Results on Atangana-Baleanu fractional semilinear neutral delay integro-differential systems in Banach space

1 Introduction

Fractional calculus has procured expansive importance during the past long time since the fractional derivative gives an imminent execution to the depiction of the memory and characteristic properties of different strategies. Lately, researchers center around fractional derivatives, especially when a couple of utilizations in biology, financial aspects, science, and engineering have displayed up (Francesco, 2010; Mainardi, 1996; Williams et al., 2020; Bedi et al., 2021; Bedi et al., 2020; Devi et al., 2021; Bedi et al., 2019; Bedi et al., 2020; Bedi et al., 2021). For Researchers, fractional derivatives have recently become a more interesting area, particularly since numerous possible methods emerged in biology, economics, science, and engineering. Fractional derivative definitions were offered in both local and nonlocal forms. Nonlocal derivatives are more interesting because the majority of these applications are dependent on the function’s history.

Furthermore, integrodifferential equations are employed in a range of scientific domains where an aftereffect or delay must be taken into account, such as biology, control theory, ecology, and medicine. In general, integrodifferential equations are always employed to represent a model with hereditary characteristics, as the researcher’s works (Mohan Raja et al., 2020; Dineshkumar et al., 2021; Kavitha et al., 2021) demonstrate. Neutral systems arise in a wide range of applied mathematics domains, including electronics, fluid dynamics, biological models, and chemical kinetics, and as a result, this type of equation has received a lot of attention in recent years, one can refer (Bedi et al., 2020; Bedi et al., 2021; Kavitha et al., 2021; Mallika Arjunan et al., 2021).

Fractional differential equations (FDEs) in several physical phenomena are difficult to handle via singular kernels. Subsequently, fractional derivatives were created involving non-singular kernels. Hence a new fractional derivative was proposed by Caputo and Fabrizio having exponential kernel in 2015. Caputo’s and Riemann–Liouville’s fractional derivative is the most well-known amongst the distinct fractional order differential operators with a singular kernel. To counter this, a new derivative was formulated by Atangana and Baleanu through the generalization of Mittag–Leffler function involving a non-singular kernel (Atangana and Baleanu, 2016) because non-singular kernel models can depict actuality in explicit ways when contrasted with the standard fractional calculus involving singular kernel, such as the Keller-Segel model (Atangana and Alqahtani, 2018). Refer (Atangana and Koca, 2016; Owolabi and Atangana, 2019; Ravichandran et al., 2019; Saad et al., 2018; Saad et al., 2018; Kumar and Pandey, 2020; Baleanu and Fernandez, 2018; Fernandez et al., 2019) for a list of Atangana-Baleanu derivative applications in several fields. Atangana and Baleanu (2016) proposed the Atangana-Baleanu (AB) fractional derivative in both the Riemann–Liouville and Caputo senses in recent years. This derivative includes the generalised Mittag–Leffler function as a kernel. The nonlocal behaviour of the generalised Mittag–Leffler function allows for a more realistic explanation of the macroscopic behaviour and memory effects of systems with non-local exchanges. Authors developed a new strategy for calculating the global conduct of difference equations with delay of threshold dynamics of difference equations for the SEIR model lately in Bentout et al. (2021). Bentout et al., 2021; Bentout et al., 2021; Djilali and Bentout, 2021; Djilali and Ghanbari, 2020; Khan et al., 2021; Zeb et al., 2021 for more information.

Furthermore, as shown in (Atangana and Koca, 2016; Mallika Arjunan et al., 2021; Owolabi and Atangana, 2019; Balasubramaniam, 2021), the Atangana-Baleanu (AB) fractional derivative retains all of the properties of previously known fractional derivatives. In a recent paper (Ravichandran et al., 2019), the authors used a fixed point approach to investigate the existence of AB fractional integro-differential and neutral systems. The authors of Mallika Arjunan et al. (2021) employed the fixed point approach given in Ravichandran et al. (2019) to show that the Atangana-Baleanu fractional neutral integro-differential and Volterra systems with or without delay exist. Motivated by these papers, the authors of Williams and Vijayakumar (2021) utilised fractional calculus, non-instantaneous impulses, the integro-differential equation, and the Darbo fixed point approach to cover the controllability and existence outcomes for fractional neutral impulsive Atangana-Baleanu integro-differential systems with delay. Recently, in Aimene et al. (2019), authors investigated the controllability of Atangana-Baleanu semilinear differential equations of fractional order with impulses and delay through semigroup theory and Darbo fixed point theorem along with measures of non-compactness. One can also refer (Mallika Arjunan et al., 2021; Williams and Vijayakumar, 2021) for the result of Atangana-Baleanu with delay.

For analysing nonlinear system existence of mild solutions, the fixed point technique can be regarded a useful and valuable tool. The fixed point technique appears to be appropriate for the solution of many problems in existence of solutions, since it is constructive and incorporates a convergence theory. The fixed point approach has yet to be widely used to stochastic impulsive control systems, despite its widespread use in both theory and numerical aspects of differential equations. Using this method, the problem is turned into a fixed point problem in a function space for an appropriate nonlinear operator. This technique relies heavily on the existence of a fixed point for the appropriate operator. The fixed point approach is the most successful method for examining the existence and controllability of differential systems with integer and fractional orders. Because of its usefulness, a number of academics have used various sorts of fixed point theorems to investigate the issues provided by evolution equations. The Mönch fixed point theorem is used to study the existence of mild solutions for Atangana-Baleanu semilinear neutral fractional integro-differential equations with finite delay.

Roused by the works above, we think about the accompanying issue of fractional semilinear differential equations in Banach space of the type

2 Preliminaries

3 Main Results

Now, let us look into the existence of (1). Suppose $\widehat{\mathcal{U}}\in U^{\zeta }\left({\alpha }_0,l_0\right)$ then $\Vert P_{\zeta }\left(\delta \right)\Vert ⩽\mathcal{T}{e}^{l\delta }$ and $\Vert Q_{\zeta }\left(\delta \right)\Vert ⩽\mathcal{C}{e}^{l\delta }(1+{\delta }^{\zeta -1}),\forall$ $\delta >0,l>l_0$ . Thus, $\widehat{\mathcal{T}}={\sup }_{\delta ⩾0}\Vert P_{\zeta }\left(\delta \right)\Vert ,{\widehat{\mathcal{T}}}_1={\sup }_{\delta ⩾0}\mathcal{C}{e}^{l\delta }(1+{\delta }^{\zeta -1})$ . So we get $\Vert P_{\zeta }\left(\delta \right)\Vert ⩽\widehat{\mathcal{T}}$ and $\Vert Q_{\zeta }\left(\delta \right)\Vert ⩽{\delta }^{\zeta -1}{\widehat{\mathcal{T}}}_1$ . One can also refer (Shu et al., 2011).

Now we assume the following assumptions.

• $\left({\mathbf{H}}_{\mathbf{1}}\right)$

  ${\mathcal{S}}_2:\mathcal{J}\times \mathcal{U}\times \mathcal{X}\to \mathcal{X}$ is a function that fits the following requirements

  1. It satisfies Carathedory condition i.e. ${\mathcal{S}}_2\left(·,·,·,w\right)$ is Lebesgue measurable and ${\mathcal{S}}_2\left(\delta ,·,·,·\right)$ is continuous.

  2. $\exists$ a non decreasing continuous function ${\mathcal{T}}_{{\mathcal{S}}_2}:[0,\infty )\to (0,\infty )$ and a function $\phi \in L^{\frac{1}{{\zeta }_1}}(U,{\mathbb{R}}^{+})$ , where ${\zeta }_1\in (0,\zeta )\ni$ $$\Vert {\mathcal{S}}_2(\delta ,u_1,u_2)\Vert ⩽\phi \left(\delta \right){\mathcal{T}}_{{\mathcal{S}}_2}(\Vert u_1\Vert +\Vert u_2\Vert ),$$ and $$\lim {\displaystyle \underset{n\to \infty }{\inf }}\frac{{\mathcal{T}}_{{\mathcal{S}}_2}\left(n\right)}{n}=\chi <\infty \text{.}$$

  3. $\exists$ constants ${\mathcal{L}}_1\in L^{\frac{1}{{\zeta }_1}}\left(\right[0,\mathcal{P}\left]\text{;}{R}^{+}\right)$ for any countable set $u_1,u_2\subset \mathcal{X}$ , $$\alpha \left({\mathcal{S}}_2\right(\delta ,u_1,u_2\left)\right)⩽{\mathcal{L}}_1\left[\alpha \right(u_1)+\alpha (u_2\left)\right],\forall \delta \in [0,\mathcal{P}]\text{.}$$

• $\left({\mathbf{H}}_{\mathbf{2}}\right)$

  For each $(\delta ,\sigma )\in \mathrm{\Lambda },\mathcal{R}:\mathrm{\Lambda }\times \mathcal{U}\to \mathcal{X}$ is a continuous function and it fits the following requirements

  1. there exist constants ${\mathcal{T}}_{\mathcal{R}}$ , such that, $$\Vert \mathcal{R}(\delta ,\sigma ,u_1)\Vert ⩽{\mathcal{T}}_{\mathcal{R}}[1+\Vert u_1\Vert ],$$ for $u_1\in \mathcal{X},\delta ,\sigma \in \mathcal{J}$ .

  2. $\exists {\mathcal{L}}_2\in L^{\frac{1}{{\zeta }_1}}(U^2,{\mathbb{R}}^{+})$ , for any bounded subset $u_2\subset \mathcal{X}\to \mathcal{X}$ $$\alpha \left({\mathcal{T}}_{\mathcal{R}}\left(\delta ,\sigma ,u_2\right)\right)⩽{\mathcal{L}}_2\left(\delta ,\sigma \right)\left[\alpha \left(u_2\right)\right]\mathrm{for}\mathrm{a}.\mathrm{e}.\delta \in U,$$ with ${\mathcal{L}}_2^{\ast }={\int }_0^{\sigma }{\mathcal{L}}_2(\delta ,\varrho )<\infty$ .

• $\left({\mathbf{H}}_{\mathbf{3}}\right)$

  For a function ${\mathcal{S}}_1:\mathcal{J}\times \mathcal{U}\to \mathcal{X}$ is continuous then it should fulfill the following

  1. $\exists$ a constant ${\mathcal{T}}_{{\mathcal{S}}_1},{\widehat{\mathcal{T}}}_{{\mathcal{S}}_1}\ni$ $$\Vert {\mathcal{S}}_1(\delta ,u_1)\Vert ⩽{\mathcal{T}}_{{\mathcal{S}}_1}(1+\Vert u_1\Vert )\mathrm{for}\delta \in U,\vartheta \in \mathcal{X},$$ $$\Vert {\mathcal{S}}_1(\delta ,u_1)-{\mathcal{S}}_1(\delta ,u_2)\Vert ⩽{\widehat{\mathcal{T}}}_{{\mathcal{S}}_1}\Vert u_1-u_2\Vert \forall u_1,u_2\in \mathcal{X}\text{.}$$

  2. $\exists$ constants ${\mathcal{L}}_3\ni$ for any countable set $u_3\subset \mathcal{X}$ , $$\alpha \left({\mathcal{S}}_1\right(\delta ,u_3\left)\right)⩽{\mathcal{L}}_3\alpha \left(u_3\right),\forall \delta \in U\text{.}$$

• $\left({\mathbf{H}}_{\mathbf{4}}\right)$

  For a bounded linear operators $\mathcal{G}$ and $\mathcal{K}$ from $\mathcal{X}$ $\exists$ positive constants $\upsilon$ and $\mu$ fulfilling $$\Vert \mathcal{G}\Vert ⩽\upsilon \mathrm{and}\Vert \mathcal{K}\Vert ⩽\mu \text{.}$$

We define an operator ${\mathcal{U}}^{\prime \prime }=\{\mathcal{M}\in {\mathcal{U}}^{\prime }:{\mathcal{M}}_0\in \mathcal{U}\}$ . Let ${\mathcal{B}}_{\xi }=\left\{\mathcal{M}\in {\mathcal{U}}^{\text{'}\text{'}}:{||\mathcal{M}||}_{{\mathcal{U}}^{\text{'}}}⩽\xi \right\}$ for some $\xi >0$ , then ${\mathcal{B}}_{\xi }\subseteq {\mathcal{U}}^{\prime \prime }$ is uniformly bounded, we have.

$\Vert {\mathcal{M}}_{\delta }+{\widehat{\mathrm{\Theta }}}_{\delta }{\Vert }_{\mathcal{U}}⩽\Vert {\mathcal{M}}_{\delta }\Vert +\Vert {\widehat{\mathrm{\Theta }}}_{\delta }\Vert ⩽\xi +{||{\widehat{\mathrm{\Theta }}}_{\delta }||}_{{\mathcal{U}}^{\text{'}}}={\xi }^{\text{'}}$ .

Define an operator $\widehat{\mathcal{E}}:{\mathcal{U}}^{\prime \prime }\to {\mathcal{U}}^{\prime \prime }$ by $$\widehat{\mathcal{E}}w\left(\delta \right)=\left\{\begin{array}{c}0,\delta \in \left[-q,0\right],\\ \mathcal{G}{P}_{\zeta }\left(\delta \right)\left[-{\mathcal{S}}_1\left(0,w_0\right)\right]\\ +\frac{\mathcal{KG}\left(1-\zeta \right)}{B\left(\zeta \right)\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)d\iota \\ +\frac{\mathcal{KG}\left(1-\zeta \right)}{B\left(\zeta \right)\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}{\sigma }^{\ast }{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)d\iota \\ +\frac{\zeta {\mathcal{G}}^2}{B\left(\zeta \right)}{\int }_0^{\delta }{Q}_{\zeta }\left(\delta -\iota \right){\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)d\iota \\ +\frac{\zeta {\mathcal{G}}^2}{B\left(\zeta \right)}{\int }_0^{\delta }{Q}_{\zeta }\left(\delta -\iota \right)\widehat{\mathcal{U}}{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)d\iota ,\delta \in \left[0,\mathcal{P}\right]\text{.}\end{array}\right.$$ Visibly, the operator $\mathcal{E}$ has a fixed point that is identical to $\widehat{\mathcal{E}}$ has one. Therefore, it is enough to prove $\widehat{\mathcal{E}}$ has fixed point.

Step 1: For a positive number $\xi >0,\widehat{\mathcal{E}}\left({\mathcal{B}}_{\xi }\right)\subseteq {\mathcal{B}}_{\xi }$ .

We assume the contrary, i.e, $\forall \xi ,\exists {\mathcal{M}}^{\xi }\in {\mathcal{B}}_{\xi }$ but $\widehat{\mathcal{E}}\left({\mathcal{M}}^{\xi }\right)\notin {\mathcal{B}}_{\xi }$ , i.e, $\Vert \widehat{\mathcal{E}}\left({\mathcal{M}}^{\xi }\right)\left(\delta \right)\Vert >\xi$ for some $\delta \in \mathcal{J}$ .

Applying $\left({\mathbf{H}}_{\mathbf{1}}\right)-\left({\mathbf{H}}_{\mathbf{4}}\right)$ , we have

This is a contraction. Hence, for some positive integer $\xi ,\widehat{\mathcal{E}}\left({\mathcal{B}}_{\xi }\right)\subseteq {\mathcal{B}}_{\xi }$ .

Step 2: $\widehat{\mathcal{E}}$ is continuous on ${\mathcal{B}}_{\xi }$ . $$\begin{array}{c}\Vert {\mathcal{E}}_2{w}_n\left(\delta \right)-{\mathcal{E}}_{2}w\left(\delta \right)\Vert ⩽\Vert \frac{\mathcal{KG}\left(1-\zeta \right)}{B\left(\zeta \right)\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}\\ \left[{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{n\theta }+{\widehat{\mathrm{\Theta }}}_{n\theta }\right)d\theta \right)-{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)\right]d\iota \Vert \\ +\Vert \frac{\mathcal{KG}\left(1-\zeta \right)}{B\left(\zeta \right)\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}{\sigma }^{\ast }\left[{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota }\right)-{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)\right]d\iota \Vert \\ +\Vert \frac{\zeta {\mathcal{G}}^2}{B\left(\zeta \right)}{\int }_0^{\delta }{Q}_{\zeta }\left(\delta -\iota \right)\\ \left[{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{n\theta }+{\widehat{\mathrm{\Theta }}}_{n\theta }\right)d\theta \right)-{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)\right]d\iota \Vert \\ +\Vert \frac{\zeta {\mathcal{G}}^2}{B\left(\zeta \right)}{\int }_0^{\delta }{Q}_{\zeta }\left(\delta -\iota \right)\widehat{\mathcal{U}}\left[{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota }\right)-{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)\right]d\iota \Vert \\ ⩽\frac{\mu \upsilon \left(1-\zeta \right)}{B\left(\zeta \right)\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}\\ \Vert {\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{n\theta }+{\widehat{\mathrm{\Theta }}}_{n\theta }\right)d\theta \right)-{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)\Vert \\ +\frac{\mu \upsilon }{\mathrm{\Gamma }\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}\Vert {\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota }\right)-{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)\Vert +\frac{\zeta {\upsilon }^2{\widehat{\mathcal{T}}}_1}{B\left(\zeta \right)}{\int }_0^{\delta }{\left(\delta -\iota \right)}^{\zeta -1}\\ \Vert {\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{n\theta }+{\widehat{\mathrm{\Theta }}}_{n\theta }\right)d\theta \right)-{\mathcal{S}}_2\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota },{\int }_0^{\iota }\mathcal{R}\left(\iota ,\theta ,{\mathcal{M}}_{\theta }+{\widehat{\mathrm{\Theta }}}_{\theta }\right)d\theta \right)\Vert \\ +\mu \upsilon {\int }_0^{\delta }{Q}_{\zeta }\left(\delta -\iota \right)\Vert {\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota }\right)-{\mathcal{S}}_1\left(\iota ,{\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }\right)\Vert \text{.}\end{array}$$ We acquire ${\lim }_{n\to \infty }\mathcal{E}({\mathcal{M}}_{n\iota }+{\widehat{\mathrm{\Theta }}}_{n\iota })=\mathcal{E}({\mathcal{M}}_{\iota }+{\widehat{\mathrm{\Theta }}}_{\iota }))$ in ${\mathcal{B}}_{\xi }$ , since the functions ${\mathcal{S}}_1,{\mathcal{S}}_2$ are continuous.