Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction with an application

1.   Introduction

Fractional differential equations are becoming a considerably more important and popular topic. In order to specify the so-called fractional differential equations, the conventional integer order derivative is generalized to arbitrary order. Fractional differential equations have been extensively employed to explain a variety of physical processes because of the effective memory function of the fractional derivative, such as seepage flow in porous media, fluid dynamic, and traffic models. Also, there are several applications of fractional differential equations in control theory, polymer rheology, aerodynamics, physics, chemistry, biology, and other exciting conceptual advancements (see ^[1,2,3,4] and their references).

In the real world, there are numerous processes and phenomena that are influenced by transient external factors as they evolve. When compared to the whole duration of the occurrences and processes being researched, their duration is tiny. As a result, it is reasonable to believe that these exterior impacts are "instantaneous", or take the shape of impulses. Differential equations including the impulse effect, or impulsive differential equations, seem to be a plausible explanation of the known evolution processes of various real-world issues. The impulsive differential equations have been studied as the subject of numerous excellent monographs ^[5,6].

Differential equations are used as representations for many processes in applied sciences research. There are a variety of classical mechanics that experience abrupt changes in their states, such as biological systems (heartbeats and blood flow), mechanical systems with impact, radiophysics, pharmacokinetics, population dynamics, mathematical economics, ecology, industrial robotics, biotechnology processes, etc ^[7,8]. The systems of differential equations with impulses are suitable mathematical models for such phenomena. Impulsive differential equations essentially have three parts: An impulse equation simulates an impulsive leap that is described by a jump function at the moment of impulse occurs, a continuous-time differential equation determines the state of the system between impulses, and jump criteria identifies a set of jump occurrences ^[9,10,11].

Furthermore, there are various models that have been developed in many fields like biology, economics, and materials science where the rate of change at time $t$ depends not only on the system's current state but also on its history over a period of time $[t-\tau, t]$ ^{[12,13,14,15]}. These models have evolutionary equations with delay, which describe them mathematically. Equations with infinite delay are produced by the more generic if we take $\tau = \infty$.

Physics provides a compelling justification for studying the nonlocal partial differential equation. Fractional derivatives in space and time are used in abstract partial differential equations such as fractional diffusion equations. They may be used to simulate anomalous diffusion, in which a particle plume spreads differently than the traditional diffusion equation would suggest. By replacing the second-order space derivative in the classic diffusion equation by an infinitesimal generator operator of strongly continuous $C_0$ semi-group or cosine functions, the time fractional evolution equation is derived ^[16,17,18].

In ^[19], Kumar and Pandey attempted to examine the results of the existence of a solution to a class of FDEs (the fractional calculus due to Atangana-Baleanu) of the sort

for all $\mathfrak{v}\in D(\mathcal{F})$ (the domain of $\mathcal{F}$), where $^{ABC}D^{{\rho}}$ is the ABC fractional derivative of order ${{\rho}}\in(0, 1)$, $\mathcal{F}\colon D(\mathcal{F})\subset {\mathrm{X}}\to {\mathrm{X}}$ is a generator of ${{\rho}}$-resolvent operator $\{S_{{{\rho}}}(r)\}_{r\geq 0}$ on a Banach space $({\mathrm{X}}, \|\cdot\|)$, $J = [0, d]$, $0 = r_{0} < r_{1} < r_{2} < \cdots < r_{m} < r_{m+1}$ and $s_{j}\in(r_{j}, r_{j+1})$ for all $j = 1, 2, \ldots, m;\; m\in \mathbb{N}$. The functions $\mathfrak{h}\colon \cup_{k = 0}^{m}(s_{j}, r_{j+1}]\times {\mathrm{X}}\to {\mathrm{X}}$ and $f\colon{\mathrm{X}}\to {\mathrm{X}}$ are given continuous functions and $\delta_{j}\colon (r_{j}, s_{j}]\times {\mathrm{X}}\to {\mathrm{X}}$ are non-instantaneous impulsive functions for each $j = 1, 2, \ldots, m;\; m\in \mathbb{N}$ and $\mathfrak{v}_{o}\in {\mathrm{X}}$.

Recently, Kavitha et al. ^[20] examined the existence of solutions for a class of non-instantaneous impulses and infinite delay of fractional differential equations within the Mittag-Leffler kernel of the kind

where the fractional order ${\mathfrak{v}}\in(0, 1)$, $B\colon D(B)\subset E\to E$ is infinitesimal generator of an ${{\rho}}$-resolvent operator $\{S_{{{\rho}}}(\xi)\}_{\xi\geq 0}$ on a Banach space $(E, \|\cdot\|$, $K = [0, b]$, $0 = \xi_{0} = s_{0} < \xi_{1}\leq s_{1} < \xi_{2} < \cdots < \xi_{q} < \xi_{q+1} = b$ and $s_{l}\in(\xi_{l}, \xi_{l+1})$ for all $l = 1, 2, \ldots, q;q\in\mathbb{N}$. The function $\mathcal{F}\colon \cup_{l = 0}^{q}(s_{l}, \xi_{l+1}]\times \mathcal{A}\to E$ satisfies Caratheodory conditions and the functions $\kappa_{l}\colon (\xi_{l}, s_{l}]\times \mathcal{A}\to E$ are non-instantaneous impulsive functions for each $l = 1, 2, \ldots, q$. They considered that $p_{\xi}\colon (-\infty, 0]\to E$ such that $p_{\xi}(x) = p(\xi+x)$ for all $x\leq 0$ and $\phi\in\mathcal{A}$ where $\mathcal{A}$ is an abstract phase space.

In light of the foregoing, in this publication, we examine the existence results for a class of fractional-order non-instantaneous impulses functional evolution equations.

Consider the fractional semilinear evolution of the form

where $^cD_{t}^{{\alpha}}$ is the fractional derivative due to Caputo of order $1 < \alpha < 2$ and $J = [0, a]$ is operational interval. Here, $h\colon J\times\mathcal{P}_{\mathfrak{h}}\to \mathbb{X}$ is a given function satisfying some assumptions that will be determined later, where $\mathcal{P}_{\mathfrak{h}}$ is an abstract phase space and $\mathbb{X}$ is a Banach space. The functions $\mu_{k}, \; \xi_{k}\in C\left((t_{k}, s_{k}]\times \mathbb{X}; \mathbb{X}\right)$ for all $k = 1, 2, \ldots m; \; m\in \mathbb{N}$ reflect the impulsive circumstances and $0 = t_{0} = s_{0} < t_{1}\leq s_{1}\leq t_{2} < \dots < t_{m}\leq s_{m}\leq t_{m+1} = a$ are pre-fixed numbers. The history function $u_{t}\colon (-\infty, 0]\to X$ is an element of $\mathcal{P}_{\mathfrak{h}}$ and defined by $u_{t}(\theta) = u(t+\theta), \theta\in(-\infty, 0]$.

The closed operator $A$ is an infinitesimal generator of a uniformly bounded family of strongly continuous cosine operators $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$, which is defined on a Banach space $\mathbb{X}$. The Banach space of continuous and bounded functions from $(-\infty, a]$ into $\mathbb{X}$ provided with the topology of uniform convergence is denoted by $\mathcal{C} = \mathcal{C}_{a}\left((-\infty, a], \mathbb{X}\right)$ with the norm

As $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$ is a cosine family on $\mathbb{ X}$, then there exists $\varpi\geq 1$ ^[21] such that

The rest of the text is structured as follows. In section 2, we give some fundamental concepts and lemmas related to our study. In section 3, we formulate the mild solution of $(1.1)$ by considering that operator $A$ is an infinitesimal generator of strongly continuous cosine functions $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$. By using the fixed-point theorem, Section 4 presents our study outcomes. An instance is provided in Section 5 to be an application.

2.   Preliminaries

In this section, we present some concepts and definitions related to the components of the research paper such as fractional calculus, cosine and sine families operators, and abstract phase space. Also, some lemmas that give helpful results to prove the main results of this contribution, are provided.

2.1.   Fractional calculus

The definitions of R-L integral and Caputo derivative and the important related lemma are introduced as follow.

Definition 2.1. (Caputo derivative ^[22]) Let $\ell-1 < \mathfrak{q}\leq\ell; \; \ell\in \mathbb{N}$ and $x\colon [a, b]\to \mathbb{R}$ $(-\infty < a < b < \infty)$ be $n$th continuously differentiable function. Then, the left derivative of fractional order $\mathfrak{q}$ due to Caputo is presented as

Definition 2.2. (Riemann-Liouville fractional integral ^[22]) The left R-L fractional integral of the integrable function $x$ over the interval $[a, b]$ is derived as

Lemma 2.1. ^[23] Let $\ell\in \mathbb{N}, \; \ell-1 < \mathfrak{q}\leq \ell$ and $x(s)$ be $n$th continuously differentiable function over the interval $[a, b]$. Then,

Definition 2.3. (Atangana-Baleanu fractional derivative in Caputo sense ^[19]) ABC-derivative for the order $\alpha\in[0, 1]$ and $x(s)\in H^1(a, b), a < b$ is given by

where $E_\alpha(\cdot)$ and $H^1$ are the Mittag-Leffler function and the non-typical Banach space defined, respectively, as

2.2.   Cosine family operator

Let $A$ be an infinitesimal generator of a uniformly bounded family of strongly continuous cosine operators $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$ which is defined on a Banach space $\mathbb{X}$. We collect some basic properties of a cosine family and its relations with the operator $A$ and the associated sine family.

Definition 2.4. ^[24] Consider $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$ is a one parameter family of bounded linear operators mapping the Banach space $\mathbb{X}\to \mathbb{X}$. It is referred to a strongly continuous cosine family if and only if

(ⅰ) $\mathfrak{R}(0) = I$;

(ⅱ) $\mathfrak{R}(s+t)+\mathfrak{R}(s-t) = 2\mathfrak{R}(s)\mathfrak{R}(t)$ for all $s, t\in \mathbb{R}$;

(ⅲ) The function $t\mapsto\mathfrak{R}(t)x$ is a continuous on $\mathbb{R}$ for any $x\in\mathbb{X}$.

The sine family $\{\mathfrak{T}(t)\}_{ t\in R}$ is correlated to the strongly continuous cosine family $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$, it is characterized by

Lemma 2.2. ^[24] Consider $A$ is an infinitesimal generator of a strongly continuous cosine family $\{\mathfrak{R}(t)\}_{t\in \mathbb{R}}$ on Banach space $\mathbb{X}$ such that $\|\mathfrak{R}(t)\|\leq Me^{\xi|t|}, t\in \mathbb{R}$. Then, for $\lambda > \xi$ and $(\xi^{2}, \infty)\subset\rho(A)$ (the resolvent set of $A$), we have

where the operator $R(\lambda; A) = (\lambda I-A)^{-1}$ is the resolvent of the operator $A$ and $\lambda\in\rho(A)$.

In this case, the operator $A$ is defined by

where $\mathcal{D}(A) = \{x\in\mathbb{X}\colon \mathfrak{R}(t)x\in \mathcal{C}^{2}(\mathbb{R}, \mathbb{X})\}$ is the domain of the operator $A$. Clearly the infinitesimal generator $A$ is densely defined operator in $\mathbb{X}$ and closed.

In the sequel to present our results, we need the following:

Definition 2.5. Suppose that $\tau > 0$, the Mainardi's Wright-type function is defined as

and achieves

2.3.   Abstract phase space $\mathcal{P}_{\mathfrak{h}}$

The abstract phase space $\mathcal{P}_{\mathfrak{h}}$ is demonstrated by convenient way ^[25,26]. Let $\mathfrak{h} = \mathcal{C}\left((-\infty, 0], [0, \infty)\right)$ with $\int_{-\infty}^{0}\mathfrak{h}(t)dt < \infty$. Then, for any $c > 0$, we can define the set

and establish the space $\mathcal{P}$ with the norm

Let us define the space

If $\mathcal{P}_{\mathfrak{h}}$ is furnished with the norm

then $\left(\mathcal{P}_{\mathfrak{h}}, \|\cdot\|_{\mathcal{P}_{\mathfrak{h}}} \right)$ is a Banach space. Next, we introduce the available space

which has the norm

Definition 2.6. ^[27] If $v\colon (-\infty, a]\to\mathbb{X}, \; a > 0$, such that $\phi\in\mathcal{P}_{\mathfrak{h}}$. The situations listed below are accurate for all $\tau\in[0, a]$,

1) $v_{\tau}\in \mathcal{P}_{\mathfrak{h}}$;

2) Two functions, $\zeta_{1}(\tau), \zeta_{2} (\tau) > 0$, are such that $\zeta_{1}(\tau)\colon [0, \infty)\to[0, \infty)$ is a continuous function and $\zeta_{2}(\tau)\colon [0, \infty)\to[0, \infty)$ is a locally bounded function which are independent to $v(\cdot)$ whereas

3) $\|v(\tau)\|\leq H \|v_{\tau}\|_{\mathcal{P}_{\mathfrak{h}}}$, where $H > 0$ is a constant.

3.   Structure of mild solution

Before introducing the mild solution of evolution Eq (1.1), we have to establish the following Lemmas.

Lemma 3.1. Let $I_{s}^{{\alpha}}$ be the left R-L integral of order ${{\alpha}}$ and $f(t)$ is integrable function defined for $t\geq s\geq0$. Then,

Proof. From the Definition 2.2 and the rule of converting double integral to single integral, we get

The proof is over.

Lemma 3.2. Let $1 < {{\alpha}}\leq 2$ and $h\colon J\to X$ be an integrable function. Then, the mild solution to our problem $(1.1)$ possess the form

where $1/2 < q = \frac{{{\alpha}}}{2}\leq1$,

and $M_{q}$ is a probability density function defined by Definition 2.5.

Proof. Using Lemma $2.1$ with operating by $I_{s_{r}}^{{{\alpha}}}$ on both sides to the fractional differential equation in (1.1), we arrive at

where $c_{1, k}, c_{0, k}\in \mathbb{R}, \; k = 0, 1, \ldots, m$ are constants to be determined.

● For $t\in[0, t_{1}]$: By taking $\rho\to 1$ to the results given in Lemma 5 in ^[28], we have

● For $t\in(t_{1}, s_{1}]$: We obtain

● For $t\in(s_{1}, t_{2}]$: The problem $(1.1)$ becomes

In this interval, Eq $(3.1)$ becomes

Considering the past impulsive conditions, we get

which imply that

Multiplying both sides by $e^{-\lambda t}$ followed by integrating from $s_1$ to $\infty$, we achieve

where

Given that $(\lambda^{{{\alpha}}}I-A)^{-1}$ exists, then $\lambda^{{{\alpha}}}\in{{\rho}}(A)$. We obtain

Let $\Psi_{q}(\theta) = \frac{q}{\theta^{q+1}}M_{q}(\theta^{-q})$ be defined for $\theta\in(0, \infty)$ and $q\in(\frac{1}{2}, 1)$. Then,

which can be used to calculate the first term with replacing $t$ by $s^q$ as

By using Lemma 3.1 with ${{\alpha}} = 1$, we get

Finally, we can write

In conclusion, we can write

Therefore, by taking the inverse Laplace transform, we have

● For $t\in(s_{k}, t_{k+1}], \; k = 2, 3, \ldots, m$: In a similar manner, we can write

Consequently, we get the solution from the earlier $(1.1)$. Direct calculations show that the opposite results are true. The proof is completed.

Remark 3.1. ^[28] From linearity of $\mathfrak{R}(t)$ and $\mathfrak{T}(t)$ for all $t\geq 0$, it is clearly to deduce that $\mathfrak{R}_{q}(t)$ and $\mathfrak{T}_{q}(t, s)$ are also linear operators where $0 < s < t$. Therefore, the proofs of all next Lemmas are same when taking ${{\rho}}$ approaches $1$.

Lemma 3.3. ^[28] The following estimates for $\mathfrak{R}_{q}(t)$ and $\mathfrak{T}_{q}(t, s)$ are verified for any fixed $t\geq 0$ and $0 < s < t$

Lemma 3.4. ^[28] The operators $\mathfrak{R}_{q}(t)$ and $\mathfrak{T}_{q}(s, t)$ are strongly continuous for every $0 < s < t$ and $t > 0$.

Lemma 3.5. ^[28] Assume that $\mathfrak{R}(t)$ and $\mathfrak{T}(t, s)$ are compact for every $0 < s < t$. Then, the operators $\mathfrak{R}_{q}(t)$ and $\mathfrak{T}_{q}(s, t)$ are compact for every $0 < s < t$.

4.   The main results

Define the operator $\mathfrak{N}\colon \overline{\mathcal{P}}_{\mathfrak{h}}\to\overline{\mathcal{P}}_{\mathfrak{h}}$ as follows

Let $\varkappa(\cdot)\colon (-\infty, a]\to \mathbb{X}$ be the function denoted by

Plainly, $\varkappa(0) = \phi(0)$. For each $z\in \mathcal {C}([0, a], \mathbb{X})$ with $z(0) = 0$, we indicate by $\vartheta$ to the function defined as

If $u(\cdot)$ satisfies that $u(t) = \mathfrak{N}(u)(t)$ for all $t\in (-\infty, a]$, we can decompose that $u(t) = \vartheta(t)+\varkappa(t)$, $t\in(-\infty, a]$, it denotes $u_{t} = \vartheta_{t}+\varkappa_{t}$ for every $t\in(-\infty, a]$ and the function $z(\cdot)$ satisfies

Set the space $\Upsilon = \{z\in\mathcal{C}([0, a], \mathbb{X}), z(0) = 0\}$ equipped the norm

Therefore, $(\Upsilon, \|\cdot\|_{\Upsilon})$ is a Banach space. Assume that the operator $\mathscr{G}\colon \Upsilon\to\Upsilon$ is formulated as follows:

The operator $\mathfrak{N}$ seems to have a fixed point is equivalent to $\mathscr{G}$ has a fixed point. Thus, we proceed to prove that $\mathscr{G}$ has a fixed point.

Now, we make the following assumptions:

$\left(\mathcal{E}_1 \right)$ The function $h\colon [0, a]\times{\mathcal{P}}_{\mathfrak{h}}\to\mathbb{X}$ is a continuous and $\mu_{k}, \xi_{k}\colon [t_{k}, s_{k}]\times\mathbb{X}\to\mathbb{X}$ are continuous functions for all $k = 1, 2, \ldots, m;\; m\in\mathbb{N}$.

$\left(\mathcal{E}_2 \right)$ There is a constant $\Omega > 0$ satisfying

$\left(\mathcal{E}_3 \right)$ There exist $\delta_{k}, \delta^{*}_{k} > 0;\; k = 1, 2, \ldots, m;\; m\in\mathbb{N}$ such that

$\left(\mathcal{E}_4 \right)$ There are positive constants $D_{k}, D^*_{k}, \; k = 1, 2, \ldots, m;\; m\in\mathbb{N}$ such that

$\left(\mathcal{E}_5 \right)$ There exists a continuous function $\mathfrak{g}(t)\colon [0, a]\to[0, \infty)$ such that, for any $(t, u_{t})\in[0, a]\times{\mathcal{P}}_{\mathfrak{h}}$, it satisfies

The brief constants that will be utilized later to streamline handling, are listed as follow

where $k = 1, 2, \ldots, m;\; m\in\mathbb{N}$.

Lemma 4.1. Assume that the requirement $\left(\mathcal {E }_2 \right)$ is met by $c = \max\limits_{t\in[0, a]}|h(t, 0)|$. Ponder about the expressions $\zeta^*_{1} = \sup\limits_{t\in[0, a]}\zeta_{1}(t)$ and $\zeta^*_{2} = \sup\limits_{t\in[0, a]}\zeta_{2}(t)$ where $\zeta_{1}(\cdot)$ and $\zeta_{2}(\cdot)$ are established in Definition 2.6. Then,

Proof. Regarding Definition $2.6$ and the presumption $\left(\mathcal{E}_2 \right)$. Then,

This ends the proof.

Lemma 4.2. Suppose that the statement $\left(\mathcal{E}_5 \right)$ is satisfied with $\mathfrak{g} = \sup\limits_{t\in[0, a]}\mathfrak{g}(t)$. Let $\zeta^*_{1} = \sup\limits_{t\in[0, a]}\zeta_{1}(t)$ and $\zeta^*_{2} = \sup\limits_{t\in[0, a]}\zeta_{2}(t)$ where $\zeta_{1}(\cdot)$ and $\zeta_{2}(\cdot)$ are outlined in Definition 6. Then,

where

Proof. By the same way in Lemma 4.1, we can easily reach the desired result.

Theorem 4.1. Consider the assertions $\left(\mathcal{E}_1\right)-\left(\mathcal{E}_4\right)$ hold and

Then, the fractional evolution equation with non-instantaneous impulsive (1.1) has a unique mild solution on $(-\infty, a]$ if $\Lambda < 1$.

Proof. To show that the operator $\mathscr{G}$ maps bounded subset of $\Upsilon$ into bounded subset in $\Upsilon$, we set

where

Then, for any $z\in\Upsilon_{r}$ and in spite of $\left(\mathcal{E}_2\right)$ and $\left(\mathcal{E}_3\right)$ and Lemma $4.1$. Correspondingly, three situations are taken into consideration.

● Case Ⅰ. Whenever $t\in[0, t_{1}]$, we have

● Case Ⅱ. Whenever $t\in(t_{k}, s_{k}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

● Case Ⅲ. Whenever $t\in(s_{k}, s_{k+1}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

For the aforementioned, we acquire $\|\mathscr{G}(z)(t)\|_{\Upsilon}\leq r$. Thus, the operator $\mathscr{G}$ maps bounded subset into bounded subset in $\Upsilon$.

Now, we prove that the operator $\mathscr{G}$ is a contraction mapping. Certainly, consider $z, z^*\in\Upsilon$. Then, there still are the subsequent situations.

● Case Ⅰ. For any $t\in[0, t_{1}]$, we have

● Case Ⅱ. For any $t\in(t_{k}, s_{k}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

● Case Ⅲ. For any $t\in(s_{k}, s_{k+1}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

For the aforementioned, we may write

Amid the existing circumstances, $\Lambda < 1$ shows that the operator $\mathscr{G}$ is a contraction. This suggests that the problem $(1.1)$ has a unique solution on $(-\infty, a]$ relying on the Banach contraction mapping principle.

Remark 4.1. In viewing our problem, it is very difficult to obtain the exact solution and so it is useful to investigate some properties of the solutions, especially the uniqueness. The previous theorem shaw that the mild solution of the problem (1.1) is unique under the assumptions $\left(\mathcal{E}_1\right)-\left(\mathcal{E}_4\right)$ and $\Lambda < 1$. This enables us to apply our results to real-life problem or phenomena as in the last section.

Assume that the operator $\mathscr{G}$ is divided as a sum of the two operators $\mathscr{G}_{i}\colon \Upsilon\to\Upsilon, \; i = 1, 2$ as

where,

and

Theorem 4.2. Suppose the hypotheses $\left(\mathcal{E}_1\right) and \left(\mathcal{E}_3\right)-\left(\mathcal{E}_5\right)$ are correct. Then the fractional evolution equation with non-instantaneous impulsive (1.1) has at least one mild solution on $(-\infty, a]$ if $\Delta < 1$ where $\Delta$ is given by

Proof. Let the operators $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$ be defined as $(4.1)$. Setting $\mathfrak{g} = \underset{t\in[0, a]}{\sup}|\mathfrak{g}(t)|$. Let us define the closed ball $\Upsilon_{{{\rho}}} = \left\{z\in\Upsilon\colon \|z\|_{\Upsilon}\leq{{\rho}} \right\}$ with radius

Then, for $u, v\in\Upsilon_{{{\rho}}}$, we claim that $\|\mathscr{G}_{1}(z)(u)+\mathscr{G}_{2}(z)(v)\|\leq{{\rho}}$ which concludes that $\mathscr{G}_{1}(u)+\mathscr{G}_{1}(v)\in\Upsilon_{{{\rho}}}$. To verify our claiming, we show that $\mathscr{G}$ maps bounded sets of $\Upsilon$ into bounded sets in $\Upsilon$, for any ${{\rho}}\geq0$. Then for any $z\in\Upsilon_{{{\rho}}}$ and in light of $\left(\mathcal{E}_3\right), \left(\mathcal{E}_5\right)$ and Lemma $4.2$, we have three cases

● Case Ⅰ. For any $t\in[0, t_{1}]$, we have

● Case Ⅱ. For any $t\in(t_{k}, s_{k}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

● Case Ⅲ. For any $t\in(s_{k}, s_{k+1}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

By virtue of the above, we obtain $\|\mathscr{G}(z)(t)\|_{\Upsilon}\leq {{\rho}}$. Thus the operator $\mathscr{G}$ maps bounded sets into bounded sets in $\Upsilon$.

The following step is to confirm that the operator $\mathscr{G}_{2}$ maps bounded sets into equicontinuous sets in $\Upsilon$. In light of the situation $\left(\mathcal{E}_1\right)$, $\mathscr{G}_{2}$ is continuous. The following scenarios are therefore possible.

● Case Ⅰ. For each $t_{k}\leq\gamma_{1} < \gamma_{2}\leq s_{k}$ and $z\in\Upsilon{{{\rho}}}$, we have

Due to the continuity of $\mu(t, u(t))$. It is clear that the above inequality approaches zero when letting $\gamma_{2}\to\gamma_{1}$.

● Case Ⅱ. For any $s_{k}\leq\gamma_{1} < \gamma_{2}\leq t_{k+1}, \; k = 1, \ldots, m;\; m\in\mathbb{N}$ and $z\in\Upsilon{{{\rho}}}$, we have

Due to compactness of operator $\mathfrak{R}_{q}(y)$ and $\mathfrak{T}_{q}(t, y)$ (see Lemma 3.5), we infer that $\|\mathscr{G}_{2}(z)(\gamma_{1})-\mathscr{G}_{2}(z)(\gamma_{2})\|\to 0$ as $\gamma_{2}\to \gamma_{1}$. Thus, $\mathscr{G}_{2}$ is a relatively compact on $\Upsilon$. By Arezela Ascoli Theorem the operator $\mathscr{G}_{2}$ is completely continuous on $\Upsilon_{{{\rho}}}$. The only thing left to do is provide evidence that $\mathscr{G}_{1}$ is a contraction mapping. Thus, two cases are thought about.

● Case Ⅰ. For any $t\in[0, t_{1}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$ and $z\in\Upsilon{{{\rho}}}$, we have

● Case Ⅱ. For any $t\in(s_{k}, t_{k+1}], \; k = 1, \ldots, m;\; m\in\mathbb{N}$, we have

As a sense, the fractional evolution equation with non-instantaneous impulsive (1.1) has at least one mild solution on $\Upsilon$, according to the Krasnoselskii Theorem. The evidence is now complete.

Remark 4.2. Also, it is useful to investigate the existence of the solution instead of the its uniqueness. The theorem above shaw that the mild solution of the problem (1.1) exists under the assumptions $\left(\mathcal{E}_1\right)-\left(\mathcal{E}_3\right)$ and $\left(\mathcal{E}_5\right)$ with $\Delta < 1$.

5.   An application

Presume the following fractional wave equation with impulsive effect and infinite delay

Consider that

Case Ⅰ. Banach fixed point theorem.

In order to explain Theorem $4.1$, we obtain:

Clearly, $h\colon [0, 1]\times\mathcal{P}_{\mathfrak{h}}\to\mathbb{R}$ is continuous and satisfying, for $u_{t}, v_{t}\in\mathcal{P}_{\mathfrak{h}}$, that

it suggests that $\Omega = {1\over 9}$. For all $t\in({2\over 5}, {4\over 5}]$ and $u, v\in\mathbb{R}$, we get

As you can see, the Theorem $4.1$ condition $\left(\mathcal{E}_4 \right)$ is satisfied with

In summary, we have

Thus all assumptions of this theorem are verified. Therefore, the problem $(1.1)$ has a unique mild solution on $(-\infty, 1].$

Case Ⅱ. Krasnoselskii's theorem.

To realize Theorem $4.2$, take $h(t, u_t)$ as given in $(5.1)$. Therefore, $\mathfrak{g}(t) = {t^3\over 8\sqrt{t+1} }$ is increasing function which admits the hypothesis $\left(\mathcal{E}_5 \right)$ with

These calculate that

Since every requirements of Theorem $4.2$ are met, it follows that there exists at least one mild solution of $(1.1)$ on $(-\infty, 1]$.

6.   Conclusions

We analyzed a set of impulsive fractional evolution equations with infinite delay in the current work. Current functional analysis methodologies serve as the foundation for our conclusions. By using the unbounded operator $A$ as the generator of the strongly continuous cosine family, we were able to suggest a mild solution for the suggested problem. In the instance of problem $(1.1)$, we had two successful outcomes: While the second argument focuses on whether there are solutions for the given problem, the first argument concentrated on the existence and uniqueness of the solution.

The first result, which is built on a Banach fixed point theorem, provides criteria for ensuring that the problem at hand has no prior solutions by requiring the usage of $h(t, u_{t})$ to satisfy the classic Lipschitz condition.

The second argument was based on a Krasnoselskii's theorem, which allows $h(t, u_{t})$ to behave as $\|h(t, u_{t})\|\leq \mathfrak{g}(t)\|u_{t}\|_{{\mathcal{P}}_{\mathfrak{h}}}$. The instruments used by fixed point theory in the scenario with simple assumptions. Finally, a numerical example that examines a function that satisfies all the prerequisites was provided to illustrate our conclusion.

In the next paper, we will study the controllability of mild solution to fractional evolution equations with an infinite time-delay and nonlocal condition by applying Krasnoselskii's theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness.

Acknowledgments

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia has funded this project, under grant no. (KEP-PhD: 34-130-1443).

Conflict of interest

The authors declare that they have no conflicts of interest.