1 Introduction

Fractional calculus (FC) has attracted much attention for many years. It is well known that FC and fractional differential equations (FDEs) have found useful applications in a wide range of disciplines, including physics and engineering [1, 2]. FC can be applied to represent many engineering, physical, and biological problems with better outcomes than those obtained with conventional calculus. Compared to traditional differential equations, the solutions of FDEs and integral equations are more efficient to describe a real-world process (see examples [3, 4]). Numerous papers have been published based on the existence and uniqueness of solutions to FDEs (we refer to [5]).

It is remarkable that there are many definitions that have been introduced by researchers in the literature of FC. On the basis of kernels, the aforementioned area contains two major kinds of operators for derivatives including singular and nonsingular, although all fractional differential operators are nonlocal and have a greeter degree of freedom. Nonsingular operators are those that contain power-law-type kernels. On the other hand, those containing exponential and Mittag–Leffler (M–L)-type kernels are called nonsingular operators. In this regard, we refer to some books where the details can be traced, e.g., [6–8]. The differential operators with M–L kernels were introduced in 2016 by Atangana and Baleanu [9]. The aforementioned operators attracted great attention and therefore were used in many research studies. One of the shortcomings of the mentioned operators was noted that the said operators suffer from an initilization condition and therefore an extra restriction needs to be implemented on the problem. Therefore, Al-Refai and coauthor [10] extended the said operators and the shortcoming was removed in 2022. Recently, the said operators in the modified form have been used in various articles. We refer to a few articles here [11–14].

The qualitative theory of the aforementioned area is an important field of research. Researchers have used classical fixed-point theory to investigate various problems of FC from different aspects including the existence of solutions, stability, and numerical analysis. Here, we remark that in the case of classical differential equations, researchers have used various degree theories also. Using the classical fixed-point theory, or Leray degree theory, we need a strong compact condition that restricts the usability of such analysis tools. Therefore, Leray created an example in 1936 to demonstrate that a degree theory for mappings cannot be defined using merely a continuity condition. Thus, it makes sense to ask: for what kinds of mappings in infinite-dimensional spaces is it possible to develop a degree theory? It has been demonstrated by Browder, Nussbaum, Sadovski, Vath, and others that a comprehensive counterpart of the Leray–Schauder theory may be defined for condensing-type mappings (see details in [15]). Hence, Mawhin [16] has successfully used the coincidence degree tools for analysis of some problems. For more details, we refer to [17]. The aforementioned degree theory is a powerful tool to use for investigating various problems for existence theory. The said theory has been increasingly used in the last several years. Here, some useful results are cited [18, 19]). The aforementioned degree theory was used to study a class of nonlinear integral problem by Isia [20].

Inspired from the importance of the mentioned degree theory, we investigate the following class of fractional-order evolution control systems using the controllability criteria with \(t\in \mathrm{J}=[0, T]\) as

$$ \begin{aligned}& _{0}^{mABC}D^{\theta} \bigl[\mathrm{u}(t)-\mathrm{\phi}\bigl(t, \mathrm{u}(t)\bigr)\bigr]= \mathrm{A} \mathrm{u}(t)+\mathrm{B}\mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr), \quad 0< \theta \leq 1, \\ & \mathrm{u}(0)=\mathrm{u}_{0}, \end{aligned} $$
(1)

where A is the infinite small generator of an analytical semigroup of bounded linear operators on the Hilbert space say \(\mathscr{H}\), x is the control variable function on \(L^{2}[\mathrm{J}, \mathscr{H}]\), while B is also a linear bounded operator from \(\mathscr{H}\) to \(\mathscr{H}\). In addition,

$$ \mathrm{\phi}, \mathrm{\psi}: \mathrm{J}\times \mathrm{R} \rightarrow \mathrm{R}. $$

We develop the existence theory by using the degree theory of the measure of noncompactness for the considered problem. In addition, U–H and generalized U–H stability results are also deduced for the aforementioned problem. The aforesaid stability analysis has been studied for various problems of fractional calculus recently [21, 22]. Finally, some sophisticated examples are discussed to demonstrate our results.

2 Preliminaries

Definition 2.1

[9] For \(\theta \in (0,1)\), and \(\mathrm{u}\in \mathscr{H}(0, T)\), the modified \(ABC\) derivative can be defined as

$$\begin{aligned} ^{mABC}D^{\theta}_{0}\mathrm{u}(t) =& \frac{\mathrm{M}(\theta )}{1-\theta} \biggl[\mathrm{u}(t)-E_{\theta}\bigl(- \mu _{\theta}t^{\theta}\bigr)\mathrm{u}(0) \\ &{}-\mu _{\theta} \int _{0}^{t}(t-\vartheta )^{\theta -1}E_{\theta , \theta} \bigl(-\mu _{\theta}(t-\vartheta )^{\theta}\bigr)\mathrm{u}(s)\,d \vartheta \biggr], \end{aligned}$$

where \(\mu _{\theta}=\frac{\theta}{1-\theta}\).

We see that \(^{mABC}D^{\theta}_{0} C=0\). The integral is defined as follows:

Definition 2.2

[9] Let \(\theta \in (0,1)\), and \(\mathrm{u}\in L^{1}(\mathrm{J})\), then one has

$$ ^{mAB}D^{\theta}_{0}\mathrm{u}(t)= \frac{\mathrm{M}(1-\theta )}{\mathrm{M}(\theta )} \bigl[\mathrm{u}(t)- \mathrm{u}(0) \bigr]+\mu _{\theta} \bigl[ ^{RL}I^{\theta}_{0}\bigl( \mathrm{u}(t)- \mathrm{u}(0)\bigr) \bigr]. $$
(2)

Lemma 2.3

[9] If \(\mathrm{u}\in \mathscr{H}(0, T)\), \(\mathrm{\zeta}\in L(\mathrm{J}\), the solution of

$$\begin{aligned} &{}^{mABC}{_{0} D}_{t}^{\mathrm{p}} \bigl[\mathrm{u}(t)\bigr]=\mathrm{\zeta}(t), \\ &\mathrm{u}(0)=\mathrm{u}_{0} \end{aligned}$$

is deduced as follows:

$$\begin{aligned} \mathrm{u}(t) =&\mathrm{u}_{0}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\bigl( \mathrm{\zeta}(t)-\mathrm{\zeta}(0)\bigr)+ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl[\mathrm{\mathrm{\zeta}}(\vartheta )-\mathrm{\mathrm{\psi}}(0)\bigr]\,d \vartheta \biggr] \\ =&\mathrm{u}_{0}+\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\mathrm{\mathrm{ \zeta}}(t)-\mathrm{\mathrm{\zeta}}(0) \biggl(1+ \frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr)+ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \mathrm{\mathrm{\zeta}}(\vartheta )\,d\vartheta \biggr]. \end{aligned}$$

Here, some results we need for onward analysis are recollected from [20].

Definition 2.4

If \(\mathbf{Z}=C(\mathrm{J})\) is the Banach space with norm \(\|\mathrm{u}\|=\max_{t\in [0, T]} \vert \mathrm{u}(t) \vert \), and \(\Omega _{b}\subset P(\mathbf{Z})\) is the family of all its bounded sets, then the measure of noncompactness due to Kuratowski is given by

$$\begin{aligned} \mu (S) =& \inf \bigl\{ r>0, S \text{ admits a finite cover by sets such } \delta (S)\leq r\bigr\} , \end{aligned}$$

where \(\delta (S)\) is a denoted diameter of S, and \(\mu : \Omega _{b}\rightarrow \mathscr{R}_{+}\). For further properties of the aforesaid measure, see [20].

Definition 2.5

Let , be a continuous and bounded operator, then it is said to be Lipschitz with a constant \(\lambda >0\) if

Moreover, if \(\lambda <1\), then the said operator is called a strict contraction.

Proposition 2.6

If , are μ-Lipschitz with constants \(\lambda _{1}\), and \(\lambda _{2}\), respectively, then their sum is also μ-Lipschitz with constant \(\lambda _{1}+\lambda _{2}\).

Proposition 2.7

If is compact, then is μ-Lipschitz with constants 0.

Theorem 2.8

[20] If is a μ-condensing operator and

If \(\Theta \subset \mathbf{Z}\) is bounded, ∋ \(\Theta \subset \mathbf{B}_{r}(0)\), then

Therefore, has a minimum of one fixed point, and the location of the fixed-point set is \(\mathbf{B}_{r}(0)\).

3 Main results

This section is devoted to define some results we need in the main contribution.

\((A_{1})\):

∃ constants \(\kappa _{\phi}, \kappa _{\psi}>0\), \(\ni \mathrm{u}, \hat{\mathrm{u}}\in \mathbf{Z}\), one has

$$\begin{aligned} & \bigl\vert \phi (t, \mathrm{u})-\phi (t, \hat{\mathrm{u}}) \bigr\vert \leq \kappa _{\phi} \vert \mathrm{u}-\hat{\mathrm{u}} \vert ,\\ & \bigl\vert \psi (t, \mathrm{u})-\psi (t, \hat{\mathrm{u}}) \bigr\vert \leq \kappa _{\psi} \vert \mathrm{u}-\hat{\mathrm{u}} \vert . \end{aligned}$$
\((A_{2})\):

For constants \(\rho _{\phi}, \rho _{\psi}>0\), and \(\varrho _{\phi}, \varrho _{\psi}>0\), we have

$$\begin{aligned} & \bigl\vert \phi \bigl(t, \mathrm{u}(t)\bigr) \bigr\vert \leq \rho _{\phi} \bigl\vert \mathrm{u}(t) \bigr\vert +\varrho _{ \phi},\\ & \bigl\vert \psi \bigl(t, \mathrm{u}(t)\bigr) \bigr\vert \leq \rho _{\psi} \bigl\vert \mathrm{u}(t) \bigr\vert +\varrho _{ \psi}. \end{aligned}$$
\((A_{3})\):

For constants \(\sigma _{\mathrm{A}}, \sigma _{\mathrm{B}}>0\), we have

$$ \bigl\vert \mathrm{A}\mathrm{u}(t) \bigr\vert \leq \sigma _{\mathrm{A}} \bigl\vert \mathrm{u}(t) \bigr\vert , \qquad \bigl\vert \mathrm{B} \mathrm{u}(t) \bigr\vert \leq M_{\mathrm{B}}. $$

Theorem 3.1

The solution of (1) is given by using \(\Omega _{\mathrm{A}, \mathrm{B}, \psi}=\mathrm{A}\mathrm{u}(0)+ \mathrm{B}\mathrm{x}(0)+\mathrm{\psi}(0, \mathrm{u}(0))\)

$$\begin{aligned} \mathrm{u}(t) =&\mathrm{u}_{0}+\phi \bigl(t, \mathrm{u}(t)\bigr)+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\mathrm{A} \mathrm{u}(t)+ \mathrm{B}\mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr) \\ &{} -\Omega _{\mathrm{A}, \mathrm{B}, \psi} \biggl(1+ \frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr) \\ &{} + \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl[\mathrm{A}\mathrm{u}(\vartheta )+\mathrm{B} \mathrm{x}(\vartheta )+\mathrm{ \psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr) \bigr]\,d\vartheta \biggr]. \end{aligned}$$
(3)

Proof

By following Lemma 2.3, the said solution can be obtained. □

Here, we define the operators denoted as \(\mathscr{P}, \mathscr{Q}:\mathbf{Z}\rightarrow \mathbf{Z}\), by

$$\begin{aligned} \begin{aligned} \mathscr{P}\mathrm{u}(t)&=\mathrm{u}_{0}+ \phi \bigl(t, \mathrm{u}(t)\bigr) \\ &\quad + \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\mathrm{A} \mathrm{u}(t)+\mathrm{B} \mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr) - \Omega _{\mathrm{A}, \mathrm{B}, \psi} \biggl(1+ \frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr) \biggr] \end{aligned} \end{aligned}$$
(4)

and

$$\begin{aligned} \mathscr{Q}\mathrm{u}(t)= \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl[\mathrm{A}\mathrm{u}(\vartheta )+\mathrm{B} \mathrm{x}(\vartheta )+\mathrm{ \psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr) \bigr]\,d\vartheta \biggr]. \end{aligned}$$
(5)

Lemma 3.2

Under the assumption \((A_{1}-A_{3})\), the operator \(\mathscr{P}\) is μ-Lipschitz, which fulfills the given condition

$$\begin{aligned} \bigl\Vert \mathscr{P}(\mathrm{u}) \bigr\Vert \leq \vert \mathrm{u}_{0} \vert + \biggl(\rho _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )}(\rho _{\psi}+ \sigma _{\mathrm{A}}) \biggr) \Vert \mathrm{u} \Vert +\varrho _{\phi}+\varrho _{ \psi}+M_{\mathrm{B}}. \end{aligned}$$
(6)

Proof

Let us take \(\mathrm{u}, \hat{\mathrm{u}}\in \mathbf{Z}\), then consider from (4)

$$\begin{aligned} \bigl\Vert \mathscr{P}(\mathrm{u})-\mathscr{P}(\hat{ \mathrm{u}}) \bigr\Vert \leq & \kappa _{\phi} \vert \mathrm{u}- \hat{\mathrm{u}} \vert + \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \bigl[\sigma _{ \mathrm{A}} \vert \mathrm{u}-\hat{\mathrm{u}} \vert +\kappa _{\psi} \vert \mathrm{u}- \hat{\mathrm{u}} \vert \bigr] \\ \leq & \biggl(\kappa _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} [\sigma _{ \mathrm{A}}+\kappa _{\psi} ] \biggr) \Vert \mathrm{u}-\hat{ \mathrm{u}} \Vert \\ =&\mathscr{K} \Vert \mathrm{u}-\hat{\mathrm{u}} \Vert , \end{aligned}$$
(7)

where \(\mathscr{K}= (\kappa _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} [\sigma _{ \mathrm{A}}+\kappa _{\psi} ] )\). Hence, the operator \(\mathscr{P}\) is μ-Lipschitz. Furthermore, we can easily obtain the growth condition as follows:

$$\begin{aligned} \bigl\Vert \mathscr{P}(\mathrm{u}) \bigr\Vert \leq & \vert \mathrm{u}_{0} \vert +\rho _{\phi} \Vert \mathrm{u} \Vert +\varrho _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \bigl[\rho _{\psi} \Vert \mathrm{u} \Vert +\varrho _{\psi}+\sigma _{\mathrm{A}} \Vert \mathrm{u} \Vert +M_{ \mathrm{B}} \bigr] \\ \leq & \vert \mathrm{u}_{0} \vert + \biggl(\rho _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )}(\rho _{\psi}+ \sigma _{\mathrm{A}}) \biggr) \Vert \mathrm{u} \Vert +\varrho _{\phi}+ \varrho _{ \psi}+M_{\mathrm{B}}. \end{aligned}$$

 □

Lemma 3.3

The operator \(\mathscr{Q}:\mathbf{Z}\rightarrow \mathbf{Z}\) is ongoing and satisfies the following growth requirements:

$$\begin{aligned} \Vert \mathscr{Q}\mathrm{u} \Vert \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{T^{\theta}\mu _{\theta}}{\Gamma (\theta +1)} \bigl[(\sigma _{ \mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+\varrho _{\psi } \bigr]. \end{aligned}$$
(8)

Proof

Let us define a bounded set \(\triangle =\{\mathrm{u}\in \mathbf{Z}: \|\mathrm{u}\|\leq r\}\) and take a sequence \(\{\mathrm{u}_{n}\}\) in △, such that \(\mathrm{u}_{n}\rightarrow \mathrm{u}, \text{ as } n\rightarrow \infty \). As we know that A, B, ψ are continuous mappings, one has

$$\begin{aligned} &\mathrm{A}\mathrm{u}_{n}(\vartheta )\rightarrow \mathrm{A} \mathrm{u}(\vartheta ),\qquad \mathrm{B}\mathrm{x}_{n}(\vartheta ) \rightarrow \mathrm{B}\mathrm{x}_{n}( \vartheta ), \quad \text{as } n\rightarrow \infty , \\ &\mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \rightarrow \mathrm{\psi}\bigl(\vartheta , \mathrm{u}(\vartheta )\bigr)\quad \text{as } n \rightarrow \infty . \end{aligned}$$
(9)

Using \(A_{1}\), we have

$$\begin{aligned} &\mathrm{A}\mathrm{u}_{n}(\vartheta )\rightarrow \mathrm{A} \mathrm{u}(\vartheta ),\qquad \mathrm{B}\mathrm{x}_{n}(\vartheta ) \rightarrow \mathrm{B}\mathrm{x}_{n}( \vartheta ),\quad \text{as } n\rightarrow \infty , \\ &\mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \rightarrow \mathrm{\psi}\bigl(\vartheta , \mathrm{u}(\vartheta )\bigr)\quad \text{as } n \rightarrow \infty , \end{aligned}$$
(10)
$$\begin{aligned} &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \bigl\vert \mathrm{A}\mathrm{u}_{n}(\vartheta )- \mathrm{A}\mathrm{u}( \vartheta ) \bigr\vert \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \sigma _{\mathrm{A}} \bigl\vert \mathrm{u}_{n}( \vartheta )-\mathrm{u}(\vartheta ) \bigr\vert \biggr] \\ &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \bigl\vert \mathrm{B}\mathrm{x}_{n}(\vartheta )-\mathrm{B} \mathrm{x}( \vartheta ) \bigr\vert \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \sigma _{\mathrm{A}} \bigl\vert \mathrm{x}_{n}( \vartheta )-\mathrm{x}(\vartheta ) \bigr\vert \biggr] \end{aligned}$$
(11)

and

$$\begin{aligned} &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \bigl\vert \mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr)- \mathrm{\psi}\bigl(\vartheta , \mathrm{u}(\vartheta )\bigr) \bigr\vert \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \kappa _{\psi} \bigl\vert \mathrm{u}_{n}( \vartheta )-\mathrm{u}(\vartheta ) \bigr\vert . \end{aligned}$$
(12)

We see that both sides of (11), and (12) are integrable on using the Lebesgue dominated convergent theorem, hence we have

$$\begin{aligned} &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl\vert \mathrm{A}\mathrm{u}_{n}(\vartheta )-\mathrm{A} \mathrm{u}(\vartheta ) \bigr\vert \,d\vartheta \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \sigma _{\mathrm{A}} \bigl\vert \mathrm{u}_{n}(\vartheta )- \mathrm{u}( \vartheta ) \bigr\vert \,d\vartheta \biggr]\rightarrow 0, \quad \text{as } n\rightarrow \infty , \\ &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl\vert \mathrm{B}\mathrm{x}_{n}(\vartheta )-\mathrm{B} \mathrm{x}(\vartheta ) \bigr\vert \,d\vartheta \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} (t-\vartheta )^{\theta -1} \sigma _{\mathrm{A}} \bigl\vert \mathrm{x}_{n}( \vartheta )-\mathrm{x}(\vartheta ) \bigr\vert \,d \vartheta \biggr]\rightarrow 0, \quad \text{as } n\rightarrow \infty . \end{aligned}$$
(13)

In addition, in the same way, we can also show that

$$\begin{aligned} &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl\vert \mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr)- \mathrm{\psi}\bigl(\vartheta , \mathrm{u}(\vartheta )\bigr) \bigr\vert \,d\vartheta \\ &\quad \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \kappa _{\psi} \bigl\vert \mathrm{u}_{n}(\vartheta )- \mathrm{u}( \vartheta ) \bigr\vert \,d\vartheta \rightarrow 0, \quad \text{as } n\rightarrow \infty . \end{aligned}$$
(14)

Thus, from (13) and (14), we conclude the continuity of \(\mathscr{Q}\). For the growth relation, we have by using \((A_{2}, A_{3})\)

$$\begin{aligned} \bigl\vert \mathscr{Q}\mathrm{u}(t) \bigr\vert =& \biggl\vert \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl[\mathrm{A}\mathrm{u}(\vartheta )+\mathrm{B} \mathrm{x}(\vartheta )+\mathrm{ \psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr) \bigr]\,d\vartheta \biggr] \biggr\vert \\ \leq & \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{t^{\theta}\mu _{\theta}}{\Gamma (\theta +1)} \bigl[(\sigma _{ \mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+\varrho _{\psi } \bigr], \end{aligned}$$

which further yields

$$\begin{aligned} \Vert \mathscr{Q}\mathrm{u} \Vert \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{T^{\theta}\mu _{\theta}}{\Gamma (\theta +1)} \bigl[(\sigma _{ \mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+ \varrho _{\psi } \bigr]. \end{aligned}$$

 □

Lemma 3.4

The operator \(\mathscr{Q}\) is compact, and hence completely continuous.

Proof

Corresponding to Lemma 3.3, the operator \(\mathscr{Q}\) is bounded and continuous. For equi-continuity, let \(t_{p}>t_{q}\) in J, and consider by using \(\nabla =\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )}\)

$$\begin{aligned} & \bigl\vert \mathscr{Q}\mathrm{u}_{n}(t_{p})- \mathscr{Q}\mathrm{u}_{n}(t_{q}) \bigr\vert \\ &\quad = \nabla \biggl\vert \int _{0}^{t_{p}} (t_{p}-\vartheta )^{\theta -1} \bigl[ \mathrm{A}\mathrm{u}_{n}(\vartheta )+ \mathrm{B}\mathrm{x}_{n}( \vartheta )+\mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \bigr]\,d\vartheta \\ &\qquad - \int _{0}^{t_{q}} (t_{q}-\vartheta )^{\theta -1} \bigl[\mathrm{A} \mathrm{u}_{n}(\vartheta )+ \mathrm{B}\mathrm{x}_{n}(\vartheta )+ \mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \bigr]\,d \vartheta \biggr\vert \\ &\quad \leq \nabla \biggl[ \int _{0}^{t_{q}}\bigl[(t_{q}- \vartheta )^{\theta -1}-(t_{p}- \vartheta )^{\theta -1}\bigr] \bigl\vert \mathrm{A}\mathrm{u}_{n}(\vartheta )+ \mathrm{B} \mathrm{x}_{n}(\vartheta )+\mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \bigr\vert \,d\vartheta \\ &\qquad + \int _{t_{q}}^{t_{p}}(t_{p}-\vartheta )^{\theta -1} \bigl\vert \mathrm{A}\mathrm{u}_{n}(\vartheta )+ \mathrm{B}\mathrm{x}_{n}( \vartheta )+\mathrm{\psi}\bigl(\vartheta , \mathrm{u}_{n}(\vartheta )\bigr) \bigr\vert \,d\vartheta \biggr] \\ &\quad \leq \nabla \bigl((\sigma _{\mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+ \varrho _{\psi}\bigr) \biggl[ \int _{0}^{t_{q}}\bigl[(t_{q}- \vartheta )^{\theta -1}-(t_{p}- \vartheta )^{\theta -1}\bigr] \,d\vartheta \\ &\qquad + \int _{t_{q}}^{t_{p}}(t_{p}-\vartheta )^{\theta -1}\,d\vartheta \biggr] \\ &\quad =\nabla \bigl((\sigma _{\mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+ \varrho _{\psi}\bigr) \biggl( \frac{t_{q}^{\vartheta}-t_{p}^{\vartheta}+2(t_{p}-t_{q})^{\vartheta }}{\vartheta} \biggr). \end{aligned}$$

We see that at \(t_{q}\rightarrow t_{p}\), the relation given by \(\nabla ((\sigma _{\mathrm{A}}+\rho _{\psi})r+M_{\mathrm{B}}+\varrho _{ \psi}) ( \frac{t_{q}^{\vartheta}-t_{p}^{\vartheta}+2(t_{p}-t_{q})^{\vartheta }}{\vartheta} )\) tends to zero. Hence,

$$ \bigl\vert \mathscr{Q}\mathrm{u}_{n}(t_{p})- \mathscr{Q}\mathrm{u}_{n}(t_{q}) \bigr\vert \rightarrow 0, \quad \text{as } t_{q}\rightarrow t_{p}. $$

Moreover, on using the boundedness and continuity of \(\mathscr{Q}\), we have

$$ \bigl\Vert \mathscr{Q}\mathrm{u}_{n}(t_{p})- \mathscr{Q}\mathrm{u}_{n}(t_{q}) \bigr\Vert \rightarrow 0. $$

Thus, \(\mathscr{Q}\) is uniform continuous, which yields that it is relatively compact and hence completely continuous. Therefore, \(\mathscr{Q}\) is μ-Lipschitz with constant zero, in view of Proposition 2.7. □

Theorem 3.5

The problem (1) has a bounded set of solutions, with at least one solution.

Proof

According to Lemma 3.2, the operator \(\mathscr{P}\) is μ-Lipschitz with constant \(\mathscr{K}\) and bounded. Also, the operator \(\mathscr{Q}\) is μ-Lipschitz and bounded. Hence, using Proposition 2.6, \(\mathscr{P}+\mathscr{Q}\) is μ-Lipschitz with constant \(\mathscr{K}\). We describe a set of solutions for (1) as

$$\begin{aligned} &\mathbf{D}=\bigl\{ \mathrm{u}\in \mathbf{Z}: \exists \partial \in [0, 1],\qquad \mathrm{u}=\partial [\mathscr{P}+\mathscr{Q}]\mathrm{u}\bigr\} ,\\ &\mathrm{u}=\partial [\mathscr{P}+\mathscr{Q}] \mathrm{u}, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert \mathrm{u} \Vert \leq &\partial \bigl[ \Vert \mathscr{P}\mathrm{u} \Vert + \Vert \mathscr{Q}\mathrm{u} \Vert \bigr] \\ \leq & \vert \mathrm{u}_{0} \vert + \biggl(\rho _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )}(\rho _{\psi}+ \sigma _{\mathrm{A}}) \biggr) \Vert \mathrm{u} \Vert +\varrho _{\phi}+ \varrho _{ \psi}+M_{\mathrm{B}} \\ &{}+\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{T^{\theta}\mu _{\theta}}{\Gamma (\theta +1)} \bigl[(\sigma _{ \mathrm{A}}+\rho _{\psi}) \Vert \mathrm{u} \Vert +M_{\mathrm{B}}+\varrho _{ \psi } \bigr]. \end{aligned}$$
(15)

Let the set D be not bounded and consider dividing (15) by \(\|\mathrm{u}\|\) both sides, we have

$$\begin{aligned} 1 \leq & \lim_{ \Vert \mathrm{u} \Vert \rightarrow \infty} \frac{1}{ \Vert \mathrm{u} \Vert } \biggl\{ \vert \mathrm{u}_{0} \vert + \biggl(\rho _{\phi}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )}(\rho _{\psi}+ \sigma _{\mathrm{A}}) \biggr) \Vert \mathrm{u} \Vert +\varrho _{\phi}+\varrho _{ \psi}+M_{\mathrm{B}} \\ &{}+\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{T^{\theta}\mu _{\theta}}{\Gamma (\theta +1)} \bigl[(\sigma _{ \mathrm{A}}+\rho _{\psi}) \Vert \mathrm{u} \Vert +M_{\mathrm{B}}+\varrho _{ \psi } \bigr] \biggr\} =0, \end{aligned}$$
(16)

which is impossible. Hence, set D is bounded. Therefore, \(\mathscr{P}+\mathscr{Q}\) has at least one solution. □

Theorem 3.6

Using the assumption \((A_{1}, A_{3})\), the problem (1) has a unique solution if

$$ \Upsilon _{\theta , \mu _{\theta}}=\kappa _{\phi}+ \frac{\mathrm{M}(1-\theta )(\sigma _{\mathrm{A}}+\kappa _{\psi})}{{\mathrm{M}}(\theta )}+ \frac{\mathrm{M}(1-\theta )(\sigma _{\mathrm{A}} +\kappa _{\psi})}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)}< 1 $$

holds.

Proof

Let us define an operator \(\mathscr{L}: \mathbf{Z}\rightarrow \mathbf{Z}\) by

$$\begin{aligned} \mathscr{L}\bigl[\mathrm{u}(t)\bigr] =&\mathrm{u}_{0}+ \phi \bigl(t, \mathrm{u}(t)\bigr)+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\mathrm{A} \mathrm{u}(t)+\mathrm{B}\mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr) \\ &{}-\Omega _{\mathrm{A}, \mathrm{B}, \psi} \biggl(1+ \frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr) \\ &{}+ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl[\mathrm{A}\mathrm{u}(\vartheta )+\mathrm{B} \mathrm{x}(\vartheta )+\mathrm{ \psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr) \bigr]\,d\vartheta \biggr]. \end{aligned}$$
(17)

Taking \(\mathrm{u}, \hat{\mathrm{u}}\in \mathbf{Z}\), and consider

$$\begin{aligned} \Vert \mathscr{L}\mathrm{u}-\mathscr{L}\hat{\mathrm{u}} \Vert \leq &\max_{t \in \mathrm{J}} \biggl\{ \bigl\vert \phi \bigl(t, \mathrm{u}(t)\bigr)-\phi \bigl(t, \hat{\mathrm{u}}(t)\bigr) \bigr\vert \\ &{}+\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[ \bigl\vert \mathrm{A}\mathrm{u}(t)- \mathrm{A}\hat{\mathrm{u}}(t) \bigr\vert + \bigl\vert \mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr)-\mathrm{\psi}\bigl(t, \hat{\mathrm{u}}(t)\bigr) \bigr\vert \\ &{}+\frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \\ &{}\times \bigl[ \bigl\vert \mathrm{A}\mathrm{u}(\vartheta )-\mathrm{A} \hat{ \mathrm{u}}(\vartheta ) \bigr\vert + \bigl\vert \mathrm{\psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr)-\mathrm{\psi}\bigl(\vartheta , \hat{\mathrm{u}}( \vartheta )\bigr) \bigr\vert \bigr]\,d\vartheta \biggr] \biggr\} . \end{aligned}$$
(18)

On applying \((A_{1}, A_{3})\), (18) yields

$$\begin{aligned} \Vert \mathscr{L}\mathrm{u}-\mathscr{L}\hat{\mathrm{u}} \Vert \leq &\Upsilon _{ \theta , \mu _{\theta}} \Vert \mathrm{u}-\hat{\mathrm{u}} \Vert . \end{aligned}$$
(19)

Thus, (1) has a unique solution. □

4 Stability analysis

Here, we deduce some results for stability theory. The stability definition given by Ulam and Hyers has attracted much attention in the last several decades. The mentioned stability is easy to implement and simple to investigate for various problems. According to Ulam near to every function there is a best approximate function. The concept was pointed out during a seminar by Ulam in 1940 and later the stability was explained by Hyers [21, 22]. Here, we recollect some definitions as follows:

Definition 4.1

Consider

$$\begin{aligned} &{}_{0}^{mABC}D^{\theta}\mathrm{u}(t) = \mathrm{f}\bigl(t, \mathrm{u}(t)\bigr),\quad t \in \mathrm{J}, \\ &\mathrm{u}(0)=\mathrm{u}_{0}. \end{aligned}$$
(20)

For any \(\varepsilon >0\), if the inequality satisfies

$$\begin{aligned} \bigl\vert _{0}^{mABC}D^{\theta} \mathrm{u}(t)-\mathrm{f}\bigl(t, \mathrm{u}(t)\bigr) \bigr\vert \leq \varepsilon , \quad t\in \mathrm{J}, \end{aligned}$$
(21)

then we say that (21) is U–H stable if ∃ a constant \(\mathscr{C}_{\mathrm{f}}>0\), and a unique solution \(\hat{\mathrm{u}}\in \mathbf{Z}\), ∋

$$\begin{aligned} \bigl\vert \mathrm{u}(t)-\hat{\mathrm{u}}(t) \bigr\vert \leq \mathscr{C}_{\mathrm{f}} \varepsilon ,\quad t\in \mathrm{J}. \end{aligned}$$
(22)

In addition, if ∃ is a nondecreasing mapping \(\chi :(0, T)\rightarrow \mathrm{R}\), such that \(\chi (0)=0\), then we say that (21) is generalized U–H stable if

$$\begin{aligned} \bigl\vert \mathrm{u}(t)-\hat{\mathrm{u}}(t) \bigr\vert \leq \mathscr{C}_{\mathrm{f}}\chi ( \varepsilon ), \quad t\in \mathrm{J}. \end{aligned}$$

Remark 4.2

Let, for a mapping ξ independent of u, such that for any \(\varepsilon >0\):

\((i)\):

\(|\xi (t)|\leq \varepsilon \), \(t\in \mathrm{J}\),

•:

The problem (1) is considered as follows:

$$ \begin{aligned}& _{0}^{mABC}D^{\theta} \bigl[\mathrm{u}(t)-\mathrm{\phi}\bigl(t, \mathrm{u}(t)\bigr)\bigr]= \mathrm{A} \mathrm{u}(t)+\mathrm{B}\mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr)+ \xi (t), \quad 0< \theta \leq 1, \\ & \mathrm{u}(0)=\mathrm{u}_{0}. \end{aligned} $$
(23)

The solution of (23) is given as follows:

$$\begin{aligned} \mathrm{u}(t) =&\mathrm{u}_{0}+\phi \bigl(t, \mathrm{u}(t)\bigr) \\ &{}+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\mathrm{A} \mathrm{u}(t)+\mathrm{B} \mathrm{x}(t)+\mathrm{\psi}\bigl(t, \mathrm{u}(t)\bigr)+ \xi (t) -\Theta _{\mathrm{A}, \mathrm{B}, \psi} \biggl(1+ \frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr) \\ &{}-\xi (0) \biggl(1+\frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr)+\frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t- \vartheta )^{\theta -1} \bigl[\mathrm{A}\mathrm{u}(\vartheta )+ \mathrm{B}\mathrm{x}(\vartheta )+\mathrm{ \psi}\bigl(\vartheta , \mathrm{u}( \vartheta )\bigr) \bigr]\,d\vartheta \\ &{}+\frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1}\xi (\vartheta ) \,d\vartheta \biggr]. \end{aligned}$$
(24)

In view of Theorem 3.6, we may write (24) as follows in terms of the operator

$$\begin{aligned} \mathrm{u}(t) =&\mathscr{L}\mathrm{u}(t)+ \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\xi (t) \\ &{}-\xi (0) \biggl(1+\frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr)+\frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t- \vartheta )^{\theta -1}\xi (\vartheta ) \,d\vartheta \biggr]. \end{aligned}$$
(25)

Lemma 4.3

From the solution (25) in view of Remark 4.2\((i)\), we have

$$ \bigl\vert \mathrm{u}(t)-\mathscr{L}\mathrm{u}(t) \bigr\vert \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[1+ \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} \biggr]\varepsilon . $$

Proof

Consider the solution (25)

$$\begin{aligned} \bigl\vert \mathrm{u}(t)-\mathscr{L}\mathrm{u}(t) \bigr\vert =& \biggl\vert \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\xi (t) \\ &{}-\xi (0) \biggl(1+\frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} \biggr)+\frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t- \vartheta )^{\theta -1}\xi (\vartheta ) \,d\vartheta \biggr] \biggr\vert . \end{aligned}$$
(26)

Since \(\xi (0) (1+\frac{\mu _{\theta}t^{\theta}}{\Gamma (\theta +1)} )>0\), therefore, from (26), we have

$$\begin{aligned} \bigl\vert \mathrm{u}(t)-\mathscr{L}\mathrm{u}(t) \bigr\vert \leq & \biggl\vert \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[\xi (t)+ \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1}\xi (\vartheta ) \,d\vartheta \biggr] \biggr\vert \\ \leq &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \bigl\vert \xi (t) \bigr\vert + \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}}{\Gamma (\theta )} \int _{0}^{t} (t-\vartheta )^{ \theta -1} \bigl\vert \xi (\vartheta ) \bigr\vert \,d\vartheta \\ \leq &\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[1+ \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} \biggr]\varepsilon . \end{aligned}$$

 □

Theorem 4.4

The solution of (1) is U–H stable and generalized U–H stable if \(\Upsilon _{\theta , \mu _{\theta}}<1\).

Proof

Consider any solution u of (1), and let û be the unique result, then we take

$$\begin{aligned} \bigl\vert \hat{\mathrm{u}}(t)-\mathrm{u}(t) \bigr\vert =& \bigl\vert \hat{\mathrm{u}}(t)- \mathscr{L}\mathrm{u}(t) \bigr\vert \\ =& \bigl\vert \hat{\mathrm{u}}(t)-\mathscr{L}\hat{\mathrm{u}}(t)+ \mathscr{L} \hat{\mathrm{u}}(t)-\mathscr{L}\mathrm{u}(t) \bigr\vert \\ \leq & \bigl\vert \hat{\mathrm{u}}(t)-\mathscr{L}\hat{\mathrm{u}}(t) \bigr\vert + \bigl\vert \mathscr{L}\hat{\mathrm{u}}(t)-\mathscr{L}\mathrm{u}(t) \bigr\vert . \end{aligned}$$

Using Lemma 4.3, and Theorem 3.6, we obtain

$$\begin{aligned} \bigl\vert \hat{\mathrm{u}}(t)-\mathrm{u}(t) \bigr\vert \leq \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[1+ \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} \biggr]\varepsilon + \Upsilon _{\theta , \mu _{\theta}} \bigl\vert \hat{\mathrm{u}}(t)- \mathrm{u}(t) \bigr\vert , \end{aligned}$$

which further yields

$$\begin{aligned} \Vert \hat{\mathrm{u}}-\mathrm{u} \Vert \leq & \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \biggl[1+ \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} \biggr]\varepsilon + \Upsilon _{\theta , \mu _{\theta}} \Vert \hat{\mathrm{u}}-\mathrm{u} \Vert \end{aligned}$$

and after simplification, we obtain

$$\begin{aligned} \Vert \hat{\mathrm{u}}-\mathrm{u} \Vert \leq \frac{\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} [1+\frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} ]}{1-\Upsilon _{\theta , \mu _{\theta}}} \varepsilon . \end{aligned}$$
(27)

Hence, the solution is U–H stable. Further, if we have a nondecreasing mapping \(\chi :(0, T)\rightarrow \mathrm{R}\) defined by \(\chi (\varepsilon )=\varepsilon \), such that \(\chi (0)=0\), then from (28), one has

$$\begin{aligned} \Vert \hat{\mathrm{u}}-\mathrm{u} \Vert \leq \frac{\frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} [1+\frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} ]}{1-\Upsilon _{\theta , \mu _{\theta}}} \chi (\varepsilon ). \end{aligned}$$

Therefore, the solution is generalized U–H stable. □

5 Example

To demonstrate our results, we consider an example as follows:

Example 5.1

$$ \begin{aligned}& _{0}^{mABC}D^{\theta} \biggl[\mathrm{u}(t)- \frac{\sin \vert \mathrm{u}(t)) \vert }{t^{2}+100} \biggr]= \frac{\exp (- \vert \mathrm{u}(t) \vert )}{t+50}+ \frac{t}{2}+ \frac{ \vert \mathrm{u}(t) \vert }{100+ \vert \mathrm{u}(t) \vert }, \quad 0< \theta < 1, \\ & \mathrm{u}(0)=0.5, \end{aligned} $$
(28)

we have from (28) by using \(\mathrm{J}=[0, 1]\)

$$\begin{aligned} &\phi \bigl(t, \mathrm{u}(t)\bigr)= \frac{\sin \vert \mathrm{u}(t)) \vert }{t^{2}+100},\qquad \psi \bigl(t, \mathrm{u}(t)\bigr)=\frac{ \vert \mathrm{u}(t) \vert }{100+ \vert \mathrm{u}(t) \vert }, \\ &\mathrm{A}\mathrm{u}(t)=\frac{\exp (- \vert \mathrm{u}(t) \vert )}{t+50},\qquad \mathrm{B}\mathrm{x}(t)= \frac{t}{2}, \quad \mathrm{u}_{0}=0.5. \end{aligned}$$

According to assumptions \((A_{1}-A_{3})\), we have

$$\begin{aligned} &\kappa _{\phi}=\frac{1}{100}, \qquad \kappa _{\psi}=\frac{1}{100},\qquad \sigma _{\mathrm{A}}= \frac{1}{50},\qquad M_{\mathrm{B}}=\frac{1}{2}, \\ & \rho _{ \phi}=\rho _{\psi}=\frac{1}{100},\qquad \varrho _{\phi}=\varrho _{\psi}=0. \end{aligned}$$

In view of Theorem 3.6, we have

$$\begin{aligned} \Upsilon _{\theta , \mu _{\theta}} =&\kappa _{\phi}+ \frac{\mathrm{M}(1-\theta )(\sigma _{\mathrm{A}}+\kappa _{\psi})}{{\mathrm{M}}(\theta )}+ \frac{\mathrm{M}(1-\theta )(\sigma _{\mathrm{A}} +\kappa _{\psi})}{{\mathrm{M}}(\theta )} \frac{\mu _{\theta}T^{\theta}}{\Gamma (\theta +1)} \\ =&\frac{1}{100}+ \frac{3\mathrm{M}(1-\theta )}{100{\mathrm{M}}(\theta )}+ \frac{3}{100} \frac{\mathrm{M}(1-\theta )}{{\mathrm{M}}(\theta )} \frac{\theta}{(1-\theta )\Gamma (\theta +1)}< 1, \quad \forall \theta \in (0, 1]. \end{aligned}$$

For instance, if we put \(\theta =0.5\), we have \(\Upsilon _{\theta , \mu _{\theta}}=0.12345<1\). Hence, the given example has a unique solution. Moreover, according to Theorem 4.4, the solution is also U–H stable and consequently generalized U–H stable.

6 Conclusion

In this paper, we have investigated a class of evolution problem with modified \(ABC\) fractional-order derivative. We have determined adequate conditions for the existence of at least one solution and its uniqueness using topological degree theory. In addition, the stability results were derived by using the U–H criteria. We concluded that

  • A modified type \(ABC\) derivative of fractional order has the capability to remove the shortcoming of a normal \(ABC\) derivative called the initialization condition.

  • The \(ABC\) derivative of fractional order of a constant function is zero in the Caputo sense.

  • No extra assumptions were needed to be imposed on the function by using the aforesaid operator.

  • Topological degree theory relaxes the strong compact condition by a weaker one.

  • Classical fixed-point theorems like Schauder and Brouwer’s fixed-point theorems need strong compact condition for operator to have at least one fixed point. However, in the case of topological degree theory, we need a part of the operator to be compact.

From the above outcomes, we concluded that topological degree theory can be applied to study various nonlinear problems, in particular impulsive fractional-order problems under a modified \(ABC\) derivative in the near future.