Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

1.   Introduction

Research work in the area of fractional differential equation is multidisciplinary such as control systems, elasticity, circuit systems, heat transfer, fluid mechanics, signal analysis, traffic flow, pollution control, etc. It is considered as an alternative model to nonlinear differential equation. Fractional differential equations are a useful tool in modelling several events. In ^[1], controllability of Hilfer fractional neutral differential systems with infinite delay is obtained. An article obtained on Neutral fractional stochastic partial differential equations with Clarke subdifferential is investigated using fractional calculus and fixed point theorems in ^[2]. Nisar et al. ^[3], in their publication, briefly discussed the analysis of controllability for nonlinear Hilfer neutral fractional derivatives via fractional calculus and Banach contraction principle. Numerous credible theoretical studies on fractional differential equation can be referred to books in ^[4,5,6,7,8] and the research articles are ^{[9,10,11,12,13,14,15]}. The fractional differential equation has many solutions with nonlocal conditions, impulsive type and sobolev type. In which, the nonlocal conditions is a generalization of the classical conditions, was motivated by the physical phenomena. The pioneering work on nonlocal condition is due to Byszewski ^[16]. Papers related on nonlocal conditions, we may refer ^{[17,18,19,20]}.

Moreover, the theory of fractional impulsive differential equations has been entirely developed during the past decades. Since 1990's many mathematician have derived lots of results on differential equations undergoing impulsive effects. It focuses on the analysis of dynamic processes that experience abrupt changes. Impulses have a relatively shorter time difference between changes than the overall length of the process. The following may act as motivation to investigate such systems using impulsive differential equations. Consider the simplest scenario for a person's hemodynamic equilibrium. Some injectable medications, such as insulin, may be provided in the case of a de-compensation, such as high or low glucose levels. It is clear that the entrance of medications into the circulation and the body's subsequent absorption are slow and ongoing processes. This circumstance might be seen as an impulsive activity that begins suddenly and lasts for a set amount of time. For detailed information about the impulsive fractional differential equation and its applications, we refer to the readers ^{[21,22,23,24,25]}.

On the other hand, the Sobolev type differential system is typically seen in the mathematical structure of numerous physical events, such as fluid flow through fractured rocks, thermodynamics. Additionally, Sobolev type differential equations are utilised to describe the attributes of systems and processes in mathematical modelling and simulations. For more literature on sobolev type differential equation, see ^{[26,27,28,29,30]} and references therein. In addition, one of the most effective methods for determining out approximations of solutions to a given differential equation in an abstract space is the Faedo-Galerkin approach. The Faedo-Galerkin method may be used within a vartional formulation in order to provide solutions of the equations under possibly weaker assumption on the data and may also prove a very useful tool for numerical approximation of equations. A detailed view on Faedo-Galerkin approximation we refer ^{[31,32,33,34,35]}.

In accordance with the aforementioned literature survey, there are relatively few work outcomes that explore the existence and uniqueness of a mild solution to a Sobolev type fractional differential equation with Impulses applying a fixed point technique. This fact is the fundamental motivator behind our current progress. This article ^[36,37], outlines the nonlocal Sobolev type fractional differential impulsive system as follows:

Where $0 < \beta < 1, \; T\in(0, \infty)$ $^{c}D_{\sigma}^{\beta}$ is the Caputo fractional derivative, $0 = \sigma_{0} < \sigma_{1} < ... < \sigma_{q} < \sigma_{q+1} = T$ are pre-fixed numbers, $\Delta x|_{\sigma = \sigma_{i}} = x(\sigma_{i}^{+})-x(\sigma_{i}^{-})$ and $x(\sigma_{i}^{+}) = \lim_{h\rightarrow 0+}x(\sigma_{i}+h)$ and $x(\sigma_{i}^{-}) = \lim_{h\rightarrow 0-}x(\sigma_{i}+h)$ denote the right and left limits of $x(\sigma)$ at $\sigma = \sigma_{i}$, respectively. From (1.1), assume $\mathcal{L}:D(\mathcal{L})\subset\mathbb{H}_{1}\rightarrow \mathbb{H}_{2}$ $\mathcal{M}:D(\mathcal{M})\subset\mathbb{H}_{1}\rightarrow \mathbb{H}_{2}$ are closed (unbounded), positive and self-adjoint operators, where $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ are Hilbert spaces and the appropriate functions are $\mathcal{F}:[0, T]\times \mathbb{H}_{1}\rightarrow \mathbb{H}_{1}$ and $\mathrm{g}:C([0, T], \mathbb{H}_{1})\rightarrow \mathbb{H}_{1}$, $\mathrm{h} : [0, T]\rightarrow [0, T]$, $I_{i}:\mathbb{H}_{1}\rightarrow \mathbb{H}_{1}$.

This articles is organized as:

Basic concepts and lemmas are covered in Section 2. In Section 3, the fixed point theorem is used to determine the existence and uniqueness of an approximate solution. In Section 4, the convergence of the approximate solutions is obtained. In Section 5, the convergence of approximate Faedo-Galerkin solutions is proved. Finally, we provide a theoretical application to assist in the effectiveness of our result.

2.   Basic results

The upcoming segment recalls the necessary things to obtain the primary facts of our discussion.

Let $(\mathbb{H}_{1}, \|\cdot\|, < \cdot, \cdot > )$, $(\mathbb{H}_{2}, \|\cdot\|, < \cdot, \cdot > )$ be Separable Hilbert spaces. Assume $C([0, T], \mathbb{H}_{1})$ from $[0, T]$ into $\mathbb{H}_{1}$ with $\|x\|_{[0, T]}: = \sup \{\|x(\sigma)\|:\sigma \in [0, T]\}$ be a Banach space of continuous function and boundedness of linear operator $L(\mathbb{H}_{1})$ equipped with $\|\mathrm{f}\|_{L(\mathbb{H}_{1})} = \sup \{\|\mathrm{f}(x)\|:\|x\| = 1\}$.

Definition 2.1. ^[37] The R-L integral of order $\beta > 0$ is

where $\mathcal{F}\in L^{1}((0, T), \mathbb{H}_{1})$.

Definition 2.2. ^[37] The R-L derivative is

where $D^{\delta}_{\sigma} = \frac{d^{\delta}}{d\sigma^{\delta}}$, $\mathcal{F}\in L^{1}((0, T), \mathbb{H}_{1})$, $\mathcal{J}^{\delta-\beta}_{\sigma} \mathcal{F}\in \; W^{\delta, 1}((0, T), \mathbb{H}_{1})$.

Definition 2.3. ^[37] The Caputo derivative is

where $\mathcal{F}\in C^{\delta-1}((0, T), \mathbb{H}_{1})\bigcap L^{1} ((0, T), \mathbb{H}_{1})$,

Operators $\mathcal{L}$, $\mathcal{M}$ impose the following conditions:

(a) $\mathcal{L}$ and $\mathcal{M}$ are closed linear operators;

(b) $D(\mathcal{M})\subset D(\mathcal{L})$ and $\mathcal{M}$ is bijective;

(c) $\mathcal{M}^{-1}:\mathbb{H}_{2}\rightarrow D(\mathcal{M})\subset\mathbb{H}_{1}$ is compact.

Conditions (a)-(c) and closed graph theorem imply $\mathcal{L}\mathcal{M}^{-1}:\mathbb{H}_{2}\rightarrow \mathbb{H}_{2}$ is the boundedness of the linear operator. Therefore, an infinitesimal generator $\mathcal{E} = \mathcal{L}\mathcal{M}^{-1}$ of semigroup $\mathcal{S}(\sigma): = e^{\mathcal{L}\mathcal{M}^{-1}}$ and so $\max_{\sigma\in I}\|\mathcal{S}(\sigma)\|$ is finite. We have the following integral as per prior definition,

The above (2.4) exists a.e. As a result, aforementioned equalization is equal to the impulsive differential equation of Sobolev type. Therefore, there exists $N_{0}\geq 1$ a positive constant such that $\|\mathcal{S}(\sigma)\|\leq N_{0}$. Let the resolvent set of $\mathcal{E}$ is $\rho(\mathcal{E})$. Hence, $\mathcal{E}^{\alpha}$, $0 < \alpha \leq 1$ be the fractional power which is a closed linear operator and $D(\mathcal{E}^{\alpha})$ is a subspace, in such a way its simple to show that it is a Banach space with supremum norm and is represented as $(\mathbb{H}_{1})_{\alpha}$ with $(\|\cdot\|_{\alpha})$. We have $(\mathbb{H}_{1})_{\eta}\hookrightarrow(\mathbb{H}_{1})_{\alpha}, 0 < \alpha < \eta$ so the embedding is continuous. Then, we define $(\mathbb{H}_{1})_{-\alpha} = ((\mathbb{H}_{1})_{\alpha})^{\ast}, \alpha > 0$, dual space of $(\mathbb{H}_{1})_{\alpha}$, is a Banach space equipped with $\|x\|_{-\alpha} = \|\mathcal{E}^{-\alpha}x\|, x\in (\mathbb{H}_{1})_{-\alpha}$.

Proposition 2.1. ^[38] Assume $\mathcal{E}$ of $\mathcal{S}(\sigma)$, $\sigma\geq 0$, $0\in \rho(\mathcal{E})$ is an infinitesimal generator. We get

(ⅰ) For $\sigma > 0$, $\alpha \geq 0$, $\mathcal{S}(\sigma) \; \mathit{{\rm{maps from}}}\; \mathbb{H}_{1}\rightarrow D(\mathcal{E}^{\alpha})$.

(ⅱ) For each $x \in D(\mathcal{E}^{\alpha})$, $\mathcal{S}(\sigma) \mathcal{E}^{\alpha}x = \mathcal{E}^{\alpha}\mathcal{S}(\sigma)x$.

(ⅲ) Let $\bigg\|\frac{d^{j}}{d\sigma^{j}}\mathcal{S}(\sigma)\bigg\| \leq N_{j}, \; j = 1, 2, \; \sigma > 0,$ where $N_{j}$ is a positive constant.

(ⅳ) A bounded operator $\mathcal{E}^{\alpha}\mathcal{S}(\sigma)$, $\|\mathcal{E}^{\alpha}\mathcal{S}(\sigma)\|\leq N_{\alpha}\sigma^{-\alpha}e^{-\delta \sigma}$, $\sigma > 0$.

(ⅴ) If $x\in D(\mathcal{E}^{\alpha})$, $\alpha \in (0, 1]$ implies $\|\mathcal{S}(\sigma)x-x\|\leq C_{\alpha}\sigma^{\alpha}\|\mathcal{E}^{\alpha}x\|$.

Remark 2.1. ^[38] The boundedness of the linear operator $\mathcal{E}^{-\alpha}$ in $\mathbb{H}_{1}$ such that $D(\mathcal{E}^{\alpha}) = \operatorname{Im}(\mathcal{E}^{-\alpha}).$ Let's denote $(\mathbb{H}_{1})_{\alpha}(T) = C([0, T], (\mathbb{H}_{1})_{\alpha})$ be Banach space of all $(\mathbb{H}_{1})_{\alpha}$-valued continuous function equipped with $\|x\|_{(\mathbb{H}_{1})_{\alpha}(T)} = \sup_{\sigma\in[0, T]}\|x(\sigma)\|_{\alpha}$, such that $x(\sigma)$ is continuous on $\sigma\neq\sigma_{i}$, left continuous at $\sigma = \sigma_{i}$ and right limit $x(\sigma_{i}^{+})$ exists for $i = 1, 2, ..., q.$

3.   Existence of solutions

We inspect the existence of (1.1)–(1.3) as well as their uniqueness. The respective assumptions on $\mathcal{E}, \mathcal{F}, h, I_{i} (i = 1, 2, ..., q)$ is presented as:

$(1)$Let $\mathcal{E}$ be closed, positive definite and self adjoint linear operator $: D(\mathcal{E})\subset\mathbb{H}_{2} \rightarrow \mathbb{H}_{2}$. A pure point spectrum $\mathcal{E}$ has

with $\lambda_{m}\rightarrow \infty$, $m\rightarrow \infty$ and complete orthonormal system $\{\phi_{j}\}$,

where

$(2)$ The continuous mapping $\mathcal{F}:[0, \infty)\times(\mathbb{H}_{1})_{\alpha}\times(\mathbb{H}_{1})_{\alpha}\rightarrow \mathbb{H}_{2}$ and $m_{R}:[0, \infty)\rightarrow (0, \infty)$ an increasing function exists, on $R > 0$ such that

for all $(\sigma, z, w), (\sigma_{1}, z_{1}, w_{1}), (\sigma_{2}, z_{2}, w_{2}) \in [0, \infty)\times\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha})\times \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha})$ where $\mathbb{B}_{R}(\mathbb{H}_{1}) = \{z\in\mathbb{H}_{1}:\|z\|_{\mathbb{H}_{1}}\leq R\}$ and $\theta_{1}\in(0, 1)$.

$(3)$ Let nonlinear function $h:[0, T]\rightarrow [0, T]$ such that $h(\sigma)\leq \sigma$, $0 \leq \sigma\leq T$ and $\exists$ constants $L_{h} > 0$ such that

$(4)$ There exist $\chi\in C([0, T], \; (\mathbb{H}_{1})_{\alpha})$ such that $\mathrm{g}(\chi) = \phi$ and $\chi(\sigma)$ is locally Lipschitz continuous.

$(5)$ All the function $I_{i}:\mathbb{H}_{1}\rightarrow \mathbb{H}_{1}$ $(i = 1, 2, ..., q)$ are continuous function such that

where $\mathcal{O}_{i}, \mathcal{N}_{i},$ $i = 1, 2, ..., q$ are positive constants.

Definition 3.1. ^[39] Let $x:[0, T]\rightarrow (\mathbb{H}_{1})_{\alpha}$ be a continuous function, if $x(0) = x_{0}$ and $x(\cdot)$ satisfies the following integral equation

is known as mild solution of (1.1)–(1.3), where

and PDF $\zeta_{\beta}(\xi)$. i.e., $\zeta_{\beta}(\xi)\geq 0$, $\int\limits^{\infty}_{0}\zeta_{\beta}(\xi)d\xi = 1$.

Remark 3.1. ^[38] Let $0\leq v\leq 1$,

Proposition 3.1. ^[26] Let $\mathcal{S}(\sigma)$ be a uniformly continuous semigroup and $\mathcal{E}$ be its infinitesimal generator. Then, $\mathcal{S}_{\beta}(\sigma)$ and $\mathcal{T}_{\beta}(\sigma)$ are boundedness of the linear operator such that

(ⅰ) $\|\mathcal{S}_{\beta}(\sigma)x\|\leq W_{1}N_{0}\|x\|$ and $\|\mathcal{T}_{\beta}(\sigma)x\|\leq\frac{W_{1}N_{0}}{\Gamma(\beta)}\|x\|$, $x\in \mathbb{H}_{1}$.

(ⅱ) The strong continuity of $\{\mathcal{T}_{\beta}(\sigma), \sigma\geq 0\}$ and $\{\mathcal{S}_{\beta}(\sigma), \sigma\geq 0\}$ $0\leq\tau_{1} < \tau_{2}\leq T$, for $x\in\mathbb{H}_{1}$, we have $\|\mathcal{T}_{\beta}(\tau_{2})x-\mathcal{T}_{\beta}(\tau_{1})x\|\rightarrow 0$ and $\|\mathcal{S}_{\beta}(\tau_{2})x-\mathcal{S}_{\beta}(\tau_{1})x\|\rightarrow 0$ as $\tau_{2}\rightarrow \tau_{1}$.

(ⅲ) Suppose $\mathcal{S}(\sigma)$, $\sigma\geq 0$ is compact, then $\mathcal{S}_{\beta}(\sigma)$ and $\mathcal{T}_{\beta}(\sigma)$ are compact operators.

(ⅳ) For each $x\in\mathbb{H}_{1}$, we have $\mathcal{E}\mathcal{T}_{\beta}(\sigma)x = \mathcal{E}^{1-\eta}\mathcal{T}_{\beta}\mathcal{E}^{\eta}x$, $\sigma\in [0, T], \; \eta \in(0, 1).$ We have $\|\mathcal{E}^{\alpha}\mathcal{T}_{\beta}(\sigma)\|\leq\frac{\beta W_{1}N_{\alpha}\Gamma(2-\alpha)}{\Gamma(1+\beta(1-\alpha))}\sigma^{-\alpha \beta}$, $\sigma\in[0, T], \; \alpha\in (0, 1)$.

(ⅴ) For any $x\in\mathbb{X}_{\alpha}$ and fixed $\sigma\geq 0$, we have $\|\mathcal{S}_{\beta}(\sigma)x\|_{\alpha}\leq W_{1}N_{0}\|x\|_{\alpha}$ and $\|\mathcal{T}_{\beta}(\sigma)x\|_{\alpha}\leq \frac{W_{1}N_{0}}{\Gamma(\beta)}\|x\|_{\alpha}$.

Arbitrarily fixed point $T_{0} > 0$ such that $0 < T < T_{0} < \infty$,

Let $\mathcal{H}_{n}$ be finite dimensional subspace, spanned by $\{\phi_{0}, \phi_{1}, ..., \phi_{n}\}$ and a projection operator $P^{n}:\mathbb{H}_{1}\rightarrow \mathcal{H}_{n}$, $n = 0, 1, ...$. Assume $\mathcal{F}_{n}:[0, T]\times (\mathbb{H}_{1})_{\alpha}\rightarrow \mathbb{H}_{1}$ and $I_{i, n}:\mathbb{H}_{1}\rightarrow \mathbb{H}_{1}$, is defined by

and the operator $\mathbb{Q}_{n}$ on $\mathbb{B}$ as follows

Theorem 3.1. Assume (1)–(5) holds, then $x_{n}\in\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ be a unique fixed point of $\mathbb{Q}_{n}$ exists i.e., $\mathbb{Q}_{n}x_{n} = x_{n}$ for each $n = 0, 1, 2, ...$ and $x_{n}$ fulfills the approximate integral equation,

Proof. Let $\mathbb{Q}_{n}:\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))\rightarrow \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ is defined by

We will demonstrate that $\mathbb{Q}_{n}$ is well defined. This is sufficient to demonstrate that the map $\sigma \longmapsto (\mathbb{Q}_{n}x)(\sigma): [0, T]$ into $(\mathbb{H}_{1})_{\alpha}$ w.r.t. $\alpha$ norm is continuous.

Let $\sigma_{1}, \sigma_{2}\in [0, T]$ with $\sigma_{2} > \sigma_{1},$ we get

For $x\in\mathbb{H}_{1},$ we have,

Thus, we get

where $R_{1} = N_{1}W_{1}\|\mathcal{M}\|\; \|\chi(0)\|_{\alpha}$.

From (3.11), we have

where $h = [1-(\frac{v}{p_{1}})^{\frac{1}{v p_{1}}}]$, $p_{1} = 1-\beta\alpha$, $v = \frac{(1-\beta)}{1-\beta\alpha}$ and $0 < d_{1}\leq 1.$

where $h_{1} = (1-(\frac{\alpha}{\beta})^{\frac{1}{\alpha\beta}}), \; 0 < b_{1}\leq1$ and $N_{1+\alpha}$ is some positive constant with $\|\mathcal{E}^{\alpha+1}\mathcal{S}(\sigma)\|$$\leq N_{1+\alpha}\sigma^{-1-\alpha}$, $\forall$ $\sigma > 0.$ Thus, from the inequalities (3.11)–(3.14), (2).

We conclude that $\sigma\longmapsto\mathcal{F}_{n}(\sigma, x(\sigma))$ map is uniformly $H\ddot{o}lder's$ continuous. We justify $\mathbb{Q}_{n}(\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T)))\subseteq \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$. Let $x\in \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$, $0\leq \sigma \leq T$. We get

We may now take $R$ as a positive integer such that,

Therefore, we deduce that $\mathbb{Q}_{n}(\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T)))\subseteq \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$. Lastly, we demonstrate $\mathbb{Q}_{n}$ is a strict contraction map. For $x_{1}, x_{2}\in \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$, $0\leq \sigma\leq T$.

In Eq (3.16), $\wedge = \frac{ W_{1}N_{\alpha}\Gamma(2-\alpha)}{\Gamma(1+\beta(1-\alpha))}m_{R}(T_{0})\frac{T^{\beta(1-\alpha)}}{(1-\alpha)}+ W_{1}N_{\alpha}\sum_{i = 1}^{q}\mathcal{N}_{i} < 1.$

As a result, $\mathbb{Q}_{n}$ is determined to be a contraction mapping. Thus, a unique $x_{n}\in \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ exists such that $\mathbb{Q}_{n}x_{n} = x_{n}$.

Lemma 3.1. Assume (1)–(5) holds.

(ⅰ) Let $\chi(0)\in D(\mathcal{E}^{\alpha}), \; \alpha\in(0, 1)$ implies $x_{n}(\sigma)\in D(\mathcal{E}^{\upsilon})\; \forall\; 0 < \sigma \leq T, \upsilon\in [0, 1)$.

(ⅱ) If $\chi(0)\in D(\mathcal{E})$, implies $x_{n}(\sigma)\in D(\mathcal{E}^{\upsilon}) \; \forall\; 0\leq \sigma \leq T, \upsilon\in [0, 1)$.

Proof. We get a unique $x_{n}\in \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ that satisfies (3.10) by using the previous theorem. In ^[38], for $\sigma > 0, \; 0\leq \upsilon < 1$ we get $\mathcal{S}(\sigma):\mathbb{H}_{1}\rightarrow D(\mathcal{E}^{\upsilon})$, $D(\mathcal{M}^{\alpha})\subseteq D(\mathcal{M}^{\upsilon})$. Also $\mathcal{S}(\sigma)x\in D(\mathcal{E}).$ As a outcome, of all these facts, $D(\mathcal{E})\subseteq D(\mathcal{E}^{\upsilon})$, $0\leq\upsilon \leq 1$.

Lemma 3.2. If (1)–(5) holds.

(ⅰ) Let $\chi(0)\in D(\mathcal{E}^{\alpha})$, $\alpha\in(0, 1)$, $0 < \sigma_{0}\leq T$, then a constant $S_{\sigma_{0}}$ exist,

(ⅱ) Let $\chi(0)\in D(\mathcal{E})$, then a constant $S_{0} > 0$ exists,

Proof. Let $\chi(0)\in D(\mathcal{A}^{\alpha})$. In (3.10), we apply $\mathcal{E}^{\upsilon}$ on both sides, we have

Again, if $\chi(0)\in D(\mathcal{E}) \Rightarrow \chi(0)\in D(\mathcal{E}^{\upsilon})$, $0\leq\upsilon\leq 1$ and we get

4.   Convergence of solutions

Now, to investigate the convergence of solution $x_{n}(\sigma)$ of the approximate integral Eq (3.10) to a unique solution $x(\cdot)$ of the Eq (3.5).

Theorem 4.1. Suppose (1)–(5) holds, $\chi(0)\in D(\mathcal{A}^{\alpha})$, $\alpha\in(0, 1).$ Then,

Proof. Let $n\geq m$, we have

For $0 < \alpha < \upsilon < 1,$ we get

Thus, from (4.2), (4.3) we obtain

Similarly, we estimate

We choose $\sigma_{0}^{\prime}$ such that $0 < \sigma_{0}^{\prime} < \sigma_{0} < T$,

$1^{st}$ integral of inequality (4.4),

$2^{nd}$ integral of (4.4) is evaluated as

Thus, from (4.4)–(4.6) we conclude

where

Since $1-W_{1}N_{\alpha}\sum_{i = 1}^{q}\mathcal{N}_{i} > 0,$ we have

By lemma 5.6.7 in ^[38], we have that there exist a constant $\mathcal{K}$ such that

Taking supremum over $[\sigma_{0}, T]$ and let $m\rightarrow \infty$, we obtain

Because $\sigma_{0}^{\prime}$ is arbitrary, the right side of the expression could be made as tiny as required simply reducing $\sigma_{0}^{\prime}$.

Proposition 4.1. Assume $\chi(0)\in D(\mathcal{E})$. Then

Theorem 4.2. Assume (1)–(5) holds, $\chi(0)\in D(\mathcal{E}^{\alpha})$. Then, $\exists$ $x_{n}(\sigma)\in (\mathbb{H}_{1})_{\alpha}(T)$ a unique function satisfying,

and $x\in (\mathbb{H}_{1})_{\alpha}(T)$ satisfying

such that $x_{n}$ converges to $x$ in $(\mathbb{H}_{1})_{\alpha}(T)$ i.e., $x_{n}\rightarrow x$ as $n\rightarrow \infty$.

Proof. Suppose $\chi(0)\in D(\mathcal{E})$. From previous preposition, there is a $x\in(\mathbb{H}_{1})_{\alpha}(T)$ such that $\lim_{n\rightarrow \infty}x_{n}(\sigma) = x(\sigma)$. Since $x_{n}\in\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ $\forall$ $n$, we get $x\in\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ $\forall$ $\sigma_{0}\in(0, T]$,

Taking supremum over $[0, T]$, we have

Thus, we get

Now, let $\chi(0)\in D(\mathcal{E}^{\alpha})$. Since $\mathcal{E}^{\alpha}x_{n}(\sigma)$ converges to $\mathcal{E}^{\alpha}x(\sigma)$ for each $\sigma\in(0, T]$ and $x_{n}(0) = x(0) = \chi(0)$. Then, $\mathcal{E}^{\alpha}x_{n}(\sigma)$ converges to $\mathcal{E}^{\alpha}x(\sigma)$ in $\mathbb{H}_{1}$. Furthermore, we have that $x_{n}\in \mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$ for each $n$. Also $x\in\mathbb{B}_{R}((\mathbb{H}_{1})_{\alpha}(T))$. From previous theorem, we get

Also, we have

Therefore, $0 < \sigma_{0} < \sigma$, Eq (3.10) can be reframed as

we estimate the $1^{st}$ integral of (4.9) as

Thus, we deduce that

Letting $n\rightarrow \infty$ and getting

Since, $\sigma_{0}$ is arbitrary, we deduce $x(\sigma)$ satisfies the integral Eq (3.5). Now, we shall show the uniqueness. Let $x_{1}$ and $x_{2}$ be the two solutions of integral Eq (3.5). Thus, we have

Taking supremum on $[0, \sigma]$ and obtaining

From Gronwall's inequality and the fact that

We deduce that $x_{1} = x_{2}$ on $[0, T]$.

5.   Faedo-Galerkin approximation

Additionally, the convergence findings were accomplished using the Faedo-Galerkin approximation technique.

There is a unique $x\in(\mathbb{H}_{1})_{\alpha}(T)$, $T\in(0, T_{0})$, that fulfils the integral equation,

An approximate integral equation has an unique solution $x_{n}\in(\mathbb{H}_{1})_{\alpha}(T)$,

The Faedo-Galerkin Approximation is produced by applying the projection on (5.2) as $v_{n}(\sigma) = P^{n}x_{n}(\sigma)$,

Let solution $x(\cdot)$ of (5.1) and $v_{n}(\cdot)$ of (5.3) have the following representation:

Using (5.5) in (5.3) and taking inner product with $\phi_{i}$, we obtain a system of fractional order integro-differential equation of the form.

Where,

We also have the following convergence theorem.

Theorem 5.1. If the hypothesis (1)–(5) holds. The results follows:

(ⅰ) If $x_{0}\in D(\mathcal{E}^{\alpha})$, then for any $\sigma_{0}\in(0, T]$

(ⅱ) If $x_{0}\in D(\mathcal{E})$, then

Proof. If $n\geq m, \; 0\leq\alpha < \upsilon$. Then, we have

By the Theorem 4.1 and Proposition 4.1, we have that $x_{n}\rightarrow x_{m}$ and $\lambda_{m}\rightarrow \infty$ as $m\rightarrow \infty$.

Theorem 5.2. Suppose (1)–(5) is fulfilled, $x_{0}\in D(\mathcal{E}^{\alpha})$, unique function $v_{n}\in(\mathbb{H}_{1})_{\alpha}(T)$ exists, satisfying the following equation:

Proof. For $x_{0}\in D(\mathcal{E}^{\alpha})$ and $\sigma\in[0, T]$. We have

We have $v_{n}\rightarrow x$ as $n\rightarrow \infty$ according to the Theorem 4.2. As a result, the Theorem 4.2 leads to the conclusion. The preceding theorem can be used to show $\alpha_{i}^{n}$ to $\alpha_{i}$'s convergence.

Theorem 5.3. Suppose (1)–(5) holds. Then,

(ⅰ) If $x_{0}\in D(\mathcal{E}^{\alpha})$, then for any $0 < \sigma_{0}\leq T$

(ⅱ) If $x_{0}\in D(\mathcal{E})$, then

Proof. The system (*) determines the $\alpha_{i}^{n}$'s. It can be easily investigated that

Thus,

As a result, the Theorems 5.1 and 5.2 lead to the conclusion.

6.   Applications

Let the fractional impulsive differential system of sobolev type is of the form:

Where $^{c}D_{\sigma}^{\beta}$ is Caputo derivative, $\beta\in(0, 1)$. Suppose $w(\sigma)(x) = w(\sigma, x)$ and $f(\sigma, \cdot) = \mathcal{F}(\sigma, \cdot)$. Let $\Delta w(\sigma_{i}, x) = w(\sigma_{i}^{+}, x)-w(\sigma_{i}^{-}, x),$ where $w(\sigma_{i}^{+}, x)$ and $w(\sigma_{i}^{-}, x)$ are respectively the right and the left hand limit of $w$ at $(\sigma, x) = (\sigma_{i}, x).$

Now, we take $\mathbb{H}_{1} = \mathbb{H}_{2} = L^{2}(0, \pi)$ and consider the operator $\mathcal{L}, \mathcal{M}$ on domains and ranges contained in $L^{2}(0, \pi)$ defined by

An infinitesimal generator of an analytic semigroup is denoted by $\mathcal{E} = \mathcal{L}\mathcal{M}^{-1}$, such that

with the domain

If we take $\beta = \frac{1}{2}$, then $D(\mathcal{E}^{\frac{1}{2}})$ which is denoted by $\beta_{\frac{1}{2}}$ is the Banach space endowed with the norm,

Also, for $\sigma\in[0, T]$. we define $D^{\frac{1}{2}}_{\sigma} = \{y:y$ is a map from $[0, T]$ into $\beta_{\frac{1}{2}} \ni x(\sigma)$ is continuous at $\sigma\neq\sigma_{i}$ left continuous at $\sigma = \sigma_{i}$ and right limit $x(\sigma_{i}^{+})$ exists for $i = 1, 2, ..., q\}$.

The spectrum of $\mathcal{E}$ is given by $\mathcal{E}y = -y^{\prime\prime} = \alpha y.$ The general solution $y$ of $\mathcal{E}y = \alpha y$ is

Using the boundary conditions $y(0) = y(\pi) = 0$. we obtain $C = 0, \; \alpha = \alpha_{n} = n^{2}, \; n\in\mathbb{N}.$ Thus for each $n \in \mathbb{N}$, the solution is given by $y_{n}(x) = D\sin nx.$ If we take $D = \frac{\sqrt{2}}{\sqrt{\pi}},$ then $< y_{n}, y_{m} > = 0, \; n\neq m$ and $< y_{n}, y_{m} > = 1, \; n = m$. Thus $\mathcal{E}$ has pure point spectrum and eigenvalues $y_{n}$ are orthonormal.

Suppose, $I_{i}(w(\sigma_{i}, x)) = \frac{2w(\sigma_{i}, x)}{2+w(\sigma_{i}, x)}.$ Let us define $y(\sigma)(x) = w(\sigma, x)$ and $I_{i}(w(\sigma_{i}, x)) = I_{i}(x(\sigma_{i}))(y)$ then $I_{i}(y(\sigma_{i})) = \frac{2y(\sigma_{i})}{2+y(\sigma_{i})}.$ For $y_{1}, y_{2}\in D(\mathcal{A}^{\frac{1}{2}})$, we have

Now, we define

then problem (6.1)–(6.4) reduces to

It is easy to see that the operator $\mathcal{E}$ fulfils (1). Also, by $H\ddot{o}lder$ continuity of $h$, fulfils (3). Then, from the definitions, it can be easily shown that $\chi$ and $I_{i}$ are satisfies (4) and (5).

Now we prove that $\mathcal{F}$ satisfies the condition (2):

Hence (2) holds.

Thus, all the assumptions of Theorem 5.2 are satisfied. So, Theorem 5.2 guarantees the existence of Faedo-Galerkin approximations and their convergence to the unique solution of (6.1)–(6.4).

7.   Conclusions

The Faedo-Galerkin approximation outcomes for Caputo fractional impulsive derivative of Sobolev type with nonlocal condition are the main subject of this paper. The major ideas were developed by utilising the analytic semigroup and the Banach fixed point theorem. Finally, we give examples to back up our abstract conclusion. In future, it might be used to find the generalization in fractional differential equations.

Acknowledgments

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (Grant number B05F650018).

Conflict of interest

The authors declare no conflicts of interest.