Some integral inequalities for harmonical $cr$-$h$-Godunova-Levin stochastic processes

1.   Introduction

Fractional calculus has shown great effectiveness in describing and modeling many phenomena in many fields such as physics, chemistry, biology, electricity, economics, etc ^[1,2,3,4]. Recently, many researchers focus to introduce and develop fractional operators such as Caputo-Fabrizio, Atangana-Baleanu and conformable derivatives ^{[5,6,7,8,9,10,11]}.

A new extended fractional operator was proposed in ^[12] as a combination between conformable and Caputo derivatives. The novel fractional derivative has attracted attention in limited papers, see ^{[13,14,15,16]}. For the first time authors in ^[13] investigated the existence, uniqueness and Ulam-Hyers stability of solutions for conformable derivatives in Caputo setting with four-point integral conditions by applying suitable fixed point theorems. Baleanu et al. ^[14] discussed Caputo fractional conformable differential inclusion subject to four-point conditions using some analytical techniques on the $\alpha$-$\psi$ contractive mappings. In ^[17] authors applied this fractional derivative to describe the behavior of an electrical circuit model.

Delay differential equations and integro-differential equations in classical and fractional order have been used in modeling many situations from science and engineering. For this reason they have attracted great attention in the last two decades and have been investigated theoretically in many papers ^{[18,19,20,21,22,23,24,25,26,27,31,32,33,34,35]}. In a previous study ^[36], Kavitha et al. established the existence results of Hilfer fractional neutral evolution equation with infinite delay by utilizing the semigroup theory, fractional calculus and Mönch fixed point theorem. In ^[20] authors studied the existence of mild solutions for a class of non local fractional integro-differential equation in neutral type with infinite delay, using the theory of resolvent operators. Valliammal et al. in ^[31] established some sufficient conditions for the existence of solutions for neutral delay fractional integro-differential systems, where the authors in ^[37] studied controllability of nonlocal neutral impulsive differential equations with measure of noncompactness.

Inspired by the above research, we consider the following nonlocal fractional integro-differential evolution equation with finite delay:

where $D^{\alpha, \varrho}_{0^{+}}$ is the Caputo conformable fractional derivative of order $0 < \alpha < 1$ and type $0 < \varrho\leq1$, and $A$ is infinitesimal generator of a fractional $C_{0}$-$\varrho$-semigroup $\{T_{\varrho}(t)\}_{t\geq0}$ of bounded operators on a Banach space $X$ with the norm $\|.\|$. The functions $\varphi \in C\left([-\delta, 0], X\right)$, $f: [0, b)\times C\left([-\delta, 0], X\right)\times X\rightarrow X$, $h: [0, b)\times [0, +\infty)\times C\left([-\delta, 0], X\right)\rightarrow X$ and $g: C\left([-\delta, b], X\right)\rightarrow C\left([-\delta, 0], X\right)$ are given abstract functions and $u_{t}: [-\delta, 0]\rightarrow X$ is defined by $u_{t}(\theta) = u(t+\theta)$.

In this paper, we try to refine the conditions imposed in some previous works^{[20,31,36,38,39]} and prove the existence of a mild solution under weaker and more general conditions than those mentioned in the previous works. For example, when we chose $\varrho = 1, \delta = 0$ and $h\equiv 0$, problem (1.1) improves the results of existence of a mild solution for the nonlocal Caputo fractional evolution equation discussed in ^[39]:

We study problem (1.1) by converting it to an equivalent integral equation using a suitable fractional Laplace transform, then we define the mild solution of (1.1) in terms of two new families of operators. Each family of operators is associated with the wright function and a suitable fractional semigroup. In the second part of this paper, we apply the Mönch fixed point theorem to establish our main results on the existence of mild solutions with the help of the general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non-compactness.

We organize the contents of our paper as follows: In Section 2, we state some basic definitions, concepts and preliminary results which are used throughout this paper. The representation of mild solution of (1.1) using fractional Laplace transform is given in Section 3. In Section 4, under some sufficient conditions, we prove the existence theorem of a mild solution of (1.1), based on Mönch fixed point theorem. An application of our abstract results is given in the last section.

2.   Preliminaries

In this section, we introduce fractional integral, fractional derivative, fractional semigroup and then give the definition of the fractional Laplace transform. Finally, we will give some definitions and lemmas which are used throughout this paper.

Let $X$ be a Banach space with the norm $\|.\|$. Let $\delta, b \in \mathbb{R}^{+}$. Denote By $\mathcal{C} = C\left([-\delta, 0], X\right)$ and $C\left([-\delta, b], X\right)$ the spaces of continuous $X$-valued functions on $[-\delta, 0]$ and $[-\delta, b]$ with the norms

and

respectively, and $B(X)$ be the space of all bounded linear operators from $X$ into $X$ with the norm $\|T\|_{B(X)} = \sup_{x\in X, \|X\|\leq1}\|Tx\|$. Define $L^{\varrho}_{p}([0, b], X)$ to be the space of $X$-valued Bochner functions on $[0, b]$ with the norm $\|u\|_{p}^{\varrho} = \left(\int_{0}^{b}\|u(s)\|^{p}\frac{ds}{s^{1-\varrho}}\right)^{\frac{1}{p}}$, $1\leq p < \infty$.

Definition 2.1. ^[5,10] The left conformable derivative with lower point $a$ of the function $f: [a, +\infty)\rightarrow \mathbb{R}$ of order $0 < \varrho \leq 1$ is defined by

If $D^{\varrho}_{a}f(t)$ exists on $(a, b)$, then $D^{\varrho}_{a}f(a) = \lim_{t\rightarrow a^{+}}D^{\varrho}_{a}f(t)$. If $f$ is differentiable, then

Definition 2.2. ^[12] Let $\alpha, \varrho > 0$. The left-sided Riemann-Liouville conformablde fractional integral operator of order $\alpha$ and type $\varrho$ with lower limit $a$ for a function $f: [a, +\infty)\rightarrow \mathbb{R}$ is defined by

Definition 2.3. ^[12] Let $\alpha > 0$, $0 < \varrho \leq1$ and $n = [\alpha]+1$. The left Caputo conformable fractional derivative with lower limit $a$ of order $\alpha$ and type $\varrho$ of a function $f \in C^{n}_{\varrho, a}\left([a, b], X\right)$ is defined by

where $\; ^{n}\!D^{\varrho}_{a} = \underbrace{D^{\varrho}_{a}D^{\varrho}_{a}...D^{\varrho}_{a}}_{n-{\rm{times}}}$, and $D^{\varrho}_{a}$ is the left conformable derivative given in Definition 2.1.

Lemma 2.1. ^[12]Let $\alpha, \beta, \gamma > 0$. Then

and

Theorem 2.2. ^[12]Let $\alpha > 0$, $0 < \varrho\leq1$ and $n = [\alpha]+1$. Then

(1) If $\alpha \notin N$ and $f \in C\left([a, b], X\right)$, then

(2) If $\alpha \in N$ and $f \in C\left([a, b], X\right)$, then

(3) If $f \in C^{n}_{\varrho, a}\left([a, b], X\right)$, then

Definition 2.4. (^[5]) Let $a\in \mathbb{R}$, $0 < \varrho\leq 1$ and $f: [a, \infty)\rightarrow X$ be an $X$-valued Bochner function. The fractional Laplace transform of order $\varrho$ started from $a$ is given by

Definition 2.5. Let $0 < \varrho\leq1$, and $u, v$ be two $X$-valued Bochner functions. We define the fractional convolution of $u$ and $v$ of order $\varrho$ by

Proposition 2.1. Let $0 < \varrho\leq1$ and $u, v$ be two $X$-valued functions which are piecewise continuous at each interval $[a, b]$and of $\varrho$-exponential order ($u(t)\leq Me^{ct^{\varrho}}$). Then

(1) For any $c_{1}, c_{2}\in \mathbb{R}$, $\mathcal{L}_{a}^{\varrho}\{c_{1}u+c_{2}v\}(s) = c_{1}\mathcal{L}_{a}^{\varrho}\{u\}(s)+c_{2}\mathcal{L}_{a}^{\varrho}\{v\}(s).$

(2) $\mathcal{L}_{a}^{\varrho}\{\left(\frac{(t-a)^{\varrho}}{\varrho}\right)^{\alpha}\}(s) = \frac{\Gamma(\alpha+1)}{s^{\alpha+1}}.$

(3) $\mathcal{L}_{0}^{\varrho}\left(u\ast_{\varrho}v\right)(s) = \mathcal{L}_{0}^{\varrho}\{u\}(s) \mathcal{L}_{0}^{\varrho}\{v\}(s).$

(4) For $\alpha > 0$, $\mathcal{L}_{0}^{\varrho}\{I^{\alpha, \varrho}_{a}u\}(s) = \frac{\mathcal{L}_{0}^{\varrho}{u}(s)}{s^{\alpha}}.$

Proof. The proof follows from the argument of ^[40] by letting $\psi(t) = \frac{t^{\varrho}}{\varrho}$.

Lemma 2.3. ^[41]Let $\alpha\geq0$, $0 < \varrho\leq1$ Assume that $u, v$ are two nonnegative locally integrable real valued functions on$[0, b]$ and $h$ is a nonnegative and nondecreasing real valued function on $[0, b]$.If

then

Now, we define and give some results about the so called fractional $\varrho$-semigroup of bounded linear operators, these results can be found in ^[5,42,43,44]

Definition 2.6. ^[5] Let $0 < \varrho\leq1$. A family $\{T_{\varrho}(t)\}_{t\geq 0} \subseteq B(X)$ is called a fractional $\varrho$-semigroup of bounded linear operators on Banach space $X$ if

(i) $T_{\varrho}(0) = I$,

(ii) $T_{\varrho}(t+s)^{\frac{1}{\varrho}} = T_{\varrho}(t^{\frac{1}{\varrho}}) T_{\varrho}(s^{\frac{1}{\varrho}})$ for all $t, s \in [0, \infty).$

An $\varrho-$semigroup $T_{\varrho}(t)$ is called a $C_{0}$-$\varrho$-semigroup, if for each $x \in X$, $T_{\varrho}(t)x\rightarrow x$ as $t\rightarrow 0^{+}$.

A linear operator $A$ defined by

and

is called the $\varrho-$infinitesimal generator of the fractional $\varrho-$semigroup $T_{\varrho}(t)$.

Theorem 2.4. ^[42,43]Let $A$ be the $\varrho$-infinitesimal generator of a fractional$C_{0}$-$\varrho$-semigroup $\{T_{\varrho}(t)\}_{t\geq 0}$ where $0 < \varrho\leq1$.Then

(1) There exist $M\geq1$ and $\omega\geq0$, such that $\|T_{\varrho}(t)\|\leq Me^{\omega t^{\varrho}}$, $t\geq0$.

(2) $A$ is closed densely defined.

(3) The resolvent set $\rho(A)$ of $A$ contains the interval $(\omega, +\infty)$ and for any $\lambda > \omega$, we have $\|R(\lambda, A)\|_{B(X)}\leq\frac{M}{\lambda-\omega}$, where the resolvent operator $R(\lambda, A)$ is defined by

Throughout this paper, assume that A is the infinitesimal generator of a uniformly bounded $C_{0}$-$\varrho$-semigroup $\{T_{\varrho}(t)\}_{t\geq0}$ on $X$. i.e, there exists $M\geq1$ such that $M = \sup_{t \in [0, +\infty)}\|T_{\varrho}(t)\|$.

Next, we define the Kurtawoski measure of noncompact $\mu(.)$ on each bounded subset $\Lambda$ of Banach space $X$ by

where $diam\left(B\right)$ is the diameter of $B$. The Kurtawoski measure of noncompact (KMN) $\mu$ satisfies the following basic properties (see ^[45,46]):

(1) $\mu\left(\Lambda_{1}\right)\leq\mu\left(\Lambda_{2}\right)$, for any bounded subsets $\Lambda_{1}, \Lambda_{2} \in X$ such that $\Lambda_{1}\subseteq \Lambda_{2}$.

(2) $\mu\left(\Lambda\right) = 0$ if and only if $\Lambda$ is relatively compact in $X$.

(3) $\mu\left(\{x\}\cup \Lambda\right) = \mu\left(\Lambda\right)$ for all $x \in X$ and $\Lambda \subset X$.

(4) $\mu\left(\Lambda_{1}\cup \Lambda_{2}\right)\leq \max\{\mu\left(\Lambda_{1}\right), \mu\left(\Lambda_{2}\right)\}$.

(5) $\mu\left(\Lambda_{1}+\Lambda_{2}\right)\leq \mu\left(\Lambda_{1}\right)+\mu\left(\Lambda_{2}\right)$.

(6) $\mu\left(\eta\Lambda\right)\leq|\eta|\mu\left(\Lambda\right)$ for $\eta \in \mathbb{R}$.

(7) If $\Theta : \Omega \subset X\rightarrow X$ is a Lipschitz map with constant $K$, then $\mu\left(\Theta\left(\Lambda\right)\right)\leq K\mu\left(\Lambda\right)$ for any bounded subset $\Lambda \subset \Omega$.

Lemma 2.5. ^[47]If $\Lambda \subseteq C([0, b], X)$ is bounded, then $\mu\left(\Lambda(t)\right)\leq\mu\left(\Lambda\right)$ forevery $t \in [0, b]$, where $\Lambda(t) = \{u(t), u \in \Lambda\}$. Moreover if $\Lambda$ is equicontinuouson $[0, b]$, then $t \rightarrow \mu\left(\Lambda(t)\right)$ is continuous real valued function on $[0, b]$, $\mu\left(\Lambda\right) = \sup\{\mu\left(\Lambda(t)\right), t \in [0, b]\}$. Furthermore$\mu\left(\int_{0}^{t}\Lambda(s)ds\right)\leq \int_{0}^{t}\mu\left(\Lambda(s)\right)ds$.

Lemma 2.6. ^[48]If $\{u_{n}\}_{n = 1}^{\infty}$ is a sequence of Bochner integrable $X$-valued functions on$[0, b]$ satisfies $\|u_{n}(t)\|\leq\phi(t)$ for almost all $t \in [0, b]$ and every $n\geq 1$, where$\phi \in L^{1}\left([0, b], \mathbb{R}\right)$, then the function$\psi(t) = \mu\left(\{u_{n}(t): n\geq1\}\right) \in L^{1}\left([0, b]\right)$ and satisfies$\mu\left(\{\int_{0}^{t}u_{n}(t)ds: n\geq1\}\right)\leq 2\int_{0}^{t}\psi(s)ds$.

Lemma 2.7. Let $\Omega$ be a closed convex subset of a Banach space $X$, and $G: \Omega \rightarrow \Omega$ be continuous satisfying Mönch's condition, i.e.,

where $conv\left(\Lambda\right)$ denotes the convex hall of $\Lambda$. Then $G$ has a fixed point.

3.   Representation of mild solution using fractional Laplace transform

According to Theorem 2.2, we can rewrite the nonlocal problem (1.1) in the following equivalent integral equation:

where $Bu(t) = \int_{0}^{t}h(t, s, u_{s})ds$, provided that the integral in (3.1) exists.

To introduce the mild solution of (1.1) we need to define the two families of operators $\{\mathrm{S}_{\alpha, \varrho}(t)\}_{t\geq0}$ and $\{\mathrm{P_{\alpha, \varrho}}(t)\}_{t\geq0}$ by

where $0\leq\alpha, \varrho\leq1$ and for $\theta\geq0$

is the wright type function defined on $(0, \infty)$ which is positive and satisfies

where $\phi_{\alpha}(\theta) = \alpha t^{-1-\alpha}\psi_{\alpha}(t^{-\alpha})$.

Lemma 3.1. If (3.1) holds, then we have

where the operators $\mathrm{S}_{\alpha, \varrho}(t)$ and $\mathrm{P_{\alpha, \varrho}}(t)$are defined in (3.2).

Proof. Let $\lambda > 0$. By applying the fractional Laplace transform of order $0 < \varrho\leq1$ to (3.1) for $t\geq0$, we get:

where $U(\lambda) = \mathcal{L}_{0}^{\varrho}(u)(\lambda)$ and $F(\lambda) = \mathcal{L}_{0}^{\varrho}\left(f(t, u_{t}, Bu(t))\right)(\lambda)$.

Now, differentiating (3.5) with respect to $\lambda$, we obtain

Using (3.7) and from Theorem 2.4, we get

By using the substitution $\frac{\tau^{\varrho}}{\varrho} = \left(\frac{s^{\varrho}}{\varrho}\right)^{\frac{1}{\alpha}}\theta$ and $\vartheta = \theta^{-\frac{\alpha}{\varrho}}$ in the last equation, we obtain

Similarly, from (3.5) and by using the substitution $\frac{\tau^{\varrho}}{\varrho} = \left(\frac{s^{\varrho}}{\varrho}\right)^{\frac{1}{\alpha}}\theta$, we get

Applying the property $3$ of Proposition 2.1 yields

According to (3.9) and (3.11), we have

Now, by inverting the inverse fractional Laplace transform, we obtain

Definition 3.1. An $X$-valued function $u \in C\left([-\delta, b], X\right)$ is called a mild solution of the nonlocal Cauchy problem (1.1), if it satisfies:

Lemma 3.2. The family of operators$\{\mathrm{S}_{\alpha, \varrho}(t)\}_{t\geq0}$ and$\{\mathrm{P_{\alpha, \varrho}}(t)\}_{t\geq0}$ satisfy:

(i) For any fixed $t\geq0$, $\mathrm{S}_{\alpha, \varrho}(t)$ and $\mathrm{P_{\alpha, \varrho}}(t)$ are linear and bounded.

(ii) For any $x\in X$, the $X$-valued functions $t \rightarrow \mathrm{S}_{\alpha, \varrho}(t)x$ and $t \rightarrow \mathrm{P_{\alpha, \varrho}}(t)x$ are continuous on $[0, +\infty)$.

Proof. The linearity is obvious. Since $\|T_{\varrho}(t)\|\leq M$ for any $t\geq0$ and from (3.3), we get

Similarly:

For the part (ii), let $t_{1}, t_{2}\geq0$. Then

From the strong continuity of $T_{\varrho}(t)$ and by using Lesbegue dominated convergence we obtain $\lim_{t_{2}\rightarrow t_{1}}\|\mathrm{S}_{\alpha, \varrho}(t_{1})x- \mathrm{S}_{\alpha, \varrho}(t_{2})x\| = 0$, which implies that $\{\mathrm{S}_{\alpha, \varrho}(t)\}_{t\geq0}$ is strongly continuous. A similar argument enables us to prove the strong continuity of $\{\mathrm{P_{\alpha, \varrho}}(t)\}_{t\geq0}$.

4.   Existence results of a mild solution

In this section we will establish the existence results by using the Hausdorff measure of noncompactness. To state and prove our main results for the existence of mild solutions of problem (1.1), we need the following hypotheses:

$(H_{1})$ The uniformly bounded $C_{0}$-$\varrho$-semigroup $\{T_{\varrho}(t)\}_{t\geq0}$ generated by $A$ is continuous in the uniform operator topology for $t > 0$.

$(H_{2})$ The function $f: [0, b]\times \mathcal{C}\times X\rightarrow X$ satisfies the following

(i) For each $(v, x)\in \mathcal{C}\times X$, $f(., v, x)$ is strongly measurable, $f(t, ., .)$ is continuous a.e. for $t\in [0, b]$.

(ii) There exists $m\in L_{1}\left([0, b], \mathbb{R}^{+}\right)$ such that $I_{0}^{\alpha, \varrho}m\in C\left((0, b], \mathbb{R}^{+}\right)$ and $\lim_{t\rightarrow 0^{+}}I_{0}^{\alpha, \varrho}m(t) = 0$, satisfying: $\|f(t, x, v)\|\leq m(t)$ for all $(x, v)\in X\times\mathcal{C}$ and almost all $t\in [0, b]$.

(iii) There exists a constant $L\geq0$ such that for any bounded sets $\Lambda_{1} \subset \mathcal{C}$, $\Lambda_{2} \subset X$

($(H_{3})$) The function $h: [0, b]\times[0, b]\times \mathcal{C}\rightarrow X$ satisfies the following

(i) For each $v\in \mathcal{C}$, $h(., ., v)$ is strongly measurable, $h(t, s, .)$ is continuous a.e. for $(t, s)\in [0, b]\times[0, b]$.

(ii) There exists a function $m_{1}: [0, b]\times[0, b]\rightarrow \mathbb{R}^{+}$, such that $\sup_{t \in [0, b]}\int_{0}^{t}m_{1}(t, s)ds = m_{1}^{\ast} < \infty$ and $\|h(t, s, v)\|\leq m_{1}(t, s)\|v\|_{\mathcal{C}}$, for all $t, s \in [0, b]$ and $v \in \mathcal{C}$.

(iii) There exists a function $\gamma: [0, b]\times[0, b]\rightarrow \mathbb{R}^{+}$, such that $\sup_{t \in [0, b]}\int_{0}^{t}\gamma(t, s)ds = \gamma^{\ast} < \infty$ and

for each bounded subset $\Lambda \in \mathcal{C}$ and almost all $t, s \in [0, b]$.

$H_{4})$) The operator $g: C\left([-\delta, b], X\right)\rightarrow \mathcal{C}$ satisfies

(i) For each $t \in[-\delta, 0]$, the operator $\Upsilon_{t}: C\left([-\delta, b], X\right)\rightarrow X$ defined by $\Upsilon_{t}(u) = (gu)(t)$ is continuous. There exists a constant $L_{3}\in \left(0, \frac{1}{M}\right)$ such that $\|g(u)\|_{\mathcal{C}}\leq L_{3}\|u\|_{\infty}$ for all $u \in C\left([-\delta, b], X\right)$, and the subset $g\left(\Lambda\right)\subset \mathcal{C}$ is equicontinuous for each bounded set $\Lambda \subset C\left([-\delta, b], X\right)$

(ii) There exists a constant $L_{4}\in [0, 1)$ such that $\mu\left(\Upsilon_{t}\left(\Lambda\right)\right)\leq L_{4}\mu\left(\Lambda(t)\right)$ for each bounded set $\Lambda \subset C\left([-\delta, b], X\right)$ and all $t \in [-\delta, 0]$.

Lemma 4.1. If $(H_{1})$ holds, then the family of operators $\{\mathrm{S}_{\alpha, \varrho}(t)\}_{t\geq0}$ and$\{\mathrm{P_{\alpha, \varrho}}(t)\}_{t\geq0}$ are continuous in the uniform operator topology for $t > 0$.

Proof. Let $t_{1}, t_{2}\geq0$. For $\varepsilon > 0$ we have

Applying the Lebesgue dominated convergence by using the continuity of $T_{\varrho}$ in the uniform operator topology and the Eq (3.3), we obtain

Then for any $x\in X$, $\|x\|\leq1$

From (3.3), and since $\varepsilon$ is arbitrary, then

and therefore

which implies that the continuity in the uniform operator topology of $\mathrm{S}_{\alpha, \varrho}(t)$ for $t > 0$.

Using the similar argument we can prove that $\mathrm{P_{\alpha, \varrho}}(t)$ is continuous in the uniform operator topology for $t > 0$.

Let $\bf{B}_{r} = \{u\in C\left([-\delta, b], X\right), \|u\|_{\infty}\leq r\}$, where $r\geq0$. Then $\bf{B}_{r}$ is clearly a bounded closed and convex subset in $C\left([-\delta, b], X\right)$. We define the operator $\Phi$ by

Obviously, $u\in \bf{B}_{r}$ is a mild solution of (1.1) if and only if the operator $\Phi$ has a fixed point on $\bf{B}_{r}$, i.e., there exists $u\in \bf{B}_{r}$ satisfies $u = \Phi u$.

Lemma 4.2. If $(H_{1})$–$(H_{4})$ hold, then $\left\{\Phi u, \; u\in \bf{B}_{r}\right\}$ is equicontinuous.

Proof. Let $u\in \bf{B}_{r}$. For $-\delta\leq t_{1}\leq t_{2}\leq0$, we have

Since $\varphi \in \mathcal{C}$ and from $(H_{4})$(i), we obtain

For $-\delta\leq t_{1}\leq0 < t_{2}\leq b$, then from $(H_{2})$(ii) and Lemma 3.2, we get

Since $\varphi \in \mathcal{C}$ and from $(H_{2})$(ii), $(H_{4})$(i) and Lemma 4.1, we find $I_{1}, ..., I_{5}\rightarrow 0$ as $t_{1}, t_{2}\rightarrow0$ and hence

For $0 < t_{1}\leq t_{2}\leq b$, we have

Applying Lemma 4.1, we get $J_{1}\rightarrow 0$ as $t_{2}\rightarrow t_{1}$. According to $(H_{2})$(ii), we find $J_{2}\rightarrow 0$ as $t_{2}\rightarrow t_{1}$. For $t_{1} < t_{2}$ and since

then from Lebesgue dominated convergence, we get $J_{3}\rightarrow 0$ as $t_{2}\rightarrow t_{1}$. For $\varepsilon > 0$ small enough, we have

Since

and from Lemma 4.1, $\mathrm{P_{\alpha, \varrho}}(t)$ is continuous in the uniform operator topology, then by using Lebesgue dominated convergence we find $J_{41}\rightarrow0$ as $t_{2}\rightarrow t_{1}$. Using the same manner in $J_{2}$ and $J_{3}$ we get $J_{42}, J_{43}\rightarrow 0$ as $\varepsilon\rightarrow0$, and consequently $J_{4}$ converges to zero independently for $u \in \bf{B}_{r}$ as $t_{2}\rightarrow t_{1}$. Therefore

which means that $\left\{\Phi u, \; u \in \bf{B}_{r}\right\}$ is equicontinuous.

Lemma 4.3. If $(H_{1})$–$(H_{4})$ hold, then $\Phi$ is continuous in $\bf{B}_{r}$ and maps $\bf{B}_{r}$ into $\bf{B}_{r}$ for any $r\geq0$ satisfies

Proof. Claim: $\Phi$ maps $\bf{B}_{r}$ into $\bf{B}_{r}$.

Obviously, from Lemma 4.2, $\Phi u \in C\left([-\delta, b], X\right)$. For $t \in [0, b]$ and for any $u \in \bf{B}_{r}$, by using $(H_{1})$, $(H_{2})$(ii) and $(H_{4})$(i), we get

For $t\in [-\delta, 0]$, we get

Hence, $\|\Phi u\|_{\infty}\leq r$ for all $u \in \bf{B}_{r}$.

Claim: $\Phi$ is continuous in $\bf{B}_{r}$.

Let $\left\{u_{n}\right\}_{n = 0}^{\infty}\subset \bf{B}_{r}$ such that $\lim_{n\rightarrow \infty}\|u_{n}-u\|_{\infty} = 0$.

For $t \in [0, b]$, we have

From conditions $(H_{2})$(i), (ii) and $(H_{3})$(i), (ii) we get

and

Then by using Lesbegue dominated convergence, we obtain

From $(H_{4})$(i), we obtain

Hence

From $(H_{4})$(i), we find $\Phi u_{n}\rightarrow \Phi u$ pointwise on $[-\delta, 0]$ and consequently $\Phi u_{n}\rightarrow \Phi u$ pointwise on $[-\delta, b]$, so the sequence $\left\{\Phi u_{n}\right\}_{n = 0}^{\infty}$ is pointwise relatively compact on $[-\delta, b]$. From Lemma 4.2, $\left\{\Phi u_{n}\right\}_{n = 0}^{\infty}$ is equicontinuous, then by Ascoli-Arzela theorem, $\left\{\Phi u_{n}\right\}_{n = 0}^{\infty}$ is relatively compact, i.e., there exists subsequence of $\left\{\Phi u_{n}\right\}_{n = 0}^{\infty}$ converge uniformly, clearly, to $\Phi u$ as $n\rightarrow \infty$, and since $C\left([-\delta, b]\right)$ is compete, $\Phi u_{n}\rightarrow \Phi u$ uniformly on $[-\delta, b]$, as $n\rightarrow \infty$, and so $\Phi$ is continuous.

Theorem 4.4. Assume that $(H_{1})$–$(H_{4})$ are hold. Then the nonlocal Cauchy problem (1.1) has at least a mild solution on $\bf{B}_{r}$, where $r$ satisfies (4.7).

Proof. We know that $\bf{B}_{r}$ is closed and convex. From Lemmas 4.2 and 4.3, we know that $\Phi$ is a continuous map from $\bf{B}_{r}$ into $\bf{B}_{r}$ and the set $\left\{\Phi u, \; u \in \bf{B}_{r}\right\}$ is equicontinuous. We shall prove that $\Phi$ satisfies the Mönch condition $\bf{B}_{r}$. Let $\Lambda = \left\{u_{n}\right\}_{n = 0}^{\infty}$ be a countable subset of $\bf{B}_{r}$ such that $\Lambda \subseteq conv\left({0}\cup\Phi\left(\Lambda\right)\right)$. Then $\Lambda$ is bounded and equicontinuous and therefore the function $t\rightarrow \varpi(t) = \mu\left(\Lambda(t)\right)$ is continuous on $[-\delta, b]$. From $(H_{4})$(ii), we have, for any $t \in [-\delta, 0]$,

Since $L_{4} < 1$, then $\varpi(t) = 0$ for all $t \in [-\delta, 0]$. For $t \in [0, b]$, then from $(H_{2})$(iii), $(H_{3})$(iii), $(H_{4})$(ii) and by using Lemma 2.6 and properties of the measure $\mu$, we obtain

From the last equation and by using the properties of supremum, we get

Then from Lemma 2.3, we obtain $\sup_{0\leq\theta\leq t}\varpi(\theta) = 0$ for all $t \in [0, b]$. Hence $\varpi\equiv0$ on $[-\gamma, b]$. This implies that $\Lambda(t)$ is relatively compact for each $t \in [-\gamma, b]$. From Ascoli-Arzela theorem, $\Lambda$ is relatively compact on $\bf{B}_{r}$. Hence from Lemma 2.7, $\Phi$ has a fixed point in $\bf{B}_{r}$, i.e., the nonlocal Cauchy problem (1.1) has at least mild solution on $\bf{B}_{r}$.

5.   Applications

Consider the following nonlocal integro-differential equation:

where $0 < \varrho\leq1$, $\delta > 0$, and $v_{t}(\theta, x) = v(t+\theta, x)$. The following conditions hold:

(1) The function $\zeta: [-\delta, 0]\rightarrow \mathbb{R}$ is integrable, i.e., $\int_{-\delta}^{0}|\zeta(\theta)|d\theta < \infty$.

(2) The function $\zeta_{1}: [0, b]\rightarrow \mathbb{R}$ is integrable, and $\int_{0}^{b}|\zeta_{2}(\theta)|d\theta < 1.$

(3) The function $\psi: [-\delta, 0]\times[0, \pi]\rightarrow \mathbb{R}$ is measurable and saisfies

for all $t_{1}, t_{2}\in [-\delta, 0]$.

Let $X = L^{2}\left([0, \pi]\right)$. Consider the operator $A = -\frac{\partial^{2}}{\partial x^{2}}$ in $X$ with domain

where $H^{2}([0, \pi])$ and $H^{1}_{0}([0, \pi])$ are the classical Sobolev spaces. Eigenvalues and the corresponding normalized eigenfunctions of $A$ are given by $n^{2}$, $\upsilon_{n} = \sqrt{\frac{2}{\pi}}\sin nx$, $n\in \mathbb{N}$. The family of eigenfunctions $\{\upsilon_{n}\}_{n = 0}^{\infty}$ forms an orthonormal basis in $X$ with inner product $(\omega, \nu) = \int_{0}^{1}\omega(x)\overline{\nu(x)}dx$.

Define the family of linear operators $\left\{T_{\varrho}(t)\right\}_{t\geq0}$ by

for $\omega \in X$ given by $\omega = \Sigma_{n = 1}^{\infty}(\omega, \upsilon_{n})\upsilon_{n}$. This family satisfies the following

(1) $T_{\varrho}(t)$ is a bounded linear operator, with $\|T_{\varrho}(t)\|\leq 1$ for $t\geq0$.

(2) For $s, t\geq0$ and $\omega\in X$ we get the semigroup property $T_{\varrho}(t^{\frac{1}{\varrho}})T_{\varrho}(s^{\frac{1}{\varrho}})\omega = T_{\varrho}(t^{\varrho}+s^{\varrho})^{\frac{1}{\varrho}}\omega.$

(3) For $s, t\geq0$, $\|T_{\varrho}(s)-T_{\varrho}(t)\| \rightarrow 0$ when $s \rightarrow t$.

(4) For $\omega \in D(A)$, $D_{0^{+}}^{\varrho}T_{\varrho}(t)\omega = AT_{\varrho}(t)\omega$. In particular $\lim_{t \rightarrow 0^{+}}D_{0^{+}}^{\varrho} T_{\varrho}(t)\omega = A\omega$.

Clearly, $\left\{T_{\varrho}(t)\right\}_{t\geq0}$ is a uniformly bounded $C_{0}$-$\varrho$-semigroup which is continuous in the uniform operator topology for $t\geq0$, and $A$ its generator. For $x\in [0, \pi]$ and $\phi \in C\left([-\delta, 0], X\right)$, we set

Then Eq (5.1) can be transformed to the abstract form (1.1).

For $t\in [0, b]$, we can obtain

where $I_{0}^{\alpha, \varrho}m\in C\left((0, b], \mathbb{R}^{+}\right)$ and $\lim_{t\rightarrow 0^{+}}I_{0}^{\alpha, \varrho}m(t) = 0$.

For any $\phi, \tilde{\phi} \in \mathcal{C}$ and $\omega, \tilde{\omega}\in X$, by straightforward calculations we get

Then for any bounded sets $\Lambda_{1}\subset \mathcal{C}$, $\Lambda \subset X$

where $L = \delta+\pi$ and $t\in [0, b]$.

For each $t, s\in [0, b]$, $\phi \in \mathcal{C}$, we obtain

where $m_{1}(t, s) = \left(\frac{t^{\varrho}-s^{\varrho}}{\varrho}\right)^{-\frac{4}{5}} \left(\frac{s^{\varrho}}{\varrho}\right)^{-\frac{1}{5}}s^{1-\varrho} \int_{-\delta}^{0}|\zeta(\theta)|d\theta$ satisfies

For any $t, s\in [0, b]$, $\phi, \tilde{\phi} \in C\left([-\delta, 0], X\right)$

Hence, for any bounded set $\Lambda \subset \mathcal{C}$,

where $\gamma(s, t) = 2\left(\frac{t^{\varrho}-s^{\varrho}}{\varrho}\right)^{-\frac{4}{5}} \left(\frac{s^{\varrho}}{\varrho}\right)^{-\frac{1}{5}}s^{1-\varrho} \int_{-\delta}^{0}|\zeta(\theta)|d\theta$, and $\gamma^{\ast} = 2 \beta\left(\frac{1}{5}, \frac{4}{5}\right)\int_{-\delta}^{0}|\zeta(\theta)|d\theta$.

For all $t \in [-\delta, 0]$, $\phi, \tilde{\phi} \in C\left([-\delta, b], X\right)$, we have

and

where $L_{3} = L_{4} = \int_{0}^{b}\left|\zeta_{2}(\theta)\right|d\theta$. Then $g(.)(t): C\left([-\delta, b], X\right)\rightarrow X$ is continuous for any $t \in [-\delta, 0]$, and therefore

Since all conditions of Theorem 4.4 are satisfied, problem (5.1) has at least a mild solution.

6.   Conclusions

In this manuscript, the existence results of mild solutions for non local fractional evolution equations with finite delay in the sense of Caputo conformable fractional derivative have been successfully investigated under some sufficient conditions on Kuratowski measure of non compactness. To the best of our knowledge, this type of problems supplemented with newly defined Caputo conformable fractional operator has not been investigated in any literature. All the obtained results are supported by an application showing the applicability of the presented theory.

Conflict of interest

The authors declare no conflict of interest