Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects

Abstract

We prove the global solvability and weakly asymptotic stability for a semilinear fractional differential inclusion subject to impulsive effects by analyzing behavior of its solutions on the half-line. Our analysis is based on a fixed point principle for condensing multi-valued maps, which is employed for solution operator acting on the space of piecewise continuous functions. The obtained results will be applied to a lattice fractional differential system.

1 Introduction

Let $(X, \Vert \cdot \Vert )$ be a Banach space. Consider the following problem:

$$\begin{aligned} D_0^\alpha u(t)&\in Au(t) + F(t, u(t), u_t),\quad \; t>0, t \ne t_k, k\in \Lambda , \end{aligned}$$

(1.1)

$$\begin{aligned} \Delta u(t_k)&= I_k(u(t_k)), \end{aligned}$$

(1.2)

$$\begin{aligned} u(s)&= \varphi (s),\quad \;s\in [-h,0], \end{aligned}$$

(1.3)

where $D_0^{\alpha }, \alpha \in (0,1)$, is the fractional derivative in the Caputo sense, A is a closed linear operator in X which generates a strongly continuous semigroup $W(\cdot )$, $F: \mathbb R^+\times X\times C([-h,0];X)\rightarrow \mathcal P(X)$ is a multi-valued map, $\Delta u(t_k) = u(t_k^+)-u(t_k^-)$, $k\in \Lambda \subset \mathbb N$, $I_k$ and g are the functions which will be specified in Sect. 3. Here, $u_t$ stands for the history of the state function up to the time t, i.e., $u_t(s) = u(t+s), s\in [-h,0]$.

The system (1.1)–(1.3) is a generalized Cauchy problem which involves impulsive effect expressed by (1.2). It should be mentioned that the multi-valued nonlinearity in (1.1) appears frequently in control theory, where control factor is taken in the form of multi-valued feedback (see, e.g., [9]) and the delay term is concerned as an inherence in control problems. On the other hand, the impulsive condition (1.3) is employed to describe processes subject to abrupt changes happening in biology, mechanics, electrical engineering, etc. A systematic study on impulsive differential equations can be found in [18, 20].

In the last decade, there have been extensive studies devoted to particular cases of our problem in the literature. We refer to some typical results on existence and properties of solution set presented in [4, 21,22,23], in which the solvability on compact intervals and the structure of solution set like $R_\delta$-set were proved. Regarding related control problems, it should be mentioned the results on controllability given in [13,14,15,16,17, 24, 25, 29], where the fixed point approach was employed as a fruitful method. More results on solvability and controllability can be found in the reference quoted in these works.

One of the most important questions associated with problem (1.1)–(1.3) is to analyze the stability of its solutions. Unfortunately, the results on this direction are less known. In [11], we proved a stability result for (1.1)–(1.3) in the case when F is single valued and Lipschitzian. However, the technique used in [11] does not work when F is a multi-valued map. Moreover, the classical concept of stability due to Lyapunov is inappropriate for applying to multi-valued cases. Therefore, we adopt the following concept of weakly asymptotic stability of zero solution to inclusion (1.1): Let $\Sigma (\varphi )$ be the solution set of (1.1)–(1.3) with respect to the initial datum $\varphi$ such that $0\in \Sigma (0)$. The zero solution of (1.1) is said to be weakly asymptotically stable if it is

• (1) stable: for every $\epsilon >0$, there exists $\delta >0$ such that if $\Vert \varphi \Vert _{h}<\delta$ then $\Vert u_t\Vert _{h}<\epsilon$ for any $u\in \Sigma (\varphi )$, here $\Vert \cdot \Vert _h$ denotes the norm in $C([-h,0];X)$;

• (2) weakly attractive: for any $\varphi \in \mathcal B$, there exists $u\in \Sigma (\varphi )$ satisfying $\Vert u_t\Vert _{h}\rightarrow 0$ as $t\rightarrow +\infty$.

It should be noted that this concept will coincide with the classical one in the Lyapunov sense if the solution to the Cauchy problem is unique, since the only difference is the weak attractivity. It requires that at least one trajectory (rather than all trajectories) at each starting point approaches the equilibrium point. So this concept is useful in the control theory and has a relation to the viability theory (see, e.g., [1]).

Our main aim in this work is to prove, for the first time, the weakly asymptotic stability of the zero solution to (1.1). To this end, we make use of fixed point approach. More precisely, by constructing a new measure of noncompactness (MNC), denoted by $\chi ^*$, on weighted spaces of piecewise continuous functions on the half-line, denoted by $\mathrm{PC}_\varrho$, we are able to show that the solution operator is $\chi ^*$-condensing, which implies the existence of exponentially bounded solutions. Consequently, when the semigroup $\{\mathrm{e}^{tA}\}_{t\ge 0}$ is exponentially stable, we obtain the weakly asymptotic stability as aforementioned. The features of our work include

• Constructing a new MNC on the space $\mathrm{PC}_\varrho$ in order to present a compactness condition on this space.

• Analyzing the fixed point set of the solution operator associated with (1.1)–(1.3) on PC$_\varrho$ to derive an asymptotic estimate of solutions to (1.1)–(1.3). Based on this estimate, the weakly asymptotic stability will be proved.

The rest of our work is as follows. In the next section, we introduce the MNC $\chi ^*$ on PC$_\varrho$ which can be used to characterize the compactness on this space. We also give some estimates via MNCs which will be used in Sect. 3. Section 3 is devoted to proving the solvability of (1.1)–(1.3) on PC$_\varrho$. In Sect. 4, the result on weakly asymptotic stability of the zero solution is derived when the semigroup $\{\mathrm{e}^{tA}\}_{t\ge 0}$ is exponentially stable. The last section shows an application of the obtained results to a lattice differential system.

2 Preliminaries

2.1 Measure of noncompactness and condensing operators

Let $\mathcal {E}$ be a Banach space. Denote

$$\begin{aligned}&\mathcal {P}(\mathcal E)=\{Y\subset \mathcal E: Y\ne \emptyset \},\\&\mathcal {P}_{b}(\mathcal E)=\{Y\in \mathcal {P}(\mathcal E): Y \text{ is } \text{ bounded }\},\\&K_v(\mathcal E)=\{Y\in \mathcal P(\mathcal E): Y \text{ is } \text{ compact } \text{ and } \text{ convex }\}. \end{aligned}$$

We will use the following definition of measure of noncompactness (see [9]).

Definition 2.1

A function $\beta : \mathcal {P}_b(\mathcal {E})\rightarrow \mathbb R^+$ is called a measure of noncompactness (MNC) in $\mathcal {E}$ if

$$\begin{aligned} \beta (\overline{\mathrm {co}}\;\Omega ) = \beta (\Omega ) \quad \;\;\text {for every}\;\;\Omega \in \mathcal {P}_b(\mathcal {E}), \end{aligned}$$

where $\overline{\mathrm {co}}\;\Omega$ is the closure of the convex hull of $\Omega$. An MNC $\beta$ is called

• (i) monotone if $\Omega _0,\Omega _1\in \mathcal {P}_b(\mathcal {E}),$  $\Omega _0\subset \Omega _1$ implies $\beta (\Omega _0)\le \beta (\Omega _1)$;

• (ii) nonsingular if $\beta (\{a\}\cup \Omega ) = \beta (\Omega )$ for any $a\in \mathcal {E},\Omega \in \mathcal {P}_b(\mathcal {E})$;

• (iii) invariant with respect to union with compact set if $\beta (K\cup \Omega ) = \beta (\Omega )$ for every relatively compact set $K\subset \mathcal {E}$ and $\Omega \in \mathcal {P}_b(\mathcal {E})$;

• (iv) algebraically semi-additive if $\beta (\Omega _0+\Omega _1)\le \beta (\Omega _0) + \beta (\Omega _1)$ for any $\Omega _0,\Omega _1\in \mathcal {P}_b(\mathcal {E})$;

• (v) regular if $\beta (\Omega )=0$ is equivalent to the relative compactness of $\Omega$.

An important example of MNC is the Hausdorff MNC $\chi (\cdot )$, which is defined as follows, for $\Omega \in \mathcal P_b(\mathcal E)$,

$$\begin{aligned} \chi (\Omega ) = \inf \{\varepsilon >0:\; \Omega \;\;\;\text {has a finite}\,\, \varepsilon \text {-net}\}. \end{aligned}$$

The Hausdorff MNC satisfies all properties stated in Definition 2.1. If $\mathcal T\in L(\mathcal E)$, the space of bounded linear operators on X, we define the $\chi$-norm of $\mathcal T$ as follows:

$$\begin{aligned} \Vert \mathcal T\Vert _\chi = \inf \{\eta >0: \chi (\mathcal T(B))\le \eta \cdot \chi (B),\;\;\text { for all bounded set } B\subset \mathcal E\}. \end{aligned}$$

(2.1)

It is clear that

$$\begin{aligned} \chi \left( \mathcal T(B)\right) \le \Vert \mathcal T\Vert _\chi \cdot \chi (B), \;\;\;\;\forall B\subset \mathcal E. \end{aligned}$$

In addition, $\Vert \mathcal T\Vert _\chi \le \Vert \mathcal T\Vert$ and $\mathcal T$ is a compact operator iff $\Vert \mathcal T\Vert _\chi =0$.

Let $\mathcal E=\mathrm{PC}(J;X)$, the space of X-valued functions defined on $J\subset \mathbb R$, satisfying that, for every $u\in \mathrm{PC}(J;X)$

• u is continuous on $J\setminus \{t_k: k\in \Lambda \}$;

• there exist $u(t_k^+)=\lim \nolimits _{t\rightarrow t_k^+} u(t)$ and $u(t_k^-)=\lim \nolimits _{t\rightarrow t_k^-}u(t)$ such that $u(t_k^-)=u(t_k)$.

If J is a compact interval, PC(J; X) with the norm

$$\begin{aligned} \Vert u\Vert _{\mathrm{PC}} = \sup _{t\in J}\Vert u(t)\Vert , \end{aligned}$$

becomes a Banach space. Let $\chi _{\mathrm{PC}}$ be the Hausdorff MNC in $\mathrm{PC}(J;X)$, then one knows that (see [8]), for any bounded set $D\subset \mathrm{PC}(J;X)$,

• $\sup _{t\in J}\chi (D(t))\le \chi _{\mathrm{PC}}(D)$, where $D(t)=\{u(t): d\in D\}$;

• If D is equicontinuous on each interval $(t_{k-1}, t_k]\subset J$ then $\chi _{\mathrm{PC}}(D)=\sup _{t\in J}\chi (D(t))$.

In the case J is the half-line, i.e., $J=[0,+\infty )$, we consider the following space:

$$\begin{aligned} \mathrm{PC}_\varrho :=\mathrm{PC}_\varrho ([0,+\infty );X)&= \{u\in \mathrm{PC}([0,+\infty );X): \lim \limits _{t\rightarrow +\infty }\frac{u(t)}{\varrho (t)}=0\}, \end{aligned}$$

where $\varrho : \mathbb R^+\rightarrow [1,+\infty )$ is a continuous and nondecreasing function. We see that PC$_\varrho ([0,+\infty );X)$ with the norm

$$\begin{aligned} \Vert u\Vert _{\varrho } = \sup _{t\ge 0}\frac{\Vert u(t)\Vert }{\varrho (t)}, \end{aligned}$$

is a Banach space. In fact, we have no formulation of the Hausdorff MNC in PC$_\varrho$. We will define a new MNC in this space, which is monotone, nonsingular and regular. For $u\in \mathrm{PC}_\varrho$, we denote by $\pi _T(u)$ the restriction of u to [0, T], i.e., $\pi _T(u)\in \mathrm{PC}([0,T];X)$. For $D\subset \mathrm{PC}_\varrho$, put

$$\begin{aligned}&\chi _\infty (D) = \sup _{T>0}\chi _{\mathrm{PC}}(\pi _T(D)),\end{aligned}$$

(2.2)

$$\begin{aligned}&d_\infty (D) = \lim _{T\rightarrow +\infty }\sup _{u\in D}\sup _{t\ge T}\frac{\Vert u(t)\Vert }{\varrho (t)},\end{aligned}$$

(2.3)

$$\begin{aligned}&\chi ^*(D) = \chi _\infty (D) + d_\infty (D). \end{aligned}$$

(2.4)

It is easily seen that $\chi _\infty (\cdot )$ and $d_\infty (\cdot )$ are monotone and nonsingular MNCs, so is $\chi ^*(\cdot )$. We will prove an important property of $\chi ^*(\cdot )$ in the next lemma.

Lemma 2.1

Let $\Omega \subset \mathrm{PC}_\varrho ([0,+\infty );X)$ be a bounded set such that $\chi ^*(\Omega )=0$. Then, $\Omega$ is relatively compact.

Proof

Let $\epsilon >0$. Since $d_\infty (\Omega )=0$, one can choose $T>0$ such that

$$\begin{aligned} \left\| \frac{u(t)}{\varrho (t)}\right\| < \frac{\epsilon }{3},\quad \; \forall t\ge T, \quad \forall u\in \Omega . \end{aligned}$$

(2.5)

Let $\{u_n\}$ be a sequence in $\Omega$. Then, $\chi _\infty (\{u_n\})=0$. This implies $\chi _{\mathrm{PC}}(\pi _T(\{u_n\}))=0$, and hence $\{u_n|_{[0,T]}\}$ has a convergent subsequence in PC([0, T]; X) (still indexed by n). So there exists $N(\epsilon )\in \mathbb N$ such that

$$\begin{aligned} \sup _{t\in [0,T]}\Vert u_n(t)-u_m(t)\Vert <\frac{\epsilon }{3},\quad \; \forall n, m\ge N(\epsilon ). \end{aligned}$$

Accordingly,

$$\begin{aligned} \sup _{t\in [0,T]}\left\| \frac{u_n(t)}{\varrho (t)}-\frac{u_m(t)}{\varrho (t)} \right\| <\frac{\epsilon }{3},\; \;\;\forall n,\;m\ge N(\epsilon ). \end{aligned}$$

(2.6)

Combining (2.5) and (2.6), one gets

$$\begin{aligned}&\Vert u_n-u_m\Vert _\varrho = \sup _{t\ge 0}\left\| \frac{u_n(t)}{\varrho (t)}-\frac{u_m(t)}{\varrho (t)}\right\| \\&\le \sup _{t\in [0,T]}\left\| \frac{u_n(t)}{\varrho (t)}-\frac{u_m(t)}{\varrho (t)}\right\| + \sup _{t\ge T}\left\| \frac{u_n(t)}{\varrho (t)}-\frac{u_m(t)}{\varrho (t)}\right\| \\&\le \sup _{t\in [0,T]}\left\| \frac{u_n(t)}{\varrho (t)}-\frac{u_m(t)}{\varrho (t)}\right\| + \sup _{t\ge T}\left\| \frac{u_n(t)}{\varrho (t)}\right\| +\sup _{t\ge T}\left\| \frac{u_m(t)}{\varrho (t)}\right\| \\&\le \frac{\epsilon }{3}+\frac{\epsilon }{3}+\frac{\epsilon }{3} = \epsilon , \end{aligned}$$

for all $n,m\ge N(\epsilon )$. Therefore, $\{u_n\}$ is a Cauchy sequence in PC$_\varrho ([0,+\infty );X)$. The proof is complete. $\square$

We now recall some notions of set-valued analysis and fixed point theory for condensing multi-valued maps. Let Y be a metric space.

Definition 2.2

[9] A multi-valued map (multimap) $\mathcal {F}: Y\rightarrow \mathcal {P}(\mathcal E)$ is said to be:

• (i) upper semicontinuous (u.s.c) if $\mathcal {F}^{-1}(V):=\{y\in Y: \mathcal {F}(y)\cap V\ne \emptyset \}$ is a closed subset of Y for every closed set $V\subset \mathcal E$;

• (ii) closed if its graph $\Gamma _{\mathcal {F}}:=\{(y, z): z\in \mathcal {F}(y)\}$ is a closed subset of $Y\times \mathcal E$.

Definition 2.3

A multimap $\mathcal {F}: Z\subseteq E\rightarrow \mathcal {P}(\mathcal E)$ is said to be condensing with respect to an MNC $\beta$ ($\beta$-condensing) if for any bounded set $\Omega \subset Z$, the inequality

$$\begin{aligned} \beta (\Omega )\le \beta (\mathcal {F}(\Omega )) \end{aligned}$$

implies the relative compactness of $\Omega$.

Let $\beta$ be a monotone nonsingular MNC in $\mathcal E$. We recall a fixed point principle for condensing multi-valued maps (see, e.g., [9]), which is the main tool for our purpose.

Theorem 2.2

[9, Corollary 3.3.1] Let $\mathcal {M}$ be a bounded convex closed subset of $\mathcal E$ and let $\mathcal {F}: \mathcal {M}\rightarrow K_v(\mathcal {M})$ be a closed and $\beta$-condensing multimap. Then, $\mathrm{Fix}(\mathcal {F}):=\{x\in \mathcal M: x\in \mathcal {F}(x)\}$ is nonempty.

2.2 Fractional calculus

Let $L^p(0,T;X), p\in (1,+\infty )$ be the space of X-valued functions u defined on [0, T] such that the function $t\mapsto \Vert u(t)\Vert ^p$ is integrable. The integrals appeared in this work will be understood in the Bochner sense. The notation $L^p(0,T)$ stands for $L^p(0,T;\mathbb R)$. We now recall some notions in fractional calculus (see, e.g., [10, 27]).

Definition 2.4

The fractional integral of order $\beta >0$ of a function $f\in L^1(0,T;X)$ is defined by

$$\begin{aligned} I^{\beta }_0 f(t) = \frac{1}{\Gamma (\alpha )}\int _0^t (t-s)^{\beta -1}f(s)\mathrm{d}s, \end{aligned}$$

where $\Gamma$ is the Gamma function, provided the integral converges.

Definition 2.5

For a function $f\in C^1([0,T];X)$, the Caputo fractional derivative of order $\alpha \in (0,1)$ is defined by

$$\begin{aligned} D^{\alpha }_0 f(t) = \frac{1}{\Gamma (1-\alpha )}\int _0^t (t-s)^{-\alpha } f^{\prime }(s) \mathrm{d}s. \end{aligned}$$

Consider the following problem:

$$\begin{aligned} D^\alpha _0 u(t)&= Au(t) + f(t),\quad t>0,\,\, t\ne t_k\in (0,+\infty ),\,\, k\in \Lambda ,\end{aligned}$$

(2.7)

$$\begin{aligned} \Delta u(t_k)&= I_k(u(t_k)),\end{aligned}$$

(2.8)

$$\begin{aligned} u(s)&= \varphi (s),\,\,\,\, s\in [-h,0], \end{aligned}$$

(2.9)

where $\alpha \in (0,1)$ and $f\in L^p(0,T;X)$. In this note, we assume that the $C_0$-semigroup $W(\cdot )$ generated by A is globally bounded, i.e.

$$\begin{aligned} \Vert W(t)x\Vert \le M_A\Vert x\Vert , \,\,\,\,\forall t\ge 0,\,\, x\in X. \end{aligned}$$

(2.10)

for some $M_A\ge 1$. By the arguments in [11] and [28], we have the following presentation:

$$\begin{aligned} u(t)=S_\alpha (t)\varphi (0)&+\sum _{0<t_k< t}S_\alpha (t-t_k)I_k (u(t_k))\nonumber \\&+\int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)f(s)\mathrm{d}s,\,\,\,\, t > 0, \end{aligned}$$

(2.11)

where

$$\begin{aligned}&S_{\alpha } (t)x = \int _0^{\infty } \phi _{\alpha }(\theta )W(t^{\alpha }\theta )x \mathrm{d}\theta ,\end{aligned}$$

(2.12)

$$\begin{aligned}&P_{\alpha }(t)x = \alpha \int _0^{\infty } \theta \phi _{\alpha }(\theta )W(t^{\alpha }\theta )x \mathrm{d}\theta ,\,\,\,\, x\in X, \end{aligned}$$

(2.13)

$$\begin{aligned}&\phi _{\alpha }(\theta ) = \frac{1}{\pi }\sum _{n=1}^{\infty }\frac{ (-\theta )^{n-1}}{(n-1)!}\Gamma (n\alpha )\sin (n\pi \alpha ). \end{aligned}$$

Following [28], we have the following estimates:

$$\begin{aligned}&\Vert S_\alpha (t)x\Vert \le M_A\Vert x\Vert ,\end{aligned}$$

(2.14)

$$\begin{aligned}&\Vert P_\alpha (t)x\Vert \le \frac{M_A}{\Gamma (\alpha )}\Vert x\Vert ,\,\,\,\,\forall x\in X. \end{aligned}$$

(2.15)

Let $\Phi (t,s)$ be a family of bounded linear operators on X for $t,s\in [0,T], s\le t$. The following result was proved in [19, Lemma 1].

Proposition 2.3

Assume that $\Phi$ satisfies the following conditions:

• $(\Phi 1)$ there exists a function $\rho \in L^q(0,T), q > 1$ such that $\Vert \Phi (t,s)\Vert \le \rho (t-s)$ for all $t,s\in [0,T], s\le t$;

• $(\Phi 2)$ $\Vert \Phi (t,s)-\Phi (r,s)\Vert \le \epsilon$ for $0\le s\le r-\epsilon , r<t=r+h\le T$ with $\epsilon = \epsilon (h)\rightarrow 0$ as $h\rightarrow 0$.

Then, the operator $\mathbf {S}: L^{q^\prime }(0,T;X)\rightarrow C([0,T];X)$ defined by

$$\begin{aligned} (\mathbf {S}g)(t) := \int _0^t \Phi (t,s)g(s) \mathrm{d}s \end{aligned}$$

sends any bounded set to an equicontinuous one, where $q^\prime$ is the conjugate of q, i.e., $\frac{1}{q} + \frac{1}{q^\prime }=1$.

Let $p>\frac{1}{\alpha }$, we define a linear operator

$$\begin{aligned}&Q_{\alpha }: L^p(0,T;X)\rightarrow C([0,T];X),\nonumber \\&Q_{\alpha }(f)(t) = \int _0^t (t-s)^{\alpha -1}P_{\alpha }(t-s)f(s)\mathrm{d}s. \end{aligned}$$

(2.16)

Before proving some properties of the operator $Q_\alpha$, we need the following result.

Proposition 2.4

[12] Let $D\subset L^1(0,T;X)$ be such that

• (1) $\Vert f(t)\Vert \le \nu (t)$ for a.e. $t\in [0,T]$ and for all $f\in D$;

• (2) $\chi (D(t))\le \mu (t)$ for a.e. $t\in [0,T]$,

where $\nu , \mu \in L^1(0,T)$ are nonnegative functions. Then, we have

$$\begin{aligned} \chi \left( \int _0^t D(s)\mathrm{d}s\right) \le 4\int _0^t \chi (D(s))\mathrm{d}s,\quad t\in [0,T], \end{aligned}$$

here

$$\begin{aligned} \int _0^t D(s)\mathrm{d}s =\left\{ \int _0^t f(s)\mathrm{d}s: f\in D\right\} . \end{aligned}$$

In what follows, with a sequence $\{z_n\}$ in a specific space, we use the notation $z_n\rightharpoonup z$ to indicate the weak convergence, and write $z_n\rightarrow z$ if $\{z_n\}$ converges strongly to z.

Definition 2.6

Let $p\ge 1$. A sequence $\{f_n\}\subset L^p(0,T;X)$ is said to be semicompact if there exist a function $\nu \in L^p(0,T)$ and a family of compact set $K(t), t\in [0,T]$, such that $\{f_n(t)\}\subset K(t)$ and $\Vert f_n(t)\Vert \le \nu (t)$ for a.e. $t\in [0,T]$.

Proposition 2.5

Assume that the semigroup $W(\cdot )$ generated by A is norm-continuous, i.e., the map $(0,\infty )\ni t\mapsto W(t)\in L(X)$ is continuous. Then

• (1) For each bounded set $\Omega \subset L^p(0,T;X)$, $Q_{\alpha }(\Omega )$ is an equicontinuous set in C([0, T]; X). Moreover, we have the following estimate:

  $$\begin{aligned} \chi _{\mathrm{PC}}(Q_\alpha (\Omega ))\le 4\sup _{t\in [0,T]}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi \cdot \chi (\Omega (s))\mathrm{d}s, \end{aligned}$$

  where $\Vert \cdot \Vert _\chi$ is the $\chi$-norm given by (2.1).

• (2) If $\{f_n\}\subset L^p(0,T;X), p>1,$ is a semicompact sequence, then $\{Q_\alpha (f_n)\}$ is relatively compact in C([0, T]; X). Moreover, if $f_n\rightharpoonup f$ in $L^p(0,T;X)$, then $Q_\alpha (f_n)\rightarrow Q_\alpha (f)$ in C([0, T]; X).

Proof

• (1) Since $W(\cdot )$ is norm-continuous, so is $P_{\alpha }(\cdot )$ (see, e.g., [26]). Then, we deduce that $\Phi (t,s) = (t-s)^{\alpha -1}P_{\alpha }(t-s)$ satisfies $(\Phi 1)-(\Phi 2)$ in Proposition 2.3, which ensures the equicontinuity of $Q_\alpha (\Omega )$. Hence

  $$\begin{aligned} \chi _{\mathrm{PC}}(Q_\alpha (\Omega )) = \sup _{t\in [0,T]}\chi (Q_\alpha (\Omega )(t)). \end{aligned}$$

  In view of Proposition 2.4, we have

  $$\begin{aligned} \chi _{\mathrm{PC}}(Q_\alpha (\Omega ))&= \sup _{t\in [0,T]}\chi \left( \int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)\Omega (s)\mathrm{d}s\right) \\&\le 4 \sup _{t\in [0,T]}\int _0^t \chi \left( (t-s)^{\alpha -1}P_\alpha (t-s)\Omega (s)\right) \mathrm{d}s\\&\le 4 \sup _{t\in [0,T]}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi \cdot \chi \left( \Omega (s)\right) \mathrm{d}s. \end{aligned}$$

• (2) By the first part of this proposition, the sequence $\{Q_\alpha (f_n)\}$ is equicontinuous. In addition, one has

  $$\begin{aligned} \chi \left( \{Q_\alpha (f_n)(t)\}\right)&= \chi \left( \left\{ \int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)f_n(s)\mathrm{d}s\right\} \right) \\&\le 4 \int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi \cdot \chi \left( \left\{ f_n(s)\right\} \right) \mathrm{d}s\\&= 0, \end{aligned}$$

  thanks to Proposition 2.4. So $\{Q_\alpha (f_n)(t)\}$, for each $t\in [0,T]$, is a relatively compact set. By the Arzelà–Ascoli theorem, $\{Q_\alpha (f_n)\}$ is relatively compact in C([0, T]; X). Then, the last assertion is justified as follows. At first, by the Hölder inequality, one sees that $Q_\alpha : L^p(0,T;X){\rightarrow }C([0,T];X)$ is bounded and hence continuous. So it is continuous with respect to weak topology (see, e.g., [3, Theorem 3.10]). This implies $Q_\alpha (f_n)\rightharpoonup Q_\alpha (f)$ in C([0, T]; X). By the relative compactness of $\{Q_\alpha (f_n)\}$, the last convergence is in the norm of C([0, T]; X). The proof is complete.

$\square$

3 Existence of solutions on the half-line

In this section, we take $\varrho (t) = \mathrm{e}^{\delta t}$ for a fixed $\delta >0$. Concerning problem (1.1)–(1.3), we give the following assumptions:

• (A) The $C_0$-semigroup $\{W(t)\}_{t \ge 0}$ generated by A is norm-continuous and globally bounded, i.e.,

  $$\begin{aligned} \Vert W(t)x\Vert \le M_A\Vert x\Vert ,\quad \forall t\ge 0,\quad x\in X. \end{aligned}$$

• (F) The nonlinearity $F: \mathbb R^+ \times X \times C([-h,0];X) \rightarrow K_v(X)$ satisfies:

  • (1) The multi-valued map $(v,w)\mapsto F(t,v,w)$ is u.s.c for each $t\in \mathbb R^+$;

  • (2) The multi-valued map $t\mapsto F(t, u(t), u_t)$ admits a strongly measurable selection for each $u\in \mathrm{PC}_\varrho$;

  • (3) There exists a function $m\in L^p_{\mathrm{loc}}(\mathbb R^+)$ such that

    $$\begin{aligned} \qquad \quad \Vert F(t,v,w)\Vert = \sup \{\Vert \xi \Vert : \xi \in F(t,v,w)\}\le m(t)(\Vert v\Vert + \Vert w\Vert _h), \end{aligned}$$

    for all $(t,v,w)\in \mathbb R^+\times X\times C([-h,0];X)$, here $\Vert w\Vert _h:=\sup \nolimits _{s\in [-h,0]}\Vert w(s)\Vert$ is the norm in $C([-h,0];X)$;

  • (4) If $W(\cdot )$ is noncompact, there is a function $k\in L^p_{\mathrm{loc}}(\mathbb R^+)$ such that

    $$\begin{aligned} \chi (F(t, V,\Omega ))\le k(t)\left[ \chi (V) + \sup _{t\in [-h,0]}\chi (\Omega (t))\right] , \end{aligned}$$

    for a.e. $t\in \mathbb R^+$, and for all bounded sets $V\subset X, \Omega \subset C([-h,0];X)$.

• (I) The function $I_k: X \rightarrow X, k\in \Lambda ,$ is continuous and satisfies:

  • (1) There exists a nonnegative sequence $\{l_k\}_{k \in \Lambda }$ such that $\sum \nolimits _{k\in \Lambda }l_k<\infty$ and

    $$\begin{aligned} \Vert I_k(x)\Vert \le l_k\,\Vert x\Vert ,\quad \text { for all }\,\, x\in X, \,\,k \in \Lambda . \end{aligned}$$

  • (2) There exists a nonnegative sequence $\{\mu _k\}_{k \in \Lambda }$ such that

    $$\begin{aligned} \chi (I_k(B)) \le \mu _k\chi (B), \end{aligned}$$

    for all bounded subsets $B \subset X$;

  • (3) The sequence $\{t_k\}_{k\in \Lambda }$ satisfies $\inf _{k\in \Lambda }(t_{k+1}-t_k)>0$.

For given $\varphi \in C([-h,0];X)$, we define the space

$$\begin{aligned} \mathrm{PC}_\varrho ^\varphi = \{v\in \mathrm{PC}_\varrho : v(0) = \varphi (0)\}. \end{aligned}$$

For $v\in \mathrm{PC}_\varrho ^\varphi$, let $v[\varphi ]$ be a function given by

$$\begin{aligned} v[\varphi ](t) = {\left\{ \begin{array}{ll} \varphi (t)\quad \text { if }\,\, t\in [-h,0],\\ v(t)\quad \text { if }\,\, t>0. \end{array}\right. } \end{aligned}$$

Now for $v\in \mathrm{PC}_\varrho ^\varphi$, we denote

$$\begin{aligned} \mathcal P_F^p(v) = \{f\in L^p_{\mathrm{loc}}(\mathbb R^+;X): f(t)\in F(t, v(t), v[\varphi ]_t) \quad \text { for a.e. }t\in \mathbb R^+\}. \end{aligned}$$

Definition 3.1

A function $u: [-h,+\infty )\rightarrow X$ is said to be an integral solution of problem (1.1)–(1.3) if and only if $u(t)=\varphi (t)$ for $t\in [-h,0]$, and there exists $f\in \mathcal P_F^p(u)$ such that

$$\begin{aligned} u(t) = S_\alpha (t)\varphi (0)&+ \sum _{0<t_k<t}S_\alpha (t - t_k)I_k(u(t_k)) + \int _0^t (t-s)^{\alpha - 1}P_\alpha (t-s)f(s)\mathrm{d}s, \end{aligned}$$

for any $t >0$.

Let $\mathcal F: \mathrm{PC}_\varrho ^\varphi \rightarrow \mathcal P(\mathrm{PC}_\varrho ^\varphi )$ be the multi-valued map defined by

$$\begin{aligned} \mathcal F(v)(t)&= S_\alpha (t)v(0) + \sum \limits _{0<t_k<t}S_\alpha (t - t_k)I_k(v(t_k))\\&\quad +\left\{ \int _0^t (t-s)^{\alpha - 1}P_\alpha (t-s)f(s)\mathrm{d}s: f\in \mathcal P_F^p(v)\right\} , \quad t >0. \end{aligned}$$

Then, v is a fixed point of the solution operator $\mathcal {F}$ iff $u=v[\varphi ]$ is an integral solution of (1.1)–(1.3). To check the closedness of $\mathcal F$, we prove the following lemma.

Lemma 3.1

Let (F) hold. If $\{v_n\}\subset \mathrm{PC}_\varrho ^\varphi$ with $v_n\rightarrow v^*$ and $f_n\in \mathcal P_F^p(v_n)$ then $f_n \rightharpoonup f^*$ in $L_{\mathrm{loc}}^p(\mathbb R^+;X)$ with $f^* \in \mathcal P_F^p(v^*)$.

Proof

Let $\{v_n\}\subset \mathrm{PC}_\varrho ^\varphi$ be such that $v_n\rightarrow v^*, f_n\in \mathcal P_F^p(v_n)$. We see that $\{f_n(t)\}\subset C(t):=\overline{F(t, \{v_n(t), v_n[\varphi ]_t\})}$, which is a compact set for a.e. $t\in \mathbb R^+$, thanks to (F)(1). Let $T>0$ be given. By (F)(3), we see that $\{f_n|_{[0,T]}\}$ is bounded by an $L^p$-integrable function. Thus, $\{f_n\}$ is a semicompact sequence and by [6, Corollary 3.3], it is weakly compact in $L^p(0, T; X)$. So one can assume that $f_n\rightharpoonup f^{1*} \in L^p(0, T; X)$. By Mazur’s lemma (see, e.g., [3]), there exists a sequence $\tilde{f_n}\in \mathrm{co}\{f_{i}: i\ge n\}$ such that $\tilde{f_n}\rightarrow f^{1*}$ in $L^p(0, T; X)$ and then $\tilde{f_n}(t)\rightarrow f^{1*}(t)$ for a.e. $t\in [0, T]$. Since F has compact values, the upper semicontinuity of $F(t,\cdot , \cdot )$ means that for $\epsilon >0$

$$\begin{aligned} F(t, v_n(t), v_n[\varphi ]_t)\subset F(t, v^*(t), v^*[\varphi ]_t)+B_{\epsilon } \end{aligned}$$

for all large n, here $B_{\epsilon }$ denotes the ball in X centered at origin with radius $\epsilon$. So

$$\begin{aligned} f_n(t)\in F(t, v^*(t), v^*[\varphi ]_t)+B_{\epsilon },\quad \text{ for } \text{ a.e. } t\in [0, T]. \end{aligned}$$

By the convexity of $F(t, v^*(t), v^*[\varphi ]_t)+B_{\epsilon }$, the last inclusion still holds for $\tilde{f_n}(t)$ instead of $f_n(t)$. Consequently, $f^{1*}(t)\in F(t, v^*(t), v^*[\varphi ]_t)+B_{\epsilon }$ for a.e. $t\in [0, T]$. Since $\epsilon$ is arbitrary, we obtain the inclusion $f^{1*}(t)\in F(t, v^*(t), v^*[\varphi ]_t)$ for a.e. $t\in [0,T]$.

Repeating the above arguments for $t\in [(j-1)T,jT], j=1,2,...$, we get that $f_n\rightharpoonup f^{j*}$ in $L^p((j-1)T,jT;X)$ with $f^{j*}(t)\in F(t, v^*(t), v^*[\varphi ]_t)$ for a.e. $t\in [(j-1)T, jT]$. Defining the function $f^*\in L_{\mathrm{loc}}^p(\mathbb R^+;X)$ as follows:

$$\begin{aligned} f^*(t) = f^{j*}(t) \quad \text { if } t\in [(j-1)T,jT], \end{aligned}$$

we obtain the conclusion of the lemma. $\square$

We are now able to show the closedness of the solution operator.

Lemma 3.2

Assume that (A), (F) and (I) are satisfied. Then, the solution operator $\mathcal F$ is closed.

Proof

Let $\{v_n\}\subset \mathrm{PC}_\varrho ^\varphi$ be a sequence converging to $v^*$ and $z_n\in \mathcal F(v_n)$ be such that $z_n\rightarrow z^*$. We prove that $z^*\in \mathcal F(v^*)$. By the definition of $\mathcal F$, one can take $f_n\in \mathcal P_F^p(v_n)$ such that

$$\begin{aligned} z_n(t) = S_\alpha (t)\varphi (0) + \sum \limits _{0<t_k<t}S_\alpha (t - t_k)I_k(v_n(t_k)) + Q_\alpha (f_n)(t),\quad \; t>0, \end{aligned}$$

(3.1)

where $Q_\alpha$ is defined in (2.16). By Lemma 3.1, $f_n\rightharpoonup f^*$ in $L^p_{\mathrm{loc}}(\mathbb R^+;X)$ with $f^*\in \mathcal P_F^p(v^*)$. We will show that

$$\begin{aligned} z^*(t) = S_\alpha (t)\varphi (0) + \sum \limits _{0<t_k<t}S_\alpha (t - t_k)I_k(v^*(t_k)) +Q_\alpha (f^*)(t), \quad \; t>0. \end{aligned}$$

(3.2)

Let $t>0$, take $T>0$ such that $t\le T$ and consider the sequence $\{f_n|_{[0,T]}\}$. As argued in the proof of Lemma 3.1, this sequence is semicompact. Then by Proposition 2.5, $Q_\alpha (f_n)\rightarrow Q_\alpha (f^*)$ in C([0, T]; X) and, in particular, $Q_\alpha (f_n)(t)\rightarrow Q_\alpha (f^*)(t)$ in X. Thanks to the continuity of g and $I_k$, one can pass to the limit in (3.1) to get (3.2). The proof is complete. $\square$

In the sequel, we show the condensivity of the solution operator.

Lemma 3.3

Let the hypotheses of Lemma 3.2 hold. If

$$\begin{aligned}&\ell := M_A\sum \limits _{k\in \Lambda }\mu _k+8 \sup _{t>0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi k(s)\mathrm{d}s <\infty , \end{aligned}$$

(3.3)

then

$$\begin{aligned} \chi _\infty (\mathcal F(D))\le \ell \cdot \chi _\infty (D), \end{aligned}$$

for all bounded sets $D\subset \mathrm{PC}_\varrho ^\varphi$.

Proof

Let $D\subset \mathrm{PC}_\varrho ^\varphi$ be a bounded set. For $v\in D$, one can write $\mathcal F(v) = \mathcal F_1(v)+\mathcal F_2(v)$, where

$$\begin{aligned} \mathcal {F}_1(v)(t)&= S_\alpha (t)\varphi (0) + \sum \limits _{0<t_k<t}S_\alpha (t-t_k)I_k(v(t_k)),\\ \mathcal {F}_2(v)(t)&= Q_\alpha \circ \mathcal P_F^p(v)(t). \end{aligned}$$

Using the same arguments as those in [11, Lemma 3.1], we get

$$\begin{aligned}&\chi _\infty (\mathcal F_1(D))\le \left( M_A\sum \limits _{k\in \Lambda }\mu _k\right) \chi _\infty (D), \end{aligned}$$

(3.4)

where $\chi _\infty$ is the MNC defined in (2.2). Regarding $\mathcal F_2(D)$, we observe that $\Omega =\mathcal P_F^p(D)|_{[0,T]}$ is bounded in $L^p(0,T;X)$, then $\pi _T(\mathcal F_2(D))=Q_\alpha (\Omega )$ obeys the following estimate:

$$\begin{aligned} \chi _{\mathrm{PC}}(\pi _T(\mathcal F_2(D))) \le 4 \sup _{t\in [0,T]}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi \cdot \chi (\Omega (s))\mathrm{d}s, \end{aligned}$$

(3.5)

due to Proposition 2.5. Now deploying (F)(4), one has

$$\begin{aligned} \chi (\Omega (s))&\le \chi (F(s, D(s), D[\varphi ]_s))\\&\le k(s)[\chi (D(s)) + \sup _{r\in [-h,0]}\chi (D(s+r))]\\&\le 2k(s)\chi _{\mathrm{PC}}(\pi _T(D)). \end{aligned}$$

Putting the last estimate in (3.5) yields

$$\begin{aligned} \chi _{\mathrm{PC}}(\pi _T(\mathcal F_2(D))) \le \left( 8 \sup _{t\in [0,T]}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi k(s)\mathrm{d}s\right) \chi _{\mathrm{PC}}(\pi _T(D)). \end{aligned}$$

Therefore

$$\begin{aligned} \chi _\infty (\mathcal F_2(D))\le \left( 8 \sup _{t\ge 0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi k(s)\mathrm{d}s\right) \chi _\infty (D). \end{aligned}$$

(3.6)

Combining (3.4) and (3.6), we arrive at

$$\begin{aligned} \chi _\infty (\mathcal F(D))\le \ell \cdot \chi _\infty (D), \end{aligned}$$

(3.7)

where $\ell$ is defined by (3.3). The proof is complete. $\square$

Lemma 3.4

Let the hypotheses of Lemma 3.2 hold. If

$$\begin{aligned}&\vartheta :=\sup _{t>0}\int _0^{\sigma t}\frac{\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s<\infty , \end{aligned}$$

(3.8)

$$\begin{aligned}&\kappa :=\sup _{t>0}\int _{\sigma t}^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s < \infty , \end{aligned}$$

(3.9)

for some $\sigma \in (0,1)$, then

$$\begin{aligned} d_\infty (\mathcal F(D))\le 2\kappa \cdot d_\infty (D), \end{aligned}$$

(3.10)

for all bounded sets $D\subset \mathrm{PC}_\varrho ^\varphi$.

Proof

Let $D\subset \mathrm{PC}_\varrho ^\varphi$ be a bounded set. Using the decomposition $\mathcal F=\mathcal F_1+\mathcal F_2$ as in the proof of Lemma 3.3, we first demonstrate that

$$\begin{aligned} d_\infty (\mathcal F_1(D)) = 0. \end{aligned}$$

To this end, for given $\epsilon >0$, we have to prove the existence of $T>0$ such that

$$\begin{aligned} \frac{\Vert \mathcal F_1(v)(t)\Vert }{\varrho (t)}<\epsilon , \quad \;\forall t\ge T, \quad v\in D. \end{aligned}$$

Take $R>0$ such that $\sup \{\Vert v\Vert _\varrho : v\in D\}\le R$ and $T_1>0$ such that

$$\begin{aligned} \frac{1}{\varrho (t)}\Vert S_\alpha (t)\Vert \Vert \varphi \Vert _h < \frac{\epsilon }{3}, \quad \forall t\ge T_1. \end{aligned}$$

For $\mathcal F_1$, we choose $N_0\in \Lambda$ such that

$$\begin{aligned} R M_A\sum _{k>N_0}l_k < \frac{\epsilon }{3}, \end{aligned}$$

(3.11)

and, in addition, take $T_2>0$ such that

$$\begin{aligned} \frac{\Vert S_\alpha (t)\Vert }{\varrho (t)}R\sum _{k\le N_0}l_k<\frac{\epsilon }{3}, \forall t\ge T_2. \end{aligned}$$

(3.12)

Then for any $v\in D$, using (I)(2) we get

$$\begin{aligned} \frac{\Vert \mathcal F_1(v)(t)\Vert }{\varrho (t)}&\le \frac{1}{\varrho (t)}\Vert S_\alpha (t)\Vert \Vert \varphi \Vert _h + \frac{1}{\varrho (t)}\sum _{k\in \Lambda }\Vert S_\alpha (t-t_k)\Vert \Vert I_k(v(t_k))\Vert \\&\le \frac{\epsilon }{3} +\frac{1}{\varrho (t)}\sum _{k\in \Lambda }\Vert S_\alpha (t-t_k)\Vert l_k \Vert v(t_k)\Vert \\&\le \frac{\epsilon }{3} +\frac{R}{\varrho (t)}\sum _{k\le N_0}\Vert S_\alpha (t-t_k)\Vert l_k + \frac{R M_A}{\varrho (t)}\sum _{k>N_0}l_k\\&\le \frac{\epsilon }{3} +R\sum _{k\le N_0}\frac{\Vert S_\alpha (t-t_k)\Vert }{\varrho (t-t_k)}l_k + R M_A\sum _{k>N_0}l_k\\&< \frac{\epsilon }{3} + \frac{\epsilon }{3}+\frac{\epsilon }{3} = \epsilon ,\quad \forall t\ge \max \{T_1,T_2+t_{N_0}\}, \end{aligned}$$

thanks to (3.11) and (3.12) and the fact that $\varrho (\cdot )$ is nondecreasing and $\varrho (t)\ge 1, \forall t\ge 0$.

We are in a position to evaluate $d_\infty (\mathcal F_2(D))$. Let $z\in \mathcal F_2(v), v\in D$. Taking $f\in \mathcal P_F^p(v)$ such that

$$\begin{aligned} z(t) = \int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)f(s)\mathrm{d}s,\quad t>0, \end{aligned}$$

we have

$$\begin{aligned} \frac{\Vert z(t)\Vert }{\varrho (t)}&\le \int _0^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}\frac{\Vert f(s)\Vert }{\varrho (s)}\mathrm{d}s\nonumber \\&\le \left( \int _0^{\sigma t} +\int _{\sigma t}^t \right) \Theta (t,s) \mathrm{d}s, \end{aligned}$$

(3.13)

where

$$\begin{aligned} \Theta (t,s) = \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert m(s)}{\varrho (t-s)}\frac{\Vert v(s)\Vert + \Vert v[\varphi ]_s\Vert _h}{\varrho (s)}, \end{aligned}$$

thanks to (F)(3). Let $t>0$ such that $\sigma t - h > 0$. Then, we get, for $s\in [0,\sigma t]$,

$$\begin{aligned} \frac{\Vert v(s)\Vert + \Vert v[\varphi ]_s\Vert _h}{\varrho (s)}&= \frac{1}{\varrho (s)}(\Vert v(s)\Vert + \sup _{\tau \in [-h,0]}\Vert v[\varphi ](s+\tau )\Vert )\\&\le \frac{1}{\varrho (s)}(\Vert v(s)\Vert + \sup _{\tau \in [-h,0]}\Vert \varphi (\tau )\Vert + \sup _{\tau \in [0,s]}\Vert v(\tau )\Vert )\\&\le \Vert \varphi \Vert _h +\frac{1}{\varrho (s)}(\Vert v(s)\Vert + \sup _{\tau \in [0,s]}\Vert v(\tau )\Vert )\\&\le \Vert \varphi \Vert _h +\frac{\Vert v(s)\Vert }{\varrho (s)} + \sup _{\tau \in [0,s]}\frac{\Vert v(\tau )\Vert }{\varrho (\tau )}\\&\le \Vert \varphi \Vert _h+2R. \end{aligned}$$

Hence

$$\begin{aligned} \int _0^{\sigma t} \Theta (t,s)\mathrm{d}s&\le 2R\int _0^{\sigma t} \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s\nonumber \\&\le \frac{\Vert \varphi \Vert _h+2R}{[(1-\sigma ) t]^{1-\alpha }} \int _0^{\sigma t}\frac{\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s\nonumber \\&\le \frac{(\Vert \varphi \Vert _h+2R)\vartheta }{[(1-\sigma ) t]^{1-\alpha }}, \end{aligned}$$

(3.14)

where $\vartheta$ is given in (3.8). On the other hand, for $s \ge \sigma t$ we see that

$$\begin{aligned} \frac{\Vert v(s)\Vert + \Vert v[\varphi ]_s\Vert _h}{\varrho (s)}&= \frac{1}{\varrho (s)}(\Vert v(s)\Vert + \sup _{\tau \in [-h,0]}\Vert v[\varphi ](s+\tau )\Vert )\\&\le \frac{\Vert v(s)\Vert }{\varrho (s)} + \sup _{\tau \in [-h,0]}\frac{\Vert v[\varphi ](s+\tau )\Vert }{\varrho (s+\tau )}\\&\le \sup _{r\ge \sigma t}\frac{\Vert v(r)\Vert }{\varrho (r)} + \sup _{r\ge \sigma t-h}\frac{\Vert v(r)\Vert }{\varrho (r)}\le 2\sup _{r\ge \sigma t-h}\frac{\Vert v(r)\Vert }{\varrho (r)}. \end{aligned}$$

Then

$$\begin{aligned} \int _{\sigma t}^t \Theta (t,s)\mathrm{d}s&\le \left( \int _{\sigma t}^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s\right) 2\sup _{r\ge \sigma t-h}\frac{\Vert v(r)\Vert }{\varrho (r)}\nonumber \\&\le 2\kappa \sup _{r\ge \sigma t-h}\frac{\Vert v(r)\Vert }{\varrho (r)}, \end{aligned}$$

(3.15)

where $\kappa$ is defined by (3.9). Now using (3.14) and (3.15) in (3.13), we have

$$\begin{aligned} \frac{\Vert z(t)\Vert }{\varrho (t)}&\le \frac{(\Vert \varphi \Vert _h+2R)\vartheta }{[(1-\sigma ) t]^{1-\alpha }} + 2\kappa \sup _{r\ge \sigma t-h}\frac{\Vert v(r)\Vert }{\varrho (r)}, \end{aligned}$$

for all $t>\frac{h}{\sigma }$, $v\in D, z\in \mathcal F_2(v)$. The last inequality implies

$$\begin{aligned} d_\infty (\mathcal F_2(D))\le 2\kappa \cdot d_\infty (D). \end{aligned}$$

The proof is complete. $\square$

Combining Lemmas 3.3 and 3.4, we arrive at the following assertion.

Lemma 3.5

Let the assumptions (A), (F) and (I) hold. Then, the solution operator $\mathcal F$ is $\chi ^*$-condensing provided that

$$\begin{aligned}&\ell = M_A\sum \limits _{k\in \Lambda }\mu _k+8 \sup _{t> 0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi k(s)\mathrm{d}s<1,\end{aligned}$$

(3.16)

$$\begin{aligned}&\vartheta =\sup _{t>0}\int _0^{\sigma t}\frac{\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s<\infty , \end{aligned}$$

(3.17)

$$\begin{aligned}&\kappa =\sup _{t>0}\int _{\sigma t}^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s < \frac{1}{2}, \end{aligned}$$

(3.18)

for some $\sigma \in (0,1)$.

Proof

Using Lemmas 3.3 and 3.4, one gets, for all bounded sets $D\subset \mathrm{PC}_\varrho ^\varphi$,

$$\begin{aligned} \chi _\infty (\mathcal F(D)) + d_\infty (\mathcal F(D))\le \max \{\ell , 2\kappa \}\cdot (\chi _\infty (D) + d_\infty (D)), \end{aligned}$$

that is,

$$\begin{aligned} \chi ^*(\mathcal F(D))\le \max \{\ell , 2 \kappa \}\cdot \chi ^*(D). \end{aligned}$$

If $\chi ^*(\mathcal F(D))\ge \chi ^*(D)$ then $\chi ^*(D)\le \max \{\ell , 2\kappa \}\cdot \chi ^*(D)$, which ensures that $\chi ^*(D)=0$. Therefore, D is relatively compact due to Lemma 2.1. We get the conclusion as desired. $\square$

Theorem 3.6

Let the hypotheses of Lemma 3.5 hold. Assume that

$$\begin{aligned} M_A \sum _{k\in \Lambda }l_k + 2\sup _{t>0}\int _0^t\frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert \,m(s)}{\varrho (t-s)}\mathrm{d}s <1. \end{aligned}$$

(3.19)

Then, problem (1.1)–(1.3) has at least one integral solution in PC$_\varrho$.

Proof

By Lemma 3.2, $\mathcal F$ is closed. In addition, $\mathcal F$ is $\chi ^*$-condensing due to Lemma 3.5. Moreover, $\mathcal F$ has compact values. Indeed, for $v\in \mathrm{PC}_\varrho ^\varphi$, we have

$$\begin{aligned} \chi ^*(\mathcal F(v))\le \max \{\ell , 2\kappa \}\cdot \chi ^*(\{v\})=0. \end{aligned}$$

It follows that $\chi ^*(\mathcal F(v))=0$ and then $\mathcal F(v)$ is a relatively compact set due to Lemma 2.1. Thanks to the closedness of $\mathcal F$, $\mathcal F(v)$ is compact.

To apply Theorem 2.2, it suffices to show that there exists $R>0$ such that

$$\begin{aligned} \mathcal F(\mathbf B _R)\subset \mathbf B _R, \end{aligned}$$

where $\mathbf B _R$ is the closed ball in PC$_\varrho ^\varphi$, centered at origin with radius R.

We first check that $\mathcal F(\mathrm{PC}_\varrho ^\varphi )\subset \mathrm{PC}_\varrho ^\varphi$. Let $v\in \mathrm{PC}_\varrho ^\varphi$, then $d_\infty (\{v\})=0$. Using (3.10), we have $d_\infty (\mathcal F(v))=0$. Then it follows that $\mathcal F(v)\subset \mathrm{PC}_\varrho ^\varphi$.

Now, we prove that $\mathcal F(\mathbf B _R)\subset \mathbf B _R$ for some $R>0$. Assume to the contrary that for each $n\in \mathbb N$, there exists $v_n\in \mathbf B _n$ and $z_n\in \mathcal F(v_n)$ such that $\Vert z_n\Vert _\varrho >n$. Taking $f_n\in \mathcal P_F^p(v_n)$ such that

$$\begin{aligned} z_n(t)&= S_\alpha (t)\varphi (0) + \sum \limits _{0<t_k<t}S_\alpha (t-t_k)I_k(v_n(t_k))\\&\quad +\int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)f_n(s)\mathrm{d}s, \end{aligned}$$

we observe that, for all $t\ge 0$,

$$\begin{aligned} \frac{\Vert z_n(t)\Vert }{\varrho (t)}&\le \frac{\Vert S_\alpha (t)\Vert }{\varrho (t)}\Vert \varphi \Vert _h + \sum _{0<t_k<t} \frac{\Vert S_\alpha (t-t_k) \Vert }{\varrho (t-t_k)} \frac{\Vert v_n(t_k)\Vert }{\varrho (t_k)}l_k\\&\quad + \int _0^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)} \frac{\Vert v_n(s)\Vert +\Vert v_n[\varphi ]_s\Vert _h}{\varrho (s)}m(s)\mathrm{d}s, \end{aligned}$$

thanks to (I)(2) and (F)(3). Noting that $\dfrac{\Vert S_\alpha (t)\Vert }{\varrho (t)} \le \Vert S_\alpha (t)\Vert \le M_A, \;\forall t\ge 0,$ and

$$\begin{aligned} \frac{\Vert v_n(s)\Vert }{\varrho (s)}&\le n,\quad \;\forall s\ge 0,\\ \frac{\Vert v_n[\varphi ]_s\Vert _h}{\varrho (s)}&\le \frac{1}{\varrho (s)}\sup _{r\in [s-h,s]}\Vert v_n[\varphi ](r)\Vert \\&\le \frac{1}{\varrho (s)}\left( \sup _{r\in [-h,0]}\Vert \varphi (r)\Vert + \sup _{r\in [0,s]}\Vert v_n(r)\Vert \right) \\&\le \Vert \varphi \Vert _h + \sup _{r\in [0,s]}\frac{1}{\varrho (r)}\Vert v_n(r)\Vert \\&\le \Vert \varphi \Vert _h + \Vert v_n\Vert _\varrho \le \Vert \varphi \Vert _h + n, \end{aligned}$$

we get

$$\begin{aligned} \frac{\Vert z_n(t)\Vert }{\varrho (t)} \le M_A \Vert \varphi \Vert _h&+ nM_A\sum _{k\in \Lambda }l_k \\&+ (2n + \Vert \varphi \Vert _h) \int _0^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s. \end{aligned}$$

This implies

$$\begin{aligned} 1<\frac{\Vert z_n\Vert _\varrho }{n}&= \frac{1}{n}\sup _{t>0}\frac{\Vert z_n(t)\Vert }{\varrho (t)}\\&\le \frac{\Vert \varphi \Vert _h }{n}\left( M_A + \sup _{t>0} \int _0^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s\right) \\&\qquad \qquad + M_A\sum _{k\in \Lambda }l_k + 2\sup _{t>0} \int _0^t \frac{(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert }{\varrho (t-s)}m(s)\mathrm{d}s. \end{aligned}$$

Passing to the limit in the last inequality, we get a contradiction with (3.19). The proof is complete. $\square$

4 Weak stability result

In this section, we replace the assumptions (A) and (F) by stronger ones:

• (A*) The semigroup $W(\cdot )$ generated by A is norm-continuous and exponentially stable, i.e., there exists $\beta >0$ such that

  $$\begin{aligned} \Vert W(t)x\Vert \le M_A \mathrm{e}^{-\beta t}\Vert x\Vert ,\quad \forall t\ge 0,\,\, x\in X. \end{aligned}$$

• (F*) The multi-valued nonlinearity function F satisfies (F) with $m\in L^1(\mathbb R^+)\cap L^p_{\mathrm{loc}}(\mathbb R^+)$.

It is proved in [11] that, by (A*) the fractional resolvent operators $S_\alpha (\cdot ), P_\alpha (\cdot )$ are asymptotically stable, i.e.

$$\begin{aligned} \Vert S_\alpha (t)\Vert , \Vert P_\alpha (t)\Vert \rightarrow 0 \quad \text { as }\quad t\rightarrow +\infty . \end{aligned}$$

By choosing $\varrho (t)\equiv 1$, we now consider the solution operator $\mathcal F$ on the space

$$\begin{aligned} \mathrm{PC}_0 = \left\{ u\in \mathrm{PC}(\mathbb R^+;X): \lim \limits _{t\rightarrow +\infty }u(t)=0\right\} , \end{aligned}$$

endowed with the norm $\Vert u\Vert _\infty = \sup \nolimits _{t\ge 0}\Vert u(t)\Vert$.

Following the same arguments as in Sect. 3, we get the existence of global attracting solutions as follows.

Theorem 4.1

Let (A*), (F*) and (I) hold. Then, the problem (1.1)–(1.3) admits an integral solution such that $\Vert u(t)\Vert = o(1)$ as $t\rightarrow +\infty$, provided that

$$\begin{aligned}&\ell = M_A\sum \limits _{k\in \Lambda }\mu _k+8 \sup _{t\ge 0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert _\chi k(s)\mathrm{d}s<1,\end{aligned}$$

(4.1)

$$\begin{aligned}&\varpi = M_A \sum _{k\in \Lambda }l_k + 2\sup _{t>0}\int _0^t(t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert m(s)\mathrm{d}s <1. \end{aligned}$$

(4.2)

Proof

We verify the assumptions of Lemma 3.5 and Theorem 3.6. Since $m\in L^1(\mathbb R^+)$, one sees that the condition (3.17) is fulfilled. In addition, condition (3.18) follows from (4.2), while condition (4.2) is exactly (3.19). $\square$

Now, we state the main result of this section.

Theorem 4.2

Let the assumptions of Theorem 4.1 hold. Then, the zero solution of (1.1) is weakly asymptotically stable.

Proof

Let $\Sigma (\varphi )$ be the set of integral solutions of (1.1)–(1.3) with respect to the initial datum $\varphi$. Obviously, we have $0\in \Sigma (0)$ thanks to the fact that $F(t,0,0)=0$ and $I_k(0)=0, k\in \Lambda$. By Theorem 4.1 we observe that, for each $\varphi \in C([-h,0];X)$ there exists $u\in \Sigma (\varphi )$ such that $\Vert u(t)\Vert \rightarrow 0$ as $t\rightarrow +\infty$. Then, we get $\Vert u_t\Vert _h \rightarrow 0$ as $t\rightarrow +\infty$, that is, the zero solution is weakly attractive. It remains to show that this solution is stable.

Let $\varphi \in C([-h,0];X)$ and $u\in \Sigma (\varphi )$. Then, there exists $f\in \mathcal P_F^p(u)$ such that

$$\begin{aligned} u(t)&= S_\alpha (t)\varphi (0) + \sum _{0<t_k<t}S_\alpha (t-t_k)I(u(t_k))\\& + \int _0^t (t-s)^{\alpha -1}P_\alpha (t-s)f(s)\mathrm{d}s,\quad \;t>0. \end{aligned}$$

Now, using the assumptions (F*) and (I), we get

$$\begin{aligned} \Vert u(t)\Vert&\le M_A \Vert \varphi \Vert _h + M_A\Vert u\Vert _\infty \sum _{k\in \Lambda }l_k\\&\quad + (2\Vert u\Vert _\infty +\Vert \varphi \Vert _h)\sup _{t>0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert m(s)\mathrm{d}s,\quad \; t>0. \end{aligned}$$

The last estimate implies

$$\begin{aligned} \Vert u\Vert _\infty&\le \bigg [M_A\sum _{k\in \Lambda }l_k + 2\sup _{t >0}\int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert m(s)\mathrm{d}s\bigg ]\Vert u\Vert _\infty \\&\quad + (M_A+1)\Vert \varphi \Vert _h. \end{aligned}$$

Hence

$$\begin{aligned} \Vert u_t\Vert _h \le \Vert \varphi \Vert _h + \Vert u\Vert _\infty \le \left( 1+\frac{1+M_A}{1-\varpi }\right) \Vert \varphi \Vert _h,\quad \forall t>0, \end{aligned}$$

where $\varpi$ is given by (4.2). The last inequality ensures the stability of the zero solution. The proof is complete. $\square$

5 Application

In this section, the obtained abstract results will be demonstrated in the following lattice differential system

$$\begin{aligned} \frac{\mathrm{d}^\alpha }{\mathrm{d}t^\alpha }u_i (t)&= (\mathrm{Au}(t))_i + f_i(t),\; \;\;t>0, \;\;t\ne t_k,\;\; k\in \mathbb N,\end{aligned}$$

(5.1)

$$\begin{aligned} f_i(t)&\in [f_{1i}(t, u_i(t), u_i(t-\rho (t))), f_{2i}(t, u_i(t), u_i(t-\rho (t)))],\end{aligned}$$

(5.2)

$$\begin{aligned} \Delta u_i(t_k)&= I_{ik}(u_i(t_k)),\end{aligned}$$

(5.3)

$$\begin{aligned} u_i(s)&= \varphi _i(s),\;\;\;\; s\in [-h,0], \end{aligned}$$

(5.4)

where $u=(u_i): [-h,+\infty ) \rightarrow \mathbb \ell ^2$ is the unknown function, $\dfrac{\mathrm{d}^\alpha }{\mathrm{d}t^\alpha }$ is the Caputo derivative of order $\alpha \in (0,1)$, $A:\ell ^2\rightarrow \ell ^2$ is the linear operator given by

$$\begin{aligned} (Av)_i = v_{i+1}-(2+\lambda ) v_i + v_{i-1}, \;\;\;\;v\in \ell ^2, \end{aligned}$$

$\rho : \mathbb R^+\rightarrow [0,h]$ is a continuous function, and $\lambda$ is a positive number. Here, $\ell ^2$ is the space of sequences $(v_i)_{i\in \mathbb Z}$ satisfying $\sum _{i\in \mathbb Z}v_i^2<\infty$, which becomes a Hilbert space with the inner product $(u,v)_{\ell ^2} = \sum _{i\in \mathbb Z} u_iv_i$, and $[f_1, f_2] = \{\tau f_1 + (1-\tau )f_2: \tau \in [0,1]\}$.

Lattice differential systems like (5.1)–(5.4) come from various problems such as image processing, pattern recognition, electrical engineering, ... In particular, (5.1)–(5.4) can be seen as a model of semi-discretization for the fractional partial differential inclusion

$$\begin{aligned} \frac{\partial ^\alpha }{\partial t^\alpha } u(x,t)&= \frac{\partial ^2}{\partial x^2} u(x,t) -\lambda u(x,t)+ f(x,t),\; \;\;x\in \mathbb R,\;\;\; t>0,\\ f(x,t)&\in [f_1(x,t, u(x,t), u(x, t-\rho (t))), f_2(x,t, u(x,t), u(x, t-\rho (t)))],\\ \Delta u(x,t_k)&= I_k(x, u(x,t_k)),\\ u(x,s)&= \varphi (x,s),\;\;\;\; s\in [-h,0], \end{aligned}$$

where the discretizing is made in spatial variable x.

Let $B:\ell ^2 \rightarrow \ell ^2$ be the linear operator defined by $(Bv)_i = v_{i+1}-v_i$, then its adjoint operator $B^*$ is given by $(B^*v)_i = v_{i-1}-v_i$. In addition, if $\tilde{A}:\ell ^2\rightarrow \ell ^2$ is the operator defined by $(\tilde{A} v)_i = v_{i+1}-2v_i+v_{i-1}$ then

$$\begin{aligned} -\tilde{A} = BB^*= B^*B. \end{aligned}$$

It should be noted that $A=\tilde{A} - \lambda I$ is a bounded operator on $\ell ^2$. Then, the semigroup $\{\mathrm{e}^{tA}: t\ge 0\}$ is uniformly continuous (see, e.g., [7]) and hence norm-continuous. However, this semigroup is non-compact, since it can be extended to a group $\{\mathrm{e}^{tA}: t\in \mathbb R\}$ and the identity operator $I=\mathrm{e}^{tA}\mathrm{e}^{-tA}$ is non-compact.

To obtain the exponential stability of $\{\mathrm{e}^{tA}: t\ge 0\}$, we consider the system

$$\begin{aligned} \frac{\mathrm{d}v(t)}{\mathrm{d}t} = \tilde{A} v(t) - \lambda v(t), \;\;\;\;\; v(t)\in \ell ^2. \end{aligned}$$

Multiplying by v, we get

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert v(t)\Vert ^2&= (\tilde{A} v(t), v(t)) - \lambda \Vert v(t)\Vert ^2\\&= - (B^*B v(t), v(t)) - \lambda \Vert v(t)\Vert ^2 = -\Vert Bv(t)\Vert ^2 - \lambda \Vert v(t)\Vert ^2\\&\le - \lambda \Vert v(t)\Vert ^2. \end{aligned}$$

Then by the Gronwall lemma, we get

$$\begin{aligned} \Vert v(t)\Vert \le \mathrm{e}^{-\lambda t}\Vert v(0)\Vert , \end{aligned}$$

and, therefore, one has the estimate $\Vert \mathrm{e}^{tA}\Vert \le \mathrm{e}^{-\lambda t}, t\ge 0$, i.e., the semigroup $\{\mathrm{e}^{tA}: t\ge 0\}$ is exponentially stable. The assumption (A*) is verified.

Before going to the further settings for (5.1)–(5.4), we recall a result on the Hausdorff MNC in $\ell ^2$ (see, e.g., [2, Theorem 4.2]). Let $R_n:\ell ^2\rightarrow \ell ^2$ be the linear operator defined by

$$\begin{aligned} R_n (v) =\sum _{|i|>n} v_i e_i, \end{aligned}$$

where $e_i = (\delta _{ij})_{j\in \mathbb Z}$. Then, the map $\chi : 2^{\ell ^2}\rightarrow \mathbb R^+$ defined by

$$\begin{aligned} \chi (B) = \limsup _{n\rightarrow +\infty }[\sup _{v\in B}\Vert R_n(v)\Vert ]=\limsup _{n\rightarrow +\infty }\bigg [\sup _{v\in B}\bigg (\sum _{|i|>n}|v_i|^2\bigg )^{\frac{1}{2}}\bigg ] \end{aligned}$$

is the Hausdorff MNC in $\ell ^2$.

Now, we give the following assumptions:

• (N1) The functions $f_{1i}, f_{2i}: \mathbb R^+\times \mathbb R^2\rightarrow \mathbb R, i\in \mathbb Z,$ are continuous and satisfy

  $$\begin{aligned} \quad \;\;\max \{|f_{1i}(t,y, z)|^2, |f_{2i}(t,y, z)|^2\}\le m^2(t)(|y|^2 + |z|^2), \forall (t,\eta ,z)\in \mathbb R^+\times \mathbb R^2, \end{aligned}$$

  where $m\in C(\mathbb R^+;\mathbb R^+)$ satisfies

  $$\begin{aligned} m(t)\le \frac{C_m}{1+t^{\alpha +1}}\quad \text { for some } C_m>0. \end{aligned}$$

• (N2) $I_{ik}:\mathbb R\rightarrow \mathbb R$, $i\in \mathbb Z, k\in \mathbb N$, are continuous functions such that

  $$\begin{aligned} |I_{ik}(y)|\le l_k |y|, \end{aligned}$$

  where $\{l_k: k\in \mathbb N\}$ is a sequence of non-negative numbers such that $\sum _{k\in \mathbb N}l_k<\infty$.

Let $f_1, f_2: \mathbb R^+\times \ell ^2\times C([-h,0];\ell ^2)\rightarrow \ell ^2$ be functions defined by

$$\begin{aligned} f_1(t, v, w)&= (f_{1i}(t, v_i, w_i(-\rho (0))))_{i\in \mathbb Z},\\ f_2(t, v, w)&= (f_{2i}(t, v_i, w_i(-\rho (0))))_{i\in \mathbb Z}. \end{aligned}$$

Then, one can check that $f_1, f_2$ are continuous. Furthermore, it follows from (N1) that

$$\begin{aligned} \Vert f_1(t,v,w)\Vert ^2&= \sum _{i\in \mathbb Z}|f_{1i}(t, v_i, w_i(-\rho (0)))|^2\\&\le m^2(t)\sum _{i\in \mathbb Z}(|v_i|^2 + |w_i(-\rho (0))|^2)\\&= m^2(t)(\Vert v\Vert ^2 + \Vert w(-\rho (0))\Vert ^2)\\&\le m^2(t)(\Vert v\Vert ^2 + \sup _{s\in [-h,0]}\Vert w(s)\Vert ^2). \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert f_2(t,v,w)\Vert ^2\le m^2(t)(\Vert v\Vert ^2 + \sup _{s\in [-h,0]}\Vert w(s)\Vert ^2). \end{aligned}$$

Now putting

$$\begin{aligned} F(t,v,w) = [f_1(t,v,w), f_2(t,v,w)], v\in \ell ^2, w\in C([-h,0];\ell ^2), \end{aligned}$$

we see that

$$\begin{aligned} \Vert F(t,v,w)\Vert \le m(t)(\Vert v\Vert + \Vert w\Vert _h). \end{aligned}$$

In addition, F is a multimap with convex and compact values. Indeed, it is easily seen that for each $(t,v,w)\in \mathbb R^+\times \ell ^2\times C([-h,0];\ell ^2)$, F(t, v, w) is a closed convex set. Moreover, $F(t,v,w)\subset \text {span}\{f_1(t,v,w), f_2(t,v,w)\}$, i.e., F(t, v, w) is a bounded set lying in a two-dimensional subspace of $\ell ^2$, thus F(t, v, w) is a compact set. Since $f_1, f_2$ are continuous, the multimap $(v,w)\mapsto F(t,v,w)$ is closed. This implies that $F(t,\cdot ,\cdot )$ is u.s.c for each $t\in \mathbb R$. Noting that for each $u\in \mathrm{PC}_0$, $\tau \in [0,1]$, the function

$$\begin{aligned} f(t) = \tau f_1(t,u(t), u(t-\rho (t))) + (1-\tau )f_2(t,u(t), u(t-\rho (t))), \tau \in [0,1] \end{aligned}$$

is a (strongly) measurable selection of F. Thus, (F)(1)–(F)(3) are fulfilled.

We now evaluate $\chi (F(t,V,W))$ for bounded sets $V\subset \ell ^2, W\subset C([-h,0];\ell ^2)$. One sees that

$$\begin{aligned}&\sup _{(v,w)\in V\times V}\Vert R_n[f_1(t,v,w)]\Vert = \left( \sum _{|i|>n} |f_{1i}(t, v_i, w_i(-\rho (0)))|^2\right) ^{\frac{1}{2}}\\&\quad \le m(t) \sup _{(v,w)\in V\times V}\left( \sum _{|i|>n} \big [|v_i|^2 + |w_i(-\rho (0))|^2\big ]\right) ^{\frac{1}{2}}\\&\quad \le m(t)\sup _{(v,w)\in V\times V}\bigg [\left( \sum _{|i|>n}|v_i|^2\right) ^{\frac{1}{2}} + \left( \sum _{|i|>n}|w_i(-\rho (0))|^2\right) ^{\frac{1}{2}}\bigg ]\\&\quad \le m(t)\bigg [\sup _{v\in V}\left( \sum _{|i|>n}|v_i|^2\right) ^{\frac{1}{2}} + \sup _{w\in W}\left( \sum _{|i|>n}|w_i(-\rho (0))|^2\right) ^{\frac{1}{2}}\bigg ]\\&\quad = m(t)\bigg [\sup _{v\in V}\Vert R_n(v)\Vert + \sup _{w\in W}\Vert R_n(w(-\rho (0)))\Vert \bigg ]. \end{aligned}$$

Passing to the limit in the last inequality yields

$$\begin{aligned} \chi (f_1(t,V,W))&\le m(t)\big [\chi (V) + \chi (W(-\rho (0)))\big ]\\&\le m(t)\big [\chi (V) + \sup _{s\in [-h,0]}\chi (W(s))\big ]. \end{aligned}$$

By the same arguments for $f_2$, we have

$$\begin{aligned} \chi (f_2(t,V,W)) \le m(t)\bigg [\chi (V) + \sup _{s\in [-h,0]}\chi (W(s))\bigg ]. \end{aligned}$$

Observing that

$$\begin{aligned} F(t,V,W)\subset \text {co}\{f_1(t,V,W)\cup f_2(t,V,W)\}, \end{aligned}$$

one gets

$$\begin{aligned} \chi (F(t,V,W))&\le \chi \left( f_1(t,V,W)\cup f_2(t,V,W)\right) \\&\le \max \{\chi \left( f_1(t,V,W)\right) , \chi \left( f_2(t,V,W)\right) \}\\&\le m(t)\bigg [\chi (V) + \sup _{s\in [-h,0]}\chi (W(s))\bigg ]. \end{aligned}$$

Then, (F)(4) is satisfied and, therefore, (F*) is testified with $k=m$.

Now, consider the maps $I_k:\ell ^2\rightarrow \ell ^2, k\in \mathbb N$, defined by

$$\begin{aligned} I_k(v) = (I_{ik}(v_i))_{i\in \mathbb Z}. \end{aligned}$$

Then, by the continuity of $I_{ik}$, one gets the continuity of $I_k$. Moreover, by (N2) we have

$$\begin{aligned} \Vert I_k(v)\Vert&= \left( \sum _{i\in \mathbb Z}|I_{ik}(v_i)|^2 \right) ^{\frac{1}{2}} \le l_k \left( \sum _{i\in \mathbb Z}|v_i|^2\right) ^{\frac{1}{2}} \\&= l_k\Vert v\Vert . \end{aligned}$$

Hence, (I)(1) is satisfied. Now, let $V\subset \ell ^2$ be a bounded set. Then

$$\begin{aligned} \sup _{v\in V}\Vert R_n(I_k(v))\Vert&= \sup _{v\in V}\left( \sum _{|i|>n}|I_{ik}(v_i)|^2\right) ^{\frac{1}{2}}\\&\le l_k\sup _{v\in V}\left( \sum _{|i|>n}|v_i|^2\right) ^{\frac{1}{2}} = l_k \sup _{v\in V}\Vert R_n(v)\Vert . \end{aligned}$$

Taking the limit of the last inequality as $n\rightarrow +\infty$, we arrive at

$$\begin{aligned} \chi (I_k(V))\le l_k\chi (V). \end{aligned}$$

So the assumption (I) holds with $\mu _k=l_k, k\in \mathbb N,$ provided that $\inf \{t_{k+1}-t_k: k\in \mathbb N\}>0$.

Finally, we give an estimate for the integral

$$\begin{aligned} I(t) = \int _0^t (t-s)^{\alpha -1}\Vert P_\alpha (t-s)\Vert m(s)\mathrm{d}s. \end{aligned}$$

Observe that in our case $\Vert \mathrm{e}^{tA}\Vert \le 1,$ then $\Vert P_\alpha (t)\Vert \le \frac{1}{\Gamma (\alpha )}, \forall t\ge 0$. Then

$$\begin{aligned} I(t)&\le \frac{C_m}{\Gamma (\alpha )}\left( \int _0^{\frac{t}{2}}\frac{(t-s)^{\alpha -1}}{1+s^{\alpha +1}}\mathrm{d}s + \int _{\frac{t}{2}}^t\frac{(t-s)^{\alpha -1}}{1+s^{\alpha +1}}\mathrm{d}s\right) \\&\le \frac{C_m}{\Gamma (\alpha )}\left( \left( \frac{t}{2}\right) ^{\alpha -1}\int _0^{\frac{t}{2}}\frac{\mathrm{d}s}{1+s^{\alpha +1}}+\frac{1}{1+\left( \frac{t}{2}\right) ^{\alpha +1}}\int _{\frac{t}{2}}^t(t-s)^{\alpha -1}\mathrm{d}s\right) \\&\le \frac{C_m}{\Gamma (\alpha )}\left( J(t) + \frac{1}{\alpha }\right) , \end{aligned}$$

where

$$\begin{aligned} J(t) = \left( \frac{t}{2}\right) ^{\alpha -1}\int _0^{\frac{t}{2}}\frac{\mathrm{d}s}{1+s^{\alpha +1}}. \end{aligned}$$

Noting that

$$\begin{aligned} \lim _{t\rightarrow 0}J(t) = \lim _{t\rightarrow +\infty }J(t) = 0, \end{aligned}$$

we get $\sup _{t > 0}J(t)<\infty$, then $\sup _{t>0}I(t)<\infty$. It follows that the conditions (4.1) and (4.2) are testified with $C_m, l_k$ small, and we get the weakly asymptotic stability of zero solution to (5.1) and (5.2).

6 Conclusion

A unified approach is proposed to prove the global solvability and weakly asymptotic stability for the semilinear fractional differential inclusion involving impulsive effects given by (1.1)–(1.3). That is, we analyze the fixed point set of the solution operator $\mathcal F$ associated with our problem on PC$_\varrho$, the space of piecewise continuous functions on the half-line with weighted function $\varrho$, on which the MNC $\chi ^*$ is defined to ensure a compactness condition. In the case when the semigroup $\{\mathrm{e}^{tA}\}_{t\ge 0}$ is exponentially stable, the existence of decay solutions to (1.1)–(1.3) implies the weak attractivity of the zero solution that leads to the weakly asymptotic stability as mentioned. In further analysis, one can estimate the decay rate of solutions provided that the impulsive condition (1.2) is relaxed or the Caputo derivative in (1.1) is replaced by a substantial Caputo derivative (see, e.g [5]).