On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

Abstract

In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$, $t\in [0,\tau ]$ subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$, where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ on a Banach space X. The elements $x_{0}$ and $x_{1}$ are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ and any Lipschitz conditions on the functions g and h.

1 Introduction

Classical derivatives appear in several mathematical models in various areas of science such as physics, engineering, biology, finance, and so on. However, there are many phenomena that may not depend only on the time moment but also on the former time history, which cannot be modeled utilizing the classical derivatives. For this reason, many authors try to replace the classical derivatives with the so-called fractional derivatives in numerous contributions [2–17], because it has been proven that this last kind of derivatives is a very good way to describe processes with memory. According to the literature of fractional calculus, it is remarkable that there are many approaches to defining fractional derivatives, and each definition has advantages compared to others [18–22]. In consequence, many researchers have paid attention to propose new fractional derivatives in order to deal better with modeling of evolutionary phenomena [23, 24]. In the work [23], the authors proposed the so-called fractional conformable derivative, which quickly became the subject of many research papers [1, 25–54]. For example, in [1] the authors proved the existence of mild solutions for the following nonlocal conformable fractional Cauchy problem:

$$ \textstyle\begin{cases} \frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t)),\quad t\in [0,\tau ], \\ x(0)=x_{0}+g(x), \\ \frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x), \end{cases} $$

(1.1)

which is a natural extension of the works [55, 56] in the frame of the conformable fractional derivative, where A is the infinitesimal generator of a cosine family $\{C(t),S(t)\}_{t\in \mathbb{R}}$ on a Banach space $(X,\|\cdot\| )$ and $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ presents the conformable fractional derivative of order $\alpha \in\, ]0,1]$. The elements $x_{0}$ and $x_{1}$ are two fixed vectors in X, and $f:[0,\tau ]\times X\longrightarrow X$, $g:\mathcal{C}\longrightarrow X$, $h:\mathcal{C}\longrightarrow X$ are given functions, where $(\mathcal{C},|\cdot| _{c})$ is the space of continuous functions from $[0,\tau ]$ into X with $|\cdot| _{c}$ being the uniform norm topology in $\mathcal{C}$. The expressions $x(0)=x_{0}+g(x)$ and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ mean the nonlocal conditions, which can be applied in physics with better effects than the classical initial conditions [57–59]. We note that the existence result given in [1, Theorem 3.1] for Cauchy problem (1.1) has been proved by using the Krasnoselskii fixed point theorem, under the following assumptions:

$(A_{1})$:

There exists a constant $L_{1}$ such that $\| g(y)- g(x)\| \leq L_{1}| y-x| _{c}$ for all $y, x\in \mathcal{C}$.

$(A_{2})$:

There exists a constant $L_{2}$ such that $\| h(y)- h(x)\| \leq L_{2}| y-x| _{c}$ for all $y, x\in \mathcal{C}$.

$(A_{3})$:

The family $(S(t))_{t\in \mathbb{R}}$ is compact for all $t>0$.

However, there are many concrete applications in which the above assumptions are difficult to realize. Indeed, for $X=L^{2}(\Omega , \mathbb{R})$ with Ω is a bounded domain in $\mathbb{R}^{n}$ ($n\geq 1$), many authors [60–62] have considered the following nonlocal condition:

$$ g(x)= \int _{\Omega } \int _{0}^{T}K\bigl(t,\cdot ,\sigma ,x(t)(\sigma) \bigr)\,dt\,d\sigma , $$

(1.2)

which does not satisfy Lipschitz condition $(A_{1})$, where K is a kernel with the following conditions:

$(C_{1})$:

$K(t,\xi ,\sigma ,\cdot)$ is a continuous function for almost every $(t,\xi ,\sigma )\in [0,T]\times \Omega \times \Omega$.

$(C_{2})$:

$K(\cdot,\cdot ,\cdot ,r)$ is a measurable function for each fixed $r\in \mathbb{R}$.

$(C_{3})$:

$| K(t,\xi _{1},\sigma ,r)-K(t,\xi _{2},\sigma ,r)|\leq m_{k}(t,\xi _{1}, \xi _{2},\sigma )$ for all $(t,\xi _{1},\sigma ,r), (t,\xi _{2},\sigma ,r) \in [0,T]\times \Omega \times \Omega \times \mathbb{R}$ with $|r|\leq k$, where $m_{k}\in L^{1}([0,T]\times \Omega \times \Omega \times \mathbb{R}, \mathbb{R}^{+})$ and ${\displaystyle{\lim_{\xi _{1}\longrightarrow \xi _{2}}}}\int _{\Omega } \int _{0}^{T}m_{k}(t,\xi _{1},\xi _{2},\sigma )\,dt\,d\sigma =0$, uniformly in $\xi _{2}\in \Omega$.

$(C_{4})$:

$|K(t,\xi ,\sigma ,r)|\leq \frac{\delta }{Tm(\Omega )}|r|+\eta (t,\xi , \sigma )$ for all $r\in \mathbb{R}$ and $\delta >0$, where $\eta \in L^{2}([0,T]\times \Omega \times \Omega , \mathbb{R}^{+})$.

Motivated by this discussion, in the present work we use the Darbo–Sadovskii fixed point theorem in order to prove the existence of mild solutions for Cauchy problem (1.1) without assuming the Lipschitz conditions imposed in $(A_{1})$, $(A_{2})$ and the compactness of the family $(S(t))_{t>0}$.

The rest of this paper is organized as follows. In Sect. 2, we briefly recall some tools related to the conformable fractional calculus, the cosine family of linear operators, and the Hausdorff measure of noncompactness. Section 3 is devoted to proving the main result.

2 Preliminaries

We recall some preliminary facts on the conformable fractional calculus.

Definition 2.1

([23])

For $\alpha \in\, ]0,1]$, the conformable fractional derivative of order α of a function $x(\cdot): [0,+\infty [\,\longrightarrow \mathbb{R}$ is defined as

$$ \frac{d^{\alpha }x(t)}{dt^{\alpha }}= {\lim_{\varepsilon \longrightarrow 0} \frac{x(t+\varepsilon t^{1-\alpha })-x(t)}{\varepsilon }} \quad \text{for } t>0 \quad \text{and}\quad \frac{d^{\alpha }x(0)}{d t^{\alpha }}= {\lim _{t \longrightarrow 0^{+}}\frac{d^{\alpha }x(t)}{dt^{\alpha }}}, $$

provided that the limits exist.

The conformable fractional integral $I^{\alpha }(\cdot)$ of a function $x(\cdot)$ is defined by

$$ I^{\alpha }(x) (t)= \int _{0}^{t}s^{\alpha -1}x(s)\,ds \quad \text{for } t>0. $$

Theorem 2.1

([23])

If $x(\cdot)$ is a continuous function in the domain of $I^{\alpha }(\cdot)$, then we have

$$ \frac{d^{\alpha }(I^{\alpha }(x)(t))}{dt^{\alpha }}=x(t). $$

Theorem 2.2

([25])

If $x(\cdot)$ is a differentiable function, then we have

$$ I^{\alpha }\biggl(\frac{d^{\alpha } x(\cdot)}{dt^{\alpha }}\biggr) (t)=x(t)-x(0). $$

Now, we present some definitions concerning the cosine family of linear operators.

Definition 2.2

([55])

A one parameter family $(C(t))_{t\in \mathbb{R}}$ of bounded linear operators on a Banach space X is called a strongly continuous cosine family if and only if:

• 1. $C(0)=I$, where I is the identity operator in the space X.

• 2. $C(s+t)+C(s-t)=2C(s)C(t)$ for all $t,s\in \mathbb{R}$.

• 3. The function $t\longmapsto C(t)x$ is strongly continuous for each $x\in X$.

We also define the sine family $(S(t))_{t\in \mathbb{R}}$ associated with the cosine family $(C(t))_{t\in \mathbb{R}}$ as follows:

$$ S(t)x= \int _{0}^{t}C(s)x\,ds, \quad x\in X. $$

The infinitesimal generator A of a strongly continuous cosine family $((C(t)),(S(t)))_{t\in \mathbb{R}}$ on X is defined by

$$ D(A)=\bigl\{ x\in X, t\longmapsto C(t)x \text{ is a twice continuously differentiable function} \bigr\} $$

and

$$ Ax=\frac{d^{2}C(0)x}{dt^{2}},\quad x\in D(A). $$

We end these preliminaries by some concepts on the Hausdorff measure of noncompactness.

Definition 2.3

([63, 64])

The Hausdorff measure of noncompactness σ of a bounded set B in a Banach space X is defined as follows:

$$ \sigma (B) = \inf \Biggl\{ \varepsilon > 0: B= {\bigcup _{i=1}^{n}}B_{i}, \text{with } \operatorname{diam}(B_{i})\leq \varepsilon \text{ for } i=1, \ldots,n\Biggr\} . $$

The following lemma presents some basic properties of the Hausdorff measure of noncompactness.

Lemma 2.1

([63, 64])

Let X be a Banach space and $B, C\subseteq X$ be bounded sets. Then the following properties hold:

• (1) B is pre-compact if and only if $\sigma (B) = 0$.

• (2) $\sigma (B)=\sigma (\overline{B})=\sigma (\operatorname{conv}(B))$, where B̅ and $\operatorname{conv}(B)$ mean the closure and convex hull of B, respectively.

• (3) $\sigma (B)\leq \sigma (C)$, where $B\subseteq C$.

• (4) $\sigma (B+C)\leq \sigma (B)+\sigma (C)$, where $B+C =\{x+y: x \in B, y \in C\}$.

• (5) $\sigma (B\cup C)\leq \max \{\sigma (B), \sigma (C) \}$.

• (6) $\sigma (\lambda B)=|\lambda |\sigma (B)$ for any $\lambda \in \mathbb{R}$, when X is a real Banach space.

• (7) If the operator $Q: D(Q)\subseteq X\longrightarrow Y$ is Lipschitz continuous with constant k, then we have $\rho (Q(B)) \leq k\sigma (B)$ for any bounded subset $B\subseteq D(Q)$, where Y is another Banach space and ρ represents the Hausdorff measure of noncompactness in Y.

Definition 2.4

([64])

The operator $Q: D(Q)\subseteq X\longrightarrow X$ is said to be an σ-contraction if there exists a positive constant $k<1$ such that $\sigma (Q(B)) \leq k\sigma (B)$ for any bounded closed subset $B\subseteq D(Q)$.

Lemma 2.2

(Darbo–Sadovskii theorem, [63, 64])

Let $B\subset X$ be a bounded, closed, and convex set. If $Q: B\longrightarrow B$ is a continuous and σ-contraction operator, then Q has at least one fixed point in B.

Lemma 2.3

([65, 66])

Let $D\subset X$ be a bounded set, then there exists a countable set $D_{0}\subset D$ such that $\sigma (D)\leq 2 \sigma (D_{0})$.

In the sequel, we denote by $\sigma _{c}$ the Hausdorff measure of noncompactness in the space $\mathcal{C}$ of continuous functions $x(\cdot)$ defined from $[0,\tau ]$ into X equipped with the norm $| x| _{c}={\displaystyle{\sup_{t\in [0,\tau ]}}\| x(t) \| }$. It is well known that the space $(\mathcal{C},|\cdot| _{c})$ is a Banach space.

Lemma 2.4

([67])

Let $D_{0}:=\{x_{n}\}\subset \mathcal{C}$ be a countable set, then we have

• (1) $\sigma (D_{0}(t)):=\sigma (\{x_{n}(t)\})$ is Lebesgue integrable on $[0,\tau ]$.

• (2) $\sigma (\int _{0}^{\tau }D_{0}(s)\,ds)\leq 2\int _{0}^{\tau }\sigma (D_{0}(s))\,ds$, where $\sigma (\int _{0}^{\tau }D_{0}(s)\,ds):=\sigma (\{\int _{0}^{\tau }x_{n}(s)\,ds \})$.

Lemma 2.5

([63])

Let $D\subset \mathcal{C}$ be bounded and equicontinuous, then we have

• (1) $\sigma (D(t))$ is continuous on $[0,\tau ]$.

• (2) $\sigma _{c}(D)={\displaystyle{\max_{t\in [0,\tau ]}}}(\sigma (D(t)))$.

3 Main result

According to [1], we have the following definition.

Definition 3.1

A function $x\in \mathcal{C}$ is called a mild solution of Cauchy problem (1.1) if

$$ x(t) =C\biggl(\frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{0}+g(x) \bigr]+S\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{1}+h(x)\bigr]+ \int _{0}^{t}s^{\alpha -1} S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,x(s)\bigr)\,ds. $$

To obtain the existence of mild solutions, we will need the following assumptions:

$(H_{1})$:

The function $f(t,\cdot): X\longrightarrow X$ is continuous, and for all $r>0$, there exists a function $\varphi _{r}\in L^{\infty }([0,\tau ],\mathbb{R}^{+})$ such that ${\displaystyle{\sup_{\| x\| \leq r}}}\| f(t,x) \| \leq \varphi _{r}(t)$ for all $t\in [0,\tau ]$.

$(H_{2})$:

The function $f(\cdot,x):[0,\tau ] \longrightarrow X$ is continuous for all $x\in X$.

$(H_{3})$:

The function $g:\mathcal{C} \longrightarrow X$ is continuous and compact.

$(H_{4})$:

The function $h:\mathcal{C} \longrightarrow X$ is continuous and compact.

$(H_{5})$:

There exist positive constants a and b such that $\| g(x)\| \leq a| x| _{c}+b$ for all $x\in \mathcal{C}$.

$(H_{6})$:

There exist positive constants c and d such that $\| h(x)\| \leq c| x| _{c}+d$ for all $x\in \mathcal{C}$.

$(H_{7})$:

There exists a positive constant L such that $\sigma (f(t,D_{0}))\leq L\sigma (D_{0})$ for any countable set $D_{0}\subset X$ and $t\in [0,\tau ]$.

$(H_{8})$:

The family $(C(t))_{t\in \mathbb{R}}$ is uniformly continuous, that is, ${\displaystyle{\lim_{t\longrightarrow s}}}|C(s)-C(t)|=0$, where $|\cdot|$ represents the norm in the space of bounded operators defined from X into itself.

Remark 3.1

The nonlocal condition given in (1.2) satisfies assumption $(H_{3})$.

Indeed, the continuity and compactness of the function g are guaranteed by using [68, Theorem 4.2] in view of conditions $(C_{1})-(C_{4})$ assumed for the kernel K. Moreover, by using condition $(C_{4})$, we get $a=\frac{\delta }{\sqrt{m(\Omega )}}$ and $b=\sqrt{Tm(\Omega )}\| \eta \| _{L^{2}([0,T] \times \Omega \times \Omega , \mathbb{R}^{+})}$.

Theorem 3.1

Assume that $(H_{1})$–$(H_{8})$ hold, then Cauchy problem (1.1) has at least one mild solution provided that

$$ \max \biggl(a {\sup_{t\in [0,\tau ]}} \biggl\vert C\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert +c {\sup_{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert , \frac{4L\tau ^{\alpha }}{\alpha } {\sup_{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \biggr)< 1. $$

Proof

In order to use the Darbo–Sadovskii fixed point theorem, we define the operator $\Gamma :\mathcal{C}\longrightarrow \mathcal{C}$ by

$$ \Gamma (x) (t) =C\biggl(\frac{t^{\alpha }}{\alpha }\biggr) \bigl[x_{0}+g(x)\bigr]+S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \bigl[x_{1}+h(x)\bigr]+ \int _{0}^{t}s^{\alpha -1} S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,x(s)\bigr)\,ds. $$

We also consider the ball $B_{r}:=\{x\in \mathcal{C}, | x | _{c}\leq r\}$, where

$$ r\geq \frac{ {\displaystyle{\sup_{t\in [0,\tau ]}}} \vert C(\frac{t^{\alpha }}{\alpha }) \vert [ \Vert x_{0} \Vert +b]+ {\displaystyle{\sup_{t\in [0,\tau ]}}} \vert S(\frac{t^{\alpha }}{\alpha }) \vert [ \Vert x_{1} \Vert +d+\frac{\tau ^{\alpha }}{\alpha } \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})}]}{1-a {\displaystyle{\sup_{t\in [0,\tau ]}}} \vert C(\frac{t^{\alpha }}{\alpha }) \vert -c {\displaystyle{\sup_{t\in [0,\tau ]}}} \vert S(\frac{t^{\alpha }}{\alpha }) \vert }. $$

The proof will be given in four steps.

Step 1: Prove that $\Gamma (B_{r})\subset B_{r}$.

For $x\in \mathcal{C}$, we have

$$\begin{aligned} \bigl\Vert \Gamma (x) (t) \bigr\Vert \leq &\biggl\Vert C\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{0}+g(x)\bigr] \biggr\Vert + \biggl\Vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{1}+h(x) \bigr] \biggr\Vert \\ &{}+ \int _{0}^{t}s^{ \alpha -1} \biggl\Vert S \biggl(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,x(s)\bigr) \biggr\Vert \,ds. \end{aligned}$$

Using assumptions $(H_{1})$, $(H_{5})$, and $(H_{6})$, we get

$$\begin{aligned} \bigl\Vert \Gamma (x) (t) \bigr\Vert \leq& \sup _{t\in [0,\tau ]} \biggl\vert C\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \bigl[ \Vert x_{0} \Vert + a \vert x \vert _{c}+b\bigr] \\ &{}+ \sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \biggl[ \Vert x_{1} \Vert +c \vert x \vert _{c}+d+\frac{\tau ^{\alpha }}{\alpha } \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})}\biggr] \\ \leq& \sup_{t\in [0,\tau ]} \biggl\vert C\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \bigl[ \Vert x_{0} \Vert + ar+b\bigr] \\ &{}+\sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \biggl[ \Vert x_{1} \Vert +cr+d+ \frac{\tau ^{\alpha }}{\alpha } \vert \varphi _{r} \vert _{L^{\infty }([0, \tau ],\mathbb{R}^{+})}\biggr] \\ \leq& r. \end{aligned}$$

Taking the supremum, we obtain $| \Gamma (x)| _{c}\leq r$, and this shows that $\Gamma (B_{r})\subset B_{r}$.

Step 2: Prove that $\Gamma :B_{r}\longrightarrow B_{r}$ is continuous.

Let $(x_{n})\subset B_{r}$ such that $x_{n}\longrightarrow x$ in $B_{r}$. We have

$$\begin{aligned} \Gamma (x_{n}) (t)-\Gamma (x) (t) =&C\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[g(x_{n})-g(x)\bigr]+S\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[h(x_{n})-h(x)\bigr] \\ &{} + \int _{0}^{t}s^{\alpha -1}S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)\bigl[f\bigl(s,x_{n}(s)\bigr)-f \bigl(s,x(s)\bigr)\bigr]\,ds. \end{aligned}$$

Using a simple computation, we obtain

$$\begin{aligned} \bigl\vert \Gamma (x_{n})-\Gamma (x) \bigr\vert _{c} \leq& \sup_{t\in [0,\tau ]} \biggl\vert C\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \bigl\Vert g(x_{n})-g(x) \bigr\Vert +\sup_{t \in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \bigl\Vert h(x_{n})-h(x) \bigr\Vert \\ &{}+\sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \int _{0}^{ \tau }s^{\alpha -1} \bigl\Vert f \bigl(s,x_{n}(s)\bigr)-f\bigl(s,x(s)\bigr) \bigr\Vert \,ds. \end{aligned}$$

In view of assumption $(H_{1})$, we get $\| s^{\alpha -1}[f(s,x_{n}(s))-f(s,x(s))]\| \leq 2s^{ \alpha -1}\varphi _{r}(s)$ and $f(s, x_{n}(s))\longrightarrow f(s,x(s))$ as $n\longrightarrow +\infty$. Then the Lebesgue dominated convergence theorem proves that

$$ \int _{0}^{\tau }s^{\alpha -1} \bigl\Vert f \bigl(s,x_{n}(s)\bigr)-f\bigl(s,x(s)\bigr) \bigr\Vert \,ds \longrightarrow 0\quad \text{as } n\longrightarrow +\infty . $$

By using the continuity of the functions g and h, we deduce that ${\displaystyle{\lim_{n\longrightarrow +\infty }}}\| g(x_{n})-g(x) \| =0$ and ${\displaystyle{\lim_{n\longrightarrow +\infty }}}\| h(x_{n})-h(x) \| =0$. Hence the operator Γ is continuous.

Step 3: Prove that $\Gamma (B_{r})$ is equicontinuous.

For $x\in B_{r}$ and $\mu , \nu \in [0,\tau ]$ such that $\mu <\nu$, we have

$$\begin{aligned} \Gamma (x) (\nu )-\Gamma (x) (\mu ) =&\biggl[C\biggl( \frac{\nu ^{\alpha }}{\alpha }\biggr)-C\biggl( \frac{\mu ^{\alpha }}{\alpha }\biggr)\biggr] \bigl(x_{0}+g(x)\bigr) \\ &{}+ \biggl[S\biggl( \frac{\nu ^{\alpha }}{\alpha } \biggr)-S\biggl(\frac{\mu ^{\alpha }}{\alpha }\biggr)\biggr]\bigl(x_{1}+h(x) \bigr) \\ &{} + \int _{0}^{\mu }s^{\alpha -1}\biggl[S\biggl( \frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }\biggr)-S\biggl( \frac{\mu ^{\alpha }-s^{\alpha }}{\alpha }\biggr)\biggr]f \bigl(s,x(s)\bigr)\,ds \\ &{} + \int _{\mu }^{\nu }s^{\alpha -1}S\biggl( \frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,x(s)\bigr)\,ds. \end{aligned}$$

Since $S(t)={\int _{0}^{t}}C(\theta )\,d\theta$, then the above equation can be rewritten as follows:

$$\begin{aligned} \Gamma (x) (\nu )-\Gamma (x) (\mu )&=\biggl[C\biggl( \frac{\nu ^{\alpha }}{\alpha }\biggr)-C\biggl( \frac{\mu ^{\alpha }}{\alpha }\biggr)\biggr] \bigl(x_{0}+g(x)\bigr)+ \biggl[ \int _{ \frac{\mu ^{\alpha }}{\alpha }}^{\frac{\nu ^{\alpha }}{\alpha }}C(\theta ) \bigl(x_{1}+h(x) \bigr)\,d\theta \biggr] \\ &+ \int _{0}^{\mu }s^{\alpha -1}\biggl[ \int _{ \frac{\mu ^{\alpha }-s^{\alpha }}{\alpha }}^{ \frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }}C(\theta )f\bigl(s,x(s)\bigr)\,d\theta \biggr]\,ds \\ &+ \int _{\mu }^{\nu }s^{\alpha -1}S\biggl( \frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,x(s)\bigr)\,ds. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \bigl\Vert \Gamma (x) (\nu )-\Gamma (x) (\mu ) \bigr\Vert \leq& \biggl\vert C\biggl( \frac{\nu ^{\alpha }}{\alpha }\biggr)-C\biggl(\frac{\mu ^{\alpha }}{\alpha } \biggr) \biggr\vert \bigl\Vert x_{0}+g(x) \bigr\Vert \\ &{}+\sup _{t\in [0, \frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \biggl[ \int _{ \frac{\mu ^{\alpha }}{\alpha }}^{\frac{\nu ^{\alpha }}{\alpha }} \bigl\Vert x_{1}+h(x) \bigr\Vert \,d\theta \biggr] \\ &{} +\sup_{t\in [0,\frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \int _{0}^{\mu }s^{ \alpha -1}\biggl[ \int _{\frac{\mu ^{\alpha }-s^{\alpha }}{\alpha }}^{ \frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }} \bigl\Vert f\bigl(s,x(s)\bigr) \bigr\Vert \,d\theta \biggr]\,ds \\ &{} +\sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \int _{\mu }^{ \nu }s^{\alpha -1} \bigl\Vert f \bigl(s,x(s)\bigr) \bigr\Vert \,ds. \end{aligned}$$

By using assumptions $(H_{1})$, $(H_{5})$, and $(H_{6})$, we get

$$\begin{aligned} \bigl\Vert \Gamma (x) (\nu )-\Gamma (x) (\mu ) \bigr\Vert \leq& \biggl\vert C\biggl( \frac{\nu ^{\alpha }}{\alpha }\biggr)-C\biggl(\frac{\mu ^{\alpha }}{\alpha } \biggr) \biggr\vert \bigl( \Vert x_{0} \Vert +a \vert x \vert _{c}+b\bigr) \\ &{} +\sup_{t\in [0,\frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \bigl( \Vert x_{1} \Vert +c \vert x \vert _{c}+d\bigr) \biggl[ \int _{\frac{\mu ^{\alpha }}{\alpha }}^{ \frac{\nu ^{\alpha }}{\alpha }}1\,d\theta \biggr] \\ &{} +\sup_{t\in [0,\frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})} \int _{0}^{\mu }s^{\alpha -1}\biggl[ \int _{\frac{\mu ^{\alpha }-s^{\alpha }}{\alpha }}^{\frac{\nu ^{\alpha }-s^{\alpha }}{\alpha }}1\,d\theta \biggr] \,ds \\ &{} +\sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})} \int _{\mu }^{\nu }s^{ \alpha -1}\,ds. \end{aligned}$$

An easy computation now shows that

$$\begin{aligned} \bigl\Vert \Gamma (x) (\nu )-\Gamma (x) (\mu ) \bigr\Vert \leq& \bigl( \Vert x_{0} \Vert +ar+b\bigr) \biggl\vert C\biggl( \frac{\nu ^{\alpha }}{\alpha }\biggr)-C\biggl( \frac{\mu ^{\alpha }}{\alpha }\biggr) \biggr\vert \\ &{}+\sup_{t\in [0,\frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \bigl( \Vert x_{1} \Vert +cr+d\bigr)\biggl[\frac{\nu ^{\alpha }-\mu ^{\alpha }}{\alpha }\biggr] \\ &{}+\sup_{t\in [0,\frac{\tau ^{\alpha }}{\alpha }]} \bigl\vert C(t) \bigr\vert \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})} \biggl[ \frac{\tau ^{\alpha }}{\alpha }\biggr] \biggl[ \frac{\nu ^{\alpha }-\mu ^{\alpha }}{\alpha }\biggr] \\ &{}+\sup_{t\in [0,\tau ]} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \vert \varphi _{r} \vert _{L^{\infty }([0,\tau ],\mathbb{R}^{+})} \biggl[ \frac{\nu ^{\alpha }-\mu ^{\alpha }}{\alpha }\biggr]. \end{aligned}$$

The above inequality combined with assumption $(H_{8})$ shows that $\Gamma (B_{r})$ is equicontinuous on $[0,\tau ]$.

Step 4: Prove that $\Gamma :B_{r}\longrightarrow B_{r}$ is a $\sigma _{c}$-contraction operator.

Let $D\subset B_{r}$, then by Lemma 2.3, there exists a countable set $D_{0}$ such that $D_{0}=\{x_{n}\}\subset D$. Hence, $\Gamma (D_{0})$ becomes a countable subset of $\Gamma (D)$. Thus, Lemma 2.3 proves that $\sigma _{c}(\Gamma (D))\leq 2\sigma _{c}(\Gamma (D_{0}))$. Since $\Gamma (D_{0})$ is bounded and equicontinuous, then by using the second point of Lemma 2.5, we obtain

$$ \sigma _{c}\bigl(\Gamma (D_{0})\bigr)= {\max _{t\in [0,\tau ]}}\bigl( \sigma \bigl(\Gamma (D_{0}) (t) \bigr)\bigr). $$

Accordingly, we deduce that

$$\begin{aligned} \sigma _{c}\bigl(\Gamma (D)\bigr) \leq& 2\sigma _{c}\bigl(\Gamma (D_{0})\bigr) \\ =&2\max_{t\in [0,\tau ]}\bigl(\sigma \bigl(\Gamma (D_{0}) (t)\bigr)\bigr) \\ =&2\max_{t\in [0,\tau ]} \biggl(\sigma \biggl(C\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{0}+g(D_{0})\bigr]+S \biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{1}+h(D_{0}) \bigr] \\ &{}+ \int _{0}^{t}s^{\alpha -1} S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,D_{0}(s)\bigr)\,ds\biggr) \biggr). \end{aligned}$$

By using point (4) of Lemma 2.1, we get

$$\begin{aligned} \sigma _{c}\bigl(\Gamma (D)\bigr) \leq& 2\max _{t\in [0,\tau ]} \biggl(\sigma \biggl(C\biggl( \frac{t^{\alpha }}{\alpha } \biggr)\bigl[x_{0}+g(D_{0})\bigr]\biggr) \\ &{}+\sigma \biggl(S\biggl( \frac{t^{\alpha }}{\alpha }\biggr)\bigl[x_{1}+h(D_{0}) \bigr]\biggr) +\sigma \biggl( \int _{0}^{t}s^{ \alpha -1}S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,D_{0}(s)\bigr)\,ds\biggr) \biggr). \end{aligned}$$

Since g and h are compact, then the sets $C(\frac{t^{\alpha }}{\alpha })[x_{0}+g(D_{0})]$ and $S(\frac{t^{\alpha }}{\alpha })[x_{1}+h(D_{0})]$ are relatively compact. According to the above equation and the first point of Lemma 2.1, we obtain

$$ \sigma _{c}\bigl(\Gamma (D)\bigr)\leq 2\max _{t\in [0,\tau ]} \biggl(\sigma \biggl( \int _{0}^{t}s^{\alpha -1} S\biggl( \frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,D_{0}(s)\bigr)\,ds\biggr) \biggr). $$

In view of Lemma 2.4, we get

$$ \sigma _{c}\bigl(\Gamma (D)\bigr)\leq 4\max _{t\in [0,\tau ]} \biggl( \int _{0}^{t}s^{ \alpha -1}\sigma \biggl( S \biggl(\frac{t^{\alpha }-s^{\alpha }}{\alpha }\biggr)f\bigl(s,D_{0}(s)\bigr)\,ds\biggr) \biggr). $$

Next, point (7) of Lemma 2.1 shows that

$$ \sigma _{c}\bigl(\Gamma (D)\bigr)\leq 4 {\sup _{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \max_{t\in [0,\tau ]} \biggl( \int _{0}^{t}s^{ \alpha -1}\sigma \bigl( f \bigl(s,D_{0}(s)\bigr)\bigr)\,ds \biggr). $$

According to assumption $(H_{7})$, we get

$$ \sigma _{c}\bigl(\Gamma (D)\bigr)\leq 4L {\sup _{t\in [0,\tau ]}} \biggl( \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha } \biggr) \biggr\vert \biggr)\max_{t\in [0,\tau ]} \biggl( \int _{0}^{t}s^{ \alpha -1}\sigma \bigl(D_{0}(s)\bigr)\,ds \biggr). $$

Hence, by using a simple computation combined with point $(2)$ of Lemma 2.5, we obtain

$$\begin{aligned} \sigma _{c}\bigl(\Gamma (D)\bigr)&\leq 4L {\sup _{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \sigma _{c}(D) \int _{0}^{\tau }s^{\alpha -1}\,ds \\ &=4L {\sup_{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \sigma _{c}(D) \frac{\tau ^{\alpha }}{\alpha }. \end{aligned}$$

Then, we have

$$ \sigma _{c}\bigl(\Gamma (D)\bigr)\leq \frac{4L\tau ^{\alpha }}{\alpha } {\sup_{t\in [0,\tau ]}} \biggl\vert S\biggl( \frac{t^{\alpha }}{\alpha }\biggr) \biggr\vert \sigma _{c}(D). $$

Since $\frac{4L\tau ^{\alpha }}{\alpha }{\displaystyle{\sup_{t\in [0,\tau ]}}}|S( \frac{t^{\alpha }}{\alpha })|<1$, then Γ is a $\sigma _{c}$-contraction operator.

In conclusion, Lemma 2.2 shows that the operator Γ has at least one fixed point, which is a mild solution of Cauchy problem (1.1). □

4 Conclusion

Without assuming the Lipschitz condition on the nonlocal conditions and the compactness of the cosine family generated by the linear part, we have proved the existence of mild solutions for a class of nonlocal differential equations of the second order with conformable fractional derivative. The main result is obtained by means of the Darbo–Sadovskii fixed point theorem combined with theory of cosine family of linear operators. The equation studied in the present work can be viewed as an abstract version of the nonlocal conformable fractional telegraph equation considered in the work [31]. As a future work, we will be interested in studying Cauchy problem (1.1) with the non-sequential operator $\frac{d^{2\alpha }x(t)}{dt^{2\alpha }}$ instead of the sequential one $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]$. This purpose is not easy! Indeed, in this case the cosine family $((C(t)),(S(t)))_{t\in \mathbb{R}}$ must be extended to the complex space $\mathbb{C}$, for more details about this point we refer to the work [1].