1 Introduction

Fractional differential equations have arisen as a new branch of applied mathematics that has been utilized to build a variety of mathematical models in science, signal, image processing, biological, control theory, engineering problems, etc. The reason for this is because fractional calculus may be used to create a realistic model of a physical occurrence that is dependent not only on the current instant, but also on the prior time history. Many authors have addressed the theory of the existence of solutions for fractional differential equations. For more specifics, refer to books [1–6] and the research articles [7–29].

In mathematical control theory, the concept of controllability is very important. Under the assumption that the system is controllable, many fundamental problems in control theory can be solved, such as pole assignment, stabilizability, and optimum control. It indicates that an acceptable control can be used to steer any system’s beginning state to any final state in a finite amount of time. Controllability is important in systems described by ordinary differential equations and partial differential equations in both finite and infinite dimensional environments. Significant progress has been achieved in the controllability of linear and nonlinear deterministic systems in recent years [30–41]. Physical issues prompted the concept of nonlocal situations. Byszewski established for the first time mild solutions to nonlocal differential equations for existence and uniqueness results in [42, 43]. In [44, 45] the authors developed the ideas in fractional evolution equations. Recently, the researchers established the nonlocal fractional differential systems with or without delay by referring to the nondense domain, semigroup, cosine families, several fixed point techniques, and a measure of noncompactness. Refer to the articles for more information [46–50].

In addition, integrodifferential equations are used in a variety of scientific fields where an aftereffect or delay must be considered, for example, in biology, control theory, ecology, and medicine. In practice, integrodifferential equations are always used to describe a model that has hereditary features, one can refer to the researcher’s articles [51–55].

In recent years, authors have signified controllability results of Caputo fractional evolution systems with order \(\alpha \in (1,2)\) referring to the cosine families, Laplace transforms, and different fixed point techniques [56]. Likewise, the researchers developed nonlocal conditions in fractional evolution inclusion with order \(\alpha \in (1,2)\) using the measure of noncompactness, condensing multivalued map, and Laplace transform [46]. For fractional evolution equations of order \(r \in (1,2)\) with delay or without delay, numerous researchers have proved their existence, exact and approximate controllability by applying the nonlocal conditions, mixed Volterra–Fredholm type, cosine families, measure of noncompactness, and different fixed point techniques [41, 48, 50, 51, 54]. Furthermore, in [30, 40, 49, 53, 57] the authors used the Sobolev type, hemivariational inequalities, stochastic systems, integrodifferential systems, Clarke’s subdifferential type, and various fixed point techniques to develop approximate controllability results for fractional evolution inclusions with or without delay of order \(1< r<2\).

Controllability results for fractional differential systems with the nonlocal condition of order \(1< r<2\) by referring to the thoughts of Mainardi’s Wright-type function, the measure of noncompactness, Mönch fixed point theorem, and cosine families are still untreated in the area [58]. The preceding facts are based on the current work. Hence, consider that the semilinear fractional evolution system of order \(1< r<2\) with nonlocal conditions has the form

$$\begin{aligned} \textstyle\begin{cases} {}^{C}D_{t}^{r} z(t) = Az(t)+g(t,z(t))+Bx(t), \quad t \in V, \\ z(t)+F(z)=z_{0}, \qquad z'(0)=z_{1} \in Z, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \({}^{C}D_{t}^{r}\) is the Caputo fractional derivative of order \(1< r<2\); A is the infinitesimal generator of a strongly continuous cosine family \(\{C(t)\}_{t \ge 0}\) in a Banach space Z. Let Y be another Banach space; the state z takes values in Z and the control function x is given in \(L^{2}(V,U)\), with U as a Banach space; B is a bounded linear operator from U into Z; \(g:V \times Z \rightarrow Z\) is a given Z-valued function, and nonlocal term \(F:C(V,Z) \rightarrow Z\) and \(z_{0}\), \(z_{1}\) are elements of space Z.

We partition our article into the following sections: We recall a few fundamental definitions and preparation results in Sect. 2. In Sect. 3, we present the controllability results for system (1.1). Further, we discuss another fixed point theorem for fractional integrodifferential evolution system in Sect. 4. Finally, an application is presented for drawing the law of the main results.

2 Preliminaries

Here, we present well-known essential facts, basic definitions, lemmas, and results.

Throughout this paper, we denote by \(\mathcal{C}\) the Banach space \(C(V,Z):V \rightarrow Z\) equipped with the sup-norm \(\|z\|_{\mathcal{C}}=\sup_{t \in V}\|z(t)\|\) for \(z \in \mathcal{C}\). \(L_{c}(Z,Y)\) stands for the space of all bounded linear operators from Z to Y equipped with \(\|\cdot \|_{L_{c}(Z,Y)}\).

The domain and range of an operator A are defined by \(D(A)\) and \(R(A)\) respectively, the resolvent set of A is denoted by \(\rho ({A})\) and the resolvent of A is defined by

$$ R(\Lambda ,{A})=(\Lambda I-{A})^{-1}\in L_{c}(Z). $$

Consider that \(\|g\|_{L^{\nu }(V,\mathbb{R}^{+})}\) denotes the \(L^{\nu }(V,\mathbb{R}^{+})\) norm of g whenever g in \(L^{\nu }(V,\mathbb{R}^{+})\), \(\nu \ge 1\). Let \(L^{\nu }(V,Z)\) denote the Banach space of function \(g: V \rightarrow Z\) is Bochner integrable normed by \(\|g\|_{L^{\nu }(V,Z)}\).

Definition 2.1

([3])

The Riemann–Liouville fractional integral of order γ with the lower limit zero for \(g:[0,\infty )\rightarrow \mathbb{R}\) is given by

$$\begin{aligned}& I^{\gamma }g(t)= \frac{1}{\Gamma (\gamma )} \int _{0}^{t} \frac{g(s)}{(t-s)^{1-\gamma }} \,ds, \quad t>0, \gamma \in \mathbb{R}^{+} \end{aligned}$$

if the right-hand side is point-wise defined on \([0,\infty )\).

Definition 2.2

([3])

The Riemann–Liouville derivative of order γ with the lower limit zero for \(g:[0,\infty )\rightarrow \mathbb{R}\) is given by

$$\begin{aligned}& {{}^{L}D^{\gamma }} g(t)= \frac{1}{\Gamma (n-\gamma )}\frac{d^{n}}{dt^{n}} \int _{0}^{t} \frac{g^{(n)}(s)}{(t-s)^{\gamma +1-n}}\,ds, \quad \bigl(t>0, n-1< \gamma < n, \gamma \in \mathbb{R}^{+}\bigr). \end{aligned}$$

Definition 2.3

([3])

The Caputo derivative of order γ with the lower limit zero for g is given by

$$\begin{aligned} {{}^{C}D^{\gamma }} g(t)={{}^{L}D^{\gamma }} \Biggl(g(t)-\sum_{n=0}^{n-1} \frac{g^{(n)}(0)}{n!} t^{n} \Biggr) \quad \bigl(t>0, n-1< \gamma < n, \gamma \in \mathbb{R}^{+}\bigr). \end{aligned}$$

Remark 2.4

\({\mathbf{(1)}}\) :

If \(g(t) \in C^{n}[0,\infty )\), then

$$\begin{aligned} {{}^{C}D^{\gamma }}g(t)=\frac{1}{\Gamma (n-\gamma )} \int _{0}^{t} \frac{g^{(n)}(s)}{(t-s)^{\gamma +1-n}}\,ds=I^{n-\gamma }g^{(n)}(t), \quad (t>0, n-1< \gamma < n). \end{aligned}$$
\({\mathbf{(2)}}\) :

If g is an abstract function with values in Z, then the integrals that appear in Definitions 2.2 and 2.3 are taken in Bochner’s sense.

\({\mathbf{(3)}}\) :

Caputo derivative of a constant function is equal to zero.

Definition 2.5

([59])

A one parameter family \(\{C(t)\}_{t\in \mathbb{R}}\) of bounded linear operators mapping Z into itself is said to be a strongly continuous cosine family if and only if

\(\mathbf{(a)}\):

\(C(0) = I\);

\(\mathbf{(b)}\):

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

\(\mathbf{(c)}\):

\(C(t)z\) is strongly continuous in t on \(\mathbb{R}\) for each fixed point \(z\in Z\).

The sine family \(\{S(t)\}_{t\in \mathbb{R}}\) is associated with the strongly continuous cosine family \(\{C(t)\}_{t\in \mathbb{R}}\) which is defined by

$$\begin{aligned}& S(t)z= \int _{0}^{t} C(s)z\,ds, \quad z\in Z, t\in \mathbb{R}. \end{aligned}$$
(2.1)

Further, an operator A is said to be an infinitesimal generator of \(\{C(t)\}_{t\in \mathbb{R}}\) if

$$\begin{aligned} Az=\frac{d^{2}}{dt^{2}}C(0)z\quad \text{for all } z \in D(A), \end{aligned}$$

where the domain of A is defined by

$$ D(A) = \bigl\{ z\in Z : C(t)z \in C^{2}(\mathbb{R},Z)\bigr\} . $$

Denote a set

$$ E= \bigl\{ z\in Z : C(t)z \in C^{1}(\mathbb{R},Z)\bigr\} . $$

Clearly, A is a closed, densely-defined operator in Z, there exists \(P\ge 1\) such that \(\|C(t)\|_{L_{c}(Z)}\le P\) for \(t \ge 0\). In the sequel, we always set \(b = \frac{r}{2}\) for \(r\in (1, 2)\), as stated in [5, 46].

Definition 2.6

([60])

Let \(N^{+}\) be the positive cone of an order Banach space \((N,\le )\). A function Θ defined on the set of all bounded subsets of the Banach space Z with values in \(N^{+}\) is said to be a measure of noncompactness on Z iff

$$ \Theta (\overline{co} \zeta )=\Theta (\zeta ) $$

for any bounded subsets \(\zeta \subset Z\), where \(\overline{co} \zeta \) denotes the closed convex hull of ζ.

The measure of noncompactness Θ is said to be:

\(\mathbf{(i)}\):

monotone iff for all bounded subsets \(\zeta _{1}\), \(\zeta _{2}\) of Z, we get

$$ (\zeta _{1} \subseteq \zeta _{2})\quad \Rightarrow\quad \bigl(\Theta (\zeta _{1}) \le \Theta (\zeta _{2})\bigr); $$
\(\mathbf{(ii)}\):

nonsigular iff \(\Theta (\{a\}\cup \zeta )=\Theta (\zeta )\) for any \(a \in Z\) and every nonempty subset \(\zeta \subseteq Z\);

\(\mathbf{(iii)}\):

regular iff \(\Theta (\zeta )=0\) iff ζ in Z, where ζ is relatively compact.

One of the most important examples of measure of noncompactness is the noncompactness measure of Hausdorff β defined on each bounded subset ζ of Z by

$$\begin{aligned} \beta (\zeta )&=\inf \{\epsilon >0; \zeta \text{ can be covered by a finite number of balls} \\ &\quad \text{of radii smaller than } \epsilon \}. \end{aligned}$$

For any bounded subsets ζ, \(\zeta _{1}\), \(\zeta _{2}\) of Z.

\(\mathbf{(iv)}\):

\(\beta (\zeta _{1}+\zeta _{2}) \le \beta (\zeta _{1})+\beta (\zeta _{2})\), where \(\zeta _{1}+\zeta _{2}=\{z+w:z\in \zeta _{1}, w\in \zeta _{2}\}\);

\(\mathbf{(v)}\):

\(\beta (\zeta _{1} \cup \zeta _{2}) \le \max \{\beta (\zeta _{1}), \beta (\zeta _{2})\}\);

\(\mathbf{(vi)}\):

\(\beta (\wp \zeta ) \le |\wp |\beta (\zeta )\) for any \(\wp \in \mathbb{R}\);

\(\mathbf{(vii)}\):

If the Lipschitz continuous function \(\phi : \mathcal{D}(\phi ) \subseteq Z \rightarrow X\) with constant ℓ, then \(\beta _{X}(\phi \zeta )\le \ell \beta (\zeta )\) for any bounded subset \(\zeta \subseteq \mathcal{D}(\phi )\), where X is a Banach space.

Definition 2.7

([46])

\(z \in C(V,Z)\) is said to be a mild solution of system (1.1) if \(z(0)+F(z)=z_{0}\), \(z'(0)=z_{1}\) such that

$$\begin{aligned} z(t)&=C_{b}(t) \bigl(z_{0}-F(z) \bigr)+K_{b}(t)z_{1}+ \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)g \bigl(s,z(s)\bigr)\,ds \\ &\quad{}\times \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)Bx(s)\,ds,\quad t \in V, \end{aligned}$$
(2.2)

where \(C_{b}(\cdot )\), \(K_{b}(\cdot )\), and \(T_{b}(\cdot )\) are called the characteristic solution operators and given by

$$\begin{aligned} &C_{b}(t)= \int _{0}^{\infty }S_{b}(\xi )C \bigl(t^{b} \xi \bigr)\,d\xi , \qquad K_{b}(t)= \int _{0}^{t} C_{b} (s)\,ds, \\ &T_{b}(t)= \int _{0}^{\infty }b \xi S_{b}(\xi )S \bigl(t^{b}\xi \bigr)\,d\xi , \qquad S_{b}( \xi )= \frac{1}{b}\xi ^{-1-\frac{1}{b}}\zeta _{b}\bigl(\xi ^{-\frac{1}{b}}\bigr) \ge 0, \\ &\zeta _{b}(\xi )=\frac{1}{\pi }\sum _{n=1}^{\infty }(-1)^{n-1}\xi ^{-nb-1} \frac{\Gamma (nb+1)}{n!}\sin (n\pi b),\quad \xi \in (0,\infty ), \end{aligned}$$

and \(S_{b}(\cdot )\) is the Mainardi’s Wright-type function defined on \((0,\infty )\) such that

$$ S_{b}(\xi )\ge 0 \quad \text{for } \xi \in (0,\infty ) \text{ and } \int _{0}^{\infty }S_{b}(\xi )\,d\xi =1. $$

Lemma 2.8

([46])

The operators \(C_{b}(t)\), \(K_{b}(t)\), and \(T_{b}(t)\) have the following properties:

\(\mathbf{(a)}\):

For any fixed \(t\ge 0\), the operators \(C_{b}(t)\), \(K_{b}(t)\), and \(T_{b}(t)\) are linear and bounded operators, i.e., for any \(z \in Z\), the following estimates hold:

$$ \bigl\Vert C_{b}(t)z \bigr\Vert \le P \Vert z \Vert ,\qquad \bigl\Vert K_{b}(t)z \bigr\Vert \le P \Vert z \Vert t, \qquad \bigl\Vert T_{b}(t)z \bigr\Vert \le \frac{P}{\Gamma (2b)} \Vert z \Vert t^{b}; $$
\(\mathbf{(b)}\):

\(\{C_{b}(t),t\ge 0\}\), \(\{K_{b}(t),t\ge 0\}\), and \(\{t^{b-1} T_{b}(t),t\ge 0\}\) are strongly continuous.

\(\mathbf{(c)}\):

For any \(t \in V\) and any bounded subsets \(\mathcal{D} \subset Z\), \(t \rightarrow \{C_{b}(t)z:z \in \mathcal{D}\}\), \(t \rightarrow \{K_{b}(t)z:z \in \mathcal{D}\}\) and \(t \rightarrow \{T_{b}(t)z:z \in \mathcal{D}\}\) are equicontinuous if \(\|C(t_{2}^{b}(\xi ))z-C(t_{1}^{b}(\xi ))z\| \rightarrow 0\) with respect to \(z \in \mathcal{D}\) as \(t_{2} \rightarrow t_{1}\) for any fixed \(\xi \in (0,\infty )\) and \(\|K(t_{2}^{b}(\xi ))z-K(t_{1}^{b}(\xi ))z\| \rightarrow 0\) with respect to \(z \in \mathcal{D}\) as \(t_{2} \rightarrow t_{1}\) for any fixed \(\xi \in (0,\infty )\).

Lemma 2.9

([59])

\(\mathbf{(i)}\) :

There exist \(P\ge 1\) and \(\omega \ge 0\) such that \(\|C(t)\|_{L_{c}(Z)}\le Pe^{\omega |t|}\) for all \(t\in \mathbb{R}\);

\(\mathbf{(ii)}\) :

\(\|S(t_{2})-S(t_{1})\|_{L_{c}(Z)}\le P |\int _{t_{1}}^{t_{2}}e^{ \omega |s|}\,ds |\) for all \(t_{2},t_{1} \in \mathbb{R}\).

\(\mathbf{(iii)}\) :

If \(z \in E\), then \(S(t)z \in D(A)\) and \(\frac{d}{dt}C(t)z = A S(t)z\).

Lemma 2.10

Let \(\{C(t)\}_{t \in \mathbb{R}}\) be a strongly continuous cosine family in Z, then

$$ \lim_{t\rightarrow 0} \frac{1}{t}S(t)z = z \quad \textit{for every } z \in Z. $$

Lemma 2.11

([59])

Let \(\{C(t)\}_{t\in \mathbb{R}}\) be a strongly continuous cosine family in Z satisfying \(\|C(t)\|_{L_{c}(Z)}\le P e^{\omega |t|}\), \(t \in \mathbb{R}\). Then for \(R e \Lambda > \omega \), \(\Lambda ^{2} \in \rho (A)\) and

$$ \Lambda R\bigl(\Lambda ^{2};A\bigr)z= \int _{0}^{\infty }e^{-\Lambda t}C(t)z \,dt, \qquad R \bigl(\Lambda ^{2};A\bigr)z= \int _{0}^{\infty }e^{-\Lambda t}S(t)z \,dt, \quad \forall z \in Z, $$

where A is the infinitesimal generator of \(\{C(t)\}_{t\in \mathbb{R}}\).

Theorem 2.12

([41])

If \(\{x_{n}\}_{n=1}^{\infty }\) is a sequence of Bochner integrable functions from V into Z with the estimation \(\|x_{n}(t)\| \le \delta (t)\) for almost all \(t \in V\) and for every \(n \ge 1\), where \(\delta \in L^{1}(V,\mathbb{R})\), then \(\varphi (t)=\beta (\{x_{n}(t):n \ge 1\})\) in \(L^{1}(V,\mathbb{R})\) and satisfies

$$ \beta \biggl( \biggl\{ \int _{0}^{t}x_{n}(s)\,ds:n\ge 1 \biggr\} \biggr) \le 2 \int _{0}^{t}\varphi (s)\,ds. $$

Definition 2.13

(Nonlocal controllability)

System (1.1) is called nonlocally controllable on V iff, for every \(z_{0},z_{1},y \in Z\), there exists \(x \in L^{2}(V,U)\) such that a mild solution z of system (1.1) satisfies \(z(c)+F(z)=y\).

Lemma 2.14

([61])

Let \(\mathcal{D}\) be a closed convex set of a Banach space Z and \(0 \in \mathcal{D}\). Consider that \(N:\mathcal{D} \rightarrow Z\) is a continuous map which satisfies Mönch’s condition, i.e., if

$$ \mathcal{H} \subseteq \mathcal{D} \textit{ is countable and } \mathcal{H} \subseteq \overline{co}\bigl(\{0\}\cup N (\mathcal{H})\bigr) \Rightarrow \overline{\mathcal{H}} \textit{ is compact.} $$

Then N has a fixed point in \(\mathcal{D}\).

3 Main results

We propose and demonstrate the requirements for the existence of system (1.1). In order to establish the results, we need the following hypotheses:

\(\mathbf{(H_{1})}\):
\(\mathbf{(i)}\):

\(\{C(t):t \geq 0\}\) in Z;

\(\mathbf{(ii)}\):

For any bounded subsets \(\mathcal{D} \subset Z\) and \(z\in \mathcal{D}\), \(\|C(t_{2}^{b}(\xi ))z-C(t_{1}^{b}(\xi ))z\| \rightarrow 0\) as \(t_{2} \rightarrow t_{1}\) for each fixed \(\xi \in (0,\infty )\).

\(\mathbf{(H_{2})}\):

The function \(g:V \times Z \rightarrow Z\) satisfies:

\(\mathbf{(i)}\):

Carathéodory condition: \(g(\cdot ,z)\) is measurable for every \(z \in Z\) and \(g(t,\cdot )\) is continuous for a.e. \(t \in V\);

\(\mathbf{(ii)}\):

There exist a constant \(b_{1} \in (0,b)\) and \(q \in L^{\frac{1}{b_{1}}}(V,\mathbb{R}^{+})\) and a nondecreasing continuous function \(\zeta :\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) such that

$$ \bigl\Vert g(t,z) \bigr\Vert \le q(t)\zeta \bigl( \Vert z \Vert \bigr),\quad z \in Z, t \in V, $$

where ζ satisfies \(\lim \inf_{n \rightarrow \infty } \frac{\zeta (n)}{n}=0\).

\(\mathbf{(iii)}\):

There exist a constant \(b_{2} \in (0,b)\) and \(j \in L^{\frac{1}{b_{2}}}(V,\mathbb{R}^{+})\) such that, for any bounded subset \(\mathcal{D} \subset Z\),

$$ \beta \bigl(g(t,\mathcal{D})\bigr) \le j(t) \beta (\mathcal{D}) \quad \text{for a.e. } t \in V, $$

where β is the Hausdorff measure of noncompactness.

\(\mathbf{(H_{3})}\):
\(\mathbf{(i)}\):

The linear operator \(B: L^{2}(V,U) \rightarrow L^{1}(V,Z)\) is bounded, \(W: L^{2}(V,U) \rightarrow Z\) defined by

$$\begin{aligned} Wx= \int _{0}^{c} (c-s)^{b-1} T_{b}(c-s) Bx(s)\,ds \end{aligned}$$

has an inverse operator \(W^{-1}\) which takes values in \(L^{2}(V,U)/ \ker W\), and there exist \(P_{1},P_{2} \ge 0\) such that \(\|B\|_{L_{c}(U,Z)} \le P_{1}\),

$$ \bigl\Vert W^{-1} \bigr\Vert _{L_{c}(Z,L^{2}(V,U)/ \ker W)} \le P_{2}; $$
\(\mathbf{(ii)}\):

There exist a constant \(b_{0} \in (0,b)\) and \(\mathcal{K}_{W} \in L^{\frac{1}{b_{0}}}(V,\mathbb{R}^{+})\) such that, for any bounded set \(\phi \subset Z\),

$$ \beta \bigl(\bigl(W^{-1} \phi \bigr) (t)\bigr) \le \mathcal{K}_{W}(t)\beta (\phi ). $$
\(\mathbf{(H_{4})}\):
\(\mathbf{(i)}\):

The continuous and compact operator \(F:C(V,Z) \rightarrow Z\);

\(\mathbf{(ii)}\):

F satisfies \(\lim \|v\|_{\mathcal{C} \rightarrow \infty } \frac{\|F(v)\|}{\|v\|_{\mathcal{C}}}=0\).

For our convenience, let us take

$$\begin{aligned}& \mathcal{O}_{n}:= \biggl[ \biggl(\frac{1-b_{n}}{2b-b_{n}} \biggr)c^{ \frac{2b-b_{n}}{1-b_{n}}} \biggr]^{1-b_{n}},\quad n=0,1,2; \\& P_{3}:=\mathcal{O}_{1} \Vert q \Vert _{L^{\frac{1}{b_{1}}}(V,\mathbb{R}^{+})}, \qquad P_{4}:=\mathcal{O}_{0} \Vert \mathcal{K}_{W} \Vert _{L^{\frac{1}{b_{0}}}(V, \mathbb{R}^{+})},\qquad P_{5}=\mathcal{O}_{2} \Vert j \Vert _{L^{\frac{1}{b_{2}}}(V, \mathbb{R}^{+})}. \end{aligned}$$

Theorem 3.1

If \(\mathbf{(H_{1})}\)–\(\mathbf{(H_{4})}\) are satisfied, then system (1.1) has a mild solution on V if

$$\begin{aligned} \widehat{L}= \biggl(1+\frac{2PP_{1}P_{4}}{\Gamma (2b)} \biggr) \frac{2PP_{5}}{\Gamma (2b)}< 1 \quad \textit{for some } \frac{3}{2}< b< 2. \end{aligned}$$
(3.1)

Proof

Using \(\mathbf{(H_{3})}\mathbf{(i)}\), for an arbitrary function \(z \in \mathcal{C}\), we define the control \(x_{z}(t)\) by

$$\begin{aligned} x_{z}(t)&=W^{-1} \biggl[y-F(z)-C_{b}(c) \bigl(z_{0}-F(z)\bigr) -K_{b}(c)z_{1} \\ &\quad {}- \int _{0}^{c}(c-s)^{b-1}T_{b}(c-s)g \bigl(s,z(s)\bigr)\,ds \biggr](t),\quad t \in V. \end{aligned}$$
(3.2)

Define the operator \(\Psi :\mathcal{C} \rightarrow \mathcal{C}\) such that

$$\begin{aligned} (\Psi z) (t)=C_{b}(t) \bigl(z_{0}-F(z)\bigr) +K_{b}(t)z_{1}+\Pi (g+Bx_{z}) (t), \end{aligned}$$
(3.3)

where \(\Pi (g+Bx_{z}) \in \mathcal{C}\) defined by

$$\begin{aligned} \Pi (g+Bx_{z}) (t) &= \int _{0}^{t}(t-s)^{b-1}T_{b}(t-s)g \bigl(s,z(s)\bigr)\,ds \\ &\quad{}+ \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)B W^{-1} \biggl[y-F(z)-C_{b}(c) \bigl(z_{0}-F(z) \bigr) \\ &\quad{}-K_{b}(c)z_{1} - \int _{0}^{c}(c-\iota )^{b-1}T_{b}(c- \iota )g\bigl( \iota ,z(\iota )\bigr)\,d\iota \biggr](s)\,ds \end{aligned}$$

has a fixed point z, which is a mild solution of system (1.1). Clearly, \((\Psi z)(c)=y-F(z)\); this means that \(x_{z}\) moves system (1.1) from \(z_{0}\) to y in finite time c. This implies that system (1.1) is completely controllable on V.

Now, we introduce the operators \(\Psi _{1}\) and \(\Psi _{2}\) defined by

$$ (\Psi _{1} z) (t)=C_{b}(t) \bigl(z_{0}-F(z) \bigr) +K_{b}(t)z_{1},\quad t \in V, $$

and

$$ (\Psi _{2} z) (t)=\Pi (g+Bx_{z}) (t),\quad t \in V. $$

It is clear that

$$ \Psi =\Psi _{1}+\Psi _{2}. $$

We prove that Ψ satisfies the results of Lemma 2.14.

Step 1: To demonstrate that there is \(\varrho >0\) such that

$$ \Psi (\mathcal{B}_{\varrho })\subseteq \mathcal{B}_{\varrho }, $$

where \(\mathcal{B}_{\varrho }=\{z \in \mathcal{C}:\|z\|_{\mathcal{C}} \le \varrho \}\). If not, then for each positive number ϱ, there exists \(z^{\varrho }(\cdot )\) in \(\mathcal{B}_{\varrho }\); however, \(\Psi (z^{\varrho }) \notin \mathcal{B}_{\varrho }\), i.e.,

$$ \bigl\Vert \Psi \bigl(z^{\varrho }\bigr) (t) \bigr\Vert >\varrho \quad \text{for some } t \in V. $$

Using Lemma 2.8, \(\mathbf{(H_{2})}\mathbf{(ii)}\), \(\mathbf{(H_{3})}\), and Hölder’s inequality, we have

$$\begin{aligned} & \bigl\Vert x(t) \bigr\Vert \\ &\quad =P_{2} \biggl[ \Vert y \Vert + \bigl\Vert F(z) \bigr\Vert + \bigl\Vert C_{b}(c) \bigl(z_{0}-F(z)\bigr) \bigr\Vert + \bigl\Vert K_{b}(c)z_{1} \bigr\Vert \\ &\qquad {}+ \int _{0}^{c}(c-s)^{b-1} \bigl\Vert T_{b}(c-s)g\bigl(s,z(s)\bigr) \bigr\Vert \,ds \biggr] \\ &\quad \le P_{2} \biggl[ \Vert y \Vert + \bigl\Vert F(z) \bigr\Vert +P \Vert z_{0} \Vert +P \bigl\Vert F(z) \bigr\Vert +Pc \Vert z_{1} \Vert \\ &\qquad{} +\frac{P}{\Gamma (2b)} \int _{0}^{c}(c-s)^{2b-1} \bigl\Vert g \bigl(s,z(s)\bigr) \bigr\Vert \,ds \biggr] \\ &\quad \le P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert +PP_{2} c \Vert z_{1} \Vert \\ &\qquad {}+\frac{PP_{2}}{\Gamma (2b)} \int _{0}^{c}(c-s)^{2b-1}q(s)\zeta \bigl( \Vert z \Vert \bigr)\,ds \\ & \quad \le P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert +PP_{2} c \Vert z_{1} \Vert + \frac{PP_{2}P_{3}}{\Gamma (2b)}\zeta \bigl( \Vert z \Vert _{\mathcal{C}}\bigr). \end{aligned}$$

Then

$$\begin{aligned} \bigl\Vert z^{\varrho } \bigr\Vert _{\mathcal{C}} &\le \varrho < \bigl\Vert \bigl(\Psi z^{\varrho }\bigr) (t) \bigr\Vert \\ & \le \bigl\Vert C_{b}(t) \bigl(z_{0}-F(z)\bigr) \bigr\Vert + \bigl\Vert K_{b}(t)z_{1} \bigr\Vert + \int _{0}^{t}(t-s)^{b-1} \\ &\quad {}\times \bigl\Vert T_{b}(t-s)g\bigl(s,z(s)\bigr) \bigr\Vert \,ds + \int _{0}^{t} (t-s)^{b-1} \bigl\Vert T_{b}(t-s)Bx(s) \bigr\Vert \,ds \\ & \le P \biggl[1+\frac{PP_{1}P_{2}}{\Gamma (2b)} \biggl( \frac{c^{4b-1}}{4b-1} \biggr)^{\frac{1}{2}} \biggr] \Vert z_{0} \Vert +P \biggl[1+ \frac{(1+P)P_{1}P_{2}}{\Gamma (2b)} \biggl(\frac{c^{4b-1}}{4b-1} \biggr)^{ \frac{1}{2}} \biggr] \bigl\Vert F(z) \bigr\Vert \\ &\quad {}+Pc \biggl[1+\frac{PP_{1}P_{2}}{\Gamma (2b)} \biggl( \frac{c^{4b-1}}{4b-1} \biggr)^{\frac{1}{2}} \biggr] \Vert z_{1} \Vert + \frac{PP_{1}P_{2}}{\Gamma (2b)} \biggl(\frac{c^{4b-1}}{4b-1} \biggr)^{ \frac{1}{2}} \Vert y \Vert \\ &\quad {}+\frac{PP_{3}}{\Gamma (2b)} \biggl[1+ \frac{PP_{1}P_{2}}{\Gamma (2b)} \biggl( \frac{c^{4b-1}}{4b-1} \biggr)^{ \frac{1}{2}} \biggr]\zeta \bigl( \bigl\Vert z^{\varrho } \bigr\Vert \bigr), \end{aligned}$$

dividing both sides of the above inequality \(\|z^{\varrho }\|_{\mathcal{C}}\) and taking the limit as \(\|z^{\varrho }\|_{\mathcal{C}}\) tends to ∞, one can obtain \(0 \geq 1\), which is a contradiction. Therefore, \(\varrho >0\), \(\Psi (\mathcal{B}_{\varrho })\subseteq \mathcal{B}_{\varrho }\).

Step 2: We prove that Ψ is continuous on \(\mathcal{B}_{\varrho }\).

Let \(z^{(n)} \rightarrow z\) in \(\mathcal{B}_{\varrho }\). From \(\mathbf{(H_{4})}\mathbf{(i)}\) and Lemma 2.8, we have

$$\begin{aligned} \Vert \Psi _{1} z_{n}-\Psi _{1} z \Vert \le P \bigl\Vert F(z_{n})-F(z) \bigr\Vert \rightarrow 0 \quad \text{as } n \rightarrow \infty . \end{aligned}$$
(3.4)

Using Lebesgue’s dominated convergence theorem and \(\mathbf{(H_{2})}\mathbf{(i)}\mathbf{(ii)}\), we have

$$\begin{aligned}& \int _{0}^{t}(t-s)^{b-1} \bigl\Vert \mathbb{G}_{n}(s)-\mathbb{G}(s) \bigr\Vert \,ds \rightarrow 0 \quad \text{as } n \rightarrow \infty , t \in V, \end{aligned}$$
(3.5)

where \(\mathbb{G}_{n}(s)=g(s,z_{n}(s))\) and \(\mathbb{G}(s)=g(s,z(s))\). Then

$$\begin{aligned} \Vert \Psi _{2} z_{n}-\Psi _{2} z \Vert _{\mathcal{{C}}} & \le \frac{P}{\Gamma (2b)} \int _{0}^{t}(t-s)^{2b-1} \bigl\Vert \mathbb{G}_{n}(s)- \mathbb{G}(s) \bigr\Vert \,ds \\ &\quad {}+ \biggl(\frac{c^{4b-1}}{4b-1} \biggr)^{\frac{1}{2}} \frac{P}{\Gamma (2b)} \Vert x_{z_{n}}-x_{z} \Vert _{L^{2}(V,U)}, \end{aligned}$$
(3.6)

where

$$\begin{aligned} \Vert x_{z_{n}}-x_{z} \Vert _{L^{2}(V,U)} &\le P_{2}(1+P) \bigl\Vert F(z_{n})-F(z) \bigr\Vert \\ &\quad {}+\frac{PP_{2}}{\Gamma (2b)} \int _{0}^{c}(c-s)^{2b-1} \bigl\Vert \mathbb{G}_{n}(s)-\mathbb{G}(s) \bigr\Vert \,ds. \end{aligned}$$
(3.7)

Using (3.4), (3.5), (3.6), (3.7), we easily conclude that

$$ \Vert \Psi _{2} z_{n}-\Psi _{2} z \Vert _{\mathcal{{C}}} \rightarrow 0 \quad \text{as } n \rightarrow \infty , $$

⇒ \(\Psi _{2}\) is continuous on \(\mathcal{B}_{\varrho }\).

Step 3: Mönch’s condition holds.

Let \(\mathcal{D} \subseteq \mathcal{B}_{\varrho }\) be countable and \(\mathcal{D} \subseteq \operatorname{conv} (\{0\} \cup \Psi (\mathcal{D}))\). We prove that \(\beta (\mathcal{D})=0\), where β is the Hausdorff measure of noncompactness. Without loss of generality, let \(\mathcal{D}=\{z_{n}\}_{n=1}^{\infty }\).

Now, we prove that \(\{\Psi z_{n}\}_{n=1}^{\infty }\) is equicontinuous on V, then \(\mathcal{D} \subseteq \operatorname{conv} (\{0\} \cup \Psi (\mathcal{D}))\) is also equicontinuous on V. Lastly, let \(\chi \in \Psi (\mathcal{D})\) and \(0 \le t_{1} < t_{2} \le c\), there is \(z \in \mathcal{D}\) such that

$$\begin{aligned} \bigl\Vert \chi {(t_{2})}-\chi {(t_{1})} \bigr\Vert & \le \bigl\Vert C_{b}(t_{2})z_{0}-C_{b}(t_{1})z_{0} \bigr\Vert + \bigl\Vert C_{b}(t_{2})F(z)-C_{b}(t_{1})F(z) \bigr\Vert \\ &\quad {}+ \bigl\Vert K_{b}(t_{2})z_{1}-K_{b}(t_{1})z_{1} \bigr\Vert \\ &\quad {}+ \bigl\Vert \Pi (g+Bx_{z}) (t_{2})-\Pi (g+Bx_{z}) (t_{1}) \bigr\Vert . \end{aligned}$$

From Lemma 2.8, we may readily deduce that the first, second, and third teams at the RHS of the above inequality tend to zero as \(t_{2} \rightarrow t_{1}\).

Now, we verify that the last team at the RHS of the above inequality tends to 0 as \(t_{2} \rightarrow t_{1}\).

$$\begin{aligned} &\mathcal{I}_{1}= \int _{t_{1}}^{t_{2}}(t_{2}-s)^{b-1}T_{b}(t_{2}-s) \bigl[\mathbb{G}(s)+Bx_{z} \bigr] \,ds, \\ &\mathcal{I}_{2}= \int _{t_{1}-\varepsilon }^{t_{1}}(t_{2}-s)^{b-1} \bigl[T_{b}(t_{2}-s)-T_{b}(t_{1}-s) \bigr] \bigl[\mathbb{G}(s)+Bx_{z} \bigr]\,ds, \\ &\mathcal{I}_{3}= \int _{t_{1}-\varepsilon }^{t_{1}} \bigl[(t_{2}-s)^{b-1}-(t_{1}-s)^{b-1} \bigr]T_{b}(t_{1}-s) \bigl[\mathbb{G}(s)+Bx_{z} \bigr]\,ds, \\ &\mathcal{I}_{4}= \int _{0}^{t_{1}-\varepsilon }(t_{2}-s)^{b-1} \bigl[T_{b}(t_{2}-s)-T_{b}(t_{1}-s) \bigr] \bigl[\mathbb{G}(s)+Bx_{z} \bigr]\,ds, \\ &\mathcal{I}_{5}= \int _{0}^{t_{1}-\epsilon } \bigl[(t_{2}-s)^{b-1}-(t_{1}-s)^{b-1} \bigr]T_{b}(t_{1}-s) \bigl[\mathbb{G}(s)+Bx_{z} \bigr]\,ds, \end{aligned}$$

we have

$$\begin{aligned} \bigl\Vert \Pi (g+Bx_{z}) (t_{2})-\Pi (g+Bx_{z}) (t_{1}) \bigr\Vert \le \sum _{n=1}^{5} \Vert \mathcal{I}_{n} \Vert . \end{aligned}$$

Using Lemma 2.8, one can check that \(\|\mathcal{I}_{n}\| \rightarrow 0\), as \(t_{2} \rightarrow t_{1}\), \(n=1,2,3,4,5\). Hence, \(\Psi (\mathcal{D})\) is equicontinuous on V.

Now, we need to verify \(\Psi (\mathcal{D})(t)\) is relatively compact in Z for every \(t \in V\). From the compactness condition of F, we have

$$ \beta \bigl(\bigl\{ (\Psi _{1} z_{n}) (t)\bigr\} _{n=1}^{\infty }\bigr) \le \beta \bigl(\bigl\{ C_{b}(t) \bigl(z_{0}-F(z_{n})\bigr)+K_{b}(t)z_{1} \bigr\} _{n=1}^{\infty }\bigr)=0. $$

From Theorem 2.12, we have

$$\begin{aligned} \beta \bigl(\bigl\{ x_{z_{n}}(s)\bigr\} _{n=1}^{\infty } \bigr) \le \mathcal{K}_{W}(s) \frac{2P}{\Gamma (2b)} \int _{0}^{c}(c-s)^{2b-1}j(s)\beta \bigl( \mathcal{D}(s)\bigr)\,ds. \end{aligned}$$

Further,

$$\begin{aligned}& \begin{aligned} \beta \bigl(\bigl\{ (\Psi _{2} z_{n}) (t)\bigr\} _{n=1}^{\infty }\bigr) &\le \frac{2P}{\Gamma (2b)} \biggl( \int _{0}^{c}(c-s)^{2b-1}j(s)\,ds \biggr) \beta \bigl(\mathcal{D}(t)\bigr) \\ & \quad {}+ \frac{2PP_{1}}{\Gamma (2b)} \biggl( \int _{0}^{c}(c-s)^{2b-1} \mathcal{K}_{W}(s)\,ds \biggr) \\ & \quad {}\times \biggl[\frac{2P}{\Gamma (2b)} \biggl( \int _{0}^{c}(c-s)^{2b-1}j(s)\,ds \biggr) \beta \bigl(\mathcal{D}(t)\bigr) \biggr] \\ & \le \frac{2PP_{5}}{\Gamma (2b)} \beta \bigl(\mathcal{D}(t)\bigr)+ \frac{2PP_{1}P_{4}}{\Gamma (2b)} \biggl( \frac{2PP_{5}}{\Gamma (2b)} \biggr) \beta \bigl(\mathcal{D}(t)\bigr), \end{aligned} \\& \beta \bigl(\Psi (\mathcal{D}) (t)\bigr) \le \beta \bigl(\Psi _{1}( \mathcal{D}) (t)\bigr)+ \beta \bigl(\Psi _{2}(\mathcal{D}) (t)\bigr) \le \biggl(1+ \frac{2PP_{1}P_{4}}{\Gamma (2b)} \biggr) \frac{2PP_{5}}{\Gamma (2b)} \beta \bigl( \mathcal{D}(t)\bigr). \end{aligned}$$

Then

$$ \beta \bigl(\Psi (\mathcal{D}) (t)\bigr) \le \widehat{L}\beta (\mathcal{D}), $$

where L̂ denotes equation (3.1). Then, from Mönch’s condition, we have

$$\begin{aligned} \beta (\mathcal{D}) \le \beta \bigl(\operatorname{conv}\bigl(\{0\}\cup \Psi ( \mathcal{D})\bigr) \bigr)=\beta \bigl(\Psi (\mathcal{D})\bigr) \le \widehat{L} \beta ( \mathcal{D}). \end{aligned}$$

\(\Rightarrow \beta (\mathcal{D})=0\).

Therefore, using Lemma 2.14, Ψ has a fixed point \(z \in \mathcal{B}_{\varrho }\), since z is a mild solution of system (1.1) satisfying \(z(c)+F(z)=y\). □

4 Fractional integro-differential evolution system

The nonlocal controllability results for fractional integro-differential evolution system of \(1< r<2\) under the Banach contraction principle are presented and demonstrated in this section. Consider that the fractional integro-differential evolution system of \(1< r<2\) has the form

$$\begin{aligned} \textstyle\begin{cases} {}^{C}D_{t}^{r} z(t) = Az(t)+g (t,z(t),\int _{0}^{t} h(t,s,z(s))\,ds )+Bx(t),\quad t \in V, \\ z(t)+F(z)=z_{0}, \qquad z'(0)=z_{1} \in Z, \end{cases}\displaystyle \end{aligned}$$
(4.1)

where \(g:V \times Z \times Z \rightarrow Z\) and \(h:\mathcal{Q} \times Z \rightarrow Z\) are continuous, where \(\mathcal{Q}=\{(t,s): 0 \le s \le t \le c\}\).

Definition 4.1

([46])

\(z \in C(V,Z)\) is said to be a mild solution of system (4.1) if \(z(0)+F(z)=z_{0}\), \(z'(0)=z_{1}\) such that

$$\begin{aligned} z(t)&=C_{b}(t) \bigl(z_{0}-F(z) \bigr)+K_{b}(t)z_{1} \\ &\quad{} + \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)g \biggl(s,z(s), \int _{0}^{s} h\bigl(s, \tau ,z(\tau )\bigr)\,d\tau \biggr) \,ds \\ &\quad {}+ \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)Bx(s)\,ds,\quad t \in V. \end{aligned}$$
(4.2)

Before starting and examining the main results, we assume the following:

\(\mathbf{(H_{5})}\):

The function \(g:V \times Z \times Z \rightarrow Z\) is continuous, and there exist constants \(L_{g} >0\), \(P_{g}>0\) such that

$$\begin{aligned} \bigl\Vert g(t,k_{1},w_{1})-g(t,k_{2},w_{2}) \bigr\Vert \le L_{g}\bigl( \Vert k_{1}-k_{2} \Vert + \Vert w_{1}-w_{2} \Vert \bigr) \quad \text{for all } t \in V, \end{aligned}$$

for any \(k_{1},k_{2}, w_{1},w_{2} \in Z\), and \(P_{g}=\max_{t \in V}\|g(t,0,0)\|\).

\(\mathbf{(H_{6})}\):

The function \(h:\mathcal{Q} \times Z \rightarrow Z\) is continuous, and there exist constants \(L_{h} >0\), \(P_{h}>0\) such that

$$\begin{aligned} \bigl\Vert h(t,s,k_{1})-h(t,s,k_{2}) \bigr\Vert \le L_{h}\bigl( \Vert k_{1}-k_{2} \Vert \bigr) \end{aligned}$$

for any \(k_{1},k_{2}, \tau _{1},\tau _{2} \in Z\), and \(P_{h}=\max_{t \in V}\|h(t,0,0)\|\).

Theorem 4.2

If \(\mathbf{(H_{1})}\), \(\mathbf{(H_{3})}\)–\(\mathbf{(H_{6})}\) are satisfied, then we assume that the following inequality holds:

$$\begin{aligned} \frac{PL_{g} c^{2b} (1+L_{h} c)}{\Gamma (2b+1)} \biggl[1+ \frac{P P_{1}P_{2} c^{2b}}{\Gamma (2b+1)} \biggr] < 1. \end{aligned}$$
(4.3)

Then system (4.1) is controllable V.

Proof

Using \(\mathbf{(H_{3})}\mathbf{(i)}\), for an arbitrary function \(z \in \mathcal{C}\), we define the control \(x_{z}(t)\) by

$$\begin{aligned} x_{z}(t)&=W^{-1} \biggl[y-F(z)-C_{b}(c) \bigl(z_{0}-F(z)\bigr) -K_{b}(c)z_{1} \\ &\quad{} - \int _{0}^{c}(c-s)^{b-1}T_{b}(c-s)g \biggl(s,z(s), \int _{0}^{s} h\bigl(s,\tau ,z(\tau )\bigr)\,d\tau \biggr)\,ds \biggr](t),\quad t \in V. \end{aligned}$$
(4.4)

Define that the operator \(\Phi :C(V,Z) \rightarrow C(V,Z)\) given by

$$\begin{aligned} (\Phi z) (t) &=C_{b}(t) \bigl(z_{0}-F(z) \bigr)+K_{b}(t)z_{1} \\ &\quad {}+ \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)g \biggl(s,z(s), \int _{0}^{s} h\bigl(s,\tau ,z(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad {}+ \int _{0}^{t} (t-s)^{b-1}T_{b}(t-s)Bx(s)\,ds \end{aligned}$$

has a fixed point z, which is a mild solution of system (4.1). Clearly, \((\Phi z)(c)=y-F(z)\); this means that \(x_{z}\) moves system (4.1) from \(z_{0}\) to y in finite time c. Therefore, we verify that the operator Φ has a fixed point.

Using Lemma 2.8, \(\mathbf{(H_{3})}\), \(\mathbf{(H_{5})}\), \(\mathbf{(H_{6})}\), and Hölder’s inequality, we have

$$\begin{aligned} & \bigl\Vert x(t) \bigr\Vert \\ &\quad =P_{2} \biggl[ \Vert y \Vert + \bigl\Vert F(z) \bigr\Vert + \bigl\Vert C_{b}(c) \bigl(z_{0}-F(z)\bigr) \bigr\Vert + \bigl\Vert K_{b}(c)z_{1} \bigr\Vert \\ &\qquad {}+ \int _{0}^{c}(c-s)^{b-1} \biggl\Vert T_{b}(c-s)g \biggl(s,z(s), \int _{0}^{s} h\bigl(s,\tau ,z(\tau )\bigr)\,d\tau \biggr) \biggr\Vert \,ds \biggr] \\ & \quad \le P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert +PP_{2} c \Vert z_{1} \Vert \\ &\qquad {}+\frac{PP_{2}}{\Gamma (2b)} \int _{0}^{c}(c-s)^{2b-1} \bigl[L_{g} \bigl( \Vert z \Vert +L_{h}c \Vert z \Vert +P_{h}c\bigr)+P_{g} \bigr]\,ds \\ &\quad \le P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert +PP_{2}c \Vert z_{1} \Vert \\ &\qquad {}+\frac{PP_{2} c^{2b}}{\Gamma (2b+1)} \bigl[ L_{g} (\varrho +L_{h} c \varrho +P_{h} c)+P_{g} \bigr]. \end{aligned}$$

The operator Φ maps \(\mathcal{B}_{\varrho }\) into \(\mathcal{B}_{\varrho }\). From the definition of the operator Φ and the assumptions, for \(z \in \mathcal{B}_{\varrho }\), we have

$$\begin{aligned} \bigl\Vert (\Phi z) (t) \bigr\Vert &\le \bigl\Vert C_{b}(t) \bigl(z_{0}-F(z)\bigr) \bigr\Vert + \bigl\Vert K_{b}(t)z_{1} \bigr\Vert \\ &\quad{} + \int _{0}^{t}(t-s)^{b-1} \biggl\Vert T_{b}(t-s)g \biggl(s,z(s), \int _{0}^{s} h\bigl(s,\tau ,z(\tau )\bigr)\,d\tau \biggr) \biggr\Vert \,ds \\ &\quad {}+ \int _{0}^{t} (t-s)^{b-1} \bigl\Vert T_{b}(t-s)Bx(s) \bigr\Vert \,ds \\ & \le P\bigl( \Vert z_{0} \Vert + \bigl\Vert F(z) \bigr\Vert \bigr)+ Pc \Vert z_{1} \Vert \\ &\quad{} + \frac{P}{\Gamma (2b)} \int _{0}^{t}(t-s)^{2b-1} \bigl[L_{g} \bigl( \Vert z \Vert +L_{h}c \Vert z \Vert +P_{h}c\bigr)+P_{g} \bigr] \,ds \\ &\quad{} +\frac{PP_{1}}{\Gamma (2b)} \int _{0}^{t}(t-s)^{2b-1} \biggl[P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert \\ &\quad {}+PP_{2}c \Vert z_{1} \Vert +\frac{PP_{2} c^{2b}}{\Gamma (2b+1)} \bigl[ L_{g}( \varrho +L_{h} c \varrho +P_{h} c)+P_{g} \bigr] \biggr] \,ds \\ & \le P\bigl( \Vert z_{0} \Vert + \bigl\Vert F(z) \bigr\Vert \bigr)+ Pc \Vert z_{1} \Vert + \frac{P c^{2b}}{\Gamma (2b+1)} \bigl[L_{g}(\varrho +L_{h} c \varrho +P_{h} c)+P_{g} \bigr] \\ &\quad {}+\frac{PP_{1} c^{2b}}{\Gamma (2b+1)} \biggl[P_{2} \Vert y \Vert +P_{2}(1+P) \bigl\Vert F(z) \bigr\Vert +PP_{2} \Vert z_{0} \Vert \\ &\quad {}+PP_{2}c \Vert z_{1} \Vert +\frac{PP_{2} c^{2b}}{\Gamma (2b+1)} \bigl[L_{g}( \varrho +L_{h} c \varrho +P_{h} c)+P_{g} \bigr] \biggr]. \end{aligned}$$

Therefore, by inequality (4.3) it follows that \(\|\Phi z\| \le \varrho \) and then \(\Phi (\mathcal{B}_{\varrho }) \subseteq \mathcal{B}_{\varrho }\). Now, for every \(u,v \in \mathcal{B}_{\varrho }\), we have

$$\begin{aligned} & \bigl\Vert (\Phi u) (t)-(\Phi v) (t) \bigr\Vert \\ &\quad \le \int _{0}^{t}(t-s)^{b-1} \biggl\Vert T_{b}(t-s) \biggl[g \biggl(s,u(s), \int _{0}^{s} h\bigl(s,\tau ,u(\tau )\bigr)\,d\tau \biggr) \\ &\qquad {}-g \biggl(s,v(s), \int _{0}^{s} h\bigl(s,\tau ,v(\tau )\bigr)\,d\tau \biggr) \biggr] \biggr\Vert \,ds + \int _{0}^{t}(t-s)^{b-1} \biggl\Vert T_{b}(t-s) BW^{-1} \\ &\qquad {}\times \biggl[ \int _{0}^{c}(c-\iota )^{b-1} \biggl\| T_{b}(c-\iota ) \biggl[g \biggl(\iota ,u(\iota ), \int _{0}^{\iota } h\bigl(\iota ,\tau ,u( \tau )\bigr)\,d \tau \biggr) \\ &\qquad {}-g \biggl(\iota ,v(\iota ), \int _{0}^{\iota } h\bigl(\iota ,\tau ,v( \tau )\bigr)\,d \tau \biggr) \biggr]\,d\iota \biggr](s) \biggr\Vert \,ds \\ & \quad \le \frac{P}{\Gamma (2b)} \int _{0}^{t}(t-s)^{2b-1} \biggl\Vert g \biggl(s,u(s), \int _{0}^{s} h\bigl(s,\tau ,u(\tau )\bigr)\,d\tau \biggr) \\ &\qquad {}-g \biggl(s,v(s), \int _{0}^{s} h\bigl(s,\tau ,v(\tau )\bigr)\,d\tau \biggr) \biggr\Vert \,ds +\frac{PP_{1}P_{2}}{\Gamma (2b)} \int _{0}^{t}(t-s)^{2b-1} \\ &\qquad{} \times \biggl[\frac{P}{\Gamma (2b)} \int _{0}^{c}(c-\iota )^{2b-1} \biggl\Vert g \biggl(\iota ,u(\iota ), \int _{0}^{\iota } h\bigl(\iota ,\tau ,u(\tau )\bigr)\,d \tau \biggr) \\ &\qquad{} -g \biggl(\iota ,v(\iota ), \int _{0}^{\iota } h\bigl(\iota ,\tau ,v( \tau )\bigr)\,d \tau \biggr) \biggr\Vert \,d\iota \biggr](s)\,ds \\ &\quad \le \frac{P L_{g} c^{2b}}{\Gamma (2b+1)}\bigl[ \Vert u-v \Vert +L_{h}c \Vert u-v \Vert \bigr]+ \frac{P^{2} P_{1}P_{2} L_{g} t^{2b}c^{2b}}{(\Gamma (2b+1))^{2}} \bigl[ \Vert u-v \Vert +L_{h}c \Vert u-v \Vert \bigr] \\ &\quad \le \biggl[\frac{PL_{g} c^{2b}}{\Gamma (2b+1)}+ \frac{P^{2} P_{1}P_{2} L_{g} t^{2b}c^{2b}}{(\Gamma (2b+1))^{2}} \biggr] (1+L_{h} c) \Vert u-v \Vert , \end{aligned}$$

which implies by inequality (4.3) that \(\|\Phi u-\Phi v\| < \|u-v\|\). Then, we can conclude that Φ is a contraction on \(\mathcal{B}_{\varrho }\). As a result, according to the Banach fixed point theorem, Φ has a unique fixed point z in \(C(V,Z)\). Therefore, we can see that \(z(\cdot )\) is a mild solution of system (4.1), and the proof is complete. □

5 Application

Let \(\mathcal{G} \subset \mathbb{R}^{N}\) be a bounded domain and \(U=Z=L^{2}(\mathcal{G})\). Consider the following nonlocal fractional integrodifferential evolution system:

$$\begin{aligned} \textstyle\begin{cases} \frac{\partial ^{r}}{\partial t^{r}} z(t,\eta )= \Delta z(t,\eta ) +l_{0}(\eta ) \sin z(t,\eta ) +l_{1} \int _{0}^{t} e^{-z (s,\eta )} \,ds \\ \hphantom{\frac{\partial ^{r}}{\partial t^{r}} z(t,\eta )=}{} +Bx(t),\quad t\in V=[0,1], \eta \in \mathcal{G}, \\ z(t,\eta )=0,\quad t\in [0,1], \eta \in \partial \mathcal{G}, \\ z(0,\eta )+\int _{0}^{c} j(s) \operatorname{In} (1+ \vert z(s,\eta ) \vert ^{ \frac{1}{2}})\,ds=0,\quad z'(0,\eta )=z_{1}(\eta ), \eta \in \mathcal{G}, \end{cases}\displaystyle \end{aligned}$$
(5.1)

where \(\frac{\partial ^{r}}{\partial t^{r}}\) denotes Caputo fractional derivative of order \(\frac{3}{2} \le r < 2\), \(j \in L^{1}(V,\mathbb{R}^{+})\), \(l_{0}\) is continuous on \(\mathcal{G}\) and \(l_{1}>0\).

Consider A to be the Laplace operator with Dirichlet boundary conditions given by \(A = \Delta \) and

$$ D(A)=\bigl\{ g\in H_{0}^{1}(\mathcal{G}), Ag\in L^{2}(\mathcal{G})\bigr\} . $$

Clearly, we have \(D(A) = H^{1}_{0}(\mathcal{G}) \cap H^{2}(\mathcal{G})\). A produces \(C(t)\) for \(t \ge 0\) in the view of [62]. Let \(\hbar _{n}=n^{2} \pi ^{2}\) and \(\mu _{n}(\eta )=\sqrt{(2/\pi )}\sin (n\pi \eta )\) for any \(n \in \mathbb{N}\).

Assume that \(\{-\hbar _{n},\mu _{n}\}_{n=1}^{\infty }\) is an eigensystem of the operator A, then \(0<\hbar _{1}\le \hbar _{2} \le \cdots \) , \(\hbar _{n}\rightarrow \infty \) when \(n \rightarrow \infty \), and \(\{\mu _{n}\}_{n=1}^{\infty }\) forms an orthonormal basis of Z. Further

$$ Az=-\sum_{n=1}^{\infty } \hbar _{n} (z,\mu _{n}) \mu _{n},\quad z\in D(A), $$

where \((\cdot ,\cdot )\) denotes the inner product in Z. Accordingly, \(C(t)\) is defined by

$$ C(t)z=\sum_{n=1}^{\infty }\cos (\sqrt{\hbar _{n}}t ) ( z, \mu _{n}) \mu _{n},\quad z\in Z, $$

which is connected with the sine family \(\{S(t), t\ge 0\}\) in Z defined by

$$ S(t)z=\sum_{n=1}^{\infty }\frac{1}{\sqrt{\hbar _{n}}}\sin (\sqrt{ \hbar _{n}}t ) (z,\mu _{n}) \mu _{n},\quad z \in Z, $$

and \(\|C(t)\|_{L_{c}(Z)}\le 1\) for any \(t \ge 0\).

Since \(r=\frac{3}{2}\), we know that \(t=\frac{3}{4}\), and then \(\|C_{c}(t)\|_{L_{c}(Z)}\le 1\) for any \(t \ge 0\).

The control operator \(B:U\rightarrow Z\) is defined by

$$ Bx=\sum_{n=1}^{\infty }a \hbar _{n}( \overline{x},\mu _{n}) \mu _{n},\quad a>0. $$

In the above

$$\begin{aligned} \overline{x}= \textstyle\begin{cases} x_{n},& n=1,2,\ldots,N, \\ 0, &n=N+1,N+2,\ldots, \end{cases}\displaystyle \end{aligned}$$

for N in \(\mathbb{N}\). Denote \(W:L^{2}(V,U)\rightarrow Z\) as follows:

$$ Wx= \int _{0}^{s} (1-s)^{-\frac{1}{4}}T_{\frac{3}{4}}(1-s)Bx(s)\,ds. $$

Hence, \(|x|= (\sum_{n=1}^{\infty }(x,\mu _{n})^{2} )^{\frac{1}{2}}\) for \(x \in U\), we have

$$ \vert Bx \vert = \Biggl(\sum_{n=1}^{\infty }a^{2} \hbar _{n}^{2}(\overline{x}, \mu _{n})^{2} \Biggr)^{\frac{1}{2}} \le a N \hbar _{N} \vert x \vert , $$

which implies that there exists \(P_{1}>0\) such that

$$ \Vert B \Vert _{L_{c} (U,Z)}\le P_{1}. $$

Let \(x(s,\eta )= z(\eta )\in U\) and z̅ denote \(z_{n}\) if \(n=1,2,\ldots,N\) or 0 if \(n=N+1,\ldots \) . Hence, we have

$$\begin{aligned} Wx&= \int _{0}^{1} (1-s)^{-\frac{1}{4}} \frac{3}{4} \int _{0}^{\infty }\xi S_{\frac{3}{4}}(\xi )S \bigl((1-s)^{\frac{3}{4}}\xi \bigr)Bz \,d \xi \,ds \\ &=a \int _{0}^{1} (1-s)^{-\frac{1}{4}} \frac{3}{4} \int _{0}^{\infty }\xi S_{\frac{3}{4}}(\xi )\sum _{n=1}^{N} \sqrt{\hbar _{n}} \sin \bigl(\sqrt{\hbar _{n}}(1-s)^{\frac{3}{4}}\xi \bigr) ) ( \overline{z},\mu _{n}) \mu _{n} \,d\xi \,ds \\ &=a \sum_{n=1}^{N} \int _{0}^{\infty }S_{\frac{3}{4}} (\xi ) \bigl(1-\cos ( \sqrt{\hbar _{n}}\xi )\bigr)\,d\xi (\overline{z},\mu _{n})\mu _{n} \\ &=a \sum_{n=1}^{\infty } \bigl(1-E_{\frac{3}{2},1}(- \hbar _{n}) \bigr) (z,\mu _{n}) \mu _{n}. \end{aligned}$$

In [63, 64], assume that \(v=E_{\frac{3}{2},1}(-\frac{1}{10})\), then for every \(n \in \mathbb{N}\), we have \(-1< E_{\frac{3}{2},1}(-\hbar _{n})\le v<1\), which implies

$$ 0< 1-v\le 1-E_{\frac{3}{2},1}(-\hbar _{n})< 2. $$

Then, we classify W is surjective since, for every \(z=\sum_{n=1}^{\infty }(z,\mu _{n}) \mu _{n} \in Z\), we illustrate \(W^{-1}:Z \rightarrow L^{2}(V,U)/\ker W\) by

$$ \bigl(W^{-1}z\bigr) (t,\eta )=\frac{1}{a}\sum _{n=1}^{\infty }\frac{(z,\mu _{n}\rangle \mu _{n}}{1-E_{\frac{3}{2},1}(-\hbar _{n})} $$

for \(z \in Z\) in such a way

$$\begin{aligned} \bigl\vert \bigl(W^{-1}z\bigr) (t,\cdot ) \bigr\vert \le \frac{1}{a(1-v)} \vert z \vert . \end{aligned}$$

We know that \(W^{-1}z\) is independent of \(t \in V\). Additionally, we obtain

$$ \bigl\Vert W^{-1} \bigr\Vert _{L_{c} (Z,L^{2}(V,U)/\operatorname{Ker} W)}\le \frac{1}{a(1-v)}. $$

Therefore assumption \(\mathbf{(H_{3})}\) satisfied.

Determine

$$\begin{aligned}& z(t) (\eta )=z(t,\eta ), \qquad {}^{C}D_{t}^{\frac{3}{2}}z(t) (\eta )= \frac{\partial ^{\frac{3}{2}}}{\partial t^{\frac{3}{2}}} z(t,\eta ), \\& g \biggl(t,z, \int _{0}^{t} h(t,s,z)\,ds \biggr)=l_{0}( \cdot ) \sin z(t,\cdot )+ \int _{0}^{t}h(t,s,z)\,ds, \quad h(t,s,z)=l_{1} e^{-z (s,\cdot )}, \end{aligned}$$

and F denotes \(F(z)(\eta )=\int _{0}^{c} j(s) \operatorname{In} (1+|z(s,\eta )|^{ \frac{1}{2}})\,ds\) and F is compact and satisfies hypothesis \(\mathbf{(H_{4})}\).

$$\begin{aligned} \textstyle\begin{cases} {}^{C}D_{t}^{r} z(t) = Az(t)+g (t,z,\int _{0}^{t} h(t,s,z)\,ds )+Bx(t), \quad t \in V=[0,c], r \in (1,2), \\ z(t)+F(z)=z_{0}, \qquad z'(0)=z_{1} \in Z. \end{cases}\displaystyle \end{aligned}$$
(5.2)

Therefore, every requirement of Theorem 4.2 is satisfied. Hence, using Theorem 4.2, (5.1) is nonlocal controllable on \([0,c]\).

6 Conclusion

The nonlocal controllability results for the fractional differential system of \(1< r<2\) in a Banach space are discussed in this work. Fractional computations, the measure of noncompactness, cosine families, Mainardi’s Wright-type function, and fixed point techniques are all used to test the main conclusions of this article. We begin by applying the Mönch fixed point theorem to analyze nonlocal controllability results of a mild solution for the fractional differential system. In addition, the Banach fixed point theorem is used to develop the controllability results for fractional integrodifferential evolution system with nonlocal conditions. Finally, an application for developing the theory of the key results is offered. We will develop approximate controllability results for Sobolev type fractional delay evolution inclusions of order \(1< r<2\) in the future.