qquad \|y_{t}\|_{\mathcal{B}_{v}}\leq R_{0}, \end{aligned}$$

where \(M_{f}\), \(M_{g}\) are constants. Hence,

$$\begin{aligned} &\bigl\| f(t,y_{t})\bigr\| _{[D(A)]}\leq M_{f}+(\epsilon+ \lambda_{3})\|y_{t}\| _{\mathcal{B}_{v}}, \\ &\bigl\| g(t,y_{t})\bigr\| _{[D(A)]}\leq M_{g}+(\epsilon+ \lambda_{4})\|y_{t}\| _{\mathcal{B}_{v}}, \quad\mbox{for all } y_{t}\in\mathcal{B}_{v}. \end{aligned}$$

Define the function \(\upsilon\in C((-\infty,T],X)\) such that \(\|\upsilon \|_{\ast}=0\). Immediately, for all \(y\in\mathcal{B}_{vT}\), by Lemma 2.3, we obtain

$$\begin{aligned} \|G_{y}\|_{[D(A)]}\leq{}&\bigl\| \phi(0)-\bigl(h(y_{t_{1}},y_{t_{2}}, \ldots ,y_{t_{m}})\bigr) (0)\bigr\| _{[D(A)]} \\ &{}+\bigl\| g(0,y_{0})\bigr\| _{[D(A)]}+\bigl\| g(t,y_{t}) \bigr\| _{[D(A)]} \\ \leq{}&L_{h}\|y_{t}\|_{\mathcal{B}_{v}}+2(\epsilon+ \lambda_{4})\|y_{t}\| _{\mathcal{B}_{v}} \\ &{}+\bigl\| \phi(0)\bigr\| _{[D(A)]}+\bigl\| \bigl(h(\upsilon_{t_{1}}, \upsilon_{t_{2}},\ldots ,\upsilon_{t_{m}})\bigr) (0) \bigr\| _{[D(A)]}+2M_{g} \\ \leq{}&(L_{h}+2\epsilon+2\lambda_{4})\|y_{t} \|_{\mathcal{B}_{v}} \\ &{}+\bigl\| \phi(0)\bigr\| _{[D(A)]}+\bigl\| \bigl(h(\upsilon_{t_{1}}, \upsilon_{t_{2}},\ldots ,\upsilon_{t_{m}})\bigr) (0) \bigr\| _{[D(A)]}+2M_{g} \\ \triangleq{}&(L_{h}+2\epsilon+2\lambda_{4}) \|y_{t}\|_{\mathcal {B}_{v}}+M_{G} \end{aligned}$$

and

$$\begin{aligned} \|F_{y}\|_{[D(A)]}\leq\frac{T^{q}(M_{f}+(\epsilon+\lambda_{3}) \| y_{t}\|_{\mathcal{B}_{v}})}{q\Gamma(q)}\triangleq \frac{T^{q}(\epsilon +\lambda_{3}) \|y_{t}\|_{\mathcal{B}_{v}}}{q\Gamma(q)}+M_{F} . \end{aligned}$$

In view of \(M_{\epsilon}<1\), we can select a sufficiently large constant \(R>R_{0}\) such that

$$\begin{aligned} &M_{\epsilon}R+l\frac{T^{q}M_{f}}{q\Gamma(q)}+l\bigl\| \phi(0)\bigr\| _{[D(A)]}+l\bigl\| \bigl(h(\upsilon _{t_{1}},\upsilon_{t_{2}},\ldots,\upsilon_{t_{m}})\bigr) (0) \bigr\| _{[D(A)]}\\ &\qquad{}+l(M_{G}+M_{F}) \bigl(1+\|\varphi_{A} \|_{L^{1}}\bigr) \\ &\quad< R. \end{aligned}$$

Hence, for \(y,x\in B_{R}\), we get

$$\begin{aligned} & \|\Psi_{1} y+\Psi_{2} y+\Psi_{3} x \|_{\ast} \\ &\quad\leq l\biggl((L_{h}+2\epsilon+2\lambda_{4}) \|x_{t}\|_{\mathcal {B}_{v}}+M_{G}+\frac{T^{q}(\epsilon+\lambda_{3}) \|y_{t}\|_{\mathcal {B}_{v}}}{q\Gamma(q)}+M_{F} \biggr) \bigl(1+\|\varphi_{A}\|_{L^{1}}\bigr) \\ &\qquad{} +l L_{h}\|x_{t}\|_{\mathcal{B}_{v}}+l\bigl\| \phi(0) \bigr\| _{[D(A)]}+l\bigl\| \bigl(h(\upsilon_{t_{1}},\upsilon_{t_{2}}, \ldots,\upsilon_{t_{m}})\bigr) (0)\bigr\| _{[D(A)]} \\ &\quad\leq l\biggl[\biggl(L_{h}+2\epsilon+2\lambda_{4}+ \frac{T^{q}(\epsilon +\lambda_{3})}{q\Gamma(q)}\biggr) \bigl(1+\|\varphi_{A}\|_{L^{1}}\bigr)+L_{h} \biggr]R \\ &\qquad{} +l\bigl\| \phi(0)\bigr\| _{[D(A)]}+l\bigl\| \bigl(h(\upsilon_{t_{1}}, \upsilon _{t_{2}},\ldots,\upsilon_{t_{m}})\bigr) (0) \bigr\| _{[D(A)]} \\ &\qquad{} +(M_{G}+M_{F}) \bigl(1+\|\varphi_{A} \|_{L^{1}}\bigr) \\ &\quad\leq M_{\epsilon}R +l\bigl\| \phi(0)\bigr\| _{[D(A)]}+l\bigl\| \bigl(h(\upsilon_{t_{1}}, \upsilon _{t_{2}},\ldots,\upsilon_{t_{m}})\bigr) (0) \bigr\| _{[D(A)]} \\ & \qquad{} +l(M_{G}+M_{F}) \bigl(1+\|\varphi_{A} \|_{L^{1}}\bigr) \\ &\quad< R, \end{aligned}$$

which implies \(\Psi_{1} y+\Psi_{2} y+\Psi_{3} x\in B_{R}\).

By Lemma 3.1 and Theorem 3.2, the operator Ψ has at least a fixed point \(y_{*}\in B_{R}\). \(y_{*}(t)\), \(-\infty< t\leq T\), is a mild solution of system (1.1)-(1.2). The proof is complete. □

4 Continuous dependence of mild solution

Now, we give the continuous dependence of mild solution of system (1.1)-(1.2) on the initial condition. As a matter of convenience, and in view of the characteristic of A, we have \(\|\phi _{1}-\phi_{2}\|_{[D(A)]}\leq L_{\phi}\|\phi_{1}-\phi_{2}\|_{\mathcal {B}_{v}}\), \(L_{\phi}\) is a constant.

Theorem 4.1

Suppose that (H1)-(H3) hold. Then for \(\phi_{1},\phi_{2}\in\mathcal{B}_{v}\) and the corresponding mild solutions \(y_{1}\), \(y_{2}\) of the problems (1.1)-(1.2), the following inequality holds:

$$ \|y_{1}-y_{2}\|_{\ast}\leq \wedge_{\phi}\|\phi_{1}-\phi_{2}\|_{\mathcal {B}_{v}}+ \wedge\|y_{1}-y_{2}\|_{\ast}, $$
(4.1)

where ∧ is defined in Theorem  3.1 and

$$\wedge_{\phi}=l\bigl(\|\varphi_{A}\|_{L^{1}}+2\bigr) L_{\phi}. $$

Furthermore if \(\wedge<1\), we have

$$ \|y_{1}-y_{2}\|_{\ast} \leq \frac{\wedge_{\phi }}{1-\wedge}\|\phi_{1}-\phi_{2}\|_{\mathcal{B}_{v}}. $$
(4.2)

Proof

Suppose that \(\phi_{i}\in\mathcal{B}_{v}\) (\(i=1,2\)) are arbitrary functions and that \(y_{1}\), \(y_{2}\) are the corresponding mild solutions of the problem (1.1)-(1.2). Let

$$\begin{aligned}& \begin{aligned}[b] G_{y\phi_{i}}={}&\phi_{i}(0)-\bigl(h \bigl((y_{i})_{t_{1}},(y_{i})_{t_{2}},\ldots ,(y_{i})_{t_{m}}\bigr)\bigr) (0) \\ &{}-g\bigl(0,(y_{i})_{0}\bigr)+g\bigl(t,(y_{i})_{t} \bigr),\quad i=1,2, \end{aligned} \\& F_{y\phi_{i}}=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac {f(s,(y_{i})_{s})}{(t-s)^{1-q}}\,ds,\quad i=1,2. \end{aligned}$$

Then, for \(0< t\leq T\), we have

$$\begin{aligned} y_{1}(t)-y_{2}(t)={}& \int _{0}^{t}\mathcal{S}'(t-s) \bigl[G_{y\phi _{1}}(s)-G_{y\phi_{2}}(s)\bigr]\,ds+G_{y\phi_{1}}(t)-G_{y\phi_{2}}(t) \\ &{}+\int_{0}^{t}\mathcal{S}'(t-s) \bigl[F_{y\phi_{1}}(s)-F_{y\phi_{2}}(s)\bigr]\,ds \\ &{}+F_{y\phi_{1}}(t)-F_{y\phi_{2}}(t), \end{aligned}$$
(4.3)

for \(t\leq0\),

$$\begin{aligned} y_{1}(t)-y_{2}(t)={}& \phi _{1}(t)-\bigl(h\bigl((y_{1})_{t_{1}},(y_{1})_{t_{2}}, \ldots,(y_{1})_{t_{m}}\bigr)\bigr) (t) \\ &{} -\phi_{2}(t)+\bigl(h\bigl((y_{2})_{t_{1}},(y_{2})_{t_{2}}, \ldots ,(y_{2})_{t_{m}}\bigr)\bigr) (t). \end{aligned}$$
(4.4)

Analogously to (3.4) and (3.5), we have

$$\begin{aligned} &\|G_{y\phi_{1}}-G_{y\phi_{2}}\|_{[D(A)]} \\ &\quad\leq \bigl\| \phi_{1}-\phi_{2} -h\bigl((y_{1})_{t_{1}},(y_{1})_{t_{2}}, \ldots,(y_{1})_{t_{m}}\bigr) \\ &\qquad{}+h\bigl((y_{2})_{t_{1}},(y_{2})_{t_{2}}, \ldots,(y_{2})_{t_{m}}\bigr)\bigr\| _{[D(A)]} \\ &\qquad{} +\bigl\| g\bigl(0,(y_{1})_{0}\bigr)-g \bigl(0,(y_{2})_{0}\bigr)\bigr\| _{[D(A)]} \\ &\qquad{}+\bigl\| g\bigl(t,(y_{1})_{t}\bigr)-g \bigl(t,(y_{2})_{t}\bigr)\bigr\| _{[D(A)]} \\ &\quad\leq L_{\phi}\|\phi_{1}-\phi_{2} \|_{\mathcal{B}_{v}} +(L_{h}+2L_{g})\bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal{B}_{v}}, \end{aligned}$$
(4.5)

and

$$ \|F_{y\phi_{1}}-F_{y\phi_{2}}\|_{[D(A)]}\leq \frac{L_{f}T^{q}}{q\Gamma (q)}\bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal{B}_{v}}. $$
(4.6)

From (4.3), (4.5), (4.5), and (4.6), we get, for \(0< t\leq T\)

$$\begin{aligned} l\bigl\| y_{1}(t)-y_{2}(t)\bigr\| \leq{}&l\biggl[L_{\phi}\| \phi_{1}-\phi_{2}\| _{\mathcal{B}_{v}}+(L_{h}+2L_{g}) \|y_{1}-y_{2}\|_{\mathcal{B}_{v}} \\ &{}+\frac{L_{f}T^{q}}{q\Gamma(q)}\bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal {B}_{v}}\biggr]\bigl(\|\varphi_{A}\|_{L^{1}}+1\bigr). \end{aligned}$$

Immediately,

$$\begin{aligned} l\|y_{1}-y_{2}\|_{[0,T]}\leq{}&l\bigl(\| \varphi_{A}\|_{L^{1}}+1\bigr) L_{\phi}\|\phi_{1}- \phi_{2}\|_{\mathcal{B}_{v}} \\ &{}+l\biggl(L_{h}+2L_{g}+\frac{L_{f}T^{q}}{q\Gamma(q)}\biggr) \bigl(\| \varphi_{A}\| _{L^{1}}+1\bigr)\bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal{B}_{v}}. \end{aligned}$$
(4.7)

Similarly to (3.6), from (4.4), we have

$$ \bigl\| (y_{1})_{0}-(y_{2})_{0} \bigr\| _{\mathcal{B}_{v}}\leq l L_{\phi}\|\phi _{1}-\phi_{2} \|_{\mathcal{B}_{v}}+lL_{h}\bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal{B}_{v}}. $$
(4.8)

Equations (4.7) and (4.8) imply that

$$\begin{aligned} &\|y_{1}-y_{2}\|_{\ast} \\ &\quad\leq l\bigl(\|\varphi_{A}\|_{L^{1}}+2\bigr) L_{\phi}\| \phi_{1}-\phi_{2}\| _{\mathcal{B}_{v}} \\ &\qquad{}+\biggl[lL_{h}+l\biggl(L_{h}+2L_{g}+ \frac{L_{f}T^{q}}{q\Gamma(q)}\biggr) \bigl(\|\varphi_{A}\| _{L^{1}}+1\bigr)\biggr] \bigl\| (y_{1})_{t}-(y_{2})_{t} \bigr\| _{\mathcal{B}_{v}} \\ &\quad\leq l\bigl(\|\varphi_{A}\|_{L^{1}}+2\bigr) L_{\phi}\| \phi_{1}-\phi_{2}\| _{\mathcal{B}_{v}} \\ &\qquad{}+\biggl[lL_{h}+l\biggl(L_{h}+2L_{g}+ \frac{L_{f}T^{q}}{q\Gamma(q)}\biggr) \bigl(\|\varphi_{A}\| _{L^{1}}+1\bigr)\biggr] \|y_{1}-y_{2}\|_{\ast} \\ &\quad\leq \wedge_{\phi}\|\phi_{1}-\phi_{2} \|_{\mathcal{B}_{v}}+\wedge \|y_{1}-y_{2}\|_{\ast}. \end{aligned}$$

Therefore, (4.1) holds. Inequality (4.2) is a consequence of (4.1). This completes the proof. □