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Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

Abstract

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order \(1<\alpha <2\) and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval \(J=[0,\infty )\). We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

Introduction

Fractional calculus has become a well-established branch of mathematical analysis. It has many applications in engineering and science. Much work has appeared studying various models involving fractional differential boundary value problems and providing solutions to those models using analytical methods or numerical methods. We highlight some recent work involving fractional differential equations. Agarwal et al. [1] investigated existence and uniqueness results on time scales for fractional nonlocal thermistor problems in the conformable sense. Sunarto et al. [2] developed a numerical method using a quarter-sweep and PAOR to solve a one-dimensional time-fractional mathematical physics model. Rezapour et al. [3] showed the existence and uniqueness of solutions for a general multi-term fractional BVP involving the generalized ψ-RL operators. Then they suggested two numerical algorithms, namely, the Dafterdar-Gejji and Jafari method (DGJIM) and the Adomian decomposition method (ADM) in which a series of approximate solutions converge to the exact ones. Agarwal et al. [4] discussed the existence and uniqueness of solutions for a nonlocal problem with integral transmitting condition for mixed parabolic-hyperbolic type equations with Caputo fractional derivative. Agarwal et al. [5] provided a detailed description of impulsive fractional differential equations using Lyapunov functions and overviewed results for the stability in Caputo’s sense. Khan et al. [6] focused on the existence and uniqueness of solutions and Hyers–Ulam stability for ABC-fractional DEs with p-Laplacian operator. Khan et al. [7] studied the stability and numerical simulation of a fractional order plant-nectar-pollinator model. Khan et al. [8] proved the existence and Hyers–Ulam stability of solutions to a class of hybrid fractional differential equations with p-Laplacian operator.

The problem of existence of non-constant periodic solutions for fractional order models has became one of the most interesting topics to conduct research on. This is particularly due to the differences between systems of integers order and systems of fractional orders in terms of the existence of non-constant periodic solutions. In much work, such as [914], the authors have shown that non-constant periodic solutions of fractional order systems do not exist contrary to the case where the order of the system is an integer. Therefore, the concept of an asymptotically periodic solution for fractional differential equations or inclusions is introduced and discussed in much work. For example, in [12, 1517], the authors considered semilinear differential equations of order \(\alpha \in (0,1)\) generated by a \(C_{0}\)-semigroup, while the papers [1820] addressed semilinear differential equations of order \(\alpha \in (0,1)\) generated by sectorial operators. Moreover, the asymptotically periodic solutions for delayed fractional differential equations with almost sectorial operator of order \(\alpha \in (0,1)\) are examined in [21]. Rogovchenko et al. [22] studied the asymptotic properties of solutions for a certain classes of second order nonlinear differential equations. Very recently, Wang et al. [23] discussed the asymptotic behavior of solutions to time-fractional neutral functional differential equations of order \(\alpha \in (0,1)\).

For more information regarding this subject, we refer the reader to [24, 25].

It is worth noting that the problems discussed in all cited work above do not contain impulse effects, whether it is instantaneous or non-instantaneous, and the nonlinear term is a single-valued function.

Let \(\alpha \in (1,2)\), E be a Banach space, m be a natural number, \(\omega >0\), \(J=[0,\infty )\),

$$ 0=s_{0}< \theta _{1}\leq s_{1}< \theta _{2}< \cdots < \theta _{m}\leq s_{m}= \omega < \theta _{m+1}=\omega +\theta _{1}\leq s_{m+1}=\omega +s_{1}< \cdots, $$

with \(\lim_{i\rightarrow \infty }\theta _{i}=\infty \), \(s_{m+i}=s_{i}+ \omega \), and \(\theta _{m+i}=\theta _{i}+\omega \); \(i\in \mathbb{N} =\{1,2,3,\ldots\}\) and \(A :D(A)\subseteq E\rightarrow E\) be a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\), where \(M>0\), \(\varphi \in (0,\frac{\pi }{2})\) and \(\mu \in \mathbb{R} \). Moreover, let \(F:J\times E\rightarrow 2^{E}-\{\phi \}\), be a multivalued function, \(g_{i}:[\theta _{i},s_{i}]\times E\longrightarrow E \); \(i\in \mathbb{N} \), \(x_{0}\in D(A)\) and \(x_{1}\in E\).

Motivated by the above cited work, we prove two results concerning the existence of S-asymptotically ω-periodic mild solutions to the following non-instantaneous impulsive semilinear differential inclusion:

$$ \textstyle\begin{cases} {}^{c}D_{s_{i},\theta }^{\alpha }x(\theta )\in Ax(\theta )+F(\theta ,x( \theta )),\quad \text{a.e. }\theta \in (s_{i}, \theta _{i+1}],i \in \mathbb{N} \cup \{0\}, \\ x(\theta _{i}^{+})=g_{i}(\theta _{i},x(\theta _{i}^{-})),\quad i\in \mathbb{N} , \\ x(\theta )=g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i} s_{i}],i\in \mathbb{N} , \\ x(0)=x_{0},\qquad x'(0)=x_{1}, \end{cases} $$
(1)

where \({}^{c}D_{s_{i},\theta }^{\alpha }x(\theta )\) is the Caputo derivative of the function x at the point θ and with lower limit at \(s_{i}\) [26]. In the first result, we apply a fixed point theorem for contraction multivalued functions, and, in the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval \(J=[0,\infty )\). Our work generalizes much recent work such as [1820] to the case where there are impulse effects, and the right-hand side is a multivalued function.

To the best of our knowledge, there is no work on S-asymptotic ω-periodic behavior of solutions to fractional non-instantaneous impulsive differential inclusions with order \(\alpha \in (1,2)\) and generated by sectorial operators, and this fact is the main goal in the present paper.

To clarify the advantage of this study, we mention that two methods have been provided to demonstrate the existence of S-asymptotic ω-periodic solutions for semilinear fractional differential inclusions in the presence of non-instantaneous impulse effects, and in which the nonlinear part is a multivalued function, and the linear part is a sectorial operator. Moreover, the technique presented in this paper can be used to generalize the work in [12, 1521, 2325] to the case where the linear part is a sectorial operator, the nonlinear part is a multivalued function, and there is impulse effects. In addition, Problem (1) can be investigated on time scales using the arguments in [1], and using the arguments in [3, 6, 8], one can examine the asymptotic periodic solutions for Problem (1) when the Caputo derivative is replaced by the ψ-Caputo derivative, ψ-RL derivative, Atangana–Baleanu derivative or p-Laplacian operator. Also, the technique used in this paper can be applied to study the asymptotic periodic solutions for many fractional differential equations or inclusions generated by sectorial operators or almost sectorial operators.

For more information related to fractional differential equations and inclusions with non-instantaneous impulse effects, we refer the reader to [2731]. See [3235] for more information about semilinear differential equations and inclusions with sectorial operators.

It is worth noting that Refs. [3644] contain very important and interesting topics in mathematics as well as their applications such as differential equations, fractional calculus and ABC-fuzzy-Volterra integro-differential equation.

The paper is organized as follows. Section 2 includes definitions and basic information that we need to prove our results. In Sect. 3, we provide two existence results of S-asymptotic ω-periodic solutions for Problem (1). In Sect. 4, we give three examples to illustrate our theoretical results.

Preliminaries and notations

Let \(J_{0}=[0,t_{1}]\) and \(J_{i}=(\theta _{i},\theta _{i+1}]\), \(i\in \mathbb{N} \). It is known that the vector spaces

$$ \mathit{PC}(J,E):=\bigl\{ x:J\rightarrow E,x \text{ is bounded}, x|_{J_{i}}\in C(J_{i},E) \bigr\} $$

and

$$ \mathit{SAP}_{\omega }\mathit{PC}(J,E):=\Bigl\{ x:J\rightarrow E,x \text{ is bounded}, x|_{J_{i}}\in C(J_{i},E)\text{, } \lim_{\theta \rightarrow \infty } \bigl\Vert x( \theta +\omega )-x(\theta ) \bigr\Vert =0\Bigr\} $$

are Banach spaces endowed with the norm

$$ \Vert x \Vert :=\max_{\theta \in J} \bigl\Vert x(\theta ) \bigr\Vert . $$

Definition 1

([45])

Let \(M>0\), \(\varphi \in (0,\frac{\pi }{2})\), and \(\mu \in \mathbb{R} \). A closed linear operator \(A:D(A)\subseteq E\rightarrow E\) with dense domain is called sectorial of type \(\{M,\varphi ,\alpha ,\mu \}\) if:

  1. (i)

    \(\delta ^{\alpha }\notin S_{\varphi }+\mu \Rightarrow \|R(\delta ^{ \alpha },A)\|\) exists, where \(R(\delta ^{\alpha },A)\) is the α-resolvent operator of A defined by

    $$ R\bigl(\delta ^{\alpha },A\bigr):=\bigl(\delta ^{\alpha }I-A \bigr)^{-1} $$

    and

    $$ S_{\varphi }+\mu =\bigl\{ \mu +\delta ^{\alpha }:\delta \in \mathbb{C} , \bigl\vert \operatorname{Arg}\bigl(-\delta ^{\alpha }\bigr) \bigr\vert < \varphi \bigr\} . $$
  2. (ii)
    $$ \bigl\Vert R\bigl(\delta ^{\alpha },A\bigr) \bigr\Vert \leq \frac{M}{ \vert \delta ^{\alpha }-\mu \vert }, \quad \delta ^{\alpha }\notin S_{\varphi }+\mu . $$

Remark 1

([45], Remark 2.1)

If A is a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\), then it is the infinitesimal generator of a α-resolvent family of operators \(\{T_{\alpha }(\theta ):\theta \geq 0\}\) in E defined by

$$ T_{\alpha }(\theta )=\frac{1}{2\pi i} \int _{\gamma }e^{\delta \theta }R\bigl( \delta ^{\alpha },A \bigr)\,d\delta , $$
(2)

where γ is a suitable path and \(\delta ^{\alpha }\notin S_{\varphi }+\mu \) for \(\delta \in \gamma \).

Definition 2

([45], Definition 3.1)

Let A be a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\) and \(f\in L^{1}([0,b],E)\). A continuous function \(u:[0,b]\rightarrow E\) is called a mild solution to the Cauchy problem:

$$ \textstyle\begin{cases} {}^{c}D_{0,\theta }^{\alpha }x(\theta )=Ax(\theta )+f(\theta ),\quad \theta \in {}[ 0,b], \\ x(0)=x_{0},\qquad x'(0)=x,\end{cases}$$

if

$$ x(\theta )=S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1}+ \int _{0}^{ \theta }T_{\alpha }(\theta -\tau )f( \theta )\,d\tau ,\quad \theta \in {}[ 0,b], $$

where

$$\begin{aligned}& S_{\alpha }(\theta )=\frac{1}{2\pi i} \int _{\gamma }e^{\delta \theta } \delta ^{\alpha -1}R\bigl( \delta ^{\alpha },A\bigr)\,d\delta , \\& K_{\alpha }(\theta )=\frac{1}{2\pi i} \int _{\gamma }e^{\delta \theta } \delta ^{\alpha -2}R\bigl( \delta ^{\alpha },A\bigr)\,d\delta , \end{aligned}$$

and \(T_{\alpha }(\theta )\) is given by (2).

The following lemma provides estimates on \(\|S_{\alpha }(\theta )\|\), \(\|K_{\alpha }(\theta) \|\) and \(\|T_{\alpha }(\theta) \|\).

Lemma 1

([45], Theorems 3.3, 3.4)

Let A be a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\). Suppose \(\mu <0\). Then \(L=L(M,\varphi ,\alpha )>0\) such that

$$ \begin{aligned} &\bigl\Vert S_{\alpha }(\theta ) \bigr\Vert \leq \frac{L}{1+ \vert \mu \vert \theta ^{\alpha }},\qquad \bigl\| K_{\alpha }(\theta) \bigr\| \leq \frac{L(\theta +1)}{1+ \vert \mu \vert \theta ^{\alpha }}, \quad \textit{and} \\ &\bigl\| T_{\alpha }(\theta) \bigr\| \leq \frac{L \theta ^{\alpha -1}}{1+ \vert \mu \vert \theta ^{\alpha }},\quad \forall \theta >0. \end{aligned} $$
(3)

Remark 2

([19], Remark 3)

In view of (6), we get:

  1. (i)
    $$ \bigl\Vert S_{\alpha }(\theta ) \bigr\Vert \leq L,\quad \forall \theta >0. $$
    (4)
  2. (ii)
    $$\begin{aligned} \bigl\Vert K_{\alpha }(\theta ) \bigr\Vert \leq &L+ \frac{L\theta }{1+ \vert \mu \vert \theta ^{\alpha }} \\ \leq & \textstyle\begin{cases} L+L \theta ,&\text{if } 0< \theta < 1, \\ L+\frac{L}{ \vert \mu \vert \theta ^{\alpha -1}},&\text{if } \theta \geq 1 \end{cases}\displaystyle \\ \leq &L \biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr),\quad \forall \theta >0. \end{aligned}$$
    (5)
  3. (iii)

    As in (ii), we derive

    $$\begin{aligned} \bigl\Vert T_{\alpha }(\theta ) \bigr\Vert \leq & \frac{L \theta ^{\alpha -1}}{1+ \vert \mu \vert \theta ^{\alpha }} \\ \leq &L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} ,\quad \forall \theta >0. \end{aligned}$$
    (6)

Based on Definition 2, we can give the definition of an S-asymptotically ω-periodic mild solution for Problem (1).

Definition 3

A function \(x\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\) is called an S-asymptotically ω-periodic mild solution for Problem (1) if it has the form

$$ x(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} , \end{cases} $$

where \(f(\tau )\in F(\tau ,x(\tau ))\), a.e. for \(\tau >0\).

Existence of S-asymptotically ω-periodic mild solutions for Problem (1)

In order to give the first result, we need the following lemma which is due to [46].

Lemma 2

Let \((X,d)\) be a metric space and G be a contraction multivalued function from X to the family of non-empty closed subsets of X. Then G has a fixed point.

For notations about multivalued functions we refer the reader to [47].

Theorem 1

Suppose the following assumptions are satisfied.

(HA):

\(A:D(A)\rightarrow E\) is a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\), where \(M>0\), \(\varphi \in (0,\frac{\pi }{2})\), and \(\mu \in \mathbb{R} \).

(HF):

\(F:J\times E\rightarrow P_{ck}(E)\) is a multivalued function such that

  1. (i)

    For any \(z\in E\), the multivalued function θ\(F(\cdot,z)\) is strongly measurable.

  2. (ii)

    For any \(x\in \mathit{PC}(J,E)\), the set

    $$\begin{aligned} S_{F(\cdot, x(\cdot))}^{1} :=&\bigl\{ \psi :J\rightarrow E,\psi \textit{ is locally integrable and } \psi (\tau )\in F\bigl(\tau ,x(\tau )\bigr), \\ &\textit{a.e. } \tau \in J \bigr\} \end{aligned}$$

    is not empty.

  3. (iii)

    There is a continuous function \(L_{1}:J\rightarrow (0,\infty )\) such that

    $$ h\bigl(F(\theta ,z_{1}),F(\theta ,z_{2})\bigr)\|\leq L_{1}(\theta ) \Vert z_{1}-z_{2} \Vert , \quad \forall \theta \in J, z_{1},z_{2}\in E, $$
    (7)

    where h is the Hausdorff distance.

  4. (iv)

    There is a continuous function \(L_{2}:J\rightarrow (0,\infty )\) such that

    $$ h\bigl(F(\theta +\omega ,z),F(\theta ,z)\bigr)\|\leq L_{2}(\theta ) \Vert 1+x \Vert , \quad \forall \theta \in J, z\in E. $$
    (8)
  5. (v)

    The function \(\sigma (\tau ):=\|F(\tau ,0)\|=\sup_{z\in F(\tau ,0)}\|z\|\) is continuous, bounded on J and satisfies the relation

    $$ \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }} \sigma ( \tau )\,d\tau =0. $$
    (9)
(\(Hg_{i}\)):

For any \(i\in \mathbb{N} \), \(g_{i}:[\theta _{i},s_{i}]\times E\longrightarrow E \) such that, for any \(x\in E\), the function \(\theta \mapsto g_{i}(\theta ,x)\) is differentiable at \(s_{i}\) and

  1. (i)
    $$ \lim_{\substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }} \bigl\Vert g_{i+m}(\theta +\omega ,z)-g_{i}(\theta ,z) \bigr\Vert =0,\quad \forall z\in E. $$
    (10)
  2. (ii)

    There is \(N>0\) such that for any \(i\in \mathbb{N} \)

    $$ \bigl\Vert g_{i}(\theta ,z_{1})-g_{i}( \theta ,z_{2}) \bigr\Vert \leq N \Vert z_{1}-z_{2} \Vert ,\quad \forall \theta \in J, \forall z_{1},z_{2} \in E. $$
    (11)
  3. (iii)

    There is \(\mathcal{N}>0\) such that for any \(i\in \mathbb{N} \)

    $$ \bigl\Vert g_{i}^{\prime }(s_{i},z_{1})-g_{i}^{\prime }(s_{i},z_{2}) \bigr\Vert \leq \mathcal{N} \Vert z_{1}-z_{2} \Vert ,\quad \forall z_{1},z_{2}\in E. $$
    (12)
  4. (iv)

    There is \(\kappa _{1}>0\) such that

    $$ \sup_{i\in \mathbb{N} }\sup_{\theta \in J} \bigl\Vert g_{i}(\theta ,z) \bigr\Vert \leq \kappa _{1}\bigl( \Vert z \Vert +1\bigr),\quad \forall z\in E. $$
    (13)
  5. (v)

    There is \(\kappa _{2}>0\) with

    $$ \sup_{i\in \mathbb{N} } \bigl\Vert g_{i}^{\prime }(s_{i},z) \bigr\Vert \leq \kappa _{2}\bigl( \Vert z \Vert +1\bigr),\quad \forall z\in E. $$
    (14)

Then Problem (1) has an S-asymptotically ω-periodic mild solution provided that the following conditions are verified:

$$\begin{aligned}& \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau =0, \end{aligned}$$
(15)
$$\begin{aligned}& \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{2}( \tau )\,d\tau =0, \end{aligned}$$
(16)

and

$$ L\biggl(2\xi +N_{\cdot}+\biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert } \biggr\} \biggr)\mathcal{N}\biggr)< 1, $$
(17)

where \(\xi =\sup_{\theta \in J}\int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+|\mu |(\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau \).

Proof

Due to \((HF)(\mathrm{ii})\), one can define a multivalued function Θ on \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\) in the following manner: an element \(y\in \Theta (x) \) if and only if

$$ y(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} ,\end{cases} $$
(18)

where \(f\in S_{F(\cdot,x(\cdot))}^{1}\).

Obviously, \(y\in \mathit{PC}(J_{i},E)\). We clarify that, if \(x\in \mathit{SAP}_{ \omega }\mathit{PC}(J,E) \), then \(\Theta (x)\) is a closed subset of \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\). We do this in the following steps.

Step 1. We demonstrate that, if \(y\in \Theta (x)\), we have

$$ \lim_{\theta \rightarrow \infty } \bigl\Vert y(\theta +\omega )-y(\theta ) \bigr\Vert =0. $$
(19)

Since \(x\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\),

$$ \lim_{\theta \rightarrow \infty } \bigl\Vert x(\theta +\omega )-x(\theta ) \bigr\Vert =0. $$
(20)

Now, we consider two cases.

(i) If \(\theta \in (\theta _{i},s_{i}]\) for some \(i\in \mathbb{N} \), then \(\theta +\omega \in {}[ \theta _{i}+\omega ,s_{i}+\omega ,]=[ \theta _{i+m},s_{i+m}]\), and hence, using (21) and (18), it yields

$$\begin{aligned}& \bigl\Vert g_{i+m}\bigl(\theta +\omega ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-g_{i}\bigl(\theta ,x \bigl( \theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq \bigl\Vert g_{i+m}\bigl(\theta +\omega ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-g_{i+m}\bigl( \theta ,x \bigl(\theta _{i+m}^{-}\bigr)\bigr) \bigr\Vert \\& \qquad {} + \bigl\Vert g_{i+m}\bigl(\theta ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-g_{i}\bigl(\theta ,x \bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq \bigl\Vert g_{i+m}\bigl(\theta +\omega ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-g_{i+m}\bigl( \theta ,x \bigl(\theta _{i+m}^{-}\bigr)\bigr) \bigr\Vert \\& \qquad {} +N \bigl\Vert x\bigl(\theta _{i+m}^{-}\bigr)-x\bigl( \theta _{i}^{-}\bigr) \bigr\Vert . \end{aligned}$$
(21)

(ii) If \(\theta \in {}[ s_{i},\theta _{i+1}]\) for some \(i\in \mathbb{N} \), then \(\theta +\omega \in {}[ s_{i}+\omega ,\theta _{i+1}+\omega ]=[s_{i+m}, \theta _{i+m+1}]\). Using (4) and arguing as in (21), it follows that

$$\begin{aligned}& \bigl\Vert S_{\alpha }\bigl(\theta +\omega -(s_{i}+ \omega )\bigr)g_{i+m}\bigl(s_{i}+\omega ,x\bigl( \theta _{i+m}^{-}\bigr)\bigr)-S_{\alpha }(\theta -s_{i})g_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad = \bigl\Vert S_{\alpha }(\theta -s_{i}) \bigr\Vert \bigl\Vert g_{i+m}\bigl(s_{i}+\omega ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-g_{i} \bigl(s_{i},x\bigl( \theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\& \quad \leq L \bigl\Vert g_{i+m}\bigl(s_{i}+\omega ,x\bigl( \theta _{i+m}^{-}\bigr)\bigr)-g_{i} \bigl(s_{i},x\bigl( \theta _{i}^{-}\bigr) \bigr) \bigr\Vert . \end{aligned}$$
(22)

Moreover, in view of (3) and (14), we arrive at

$$\begin{aligned}& \bigl\Vert K_{\alpha }\bigl(\theta +\omega -(s_{i}+ \omega )\bigr)g_{i+m}^{\prime }\bigl(s_{i}+ \omega ,x\bigl(\theta _{i+m}^{-}\bigr)\bigr)-K_{\alpha }( \theta -s_{i})g'_{i} \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\& \quad = \bigl\Vert K_{\alpha }(\theta -s_{i})g_{i+m}^{\prime } \bigl(s_{i+m},x\bigl(\theta _{i+m}^{-}\bigr) \bigr)-K_{ \alpha }(\theta -s_{i})g'_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq \bigl\Vert K_{\alpha }(\theta -s_{i}) \bigr\Vert \bigl\Vert g_{i+m}^{\prime }\bigl(s_{i+m},x\bigl( \theta _{i+m}^{-}\bigr)\bigr)-g'_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq 2\kappa _{2}\bigl( \Vert x \Vert +1\bigr) \bigl\Vert K_{\alpha }(\theta -s_{i}) \bigr\Vert \\& \quad \leq 2\kappa _{2}\bigl( \Vert x \Vert +1\bigr) \frac{L(\theta -s_{i}+1)}{1+ \vert \mu \vert (\theta -s_{i})^{\alpha }}. \end{aligned}$$
(23)

Next,

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta +\omega }T_{\alpha }(\theta +\omega - \tau )f( \tau )\,d\tau - \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad = \biggl\Vert \int _{-\omega }^{\theta }T_{\alpha }(\theta -\tau )f( \tau + \omega )\,d\tau - \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad \leq \int _{-\omega }^{0} \bigl\Vert T_{\alpha }( \theta -\tau ) \bigr\Vert \bigl\Vert f(\tau + \omega ) \bigr\Vert \,d\tau \\& \qquad {} + \int _{0}^{\theta } \bigl\Vert T_{\alpha }( \theta -\tau ) \bigr\Vert \bigl\Vert f(\tau + \omega )-f(\tau ) \bigr\Vert \,d\tau \\& \quad = I_{1}+I_{2}. \end{aligned}$$
(24)

Let \(\tau \in {}[ -\omega ,0]\) be fixed. Since \(F(\tau +\omega ,0)\) is compact, there is \(v_{\tau +\theta }\in F(\tau +\omega ,0)\) such that

$$\begin{aligned} \bigl\Vert f(\tau +\omega )-v_{\tau +\omega } \bigr\Vert =&d(f(\tau + \omega ),F(\tau + \omega ,0) \\ \leq &h\bigl(F\bigl(\tau +\omega ,x(\tau +\omega )\bigr),F(\tau +\omega ,0) \bigr). \end{aligned}$$
(25)

From (7), \((HF)(\mathrm{v})\) and (25), we get

$$\begin{aligned} \bigl\Vert f(\tau +\omega ) \bigr\Vert \leq &h\bigl(F\bigl(\tau +\omega ,x( \tau +\omega )\bigr),F( \tau +\omega ,0)\bigr)+ \Vert v_{\tau +\omega } \Vert \\ \leq &L_{1}(\tau +\omega ) \bigl\Vert x(\tau +\omega ) \bigr\Vert +\sigma (\tau +\omega ) \\ \leq & \Vert x \Vert L_{1}(\tau +\omega )+\sigma (\tau +\omega ). \end{aligned}$$
(26)

Since \(L_{1}\) and σ are continuous on J, there are two positive numbers \(\omega _{1}\), \(\omega _{2}\) such that

$$ \sup_{t\in {}[ 0,\omega ]} \bigl\vert L_{1}(t) \bigr\vert \leq \omega _{1},\quad \text{and}\quad \sup _{t\in {}[ 0,\omega ]} \bigl\vert \sigma (\theta ) \bigr\vert \leq \omega _{2}. $$
(27)

Then, from (3), (26) and (27), we have

$$\begin{aligned} I_{1} =& \biggl\Vert \int _{-\omega }^{0}T_{\alpha }(\theta -\tau )f( \tau + \omega )\,d\tau \biggr\Vert \\ \leq &\bigl(\omega _{1} \Vert x \Vert +\omega _{2}\bigr)L \int _{-\omega }^{0} \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\,d\tau \\ \leq &\frac{(\omega _{1} \Vert x \Vert +\omega _{2})L}{ \vert \mu \vert } \int _{-\omega }^{0}\frac{ \vert \mu \vert (\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\,d\tau \\ =&\frac{(\omega _{1} \Vert x \Vert + \omega _{2})L}{\alpha \vert \mu \vert }\ln \frac{1+ \vert \mu \vert (\theta +\omega )^{\alpha }}{1+ \vert \mu \vert \theta ^{\alpha }}. \end{aligned}$$
(28)

Next, let \(\tau \in {}[ 0,\theta ]\) be fixed. From the fact that \(F(\tau +\omega ,x(\tau ))\) is compact, there are \(z_{\tau +\omega },z_{\tau }\in F(\tau ,x(\tau +\omega ))\) such that \(d(f(\tau +\omega ),z_{\tau +\omega })=d(f(\tau +\omega ),F(\tau ,x( \tau +\omega )))\) and \(d(f(\tau ),z_{\tau })=d(f(\tau ,F(\tau ,x(\tau +\omega ))))\). Then, by (12), (13) and (25), we arrive at

$$\begin{aligned}& \bigl\Vert f(\tau +\omega )-f(\tau ) \bigr\Vert \\& \quad \leq \bigl\Vert f(\tau +\omega )-z_{\tau +\omega } \bigr\Vert + \Vert z_{\tau +\omega }-z_{ \tau } \Vert + \bigl\Vert z_{\tau }-f( \tau ) \bigr\Vert \\& \quad \leq d(f(\tau +\omega ),F\bigl(\tau +\omega ,x(\tau )\bigr)+ \Vert z_{\tau + \omega }-z_{\tau } \Vert \\& \qquad {} +d(f(\tau ,F\bigl(\tau +\omega ,x(\tau )\bigr) \\& \quad \leq h(F((\tau +\omega ),x\bigl((\tau +\omega )\bigr),F\bigl(\tau ,x(\tau + \omega )\bigr) \\& \qquad {} +2 \bigl\Vert F\bigl(\tau ,x(\tau +\omega )\bigr) \bigr\Vert +h\bigl(F\bigl( \tau ,x(\tau +\omega )\bigr),F\bigl(\tau ,x( \tau )\bigr)\bigr) \\& \quad \leq L_{2}(\tau ) \bigl\Vert 1+x(\tau +\omega ) \bigr\Vert +2 \bigl\| F(\tau ,0) \bigr\Vert +2h\bigl(F\bigl( \tau ,x(\tau +\omega )\bigr),F( \tau ,0)\bigr) \\& \qquad {} +L_{1}(\tau )\bigl\| x(\tau +\omega )-x(\tau )\bigr\| \\& \quad \leq L_{2}(\tau ) \bigl(1+ \Vert x \Vert \bigr)+2\sigma (\tau )+3L_{1}(\tau ) \Vert x \Vert . \end{aligned}$$
(29)

By (3) and (29), we arrive at

$$\begin{aligned} I_{2} =& \int _{0}^{\theta } \bigl\Vert T_{\alpha }( \theta -\tau ) \bigr\Vert \bigl\Vert f(\tau + \omega )-f(\tau ) \bigr\Vert \,d\tau \\ \leq &3L \Vert x \Vert \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau \\ &{}+2L \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }} \sigma ( \tau )\,d\tau \\ &{}+L\bigl(1+ \Vert x \Vert \bigr) \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{2}( \tau )\,d\tau . \end{aligned}$$
(30)

By arguing as in (24), (28) and (30), one can show

$$\begin{aligned}& \biggl\Vert \int _{0}^{s_{i}+\omega }T_{\alpha }(s_{i}+ \omega -\tau )f(\tau )\,d\tau - \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau \biggr\Vert \\& \quad \leq \frac{(\omega _{1} \Vert x \Vert +\omega _{2})L}{\alpha \vert \mu \vert }\ln \frac{1+ \vert \mu \vert (s_{i}+\omega )^{\alpha }}{1+ \vert \mu \vert s_{i}^{\alpha }} \\& \qquad {} +3L \Vert x \Vert \int _{0}^{s_{i}} \frac{(s_{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (s_{i}-\tau )^{\alpha }}L_{1}( \tau )\,d\tau \\& \qquad {} +2L \int _{0}^{s_{i}} \frac{(s_{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (s_{i}-\tau )^{\alpha }} \sigma ( \tau )\,d\tau \\& \qquad {} +L\bigl(1+ \Vert x \Vert \bigr) \int _{0}^{s_{i}} \frac{(s_{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (s_{i}-\tau )^{\alpha }}L_{2}( \tau )\,d\tau . \end{aligned}$$
(31)

Since \(s_{i}\rightarrow \infty \) when \(\theta \rightarrow \infty \), we can derive (19) from (9), (10), (15), (16), (20), (21)–(23), (28), (30) and (31).

Then, due to (21)–(24), (28), (30) and (31), we arrive at (19).

Step 2. In this step, we show that, if \(x\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\) and \(y\in \Theta (x)\), then y is bounded.

(i) Let \(\theta \in {}[ 0,\theta _{1}]\). Then, using (4)–(6), we get

$$\begin{aligned} \bigl\Vert y(\theta ) \bigr\Vert \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+\max \biggl\{ 1, \frac{1}{ \vert \mu \vert }\biggr\} \biggr) \\ &{}+L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \int _{0}^{\theta } \bigl\Vert f(\tau ) \bigr\Vert \,d\tau . \end{aligned}$$
(32)

On the hand, from (7), (9) and (27), we derive

$$\begin{aligned} \bigl\Vert f(\tau ) \bigr\Vert \leq & \bigl\Vert F\bigl(\tau ,x(\tau ) \bigr) \bigr\Vert \leq \bigl\Vert F(\tau ,0) \bigr\Vert +h\bigl(F( \tau ,0),F\bigl(\tau ,x(\tau )\bigr)\bigr) \\ \leq &\sigma (\tau )+L_{1}(\tau ) \Vert x \Vert \leq \omega _{1}+\omega _{2} \Vert x \Vert . \end{aligned}$$
(33)

Then, by (32) and (33), we conclude that

$$\begin{aligned} \sup_{\theta \in {}[ 0,\theta _{1}]} \bigl\Vert y(\theta ) \bigr\Vert \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+ \max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr) \\ &{}+L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \bigl[\omega _{1}+ \omega _{2} \Vert x \Vert )\bigr]\omega . \end{aligned}$$
(34)

(ii) Let \(\theta \in (\theta _{i},s_{i}];i\in \mathbb{N} \). In view of (18) and (19), it follows that

$$ \bigl\Vert g_{i}\bigl(\theta ,x\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \leq \kappa _{1}\bigl( \Vert x \Vert +1 \bigr) $$
(35)

and

$$ \bigl\Vert g_{i}^{\prime }\bigl(s_{i},x\bigl( \theta _{i}^{-}\bigr)\bigr) \bigr\Vert \leq \kappa _{2}\bigl( \Vert x \Vert +1\bigr). $$
(36)

(iii) Let \(\theta \in {}[ s_{i},\theta _{i+1}]\); \(i\in \mathbb{N} \). In view of (4), (5), (35) and (36), we have

$$\begin{aligned}& \sup_{\theta \in {}[ s_{i},\theta _{i+1}]} \bigl\Vert S_{\alpha }( \theta -s_{i})g_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr)\bigr)+K_{\alpha }(\theta -s_{i})g_{i}^{ \prime }\bigl(s_{i},x \bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq L\kappa _{1}\bigl( \Vert x \Vert +1\bigr)+L \max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \kappa _{2}\bigl( \Vert x \Vert +1\bigr). \end{aligned}$$
(37)

Furthermore, using (3) and (33), we arrive at

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad \leq L \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }} \bigl\Vert f( \tau ) \bigr\Vert \,d\tau \\& \quad \leq L \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }} \sigma ( \tau )\,d\tau \\& \qquad {} + \Vert x \Vert L \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau . \end{aligned}$$
(38)

From (9), (15) and (38), there is \(\xi >0\) with

$$ \sup_{\theta \in J} \biggl\Vert \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \leq \varsigma $$
(39)

and

$$ \sup_{i\in \mathbb{N} } \biggl\Vert \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau \biggr\Vert \leq \varsigma . $$
(40)

As a result of (34), (35), (37), (39) and (40), we conclude that y is bounded on J.

Hence, Θ is a multivalued function from \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\) to the non-empty subsets of \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\).

Step 3. The values of Θ are closed.

To show this, let \(x\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\) and \(y_{n}\in \Theta (x)\), \(\forall n\geq 1\), with \(y_{n}\rightarrow y\) in \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\). Then we have \(f_{n}\in S_{F(\cdot,x(\cdot))}^{1}\) such that

$$ y_{n}(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ +\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{n}(\tau )\,d\tau , \quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f_{n}(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{n}(\tau )\,d\tau , \quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} . \end{cases} $$
(41)

Let θ be a fixed point in \([0,\theta _{1}]\) and put \(J_{\theta }=[0,\theta ]\). Then

$$\begin{aligned} \bigl\Vert f_{n}(\tau ) \bigr\Vert \leq & \bigl\Vert F\bigl( \tau ,x(\tau )\bigr) \bigr\Vert \leq \bigl\| F(\tau ,0) \bigr\Vert +L_{1}( \tau ) \bigl\Vert x(\tau ) \bigr\Vert \\ \leq &\sigma (\tau )+L_{1}(\tau ) \Vert x \Vert ,\quad \forall \tau \in J_{\theta }. \end{aligned}$$
(42)

This relation with the fact that σ and \(L_{1}\) are continuous guarantee that the family \(\{f_{n}:n\geq 1\}\) is bounded in \(L^{2}(J_{\theta },E)\), and hence, by Mazur’s lemma, there is a sequence \((z_{n})_{n\geq 1}\) of convex combinations of \(f_{n}\) with \(z_{n}\rightarrow f \) strongly in \(L^{2}(J_{\theta },E)\). Hence, we can suppose, without loss of generality, that \(z_{n}(\theta )\rightarrow f (\theta )\), a.e. \(\theta \in J_{\theta }\). Moreover, from (6) and (42), we get

$$\begin{aligned}& \bigl\Vert T_{\alpha }(\theta -\tau )f_{n}(\tau ) \bigr\Vert \\& \quad \leq L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \bigl(\sigma (\tau )+L_{1}(\tau )\bigr) \Vert x \Vert ,\quad \forall \tau \in J_{\theta }. \end{aligned}$$

Therefore, by the Lebesgue dominated convergence theorem and the continuity of \(T_{\alpha }(\theta -\tau )\), we obtain

$$\begin{aligned}& \lim_{n\rightarrow \infty } \int _{0}^{\theta }T_{\alpha }(\theta - \tau )z_{n}(\tau )\,d\tau \\& \quad = \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau . \end{aligned}$$

Consequently,

$$ \lim_{n\rightarrow \infty }y_{n}(\theta )=S_{\alpha }( \theta )x_{0}+K_{ \alpha }(\theta )x_{1}+ \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}]. $$
(43)

Similarly, one can show that, for any \(\theta \in {}[ s_{i},\theta _{i+1}]\); \(i\in \mathbb{N} \), we have

$$\begin{aligned} \lim_{n\rightarrow \infty }y_{n}(\theta ) =&S_{\alpha }( \theta -s_{i})g_{i}\bigl(s_{i},x\bigl( \theta _{i}^{-}\bigr)\bigr)+K_{\alpha }(\theta -s_{i})g_{i}^{\prime }\bigl(s_{i},x \bigl( \theta _{i}^{-}\bigr)\bigr) \\ &{}- \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau + \int _{0}^{ \theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau . \end{aligned}$$
(44)

Remark that \((HF)(\mathrm{iv})\) implies that \(f(\theta )\in F(\theta ,x ( \theta ))\), a.e. \(\theta \in J\). So, from (43) and (44), we arrive at

$$ y(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} .\end{cases}$$

Then \(y\in \Theta (x)\).

Step 4. Θ is a contraction.

Let \(u_{1},u_{2}\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\) and \(y_{1}\in \Theta (u_{1})\). Then we have \(f_{1}\in S_{F(\cdot,u_{1}(\cdot))}^{1}\) such that

$$ y_{1}(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{1}(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,u_{1}(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ C_{q}(\theta -s_{i})g_{i}(s_{i},u_{1}(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},u_{1}(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f_{1}(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{1}(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} .\end{cases} $$
(45)

Consider the multivalued function \(\Psi :J\rightarrow 2^{E}\) defined by

$$ \Psi (\theta )=\bigl\{ z\in E: \bigl\Vert z-f_{1}(\theta ) \bigr\Vert \leq L_{1}(\theta ) \bigl\Vert u_{1}( \theta )-u_{2}(\theta ) \bigr\Vert ,\mbox{a.e. } \theta \in J\bigr\} . $$

We clarify that the values of Ψ are non-empty. Let \(\theta \in J\). From (5), we get

$$ h\bigl(F\bigl(\theta ,u_{1}(\theta )\bigr),F\bigl(\theta ,u_{2}(\theta )\bigr)\bigr)\leq L_{1}( \theta ) \bigl\Vert u_{1}(\theta )-u_{2}(\theta ) \bigr\Vert . $$

So, there is \(z_{\theta }\in F(\theta ,u_{2}(\theta ))\) such that

$$ \bigl\| f_{1}(\theta )-z_{\theta }\bigr\| \leq h\bigl(F\bigl( \theta ,u_{1}(\theta )\bigr),F\bigl(\theta ,u_{2}( \theta )\bigr)\bigr)\leq L_{1}(\theta ) \bigl\Vert u_{1}(\theta )-u_{2}(\theta ) \bigr\Vert , $$

which leads to \(\Psi (\theta )\neq \phi \); \(\theta \in J\). Moreover, the set \(\Lambda (\theta )=\Psi (\theta )\cap F(\theta ,u_{2}(\theta ))\); \(\theta \in J\) is not empty. Because the functions \(f_{1}\), \(L_{1}\), \(u_{1}\), \(u_{2}\) are measurable, Proposition 3.4 in [47] or (Corollary 1.3.1(a) in [48]) guarantees that the multivalued map \(\theta \rightarrow \Lambda (\theta )\) is measurable. Notice that the set \(\Psi (\theta )\); \(\theta \in J\) is closed. Consequently, the values of Λ are non-empty and compact, and hence, there exists a measurable selection \(f_{2}\) for Λ with

$$ \bigl\Vert f_{1}(\theta )-f_{2}(\theta ) \bigr\Vert \leq L_{1}(\theta ) \bigl\Vert u_{1}( \theta )-u_{2}(\theta ) \bigr\Vert ,\quad \mbox{a.e. }\theta \in J. $$
(46)

We define \(y_{2}:J\rightarrow E\) as follows:

$$ y_{2}(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{q}(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{2}(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,u_{2}(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ C_{q}(\theta -s_{i})g_{i}(s_{i},u_{2}(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},u_{2}(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f_{2}(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{2}(\tau )\,d\tau , \quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} .\end{cases} $$
(47)

Obviously, \(y_{2}\in \Theta (u_{1})\). Now, we estimate the quantity \(\|y_{1}-y_{2}\|\). To do this, we consider three cases.

Case 1. \(\theta \in {}[ 0,\theta _{1}]\). In view of (3), (15), (47), (46) and (47), we have

$$\begin{aligned}& \bigl\Vert y_{1}(\theta )-y_{2}(\theta ) \bigr\Vert \\& \quad \leq \Vert \int _{0}^{\theta } \bigl\Vert T_{\alpha }(\theta -\tau )\bigr\| \bigl\| f_{1}( \tau )-f_{2}(\tau )\bigr\| \,d\tau \\& \quad \leq L \Vert u_{1}-u_{2} \Vert \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau \\& \quad \leq \xi L \Vert u_{1}-u_{2} \Vert . \end{aligned}$$
(48)

Case 2. \(\theta \in (\theta _{i},s_{i}]\). Using (11), we get

$$\begin{aligned}& \bigl\Vert y_{1}(\theta )-y_{2}(\theta ) \bigr\Vert \\& \quad = \bigl\Vert g_{i}\bigl(\theta ,u_{1}\bigl(\theta _{i}^{-}\bigr)\bigr)-g_{i}\bigl(\theta ,u_{2}\bigl( \theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\& \quad \leq N \bigl\Vert u_{1}\bigl(\theta _{i}^{-} \bigr)-u_{2}\bigl(\theta _{i}^{-}\bigr) \bigr\Vert \leq N \Vert u_{1}-u_{2} \Vert . \end{aligned}$$
(49)

Case 3. \(\theta \in {}[ s_{i},\theta _{i+1}]\), \(i\in \mathbb{N} \). From (4) and (49), we derive

$$\begin{aligned}& \bigl\Vert S_{\alpha }(\theta -s_{i})g_{i} \bigl(s_{i},u_{1}\bigl(\theta _{i}^{-} \bigr)\bigr)-S_{ \alpha }(\theta -s_{i})g_{i} \bigl(s_{i},u_{2}\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\& \quad \leq L \bigl\Vert g_{i}\bigl(s_{i},u_{1} \bigl(\theta _{i}^{-}\bigr)\bigr)-g_{i} \bigl(s_{i},u_{2}\bigl( \theta _{i}^{-} \bigr)\bigr) \bigr\Vert \leq LN \Vert u_{1}-u_{2} \Vert . \end{aligned}$$
(50)

By (4) and (12), we arrive at

$$\begin{aligned}& \bigl\Vert K_{\alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},u_{1}\bigl(\theta _{i}^{-} \bigr)\bigr)-K_{ \alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},u_{2}\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\& \quad \leq \bigl\Vert K_{\alpha }(\theta -s_{i}) \bigr\Vert \bigl\Vert g_{i}^{\prime }\bigl(s_{i},u_{1} \bigl( \theta _{i}^{-}\bigr)\bigr)-g_{i}^{\prime } \bigl(s_{i},u_{2}\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\& \quad \leq L \biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr)\mathcal{N} \Vert u_{1}-u_{2} \Vert . \end{aligned}$$
(51)

Moreover, as in (48), one can show that

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{1}(\tau )\,d\tau - \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{2}(\tau )\,d\tau \biggr\Vert \\& \quad \leq L\xi \Vert u_{1}-u_{2} \Vert \end{aligned}$$
(52)

and

$$\begin{aligned}& \biggl\Vert \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f_{1}(\tau )\,d\tau - \int _{0}^{s_{i}}T_{ \alpha }(s_{i}- \tau )f_{2}(\tau )\,d\tau \biggr\Vert \\& \quad \leq L\xi \Vert u_{1}-u_{2} \Vert . \end{aligned}$$
(53)

Now, by (48)–(53), we conclude that

$$ \Vert y_{1}-y_{2} \Vert \leq \Vert u-v \Vert L \biggl(2\xi +N_{\cdot}+\biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert } \biggr\} \biggr)\mathcal{N}\biggr). $$
(54)

As a consequence of (17), Eq. (54) becomes

$$ \bigl\Vert y_{1}(\theta )-y_{2}(\theta ) \bigr\Vert < \vartheta \Vert u-v \Vert , $$
(55)

where \(\vartheta =L(2\xi +N_{\cdot}+(1+\max \{1,\frac{1}{|\mu |}\})\mathcal{N})<1\). By interchange o the role of \(y_{1}\) and \(y_{2}\) in the above discussion, we conclude that Θ is a contraction. So, applying Lemma 2, shows that Θ has a fixed point which is an S-asymptotically ω-periodic mild solution to Problem (1). □

Remark 3

If there is no impulse effect, then \(N=\mathcal{N}=0\), and hence inequality (17) becomes \(2L\xi <1\).

Now, we present another result concerning the existence of S-asymptotically ω-periodic solutions for Problem (1).

We need the following fixed point theorem for multivalued functions and a compactness criterion in \(\mathit{PC}(J,E)\).

Lemma 3

([49], Corollary 3.3.1)

Let W be a closed convex subset of a Banach space X and \(N:E\rightarrow P_{ck} ( W ) \) be a closed multifunction which is ϑ-condensing on every bounded subset of W, where ϑ is a non-singular measure of noncompactness defined on subsets of W, then the set of fixed points for N is non-empty.

Lemma 4

([50], Lemma 1.2)

Let \(D\subseteq \mathit{PC}(J,E)\). Assume that

  1. (i)

    \(\operatorname{Lim}_{\theta \rightarrow \infty }\|u(\theta )\|=0\), uniformly for \(u\in D\).

  2. (ii)

    The set \(D_{| \overline{J_{i}}}\) is equicontinuous for every \(i\in \mathbb{N} \), where

    $$ D_{| \overline{J_{i}}}=\bigl\{ y^{\ast }\in C(\overline{J_{i}},E):y^{ \ast }(t)=y(t),t \in J_{i}=(t_{i},t_{i+1}],y^{\ast }(t_{i})=y \bigl(t_{i}^{+}\bigr),y \in D\bigr\} . $$
  3. (iii)

    For any \(i\in \mathbb{N} \), and any \(\theta \in J\), the set \(\{y^{\ast }(t):y^{\ast }\in D_{| \overline{J_{i}}}\}\) is relatively compact in E.

Then D is relatively compact in \(\mathit{PC}(J,E)\).

Theorem 2

Assume that (HA) and the following conditions are verified:

\((HF)^{\ast }\):

\(F:J\times E\rightarrow P_{ck}(E)\) such that:

  1. (i)

    For every \(x\in E\), the multivalued function \(t\longrightarrow F(t,x)\) is measurable.

  2. (ii)

    For almost \(t\in J\), the multivalued function \(x\longrightarrow F(t,x)\) is upper semicontinuous.

  3. (iii)

    For any \(x\in \mathit{PC}(J,E)\), the set

    $$\begin{aligned} S_{F(\cdot,x(\cdot))}^{1} :=&\bigl\{ \psi :J\rightarrow E,\psi \textit{ is locally integrable and } \psi (\tau )\in F\bigl(\tau ,x(\tau )\bigr), \\ &\textit{a.e.} \tau \in J \bigr\} \end{aligned}$$

    is not empty.

  4. (iv)

    There exists a continuous function \(\varphi :J\rightarrow (0,\infty )\) with

    $$ \bigl\Vert F(t,z) \bigr\Vert \leq \varphi (t) \bigl(1+ \Vert z \Vert \bigr)\quad \forall (t,z) \in J\times E. $$
\((Hg_{i})^{\ast }\):

For any \(i\in \mathbb{N} \), \(g_{i}:[\theta _{i},s_{i}]\times E\longrightarrow E\) (\(i\in \mathbb{N} \)) is uniformly continuous on bounded sets and for any \(z\in E\), the function \(\theta \mapsto g_{i}(\theta ,z)\) is continuously differentiable at \(s_{i}\) such that (10), (11), (14) and the following conditions are satisfied:

  1. (i)

    There is a bounded continuous function \(h^{\ast }:J\rightarrow J\) with \(\lim_{\theta \rightarrow \infty }h^{\ast }(\theta )=0\) and

    $$ \bigl\Vert g_{i}(\theta ,z) \bigr\Vert \leq h^{\ast }( \theta ) \bigl( \Vert z \Vert +1\bigr),\quad \forall (i, \theta ,z)\in \mathbb{N} \times {}[ t_{i},s_{i}]\times E. $$
    (56)
  2. (ii)

    For any \(t\in J\), the function. \(z\rightarrow g_{i}(t,z)\) is compact.

Then Problem (1) has an S-asymptotically ω-periodic mild solution provided that the families \(\{S_{\alpha }(t):t>0\}\), \(\{K_{\alpha }(t):t>0\}\) and \(\{T_{\alpha }(t):t>0\}\) are compact,

$$\begin{aligned}& \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\bigl( \varphi (\tau +\omega )+\varphi (t)\bigr)=0, \end{aligned}$$
(57)
$$\begin{aligned}& \sup_{\theta \in J} \int _{0}^{\theta }\varphi (t)\,d\tau < \frac{1}{4L\max \{1,\frac{1}{ \vert \mu \vert }\}}, \end{aligned}$$
(58)

and

$$ \eta +L\eta +L\kappa _{2} \max \biggl\{ 1,\frac{1}{ \vert \mu \vert } \biggr\} < \frac{1}{2}, $$
(59)

where \(\eta =\sup_{t\in J}\|h^{\ast }(t)\|\), and \(\kappa _{2}\) is as defined in (14).

Proof

Due to \((HF)^{\ast }(\mathrm{iii}) \), we can consider a multioperator Θ on \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\) defined as in (18).

In the following steps we show that Θ satisfies the assumptions of Lemma 3.

Step 1. In this step, we demonstrate that, if \(x\in \mathit{SAP}_{\omega }\mathit{PC}(J,E)\), \(y\in \Theta (x)\), then \(\lim_{\theta \rightarrow \infty }\|y(\theta +\omega )-y(\theta )\|=0\).

Notice that condition \((Hg_{i})^{\ast }(\mathrm{i}) \) implies \((Hg_{i})(\mathrm{ii}) \), so Eqs. (21)–(23) are satisfied. Now, let \(f\in S_{F(\cdot,x(\cdot))}^{1}\) and \(\theta \in {}[ s_{1},\theta _{i+1}]\); \(i>1\). As in (24), we have

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta +\omega }T_{\alpha }(\theta +\omega - \tau )f( \tau )\,d\tau - \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad \leq \int _{-\omega }^{0}T_{\alpha }(\theta -\tau ) \bigl\Vert f(\tau + \omega ) \bigr\Vert \,d\tau \\& \qquad {} + \int _{0}^{\theta }T_{\alpha }(\theta -\tau ) \bigl\Vert f(\tau +\omega )-f( \tau )\,d\tau \bigr\Vert \\& \quad = I_{1}+I_{2}. \end{aligned}$$
(60)

From the continuity of φ, there is ρ such that \(\sup_{t\in {}[ 0,\omega |}\varphi (t)\leq \rho \), and hence

$$ \sup_{\tau \in {}[ -\omega ,0]} \bigl\Vert f(\tau +\omega ) \bigr\Vert \leq \rho \bigl(1+ \Vert x \Vert \bigr). $$
(61)

Then, using (3), \((HF)^{\ast }(\mathrm{iv})\) and (61), we get

$$\begin{aligned} I_{1} \leq &\bigl(1+ \Vert x \Vert \bigr)\rho L \int _{-\omega }^{0} \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\,d\tau \\ =&\frac{(1+ \Vert x \Vert )\rho L}{\alpha \vert \mu \vert }\ln \frac{1+ \vert \mu \vert (\theta +\omega )^{\alpha }}{1+ \vert \mu \vert \theta ^{\alpha }}. \end{aligned}$$
(62)

Also, in view of (3) and \((HF)^{\ast }(\mathrm{iv})\), we arrive at

$$\begin{aligned} I_{2} \leq & \int _{0}^{\theta } \bigl\Vert T_{\alpha }( \theta -\tau ) \bigr\Vert \bigl\Vert f( \tau +\omega )-f(\tau ) \bigr\Vert \,d\tau \\ \leq &2L\bigl(1+ \Vert x \Vert \bigr) \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\bigl( \varphi (\tau +\omega )+\varphi (t)\bigr)\,d\tau . \end{aligned}$$
(63)

Then, by (60), (62) and (63), it follows that

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta +\omega }T_{\alpha }(\theta +\omega - \tau )f( \tau )\,d\tau - \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad \leq \frac{(1+ \Vert x \Vert )\rho L}{\alpha \vert \mu \vert }\ln \frac{1+ \vert \mu \vert (\theta +\omega )^{\alpha }}{1+ \vert \mu \vert \theta ^{\alpha }} \\& \qquad {} +2L\bigl(1+ \Vert x \Vert \bigr) \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\bigl( \varphi (\tau +\omega )+\varphi (t)\bigr)\,d\tau . \end{aligned}$$
(64)

Similarly,

$$\begin{aligned}& \biggl\Vert \int _{0}^{s_{i}+\omega }T_{\alpha }(s_{i}+ \omega -\tau )f(\tau )\,d\tau - \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau \biggr\Vert \\& \quad \leq \frac{(1+ \Vert x \Vert )\rho L}{\alpha \vert \mu \vert }\ln \frac{1+ \vert \mu \vert (s_{i}+\omega )^{\alpha }}{1+ \vert \mu \vert s_{i}^{\alpha }} \\& \qquad {} +2L\bigl(1+ \Vert x \Vert \bigr) \int _{0}^{s_{i}} \frac{(s_{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (s_{i}-\tau )^{\alpha }}\bigl( \varphi (\tau +\omega )+\varphi (t)\bigr)\,d\tau . \end{aligned}$$
(65)

Then, from (21)–(23), (57), (64) and (65), we derive \(\lim_{\theta \rightarrow \infty }\|y(\theta +\omega )-y(\theta )\|=0 \).

Step 2. Put \(D_{\lambda }=\{u\in \mathit{SAP}_{\omega }\mathit{PC}(J,E):\|u\|\leq \lambda \}\), where

$$ \lambda = \frac{L \Vert x_{0} \Vert +L( \Vert x_{1} \Vert +\kappa _{2})(1+\max \{1,\frac{1}{ \vert \mu \vert }\})+\eta +L\eta +\frac{1}{2}}{1-[\eta +L\eta +L\kappa _{2} \max \{1,\frac{1}{ \vert \mu \vert }\}+\frac{1}{2}]}. $$
(66)

Due to Eq. (59), λ is well defined. In this step, we show that, if \(x\in D_{\lambda }\) and \(y\in \Theta (x)\), then y \(\|\leq \lambda \).

(i) Let \(\theta \in {}[ 0,\theta _{1}]\). Then from (4)–(6) and \((HF)^{\ast }(\mathrm{iv})\), we get

$$\begin{aligned} \bigl\Vert y(\theta ) \bigr\Vert \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+\max \biggl\{ 1, \frac{1}{ \vert \mu \vert }\biggr\} \biggr) \\ &{}+\bigl(1+ \Vert x \Vert \bigr)L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \int _{0}^{\theta _{1}} \varphi (t)\,d\tau \\ \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr)+ \frac{1+ \Vert x \Vert }{4} \\ \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr)+ \frac{1+\lambda }{4}. \end{aligned}$$
(67)

(ii) Let \(\theta \in (\theta _{i},s_{i}]\), for some \(i\in \mathbb{N} \). In view of (56), we have

$$ \bigl\Vert g_{i}\bigl(\theta ,x\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \leq \eta (\lambda +1), $$
(68)

where \(\eta =\sup_{t\in J}\|h^{\ast }(t)\|\). Moreover, from (15), it follows that

$$ \sup_{i\in \mathbb{N} } \bigl\Vert g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \leq \kappa _{2}( \lambda +1). $$
(69)

Next, let \(\theta \in {}[ s_{i},\theta _{i+1}]\). In view of (4), (5) and (69), we arrive at

$$\begin{aligned}& \bigl\Vert S_{\alpha }(\theta -s_{i})g_{i} \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr)+K_{ \alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\& \quad \leq \biggl(L\eta +L\kappa _{2} \max \biggl\{ 1, \frac{1}{ \vert \mu \vert }\biggr\} \biggr) (\lambda +1) . \end{aligned}$$
(70)

Furthermore, by (6) and \((HF)^{\ast }(\mathrm{iv})\), and (58), we obtain

$$\begin{aligned}& \biggl\Vert \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\& \quad \leq L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \bigl(1+ \Vert x \Vert \bigr) \int _{0}^{\theta } \varphi (t)\,d\tau \\& \quad \leq \bigl(1+ \Vert x \Vert \bigr)L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert } \biggr\} \sup_{\theta \in J} \int _{0}^{ \theta }\varphi (t)\,d\tau < \frac{\lambda +1}{4}. \end{aligned}$$
(71)

Similarly, we obtain

$$ \sup_{i\in \mathbb{N} } \biggl\Vert \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau \biggr\Vert \leq \frac{\lambda +1}{4}. $$
(72)

As a result of (67), (68), (70)- (72), we arrive at

$$\begin{aligned} \sup_{\theta \in J} \bigl\Vert y(\theta ) \bigr\Vert \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+ \max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr)+ \frac{1+\lambda }{4} \\ &{}+\eta (\lambda +1)+\biggl(L\eta +L\kappa _{2} \max \biggl\{ 1, \frac{1}{ \vert \mu \vert }\biggr\} \biggr) (\lambda +1) \\ &{}+\frac{\lambda +1}{4} \\ \leq &L \Vert x_{0} \Vert +L \Vert x_{1} \Vert \biggl(1+\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \biggr)+\eta \\ &{}+L\eta +L\kappa _{2} \max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} +\frac{1}{2} \\ &{}+\lambda {}\biggl[ \eta +L\eta +L\kappa _{2} \max \biggl\{ 1, \frac{1}{ \vert \mu \vert }\biggr\} +\frac{1}{2}\biggr]\leq \lambda. \end{aligned}$$

Therefore, our aim in this step is achieved.

Now, as a result of Steps 1 and 2, Θ is a multivalued function from \(D_{\lambda }\subseteq \mathit{SAP}_{\omega }\mathit{PC}(J,E)\) to the non-empty subsets of \(D_{\lambda }\).

Step 3. Θ is closed (its graph is closed) on \(D_{\lambda }\).

Let \((x_{n})_{n\geq 1}\), \((y_{n})_{n\geq 1}\) be two sequences in \(D_{\lambda }\) with \(x_{n}\rightarrow x\), \(y_{n}\rightarrow y\) and \(y_{n}\in \Theta (x_{n})\), \(\forall n\geq 1\). Then we have \(f_{n}\in S_{F(\cdot,x_{n}(\cdot))}^{1}\) such that

$$ y_{n}(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{n}(\tau )\,d\tau , \quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x_{n}(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x_{n}(\theta _{i}^{-}))+K_{ \alpha }(\theta -s_{i})g_{i}^{\prime }(s_{i},x_{n}(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f_{n}(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f_{n}(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} .\end{cases}$$

Let θ be a fixed point in \([0,\theta _{1}]\) and \(J_{\theta }=[0,\theta ]\). In view of \((HF)^{\ast }(\mathrm{iv})\), we have

$$ \bigl\Vert f_{n}(t) \bigr\Vert \leq \varphi (t) (1+\lambda ), \quad \mbox{a.e. }t\in {}[ 0, \theta ]. $$
(73)

Using similar arguments to Step 3 in the proof of Theorem 1, one can show, by (73), that \(f_{n}\rightharpoonup f \) weakly in \(L^{2}(J_{\theta },E)\) and there is a sequence of convex combinations \((z_{n})\) of \((f_{n}) \) such that \(z_{n}\rightarrow f\), a.e. \(t\in J_{\theta }\), and

$$ \lim_{n\rightarrow \infty } \int _{0}^{\theta }T_{\alpha }(\theta - \tau )z_{n}(\tau )\,d\tau = \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau ; \quad \theta \in {}[ 0,\theta _{1}]. $$
(74)

Now, due to the continuity of \(g_{i}(\theta ,\cdot)\), \(g_{i}^{\prime }(s_{i},\cdot)\), \(S_{\alpha }(\theta -s_{i})\) and \(K_{\alpha }(\theta -s_{i})\), we arrive at

$$ \lim_{n\rightarrow \infty }g_{i}\bigl(\theta ,x_{n} \bigl(\theta _{i}^{-}\bigr)\bigr)=g_{i} \bigl( \theta ,x\bigl(\theta _{i}^{-}\bigr) \bigr) $$
(75)

and

$$\begin{aligned}& \lim_{n\rightarrow \infty }S_{\alpha }(\theta -s_{i})g_{i} \bigl(s_{i},x_{n}\bigl( \theta _{i}^{-} \bigr)\bigr)+K_{\alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},x_{n}\bigl( \theta _{i}^{-} \bigr)\bigr) \\& \quad = S_{\alpha }(\theta -s_{i})g_{i} \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr)+K_{ \alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) . \end{aligned}$$
(76)

Noting that \((z_{n})\) is a subsequence of \((f_{n})\), and hence by (74)–(76), there is a subsequence of \((y_{n})\) that converge to

$$ y^{\ast }(t)= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} .\end{cases}$$

Because \(y_{n}\rightarrow y\), we arrive at \(y=y^{\ast }\). Moreover, \((HF)^{\ast }(\mathrm{ii})\) ensures that \(f(\tau )\in F(\tau ,x(\tau ))\), a.e. \(\tau \in J\). So, \(y\in \Theta (x)\).

Step 4. lim\(_{\theta \rightarrow \infty }\|y(\theta )\|=0\) uniformly on \(D_{\lambda }\).

Let \(x\in D_{\lambda }\) and \(y\in \Theta (x)\). According to the definition of Θ, there is \(f\in S_{F(\cdot,x(\cdot))}^{1}\) such that y is given by (18). We consider two cases:

Case 1. \(\theta \in (\theta _{i},s_{i}]\), \(i\in \mathbb{N} \). By (56), we get

$$ \bigl\Vert y(\theta ) \bigr\Vert = \bigl\Vert g_{i}\bigl( \theta ,x\bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \leq (\lambda +1) \bigl\Vert h^{ \ast }(\theta ) \bigr\Vert . $$
(77)

Case 2. \(\theta \in {}[ s_{i},\theta _{i+1}]\), \(i\in \mathbb{N} \). In view of (3), (15), (56) and \((HF)^{\ast }(\mathrm{iv})\), we find

$$\begin{aligned} \bigl\Vert y(\theta ) \bigr\Vert =& \bigl\Vert S_{\alpha }(\theta -s_{i})g_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr)\bigr) \bigr\Vert \\ &{}+ \bigl\Vert K_{\alpha }(\theta -s_{i})g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\ &{}+ \biggl\Vert \int _{0}^{s_{i}}T_{\alpha }(s_{i}- \tau )f(\tau )\,d\tau \biggr\Vert \\ &{}+ \biggl\Vert \int _{0}^{\theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau \biggr\Vert \\ \leq &(\lambda +1)\eta \frac{L}{1+ \vert \mu \vert (\theta -s_{i})^{\alpha }}+( \lambda +1)\kappa _{2} \frac{L(\theta -s_{i}+1)}{1+ \vert \mu \vert (\theta -s_{i})^{\alpha }} \\ &{}+L(1+\lambda ) \int _{0}^{s_{i}} \frac{L (s_{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (s_{i}-\tau )^{\alpha }} \varphi ( \tau )\,d\tau \\ &{}+L(1+\lambda ) \int _{0}^{\theta _{i}} \frac{L (\theta _{i}-\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta _{i}-\tau )^{\alpha }} \varphi ( \tau )\,d\tau . \end{aligned}$$
(78)

It follows from (77) (78) and (57) that \(\lim_{\theta \rightarrow \infty }\|y(\theta )\|=0\) uniformly on \(D_{\lambda }\).

Step 5. Let \(D=\Theta (D_{\lambda })\). In this step, we claim that the set \(D_{| \overline{J_{i}}}\) is equicontinuous for every \(i\in \mathbb{N} \), where

$$ D_{| \overline{J_{i}}}=\bigl\{ y^{\ast }\in C(\overline{J_{i}},E):y^{ \ast }( \theta )=y(\theta ),t\in J_{i}=\, ] \theta _{i},\theta _{i+1} ] ,y^{\ast }(\theta _{i})=y\bigl(\theta _{i}^{+}\bigr),y\in D\bigr\} . $$

Let \(y^{\ast }\in D_{| \overline{J_{i}}}\). Then we have \(x\in D\) and \(f \in S_{F(\cdot,x(\cdot))}^{1}\) such that

$$ y(\theta )= \textstyle\begin{cases} S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1} \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ 0,\theta _{1}], \\ g_{i}(\theta ,x(\theta _{i}^{-})),\quad \theta \in (\theta _{i},s_{i}],i \in \mathbb{N} , \\ S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-}))+K_{\alpha }( \theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})) \\ \quad {}-\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau \\ \quad {}+\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau ,\quad \theta \in {}[ s_{i},\theta _{i+1}],i\in \mathbb{N} ,\end{cases}$$

and \(y^{\ast }(\theta _{i})=y(\theta _{i}^{+})\).

Case 1. Let \(\theta _{1},\theta _{2}\in \overline{J_{0}}=[0,\theta _{1}]\) with \(\theta _{1}<\theta _{2}\). We have

$$\begin{aligned} \bigl\Vert y^{\ast }(\theta _{2})-y^{\ast }( \theta _{1}) \bigr\Vert =& \bigl\Vert y(\theta _{2})-y(\theta _{1}) \bigr\Vert \\ \leq & \bigl\Vert S_{\alpha }(\theta _{2})-S_{\alpha }( \theta _{1}) \bigr\Vert \Vert x_{0} \Vert + \bigl\Vert K_{\alpha }(\theta _{2})-K_{\alpha }(\theta _{1}) \bigr\Vert \Vert x_{1} \Vert \\ &{}+ \biggl\Vert \int _{0}^{\theta _{2}}T_{\alpha }(\theta _{2}-\tau )f(s)\,ds- \int _{0}^{\theta _{1}}T_{\alpha }(\theta _{1}-\tau )f(\tau )\,ds \biggr\Vert \\ \leq & \bigl\Vert S_{\alpha }(\theta _{2})-S_{\alpha }( \theta _{1}) \bigr\Vert \Vert x_{0} \Vert + \bigl\Vert K_{\alpha }(\theta _{2})-K_{\alpha }(\theta _{1}) \bigr\Vert \Vert x_{1} \Vert \\ &{}+ \int _{\theta _{1}}^{\theta _{2}} \bigl\Vert T_{\alpha }( \theta _{2}-\tau ) \bigr\Vert \bigl\Vert f(\tau ) \bigr\Vert \,d\tau \\ &{}+ \int _{0}^{\theta _{1}} \bigl\Vert T_{\alpha }( \theta _{2}-\tau )-T_{\alpha }( \theta _{1}-\tau ) \bigr\Vert \bigl\Vert f(\tau ) \bigr\Vert \,d\tau \\ =&Q_{1}+Q_{2}+Q_{3}. \end{aligned}$$
(79)

From the compactness of the families \(\{S_{\alpha }(\theta ):\theta >0\}\) and \(\{K_{\alpha }(\theta ):\theta >0\}\), we get

$$ \lim_{\theta _{2}\rightarrow \theta _{1}}Q_{1}=\lim_{\theta _{2} \rightarrow \theta _{1}} \bigl\Vert S_{\alpha }(\theta _{2})-S_{\alpha }( \theta _{1}) \bigr\Vert \Vert x_{0} \Vert + \bigl\Vert K_{\alpha }(\theta _{2})-K_{\alpha }(\theta _{1}) \bigr\Vert \Vert x_{1} \Vert =0, $$
(80)

and from the continuity of φ, we arrive at

$$\begin{aligned} \lim_{\theta _{2}\rightarrow \theta _{1}}Q_{2} =&\lim_{\theta _{2} \rightarrow \theta _{1}} \int _{\theta _{1}}^{\theta _{2}} \bigl\Vert T_{\alpha }( \theta _{2}-\tau ) \bigr\Vert \bigl\Vert f(\tau ) \bigr\Vert \,d\tau \\ \leq &(1+\lambda )L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} \lim _{\theta _{2} \rightarrow \theta _{1}} \int _{\theta _{1}}^{\theta _{2}}\varphi ( \tau )\,d\tau =0. \end{aligned}$$
(81)

Moreover, since the family \(\{T_{\alpha }(\theta ):\theta >0\}\) is compact, we get

$$\begin{aligned} \lim_{\theta _{2}\rightarrow \theta _{1}}Q_{3} =&\lim_{\theta _{2} \rightarrow \theta _{1}} \int _{0}^{\theta _{1}} \bigl\Vert T_{\alpha }( \theta _{2}- \tau )-T_{\alpha }(\theta _{1}-\tau ) \bigr\Vert \bigl\Vert f(\tau ) \bigr\Vert \,d\tau \\ \leq &(1+\lambda )\lim_{\theta _{2}\rightarrow \theta _{1}} \int _{0}^{ \theta _{{1}}} \bigl\Vert T_{\alpha }( \theta _{2}-\tau )-T_{\alpha }(\theta _{1}- \tau ) \bigr\Vert \varphi (\tau )\,d\tau \\ =&0. \end{aligned}$$
(82)

Equations (79)–(82) lead to \(\lim_{\theta _{2}\rightarrow \theta _{1}} \Vert y^{\ast }( \theta _{2})-y^{\ast }(\theta _{1}) \Vert =0\).

Case 2. Let \(\theta _{1},\theta _{2}\in (\theta _{i},s_{i}]\) (\(i\in \mathbb{N} \)) with \(\theta _{1}<\theta _{2}\). From the fact that \(g_{i}\) is uniformly continuous on bounded sets,

$$\begin{aligned} \lim_{\theta _{2}\rightarrow \theta _{1}} \bigl\Vert y^{\ast }( \theta _{2})-y^{\ast }(\theta _{1}) \bigr\Vert =& \lim_{\theta _{2} \rightarrow \theta _{1}} \bigl\Vert y(\theta _{2})-y(\theta _{1}) \bigr\Vert \\ \leq &\lim_{\theta _{2}\rightarrow \theta _{1}} \bigl\Vert g_{i}\bigl( \theta _{2},x\bigl(\theta _{i}^{-}\bigr) \bigr)-g_{i}\bigl(\theta ,x\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\ \leq &\lim_{\theta _{2}\rightarrow \theta _{1}}\sup_{ \Vert z \Vert \leq \lambda }\bigl\Vert g_{i}(\theta _{2},z)-g_{i}(\theta _{1},z) \bigr\Vert =0. \end{aligned}$$
(83)

Case 3. Let \(\theta _{1},\theta _{2}\in (s_{i},\theta _{i+1}]\) (\(i\in \mathbb{N} \)) with \(\theta _{1}<\theta _{2}\).

$$\begin{aligned} \lim_{\theta _{2}\rightarrow \theta _{1}} \bigl\Vert y^{\ast }( \theta _{2})-y^{\ast }(\theta _{1}) \bigr\Vert =& \lim_{\theta _{2} \rightarrow \theta _{1}} \bigl\Vert y(\theta _{2})-y(\theta _{1}) \bigr\Vert \\ \leq &\lim_{\theta _{2}\rightarrow \theta _{1}} \bigl\Vert S_{\alpha }( \theta _{2}-s_{i})-S_{\alpha }(\theta _{1}-s_{i}) \bigr\Vert \bigl\Vert g_{i}\bigl(s_{i},x\bigl( \theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\ &{}+\lim_{\theta _{2}\rightarrow \theta _{1}}\|K_{\alpha }(\theta _{2}-s_{i})-K_{ \alpha }( \theta _{1}-s_{i}) \bigl\Vert g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\ &{}+\lim_{\theta _{2}\rightarrow \theta _{1}} \biggl\Vert \int _{0}^{\theta _{2}}T_{ \alpha }(\theta _{2}-\tau )f(s)\,ds- \int _{0}^{\theta _{1}}T_{\alpha }( \theta _{1}-\tau )f(s)\,ds \biggr\Vert . \end{aligned}$$
(84)

Again, from the compactness of the families \(\{S_{\alpha }(\theta ):\theta >0\}\) and \(\{K_{\alpha }(\theta ):\theta >0\}\), we have

$$\begin{aligned}& \lim_{\theta _{2}\rightarrow \theta _{1}} \bigl\Vert S_{\alpha }(\theta _{2}-s_{i})-S_{ \alpha }(\theta _{1}-s_{i}) \bigr\Vert \bigl\Vert g_{i}\bigl(s_{i},x\bigl(\theta _{i}^{-} \bigr)\bigr) \bigr\Vert \\& \quad = \lim_{\theta _{2}\rightarrow \theta _{1}}\bigl\| K_{\alpha }(\theta _{2}-s_{i})-K_{ \alpha }( \theta _{1}-s_{i})\bigr\| \bigl\Vert g_{i}^{\prime } \bigl(s_{i},x\bigl(\theta _{i}^{-}\bigr) \bigr) \bigr\Vert \\& \quad = 0. \end{aligned}$$
(85)

Furthermore, by repeating the arguments employed in Case 1, one can show that

$$ \lim_{\theta _{2}\rightarrow \theta _{1}} \biggl\Vert \int _{0}^{\theta _{2}}T_{ \alpha }(\theta _{2}-\tau )f(s)\,ds- \int _{0}^{\theta _{1}}T_{\alpha }( \theta _{1}-\tau )f(s)\,ds \biggr\Vert =0. $$
(86)

Equations (84) and (86) ensure that \(\lim_{\theta _{2}\rightarrow \theta _{1}} \Vert y^{\ast }( \theta _{2})-y^{\ast }(\theta _{1}) \Vert \). Therefore, \(T_{| \overline{J_{i}}}\) is equicontinuous for any \(i\in \mathbb{N} \).

Step 6. Our goal in this step is showing that, for any \(i\in \mathbb{N} \) and any \(\theta \in J_{i}\), the set \(Z_{\theta }^{i}:=\{y(\theta ):y\in D=\Theta (D_{\lambda })\}\) is relatively compact in E.

Case 1. Let \(\theta \in J_{0}=[0,\theta _{1}]\). If \(\theta =0\), then \(Z_{\theta }^{1}=\{x_{0}\}\) is compact. Let \(\theta \in (0,\theta _{1}]\). be a fixed point. We have

$$\begin{aligned} Z_{\theta }^{1} =&\bigl\{ y(\theta ):y\in \Theta (D_{\lambda })\bigr\} \\ =&\biggl\{ S_{\alpha }(\theta )x_{0}+K_{\alpha }(\theta )x_{1}+ \int _{0}^{ \theta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau :f\in S_{F(\cdot,x(\cdot))}^{1},x \in D_{\lambda } \biggr\} . \end{aligned}$$

Now, for any \(\delta \in (0,\theta )\), let

$$ Z_{\theta ,\delta }^{1}:=\biggl\{ S_{\alpha }(\theta )x_{0}+K_{\alpha }( \theta )x_{1}+ \int _{0}^{\theta -\delta }T_{\alpha }(\theta -\tau )f( \tau )\,d\tau :f\in S_{F(\cdot,x(\cdot))}^{1},x\in D_{\lambda } \biggr\} . $$

Notice that \(\|f(\tau )\|\leq (1+\lambda )\varphi (\tau )\), for any \(\tau \in {}[ 0,\theta -\delta ]\), any \(n\geq 1\). Because φ is continuous on \([0,\theta ]\), the set \(\{f_{n}(\tau ):n\geq 1\}\) is bounded, and hence, by the compactness of \(T_{\alpha }(\theta -\tau )\); \(\tau \in {}[ 0,\theta -\delta ]\), the set \(Z_{\theta ,\delta }^{1}\) is relatively compact in E. Moreover, for any \(x\in D_{\lambda }\) and any \(f\in S_{F(\cdot,x(\cdot))}^{1}\), we get from (6), \((HF)^{\ast }(\mathrm{iv})\) and the continuity of φ,

$$\begin{aligned}& \lim_{\delta \rightarrow 0} \biggl\Vert \int _{0}^{\theta }T_{\alpha }( \theta -\tau )f(\tau )\,d\tau - \int _{0}^{\theta -\delta }T_{\alpha }( \theta -\tau )f(\tau )\,d\tau \biggr\Vert \\& \quad \leq L\max \biggl\{ 1,\frac{1}{ \vert \mu \vert }\biggr\} (1+\lambda )\lim _{\delta \rightarrow 0} \int _{\theta -\delta }^{\theta }\varphi (\tau )\,d\tau \\& \quad = 0. \end{aligned}$$

Then there exist relatively compact sets that can be arbitrarily approximated to the set \(Z_{\theta }^{i}\). Then it is relatively compact in E.

Case 2. Let \(\theta \in {}[ s_{i},\theta _{i}]\); \(i\in \mathbb{N}\) be a fixed point. Since the set \(\{x(\theta _{i}^{-}):x\in D_{\lambda }\}\) is bounded, then by \((Hg_{i})^{ \ast }(\mathrm{ii})\), the set \(\{y(\theta ):y\in \Theta (x),x\in D_{\lambda }\}=\{g_{i}(\theta ,x( \theta _{i}^{-})):x\in D_{\lambda }\}\) is relatively compact in E.

Case 3. Let \(\theta \in {}[ \theta _{i},s_{i+1}]\); \(i\in \mathbb{N}\) be a fixed point. As in Case 2, the set \(\{g_{i}(\theta ,x(\theta _{i}^{-})):x\in D_{\lambda }\}\) is relatively compact in E, and hence, by the compactness of \(S_{\alpha }(\theta -s_{i})\), we conclude that \(\{S_{\alpha }(\theta -s_{i})g_{i}(s_{i},x(\theta _{i}^{-})):x\in D_{ \lambda }\}\) is relatively compact in E. Also, due to the compactness of \(K_{\alpha }(t)\); \(t>0\), the set \(\{K_{\alpha }(\theta -s_{i})g_{i}^{\prime }(s_{i},x(\theta _{i}^{-})):x \in D_{\lambda }\}\) is relatively compact in E. Furthermore, using the same arguments as in Case 1, one can show that the sets \(\{\int _{0}^{\theta }T_{\alpha }(\theta -\tau )f(\tau )\,d\tau :f\in S_{F(\cdot,x(\cdot))}^{1},x \in D_{\lambda }\}\) and \(\{\int _{0}^{s_{i}}T_{\alpha }(s_{i}-\tau )f(\tau )\,d\tau :f\in S_{F(\cdot,x(\cdot))}^{1},x \in D_{\lambda }\}\) are relatively compact. As a result of this discussion, we conclude that, for any \(i\in \mathbb{N} \) and any \(t\in J_{i}\), the set \(Z_{\theta }^{i}=\{y(\theta ):y\in \Theta (D_{\lambda })\}\) is relatively compact in E.

Now, according to Lemma 4, Steps 4, 5 and 6 imply the set \(T=\Theta (D_{\lambda })\) is relatively compact in \(\mathit{SAP}_{\omega }\mathit{PC}(J,E)\).

Now, as a result of Steps 1–6, we conclude that Θ is closed and completely continuous from \(D_{\lambda }\) to the family on non-empty convex compact of \(D_{\lambda }\). Applying Lemma 3, Θ has a fixed point which is an S-asymptotically ω-periodic mild solution to Problem (1). □

Remark 4

For any \(t\geq 0\), we have

$$ \frac{t^{\alpha -1}}{1+ \vert \mu \vert t^{\alpha }}\leq \textstyle\begin{cases} \frac{2}{ \vert \mu \vert (1+t)};&\text{if }0< \vert \mu \vert \leq 1, \\ \frac{2}{1+t} ;&\text{if } \vert \mu \vert >1.\end{cases} $$
(87)

To clarify this, we consider the following cases:

Case 1. Let \(0<|\mu |\leq 1\). Then

$$ \frac{t^{\alpha -1}}{1+ \vert \mu \vert t^{\alpha }}\leq \textstyle\begin{cases} t^{\alpha -1}\leq 1\leq \frac{2}{ \vert \mu \vert (1+t)};&\text{if }0\leq t\leq 1, \\ \frac{t^{\alpha -1}}{ \vert \mu \vert t^{\alpha }}=\frac{1}{ \vert \mu \vert t}\leq \frac{2}{ \vert \mu \vert (1+t)};&\text{if }t>1.\end{cases}$$

So,

$$ \frac{t^{\alpha -1}}{1+ \vert \mu \vert t^{\alpha }}\leq \frac{2}{ \vert \mu \vert (1+t)}, \quad \forall t\geq 0. $$

Case 2. Let \(|\mu |>1\). In this case, we have

$$ \frac{t^{\alpha -1}}{1+ \vert \mu \vert t^{\alpha }}\leq \textstyle\begin{cases} t^{\alpha -1}\leq 1\leq \frac{2}{1+t};&\text{if }0\leq t\leq 1, \\ \frac{t^{\alpha -1}}{ \vert \mu \vert t^{\alpha }}=\frac{1}{ \vert \mu \vert t}\leq \frac{1}{t}\leq \frac{2}{1+t};&\text{if }t>1,\end{cases}$$

which yields

$$ \frac{t^{\alpha -1}}{1+ \vert \mu \vert t^{\alpha }}\leq \frac{2}{1+t},\quad \forall t \geq 0. $$

Then (87) holds.

Examples

Example 1

Let \(\alpha =\frac{3}{2}\), \(m=4\), \(\omega =2\pi \), \(J=[0,\infty )\), \(s_{i}=i\frac{\pi }{2}\); \(i\in \{0\}\cup \mathbb{N} \), and \(\theta _{i}=(2i-1)\frac{\pi }{4}\); \(i\in \mathbb{N} \). Observe that, for \(i\in \mathbb{N} \), \(s_{i+m}=s_{i+4}=(i+4)\frac{\pi }{2}=s_{i}+2\pi \), and \(\theta _{i+4}=(2(i+4)-1)\frac{\pi }{4}=\theta _{i}+2\pi \). Let \(\Omega =\{s=(s_{1},s_{2}):s_{1}^{2}+s_{2}^{2} \leq 1\}\), and \(E=L^{2}(\Omega )\). Define an operator \(A:D(A)\subseteq E\rightarrow E\) by

$$ A(u):=\Delta u-u, $$
(88)

with \(D(A)=H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\). It is known that (see [19]) A is a sectorial operator of type \(\{M,\varphi ,\alpha ,\mu \}\) with \(\mu =-1\) and \(L=3\). Let Z be a non-empty, compact and convex subset of E, \(\upsilon =\sup_{z\in Z}\|z\|\). Consider the multivalued function \(F:J\times E\rightarrow 2^{E}\) defined by

$$ F(\theta ,u):=\{v_{\theta ,u,z}:z\in Z\}, $$
(89)

where for any \(z \in Z\) any \((\theta ,u)\in J\times E\), \(v_{\theta ,u,z}:\Omega \rightarrow \mathbb{R} \); \(v_{\theta ,u,z}(s)= \frac{1+\rho \sin u(s)}{(\theta +1)\upsilon }\|z\|\); \(\rho >0\). Since Z is a non-empty, convex and compact subset, the values of F also are. Moreover,

$$\begin{aligned}& \bigl\Vert F(\theta ,0) \bigr\Vert =\sup_{z\in Z} \frac{1}{(\theta +1)\upsilon } \Vert z \Vert \leq \frac{1}{\theta +1}=\sigma (\theta ),\quad \forall \theta \in J, \end{aligned}$$
(90)
$$\begin{aligned}& \begin{aligned}[b] h\bigl(F(\theta ,u),F(\theta ,v)\bigr) &\leq \biggl( \int _{\Omega } \bigl\vert v_{\theta ,u,z}(s)-v_{ \theta ,v,z}(s) \bigr\vert ^{2}\,ds\biggr)^{\frac{1}{2}},\quad \forall z\in Z \\ &=\frac{\rho }{\theta +1} \int _{\Omega } \bigl\vert \sin u(s)-\sin v(s) \bigr\vert ^{2}dx)^{ \frac{1}{2}} \\ &\leq \frac{\rho }{\theta +1} \int _{\Omega } \bigl\vert u(s)-v(s) \bigr\vert ^{2}dx)^{ \frac{1}{2}} \\ &=\frac{\rho }{\theta +1} \Vert u-v \Vert , \end{aligned} \end{aligned}$$
(91)

and

$$\begin{aligned} h\bigl(F(\theta +2\pi ,u),F(\theta ,u)\bigr) \leq &\biggl( \int _{\Omega } \bigl\vert v_{\theta +2 \pi ,u,z}(s)-v_{\theta ,u,z}(s) \bigr\vert ^{2}\,ds\biggr)^{\frac{1}{2}},\quad \forall z\in Z \\ =& \int _{\Omega } \biggl\vert \frac{\rho \sin u(s)+1}{\theta +1+2\pi }- \frac{\rho \sin u(s)+1}{\theta +1} \biggr\vert ^{2}dx)^{\frac{1}{2}} \\ \leq &\frac{1}{(\theta +1+2\pi )(\theta +1)}\biggl( \int _{\Omega }2\pi \bigl\vert \rho \sin u(s)+1 \bigr\vert ^{2}\,ds\biggr)^{{\frac{1}{2}}} \\ \leq &\frac{2\pi (\rho +1)}{(\theta +1+2\pi )(\theta +1)}\bigl( \Vert u \Vert +1 \bigr)=L_{2}(\theta ) \bigl( \Vert u \Vert +1\bigr). \end{aligned}$$
(92)

Due to (91) and (92), Eqs. (7) and (8) are verified with \(L_{1}(\theta )=\frac{\rho }{\theta +1}\) and \(L_{2}(\theta )=\frac{2\pi (\rho +1)}{(\theta +1+2\pi )(\theta +1)}\); \(\theta \in {}[ 0,\infty )\).

Next, in view of (87), and since \(|\mu |=1\), it follows that

$$ \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }} \leq \frac{2}{1+\theta -\tau },\quad \forall \theta \geq 0,\tau \in {}[ 0,\theta ]. $$

Therefore,

$$\begin{aligned}& \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau \\& \quad \leq 2\rho \int _{0}^{\theta }\frac{1}{1+\theta -\tau } \frac{1}{\tau +1}\,d\tau \\& \quad = 2\rho \int _{0}^{\theta }\frac{1}{\theta +2}\biggl[ \frac{1}{1+\theta -\tau }+\frac{1}{\tau +1}\biggr]\,d\tau \\& \quad = 4\rho \frac{\ln \vert 1+\theta \vert }{(\theta +2)}, \end{aligned}$$

which yields \(\lim_{\theta \rightarrow \infty }\int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+|\mu |(\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau =0\). Similarly, \(\lim_{\theta \rightarrow \infty }\int _{0}^{ \theta }\frac{(\theta -\tau )^{\alpha -1}}{1+|\mu |(\theta -\tau )^{\alpha }} \sigma (\tau )\,d\tau =0\). Hence, (9) and (15) are verified. Moreover,

$$ \xi =\sup_{\theta \in J} \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{1}( \tau )\,d\tau \leq 4\rho . $$
(93)

Now, again from (87), one has

$$\begin{aligned}& \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}L_{2}( \tau )\,d\tau \\& \quad \leq (\rho +1)\lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{1}{1+\theta -\tau }\biggl( \frac{2\pi }{(\tau +1+2\pi )(\tau +1)}\biggr)\,d\tau \\& \quad \leq (\rho +1)\lim_{\theta \rightarrow \infty } \int _{0}^{\theta }\biggl[ \frac{1}{1+\theta -\tau } \frac{1}{\tau +1}-\frac{1}{1+\theta -\tau } \frac{1}{\tau +1+2\pi }\biggr]\,d\tau \\& \quad = (\rho +1)\lim_{\theta \rightarrow \infty }\biggl[\frac{1}{\theta +2}\int _{0}^{\theta }\biggl[\frac{1}{1+\theta -\tau }+ \frac{1}{\tau +1}\biggr]\,d\tau \\& \qquad {} -\frac{1}{\theta +2+2\pi } \int _{0}^{\theta }\biggl[ \frac{1}{1+\theta -\tau }+\frac{1}{\tau +1+2\pi }\biggr]\,d\tau \biggr] \\& \quad = (\rho +1)\lim_{\theta \rightarrow \infty } \frac{2\ln \vert 1+\theta \vert }{\theta +2} \\& \qquad {} -(\rho +1)\lim_{\theta \rightarrow \infty } \frac{1}{\theta +2+2\pi }\bigl[\ln \vert 1+\theta \vert +\ln \vert 1+2\pi +\theta \vert -\ln (2 \pi +1)\bigr] \\& \quad = 0, \end{aligned}$$
(94)

which means that (16) holds.

Furthermore, for any \(i\in \mathbb{N} \), define \(g_{i}:[t_{i},s_{i}]\times E\rightarrow E\) by

$$ g_{i}(\theta ,u) (s):=\frac{\varrho u(s)\sin i\theta }{i} ;\quad ( \theta ,u)\in {}[ \theta _{i},s_{i}]\times E,s\in \Omega , $$
(95)

where ϱ is a positive real number. Obviously for any \(u\in E\), \(\frac{d}{d\theta }(g_{i}(\theta ,u))(s)=\varrho (\cos i \theta )u(s)\); \(t\in J\). Notice that, for any \(u\in E\),

$$\begin{aligned}& \lim_{ \substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }} \bigl\Vert g_{i+m}( \theta +2\pi ,u)-g_{i}(\theta ,u) \bigr\Vert \\& \quad = \lim_{ \substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }}\biggl( \int _{\Omega } \biggl\vert \frac{\varrho u(s)\sin (i+m)(\theta +2\pi )}{i+m}- \frac{\varrho u(s)\sin (i\theta )}{i} \biggr\vert ^{2}\,ds\biggr)^{\frac{1}{2}} \\& \quad \leq \lim_{i\rightarrow \infty }\frac{2\varrho }{i} \Vert u \Vert =0. \end{aligned}$$

Then (10) is satisfied. Moreover, for any \(u_{1},u_{2}\in E\),

$$\begin{aligned}& \bigl\Vert g_{i}(\theta ,u_{1})-g_{i}( \theta ,u_{2}) \bigr\Vert \\& \quad = \frac{\varrho }{i}\biggl( \int _{\Omega } \bigl\vert u_{1}(s)\sin i\theta -u_{2}(s) \sin i\theta \bigr\vert ^{2}\,ds \biggr)^{\frac{1}{2}} \\& \quad \leq \varrho \Vert u_{1}-u_{2} \Vert \end{aligned}$$
(96)

and

$$\begin{aligned} \bigl\Vert g_{i}^{\prime }(s_{i},u_{1})-g_{i}^{\prime }(s_{i},u_{2}) \bigr\Vert =&\biggl( \int _{\Omega }\varrho ^{2} \bigl\vert u_{1}(s) (\cos i\theta )-u_{2}(s) (\cos i \theta ) \bigr\vert ^{2}\,ds\biggr)^{\frac{1}{2}} \\ \leq &\varrho \|u_{1}-u_{2})\|. \end{aligned}$$
(97)

From (96) and (97), it follows that (11) and (12) hold with \(N=\mathcal{N}=\varrho \). In addition,

$$ \bigl\Vert g_{i}(\theta ,u) \bigr\Vert =\biggl( \int _{\Omega } \biggl\vert \frac{\varrho u(s)\sin i\theta }{i} \biggr\vert ^{2}\,ds\biggr)^{\frac{1}{2}}\leq \frac{\varrho }{i} \Vert u \Vert \leq \varrho \bigl( \Vert u \Vert +1 \bigr) $$
(98)

and

$$ \bigl\Vert g_{i}^{\prime }(s_{i},z) \bigr\Vert =\biggl( \int _{\Omega } \bigl\vert \varrho u(s)\cos is_{i} \bigr\vert ^{2}\,ds\biggr)^{ \frac{1}{2}}\leq \varrho \Vert u \Vert \leq \varrho \bigl( \Vert u \Vert +1\bigr). $$
(99)

By (98) and (99), we arrive at (13) and (14) are verified where \(\kappa _{1}=\kappa _{1}=\varrho \).

Now, all the assumptions of Theorem 1 are satisfied, so the problem

$$ \textstyle\begin{cases} {}^{c}D_{s_{i},\theta }^{\alpha }x(\theta ,s)\in Ax(\theta ,s)+F( \theta ,x(\theta )),\quad \mbox{a.e. } \theta \in (s_{i}, \theta _{i+1}],i\in \mathbb{N} \cup \{0\},s\in \Omega , \\ x(\theta _{i}^{+},s)=g_{i}(\theta _{i},x(\theta _{i}^{-},s)),\quad i\in \mathbb{N} ,s\in \Omega , \\ x(\theta ,s)=g_{i}(\theta ,x(\theta _{i}^{-},s)),\quad \theta \in (\theta _{i} s_{i}],i\in \mathbb{N} ,s\in \Omega , \\ x(0,s)=x_{0}(s),\qquad x'(0)=x_{1}(s);\quad s\in \Omega , \\ x(\theta ,s)=0,\quad \theta \in J,s\in \partial \Omega ,\end{cases} $$

has an S-asymptotically 2π-periodic mild solution, \(x:J\rightarrow L^{2}(\Omega )\), provided that

$$ 8\rho +3\varrho < \frac{1}{3}, $$
(100)

where A, F, \(g_{i}\) are given by (88), (90) and (94). By choosing ρ and ϱ sufficiently small, we can arrive at (100).

Example 2

Let A, α, E, m, \(\omega =2\pi \), J, \(s_{i}\), \(\theta _{i}\); \(i\in \mathbb{N} \) be as in Example 1, Z be a non-empty convex compact subset of E, the families \(\{S_{\alpha }(t):t>0\}\), \(\{K_{\alpha }(t):t>0\}\) and \(\{T_{\alpha }(t):t>0\}\) are compact [19]. Define a multivalued function \(F:J\times E\rightarrow P_{ck}(E)\) by

$$ F(\theta ,u)= \frac{\rho ( \Vert u \Vert +1) }{\upsilon (1+\theta )^{\frac{3}{2}}}Z, $$
(101)

where \(\rho >0\) and \(\upsilon =\operatorname{Sup}\{ \|z \| : z\in Z\}\). Clearly for every \(x\in E\), \(\theta \rightarrow F(\theta ,x)\) is strongly measurable, \(F(\theta ,\cdot)\) is upper semicontinuous and, for any \(u\in \mathit{PC}(J,E)\), the function \(f(\theta )= \frac{\rho (\|u\|+1) }{\upsilon (1+\theta )^{\frac{3}{2}}}z_{0}\); \(z_{0}\in Z\) is locally integrable and \(f(\theta )\in F(\theta ,u(\theta ))\); \(\theta \in J\). Moreover, in view of (101), for any \(u\in E\) and any \(\theta \in J\),

$$ \bigl\Vert F(\theta ,u) \bigr\Vert \leq \frac{\rho ( \Vert u \Vert +1) }{(1+\theta )^{\frac{3}{2}}}= \varphi (\theta ) \bigl(1+ \Vert u \Vert \bigr), $$
(102)

where

$$ \varphi (\theta )=\frac{\rho }{(1+\theta )^{\frac{3}{2}}};\quad \theta \in J. $$
(103)

We show that (57) and (58) are verified. In view of (87), and by arguing as in (94), one has

$$\begin{aligned}& \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{(\theta -\tau )^{\alpha -1}}{1+ \vert \mu \vert (\theta -\tau )^{\alpha }}\bigl( \varphi (\tau +2\pi )+\varphi (\tau )\bigr)\,d\tau \\& \quad = \rho \lim_{\theta \rightarrow \infty } \int _{0}^{\theta }\biggl[ \frac{1}{1+\theta -\tau }\biggl( \frac{1}{(\tau +1+2\pi )^{\frac{3}{2}}}+ \frac{1}{(\tau +1)^{\frac{3}{2}}}\biggr)\biggr]\,d\tau \\& \quad \leq \rho \lim_{\theta \rightarrow \infty } \int _{0}^{\theta }\biggl[ \frac{1}{1+\theta -\tau }\biggl( \frac{1}{\tau +1+2\pi }+\frac{1}{\tau +1}\biggr)\biggr]\,d\tau \\& \quad \leq 2\rho \lim_{\theta \rightarrow \infty } \int _{0}^{\theta } \frac{1}{1+\theta -\tau } \frac{1}{\tau +1}\,d\tau =0. \end{aligned}$$

Then (57) holds. Furthermore,

$$ \sup_{\theta \in J} \int _{0}^{\theta }\varphi (\tau )\,d\tau =\sup _{ \theta \in J}\frac{2\rho }{\sqrt{1+\theta }}=2\rho . $$
(104)

Next, let \(K:D(K)=E\rightarrow E\) be a linear bounded compact operator and for any \(i\in \mathbb{N} \), define \(g_{i}:[\theta _{i},s_{i}]\times E\rightarrow E \) by

$$ g_{i}(\theta ,u) (s)=\frac{(Ku)(s)}{i(1+\theta )},\quad \forall s\in \Omega . $$
(105)

Notice that

$$ g_{i}^{\prime }(s_{i},u) (s)= \frac{-(Ku)(s)}{i(1+s_{i})^{2}},\quad \forall s \in \Omega . $$
(106)

In view of (105) and (106), we get

$$\begin{aligned}& \lim_{ \substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }} \bigl\Vert g_{i+m}( \theta +2\pi ,u)-g_{i}(\theta ,u) \bigr\Vert \\& \quad = \Vert K \Vert \lim_{ \substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }}( \int _{\Omega } \biggl\vert \frac{u(s)}{(i+2\pi )(1+\theta +2\pi )}- \frac{u(s)}{i(1+\theta )} \biggr\vert ^{2}\,ds) ^{\frac{1}{2}} \\& \quad = \Vert K \Vert \Vert u \Vert \lim_{ \substack{ \theta \rightarrow \infty \\ i\rightarrow \infty }} \biggl\vert \frac{1}{(i+2\pi )(1+\theta +2\pi )}-\frac{1}{i(1+\theta )} \biggr\vert =0, \\& \bigl\Vert g_{i}(\theta ,u_{1})-g_{i}( \theta ,u_{2}) \bigr\Vert \\& \quad =\biggl( \int _{\Omega } \biggl\vert \frac{(Ku_{1})(s)}{i(1+\theta )}- \frac{(Ku_{2})(s)}{i(1+\theta )} \biggr\vert ^{2}\,ds\biggr)^{\frac{1}{2}} \\& \quad \leq \frac{\|K|}{i(1+\theta )} \Vert u_{1}-u_{2} \Vert \leq \Vert K \Vert \Vert u_{1}-u_{2} \Vert , \\& \begin{aligned} \bigl\Vert g_{i}(\theta ,u) \bigr\Vert &=\biggl( \int _{\Omega } \biggl\vert \frac{(Ku)(s)}{i(1+\theta )} \biggr\vert ^{2}\,ds\biggr)^{\frac{1}{2}} \\ &\leq \frac{ \Vert K \Vert \Vert u \Vert }{1+\theta }=h^{\ast }(\theta ) \Vert u \Vert , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert g_{i}^{\prime }(s_{i},u) \bigr\Vert =&\biggl( \int _{\Omega } \biggl\vert \frac{(Ku)(s)}{i(1+s_{i})^{2}} \biggr\vert ^{2}\,ds\biggr)^{\frac{1}{2}} \\ \leq &\frac{ \Vert K \Vert \Vert u \Vert }{i(1+s_{i})^{2}}\leq \Vert K \Vert \bigl( \Vert u \Vert +1 \bigr). \end{aligned}$$

Then (10), (11), (14) and (56) are verified with \(N=\mathcal{N}=\|K\|\) and \(h^{\ast }(\theta )=\frac{\|K\|}{1+\theta }\); \(\theta \in J\) and \(\eta = \kappa _{2}=\|K\|\). Notice that, by the compactness of K, the function \(g_{i}(\theta ,\cdot)\); \(i\in \mathbb{N}\) is compact, and hence all assumptions of Theorem 2 are satisfied. So, by applying Theorem 2, Problem (1), where A as be in example (1) and F, \(g_{i}\) are given by (102) and (105), has an S-asymptotically 2π-periodic mild solution provided that \(2\rho <\frac{1}{4L}=\frac{1}{12}\) and \(\|k \|<\frac{1}{14}\).

Example 3

Let A, α, E, m, \(\omega =2\pi \), J, Ω, \(s_{i}\), \(\theta _{i}\);i \(\in \mathbb{N} \) be as in Example 1, Z a non-empty convex compact subset of E and \(x_{0}\), \(x_{1}\) two fixed elements of E. For any \((\theta ,s)\in (J,\Omega )\) and any \(x:J\rightarrow E\), we denote \(x(\theta )(s)\) by \(x(\theta ,s)\). Consider the impulsive semilinear differential inclusion:

$$ \textstyle\begin{cases} {}^{c}D_{s_{i},\theta }^{\alpha }x(\theta ,s)\in \Delta x(\theta ,s)-x( \theta ,s)+\frac{\cos x(\theta ,s)}{40(1+\theta )}Z, \\ \quad \mbox{a.e. } \theta \in (s_{i}, \theta _{i+1}],i\in \mathbb{N} \cup \{0\},s\in \Omega , \\ x(\theta _{i}^{+},s)= \frac{x(\theta _{i}^{-},s)\sin i \theta _{i}}{40i},\quad i\in \mathbb{N} ,s\in \Omega , \\ x(\theta ,s)=\frac{x(\theta _{i}^{-},s)\sin i \theta }{40i},\quad \theta \in (\theta _{i} s_{i}],i\in \mathbb{N} ,s\in \Omega , \\ x(0,s)=x_{0}(s),\qquad x'(0)=x_{1}(s);\quad s\in \Omega , \\ x(\theta ,s)=0,\quad \theta \in J,s\in \partial \Omega .\end{cases} $$
(107)

Let \(F:J\times E\rightarrow 2^{E}\) be defined by

$$ F(\theta ,u)=\frac{\cos u(s) }{40 (1+\theta )}Z. $$
(108)

Obviously, F verifies (i) and (ii) of (HF). Moreover,

$$\begin{aligned}& \bigl\Vert F(\theta ,0) \bigr\Vert \leq \frac{1}{40 (1+\theta )}=\sigma (\theta ), \quad \forall \theta \in J, \end{aligned}$$
(109)
$$\begin{aligned}& h\bigl(F(\theta ,u),F(\theta ,v)\bigr)\leq \frac{1}{40 (1+\theta )} \Vert u-v \Vert , \end{aligned}$$
(110)

and

$$\begin{aligned} h\bigl(F(\theta +2\pi ,u),F(\theta ,u)\bigr) \leq &\frac{1}{40(\theta +1+2\pi )(\theta +1)} \Vert u \Vert \\ \leq &\frac{1}{40(\theta +1+2\pi )(\theta +1)}\bigl( \Vert u \Vert +1\bigr). \end{aligned}$$
(111)

It shows that (7) and (8) are satisfied with \(L_{1}(\theta )=\frac{1}{40 (1+\theta )}\) and \(L_{2}(\theta )=\frac{1}{40(\theta +1+2\pi )(\theta +1)}\); \(\theta \in {}[ 0,\infty )\). Define

$$ g_{i}(\theta ,u) (s):=\frac{u(s)\sin i \theta }{40i};\quad (\theta ,u) \in {}[ \theta _{i},s_{i}]\times E,s\in \Omega . $$
(112)

Using the same arguments as in Example 1, one can demonstrate that (9) and (10)–(16) are obtained with \(N=\mathcal{N}=\kappa _{1}=\kappa _{1}=\frac{1}{40}\) and \(\xi \leq \frac{1}{10}\). Notice that \(L=3 \). So, \(L(2\xi +N_{\cdot}+(1+\max \{1, \frac{1}{|\mu |}\})\mathcal{N})<1\) By applying Theorem 1, Problem (107) has an S-asymptotically 2π-periodic mild solution.

Conclusion

Two existence results of S-asymptotically ω-periodic of mild solutions to non-instantaneous impulsive semilinear differential inclusions of order \(1<\alpha <2\) and generated by sectorial operators are given This work generalizes much recent work such as [1820] to the case when there are impulse effects and the right-hand side is a multivalued function. Moreover, our technique can be used to develop the work in [12, 1517, 21, 2325] to the case when the linear part is a sectorial operator, the nonlinear part is a multivalued function and we have impulse effects. There are many directions for future work, for example:

1—:

With the help of technique in [1], we study the existence of solutions for Problem (1) on a time scales.

2—:

Investigation an existence theorem for a nonlinear singular-delay-fractional differential equation considered in [42, 43], when it contains a sectorial operator as a linear term and the nonlinear term becomes a multivalued function instead of single-valued function.

3—:

With the help of technique in [3], discuss the numerical solutions for Problem (1) on a closed bounded interval.

4—:

Study the S-asymptotically periodic solutions to Problem (1) when the sectorial operator is replaced by almost sectorial.

5—:

Study the S-asymptotically periodic solutions to Problem (1) when it involves p-Laplacian operator \(\varphi _{p}\) as well as when the Caputo derivative is replaced by the ψ-Caputo or ψ-Riemann–Liouville derivative. For contributions on BVP involving the ψ-Riemann–Liouville derivative, see [3] and for references on BVP containing the p-Laplacian operator \(\varphi _{p}\), see [6, 8].

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References

  1. 1.

    Agarwal, P., Ammi, M.S., Asad, J.: Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense. Adv. Differ. Equ. 2021, 162 (2021)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Sunarto, A., Agarwal, P., Sulaiman, J., Chew, J.V., Aruchunan, E.: Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR. Adv. Differ. Equ. 2021, 147 (2021)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Rezapour, S., Etemad, S., Tellab, B., Agarwal, P., Garcia Guirao, J.L.: Numerical solutions caused by DGJIM and ADM methods for multi-term fractional BVP involving the generalized ψ-RL-operators. Symmetry 13(4), 532 (2021)

    Article  Google Scholar 

  4. 4.

    Agarwal, P., Berdyshev, A., Karimov, E.: Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative. Results Math. 71(3), 1235–1257 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Agarwal, R., Hristova, S., O’Regan, D.: A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19(2), 290–318 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Khan, H., Jarad, F., Abdeljawad, T., Khan, A.: A singular ABC-fractional differential equation with p-Laplacian operator. Chaos Solitons Fractals 129, 56–61 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Khan, A., Gomez-Aguilar, J.F., Abdeljawad, T., Khan, H.: Stability and numerical simulation of a fractional order plant-nectar-pollinator model. Alex. Eng. J. 59(1), 49–59 (2020)

    Article  Google Scholar 

  8. 8.

    Khan, H., Tunc, C., Chen, W., Khan, A.: Existence theorems and Hyers–Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator. J. Appl. Anal. Comput. 8(4), 1211–1226 (2018)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Diblik, J., Fečkan, M., Pospish, M.: Non-existence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations. Appl. Math. Comput. 257, 230–240 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kang, Y.M., Xie, Y., Lu, J.C., Jiang, J.: On the non-existence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical system. Nonlinear Dyn. 82, 1259–1267 (2015)

    MATH  Article  Google Scholar 

  11. 11.

    Kaslik, E., Srivastava, H.M., Trujillo, J.J.: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal., Real World Appl. 13, 1489–1497 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Ren, L., Wang, J.R., Fečkan, M.: Asymptotically periodic behavior solutions for Caputo type fractional evolution equations. Fract. Calc. Appl. Anal. 27(5), 1294–1312 (2018)

    MATH  Article  Google Scholar 

  13. 13.

    Tavazoel, M.S., Haeri, M.: A proof for nonexistence of periodic solutions in time invariant fractional order system. Automatica 45, 1886–1890 (2009)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wang, J.-R., Fečkan, M., Zhou, Y.: Non-existence of periodic solutions and asymptotically periodic solutions of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18, 246–256 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Li, F., Liang, J., Wang, H.: S-Asymptotically θ-periodic solution for fractional differential equations of order \(q\in (0,1)\) with finite delay. Adv. Differ. Equ. 2017, 83 (2017)

    Article  Google Scholar 

  16. 16.

    Maghsoodi, S., Neamaty, A.: Existence and uniqueness of S-asymptotically w-periodic solution for fractional semilinear problem. J. Appl. Comput. Math. 8(2), 1–5 (2019)

    Google Scholar 

  17. 17.

    Mu, J., Zhou, Y., Peng, L.: Periodic solutions and S-asymptotically periodic solutions to fractional evolution equations. Discrete Dyn. Nat. Soc. 2017, Article ID 1364532 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Cuevas, C., César de Souza, J.: S-Asymptotically T-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22, 865–870 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Ren, L., Wang, J.R., O’Regan, D.: Asymptotically periodic behavior of solutions of fractional evolution equations of order \(1<\alpha <2 \). Math. Slovaca 69, 599–610 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Wu, Z.-H.: Asymptotic periodicity for a class of fractional integro-differential equations. J. Nonlinear Sci. Appl. 9, 506–517 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Wang, H., Li, F.: S-Asymptotically θ-periodic solutions for delay fractional differential equations with almost sectorial operator. Adv. Differ. Equ. 2016, 315 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Rogovchenko, S.P., Rogovchenko, Y.V.: Asymptotic behavior of solutions of second order nonlinear differential equations. Port. Math. 57(1), 17–33 (2000)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Wang, D., Xiao, A., Sun, S.: Asymptotic behavior of solutions to time fractional neutral functional differential equations. J. Comput. Appl. Math. 382, 113086 (2021)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Abou-El-Ela, A.M., Sadek, A.-R., Mahmoud, A.M., Farghaly, E.S.: S-Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation. Turk. J. Math. 41, 576–584 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Cuevas, C., Pierri, M., Sepulveda, A.: Weighted S-asymptotically T-periodic solutions of a class of fractional differential equations. Adv. Differ. Equ. 2011, 584874 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    MATH  Book  Google Scholar 

  27. 27.

    Hernandez, E., O’Regan, D.: On a new class of abstract impulsive differential equation. Proc. Am. Math. Soc. 141, 1641–1649 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Wang, J.-R., Li, M., O’Regan, D.: Robustness for nonlinear evolution equation with non-instantaneous effects. Bull. Sci. Math. 159, 102827 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Wang, J.-R., Ibrahim, A.G., O’Regan, D.: Global attracting solutions to Hilfer fractional non-instantaneous impulsive semilinear differential inclusions of Sobolev type and with nonlocal conditions. Nonlinear Anal., Model. Control 24(5), 775–803 (2019)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Wang, J.-R., Ibrahim, A.G., O’Regan, D.: Hilfer type fractional differential switched inclusions with non-instantaneous impulsive and nonlocal conditions. Nonlinear Anal., Model. Control 23(6), 921–941 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Wang, J.-R., Ibrahim, A.G., O’Regan, D., Elmandouha, A.A.: Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order \(1<\alpha <2\). J. Nonlinear Sci. Numer. Simul. (2020). https://doi.org/10.1515/ijnsns-2019-0179

    Article  Google Scholar 

  32. 32.

    Chalishajar, D.N., Malar, K., Ilavaras, R.: Existence and controllability results of impulsive fractional neutral integro-differential equation with sectorial operator and infinite delay. AIP Conf. Proc. 2159, 030006 (2019)

    Article  Google Scholar 

  33. 33.

    Wang, J.-R., Ibrahim, A.G., Fecčkan, M.: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl. Math. Comput. 257, 103–112 (2015)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Yan, Z.: Approximate controllability of fractional impulsive partial neutral stochastic differential inclusions with state-dependent delay and fractional sectorial operators. Numer. Funct. Anal. Optim. 7(12), 1590–1639 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Yan, Z., Han, L.: Optimality of fractional impulsive partial stochastic differential systems with analytic sectorial operators and controls. Optimization 68(4), 853–894 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Agarwal, P., Agarwal, R.P., Ruzhansky, M.: Special Functions and Analysis of Differential Equations. Chapman & Hall/CRC, Boca Raton (2020)

    MATH  Book  Google Scholar 

  37. 37.

    Agarwal, P., Dragomir, S.S., Jleli, M., Samet, B.: Advances in Mathematical Inequalities and Applications. Birkhäuser, Basel (2018)

    MATH  Book  Google Scholar 

  38. 38.

    Agarwal, P., Ramadan, M., Osheba, H.S., Chu, Y.M.: Study of hybrid orthonormal functions method for solving second kind fuzzy Fredholm integral equations. Adv. Differ. Equ. 2020, 533 (2020)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Applications. Birkhäuser, Basel (2017)

    MATH  Book  Google Scholar 

  40. 40.

    Ali, M.A., Abbas, M., Budak, H., Agarwal, P., Murtaza, G., Chu, Y.M.: New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions. Adv. Differ. Equ. 2021, 64 (2021)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Wu, S., Li, C., Agarwal, P.: Relaxed modulus-based matrix splitting methods for the linear complementarity problem. Symmetry 13(3), 503 (2021)

    Article  Google Scholar 

  42. 42.

    Khan, H., Khan, A., Abdeljawad, T., Alkhazzan, A.: Existence results in Banach space for a nonlinear impulsive system. Adv. Differ. Equ. 2019, 18 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Khan, H., Tunc, C., Khan, A.: Green function’s properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete Contin. Dyn. Syst., Ser. S 13(9), 2475–2487 (2020)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Khan, H., Gomez-Aguillar, J.F., Abdeljawad, T., Khan, A.: Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation. Fractals 28(8), 2040048 (2020)

    Article  Google Scholar 

  45. 45.

    Shu, X.B., Wang, Q.: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order \(1<\alpha <2\). Comput. Math. Appl. 64, 2100–2110 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Covitz, H., Nadler, S.B.: Multivalued contraction mapping in generalized metric spaces. Isr. J. Math. 8, 5–11 (1970)

    MATH  Article  Google Scholar 

  47. 47.

    Casting, C., Vladier, M.: Convex Analysis and Measurable Multifunctions. Lect. Notes in Math., vol. 580. Springer, Berlin (1977)

    Book  Google Scholar 

  48. 48.

    Hiai, F., Umegaki, H.: Integrals conditional expectation and martingales of multivalued functions. J. Multivar. Anal. 7, 149–182 (1977)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Kamenskii, M., Obukhowskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter, Berlin (2001)

    MATH  Book  Google Scholar 

  50. 50.

    Hernandez, E.: Global solutions for abstract impulsive neutral differential equations. Math. Comput. Model. 53, 196–204 (2011)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

This research has been funded by the Scientific Research Deanship at University of Ha’il—Kingdom of Saudi Arabia through project number BA-2012.

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Alsheekhhussain, Z., Wang, J. & Ibrahim, A.G. Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators. Adv Differ Equ 2021, 330 (2021). https://doi.org/10.1186/s13662-021-03475-w

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MSC

  • 26A33
  • 34A08
  • 34A60

Keywords

  • Differential inclusions of fractional order
  • Sectorial operators
  • Asymptotically periodic solutions
  • Caputo fractional derivative
  • Non-instantaneous impulses