Optimal and total controllability approach of non-instantaneous Hilfer fractional derivative with integral boundary condition

Abstract

The focus of this work is on the absolute controllability of Hilfer impulsive non-instantaneous neutral derivative (HINND) with integral boundary condition of any order. Total controllability refers to the system’s ability to be controlled during the impulse time. Kuratowski measure and semigroup theory in Banach space yield the results. Furthermore, we talked about optimal controllability in conjunction with appropriate limitations. Our established outcomes are described using an example.

1 Introduction

The concept of differential equations with non-instantaneous impulses(NII) involves many physical processes due to its tremendous applications. Impulse is an action, that starts at an arbitrary fixed point and remains active on a finite time interval is called as NI impulse that occurs in many physical processes like harvesting, vaccination, natural disasters, and shocks subjected to unexpected change in their state. The above situations have to be modeled by impulses [1, 2] if necessary that can not be solved using ordinary differential equations. For some processes, instantaneous impulsive dynamic systems do not support a perfect description, for example, endorsement of insulin of hyperglycemia patients. The change in the above system caused by this medication will remain until the total absorption for a finite time, thanks to the evolutionary process can be modeled with NII. This theory is originated by Hernández [3]. Recently, Vipin Kumar et al. [4–7] derived the controllability results of fractional systems with and without NII for various models. To seek more about NI impulse, track and surf the articles [8–16] and cited references.

On the other hand, the existence and controllability theory extended for both DEs of integer and non-integer order with NII. Fractional calculus is the most appropriate way to evaluate the exact solutions to the given model. The results on Caputo and R-L fractional derivatives were discussed in [17–19]. Theory on HFD was introduced by Hilfer [20] and the results are discussed in [21–24]. One can refer to the monographs [25–29] to know more about fractional derivatives. In general, controllability enables directing the system from a random initial state to the desired ultimate state. The articles [30–35] discuss the controllability results of Caputo and Hilfer fractional differential system in the nondense domain. Furthermore, the existence and controllability of the Hilfer fractional system with infinite delay were examined in [36, 37]. The exact controllability for Hilfer fractional differential inclusions including nonlocal initial conditions was examined by Du et al. [38]. The approximate controllability results for the Hilfer fractional system were derived by [39, 40]. Recently, a prospective field in control systems is optimal control studied in [41–43]. Ultimately, is more appropriate to evaluate them using an optimization procedure involving fractional differential equations.

The outcome of the existence of HINND of arbitrary order was discussed in [44]. Moreover, results on total controllability fractional neutral non-instantaneous system discussed by [45]. In addition, optimization of the non-instantaneous neutral fractional system is investigated by in [46]. No article was found in the existing literature about the investigation of total controllability using semigroup theory.

We contribute this article to analyze the total controllability & optimal control results for HINND of arbitrary order as:

$$\begin{array}{ccc}& {\mathcal{D}}_{{\mathfrak{e}}_k}^{p_1,p_2}\left[\mathfrak{z}\right(\mathfrak{t})-\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right)\left)\right]=A\left[\mathfrak{z}\right(\mathfrak{t})-\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right)\left)\right]+B\mathfrak{u}\left(\mathfrak{t}\right)+\mathfrak{F}(\mathfrak{t},\mathfrak{z}(\mathfrak{t}\left)\right),\mathfrak{t}\in \bigcup _{k=0}^N({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]& \\ & \mathfrak{z}\left(\mathfrak{t}\right)=\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega ,\mathfrak{t}\in \bigcup _{k=1}^N({\mathfrak{t}}_k,{\mathfrak{e}}_k)& \\ & I_{0^{+}}^{(1-\eta )}\mathfrak{z}\left(0\right)=\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c,c\in \mathbb{R}.\end{array}$$ (1.1)

Here, $\mathcal{Y}$ be a Banach space and $A:\mathcal{D}\left(A\right)\subset \mathcal{Y}\to \mathcal{Y}$ is closed together with $\mathcal{D}\left(A\right)\in \mathcal{Y}$. ${\mathcal{D}}_{{\mathfrak{e}}_k}^{p_1,p_2}$ represents Hilfer derivative of fractional order with 0 < p₁ < 1, 0 ≤ p₂ ≤ 1. Also, η = p₁ + p₂ − p₁p₂, $\mathfrak{t}\in \mathcal{I}=[0,T],T>0$. Here $\mathcal{K}:\mathcal{I}\times \mathcal{Y}\to \mathcal{D}\left(A\right)\subset \mathcal{Y}$, $\mathfrak{F}:\mathcal{I}\times \mathcal{Y}\to \mathcal{D}\left(A\right)\subset \mathcal{Y}$, ${\mathcal{I}}_k:[{\mathfrak{t}}_k,{\mathfrak{e}}_k]\times \mathcal{Y}\to \mathcal{Y}$ are relevant functions. ${\mathfrak{t}}_k,{\mathfrak{e}}_k$ fulfills $0={\mathfrak{t}}_0={\mathfrak{e}}_0<{\mathfrak{t}}_1<{\mathfrak{e}}_1<{\mathfrak{t}}_2<\dots .<{\mathfrak{e}}_N<{\mathfrak{t}}_{N+1}=T$. Moreover, $\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right)=\underset{h\to 0^{+}}{\text{lim}}\mathfrak{z}({\mathfrak{t}}_k-h)$. $B:U\to \mathcal{Y}$ is a bounded linear operator and $\mathfrak{u}(·)\in L^2[\mathcal{I},U]$. The integral boundary condition λ = + 1 or −1. We briefly orchestrated our objective of this work:

• (i) By incorporating HFD with semigroup operator theory and LT, we have introduced the integral solution of (1.1).

• (ii) Kuratowski’s measure with κ–set-contraction theory has been supported very much to the total controllability of HINND with C₀ semigroup operator for the first time in the literature.

• (iii) The results on optimal controllability of HINND had been discussed via Lipschitz continuity.

• (iv) We have gone through with an illustration that enables our analytical outcomes existence.

2 Key notes

The space of continuous functions is defined by $C(\mathcal{I},\mathcal{Y})$ be a provided $\left|\right|\mathfrak{z}\left|\right|=\underset{\mathfrak{t}\in \mathcal{I}}{\text{sup}}\left|\right|\mathfrak{z}\left(\mathfrak{t}\right)\left|\right|.$

$C_{1-\eta }(\mathcal{I},\mathcal{Y})=\{\mathfrak{z}:\mathcal{Y}\to \mathcal{Y}\text{provided}{t}^{1-\eta }\mathfrak{z}\left(\mathfrak{t}\right)\in C(\mathcal{I},\mathcal{Y})\}$ , ${\left|\right|\mathfrak{z}\left|\right|}_{C_{1-\eta }}=\underset{0\le \mathfrak{t}\le T}{\text{sup}}\left|{\mathfrak{t}}^{1-\eta }\mathfrak{z}\left(\mathfrak{t}\right)\right|$.

Let $P{C}_{1-\eta }(({\mathfrak{t}}_k,{\mathfrak{t}}_{k+1}],\mathcal{Y})$ defines the space of piecewise functions as

$$P{C}_{1-\eta }(\mathcal{I},\mathcal{Y})=\left\{\begin{array}{c}{(\mathfrak{t}-{\mathfrak{t}}_k)}^{1-\eta }\mathfrak{z}\left(\mathfrak{t}\right)\in C_{1-\eta }(({\mathfrak{t}}_k,{\mathfrak{t}}_{k+1}],\mathbb{R})\hfill \\ \underset{\mathfrak{t}\to {\mathfrak{t}}_k}{\text{lim}}{(\mathfrak{t}-{\mathfrak{t}}_k)}^{1-\eta }\mathfrak{z}\left(\mathfrak{t}\right),k=1,2,\dots ,N,\hfill \end{array}\right\}$$

provided

$$\left|\right|\mathfrak{z}\left|\right|=\text{max}\left\{\underset{\mathfrak{t}\in \mathcal{I}}{\text{sup}}\left|\right|{\mathfrak{t}}^{1-\eta }\mathfrak{z}\left({\mathfrak{t}}^{+}\right)\left|\right|,\underset{\mathfrak{t}\in \mathcal{I}}{\text{sup}}\left|\right|{\mathfrak{t}}^{1-\eta }\mathfrak{z}\left({\mathfrak{t}}^{-}\right)\left|\right|\right\}.$$

$L\left(\mathcal{Y}\right)$, characterize the space of all bounded linear operators on $\mathcal{Y}$. A, generates the semigroup $\left\{S_{p_1,p_2}\left(\mathfrak{t}\right)\right\}$ where $\mathfrak{t}\ge 0$ with $\text{sup}\left|\right|S_{p_1,p_2}\left(\mathfrak{t}\right){\left|\right|}_{L\left(\mathcal{Y}\right)}=\mathcal{M}$. Define a convex, bounded and closed set $\ell =\{\mathfrak{z}\in P{C}_{1-\eta }(\mathcal{I},\mathcal{Y}),|\left|\mathfrak{z}\left(\mathfrak{t}\right)\right||<\mathfrak{r},\mathfrak{t}\in \mathcal{I},\mathfrak{r}>0\}$ in $P{C}_{1-\eta }(\mathcal{I},\mathcal{Y})$.

Definition 2.1 [20]. For n − 1 < p₁ < n, n ∈ N and p₂ ∈ (0, 1], HFD is defined by:

$$\begin{array}{ccc}& {\mathcal{D}}_{0^{+}}^{p_1,p_2}y\left(t\right)={\mathcal{I}}_{0^{+}}^{p_1(n-p_2)}\frac{d}{dt}{\mathcal{I}}_{0^{+}}^{(1-p_1)(n-p_2)}y\left(t\right)={\mathcal{I}}_{0^{+}}^{p_1(n-p_2)}{\mathcal{D}}_{0^{+}}^{p_2+p_{1}n-p_2{p}_1}y\left(t\right),& \end{array}$$

where ${\mathcal{D}}_{0^{+}}^{p_2+p_{1}n-p_2{p}_1}$ and ${\mathcal{I}}_{0^{+}}^{p_1(n-p_2)}$ are R-L derivative and integral respectively.

Definition 2.2 [8, 44, 47]. The Kuratowski noncompact measure ℓ(⋅) characterized as: $\ell \left(\hslash \right)≔\text{inf}\{\rho >0:\hslash =\bigcup _{i=1}^{m}withdiam\left({\hslash }_i\right)\le \rho \}$, where ℏ is a bounded set on $\mathcal{Y}$.

Lemma 2.3. (see [8, 44, 47]) For ${\hslash }_1,{\hslash }_2\subset \mathcal{Y}$, the Kuratowski noncompact measure meets:

1. $\ell \left(\hslash \right)=\ell \left(\overline{\hslash }\right)=\ell \left(conv\hslash \right);$

2. ℓ(ℏ) = 0 iff $\overline{\hslash }$ is compact;

3. for given λ ∈ R, ℓ(λℏ) ≤ |λ|ℓ(ℏ);

4. ℏ₁ ⊂ ℏ₂ implies ℓ(ℏ₁) ≤ ℓ(ℏ₂);

5. ℓ(ℏ₁∪ℏ₂) = max{ℓ(ℏ₁), ℓ(ℏ₂)};

6. ℓ(ℏ₁ + ℏ₂) ≤ ℓ(ℏ₁) + ℓ(ℏ₂), where ${\hslash }_1+{\hslash }_2=\left\{\mathfrak{z}\right|\mathfrak{z}={\mathfrak{z}}_1+{\mathfrak{z}}_2;{\mathfrak{z}}_1\in {\hslash }_1,{\mathfrak{z}}_2\in {\hslash }_2\}$;

7. The Lipschitz function $\Re :\mathcal{D}\left(\Re \right)\subset \mathcal{Y}\to \mathcal{Y}$ and the subset $W\subset \mathcal{D}\left(\Re \right)$ , ℓ(ℜ(W)) ≤ κ ℓ(W) is bounded.

Let $\mathcal{D}\subset C_{1-\eta }(\mathcal{I},\mathcal{Y})$ and $\mathfrak{t}\in \mathcal{I}$, $\mathcal{D}\left(\mathfrak{t}\right)=\left\{\mathfrak{z}\right(\mathfrak{t}\left)\right|\mathfrak{z}\in \mathcal{D}\}$ and $\ell \left(\mathcal{D}\left(\mathfrak{t}\right)\right)\le \ell C_{1-\eta }\left(\mathcal{D}\right)$.

Lemma 2.4. (see [8, 44, 47]) Let $\mathcal{D}\subset C_{1-\eta }([c_1,c_2],\mathcal{Y})$ be bounded and equicontinuous such that

$$\ell C_{1-\eta }\left(\mathcal{D}\right)=\underset{\mathfrak{t}\in [c_1,c_2]}{\text{max}}\ell \left(\mathcal{D}\left(\mathfrak{t}\right)\right),$$

and $\ell \left(\mathcal{D}\right(\mathfrak{t}\left)\right)$ is continuous on [c₁, c₂].

Lemma 2.5. (see [8, 44, 47]) Assume that $\overline{\mathcal{Y}}\subset \mathcal{Y}$ is bounded and for some ${\mathcal{D}}_0\subset \mathcal{D}$, the countable set meets $\ell \left(\mathcal{D}\right)\le 2\ell \left({\mathcal{D}}_0\right)$.

Lemma 2.6. (see [8, 44, 47]) Let $\mathcal{D}=\left\{{\mathfrak{z}}_n\right\}\subset P{C}_{1-\eta }([c_1,c_2],\mathcal{Y})$ where −∞ < c₁ < c₂ < ∞. Hence $\ell \left(\mathcal{D}\right(\mathfrak{t}\left)\right)$ on [c₁, c₂] such that:

$$\ell \left(\right\{{\int }_{c_1}^{c_2}{\mathfrak{z}}_n\left(\mathfrak{t}\right)d\mathfrak{t}\left\}\right)\le 2{\int }_{c_1}^{c_2}\ell \left(\mathcal{D}\left(\mathfrak{t}\right)\right)d\mathfrak{t},n\in \mathbb{N}.$$

Lemma 2.7. (see [21, 22, 44]) The system (1.1) becomes:

$$\begin{array}{cc}\mathfrak{z}\left(\mathfrak{t}\right)& =\frac{{\mathfrak{t}}^{\eta -1}}{\mathrm{\Gamma }\left(\eta \right)}[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\hfill \\ & +\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}\left[A\right[\mathfrak{z}\left(\omega \right)-\mathcal{K}(\omega ,\mathfrak{z}(\omega \left)\right)]+B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))]d\omega ,\mathfrak{t}\in (0,{\mathfrak{t}}_1],\hfill \\ \mathfrak{z}\left(\mathfrak{t}\right)& =\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega ,\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k],\hfill \\ \mathfrak{z}\left(\mathfrak{t}\right)& =\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\hfill \\ & +\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}\left[A\right[\mathfrak{z}\left(\omega \right)-\mathcal{K}(\omega ,\mathfrak{z}(\omega \left)\right)]\hfill \\ & +B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))]d\omega ,\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}].\hfill \end{array}$$

Definition 2.8. (see [21, 22, 44]) A function $\mathfrak{z}\in P{C}_{1-\eta }(\mathcal{I},\mathcal{Y})$ is a solution of (1.1), if

• (i) $\mathfrak{z}\left(\mathfrak{t}\right)=\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega ,\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k],k=1,2,...,N$

• (ii) $I_0^{1-\eta }\mathfrak{z}\left(0\right)=\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c$ ,

together with

$$\begin{array}{cc}\mathfrak{z}\left(\mathfrak{t}\right)& =S_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\hfill \\ & +{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in (0,{\mathfrak{t}}_1],\hfill & \end{array}$$ (2.1)

$$\begin{array}{cc}\mathfrak{z}\left(\mathfrak{t}\right)& =S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\hfill \\ & +{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],k=1,2,...,N.\hfill \end{array}$$ (2.2)

$$T_{p_1}\left(\mathfrak{t}\right)=p_1{\int }_0^{\infty }\nu {\psi }_y\left(\nu \right)\Re \left({\mathfrak{t}}^{p_1}\nu \right)d\nu ,K_{p_1}\left(\mathfrak{t}\right)={\mathfrak{t}}^{p_1-1}{T}_{p_1}\left(\mathfrak{t}\right),S_{p_1,p_2}\left(\mathfrak{t}\right)={\mathcal{I}}_{0^{+}}^{p_2(1-p_1)}{K}_{p_1}\left(\mathfrak{t}\right).$$

$$\begin{array}{ccc}\hfill W_y\left(\nu \right)& =& \frac{1}{\pi }\sum _{m=1}^{\infty }{(-1)}^{m-1}{\nu }^{-my-1}\frac{\mathrm{\Gamma }(my+1)}{m!}\text{sin}\left(m\pi \mathfrak{z}\right),\nu \in (0,\infty ),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\psi }_y\left(\nu \right)& =& \frac{1}{\mathfrak{z}}{\nu }^{(-1-\frac{1}{\mathfrak{z}})}{W}_y\left({\nu }^{-\frac{1}{\mathfrak{z}}}\right)\ge 0.\hfill \end{array}$$

Lemma 2.9. (see [8, 44, 47]) If a family $\{S_{p_1}\left(\mathfrak{t}\right),\mathfrak{t}\ge 0\}\subset \mathcal{B}\left(\mathcal{Y}\right)$ satisfies

• (i) for all $\mathfrak{z}\in D\left(A\right)$ , $S_{p_1}\left(\mathfrak{t}\right)\mathfrak{z}=\mathfrak{z}+I_{0^{+}}^{p_1}{S}_{p_1}\left(\mathfrak{t}\right)Ay$, $\mathfrak{t}\ge 0;$

• (ii) $S_{p_1}\left(\mathfrak{t}\right)$ is strongly continuous on ${\mathbb{R}}_{+}$ , $S_{p_1}\left(0\right)=I;$

• (iii) $A{S}_{p_1}\left(\mathfrak{t}\right)\mathfrak{z}=S_{p_1}A\left(\mathfrak{z}\right)$ for each, $\mathfrak{z}\in D\left(A\right),\mathfrak{t}\ge 0$, $D\subset \mathcal{Y}$.

Then, it is said to be p₁–times resolvent generator by A.

Definition 2.10. A system is defined as totally controllable on $\mathcal{I}$, if for k = 1, 2, …, N, it is controllable on $(0,{\mathfrak{t}}_1]$, $({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$ such that $\mathfrak{z}\left(0\right)={\mathfrak{z}}_0$ and $\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right)={\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}$.

For further discussions, we consider the subsequent assumptions as:

• (H1) $\mathcal{K}:J_0\times \mathcal{Y}\to \mathcal{Y},J_0=\bigcup _{k=0}^N({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$ is continuous and for ${\mathcal{L}}_p,{\mu }_p>0$ as

  $$\left|\right|\mathcal{K}(\mathfrak{t},\widehat{{\mathfrak{z}}_2})-\mathcal{K}(\mathfrak{t},\widehat{{\mathfrak{z}}_1})\left|\right|\le {\mathcal{L}}_p\left|\right|\widehat{{\mathfrak{z}}_2}-\widehat{{\mathfrak{z}}_1}\left|\right|,\widehat{{\mathfrak{z}}_2},\widehat{{\mathfrak{z}}_1}\in \mathcal{Y},\mathfrak{t}\in J_0,$$

  also $\left|\right|\mathcal{K}(\mathfrak{t},\mathfrak{z}||\le {\mu }_p,\mathfrak{z}\in \mathcal{Y},\mathfrak{t}\in J_0;$
• (H2) for any bounded set $D_1\subset \mathcal{Y}$, exist ${\mathcal{L}}_{p^{*}}>0$, such that

  $$\ell \left(\mathcal{K}(\mathfrak{t},D_1)\right)\le {\mathcal{L}}_{p^{*}}\ell \left(D_1\right);$$
• (H3) Function $\mathfrak{F}:J_0\times \mathcal{Y}\to \mathcal{Y}$ is continuous with ${\mathcal{L}}_f>0$, satisfies

  $$\left|\right|\mathfrak{F}(\mathfrak{t},\widehat{{\mathfrak{z}}_1})-\mathfrak{F}(\mathfrak{t},\widehat{{\mathfrak{z}}_2})\left|\right|\le {\mathcal{L}}_f\left|\right|\widehat{{\mathfrak{z}}_1}-\widehat{{\mathfrak{z}}_2}\left|\right|,\widehat{{\mathfrak{z}}_1},\widehat{{\mathfrak{z}}_2}\in \mathcal{Y},\mathfrak{t}\in J_0.$$

  $$\left|\right|\mathfrak{F}(\mathfrak{t},\mathfrak{z})\left|\right|\le \mathrm{\Psi }\left(\mathfrak{t}\right)\mathcal{P}\left(\right|\left|\mathfrak{z}\right|\left|\right)\text{and}\underset{l\to \infty }{\text{lim}}\text{inf}\frac{\mathcal{P}\left(l\right)}{l}=\nu <\infty ;$$

  where ℘ : [0, ∞)→[0, ∞), a non decreasing continuous function, $\psi :\mathcal{I}\to [0,\infty )$, a Lebesgue integrable function and ν > 0 such that for all $\mathfrak{z}\in \mathcal{Y}$, $\mathfrak{t}\in \mathcal{I}$ and meets ${\left|\right|\mathfrak{z}\left|\right|}_{C_{1-\eta }}\le l$.
• (H4) For $k=0,1,\dots N,{\mathcal{L}}_k$,

  $$\ell \left(\mathfrak{F}(\mathfrak{t},\mathfrak{z})\right)\le {\mathcal{L}}_k\ell \left(D\right),\mathfrak{t}\in J_0\text{with}\mathcal{L}=\underset{k}{\text{max}}{\mathcal{L}}_k,$$

  where the subset D of $\mathcal{Y}$ is a countable;
• (H5) For $J_k=[{\mathfrak{t}}_k,{\mathfrak{e}}_k],k=1,2,\dots N$, ${\mathcal{I}}_k:J_k\times \mathcal{Y}\to \mathcal{Y}$ are continuous functions, for $K_{{\mathcal{I}}_k}>0,k=1,2,\dots N$, provided for every $\widehat{{\mathfrak{z}}_1},\widehat{{\mathfrak{z}}_2}\in \mathcal{Y}$,

  $$\left|\right|{\mathcal{I}}_k(\mathfrak{t},\widehat{{\mathfrak{z}}_1})-{\mathcal{I}}_k(\mathfrak{t},\widehat{{\mathfrak{z}}_2})\left|\right|\le K_{{\mathcal{I}}_k}\left|\right|\widehat{{\mathfrak{z}}_1}-\widehat{{\mathfrak{z}}_2}\left|\right|,\text{for}\text{each}\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k],K≔\underset{k=0,1,\dots N}{\text{max}}{K}_{{\mathcal{I}}_k}.$$

  Moreover, ${\mathcal{M}}_{\mathcal{I}}$, together with $\left|\right|{\mathcal{I}}_k(\mathfrak{t},\mathfrak{z})\left|\right|\le {\mathcal{M}}_{\mathcal{I}};$
• (H6) $W:L^2(\mathcal{I},U)\to \mathcal{Y}$ defined by:

  $$W\mathfrak{u}={\int }_0^a{K}_{p_1}(a-\omega )B\mathfrak{u}\left(\omega \right)d\omega ,$$

  is invertible. Also, for ${\mathcal{M}}_b,{\mathcal{M}}_w\ge 0$, and $\Vert W^{-1}\Vert \le {\mathcal{M}}_w$, $\Vert B\Vert \le {\mathcal{M}}_b$.

• (H7) Given ${{\mathcal{L}}_{\mathfrak{u}}}^{*}>0$, for $\ell \left(\mathfrak{u}(\mathfrak{z},\mu )\right)\le {{\mathcal{L}}_{\mathfrak{u}}}^{*}{\mathfrak{t}}^{1-\eta }\sigma (\mathfrak{z},\mu )\ell \left(\mathfrak{z}\left(\mu \right)\right)$, a.e. $\mu \in \mathcal{I}$ and $\underset{\mathfrak{t}\in \mathcal{I}}{\text{sup}}{\int }_0^{\mathfrak{t}}\sigma (\mathfrak{t},\mu )d\omega ={\sigma }^{*}<\infty$.

Conveniently, we assign some notations as follows:

$$\begin{array}{ccc}& {\mathcal{C}}_2=\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^m{T}^{p_1}}{p_1};k_1=\text{max}\{\mathcal{M}\mathrm{\lambda }T+{\mathcal{L}}_p,\mathcal{M}(\frac{K_{{\mathcal{I}}_k}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mathcal{L}}_p)+{\mathcal{L}}_p,\frac{K_{{\mathcal{I}}_k}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}\};& \\ & {\mathcal{N}}_1=\mathcal{M}(\frac{{\mathcal{M}}_{\mathcal{I}}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{\mathfrak{t}}_{k+1}^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_{k+1}]};{\mathcal{C}}_1=\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^1{\mathfrak{t}}_1^{p_1}}{p_1};& \\ & {\mathcal{M}}_1=\text{sup}\left|\right|K_{p_1}{\left(\mathfrak{t}\right)\left|\right|}_{L\left(\mathcal{Y}\right)};\mathcal{N}=\mathcal{M}(\mathrm{\lambda }\left|\right|\mathfrak{z}\left|\right|+\left|c\right|+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{{\mathfrak{t}}_1}^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_1]}.\end{array}$$

3 Main sequels

Lemma 3.1. Let $\mathcal{S}\subset \mathcal{Y}$ and $\Re :\mathcal{S}\to \mathcal{Y}$ be called as κ-set-contractive for any bounded set ℵ in $\mathcal{S}$ such that and for κ ∈ [0, 1), as

$$\ell \left(\Re \right(\aleph \left)\right)\le \kappa \ell \left(\aleph \right).$$

Lemma 3.2. Let ℵ be a convex, bounded and closed subset of $\mathcal{Y}$. If ℜ : ℵ → ℵ is κ-set-contractive. Then ℜ has at least one fixed point in ℵ.

Lemma 3.3. If the assumptions (H1)–(H7) true, hence

$$\begin{array}{cc}\mathfrak{u}\left(\mathfrak{t}\right)& =W^{-1}[{\mathfrak{z}}_{{\mathfrak{t}}_1}-S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]-\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))\hfill \\ & -{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega ],\mathfrak{t}\in (0,{\mathfrak{t}}_1],\hfill \end{array}$$ (3.1)

$$\begin{array}{cc}\mathfrak{u}\left(\mathfrak{t}\right)& =W^{-1}[{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}-S_{p_1,p_2}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k)\hfill \\ & (\times )[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]-\mathcal{K}({\mathfrak{t}}_{k+1},\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right))\hfill \\ & -{\int }_{{\mathfrak{e}}_k}^{{\mathfrak{t}}_{k+1}}{K}_{p_1}({\mathfrak{t}}_{k+1}-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega ],\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],\hfill \end{array}$$ (3.2)

drives to $\mathfrak{z}\left(\mathfrak{t}\right)$ of (1.1) from $\mathfrak{z}\left({\mathfrak{t}}_1\right)={\mathfrak{z}}_{{\mathfrak{t}}_1}$ and $\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right)={\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}$, also $\left|\right|\mathfrak{u}\left(\mathfrak{t}\right)\left|\right|={\mathcal{M}}_{\mathfrak{u}}^1$, $\left|\right|\mathfrak{u}\left(\mathfrak{t}\right)\left|\right|={\mathcal{M}}_{\mathfrak{u}}^k$ with

$${\mathcal{M}}_{\mathfrak{u}}^1={\mathcal{M}}_w^1(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{N}),{\mathcal{M}}_{\mathfrak{u}}^k={\mathcal{M}}_w^m(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+{\mathcal{N}}_1),k=1,2,\dots ,N.$$

Proof. For $\mathfrak{t}={\mathfrak{t}}_1$,

$$\begin{array}{cc}\mathfrak{z}\left({\mathfrak{t}}_1\right)& =S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))\hfill \\ & +{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \hfill \\ & +{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\tau )W^{-1}[{\mathfrak{z}}_{{\mathfrak{t}}_1}-S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]\hfill \\ & -\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))-{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega ]d\tau \hfill \\ & =S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))\hfill \\ & +{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega +{\mathfrak{z}}_{{\mathfrak{t}}_1}-S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]\hfill \\ & -\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))-{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \hfill \\ & ={\mathfrak{z}}_{{\mathfrak{t}}_1},\hfill \end{array}$$

with

$$\begin{array}{cc}\left|\right|\mathfrak{u}\left(\mathfrak{t}\right)\left|\right|& \le \left|\right|W^{-1}[{\mathfrak{z}}_{{\mathfrak{t}}_1}-S_{p_1,p_2}\left({\mathfrak{t}}_1\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]-\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))\hfill \\ & -{\int }_0^{{\mathfrak{t}}_1}{K}_{p_1}({\mathfrak{t}}_1-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \left]\right||\hfill \\ & \le {\mathcal{M}}_w^1\left(\right||{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{M}\left|\right|\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))\left|\right|+\left|\right|\mathcal{K}({\mathfrak{t}}_1,\mathfrak{z}\left({\mathfrak{t}}_1\right))\left|\right|\hfill \\ & +{\mathcal{M}}_1\left|\right|{\int }_0^{{\mathfrak{t}}_1}{\omega }^{1-\eta }\mathrm{\Psi }\left(\omega \right)\mathcal{P}\left({\left|\right|\mathfrak{z}\left|\right|}_{C_{1-\eta }}\right)d\omega \left|\right|)\hfill \\ & \le {\mathcal{M}}_w^1(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{M}(\mathrm{\lambda }T\left|\right|\mathfrak{z}\left(\omega \right)\left|\right|+\left|c\right|+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{\mathfrak{t}}_1^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_1]})\hfill \\ & \le {\mathcal{M}}_w^1(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{N})\hfill \\ & ={\mathcal{M}}_{\mathfrak{u}}^1.\hfill \end{array}$$

Also, for $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$ and $\mathfrak{t}={\mathfrak{t}}_{k+1}$,

$$\begin{array}{cc}\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right)& =S_{p_1,p_2}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\hfill \\ & +\mathcal{K}({\mathfrak{t}}_{k+1},\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right))+{\int }_{{\mathfrak{e}}_k}^{{\mathfrak{t}}_{k+1}}{K}_{p_1}({\mathfrak{t}}_{k+1}-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \hfill \\ & +{\int }_{{\mathfrak{e}}_k}^{{\mathfrak{t}}_{k+1}}{K}_{p_1}({\mathfrak{t}}_{k+1}-\tau )W^{-1}[{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}-S_{p_1,p_2}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k)\hfill \\ & (\times )[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\hfill \\ & -\mathcal{K}({\mathfrak{t}}_{k+1},\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right))-{\int }_{{\mathfrak{e}}_k}^{{\mathfrak{t}}_{k+1}}{K}_{p_1}({\mathfrak{t}}_{k+1}-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega ]d\tau \hfill \\ & ={\mathfrak{z}}_{{\mathfrak{t}}_{k+1}},\hfill \end{array}$$

with

$$\begin{array}{cc}\left|\right|\mathfrak{u}\left(\mathfrak{t}\right)\left|\right|& \le {\mathcal{M}}_w^m(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+\mathcal{M}(\frac{{\mathcal{M}}_{\mathcal{I}}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{\mathfrak{t}}_{k+1}^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_{k+1}]})\hfill \\ & \le {\mathcal{M}}_w^m(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+{\mathcal{N}}_1)\hfill \\ & ={\mathcal{M}}_{\mathfrak{u}}^m.\hfill \end{array}$$

Theorem 3.4. The system (1.1) is totally controllable on $\mathcal{I}$, if it meets the assumptions (H1)–(H7) together with the conditions

$$\begin{array}{cc}& [\mathcal{M}\mathrm{\lambda }+2{\mathcal{L}}_p^{*}+{\mathcal{M}}_{\mathcal{I}}(\mathcal{M}+1)+4{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+4{\mathcal{M}}_1{\mathcal{L}}_k]<1.\end{array}$$ (3.3)

Proof. Construct $\mathcal{G}:P{C}_{1-\eta }(\mathcal{I},\mathcal{Y})\to P{C}_{1-\eta }(\mathcal{I},\mathcal{Y})$ as

$$\left(\mathcal{G}\mathfrak{z}\right)\left(\mathfrak{t}\right)=\{\begin{array}{c}{S}_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\hfill \\ +{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in (0,{\mathfrak{t}}_1];\hfill \\ \frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega ,\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k];\hfill \\ S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\hfill \\ +\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))+{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],\hfill \end{array}$$

where $\mathfrak{u}\left(\mathfrak{t}\right)$ is described in (3.1) and (3.2) for $(0,{\mathfrak{t}}_1]$ and $({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$, respectively. Moreover, by Lemma 3.1, $\mathfrak{z}\left({\mathfrak{t}}_1\right)={\mathfrak{z}}_{{\mathfrak{t}}_1}$ and $\mathfrak{z}\left({\mathfrak{t}}_{k+1}\right)={\mathfrak{z}}_{{\mathfrak{t}}_{k+1}},k=1,2,\dots ,N$. Let ${\aleph }_{\gamma }=\{\mathfrak{z}\in P{C}_{1-\eta }{(\mathcal{I},\mathcal{Y}):\left|\right|\mathfrak{z}\left|\right|}_{P{C}_{1-\eta }}\le \gamma \}\subseteq P{C}_{1-\eta }(\mathcal{I},\mathcal{Y}),\gamma >0$, and

$$\gamma >\text{max}\{\mathcal{N}+{\mathcal{C}}_1(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{N}),\underset{k=1,2,\dots ,N}{\text{max}}\{{\mathcal{N}}_1+{\mathcal{C}}_2(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+{\mathcal{N}}_1\left)\right\},\frac{{\mathcal{M}}_{\mathcal{I}}{T}^{p_1}}{\mathrm{\Gamma }(p_1+1)}\}.$$

Step 1: $\mathcal{G}:{\aleph }_{\gamma }\to {\aleph }_{\gamma }$ .

For $\mathfrak{t}\in (0,{\mathfrak{t}}_1]$, let $\mathfrak{z}\in {\aleph }_{\gamma }$

$$\begin{array}{cc}\left|\right|\left(\mathcal{G}\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|& \le \left|\right|S_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]\left|\right|+\left|\right|\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\left|\right|\hfill \\ & +\left|\right|{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \left|\right|+\left|\right|{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )B\mathfrak{u}\left(\omega \right)d\omega \left|\right|\hfill \\ & \le \mathcal{M}(\mathrm{\lambda }T\left|\right|\mathfrak{z}\left|\right|+\left|c\right|+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{\mathfrak{t}}_1^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_1]}+\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_{\mathfrak{u}}^1{\mathfrak{t}}^{p_1}}{p_1}\hfill \\ & \le \mathcal{N}+\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^1{\mathfrak{t}}^{p_1}}{p_1}[\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{N}]\hfill \\ & \le \gamma .\hfill \end{array}$$ (3.4)

Also, for $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$,

$$\begin{array}{cc}\left|\right|\left(\mathcal{G}\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|& \le \left|\right|S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\left|\right|\hfill \\ & +\left|\right|\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\left|\right|+\left|\right|{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))d\omega \left|\right|+\left|\right|{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )B\mathfrak{u}\left(\omega \right)d\omega \left|\right|\hfill \\ & \le \mathcal{M}(\frac{{\mathcal{M}}_{\mathcal{I}}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mu }_p)+{\mu }_p+\frac{{\mathcal{M}}_1{\mathfrak{t}}_1^{p_1}}{p_1}{\mathcal{P}\left(l\right)\left|\right|\mathrm{\Psi }\left|\right|}_{L[0,{\mathfrak{t}}_{k+1}]}+\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_{\mathfrak{u}}^1{T}^{p_1}}{p_1}\hfill \\ & \le {\mathcal{N}}_1+\frac{{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^m{T}^{p_1}}{p_1}(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+{\mathcal{N}}_1)\hfill \\ & \le \gamma .\hfill \end{array}$$ (3.5)

Also, for $\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k]$, and $\mathfrak{z}\in {\aleph }_{\gamma }$,

$$\begin{array}{cc}& \left|\right|\left(\mathcal{G}\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|\le \frac{{\mathcal{M}}_{\mathcal{I}}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}\le \gamma .\end{array}$$ (3.6)

Hence, from (3.4)–(3.6), for some $\mathfrak{t}\in \mathcal{I}$, gives ${\left|\right|\left(\mathcal{G}\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|}_{P{C}_{1-\eta }}\le \gamma$. Then $\mathcal{G}:{\aleph }_{\gamma }\to {\aleph }_{\gamma }$.

Construct ${\mathcal{G}}_1$, ${\mathcal{G}}_2$ as:

$$\left({\mathcal{G}}_{1}y\right)\left(\mathfrak{t}\right)=\{\begin{array}{c}{S}_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right)),\mathfrak{t}\in (0,{\mathfrak{t}}_1],\hfill \\ \frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega ,\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k],k=1,2,\dots ,N,\hfill \\ S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\hfill \\ +\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right)),\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],k=1,2,\dots ,N,\hfill \end{array}$$

and

$$\left({\mathcal{G}}_{2}y\right)\left(\mathfrak{t}\right)=\{\begin{array}{cc}{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\hfill & \mathfrak{t}\in (0,{\mathfrak{t}}_1],\hfill \\ 0,\hfill & \mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k],\hfill \\ {\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[B\mathfrak{u}\right(\omega )+\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right)\left)\right]d\omega ,\hfill & \mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}].\hfill \end{array}$$

Clearly, $\mathcal{G}={\mathcal{G}}_1+{\mathcal{G}}_2$.

Step 2: ${\mathcal{G}}_1$ is contraction.

Let $\mathfrak{z}\in {\aleph }_{\gamma }$, for any $\mathfrak{t}\in (0,{\mathfrak{t}}_1]$,

$$\begin{array}{cc}\left|\right|[\left({\mathcal{G}}_1{\mathfrak{z}}_1\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_1{\mathfrak{z}}_2\right)\left(\mathfrak{t}\right)]\left|\right|& =\left|\right|S_{p_1,p_2}\left(\mathfrak{t}\right)(\mathrm{\lambda }{\int }_0^T{\mathfrak{z}}_1\left(\omega \right)d\omega -\mathrm{\lambda }{\int }_0^T{\mathfrak{z}}_2\left(\omega \right)d\omega )\left|\right|\hfill \\ & +\left|\right|\mathcal{K}(\mathfrak{t},{\mathfrak{z}}_1\left(\mathfrak{t}\right))-\mathcal{K}(\mathfrak{t},{\mathfrak{z}}_2\left(\mathfrak{t}\right))\left|\right|\hfill \\ & \le \mathcal{M}\mathrm{\lambda }T\left|\right|{\mathfrak{z}}_1\left(\omega \right)-{\mathfrak{z}}_2{\left(\omega \right)\left|\right|}_{P{C}_{1-\eta }}+{\mathcal{L}}_p\left|\right|{\mathfrak{z}}_1\left(\omega \right)-{\mathfrak{z}}_2\left(\omega \right){\left|\right|}_{P{C}_{1-\eta }}\hfill \\ & \le (\mathcal{M}\mathrm{\lambda }T+{\mathcal{L}}_p)\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}\hfill \\ & \le k_1\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.7)

Also, for $\mathfrak{z}\in {\aleph }_{\gamma }$, $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$,

$$\begin{array}{cc}\left|\right|\left({\mathcal{G}}_1{\mathfrak{z}}_1\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_1{\mathfrak{z}}_2\right)\left(\mathfrak{t}\right)\left|\right|& \le \left(\mathcal{M}\right(\frac{K_{{\mathcal{I}}_k}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mathcal{L}}_p)+{\mathcal{L}}_p)\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}\hfill \\ & \le k_1\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.8)

Also, for $\mathfrak{t}\in ({\mathfrak{t}}_k,{\mathfrak{e}}_k]$, and $\mathfrak{z}\in {\aleph }_{\gamma }$,

$$\begin{array}{cc}& \left|\right|\left({\mathcal{G}}_1{\mathfrak{z}}_1\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_1{\mathfrak{z}}_2\right)\left(\mathfrak{t}\right)\left|\right|\le \frac{K_{{\mathcal{I}}_k}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}\le k_1\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}.\end{array}$$ (3.9)

For any $\mathfrak{t}\in \mathcal{I}$, $\left|\right|\left({\mathcal{G}}_1{\mathfrak{z}}_1\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_1{\mathfrak{z}}_2\right)\left(\mathfrak{t}\right)\left|\right|\le k_1\left|\right|{\mathfrak{z}}_1-{\mathfrak{z}}_2{\left|\right|}_{P{C}_{1-\eta }}$. Since $k_1<1,{\mathcal{G}}_1$ is contracting operator.

Step 3: By step 1, it is clear that ${\mathcal{G}}_2$ is bounded. To prove continuity, consider a sequence ${\left\{{\mathfrak{z}}^n\right\}}_{n=1}^{\infty }$ in ℵ_γ such that ${\mathfrak{z}}^n\to \mathfrak{z}$ in ℵ_γ. For $\mathfrak{t}\in (0,{\mathfrak{t}}_1]$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}^n\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|\hfill \\ \le \left|\right|{\mathfrak{t}}^{\eta -1}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}[\mathfrak{F}(\omega ,{\mathfrak{z}}^n\left(\omega \right))-\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))]d\omega \left|\right|\hfill \\ +\left|\right|{\mathfrak{t}}^{\eta -1}{\int }_0^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}[B{\mathfrak{u}}_{{\mathfrak{z}}^n}\left(\omega \right)-B{\mathfrak{u}}_{\mathfrak{z}}\left(\omega \right)]d\omega \left|\right|\hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f{\int }_0^{\mathfrak{t}}\left|\right|{\mathfrak{z}}^n\left(\omega \right)-\mathfrak{z}\left(\omega \right)\left|\right|d\omega \hfill \\ +{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^1{\int }_0^{\mathfrak{t}}\left[\right|\left|S_{p_1,p_2}\left(\mathfrak{t}\right)\right(\mathrm{\lambda }{\int }_0^T{\mathfrak{z}}^n\left(\omega \right)d\omega -\mathrm{\lambda }{\int }_0^T\mathfrak{z}\left(\omega \right)d\omega \left)\right||\hfill \\ +\left|\right|\mathcal{K}({\mathfrak{t}}_1,{\mathfrak{z}}^n\left({\mathfrak{t}}_1\right))-\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\left|\right|+{\int }_0^{{\mathfrak{t}}_1}{(\mathfrak{t}-\omega )}^{p_1-1}\left|\right|\mathfrak{F}(\omega ,{\mathfrak{z}}^n\left(\omega \right))d\omega -\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))\left|\right|d\omega ]d\tau \hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f\frac{{\mathfrak{t}}_1^{p_1}}{p_1}\left|\right|{\mathfrak{z}}^n-{\mathfrak{z}\left|\right|}_{P{C}_{1-\eta }}+{\mathcal{C}}_1[{\mathcal{L}}_p+{\mathcal{M}}_1{\mathcal{L}}_f\frac{{\mathfrak{t}}_1^{p_1}}{p_1}]\left|\right|{\mathfrak{z}}^n-\mathfrak{z}{\left|\right|}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.10)

Therefore, $\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}^n\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|\to 0$ as n → ∞. Also, for $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],k=1,2,\dots ,N$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}^n\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|\hfill \\ \le \left|\right|{\mathfrak{t}}^{1-\eta }{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}[\mathfrak{F}(\omega ,{\mathfrak{z}}^n\left(\omega \right))-\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))]d\omega \left|\right|\hfill \\ +\left|\right|{\mathfrak{t}}^{1-\eta }{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{(\mathfrak{t}-\omega )}^{p_1-1}[B{\mathfrak{u}}_{{\mathfrak{z}}^n}\left(\omega \right)-B{\mathfrak{u}}_{\mathfrak{z}}\left(\omega \right)]d\omega \left|\right|\hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}\left|\right|{\mathfrak{z}}^n\left(\omega \right)-\mathfrak{z}\left(\omega \right)\left|\right|d\omega \hfill \\ +{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^m{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}\left(\right|\left|S_{p_1,p_2}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k)\right[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}\hfill \\ \times [{\mathcal{I}}_k(\omega ,{\mathfrak{z}}^n\left({\mathfrak{t}}_k^{-}\right))-{\mathcal{I}}_k(\omega ,\mathfrak{z}\left({\mathfrak{t}}_k^{-}\right))]d\omega +[\mathcal{K}({\mathfrak{e}}_k,{\mathfrak{z}}^n\left({\mathfrak{e}}_k\right))-\mathcal{K}({\mathfrak{e}}_k,\mathfrak{z}\left({\mathfrak{e}}_k\right))]\left]\right||\hfill \\ +\left|\right|\mathcal{K}(\mathfrak{t},{\mathfrak{z}}^n\left(\mathfrak{t}\right))-\mathcal{K}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right))\left|\right|+{\int }_{{\mathfrak{e}}_k}^{{\mathfrak{t}}_{k+1}}{({\mathfrak{t}}_{k+1}-\omega )}^{p_1-1}\left|\right|\mathfrak{F}(\omega ,{\mathfrak{z}}^n\left(\omega \right))d\omega -\mathfrak{F}(\omega ,\mathfrak{z}\left(\omega \right))\left|\right|d\omega )d\tau \hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f\frac{{\mathfrak{t}}_{k+1}^{p_1}}{p_1}\left|\right|{\mathfrak{z}}^n-\mathfrak{z}{\left|\right|}_{P{C}_{1-\eta }}\hfill \\ +{\mathcal{C}}_2\left(\mathcal{M}\right(\frac{K_{{\mathcal{I}}_k}{\mathfrak{t}}_{k+1}^{p_1}}{\mathrm{\Gamma }(p_1+1)}+{\mathcal{L}}_p)+{\mathcal{L}}_p+{\mathcal{M}}_1{\mathcal{L}}_f\frac{{\mathfrak{t}}_{k+1}^{p_1}}{p_1})\left|\right|{\mathfrak{z}}^n-\mathfrak{z}{\left|\right|}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.11)

Hence, $\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}^n\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|$ approaches to 0 as n approaches to ∞. Hence from (3.10) and (3.11) and for each $\mathfrak{t}\in \mathcal{I}$, $\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}^n\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left(\mathfrak{t}\right)\left|\right|\to 0$ as n → ∞.

Step 4: ${\mathcal{G}}_2$ is equicontinuous.

Take τ₁ < τ₂ on ℵ_γ, and for ${\tau }_1,{\tau }_2\in (0,{\mathfrak{t}}_1]$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_2\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_1\right)\left|\right|\hfill \\ \le \left|\right|{\tau }_2^{\eta -1}{\int }_0^{{\tau }_2}{({\tau }_2-\omega )}^{p_1-1}\left[\mathfrak{F}\right(\omega ,\mathfrak{z}\left(\omega \right))+B\mathfrak{u}(\omega \left)\right]d\omega \hfill \\ -{\tau }_1^{\eta -1}{\int }_0^{{\tau }_1}{({\tau }_1-\omega )}^{p_1-1}\left[\mathfrak{F}\right(\omega ,\mathfrak{z}\left(\omega \right))+B\mathfrak{u}(\omega \left)\right]d\omega \left|\right|\hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f\left({\int }_0^{{\tau }_1}\right[{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}-{\tau }_1^{\eta -1}{({\tau }_1-\omega )}^{p_1-1}]d\omega +{\int }_{{\tau }_1}^{{\tau }_2}{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}d\omega )\hfill \\ +{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^1(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_1}\left|\right|+\mathcal{N})\left({\int }_0^{{\tau }_1}\right[{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}-{\tau }_1^{\eta -1}{({\tau }_1-\omega )}^{p_1-1}]d\omega \hfill \\ +{\int }_{{\tau }_1}^{{\tau }_2}{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}d\omega ).\hfill \end{array}$$ (3.12)

Similarly, For ${\tau }_1,{\tau }_2\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_2\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_1\right)\left|\right|\hfill \\ \le \left|\right|{\tau }_2^{\eta -1}{\int }_{{\mathfrak{e}}_k}^{{\tau }_2}{({\tau }_2-\omega )}^{p_1-1}\left[\mathfrak{F}\right(\omega ,\mathfrak{z}\left(\omega \right))+B\mathfrak{u}(\omega \left)\right]d\omega \hfill \\ -{\int }_{{\mathfrak{e}}_k}^{{\tau }_1}{({\tau }_1-\omega )}^{p_1-1}\left[\mathfrak{F}\right(\omega ,\mathfrak{z}\left(\omega \right))+B\mathfrak{u}(\omega \left)\right]d\omega \left|\right|\hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f\left({\int }_{{\mathfrak{e}}_k}^{{\tau }_1}\right[{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}-{\tau }_1^{\eta -1}{({\tau }_1-\omega )}^{p_1-1}]d\omega +{\int }_{{\tau }_1}^{{\tau }_2}{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}d\omega )\hfill \\ +{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w^m(\left|\right|{\mathfrak{z}}_{{\mathfrak{t}}_{k+1}}\left|\right|+{\mathcal{N}}_1)\left({\int }_{{\mathfrak{e}}_k}^{{\tau }_1}\right[{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}-{\tau }_1^{\eta -1}{({\tau }_1-\omega )}^{p_1-1}]d\omega \hfill \\ +{\int }_{{\tau }_1}^{{\tau }_2}{\tau }_2^{\eta -1}{({\tau }_2-\omega )}^{p_1-1}d\omega ).\hfill \end{array}$$ (3.13)

By (H3), $\left|\right|\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_2\right)-\left({\mathcal{G}}_2\mathfrak{z}\right)\left({\tau }_1\right)\left|\right|\to 0$ as τ₂ → τ₁. Then ${\mathcal{G}}_2$ is equicontinuous.

The countable subset $D_0={\left\{{\mathfrak{z}}_n\right\}}_{n=1}^{\infty }\subset D$, and by Lemma 2.4, we have

$$\begin{array}{cc}& \ell {\left({\mathcal{G}}_2\left(D\right)\right)}_{P{C}_{1-\eta }}\le 2\ell {\left({\mathcal{G}}_2\left(D_0\right)\right)}_{P{C}_{1-\eta }},\end{array}$$ (3.14)

where D is a bounded subset of ℵ_γ. Since ${\mathcal{G}}_2\left(D_0\right)\subset {\mathcal{G}}_2\left({\aleph }_{\gamma }\right)$ is bounded and equicontinuous, by Lemma 2.6,

$$\begin{array}{cc}& \ell {\left({\mathcal{G}}_2\left(D_0\right)\right)}_{P{C}_{1-\eta }}\le \underset{\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],k=0,1,2,\dots ,N}{\text{max}}\ell {\left({\mathcal{G}}_2\left(D_0\right)\right)}_{P{C}_{1-\eta }}\left(\mathfrak{t}\right).\end{array}$$ (3.15)

Moreover, for $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$, (H4), (H7) and ${\mathcal{G}}_2$, with Lemma 2.5, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \ell \left({\mathcal{G}}_2\left(D_0\right)\right)\left(\mathfrak{t}\right)& \le \ell \left({\mathcal{M}}_1{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}\right[B\mathfrak{u}(\omega ,{\left\{{\mathfrak{z}}_n\left(\omega \right)\right\}}_{n=1}^{\infty })+\mathfrak{F}(\omega ,{\left\{{\mathfrak{z}}_n\left(\omega \right)\right\}}_{n=1}^{\infty })\left]d\omega \right)\hfill \\ & \le [2{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+2{\mathcal{M}}_1{\mathcal{L}}_k]{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}\ell \left({\left\{{\mathfrak{z}}_n\left(\omega \right)\right\}}_{n=1}^{\infty }\right)d\omega \hfill \\ & \le [2{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+2{\mathcal{M}}_1{\mathcal{L}}_k]\ell {\left(D\right)}_{P{C}_{1-\eta }}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k).\hfill \end{array}\end{array}$$ (3.16)

Then, by (3.14)–(3.16) and (H2),

$$\begin{array}{cc}\ell {\left({\mathcal{G}}_2\left(D\right)\left(\mathfrak{t}\right)\right)}_{P{C}_{1-\eta }}& \le [2{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+2{\mathcal{M}}_1{\mathcal{L}}_k]\ell {\left(D\right)}_{P{C}_{1-\eta }}({\mathfrak{t}}_{k+1}-{\mathfrak{e}}_k)\hfill \\ & \le [4{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+4{\mathcal{M}}_1{\mathcal{L}}_k]\ell {\left(D\right)}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.17)

Now, for any $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$, on D ∈ ℵ_γ,

$$\begin{array}{cc}& \ell \left({\mathcal{G}}_1\left(D\right)\right)\le [\mathcal{M}\mathrm{\lambda }+2{\mathcal{L}}_p^{*}+{\mathcal{M}}_{\mathcal{I}}(\mathcal{M}+1)]\ell \left(D\right).\end{array}$$ (3.18)

Also,

$$\begin{array}{c}\ell \left(\mathcal{G}\left(D\right)\right)\le \ell \left({\mathcal{G}}_1\left(D\right)\right)+\ell \left({\mathcal{G}}_2\left(D\right)\right)\hfill \\ \le [\mathcal{M}\mathrm{\lambda }+2{\mathcal{L}}_p^{*}+{\mathcal{M}}_{\mathcal{I}}(\mathcal{M}+1)+4{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+4{\mathcal{M}}_1{\mathcal{L}}_k]\ell {\left(D\right)}_{P{C}_{1-\eta }}.\hfill \end{array}$$ (3.19)

Combining Lemma 3.1, and (3.3) and (3.19) it is clear that the mapping $\mathcal{G}$ from ℵ_γ to ℵ_γ is κ-set-contractive. Hence, the system $\mathcal{G}$ has a fixed point by Lemma 3.2. This completes the proof.

4 Optimal control

• (H8) (i) The Lagrange function $\mathfrak{L}:\mathcal{I}\times \mathcal{Y}\times U\to \mathbb{R}\cup \left\{\infty \right\}$ is Borel measurable;

    (ii) For $\mathfrak{t}\in \mathcal{I}$, and for every ${\mathfrak{z}}_1,{\mathfrak{z}}_2\in \mathcal{Y}$, $\mathfrak{L}(\mathfrak{t},\mathfrak{z},·)$ is convex on U;

    (iii) For almost all $\mathfrak{t}\in \mathcal{I}$, $\mathfrak{L}(\mathfrak{t},·,·)$ is sequentially lower semi continuous on $\mathcal{Y}\times U$;

    (iv) For ${\mathfrak{c}}_1\ge 0,{\mathfrak{c}}_2>0$, $\mathfrak{h}\in L^P(\mathcal{I},\mathbb{R})$,

  $$\mathfrak{L}(\mathfrak{t},\mathfrak{z},\mathfrak{u})\ge \mathfrak{h}\left(\mathfrak{t}\right)+{\mathfrak{c}}_1{\left|\right|\mathfrak{z}\left|\right|}_{P{C}_{1-\eta }}+{\mathfrak{c}}_2{\left|\right|\mathfrak{u}\left|\right|}^P.$$

This part deals with the verification of existence of optimal pair for the system (1.1) by sequencing technique as discussed in [46, 48]. Let the cost function($\mathfrak{L}$) as:

$$\mathfrak{I}({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u})={\int }_0^T\mathfrak{L}(\mathfrak{t},\mathfrak{z}\left(\mathfrak{t}\right),\mathfrak{u}\left(\mathfrak{t}\right))d\mathfrak{t},\mathfrak{u}\in {\mathcal{U}}_{ad}.$$

Define the admissible control function ${\mathcal{U}}_{ad}$ as:

$${\mathcal{U}}_{ad}=\{\mathfrak{u}\in L^P(\mathcal{I},H);\mathfrak{u}\left(\mathfrak{t}\right)\in \mathcal{P}\left(\mathfrak{t}\right),a.e.\mathfrak{t}\in \mathcal{I}\},P>1,$$

where $\mathfrak{u}\left(\mathfrak{t}\right)$ takes its values in $\mathcal{S}\subset U$. A multivalued map $\mathcal{P}:\mathcal{I}\to P{C}_{1-\eta }$, is measurable as $\mathcal{P}(·)\subset \mathcal{S}$. It is clear that ${\mathcal{U}}_{ad}$ is bounded, convex & closed with ${\mathcal{U}}_{ad}=0$. Define the solution set

$$\mathcal{T}\left(\mathfrak{u}\right)=\{{\mathfrak{z}}^{\mathfrak{u}}\in {\aleph }_{\gamma }:{\mathfrak{z}}^{\mathfrak{u}}\mathfrak{u}\in {\mathcal{U}}_{ad}\}.$$

Also, the set of all $A_{ad}=\{({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u});\mathfrak{u}\in {\mathcal{U}}_{ad};{\mathfrak{z}}^{\mathfrak{u}}\in \mathcal{T}\left(\mathfrak{u}\right)\}$.

Theorem 4.1. The system (1.1) is optimal controllable together with the assumptions (H1)-(H8) provided

$$\mathfrak{I}({\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0},{\mathfrak{u}}^0)={\int }_0^T\mathfrak{L}(\mathfrak{t},{\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0}\left(\mathfrak{t}\right),{\mathfrak{u}}^0\left(\mathfrak{t}\right))d\mathfrak{t}\le \mathfrak{I}({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u}),\forall ({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u})\in A_{ad}.$$

Proof. Define $\mathfrak{I}\left(\mathfrak{u}\right)=\underset{{\mathfrak{z}}^{\mathfrak{u}}\in \mathcal{T}\left(\mathfrak{u}\right)}{\text{inf}}\mathfrak{I}({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u})$. Initially we prove $\mathfrak{I}({\overline{\mathfrak{z}}}^{\mathfrak{u}},\mathfrak{u})=\mathfrak{I}\left(\mathfrak{u}\right),{\mathfrak{z}}^{\mathfrak{u}}\in \mathcal{T}\left(\mathfrak{u}\right)$. If $\mathfrak{I}\left(\mathfrak{u}\right)=+\infty$ or $\mathcal{T}\left(\mathfrak{u}\right)$ has finite elements, the proof is trivial. Using (H8)(iv), $\mathfrak{I}\left(\mathfrak{u}\right)>-\infty$. Let $\mathfrak{I}\left(\mathfrak{u}\right)<\infty$. By infimum properties, a sequence ${\left\{{\mathfrak{z}}_n^{\mathfrak{u}}\right\}}_{n=1}^{\infty }\in \mathcal{T}\left(\mathfrak{u}\right)$ satisfies $\mathfrak{I}({\mathfrak{z}}_n^{\mathfrak{u}},\mathfrak{u})\to \mathfrak{I}\left(\mathfrak{u}\right)$ as n → ∞. Using reflexive property, $\left\{{\mathfrak{u}}^0\right\}\in \mathcal{T}\left(\mathfrak{u}\right)$ provided ${\mathfrak{u}}^0\in {\mathcal{U}}_{ad}$.

For n ≥ 1,

$$\left({\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)=\{\begin{array}{c}{S}_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},{\mathfrak{z}}_n^{\mathfrak{u}}\left(\mathfrak{t}\right))\hfill \\ +{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in (0,{\mathfrak{t}}_1];\hfill \\ S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,{\mathfrak{z}}_n^{\mathfrak{u}}\left({\mathfrak{e}}_k\right))]\hfill \\ +\mathcal{K}(\mathfrak{t},{\mathfrak{z}}_n^{\mathfrak{u}}\left(\mathfrak{t}\right))+{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}],\hfill \end{array}$$

where

$$\left({\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)=\left({\mathcal{G}}_1{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)+\left({\mathcal{G}}_2{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)=\left(\mathcal{G}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right).$$

To prove $\left(\mathcal{B}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right):\{{\mathcal{G}}_2\left(\mathfrak{t}\right);{\mathfrak{z}}_n^{\mathfrak{u}}\in \aleph \gamma \}$ is relatively compact in PC_1−η for each $\mathfrak{t}\in \mathcal{I}$.

It is clear that $\mathcal{B}\left(0\right):\{{\mathcal{G}}_2\left(0\right);{\mathfrak{z}}_n^{\mathfrak{u}}\in \aleph \gamma \}$ is relatively compact. For any $\mathfrak{u}\in U,\mathfrak{t}\in \mathcal{I},{\mathfrak{z}}_n^{\mathfrak{u}}\in {\aleph }_{\gamma }$,

$$\left({\mathcal{G}}_2{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)={\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)\left)\right]d\omega .$$

By (H3), and the property of admissible of control functions the set ${\mathcal{W}}_{ϵ}=\left\{K_{p_1}(\mathfrak{t}-\omega )\right[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))];0\le {\mathfrak{e}}_k\le {\mathfrak{t}}_k-ϵ\}$ is relatively compact. Therefore, $\overline{{\mathcal{W}}_{ϵ}}$, the convex hull of ${\mathcal{W}}_{ϵ}$ is compact due to Lemma 2.3(ii). Using Lemma 2.5, we can conclude $\left({\mathcal{G}}_2^{ϵ}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)\in \overline{{\mathcal{W}}_{ϵ}}$ for all $\mathfrak{t}\in \mathcal{I}$. Therefore ${\mathcal{B}}_{ϵ}\left(\mathfrak{t}\right):\{\left({\mathcal{G}}_2^{ϵ}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right);{\mathfrak{z}}_n^{\mathfrak{u}}\in \aleph \gamma \}$ is relatively compact in PC_1−η. For $\mathfrak{t}\in (0,{\mathfrak{t}}_1]$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2^{ϵ}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)\left|\right|\hfill \\ \le \left|\right|{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega -{\int }_0^{\mathfrak{t}-ϵ}\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega \left|\right|\hfill \\ \le {\int }_{\mathfrak{t}-ϵ}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega \hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f{\int }_{\mathfrak{t}-ϵ}^{\mathfrak{t}}\left|\right|{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)\left|\right|d\omega +{\mathcal{M}}_1{\mathcal{M}}_b{\left|\right|\mathfrak{u}\left|\right|}_{{\mathcal{L}}_P}.\hfill \end{array}$$

Similarly, for $\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}]$,

$$\begin{array}{c}\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2^{ϵ}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)\left|\right|\hfill \\ \le \left|\right|{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega -{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}-ϵ}\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega \left|\right|\hfill \\ \le {\int }_{\mathfrak{t}-ϵ}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )\left[\mathfrak{F}\right(\omega ,{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right))+B\mathfrak{u}\left(\omega \right)]d\omega \hfill \\ \le {\mathcal{M}}_1{\mathcal{L}}_f{\int }_{\mathfrak{t}-ϵ}^{\mathfrak{t}}\left|\right|{\mathfrak{z}}_n^{\mathfrak{u}}\left(\omega \right)\left|\right|d\omega +{\mathcal{M}}_1{\mathcal{M}}_b{\left|\right|\mathfrak{u}\left|\right|}_{{\mathcal{L}}_P},\hfill \end{array}$$

implies that $\text{lim}ϵ\to 0\left|\right|\left({\mathcal{G}}_2{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)-\left({\mathcal{G}}_2^{ϵ}{\mathfrak{z}}_n^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)\left|\right|=0$. Hence ${\mathcal{B}}_{ϵ}\left(\mathfrak{t}\right)$, is a family of relatively compact sets. Moreover, ${\mathcal{G}}_1{\mathfrak{z}}_n^{\mathfrak{u}}$ is bounded and equicontinuous in ℵ_γ. By (3.16) and (3.18) we have

$$\begin{array}{cc}& \ell \left({\mathfrak{z}}_n^{\mathfrak{u}}\right)\le [\mathcal{M}\mathrm{\lambda }+2{\mathcal{L}}_p^{*}+{\mathcal{M}}_{\mathcal{I}}(\mathcal{M}+1)+4{\mathcal{L}}_{\mathfrak{u}}^{*}{\sigma }^{*}{\mathcal{M}}_1{\mathcal{M}}_b{\mathcal{M}}_w+4{\mathcal{M}}_1{\mathcal{L}}_k]\ell \left({\mathfrak{z}}_n^{\mathfrak{u}}\right),\end{array}$$

leads to $\ell \left({\left\{{\mathfrak{z}}_n^{\mathfrak{u}}\right\}}_{n=0}^{\infty }\right)=0$ by using (3.3). Hence, ${\left\{{\mathfrak{z}}_n^{\mathfrak{u}}\right\}}_{n=1}^{\infty }$ is relatively compact in PC_1−η. Assume ${\tilde{\mathfrak{z}}}^{\mathfrak{u}}$, a subsequence in PC_1−η of ${\left\{{\mathfrak{z}}_n^{\mathfrak{u}}\right\}}_{n=0}^{\infty }$ such that ${\mathfrak{z}}_n^{\mathfrak{u}}\to {\tilde{\mathfrak{z}}}^{\mathfrak{u}}$ as lim n → ∞. Moreover, by Lebesgue theorem and (H1), (H3), (H5)

$$\left({\tilde{\mathfrak{z}}}^{\mathfrak{u}}\right)\left(\mathfrak{t}\right)=\{\begin{array}{c}{S}_{p_1,p_2}\left(\mathfrak{t}\right)[\mathrm{\lambda }{\int }_0^T{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\omega \right)d\omega +c-\mathcal{K}(0,\mathfrak{z}\left(0\right))]+\mathcal{K}(\mathfrak{t},{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\mathfrak{t}\right))\hfill \\ +{\int }_0^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in (0,{\mathfrak{t}}_1];\hfill \\ S_{p_1,p_2}(\mathfrak{t}-{\mathfrak{e}}_k)[\frac{1}{\mathrm{\Gamma }\left(p_1\right)}{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{e}}_k}{({\mathfrak{e}}_k-\omega )}^{p_1-1}{\mathcal{I}}_k(\omega ,{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left({\mathfrak{t}}_k^{-}\right))d\omega -\mathcal{K}({\mathfrak{e}}_k,{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left({\mathfrak{e}}_k\right))]\hfill \\ +\mathcal{K}(\mathfrak{t},{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\mathfrak{t}\right))+{\int }_{{\mathfrak{e}}_k}^{\mathfrak{t}}{K}_{p_1}(\mathfrak{t}-\omega )[B\mathfrak{u}\left(\omega \right)+\mathfrak{F}(\omega ,{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\omega \right)\left)\right]d\omega ,\mathfrak{t}\in ({\mathfrak{e}}_k,{\mathfrak{t}}_{k+1}].\hfill \end{array}$$

Then, ${\tilde{\mathfrak{z}}}^{\mathfrak{u}}\in \mathcal{T}\left(\mathfrak{u}\right)$ is continuously embedded in $L^1(\mathcal{I},U)$, by Balder’s theorem [49] and (H8),

$$\mathfrak{I}\left(\mathfrak{u}\right)=\underset{n\to \infty }{\text{lim}}{\int }_0^T\mathfrak{L}(\mathfrak{t},{\mathfrak{z}}_n^{\mathfrak{u}}\left(\mathfrak{t}\right),\mathfrak{u}\left(\mathfrak{t}\right))d\mathfrak{t}\ge {\int }_0^T\mathfrak{L}(\mathfrak{t},{\tilde{\mathfrak{z}}}^{\mathfrak{u}}\left(\mathfrak{t}\right),\mathfrak{u}\left(\mathfrak{t}\right))d\mathfrak{t}=\mathfrak{I}({\tilde{\mathfrak{z}}}^{\mathfrak{u}},\mathfrak{u})\ge \mathfrak{I}\left(\mathfrak{u}\right),$$

which shows $\mathfrak{I}({\tilde{\mathfrak{z}}}^{\mathfrak{u}},\mathfrak{u})\to \mathfrak{I}\left(\mathfrak{u}\right)$. Therefore, $\mathfrak{I}\left(\mathfrak{u}\right)$ reaches its least value at ${\tilde{\mathfrak{z}}}^{\mathfrak{u}}\in \mathcal{T}\left(\mathfrak{u}\right)$ for every $\mathfrak{u}\in {\mathcal{U}}_{ad}$.

Also, consider ${\mathfrak{u}}^0\in {\mathcal{U}}_{ad}$ such that $\mathfrak{I}\left(\mathfrak{u}\right)=\underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right)$. By the infimum property, ${\left\{{\mathfrak{u}}_n\right\}}_{n=0}^{\infty }\subseteq {\mathcal{U}}_{ad}$ provided $\underset{n\to \infty }{\text{lim}}\mathfrak{I}\left({\mathfrak{u}}_n\right)=\underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right)$. Since ${\left\{{\mathfrak{u}}_n\right\}}_{n=0}^{\infty }$ in ${\mathcal{L}}_P(\mathcal{I},U)$ is bounded for P > 1, ${\mathfrak{u}}^0\in {\mathcal{L}}_P(\mathcal{I},U)$ and by relative compactness of ${\mathfrak{z}}_n^{\mathfrak{u}}$ there is a subsequence ${\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0}\in P{C}_{1-\eta }$ as $\underset{n\to \infty }{\text{lim}}{\mathfrak{z}}_n^{\mathfrak{u}}\to {\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0}$. Using Balder’s theorem [49] and the property that $P{C}_{1-\eta }\to \mathcal{L}(\mathcal{I},U)$ is continuous, we conclude

$$\begin{array}{cc}\underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right)& =\underset{n\to \infty }{\text{lim}}{\int }_0^T\mathfrak{L}(\mathfrak{t},{\mathfrak{z}}_n^{{\mathfrak{u}}_n}\left(\mathfrak{t}\right),{\mathfrak{u}}_n\left(\mathfrak{t}\right))d\mathfrak{t}\ge {\int }_0^T\mathfrak{L}(\mathfrak{t},{\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0}\left(\mathfrak{t}\right),{\mathfrak{u}}^0\left(\mathfrak{t}\right))d\mathfrak{t}=\mathfrak{I}({\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0},{\mathfrak{u}}^0)\hfill \\ & =\mathfrak{I}\left({\mathfrak{u}}^0\right)\ge \underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right).\hfill \end{array}$$

Therefore, $\mathfrak{I}\left(\mathfrak{u}\right)=\underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right)$, leads that $\mathcal{I}$ attains its minimum at ${\mathfrak{u}}^0\in {\mathcal{U}}_{ad}$. Subsequently, we have

$$\mathcal{I}({\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0},{\mathfrak{u}}^0)=\underset{\mathfrak{u}\in {\mathcal{U}}_{ad}}{\text{inf}}\mathfrak{I}\left(\mathfrak{u}\right)=\underset{({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u})\in A_{ad}}{\text{inf}}\mathfrak{I}({\mathfrak{z}}^{\mathfrak{u}},\mathfrak{u}).$$

Hence, $({\tilde{\mathfrak{z}}}^{{\mathfrak{u}}^0},{\mathfrak{u}}^0)\in A_{ad}$. This completes the proof.

5 Application

Consider a nonlinear equation of the form given below to validate the outcome,

$$\begin{array}{c}{\mathcal{D}}^{\frac{1}{2},\frac{2}{3}}[\mathfrak{z}(\varsigma ,\mathfrak{t})-\text{exp}(\mathfrak{z}\left(\frac{3\varsigma \mathfrak{t}}{4}\right)\left)\right]=\frac{{\partial }^2}{\partial {\mathfrak{t}}^2}[\mathfrak{z}(\varsigma ,\mathfrak{t})-\text{exp}(\mathfrak{z}\left(\frac{3\varsigma \mathfrak{t}}{4}\right)\left)\right]\hfill \\ +{\int }_0^1\mathfrak{h}(\varsigma ,\mathfrak{t})\mathfrak{u}(\varsigma ,\mathfrak{t})d\mathfrak{t}+\text{sin}[\text{exp}\left(\varsigma \mathfrak{t}\right)+\mathfrak{z}(\frac{2\varsigma \mathfrak{t}}{5}\left)\right],\varsigma \in (0,3]\backslash (1,2],\hfill \\ \mathfrak{z}(\varsigma ,0)=\mathfrak{z}(\varsigma ,\pi )=0,\varsigma \in [1,2],\hfill \\ \mathfrak{z}(\varsigma ,\mathfrak{t})=\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}\right)}{\int }_{{\mathfrak{t}}_k}^{\mathfrak{t}}\frac{1}{{(\varsigma -\mathfrak{t})}^{\frac{1}{2}}}\left(\frac{\mathfrak{z}\left({\frac{1}{2}}^{-}\right)}{20exp\left(\mathfrak{t}\right)+1}\right)d\mathfrak{t},\varsigma ,\mathfrak{t}\in [1,2],\hfill \\ I_0^{\frac{5}{6}}\mathfrak{z}(0,\mathfrak{t})={\int }_0^3\mathfrak{z}(\omega ,\mathfrak{t})d\mathfrak{t}+5,\mathfrak{t}\in [0,\pi ],\hfill \end{array}$$ (4.1)

with $B\mathfrak{u}\left(\varsigma \right)\left(\mathfrak{t}\right)={\int }_0^1\mathfrak{h}(\varsigma ,\mathfrak{t})\mathfrak{u}(\varsigma ,\mathfrak{t})d\mathfrak{t}$, & $p_1=\frac{1}{2},p_2=\frac{2}{3},\eta =\frac{5}{6}$. Assume $\mathcal{N}=L^2[0,\pi ]$ and $A:D\left(A\right)\subset \mathcal{N}\to \mathcal{N}$ by $A\mathfrak{z}=\frac{{\partial }^2}{\partial {\mathfrak{t}}^2}\left(\mathfrak{z}\right),$

$$\begin{array}{c}\hfill D\left(A\right)=\{\mathfrak{z}\in \mathcal{N},{\mathfrak{z}}_{\mathfrak{t}},{\mathfrak{z}}_{\mathfrak{t}\mathfrak{t}}\in \mathcal{N},\mathfrak{z}(\varsigma ,0)=\mathfrak{z}(\varsigma ,\pi )=0\}.\end{array}$$

It is clear that A is a strongly continuous semigroup and $\left(S\right(\varsigma \left)\mathfrak{z}\right)$ in $\mathcal{N}$,

$$\left(S\left(\varsigma \right)\mathfrak{z}\right)\left(\mathfrak{t}\right)=\{\begin{array}{cc}{\int }_0^{\pi }\mathcal{M}(\varsigma ,\mathfrak{t}-\omega )\mathfrak{z}\left(\omega \right)d\omega ,\hfill & \varsigma >0,\hfill \\ \mathfrak{z}\left(\mathfrak{t}\right),\hfill & \varsigma =0,\hfill \end{array}$$

with

$$\mathcal{M}(\varsigma ,\mathfrak{t})=\sqrt{\frac{2}{\pi }}exp(-(\frac{{\mathfrak{t}}^2}{4\varsigma }\left)\right),\varsigma >0,0<\mathfrak{t}<\pi ,$$

with $\mathfrak{z}\left(\varsigma \right)\left(\mathfrak{t}\right)=\mathfrak{z}(\varsigma ,\mathfrak{t})$. This leads to the conclusion $\left|\right|S\left(\varsigma \right)\left|\right|\le \mathcal{M}$. Let ${\mathcal{U}}_{ad}={\{\mathfrak{u}\in U|\left|\right|\mathfrak{u}\left|\right|}_{L^2[[0,3],U]}\le 1\}.$ Hence

$$\mathfrak{I}(\mathfrak{z},\mathfrak{u})={\int }_{{\mathfrak{t}}_k}^{{\mathfrak{t}}_{k+1}}{\int }_0^{\pi }{\left|\mathfrak{z}(\varsigma ,\mathfrak{t})\right|}^{2}d\mathfrak{t}d\varsigma +{\int }_{{\mathfrak{t}}_k}^{{\mathfrak{t}}_{k+1}}{\int }_0^{\pi }{\left|\mathfrak{u}(\varsigma ,\mathfrak{t})\right|}^{2}d\mathfrak{t}d\varsigma$$

related to the system (4.1) which correlates the system (1.1) with

$$\mathfrak{I}(\mathfrak{z},\mathfrak{u})={\int }_{{\mathfrak{t}}_k}^{{\mathfrak{t}}_{k+1}}({\left|\right|\mathfrak{z}\left(\varsigma \right)\left|\right|}^2+{\left|\right|\mathfrak{u}\left(\varsigma \right)\left|\right|}_U^2)d\varsigma .$$

Therefore, (H1)–(H8) satisfied. This completes the proof.

6 Conclusion

We examine the total controllability of non-instantaneous Hilfer fractional neutral system under integral boundary condition. By incorporating HFD with semigroup operator theory and Laplace transform technique, the integral solution is derived. Controllability outcomes were attained using Kuratowski’s measure with contraction theory. Furthermore, the sequencing technique has been used to discuss the existence of the optimal pair for the system. To confirm the derived consequences, an example is given. The concept can be extended to Hilfer stochastic differential equations.

Data Availability

All relevant data are within the paper.

Funding Statement

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2023/01/2189822).