Controlled Extended Branciari Quasi-b-Metric Spaces, Results, and Applications to Riesz-Caputo Fractional Differential Equations and Nonlinear Matrix Equations

Abstract

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a ${\mathcal{G}}_q$-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples.

1. Extended Branciari Quasi-b-Distance

In the process of the rapid development of fixed-point theory, many new spaces have been emerged with their applications and the study of the newly emerged spaces. Such developments have been an interesting topic among the mathematical research community. A very interesting notion of b-metric space was introduced by Bakhtin [1] (later used by Czerwik in [2,3]). Branciari [4] developed the notion of Branciari distance space via substituting triangle inequality by quadrilateral inequality, while Kamran et al. [5] introduced the notion of extended b-metric space. Mlaiki et al. [6] introduced the concept of controlled metric type space by using two controlled functions in triangle inequality, and thus extended the work of Kamran et al. [5].

Definition 1

([6]). Let $\mathsf{ϝ}\ne \varnothing$ be a set and $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$. A function $d_c:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ is said to be a controlled metric type if, for all $\varpi ,\varsigma ,\vartheta \in \mathsf{ϝ}$:

(dc1) 

$d_c(\varpi ,\varkappa )=0$ if and only if $\varpi =\varkappa$,

(dc2) 

$d_c(\varpi ,\varkappa )=d_c(\varkappa ,\varpi )$,

(dc3) 

$d_c(\varpi ,\varkappa )\le \theta (\varpi ,\vartheta )d_c(\varpi ,\vartheta )+\theta (\vartheta ,\varkappa )d_c(\vartheta ,\varkappa )$.

The pair $(\mathsf{ϝ},d_c)$ is called a controlled metric-type space.

Recently, Abdeljawad et al. [7] have defined the notion of extended Branciari b-metric space by combining extended b-metric and Branciari distances.

Definition 2

([7]). Let $\mathsf{ϝ}\ne \varnothing$ be a set and $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$. A function $d_{eb}:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ is said to be an extended Branciari b-metric ($d_{eb}$-metric, for short) if it satisfies, for all $\varpi ,\varsigma \in \mathsf{ϝ}$, $\varkappa \ne \varrho \in \mathsf{ϝ}\backslash \{\varpi ,\varsigma \}$:

(ebb1) 

$d_{eb}(\varpi ,\varsigma )=0$ if and only if $\varpi =\varsigma$,

(ebb2) 

$d_{eb}(\varpi ,\varsigma )=d_{eb}(\varsigma ,\varpi )$,

(ebb3) 

$d_{eb}(\varpi ,\varsigma )\le \theta (\varpi ,\varsigma )[d_{eb}(\varpi ,\varkappa )+d_{eb}(\varkappa ,\varrho )+d_{eb}(\varrho ,\varsigma )]$.

The pair $(\mathsf{ϝ},d_{eb})$ will be called an extended Branciari b-metric space (EBbMS for short). If $\theta (\varpi ,\varsigma )=b$, then $(\mathsf{ϝ},d_{eb})$ will be called a Branciari b-metric space.

In [8], Zubair et al. combined the work of Mlaiki et al. [6] and Abdeljawad et al. [7], and introduced the controlled b-Branciari metric-type space as follows:

Definition 3

([8]). Let $\mathsf{ϝ}\ne \varnothing$ be a set and $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$. A function $d_c:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ is said to be a controlled b-Branciari metric type if it satisfies, for all distinct $\vartheta ,\varsigma \in \mathsf{ϝ}\backslash \{\varpi ,\varkappa \}$:

(dcb1) 

$d_{cb}(\varpi ,\varkappa )=0$ if and only if $\varpi =\varkappa$,

(dcb2) 

$d_{cb}(\varpi ,\varkappa )=d_{cb}(\varkappa ,\varpi )$,

(dcb3) 

$d_{cb}(\varpi ,\varkappa )\le \theta (\varpi ,\vartheta )d_{cb}(\varpi ,\vartheta )+\theta (\vartheta ,\varsigma )d_{cb}(\vartheta ,\varsigma )+\theta (\varsigma ,\varkappa )d_{cb}(\varsigma ,\varkappa )$.

The pair $(\mathsf{ϝ},d_{cb})$ is called a controlled b-Branciari metric-type space.

On the other hand, the notion of b-metric space was generalized to quasi-b-metric space in [9], and the concepts of right and left quasi-b-metric spaces were established in [10]. Jain et al. [11] recently introduced the idea of extended Branciari quasi-b-metric spaces and right and left completeness in these spaces, which represent an extension of the quasi-b-metric spaces notion.

Definition 4.

Let $\mathsf{ϝ}\ne \varnothing$ be a set and $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$. A function $b_q:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ is called an extended Branciari quasi-b-metric, if for all $\varpi ,\varkappa \in \mathsf{ϝ}$, and all distinct $\vartheta ,\varsigma \in \mathsf{ϝ}\backslash \{\varpi ,\varkappa \}$:

(qeb1) 

$b_q(\varpi ,\varkappa )=0\Rightarrow \nu =\vartheta$,

(qeb2) 

$b_q(\varpi ,\varkappa )\le \theta (\varpi ,\varkappa )[q_c(\varpi ,\vartheta )+q_c(\vartheta ,\varsigma )+q_c(\varsigma ,\varkappa )]$.

The triplet $(\mathsf{ϝ},b_q,\theta )$ is then called an extended Branciari quasi-b-metric space (EBQbMS for short) with the coefficient $\theta (\nu ,\vartheta )$.

Inspired by the above discussion, in the following sections, we are going to introduce the controlled extended Branciari quasi-b-metric spaces and its related notions.

Definition 5.

Let $\mathsf{ϝ}\ne \varnothing$ be a set and $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$. A function $q_c:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ is said to be a controlled extended Branciari quasi-b-metric if it satisfies the following:

(q1) 

$q_c(\varpi ,\varkappa )=0\Rightarrow \varpi =\varkappa$,

(q2) 

$q_c(\varpi ,\varkappa )\le \theta (\varpi ,\vartheta )q_c(\varpi ,\vartheta )+\theta (\vartheta ,\varsigma )q_c(\vartheta ,\varsigma )+\theta (\varsigma ,\varkappa )q_c(\varsigma ,\varkappa )$

for all distinct $\vartheta ,\varsigma \in \mathsf{ϝ}\backslash \{\varpi ,\varkappa \}$. The pair $(\mathsf{ϝ},q_c,\theta )$ is called a controlled extended Branciari quasi-b-metric space (controlled EBQbMS for short).

Definition 6.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS and let $\left\{{\varpi }_n\right\}$ be a sequence in ϝ and $\varpi \in \mathsf{ϝ}$. The sequence $\left\{{\varpi }_n\right\}$ converges to ϖ if and only if ${lim}_{n\to \infty }{q}_c({\varpi }_n,\varpi )={lim}_{n\to \infty }{q}_c(\varpi ,{\varpi }_n)=0$.

Remark 1.

In a controlled EBQbMS, the uniqueness of limit is for a convergent sequence.

Definition 7.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS and let $\left\{{\varpi }_n\right\}$ be a sequence in ϝ. The sequence $\left\{{\varpi }_n\right\}$ is said to be as follows:

(i) 

left-$q_c$-Cauchy if for every $ϵ>0$, there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that $q_c({\varpi }_r,{\varpi }_s)<ϵ$ for all $r>s>N$,

(ii) 

right-$q_c$-Cauchy if for every $ϵ>0$, there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that $q_c({\varpi }_r,{\varpi }_s)<ϵ$ for all $s>r>N$,

(iii) 

$q_c$-Cauchy if for every $ϵ>0$, there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that $q_c({\varpi }_r,{\varpi }_s)<ϵ$ for all $r,s>N$.

Definition 8.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS. Then $(\mathsf{ϝ},q_c,\theta )$ is called the following:

(i) 

left-$q_c$-complete if every left-$q_c$-Cauchy sequence in ϝ is convergent,

(ii) 

right-$q_c$-complete if every right-$q_c$-Cauchy sequence in ϝ is convergent,

(iii) 

$q_c$-complete if every $q_c$-Cauchy sequence in ϝ is convergent.

Definition 9.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS. The mapping $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ is continuous if each sequence for $\left\{{\varpi }_n\right\}$ in ϝ is convergent to $\varpi \in \mathsf{ϝ}$ and the sequence $\{\Im {\varpi }_n\}$ converges to $\Im \varpi$, that is,

$$\underset{n\to \infty }{lim}{q}_c(\Im {\varpi }_n,\Im \varpi )=\underset{n\to \infty }{lim}{q}_c(\Im \varpi ,\Im {\varpi }_n)=0.$$

Example 1.

Let $A,B,C,D$ be four cities of India, and suppose that there is only a one-way inter-city connection. Then, we define a set of nodes as $\mathsf{ϝ}=\{A,B,C,D\}$. Denote $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$ as the fare between the cities, which is defined as

$$\theta (A,B)=2=\theta (B,A),\theta (A,C)=2=\theta (C,A),\theta (A,D)=3=\theta (D,A),$$

$$\theta (B,C)=2=\theta (C,B),\theta (B,D)=2=\theta (D,B),\theta (C,D)=3=\theta (D,C).$$

Next, we denote the inter-city distances as $q_c:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ and define

$$q_c(\varsigma ,\varsigma )=0,\mathit{for}\mathit{all}\varsigma \in \mathsf{ϝ},$$

$$q_c(A,B)=5,q_c(B,A)=6,q_c(A,C)=100,q_c(C,A)=98,q_c(A,D)=5,$$

$$q_c(D,A)=6,q_c(B,C)=8,q_c(C,B)=7,q_c(B,D)=28,q_c(D,B)=30,$$

$$q_c(C,D)=13,q_c(D,C)=11.$$

In Figure 1, $q_c$ is demonstrated.

Figure 1. Distance graph, $q_c$.

It can be readily ascertained that $q_c$ is a controlled EBQbMS but not an EBQbMS as

$$q_c(A,C)=100\nleqq 88=\theta (A,C)[q_c(A,B)+q_c(B,D)+q_c(D,C)].$$

Example 2.

Let $\mathsf{ϝ}=\{a,b,c,d\}$. Define $q_c:{\mathsf{ϝ}}^2\to {\mathbb{R}}_{+}$ by

$$q_c(\varsigma ,\varsigma )=0,\mathit{for}\mathit{all}\varsigma \in \mathsf{ϝ},$$

$$q_c(a,b)=q_c(b,a)=1/2,q_c(a,c)=1,q_c(a,d)=1/4,$$

$$q_c(b,c)=1/3,q_c(b,d)=2,q_c(c,a)=1/2,q_c(c,b)=1,$$

$$q_c(c,d)=1/3,q_c(d,a)=3/2,q_c(d,b)=2,q_c(d,c)=1/4.$$

Define $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$ as

$$\theta (\varsigma ,\varsigma )=1,forall\varsigma \in \mathsf{ϝ},$$

$$\theta (a,b)=\theta (b,c)=\theta (c,b)=\theta (a,c)=1,\theta (b,a)=4/3,$$

$$\theta (c,a)=2,\theta (d,b)=4/3,\theta (b,d)=1,\theta (c,d)=3,$$

$$\theta (c,b)=1,\theta (a,d)=\theta (d,a)=4/3,\theta (d,c)=2.$$

In Figure 2, $q_c$ and θ are demonstrated.

Figure 2. (a) Distance graph, $q_c$. (b) Weight graph, $\theta$.

It can be readily ascertained that $q_c$ is a controlled EBQbMS as

$$q_c(b,d)=2\le \frac{8}{3}=\theta (b,a)q_c(b,a)+\theta (a,c)q_c(a,c)+\theta (c,d)q_c(c,d)$$

but not an EBQbMS as

$$q_c(b,d)=2\nleqq \frac{11}{6}=\theta (b,d)[q_c(b,a)+q_c(a,c)+q_c(c,d)].$$

The importance of this work lies in the fact that, in the context of right completeness (or left completeness), a new implicit relation makes it easier to demonstrate fixed-point results in a controlled EBQbMS space. In the context of underlining space, we also present novel ideas such as right and left Ulam–Hyers stability, right and left weak well-posed properties, right and left weak-limit shadowing properties, as well as their associated results. The two graphics depict a novel application to nonlinear matrix equations when using two illustrations. This demonstrates that our matrix equation solution assurance criteria are “weaker" than those previously derived in the literature. In addition, we used these findings to solve the fractional differential Riesz–Caputo equations with integral antiperiodic boundary values in the underlying space, and this was followed by an example demonstrating the validity of the result.

2. ${\mathcal{G}}_{\mathit{q}}$-Implicit Relations

The following notion was inspired by [12]. Let $\mathsf{ϝ}$ be a nonempty set. As denoted by ${\Psi }_q$, the set of all functions $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ satisfy the following conditions:

(i)

$\psi$ is increasing and $\psi \left(0\right)=0$;

(ii)

there exists $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$ such that

${\sum }_{n=1}^{\infty }{\psi }^n\left(t\right){\prod }_{\mathsf{ȷ}=1}^n\theta ({\varpi }_{\mathsf{ȷ}},{\varpi }_m)<\infty$, for all $t>0$, ${\varpi }_{\mathsf{ȷ}}\in \mathsf{ϝ}$ and $m\in \mathbb{N}$, where ${\psi }^n$ denotes the n-th iterate of $\psi$.

It is clear that $\psi \left(\varpi \right)<\varpi$ and the class ${\Psi }_q\ne \varnothing$.

Example 3.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS, where $\mathsf{ϝ}=[1,\infty )$ and $\theta (\varpi ,\varsigma )=1+\frac{6}{1+ln(\varpi +\varsigma )}$. Consider the mapping $\psi \left(t\right)=\frac{\lambda t}{7}$, where $0<\lambda <1$. Note that $1+\frac{6}{1+ln(\varpi +\varsigma )}\le 7$. Thus,

$${\psi }^n\left(t\right)\prod _{\mathsf{ȷ}=1}^n\theta ({\varkappa }_{\mathsf{ȷ}},{\varkappa }_m)\le \frac{{\lambda }^{n}t}{7^n}·7^n={\lambda }^{n}t.$$

Therefore, ${\sum }_{n=1}^{\infty }{\psi }^n\left(t\right){\prod }_{\mathsf{ȷ}=1}^n\theta ({\varkappa }_{\mathsf{ȷ}},{\varkappa }_m)<\infty$ and hence ${\Psi }_q\ne \varnothing$.

We start by introducing a modified implicit relation, as in [13,14].

Let $\Omega$ be the set of all functions $\mathcal{G}:{\mathbb{R}}_{+}^5\to \mathbb{R}$ satisfying the following conditions:

(Ω₁)

$\mathcal{G}(\kappa \varpi ,\varkappa ,\varkappa ,\varpi ,\varsigma )\le 0$ for all $\varpi ,\varkappa ,\varsigma \ge 0$ and $\kappa \ge 1$, implies that there exists $\psi \in {\Psi }_q$ such that $\kappa \varpi \le \psi \left(\varkappa \right)$;

(Ω₂)

If $\mathcal{G}(\kappa \varpi ,\varpi ,0,0,\varpi )\le 0$, then $\varpi =0$.

Example 4.

Let ${\displaystyle \mathcal{G}({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4,{\varrho }_5)={\varrho }_1-amax\{{\varrho }_2,{\varrho }_3,{\varrho }_4\}+b{\varrho }_5}$, $0<b$ and $0\le a<1$.

(Ω₁)

Let $\varpi ,\varkappa ,\varsigma \ge 0$, $\kappa \ge 1$ and $\mathcal{G}(\kappa \varpi ,\varkappa ,\varkappa ,\varpi ,\varsigma )=\kappa \varpi -amax\{\varkappa ,\varkappa ,\varpi \}+b\varsigma \le 0$.

Now, if $\varkappa <\varpi$, then we obtain $\kappa \varpi \le a\varpi -b\varsigma <a\varpi$, which gives $\varpi <\varpi$ as $b>0$ and $a<1$, i.e., a contradiction. Therefore, we obtain $\kappa \varpi <\psi \left(\varkappa \right)$, where $\psi \left(\varkappa \right)=a\varkappa$ so that $\psi \left(\varkappa \right)<\varkappa$.

(Ω₂)

If $\mathcal{G}(\kappa \varpi ,\varpi ,0,0,\varpi )=\kappa \varpi -amax\{\varpi ,0,0\}+b\varpi \le 0$, that is, $(\kappa -a+b)\varpi \le 0$, then $\kappa \varpi +b\varpi \le a\varpi$ gives $(\kappa +b)\le a$, which is a contradiction. Therefore, $\varpi =0$.

Example 5.

Let ${\displaystyle \mathcal{G}({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4,{\varrho }_5)={\varrho }_1-a{\varrho }_2-b{\varrho }_3-c\frac{{\varrho }_4{\varrho }_5}{1+{\varrho }_5}}$, $0<a<1$, $0\le b,c<1$ and $a+b+c<1$.

(Ω₁)

Let $\varpi ,\varkappa ,\varsigma \ge 0$, $k\ge 1$ and $\mathcal{G}(\kappa \varpi ,\varkappa ,\varkappa ,\varpi ,\varsigma )=\kappa \varpi -a\varkappa -b\varkappa -c\frac{\varpi ,\varsigma }{1+\varsigma }\le 0$.

Then, $\kappa \varpi \le \left(\frac{\kappa (a+b)}{\kappa -c}\right)\varkappa$. Hence, $\kappa \varpi \le \phi \left(\varkappa \right)$, where $\phi \left(\varkappa \right)=h\varkappa$, $h=\frac{\kappa (a+b)}{\kappa -c}<1$. It is easy to check that $0<a<1$, $0\le b,c<1$ with $a+b+c<1$, and $\kappa \ge 1$ can be chosen so that $h<1$.

(Ω₂)

If $\mathcal{G}(\kappa \varpi ,\varpi ,0,0,\varpi )=(\kappa -a){\varpi }^2\le 0$, then it is a contradiction as $\kappa -a>0$. Therefore, $\varpi =0$.

3. Main Results

We introduced ${\mathcal{G}}_q$-implicit type mappings on a controlled EBbQMS.

Definition 10.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS and $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$. We then say that ℑ is a ${\mathcal{G}}_q$-implicit type mapping if there exists $\mathcal{G}\in \Omega$ such that for $\varpi ,\varkappa \in \mathsf{ϝ}$,

$$\begin{array}{c}\hfill \mathcal{G}\left(\begin{array}{c}\theta (\varpi ,\varkappa )q_c(\Im \varpi ,\Im \varkappa ),q_c(\varpi ,\varkappa ),\hfill \\ q_c(\varpi ,\Im \varpi ),q_c(\varkappa ,\Im \varkappa ),q_c(\varpi ,\Im \varkappa )\hfill \end{array}\right)\le 0.\end{array}$$

(1)

For a self-mapping ℑ on a set $\mathsf{ϝ}\ne \varnothing$, it is denoted as $Fix(\Im ):=\{\varpi \in \mathsf{ϝ}:\Im \varpi =\varpi \}$.

Theorem 1.

Let $(\mathsf{ϝ},q_c,\theta )$ be a right-$q_c$-complete controlled EBQbMS with $\theta :{\mathsf{ϝ}}^2\to [0,\infty )$, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be a continuous ${\mathcal{G}}_q$-implicit-type mapping where $\mathcal{G}\in \Omega$. Then, $Fix(\Im )$ is a singleton set.

Proof. 

Let ${\varpi }_0\in \mathsf{ϝ}$ and define a sequence $\left\{{\varpi }_n\right\}\subset \mathsf{ϝ}$ by ${\varpi }_n=\Im {\varpi }_{n-1}$ for all $n\in \mathbb{N}$. If ${\varpi }_n={\varpi }_{n+1}$ for some $n\in {\mathbb{N}}^{*}$, and we then come to the conclusion. Thus, assume that ${\varpi }_n\ne {\varpi }_{n-1}$ for all $n\in \mathbb{N}$. Using (1) with $\varpi ={\varpi }_{n-1}$ and $\varkappa ={\varpi }_n$, we have

$$\mathcal{G}\left(\begin{array}{c}\theta ({\varpi }_{n-1},{\varpi }_n)q_c(\Im {\varpi }_{n-1},\Im {\varpi }_n),q_c({\varpi }_{n-1},{\varpi }_n),\hfill \\ q_c({\varpi }_{n-1},\Im {\varpi }_{n-1}),q_c({\varpi }_n,\Im {\varpi }_n),q_c({\varpi }_{n-1},\Im {\varpi }_n),\hfill \end{array}\right)\le 0,$$

that is,

$$\mathcal{G}\left(\begin{array}{c}\theta ({\varpi }_{n-1},{\varpi }_n)q_c({\varpi }_n,{\varpi }_{n+1}),q_c({\varpi }_{n-1},{\varpi }_n),\hfill \\ q_c({\varpi }_{n-1},{\varpi }_n),q_c({\varpi }_n,{\varpi }_{n+1}),q_c({\varpi }_{n-1},{\varpi }_{n+1})\hfill \end{array}\right)\le 0.$$

Using $\left({\Omega }_1\right)$, we obtain that there is a $\psi \in {\Psi }_q$ such that

$$\theta ({\varpi }_{n-1},{\varpi }_n)q_c({\varpi }_n,{\varpi }_{n+1})\le \psi \left(q_c({\varpi }_{n-1},{\varpi }_n)\right),\mathrm{for}\mathrm{all}n\in \mathbb{N},$$

and so

$$q_c({\varpi }_n,{\varpi }_{n+1})\le \psi \left(q_c({\varpi }_{n-1},{\varpi }_n)\right).$$

With the successive use of (1) with $\left({\Omega }_1\right)$, we have

$$q_c({\varpi }_n,{\varpi }_{n+1})\le {\psi }^n\left(q_c({\varpi }_0,{\varpi }_1)\right),\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(2)

Similarly, if we take $\varpi ={\varpi }_{n-1}$ and $\varkappa ={\varpi }_{n+1}$, then we obtain

$$q_c({\varpi }_n,{\varpi }_{n+2})\le {\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right),\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(3)

Next, we prove $\left\{{\varpi }_n\right\}$ is a right-$q_c$-Cauchy sequence, that is, $q_c({\varpi }_n,{\varpi }_{n+p})\to 0$. We then discuss the two possible cases.

Case I. Let $p=2m$, $m\ge 2$. Then, by (q2), we have

$$\begin{array}{cc}\hfill & q_c({\varpi }_n,{\varpi }_{n+2m})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2})q_c({\varpi }_n,{\varpi }_{n+2})+\theta ({\varpi }_{n+2},{\varpi }_{n+3})q_c({\varpi }_{n+2},{\varpi }_{n+3})\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_{n+2m})q_c({\varpi }_{n+3},{\varpi }_{n+2m})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)+\theta ({\varpi }_{n+2},{\varpi }_{n+3}){\psi }^{n+2}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_m)q_c({\varpi }_{n+3},{\varpi }_{n+2m})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)+\theta ({\varpi }_{n+2},{\varpi }_{n+3}){\psi }^{n+2}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\left[\begin{array}{c}\theta ({\varpi }_{n+3},{\varpi }_{n+4})q_c({\varpi }_{n+3},{\varpi }_{n+4})\hfill \\ +\theta ({\varpi }_{n+4},{\varpi }_{n+5})q_c({\varpi }_{n+4},{\varpi }_{n+5})\hfill \end{array}\right]\hfill \\ \hfill & +\theta ({\varpi }_{n+5},{\varpi }_{n+2m})q_c({\varpi }_{n+5},{\varpi }_{n+2m})]\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)+\theta ({\varpi }_{n+2},{\varpi }_{n+3}){\psi }^{n+2}\left(q_c({\varpi }_0,{\varpi }_1)\right)+\hfill \\ \hfill & \theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta ({\varpi }_{n+3},{\varpi }_{n+4}){\psi }^{n+3}\left(q_c({\varpi }_0,{\varpi }_1)\right)+\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\hfill \\ \hfill & \theta ({\varpi }_{n+4},{\varpi }_{n+5}){\psi }^{n+4}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta ({\varpi }_{n+5},{\varpi }_{n+2m})q_c({\varpi }_{n+5},{\varpi }_{n+2m})\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)+\theta ({\varpi }_{n+2},{\varpi }_{n+3}){\psi }^{n+2}\left(q_c({\varpi }_0,{\varpi }_1)\right)+\hfill \\ \hfill & \theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta ({\varpi }_{n+3},{\varpi }_{n+4}){\psi }^{n+3}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta ({\varpi }_{n+4},{\varpi }_{n+5}){\psi }^{n+4}\left(q_c({\varpi }_0,{\varpi }_1)\right)+\hfill \\ \hfill & ⋮\hfill \\ \hfill & +\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta \left({\varpi }_{n+5;}{\varpi }_{n+2m}\right)\dots \theta ({\varpi }_{n+2m-3},{\varpi }_{n+2m})\hfill \\ \hfill & \left[\begin{array}{c}\theta ({\varpi }_{n+2m-3},{\varpi }_{n+2m-2})q_c({\varpi }_{n+2m-3},{\varpi }_{n+2m-2})\hfill \\ +\theta ({\varpi }_{n+2m-2},{\varpi }_{n+2m-1})q_c({\varpi }_{n+2m-2},{\varpi }_{n+2m-1})\hfill \end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{c}\theta ({\varpi }_{n+3},{\varpi }_{n+2m})\theta ({\varpi }_{n+5},{\varpi }_{n+2m})\dots \hfill \\ \dots \theta ({\varpi }_{n+2m-1},{\varpi }_{n+2m})q_c({\varpi }_{n+2m-1},{\varpi }_{n+2m})\hfill \end{array}\right]\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)\hfill \\ & +\sum _{i=n+2}^{n+2m-2}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & +\prod _{i=1}^{n+2m-1}\theta ({\varpi }_i,{\varpi }_{n+2m})q_c({\varpi }_{n+2m-1},{\varpi }_{n+2m})\hfill \end{array}$$

$$\begin{array}{cc}\hfill q_c({\varpi }_n,{\varpi }_{n+2m})& \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)\hfill \\ & +\sum _{i=n+2}^{n+2m-1}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+2}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_2)\right)\hfill \\ \hfill & +\sum _{i=1}^{\infty }{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & -\sum _{i=1}^{n+1}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & \to 0\mathrm{as}n\to \infty \mathrm{using}\mathrm{property}\mathrm{of}{\Psi }_q,\hfill \end{array}$$

that is, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }_{n+2m})=0$.

Case II. Let $p=2m+1$, $m\ge 1$. Then, by (q2), we have

$$\begin{array}{cc}\hfill & q_c({\varpi }_n,{\varpi }_{n+2m+1})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+1})q_c({\varpi }_n,{\varpi }_{n+1})+\theta ({\varpi }_{n+1},{\varpi }_{n+2})q_c({\varpi }_{n+1},{\varpi }_{n+2})\hfill \\ \hfill & +\theta ({\varpi }_{n+2},{\varpi }_{n+2m+1})q_c({\varpi }_{n+2},{\varpi }_{n+2m+1})\hfill \\ \hfill & \le \theta ({\varpi }_n,{\varpi }_{n+1}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_1)\right)+\theta ({\varpi }_{n+1},{\varpi }_{n+2}){\psi }^{n+1}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+2},{\varpi }_{n+2m+1})\left[\begin{array}{c}\theta ({\varpi }_{n+2},{\varpi }_{n+3})q_c({\varpi }_{n+2},{\varpi }_{n+3})\hfill \\ +\theta ({\varpi }_{n+3},{\varpi }_{n+4})q_c({\varpi }_{n+3},{\varpi }_{n+4})\hfill \\ +\theta ({\varpi }_{n+4},{\varpi }_{n+2m+1})d({\varpi }_{n+4},{\varpi }_{n+2m+1})\hfill \end{array}\right]\hfill \\ \hfill & ⋮\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \theta ({\varpi }_n,{\varpi }_{n+1}){\psi }^n\left(q_c({\varpi }_0,{\varpi }_1)\right)+\theta ({\varpi }_{n+1},{\varpi }_{n+2}){\psi }^{n+1}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+2},{\varpi }_{n+2m+1})\theta ({\varpi }_{n+2},{\varpi }_{n+3}){\psi }^{n+2}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\theta ({\varpi }_{n+2},{\varpi }_{n+2m+1})\theta ({\varpi }_{n+3},{\varpi }_{n+4}){\psi }^{n+3}\left(q_c({\varpi }_0,{\varpi }_1)\right)\hfill \\ \hfill & +\dots \hfill \\ \hfill & +\theta ({\varpi }_{n+2},{\varpi }_{n+2m+1})\theta \left({\varpi }_{n+4;}{\varpi }_{n+2m+1}\right)\dots \theta ({\varpi }_{n+2m-2},{\varpi }_{n+2m+1})\hfill \\ \hfill & \left[\begin{array}{c}\theta ({\varpi }_{n+2m-2},{\varpi }_{n+2m-1})q_c({\varpi }_{n+2m-2},{\varpi }_{n+2m-1})\hfill \\ +\theta ({\varpi }_{n+2m-1},{\varpi }_{n+2m})q_c({\varpi }_{n+2m-1},{\varpi }_{n+2m})\hfill \\ +\theta ({\varpi }_{n+2m},{\varpi }_{n+2m+1})q_c({\varpi }_{n+2m},{\varpi }_{n+2m+1})\hfill \end{array}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \sum _{i=n}^{n+2m-1}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m+1})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & +\prod _{i=1}^{n+2m}\theta ({\varpi }_i,{\varpi }_{n+2m+1})q_c({\varpi }_{n+2m},{\varpi }_{n+2m+1})\hfill \\ \hfill & \le \sum _{i=n}^{n+2m}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m+1})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \sum _{i=1}^{\infty }{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m+1})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & -\sum _{i=1}^{n-1}{\psi }^i\left(q_c({\varpi }_0,{\varpi }_1)\right)\prod _{j=1}^i\theta ({\varpi }_j,{\varpi }_{n+2m+1})\theta ({\varpi }_i,{\varpi }_{i+1})\hfill \\ \hfill & \to 0\mathrm{as}n\to \infty \mathrm{using}\mathrm{property}\mathrm{of}{\Psi }_q,\hfill \end{array}$$

that is, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }_{n+2m+1})=0$. Thus, by combining both cases, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }_{n+p})=0$ for all $p\in \mathbb{N}$; hence, the sequence $\left\{{\varpi }_n\right\}$ is a right-$q_c$-Cauchy one.

Using the right-$q_c$-completeness of $(\mathsf{ϝ},q_c,\theta )$, we can deduce that there exists $\varpi \in \mathsf{ϝ}$ such that ${\varpi }_n\to \varpi$ as $n\to \infty$, that is,

$$\underset{n\to \infty }{lim}{q}_c({\varpi }_n,\varpi )=\underset{n\to \infty }{lim}{q}_c(\varpi ,{\varpi }_n)=0.$$

Using (q2), we have

$$\begin{array}{ccc}\hfill q_c(\varpi ,\Im \varpi )& \le & \theta (\varpi ,{\varpi }_n)q_c(\varpi ,{\varpi }_n)+\theta ({\varpi }_n,{\varpi }_{n+1})q_c({\varpi }_n,{\varpi }_{n+1})\hfill \\ & & +\theta ({\varpi }_{n+1},\Im \varpi )q_c({\varpi }_{n+1},\Im \varpi ).\hfill \end{array}$$

Since ℑ is continuous, by taking $n\to \infty$, we obtain $q_c(\varpi ,\Im \varpi )=0$, that is, $\Im \varpi =\varpi$.

At last, we can show that $Fix(\Im )$ is a set of one. Assume, however, that there are different $\varpi ,{\varpi }_0\in Fix(\Im )$. Through using (1), we have

$$\mathcal{G}\left(\begin{array}{c}\theta (\varpi ,{\varpi }_0)q_c(\Im \varpi ,\Im {\varpi }_0),q_c(\varpi ,{\varpi }_0),\hfill \\ q_c(\varpi ,\Im \varpi ),q_c({\varpi }_0,\Im {\varpi }_0),q_c(\varpi ,\Im {\varpi }_0)\hfill \end{array}\right)\le 0,$$

i.e.,

$$\mathcal{G}(\theta (\varpi ,{\varpi }_0)q_c(\varpi ,{\varpi }_0),q_c(\varpi ,{\varpi }_0),0,0,q_c(\varpi ,{\varpi }_0))\le 0.$$

(4)

It follows from $\left({\mathcal{G}}_2\right)$ that $q_c(\varpi ,{\varpi }_0)=0$, which implies that $\varpi ={\varpi }_0$. □

Example 6.

Let $\mathsf{ϝ},\theta$ and $q_c$ be as in Example 1, and let us denote ℑ on ϝ as the toll tax of the city and define

$$\Im \left(A\right)=\Im \left(B\right)=A,\Im \left(C\right)=\Im \left(D\right)=B.$$

Only Condition (1) of Theorem 1 has to be checked. When using Example 4, we have to check

$$\theta (\varpi ,\varkappa )q_c(\Im \varpi ,\Im \varkappa )\le amax\{q_c(\varpi ,\varkappa ),q_c(\varpi ,\Im \varpi ),q_c(\varkappa ,\Im \varkappa )\}-b{q}_c(\varpi ,\Im \varkappa )$$

(5)

for $0<b$ and $0\le a<1$, where $\varpi ,\varkappa \in \mathsf{ϝ}$.

Consider the following cases:

Case($1^0$) If $\varpi =A$ and $\varkappa =B$ (or $\varpi =B$ and $\varkappa =A$; $\varpi =C$ and $\varkappa =D$; or $\varpi =D$ and $\varkappa =A$), then (5) holds trivially.

Case($2^0$) Let $\varpi =A$ and $\varkappa =C$. Then,

$$\theta (A,C)q_c(A,B)=20,$$

and

$$amax\{q_c(A,C),q_c(A,\Im \left(A\right)),q_c(C,\Im \left(C\right))\}-b{q}_c(A,\Im \left(C\right))=100a-10b,$$

so (5) reduces to $20<100a-10b$.

Case($3^0$) Let $\varpi =A$ and $\varkappa =D$. Then,

$$\theta (A,D)q_c(A,B)=15,$$

and

$$amax\{q_c(A,D),q_c(A,\Im \left(A\right)),q_c(D,\Im \left(D\right))\}-b{q}_c(A,\Im \left(D\right))=30a-10b,$$

so (5) reduces to $15<30a-10b$.

Case($4^0$) Let $\varpi =B$ and $\varkappa =C$. Then,

$$\theta (B,C)q_c(A,B)=10,$$

and

$$amax\{q_c(B,C),q_c(B,\Im \left(B\right)),q_c(C,\Im \left(C\right))\}-b{q}_c(B,\Im \left(C\right))=98a,$$

so (5) reduces to $10<98a$.

Case($5^0$) Let $\varpi =B$ and $\varkappa =D$. Then,

$$\theta (B,D)q_c(A,B)=10,$$

and

$$amax\{q_c(B,D),q_c(B,\Im \left(B\right)),q_c(D,\Im \left(D\right))\}-b{q}_c(B,\Im \left(D\right))=98a-6b,$$

so (5) reduces to $10<98a-6b$.

Case($6^0$) Let $\varpi =C$ and $\varkappa =A$. Then, we have

$$\theta (C,A)q_c(B,A)=6,$$

and

$$amax\{q_c(C,A),q_c(C,\Im \left(C\right)),q_c(A,\Im \left(A\right))\}-b{q}_c(C,\Im \left(A\right))=(a-b)98,$$

so (5) reduces to $6<(a-b)98$.

Case($7^0$) Let $\varpi =C$ and $\varkappa =B$. Then,

$$\theta (C,B)q_c(B,A)=12,$$

and

$$amax\{q_c(C,B),q_c(C,\Im \left(C\right)),q_c(B,\Im \left(B\right))\}-b{q}_c(C,\Im \left(B\right))=(a-b)98,$$

so (5) reduces to $12<(a-b)98$.

Case($8^0$) Let $\varpi =D$ and $\varkappa =A$. Then,

$$\theta (D,A)q_c(B,A)=18,$$

and

$$amax\{q_c(D,A),q_c(D,\Im \left(D\right)),q_c(A,\Im \left(A\right))\}-b{q}_c(D,\Im \left(A\right))=30a-6b,$$

so (5) reduces to $18<30a-6b$.

Case($9^0$) Let $\varpi =D$ and $\varkappa =B$. Then,

$$\theta (D,B)q_c(B,A)=12,$$

and

$$amax\{q_c(D,B),q_c(D,\Im \left(D\right)),q_c(B,\Im \left(B\right))\}-b{q}_c(D,\Im \left(B\right))=30a-6b,$$

so (5) reduces to $12<30a-6b$.

If we take $a=\frac{28}{30}$ and $b=\frac{1}{6}$, then in all the cases the contractive Condition (5) holds true. Hence, ℑ is a ${\mathcal{G}}_q$-implicit type mapping.

Thus, all the conditions of Theorem 1 are fulfilled and $\varpi =A$ is the unique fixed point of ℑ.

By choosing $\mathcal{G}\in \Omega$ from Examples 4 and 5, we obtain the following consequences.

Corollary 1.

Let all the conditions of Theorem 1 be satisfied, except that the ${\mathcal{G}}_q$-implicit type condition is replaced by condition of the form (for all $\varpi ,\varkappa \in \mathsf{ϝ}$, with $q_c(\Im \varpi ,\Im \varkappa )>0$). Thus, we have the following:

(I) 

$$\begin{array}{c}\hfill \theta (\varpi ,\varkappa )q_c(\Im \varpi ,\Im \varkappa )\le amax\{q_c(\varpi ,\varkappa ),q_c(\varpi ,\Im \varpi ),q_c(\varkappa ,\Im \varkappa )\}-b{q}_c(\varpi ,\Im \varkappa )\end{array}$$

where $0<b$ and $0\le a<1$, or

(II) 

$$\begin{array}{c}\hfill \theta (\varpi ,\varkappa )q_c(\Im \varpi ,\Im \varkappa )\le a{q}_c(\varpi ,\varkappa )+b{q}_c(\varpi ,\Im \varpi )+c\frac{q_c(\varkappa ,\Im \varkappa )q_c(\varpi ,\Im \varkappa )}{1+q_c(\varpi ,\Im \varkappa )},\end{array}$$

where $0<a,b,c<1$ and $a+b+c<1.$

Then, $Fix(\Im )$ is a singleton.

4. A Periodic Point Result

It is self-evident that if $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ and $\Im \varpi =\varpi$, then ${\Im }^q\varpi =\varpi$, for an arbitrary $q\in \mathbb{N}$. Nonetheless, it is evident that the opposite is not true, that is, a self-map may have a “periodic" point (a point $\varpi$ satisfying ${\Im }^q\varpi =\varpi$ for some $q\in \mathbb{N}$) that is not its fixed point.

Definition 11

([15]). A mapping $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ is said to have the property (P) if it has no periodic points, i.e., if $Fix\left({\Im }^m\right)=Fix(\Im )$ for every $m\in \mathbb{N}$.

Theorem 2.

Under the assumptions of Theorem 1, ℑ has the property (P).

Proof. 

Let $\varpi \in Fix\left({\Im }^m\right)\backslash Fix(\Im )$, then $q_c(\varpi ,\Im \varpi )>0$. If ${\varpi }_0=\varpi$, as in the proof of Theorem 1, we obtain $\left\{{\varpi }_m\right\},\varpi \in \mathsf{ϝ}$ such that ${\varpi }_m\to \varpi \in Fix(\Im )$. Using (1), we have

$$\begin{array}{cc}\hfill q_c(\varpi ,\Im \varpi )& =q_c({\Im }^m\varpi ,\Im {\Im }^m\varpi )=q_c({\varpi }_m,{\varpi }_{m+1})\hfill \\ \hfill & \le \psi \left(q_c({\varpi }_{m-1},{\varpi }_m)\right)\le {\psi }^n\left(q_c(\varpi ,\Im \varpi )\right)\hfill \\ \hfill & <q_c(\varpi ,\Im \varpi ),\hfill \end{array}$$

which is a contradiction. Therefore, $Fix\left({\Im }^m\right)=Fix(\Im )$ for all $m\in \mathbb{N}$. □

5. Results of the Ulam–Hyers Stability

This section provides a definition of the right and left Ulam–Hyers stability of the fixed-point problem (FPP) in controlled EBQbMS, which is an extension of the EBQbMS case discussed in [11] (see also [16,17,18]).

Definition 12.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS and $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$. Then, the fixed-point equation (FPE)

$$\varpi =\Im \varpi ,\varpi \in \mathsf{ϝ}$$

(6)

is said to be as follows:

1. 

The right Ulam–Hyers is stable in the setting of EBQbMS if there exists $c>0$ such that for all $\epsilon >0$ and for all ε-solutions ${\varkappa }^{*}\in \mathsf{ϝ}$, that is,

$$q_c({\varkappa }^{*},\Im {\varkappa }^{*})\le \epsilon ,$$

wherein there exists a solution ${\varpi }^{*}\in \mathsf{ϝ}$ of (6) such that

$$q_c({\varkappa }^{*},{\varpi }^{*})\le c\epsilon .$$

(7)

2. 

The left Ulam–Hyers is stable in the setting of EBQbMS if there exists $c>0$ such that for all $\epsilon >0$ and for all ε-solutions ${\varkappa }^{*}\in \mathsf{ϝ}$, that is,

$$q_c(\Im {\varkappa }^{*},{\varkappa }^{*})\le \epsilon ,$$

wherein there exists a solution ${\varpi }^{*}\in \mathsf{ϝ}$ of (6) such that

$$q_c({\varpi }^{*},{\varkappa }^{*})\le c\epsilon .$$

(8)

Theorem 3.

Let $(\mathsf{ϝ},q_c,\theta )$ be a right $q_c$-complete controlled EBQbMS with $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be a continuous mapping satisfying the following:

$$\begin{array}{cc}\hfill \theta (\varpi ,\varkappa )q_c(\Im \varpi ,\Im \varkappa )& \le amax\{q_c(\varpi ,\varkappa ),q_c(\varpi ,\Im \varpi ),q_c(\varkappa ,\Im \varkappa )\}\hfill \\ \hfill & -b{q}_c(\varpi ,\Im \varkappa )\hfill \end{array}$$

(9)

for all $\varpi ,\varkappa \in \mathsf{ϝ}$, where $0<b$ and $a\in [0,1)$. Then, the FPE (6) is right Ulam–Hyers stable.

Proof. 

Owing to Theorem 1, there is a unique solution ${\varpi }^{*}\in \mathsf{ϝ}$ of the FPE (6) with $q_c({\varpi }^{*},\Im {\varpi }^{*})=0$. Let $ϵ>0$ and ${\varkappa }^{*}\in \mathsf{ϝ}$ be an $ϵ$-solution of (6). Since $q_c({\varpi }^{*},\Im {\varpi }^{*})=q_c({\varpi }^{*},{\varpi }^{*})=0\le ϵ$, ${\varpi }^{*}$ and ${\varkappa }^{*}$ are $ϵ$-solutions, and since $\theta ({\varkappa }^{*},{\varpi }^{*})\ge 1$, then

$$\begin{array}{cc}\hfill q_c({\varkappa }^{*},{\varpi }^{*})& \le \theta ({\varkappa }^{*},\Im {\varkappa }^{*})q_c({\varkappa }^{*},\Im {\varkappa }^{*})+\theta (\Im {\varkappa }^{*},\Im {\varpi }^{*})q_c(\Im {\varkappa }^{*},\Im {\varpi }^{*})\hfill \\ \hfill & +\theta (\Im {\varpi }^{*},\Im {\varpi }^{*})q_c(\Im {\varpi }^{*},\Im {\varpi }^{*})\hfill \\ \hfill & \le ϵ\theta ({\varkappa }^{*},\Im {\varkappa }^{*})+\theta (\Im {\varkappa }^{*},\Im {\varpi }^{*})q_c(\Im {\varkappa }^{*},\Im {\varpi }^{*}).\hfill \end{array}$$

(10)

From the contractive Condition (9), we obtain

$$\begin{array}{cc}\hfill & \theta ({\varkappa }^{*},{\varpi }^{*})q_c(\Im {\varkappa }^{*},\Im {\varpi }^{*})\hfill \\ \hfill & \le amax\{q_c({\varkappa }^{*},{\varpi }^{*}),q_c({\varkappa }^{*},\Im {\varkappa }^{*}),q_c({\varpi }^{*},\Im {\varpi }^{*}),\}-b{q}_c({\varkappa }^{*},\Im {\varpi }^{*})\hfill \\ \hfill & \le amax\{q_c({\varkappa }^{*},{\varpi }^{*}),ϵ\}.\hfill \end{array}$$

(11)

Consider the following two possible cases:

• If $max\{q_c({\varkappa }^{*},{\varpi }^{*}),ϵ\}=q_c({\varkappa }^{*},{\varpi }^{*})$, then from (10) and (11), we obtain

  $$\begin{array}{cc}\hfill q_c({\varkappa }^{*},{\varpi }^{*})& \le ϵ\theta ({\varkappa }^{*},\Im {\varkappa }^{*})+a\theta (\Im {\varkappa }^{*},\Im {\varpi }^{*})q_c({\varkappa }^{*},{\varpi }^{*}),\hfill \end{array}$$

  which implies that

  $$q_c({\varkappa }^{*},{\varpi }^{*})\left[\frac{1-a\theta (\Im {\varkappa }^{*},\Im {\varpi }^{*})}{\theta ({\varkappa }^{*},\Im {\varkappa }^{*})}\right]\le ϵ,$$

  i.e.,

  $$q_c({\varkappa }^{*},{\varpi }^{*})\le cϵ\mathrm{as}c=\frac{\theta ({\varkappa }^{*},\Im {\varkappa }^{*})}{1-a\theta (\Im {\varkappa }^{*},\Im {\varpi }^{*})}>0.$$
• If $max\{q_c({\varkappa }^{*},{\varpi }^{*}),ϵ\}=ϵ$, then from (10) and (11), we obtain

  $$\begin{array}{cc}\hfill q_c({\varkappa }^{*},{\varpi }^{*})& \le ϵ\theta ({\varkappa }^{*},\Im {\varkappa }^{*})+aϵq_c({\varkappa }^{*},{\varpi }^{*}),\hfill \end{array}$$

  which implies that

  $$q_c({\varkappa }^{*},{\varpi }^{*})\left[\frac{1-aϵ}{\theta ({\varkappa }^{*},\Im {\varkappa }^{*})}\right]\le ϵ,$$

  i.e.,

  $$q_c({\varkappa }^{*},{\varpi }^{*})\le \frac{\theta ({\varkappa }^{*},\Im {\varkappa }^{*})}{1-aϵ}ϵ=cϵ,c=\frac{\theta ({\varkappa }^{*},\Im {\varkappa }^{*})}{1-aϵ}>0.$$

Therefore, (7) holds and the FPE (6) is thus right Ulam–Hyers stable. □

Theorem 4.

Let $(\mathsf{ϝ},q_c,\theta )$ be a left $q_c$-complete controlled EBQbMS with $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be a continuous mapping satisfying Condition (9) for all $\varpi ,\varkappa \in \mathsf{ϝ}$. Then, the FPE (6) is left Ulam–Hyers stable.

6. Right and Left Weak Well-Posed Properties (Right and Left Weak Limit Shadowing)

The concept of a well posited fixed-point problem (FPP) has captured the attention of numerous mathematicians, including Popa [19,20] and others. The authors of the paper [21] defined a weak well posed (wwp) property in a BbDS. In the following section, we will extend this concept to a controlled EBQbMS.

Definition 13.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$. The FPP of ℑ is said to be right $wwp$ [resp. left-$wwp$] if it satisfies the following:

1. 

$Fix(\Im )=\left\{{\varpi }^{*}\right\}$;

2. 

for any sequence $\left\{{\varpi }_p\right\}$ in ϝ with ${lim}_{p\to \infty }{q}_c({\varpi }_p,\Im {\varpi }_p)=0$ and

${lim}_{p,r\to \infty }{q}_c(\Im {\varpi }_p,\Im {\varpi }_r)=0$, one has ${lim}_{p\to \infty }{q}_c({\varpi }_p,{\varpi }^{*})=0$

[resp. for any sequence $\left\{{\varpi }_p\right\}$ in ϝ with ${lim}_{p\to \infty }{q}_c(\Im {\varpi }_p,{\varpi }_p)=0$ and

${lim}_{p,r\to \infty }{q}_c(\Im {\varpi }_r,\Im {\varpi }_p)=0$, one has ${lim}_{p\to \infty }{q}_c({\varpi }^{*},{\varpi }_p)=0$].

Theorem 5.

Let $(\mathsf{ϝ},q_c,\theta )$ be a right $q_c$-complete controlled EBQbMS with $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be continuous and satisfy Condition (9). Then, the FPP of ℑ is right wwp.

Proof. 

Let ${\varpi }^{*}$ be a (unique) fixed point of ℑ, and let $\left\{{\varpi }_n\right\}$ be a sequence in $\mathsf{ϝ}$ such that ${lim}_{n\to \infty }{q}_c({\varpi }_n,\Im {\varpi }_n)=0$ and ${lim}_{n,m\to \infty }{q}_c(\Im {\varpi }_n,\Im {\varpi }_m)=0$, for $m>n$. Owing to (q2), we have

$$\begin{array}{cc}\hfill q_c({\varpi }_n,{\varpi }^{*})& \le \theta ({\varpi }_n,\Im {\varpi }_n)q_c({\varpi }_n,\Im {\varpi }_n)+\theta (\Im {\varpi }_n,\Im {\varpi }_m)q_c(\Im {\varpi }_n,\Im {\varpi }_m)\hfill \\ \hfill & +\theta (\Im {\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*}).\hfill \end{array}$$

By letting $n\to \infty$ in the above inequality, we obtain

$$\underset{n\to \infty }{lim}{q}_c({\varpi }_n,{\varpi }^{*})\le \underset{n\to \infty }{lim}[\theta (\Im {\varpi }_n,\Im {\varpi }_m)q_c(\Im {\varpi }_n,\Im {\varpi }_m)+\theta (\Im {\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*})].$$

(12)

WLOG. It is reasonable to presume that a subsequence $\{\Im {\varpi }_{n_k}\}$ of $\{\Im {\varpi }_n\}$ having distinct elements exists. Alternatively, there exists ${\varpi }_0\in \mathsf{ϝ}$ and $n_1\in \mathbb{N}$ such that $\Im {\varpi }_n={\varpi }_0$ for $n\ge n_1$. As ${lim}_{n\to \infty }{q}_c({\varpi }_n,\Im {\varpi }_n)=0$, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }_0)=0$. If ${\varpi }_0\ne {\varpi }^{*}$, then ${\varpi }_0\ne \Im {\varpi }_0$ since there is only one fixed point of ℑ. For $n\ge n_1$, we obtain ${\varpi }_0=\Im {\varpi }_n\ne \Im {\varpi }_0$.

Using (9) for $\varpi ={\varpi }_n,\varkappa ={\varpi }_0$, we obtain

$$\begin{array}{cc}\hfill \theta ({\varpi }_n,{\varpi }_0)q_c(\Im {\varpi }_n,\Im {\varpi }_0)& \le amax\{q_c({\varpi }_n,{\varpi }_0),q_c({\varpi }_n,\Im {\varpi }_n),q_c({\varpi }_0,\Im {\varpi }_0)\}\hfill \\ \hfill & -b{q}_c({\varpi }_n,\Im {\varpi }_0)\hfill \\ \hfill & \le amax\{q_c({\varpi }_n,{\varpi }_0),q_c({\varpi }_0,\Im {\varpi }_0)\}.\hfill \end{array}$$

By letting $n\to \infty$ and using ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }_0)=0$, we have

$$\underset{n,m\to \infty }{lim}\theta ({\varpi }_n,{\varpi }_0)q_c({\varpi }_0,\Im {\varpi }_0)\le a{q}_c({\varpi }_0,\Im {\varpi }_0),$$

which is not true. Thus, there exists $m,q,n>n_0$$(m>q>n)$ such that $\Im {\varpi }_m\ne \Im {\varpi }_q\ne \Im {\varpi }_n\ne {\varpi }_n$. Then,

$$\begin{array}{cc}\hfill q_c({\varpi }_n,\Im {\varpi }_m)& \le \theta ({\varpi }_n,\Im {\varpi }_n)q_c({\varpi }_n,\Im {\varpi }_n)+\theta (\Im {\varpi }_n,\Im {\varpi }_q)q_c(\Im {\varpi }_n,\Im {\varpi }_q)\hfill \\ \hfill & +\theta (\Im {\varpi }_q,\Im {\varpi }_m)q_c(\Im {\varpi }_q,\Im {\varpi }_m)\hfill \\ \hfill & \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Therefore, from (12), we have

$$\underset{n\to \infty }{lim}{q}_c({\varpi }_n,{\varpi }^{*})\le \underset{m\to \infty }{lim}\theta (\Im {\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*}).$$

(13)

Again, by using (9), we obtain

$$\begin{array}{cc}\hfill \theta ({\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,\Im {\varpi }^{*})& \le amax\{q_c({\varpi }_m,{\varpi }^{*}),q_c({\varpi }_m,\Im {\varpi }_m),q_c({\varpi }^{*},\Im {\varpi }^{*})\}\hfill \\ \hfill & -b{q}_c({\varpi }_m,\Im {\varpi }^{*}),\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill \theta ({\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*})& \le amax\{q_c({\varpi }_m,{\varpi }^{*}),q_c({\varpi }_m,\Im {\varpi }_m),q_c({\varpi }^{*},{\varpi }^{*})\}\hfill \\ \hfill & -b{q}_c({\varpi }_m,{\varpi }^{*}).\hfill \end{array}$$

By letting $m\to \infty$ and using the continuity of ℑ, we obtain

$$\underset{m\to \infty }{lim}\theta ({\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*})\le a\underset{m\to \infty }{lim}{q}_c({\varpi }_m,{\varpi }^{*}),$$

which implies that

$$\underset{m\to \infty }{lim}{q}_c(\Im {\varpi }_m,{\varpi }^{*})\le a\underset{m\to \infty }{lim}{q}_c({\varpi }_m,{\varpi }^{*}).$$

Therefore, from (13), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{q}_c({\varpi }_n,{\varpi }^{*})& \le \underset{m\to \infty }{lim}\theta (\Im {\varpi }_m,{\varpi }^{*})q_c(\Im {\varpi }_m,{\varpi }^{*})\hfill \\ \hfill & \le a\underset{m\to \infty }{lim}{q}_c({\varpi }_m,{\varpi }^{*})\theta (\Im {\varpi }_n,{\varpi }^{*})\hfill \\ \hfill & <\underset{m\to \infty }{lim}{q}_c({\varpi }_m,{\varpi }^{*}),\hfill \end{array}$$

which is a contradiction. Therefore, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }^{*})=0$. □

Theorem 6.

Let $(\mathsf{ϝ},q_c,\theta )$ be a left $q_c$-complete controlled EBQbMS with $\theta :{\mathsf{ϝ}}^2\to [1,\infty )$, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be continuous and satisfy Condition (9). Then, the FPP of ℑ is left wwp.

There are some pieces of literature on the limit shadowing property of fpps ([22,23]). In order to investigate the right and left weak limit shadowing property (right wlsp/left wlsp) in a controlled EBQbMS, we first need to define these terms.

Definition 14.

Let $(\mathsf{ϝ},q_c,\theta )$ be a controlled EBQbMS and $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$.

1. 

The fpp of ℑ is said to have right wlsp in ϝ if assuming that $\left\{{\varpi }_n\right\}$ in ϝ satisfies $q_c({\varpi }_n,\Im {\varpi }_n)\to 0$ as $n\to \infty$ and $q_c(\Im {\varpi }_n,\Im {\varpi }_m)\to 0$ as $m,n\to \infty$. Thus, it follows that there exists $\varpi \in \mathsf{ϝ}$ such that $q_c({\varpi }_n,{\Im }^n\varpi )\to 0$ as $n\to \infty$.

2. 

The fpp of ℑ is said to have left wlsp in ϝ if assuming that $\left\{{\varpi }_n\right\}$ in ϝ satisfies $q_c(\Im {\varpi }_n,{\varpi }_n)\to 0$ as $n\to \infty$ and $q_c(\Im {\varpi }_m,\Im {\varpi }_n)\to 0$ as $m,n\to \infty$. Thus, it follows that there exists $\varpi \in \mathsf{ϝ}$ such that $q_c({\Im }^n\varpi ,{\varpi }_n)\to 0$ as $n\to \infty$.

Theorem 7.

Let $(\mathsf{ϝ},q_c,\theta )$ be a right $q_c$-complete controlled EBQbMS, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be continuous and satisfy Condition (9) Then, ℑ has the right wlsp.

Proof. 

Let ${\varpi }^{*}\in Fix(\Im )$, and let $\left\{{\varpi }_n\right\}$ in $\mathsf{ϝ}$ be such that ${lim}_{n\to \infty }{q}_c({\varpi }_n,\Im {\varpi }_n)=0$ and ${lim}_{n,m\to \infty }{q}_c(\Im {\varpi }_n,\Im {\varpi }_m)=0$. Since ${\varpi }^{*}\in Fix(\Im )$, $q_c({\varpi }^{*},\Im {\varpi }^{*})=0$, then, owing to Theorem 5, ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\varpi }^{*})=0$. Therefore, we obtain ${lim}_{n\to \infty }{q}_c({\varpi }_n,{\Im }^n{\varpi }^{*})=0$. □

Theorem 8.

Let $(\mathsf{ϝ},q_c,\theta )$ be a left $q_c$-complete controlled EBQbMS, and let $\Im :\mathsf{ϝ}\to \mathsf{ϝ}$ be continuous and satisfy (9). Then, ℑ has the left wlsp.

7. Applications to Riesz–Caputo Fractional Derivatives

In comparison to integer-order models, fractional-order models provide a more plausible account of memory and genetic processes. Riemann–Liouville and Riesz derivatives have been frequently used in the research on fractional boundary/initial value problems (FBVP/FIVP) in recent years. These fractional operators, however, are one-sided and can only alter the past or the future. The Riesz space fractional operator, in contrast to the other fractional operators, is a two-sided operator that captures both past and present non-local memory effects. This is critical because present states in the mathematical models of physical processes on finite domains are affected by both past and future memory effects. In the anomalous diffusion problem, for example, the Riesz fractional derivative has been utilized to account for memory effects in both past and future concentrations. The authors of [24,25,26] addressed the solution of Riesz–Caputo fractional type BVP.

In this section, we investigate the uniqueness of a solution for an integral-type anti-periodic boundary value problem (APBVP) that is of a Riesz–Caputo fractional type of the form

$${}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)=\Phi (\kappa ,\varpi \left(\kappa \right)),\zeta \in (1,2],0\le \kappa \le \ell$$

(14)

$$\varpi \left(0\right)+\varpi (\ell )={\int }_0^{\ell }{\hslash }_1\left(\varpi \left(\varsigma \right)\right)d\varsigma ,{\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell )={\int }_0^{\ell }{\hslash }_2\left(\varpi \left(\varsigma \right)\right)d\varsigma ,$$

(15)

where ${}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }$ is the Riesz–Caputo derivative, $\Phi :[0,\ell ]\times \mathbb{R}\to \mathbb{R}$ is a continuous function, and ${\hslash }_1,{\hslash }_2:X\to X$ (where X is a Banach space).

Before a discussion on the solutions of APBVP, we introduce the related notions.

Definition 15

([27]). Let $\zeta >0$. The left and right Riemann–Liouville fractional integral of a function $\varpi \in C[0,\ell ]$ of order ζ is defined as follows:

$$\begin{array}{c}\hfill {}^{RL}{\mathcal{I}}_0^{\zeta }\varpi \left(\kappa \right)=\frac{1}{\Gamma \left(\zeta \right)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -1}\varpi \left(\varsigma \right)d\varsigma ,\kappa \in [0,\ell ].\\ \hfill {}_{\ell }^{RL}{\mathcal{I}}^{\zeta }\varpi \left(\kappa \right)=\frac{1}{\Gamma \left(\zeta \right)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\mu -1}\varpi \left(\varsigma \right)d\varsigma ,\kappa \in [0,\ell ].\end{array}$$

Definition 16.

Let $\zeta >0$. The Riesz fractional integral of a function $\varpi \in C[0,\ell ]$ of order ζ is defined as follows:

$${}_0^R{\mathcal{I}}_{\ell }^{\zeta }\varpi \left(\kappa \right)=\frac{1}{2}\left({\mathcal{I}}_0^{\zeta }\varpi \left(\kappa \right)+{}_{\ell }{\mathcal{I}}^{\zeta }\varpi \left(\kappa \right)\right).$$

(16)

Definition 17

([27]). Let $\zeta \in (m,m+1]$, $m\in \mathbb{N}$. Then, the left and right Caputo fractional derivatives of a function $\varpi \in C^{m+1}[0,\ell ]$ of order ζ are defined as follows:

$$\begin{array}{cc}\hfill {}_0^C{\mathcal{D}}_{\kappa }^{\zeta }\varpi \left(\kappa \right)& =\frac{1}{\Gamma (m+1-\zeta )}{\int }_0^{\kappa }{(\kappa -s)}^{m-v}{\varpi }^{(m+1)}ds\hfill \\ & =\left({\mathcal{I}}_0^{m+1-\zeta }{\mathcal{D}}^{m+1}\right)\varpi \left(\kappa \right)\hfill \\ \hfill {}_{\kappa }^C{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)& =\frac{{(-1)}^{m+1}}{\Gamma (m+1-\zeta )}{\int }_{\kappa }^{\ell }{(s-\kappa )}^{m-\zeta }{\varpi }^{(m+1)}ds\hfill \\ & ={(-1)}^{m+1}\left({}_{\ell }{\mathcal{I}}^{m+1-\zeta }{\mathcal{D}}^{m+1}\right)\varpi \left(\kappa \right),\hfill \end{array}$$

where $\mathcal{D}$ is the ordinary differential operator.

Definition 18.

Let $\zeta \in (m,m+1]$, $m\in \mathbb{N}$. Then, the Riesz–Caputo fractional derivative ${}_0^{RC}{\mathcal{D}}^{\zeta }\varpi$ of order ζ of a function $\varpi \in C^{m+1}[0,\ell ]$ is defined by

$$\begin{array}{cc}\hfill {}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)& =\frac{1}{\Gamma (m+1-\zeta )}{\int }_0^{\ell }\mid \kappa -s{\mid }^{m-\zeta }{\varpi }^{(m+1)}\left(s\right)ds\hfill \\ & =\frac{1}{2}\left({}_0^C{\mathcal{D}}_{\kappa }^{\zeta }\varpi \left(\kappa \right)+{(-1)}^{m+1}{}_{\kappa }{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)\right)\hfill \\ & =\frac{1}{2}\left(\left({\mathcal{I}}_0^{m+1-\zeta }{\mathcal{D}}^{m+1}\right)\varpi \left(\kappa \right)+{(-1)}^{m+1}\left({}_{\ell }{\mathcal{I}}^{m+1-\zeta }{\mathcal{D}}^{m+1}\right)\varpi \left(\kappa \right)\right).\hfill \end{array}$$

Lemma 1

([27]). Let $\varpi \in C^m[0,\ell ]$ and $\zeta \in (m,m+1]$. Then, the following relations hold:

$$\begin{array}{cc}& {}_{\kappa }^R{\mathcal{I}}_{\ell }^{\zeta }{}_0^{RC}{\mathcal{D}}_{\kappa }^{\zeta }\varpi \left(\kappa \right)=\varpi \left(\kappa \right)-\sum _{\mathsf{ȷ}=0}^{m-1}\frac{{\varpi }^{(\mathsf{ȷ})}\left(\alpha \right)}{\mathsf{ȷ}!}{(\kappa -\alpha )}^{\mathsf{ȷ}},\hfill \\ & {}_{\ell }^R{\mathcal{I}}_{\kappa }^{\zeta }{}^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)=\varpi \left(\kappa \right)-\sum _{\mathsf{ȷ}=0}^{m-1}\frac{{(-1)}^{\mathsf{ȷ}}{\varpi }^{(\mathsf{ȷ})}\left(\alpha \right)}{\mathsf{ȷ}!}{(\alpha -\kappa )}^{\mathsf{ȷ}}.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill {}_0^R{\mathcal{I}}_{\ell }^{\zeta }{}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)& =\frac{1}{2}\left({}_0^R{\mathcal{I}}_{\kappa }^{\zeta }{}_0^C{\mathcal{D}}_{\kappa }^{\zeta }+{}_{\kappa }^R{\mathcal{I}}_{\ell }^{\zeta }{}_0^C{\mathcal{D}}_{\kappa }^{\zeta }\right)\varpi \left(\kappa \right)+{(-1)}^n\frac{1}{2}\left({}_0^R{\mathcal{I}}_{\kappa }^{\zeta }{}_{\kappa }^C{\mathcal{D}}_{\ell }^{\zeta }+{}_{\kappa }^R{\mathcal{I}}_{\ell }^{\zeta }{}_{\kappa }^C{\mathcal{D}}_{\ell }^{\zeta }\right)\varpi \left(\kappa \right)\hfill \\ & =\frac{1}{2}\left({}_0^R{\mathcal{I}}_{\kappa }^{\zeta }{}_0^C{\mathcal{D}}_{\kappa }^{\zeta }+{(-1)}^n{}_{\kappa }^R{\mathcal{I}}_{\ell }^{\zeta }{}_{\kappa }^C{\mathcal{D}}_{\ell }^{\zeta }\right)\varpi \left(\kappa \right).\hfill \end{array}$$

When $\zeta \in (1,2]$ and $\varpi \left(\kappa \right)\in C^1(0,\ell )$, we have

$$\begin{array}{cc}\hfill {}_0^R{\mathcal{I}}_{\ell }^{\zeta }{}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)=& \varpi \left(\kappa \right)-\frac{1}{2}[\varpi \left(0\right)+\varpi (\ell )]-\frac{1}{2}[{\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell )]\kappa +\frac{\ell }{2}{\varpi }^{\prime }(\ell ).\hfill \end{array}$$

(17)

Lemma 2.

Suppose that $\chi \in \Lambda :=C\left(\right[0,\ell ],\mathbb{R})$ and $\varpi \in C^1\left([0,\ell ]\right)$. Then, the fractional APBVP of order ζ in $(1,2]$ is

$${}_0^{RC}{\mathcal{D}}_{\ell }^{\zeta }\varpi \left(\kappa \right)=\chi \left(\kappa \right),\zeta \in (1,2],0\le \kappa \le \ell$$

(18)

$$\varpi \left(0\right)+\varpi (\ell )={\int }_0^{\ell }{\eta }_1\left(\varsigma \right)d\varsigma ,{\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell )={\int }_0^{\ell }{\eta }_2\left(\varsigma \right)d\varsigma ,$$

(19)

which is equivalent to the integral equation of the form

$$\begin{array}{cc}\hfill \varpi \left(\kappa \right)& =\frac{1}{2}{\int }_0^{\ell }{\eta }_1\left(\varsigma \right)d\varsigma +\frac{(2\kappa -\ell )}{4}{\int }_0^{\ell }{\eta }_2\left(\varsigma \right)d\varsigma +\frac{\ell }{2\Gamma (\zeta -1)}{\int }_0^{\ell }{(\kappa -\varsigma )}^{\zeta -2}\chi \left(\varsigma \right)d\varsigma \hfill \\ \hfill & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -1}\chi \left(\varsigma \right)d\varsigma +\frac{1}{\Gamma \left(\zeta \right)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\zeta -1}\chi \left(\varsigma \right)d\varsigma .\hfill \end{array}$$

(20)

Proof. 

Owing to (17) and (18), we conclude that

$$\begin{array}{ccc}\hfill \varpi \left(\kappa \right)& =& \frac{1}{2}[\varpi \left(0\right)+\varpi (\ell )]+\frac{\kappa }{2}({\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell ))-\frac{\ell }{2}{\varpi }^{\prime }(\ell )+{}_0{\mathcal{I}}_{\ell }^{\zeta }\chi \left(\kappa \right)\hfill \\ & =& \frac{1}{2}{\int }_0^{\ell }{\eta }_1\left(\varsigma \right)d\varsigma +\frac{\kappa }{2}({\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell ))-\frac{\ell }{2}{\varpi }^{\prime }(\ell )\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -1}\chi \left(\varsigma \right)d\varsigma +\frac{1}{\Gamma \left(\zeta \right)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\zeta -1}\chi \left(\varsigma \right)d\varsigma .\hfill \end{array}$$

(21)

Then,

$$\begin{array}{cc}\hfill {\varpi }^{\prime }\left(\kappa \right)& =\frac{1}{2}\left({\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }(\ell )\right)+\frac{1}{\Gamma (\zeta -1)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -2}\chi \left(\varsigma \right)d\varsigma \hfill \\ & -\frac{1}{\Gamma (\zeta -1)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\zeta -2}\chi \left(\varsigma \right)d\varsigma .\hfill \end{array}$$

By applying APBVP (19), we have

$$\begin{array}{cc}\hfill {\varpi }^{\prime }(\ell )& =\frac{1}{2}{\int }_0^{\ell }{\eta }_2\left(\varsigma \right)d\varsigma +\frac{1}{\Gamma (\zeta -1)}{\int }_0^{\ell }{(\ell -\varsigma )}^{\zeta -2}\chi \left(\varsigma \right)d\varsigma ,\hfill \\ \hfill {\varpi }^{\prime }\left(0\right)& =\frac{1}{2}{\int }_0^{\ell }{\eta }_2\left(\varsigma \right)d\varsigma -\frac{1}{\Gamma (\zeta -1)}{\int }_0^{\ell }{(\ell -\varsigma )}^{\zeta -2}\chi \left(\varsigma \right)d\varsigma .\hfill \end{array}$$

When we insert the quantities that we have obtained for ${\varpi }^{\prime }\left(0\right)$ and ${\varpi }^{\prime }(\ell )$ into (21), we obtain (20). □

We are now able to bring up, with the appropriate conditions for our APBVP, a unique solution. Let $C[0,\ell ]$ be the space of continuous functions $\varpi$, which are defined on $[0,\ell ]$ with the norm $\parallel \varpi \parallel ={sup}_{\varsigma \in [0,\ell ]}\left|\varpi \left(\varsigma \right)\right|$.

Theorem 9.

Let $\Phi :[0,\ell ]\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Assume that there exist non-negative real numbers ${\lambda }_i$, $i=\{1,2,3\}$, such that for all $(\varsigma ,\varpi ),(\varsigma ,\varkappa )\in {\mathbb{R}}^2$, we have the following:

(A1) 

$$|\Phi (\varsigma ,\varpi )-\frac{1}{2}\Phi (\varsigma ,\varkappa )|\le {\lambda }_1\frac{|\varpi -\frac{1}{2}\varkappa |}{{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},\forall \varsigma \in [0,\ell ].$$

(A2) 

for all $\varsigma \in [0,\ell ]$,

$$|{\hslash }_1\left(\varpi \right)-\frac{1}{2}{\hslash }_1\left(\varkappa \right)|\le \frac{{\lambda }_2|\varpi -\frac{1}{2}\varkappa |}{{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},|{\hslash }_2\left(\varpi \right)-\frac{1}{2}{\hslash }_2\left(\varkappa \right)|\le \frac{{\lambda }_3|\varpi -\frac{1}{2}\varkappa |}{{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}}.$$

(A3) 

$${\left[\frac{\ell }{2}{\lambda }_2+\frac{{\ell }^2}{2}{\lambda }_3+\frac{{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}{\lambda }_1+\frac{2{\ell }^{\zeta }}{\Gamma (\zeta +1)}{\lambda }_1\right]}^2<1.$$

Then, Problem (14) has a unique solution on $[0,\ell ]$.

Proof. 

We converted the fractional AVBVP (14) and (15) into a integral equation using the operator $\Im :\Lambda \to \Lambda$ of the form

$$\begin{array}{cc}\hfill \Im \varpi \left(\kappa \right)& =\frac{1}{2}{\int }_0^{\ell }{\hslash }_1\left(\varpi \left(\varsigma \right)\right)d\varsigma -\frac{(\ell -2\kappa )}{4}{\int }_0^{\ell }{\hslash }_2\left(\varpi \left(\varsigma \right)\right)d\varsigma \hfill \\ \hfill & -\frac{\ell }{2\Gamma (\zeta -1)}{\int }_0^{\ell }{(\kappa -\varsigma )}^{\zeta -2}\Phi (\varsigma ,\varpi \left(\varsigma \right))d\varsigma \hfill \\ \hfill & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -1}\Phi (\varsigma ,\varpi \left(\varsigma \right))d\varsigma \hfill \\ \hfill & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\zeta -1}\Phi (\varsigma ,\varpi \left(\varsigma \right))d\varsigma .\hfill \end{array}$$

(22)

For $\varpi ,\widehat{\varpi }\in \Lambda$ and for each $\kappa \in [0,\ell ]$, we have

$$\begin{array}{ccc}& & \mid \Im \varpi \left(\kappa \right)-\frac{1}{2}\Im \widehat{\varpi }\left(\kappa \right)\mid \hfill \\ & & \le \frac{1}{2}{\int }_0^{\ell }|{\hslash }_1\left(\varpi \left(\varsigma \right)\right)-\frac{1}{2}{\hslash }_1\left(\widehat{\varpi }\left(\varsigma \right)\right)|d\varsigma +\frac{(\ell -2\kappa )}{4}{\int }_0^{\ell }|{\hslash }_2\left(\varpi \left(\varsigma \right)\right)-\frac{1}{2}{\hslash }_2\left(\widehat{\varpi }\left(\varsigma \right)\right)|d\varsigma \hfill \\ & & +\frac{\ell }{2\Gamma (\zeta -1)}{\int }_0^{\ell }{(\kappa -\varsigma )}^{\zeta -2}\mid \Phi (\varsigma ,\varpi \left(\varsigma \right))-\frac{1}{2}\Phi (\varsigma ,\widehat{\varpi }\left(\varsigma \right))\mid d\varsigma \hfill \\ & & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_0^{\kappa }{(\kappa -\varsigma )}^{\zeta -1}\mid \Phi (\varsigma ,\varpi \left(\varsigma \right))-\frac{1}{2}\Phi (\varsigma ,\widehat{\varpi }\left(\varsigma \right))\mid d\varsigma \hfill \\ & & +\frac{1}{\Gamma \left(\zeta \right)}{\int }_{\kappa }^{\ell }{(\varsigma -\kappa )}^{\zeta -1}\mid \Phi (\varsigma ,\varpi \left(\varsigma \right))-\frac{1}{2}\Phi (\varsigma ,\widehat{\varpi }\left(\varsigma \right))\mid d\varsigma \hfill \\ & & \le \frac{\ell }{2}{\lambda }_2\frac{\parallel \varpi -\frac{1}{2}\widehat{\varpi }\parallel }{\left(\right|\varpi |+|\widehat{\varpi }{|+1)}^{1/2}}+\frac{{\ell }^2}{4}{\lambda }_3\frac{\parallel \varpi -\frac{1}{2}\widehat{\varpi }\parallel }{\left(\right|\varpi |+|\widehat{\varpi }{|+1)}^{1/2}}+\frac{{\lambda }_1{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}\frac{\parallel \varpi -\frac{1}{2}\widehat{\varpi }\parallel }{\left(\right|\varpi |+|\widehat{\varpi }{|+1)}^{1/2}}\hfill \\ & & +\frac{2{\ell }^{\zeta }{\lambda }_1}{\Gamma (\zeta +1)}\frac{\parallel \varpi -\frac{1}{2}\widehat{\varpi }\parallel }{\left(\right|\varpi |+|\widehat{\varpi }{|+1)}^{1/2}}\hfill \\ & & =\left[\frac{\ell }{2}{\lambda }_2+\frac{{\ell }^2}{4}{\lambda }_3+\frac{{\lambda }_1{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}+\frac{2{\ell }^{\zeta }{\lambda }_1}{\Gamma (\zeta +1)}\right]\frac{\parallel \varpi -\frac{1}{2}\widehat{\varpi }\parallel }{\left(\right|\varpi |+|\widehat{\varpi }{|+1)}^{1/2}}.\hfill \end{array}$$

Let $q_c:\Lambda \times \Lambda \to {\mathbb{R}}_{+}$ be defined by

$$q_c(\varpi ,\varkappa )={\parallel \varpi -\frac{1}{2}\varkappa \parallel }^2\mathrm{for}\mathrm{all}\varpi \ne \varkappa \in \Lambda .$$

Then, $(\Lambda ,q_c,\theta )$ is a complete controlled EBQbMS with the coefficient $\theta (\varpi ,\varkappa )=\left|\varpi \right|+\left|\varkappa \right|+1$. Then,

$$\theta (\varpi ,\widehat{\varpi })q_c(\Im \left(\varpi \right),\Im \left(\widehat{\varpi }\right))\le \gamma q_c(\varpi ,\widehat{\varpi }),\gamma <1,$$

where

$$\gamma ={\left[\frac{\ell }{2}{\lambda }_2+\frac{{\ell }^2}{4}{\lambda }_3+\frac{{\lambda }_1{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}+\frac{2{\ell }^{\zeta }{\lambda }_1}{\Gamma (\zeta +1)}\right]}^2.$$

By virtue of Example 5, ℑ is a ${\mathcal{G}}_q$-implicit type mapping for $a=\gamma$, $b=0=c$. Hence, following Theorem 1, ℑ has a unique fixed point, that is, a solution to Equation (14) with (15). □

Remark 2.

Lemma 2 and Theorem 9 generalize the work that is discussed in Wang and Wang [28]. Also, we used the generalized Banach fixed-point theorem 1 in a controlled EBQbMS to prove these results.

Example 7.

Consider the following nonlinear FDE with a Riesz–Caputo derivative:

$$\left\{\begin{array}{c}{}_0{}^{RC}\mathcal{D}_1^{\frac{3}{2}}\varpi \left(\kappa \right)=\frac{\left|\varpi \right(\kappa \left)\right|}{\sqrt{{\kappa }^2+400}{\left(\right|\varpi \left(\kappa \right)|+1)}^{1/2}},\kappa \in [0,1],\hfill \\ \varpi \left(0\right)+\varpi \left(1\right)={\int }_0^1\frac{\left|\varpi \right(\varsigma \left)\right|}{{(49+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}d\varsigma ,{\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }\left(1\right)={\int }_0^1\frac{\left|\varpi \right(\varsigma \left)\right|}{{(9+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}d\varsigma .\hfill \end{array}\right.$$

(23)

Here, $\zeta =\frac{3}{2}$, $\ell =1$, $\Phi \left(\kappa ,\varpi \left(\kappa \right)\right)=\frac{\left|\varpi \right(\kappa \left)\right|}{\sqrt{{\kappa }^2+400}{\left(\right|\varpi \left(\kappa \right)|+1)}^{1/2}}$, ${\hslash }_1\left(\varpi \right)=\frac{\left|\varpi \right(\varsigma \left)\right|}{{(49+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}$, and ${\hslash }_2\left(\varpi \right)=\frac{\left|\varpi \right(\varsigma \left)\right|}{{(9+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}.$

$$|\Phi (\varsigma ,\varpi )-\frac{1}{2}\Phi (\varsigma ,\varkappa )|\le \frac{1}{20}\frac{|\varpi -\frac{1}{2}\varkappa |}{{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},\forall \varsigma \in [0,1].$$

$$|{\hslash }_1\left(\varpi \right)-\frac{1}{2}{\hslash }_1\left(\varkappa \right)|\le \frac{|\varpi -\frac{1}{2}\varkappa |}{7{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},|{\hslash }_2\left(\varpi \right)-\frac{1}{2}{\hslash }_2\left(\varkappa \right)|\le \frac{|\varpi -\frac{1}{2}\varkappa |}{3{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}}.$$

Therefore, (A1) and (A2) are satisfied with ${\lambda }_1=\frac{1}{20}$, ${\lambda }_2=\frac{1}{7}$ and ${\lambda }_3=\frac{1}{3}$. Further,

$${\left[\frac{\ell }{2}{\lambda }_2+\frac{{\ell }^2}{2}{\lambda }_3+\frac{{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}{\lambda }_1+\frac{2{\ell }^{\zeta }}{\Gamma (\zeta +1)}{\lambda }_1\right]}^2=0.0667<1.$$

Thus, by Theorem 9, Problem (23) has a unique solution on $[0,1]$.

Example 8.

Consider the following nonlinear FDE with a Riesz–Caputo derivative:

$$\left\{\begin{array}{c}{}_0{}^{RC}\mathcal{D}_2^{\frac{5}{3}}\varpi \left(\kappa \right)=\frac{e^{2\kappa }sin\left(\right|\varpi \left(\kappa \right)\left|\right)}{(e^{3\pi }+50)\left(\right|\varpi \left(\kappa \right){|+1)}^{1/2}},\kappa \in [0,2],\hfill \\ \varpi \left(0\right)+\varpi \left(1\right)={\int }_0^1\frac{\left|\varpi \right(\varsigma \left)\right|}{{(100+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}d\varsigma ,{\varpi }^{\prime }\left(0\right)+{\varpi }^{\prime }\left(1\right)={\int }_0^1\frac{\left|\varpi \right(\varsigma \left)\right|}{{(81+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}d\varsigma .\hfill \end{array}\right.$$

(24)

Here, $\zeta =\frac{5}{3}$, $\ell =2$, $\Phi \left(\kappa ,\varpi \left(\kappa \right)\right)=\frac{e^{2\kappa }sin\left(\right|\varpi \left(\kappa \right)\left|\right)}{(e^{3\pi }+50)\left(\right|\varpi \left(\kappa \right){|+1)}^{1/2}}$, ${\hslash }_1\left(\varpi \right)=\frac{\left|\varpi \right(\varsigma \left)\right|}{{(100+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}$, and ${\hslash }_2\left(\varpi \right)=\frac{\left|\varpi \right(\varsigma \left)\right|}{{(81+|\varpi \left(\varsigma \right)\left|\right)}^{1/2}}.$

$$|\Phi (\varsigma ,\varpi )-\frac{1}{2}\Phi (\varsigma ,\varkappa )|\le \frac{e^4}{(e^{3\pi }+50)}\frac{|\varpi -\frac{1}{2}\varkappa |}{{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},\forall \varsigma \in [0,2].$$

$$|{\hslash }_1\left(\varpi \right)-\frac{1}{2}{\hslash }_1\left(\varkappa \right)|\le \frac{|\varpi -\frac{1}{2}\varkappa |}{10{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}},|{\hslash }_2\left(\varpi \right)-\frac{1}{2}{\hslash }_2\left(\varkappa \right)|\le \frac{|\varpi -\frac{1}{2}\varkappa |}{9{(\mid \varpi \mid +\mid \varkappa \mid +1)}^{1/2}}.$$

Therefore, (A1) and (A2) are satisfied with ${\lambda }_1=\frac{e^4}{e^{3\pi }+50}$, ${\lambda }_2=\frac{1}{10}$ and ${\lambda }_3=\frac{1}{9}$. Further,

$${\left[\frac{\ell }{2}{\lambda }_2+\frac{{\ell }^2}{2}{\lambda }_3+\frac{{\ell }^{\zeta }}{2\Gamma \left(\zeta \right)}{\lambda }_1+\frac{2{\ell }^{\zeta }}{\Gamma (\zeta +1)}{\lambda }_1\right]}^2=0.0563<1.$$

Thus, by Theorem 9, Problem (24) has a unique solution on $[0,2]$.

8. Application to Nonlinear Matrix Equations

A Hermitian matrix is a square matrix that is equal to its conjugate transposed matrix. Let ${\mathcal{H}}_n$ denote the set of all $n\times n$ Hermitian matrices over $\mathbb{C}$, and let ${\mathcal{M}}_n$ be the set of all $n\times n$ matrices over $\mathbb{C}$. Denote by $s\left(\mathcal{K}\right)$ any singular value of a matrix $\mathcal{K}$ (singular values are the absolute values of the eigenvalues of a matrix $\mathcal{K}$) and the trace norm of $\mathcal{K}$ by $s^{+}\left(\mathcal{K}\right)=\parallel \mathcal{K}\parallel$. On ${\mathcal{H}}_n$, we define $\mathcal{K}\ge \mathcal{L}$ ( or $\mathcal{K}>\mathcal{L}$) if and only if $\mathcal{K}-\mathcal{L}$ is a positive semi-definite matrix (both of which are positive definites).

In [29], Ran and Reurings discussed the existence of solutions for the equation of

$$\mathcal{G}+{\mathcal{D}}^{*}\hslash \left(\mathcal{K}\right)\mathcal{D}=\mathcal{K}$$

(25)

in $\mathcal{P}\left(n\right)$, where $\mathcal{D}\in \mathcal{M}\left(n\right)$, $\mathcal{K}$ is a positive definite (PD), and ℏ is a mapping from $\mathcal{K}\left(n\right)$ into $\mathcal{M}\left(n\right)$. Note that $\mathcal{K}$ is a solution of (25) if and only if it is a fixed point of the mapping $\Im \left(\mathcal{G}\right)=\mathcal{K}-{\mathcal{D}}^{*}\hslash \left(\mathcal{K}\right)\mathcal{D}$.

In [30], Sawangsup and Sintunavarat discussed the nonlinear matrix equation (NME) $\mathcal{K}=\mathcal{G}+{\sum }_{i=1}^k{\mathcal{D}}_i^{*}\hslash \left(\mathcal{K}\right){\mathcal{D}}_i$ for the spectral norm of a matrix. For other variants on NMEs, one is referred to the studies of [31,32,33,34].

Theorem 10.

Consider the equation

$$\mathcal{K}=\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}},$$

(26)

where $\mathcal{G}\in {\mathcal{P}}_n$, ${\mathcal{D}}_{\mathsf{ȷ}}\in {\mathcal{M}}_n$, $\mathsf{ȷ}=1,\cdots ,k$, and the operator $\varrho :{\mathcal{P}}_n\to {\mathcal{P}}_n$ is continuous in the trace norm. For some $M,N_1\in \mathbb{R}$, and for any $\mathcal{K}\in {\mathcal{P}}_n$ with $\parallel \mathcal{K}\parallel \le M$, $s\left(\varrho \left(\mathcal{K}\right)\right)\le N_1$, let them hold for all singular values of $\varrho \left(\mathcal{K}\right)$.

Assume the following:

1. 

$\parallel \mathcal{G}\parallel \le M-N{N}_{1}n$, where ${\displaystyle \sum _{\mathsf{ȷ}=1}^k}\parallel {\mathcal{D}}_{\mathsf{ȷ}}^{*}\parallel \parallel {\mathcal{D}}_{\mathsf{ȷ}}\parallel =N$;

2. 

For any $\mathcal{Z}\in {\mathcal{P}}_n$ with $\parallel \mathcal{Z}\parallel \le M$, ${\sum }_{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{Z}\right){\mathcal{D}}_{\mathsf{ȷ}}⪰O$ holds;

3. 

For any $\mathcal{Z}\in {\mathcal{P}}_n$ with $\parallel \mathcal{Z}\parallel \le M$, $\mathcal{Z}⪯\mathcal{G}+{\sum }_{\mathsf{ȷ}=1}^k{\mathcal{D}}_i^{*}\varrho \left(\mathcal{Z}\right){\mathcal{D}}_i$ holds;

4. 

There exists $b>0$, $0\le a<1$ so that

$${(\frac{1}{2}M+\frac{3}{2}N{N}_{1}n)}^2\le \frac{1}{(s^{+}\left(\mathcal{K}\right)+s^{+}\left(\mathcal{L}\right)+1)}{\rm Y}(\mathcal{K},\mathcal{L})$$

(27)

holds, where

$$\begin{array}{ccc}\hfill & & {\rm Y}(\mathcal{K},\mathcal{L})\hfill \\ & & =amax\left\{\begin{array}{c}|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}{\left(\mathcal{L}\right)|}^2,{|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}})|}^2,\\ |s^{+}\left(\mathcal{L}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}){|}^2\end{array}\right\},\hfill \\ \hfill & & -b|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}){|}^2,\hfill \end{array}$$

(28)

for all $\mathcal{K},\mathcal{L}\in {\mathcal{P}}_n$ with $\parallel \mathcal{K}\parallel ,\parallel \mathcal{L}\parallel \le M$ and ${\displaystyle \sum _{\mathsf{ȷ}=1}^k}{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}}\ne {\displaystyle \sum _{\mathsf{ȷ}=1}^k}{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}$.

Then, Equation (26) has a unique solution of $\widehat{\mathcal{K}}\in {\mathcal{P}}_n$ with $\parallel \widehat{\mathcal{K}}\parallel \le M$. Moreover, the solution can be obtained as the limit of the iterative sequence $\left\{{\mathcal{K}}_m\right\}$, where, for $m\ge 0$, we have

$${\mathcal{K}}_{m+1}=\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left({\mathcal{K}}_m\right){\mathcal{D}}_{\mathsf{ȷ}},$$

(29)

and ${\mathcal{K}}_0$ is an arbitrary element of ${\mathcal{P}}_n$ satisfying $\parallel {\mathcal{K}}_0\parallel \le M$.

Proof. 

Denote $\Lambda :=\{\mathcal{K}\in {\mathcal{P}}_n:\parallel \mathcal{K}\parallel \le M\}$ as being a closed subset of ${\mathcal{P}}_n$. According to (2), any solution of (26) in $\Lambda$ has to be PD. For any $\mathcal{K}\in \Lambda$, we have

$$\begin{array}{cc}\hfill \parallel \mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}}\parallel & \le \parallel \mathcal{G}\parallel +\parallel \sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}}\parallel \hfill \\ \hfill & \le \parallel \mathcal{G}\parallel +\sum _{\mathsf{ȷ}=1}^k\parallel {\mathcal{D}}_{\mathsf{ȷ}}^{*}\parallel \parallel {\mathcal{D}}_{\mathsf{ȷ}}\parallel \parallel \varrho \left(\mathcal{K}\right)\parallel \hfill \\ \hfill & =\parallel \mathcal{G}\parallel +N\parallel \varrho \left(\mathcal{K}\right)\parallel .\hfill \end{array}$$

(30)

For all $s\left(\varrho \left(\mathcal{K}\right)\right)\le N_1$, it follows that $\parallel \varrho \left(\mathcal{K}\right)\parallel \le N_{1}n$. Thus, (30) implies

$$\begin{array}{cc}\hfill \parallel \mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}}\parallel & \le \parallel \mathcal{G}\parallel +N{N}_{1}n\hfill \\ \hfill & \le M-N{N}_{1}n+N{N}_{1}n=M.\hfill \end{array}$$

Now, define an operator $\Im :\Lambda \to \Lambda$ by

$$\begin{array}{cc}\hfill \Im \left(\mathcal{K}\right)& =\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}},\hfill \end{array}$$

for $\mathcal{K}\in \Lambda$. Consequently, finding the PDS of Equation (26) is equivalent to finding the fixed point(s) of ℑ.

For all $\mathcal{K},\mathcal{L}\in \Lambda$ can be expressed as

$$\begin{array}{cc}\hfill \parallel \Im \left(\mathcal{K}\right)-\frac{1}{2}\Im \left(\mathcal{L}\right)\parallel & =\parallel (\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}})-\frac{1}{2}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{J}\right){\mathcal{D}}_{\mathsf{ȷ}}\parallel )\parallel \hfill \\ \hfill & =\parallel \frac{1}{2}\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}}-\frac{1}{2}\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}\parallel \hfill \\ \hfill & \le \frac{1}{2}\parallel \mathcal{G}\parallel +\sum _{\mathsf{ȷ}=1}^k\parallel {\mathcal{D}}_{\mathsf{ȷ}}^{*}\parallel \parallel {\mathcal{D}}_{\mathsf{ȷ}}\parallel \parallel \varrho \left(\mathcal{K}\right)-\frac{1}{2}\varrho \left(\mathcal{L}\right)\parallel ,\hfill \\ \hfill & \le \frac{1}{2}M+N(\parallel \varrho \left(\mathcal{K}\right)\parallel +\frac{1}{2}\parallel \varrho \left(\mathcal{L}\right)\parallel )\hfill \\ \hfill & \le \frac{1}{2}M+\frac{3}{2}N{N}_{1}n.\hfill \end{array}$$

Thus, for all $\mathcal{K},\mathcal{L}\in \Lambda$, we have

$$\parallel \Im \left(\mathcal{K}\right)-\frac{1}{2}\Im \left(\mathcal{L}\right)\parallel \le \frac{1}{2}M+\frac{3}{2}N{N}_{1}n.$$

(31)

For some fixed $\mathcal{K},\mathcal{L}\in \Lambda$, from (27) and (28), we have

$$\begin{array}{ccc}& & {(\frac{1}{2}M+\frac{3}{2}N{N}_{1}n)}^2(s^{+}\left(\mathcal{K}\right)+s^{+}\left(\mathcal{L}\right)+1)\hfill \\ & \le & amax\left\{\begin{array}{c}|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}{\left(\mathcal{L}\right)|}^2,{|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{K}\right){\mathcal{D}}_{\mathsf{ȷ}})|}^2,\\ |s^{+}\left(\mathcal{L}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}){|}^2\end{array}\right\},\hfill \\ & & -b|s^{+}\left(\mathcal{K}\right)-\frac{1}{2}{s}^{+}(\mathcal{G}+\sum _{\mathsf{ȷ}=1}^k{\mathcal{D}}_{\mathsf{ȷ}}^{*}\varrho \left(\mathcal{L}\right){\mathcal{D}}_{\mathsf{ȷ}}){|}^2,\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill {(\frac{1}{2}M+\frac{3}{2}N{N}_{1}n)}^2(\parallel \mathcal{K}\parallel +\parallel \mathcal{L}\parallel +1)& \le & amax\left\{\begin{array}{c}\parallel \mathcal{K}-\frac{1}{2}{\mathcal{L}\parallel }^2,{\parallel \mathcal{K}-\frac{1}{2}\Im \left(\mathcal{K}\right)\parallel }^2,\\ \parallel \mathcal{L}-\frac{1}{2}{\Im \left(\mathcal{L}\right)\parallel }^2\end{array}\right\}\hfill \\ & & -b\parallel \mathcal{K}-\frac{1}{2}{\Im \left(\mathcal{L}\right)\parallel }^2.\hfill \end{array}$$

Therefore, from (31), we have

$$\begin{array}{ccc}\hfill (\parallel \mathcal{K}\parallel +\parallel \mathcal{L}\parallel +1)\parallel \Im \left(\mathcal{K}\right)-\frac{1}{2}{\Im \left(\mathcal{L}\right)\parallel }^2& \le & amax\left\{\begin{array}{c}\parallel \mathcal{K}-\frac{1}{2}{\mathcal{L}\parallel }^2,{\parallel \mathcal{K}-\frac{1}{2}\Im \left(\mathcal{K}\right)\parallel }^2,\\ \parallel \mathcal{L}-\frac{1}{2}{\Im \left(\mathcal{L}\right)\parallel }^2\end{array}\right\}\hfill \\ \hfill & & -b\parallel \mathcal{K}-\frac{1}{2}{\Im \left(\mathcal{L}\right)\parallel }^2.\hfill \end{array}$$

(32)

Let $q_c:{\mathcal{P}}_n\times {\mathcal{P}}_n\to {\mathbb{R}}_{+}$ be defined by

$$q_c(\mathcal{K},\mathcal{L})={\parallel \mathcal{K}-\frac{1}{2}\mathcal{L}\parallel }^2\mathrm{for}\mathrm{all}\mathcal{K}\ne \mathcal{L}\in {\mathcal{P}}_n.$$

Then, $({\mathcal{P}}_n,q_c,\theta )$ is a complete controlled EBQbMS with the coefficient $\theta (\mathcal{K},\mathcal{L})=\parallel \mathcal{K}\parallel +\parallel \mathcal{J}\parallel +1$. It follows from (32) that

$$\begin{array}{ccc}\hfill \theta (\mathcal{K},\mathcal{L})q_c(\Im \left(\mathcal{K}\right),\Im \left(\mathcal{L}\right))& \le & amax\left\{q_c(\mathcal{K},\mathcal{L}),q_c(\mathcal{K},\Im \left(\mathcal{L}\right)),q_c(\mathcal{L},\Im \left(\mathcal{L}\right))\right\}\hfill \\ \hfill & & -b{q}_c(\mathcal{K},\Im \left(\mathcal{L}\right)).\hfill \end{array}$$

(33)

Let ${\displaystyle \mathcal{M}({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4,{\varrho }_5)={\varrho }_1-amax\{{\varrho }_2,{\varrho }_3,{\varrho }_4\}+b{\varrho }_5}$, $0<b$ and $0<a<1$ in (33). Thus, we have

$$\begin{array}{c}\hfill \mathcal{G}\left(\begin{array}{c}\theta (\mathcal{K},\mathcal{L})q_c(\Im \left(\mathcal{K}\right),\Im \left(\mathcal{L}\right)),q_c(\mathcal{K},\mathcal{L}),q_c(\mathcal{K},\Im \left(\mathcal{K}\right)),q_c(\mathcal{L},\Im \left(\mathcal{L}\right)),q_c(\mathcal{K},\Im \left(\mathcal{L}\right))\end{array}\right)\le 0.\end{array}$$

Then, the formulated results follow from Theorem 1. □

9. Numerical Experiments

The experiment was conducted on an macOS Mojave version 10.14.6 CPU1.6 GHz Intel core i5 8GB with the programming language MATLAB R2020b (Online). The error is the symbol for the trace norm of the residual ($\mathrm{Res}\left(\mathcal{K}\right)=\parallel {\mathcal{K}}_{n+1}-{\mathcal{K}}_n{\parallel }_{tr}$), and CPU is the estimate time. In all of the investigations, $tol={10}^{-10}$ was allocated.

Example 9.

Consider matrices with coefficients that are constructed at random by

$${\mathcal{D}}_1=(2/3^n)\times rand\left(n\right);{\mathcal{D}}_2=(2/3^n)\times rand\left(n\right);$$

where ${\mathcal{D}}_1,{\mathcal{D}}_2\in {\mathcal{C}}^{n\times n}$. For $n=4$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{D}}_1& =\left[\begin{array}{cccc}0.0243& 0.0027& 0.0064& 0.0077\\ 0.0180& 0.0224& 0.0147& 0.0040\\ 0.0085& 0.0217& 0.0006& 0.0044\\ 0.0144& 0.0202& 0.0105& 0.0104\end{array}\right],\hfill \\ \hfill {\mathcal{D}}_2& =\left[\begin{array}{cccc}0.0023& 0.0173& 0.0079& 0.0202\\ 0.0148& 0.0158& 0.0131& 0.0177\\ 0.0116& 0.0008& 0.0162& 0.0239\\ 0.0172& 0.0017& 0.0101& 0.0131\end{array}\right],\hfill \\ \hfill \mathcal{G}& =\left[\begin{array}{cccc}1.0009& 0.0008& 0.0003& 0.0004\\ 0.0008& 1.0009& 0.0003& 0.0005\\ 0.0003& 0.0003& 1.0004& 0.0002\\ 0.0004& 0.0005& 0.0002& 1.0003\end{array}\right].\hfill \end{array}$$

We used the initial values

$$\begin{array}{c}{\mathcal{K}}_0=1.0e^{-03}\times \left[\begin{array}{cccc}0.0697& 0.0622& 0.0301& 0.0552\\ 0.0622& 0.1057& 0.0665& 0.0908\\ 0.0301& 0.0665& 0.0564& 0.0613\\ 0.0552& 0.0908& 0.0613& 0.0835\end{array}\right],{\mathcal{L}}_0=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],\\ {\mathcal{J}}_0=\left[\begin{array}{cccc}1.0018& 0.0015& 0.0006& 0.0009\\ 0.0015& 1.0018& 0.0005& 0.0010\\ 0.0006& 0.0005& 1.0008& 0.0004\\ 0.0009& 0.0010& 0.0004& 1.0006\end{array}\right],\end{array}$$

where ${\mathcal{K}}_0,{\mathcal{L}}_0,{\mathcal{J}}_0\in \mathcal{P}\left(n\right)$. For $b=0.1$, $a=0.95$, the PDS is

$$\widehat{\mathcal{K}}=\left[\begin{array}{cccc}1.0028& 0.0020& 0.0014& 0.0017\\ 0.0020& 1.0029& 0.0012& 0.0016\\ 0.0014& 0.0012& 1.0014& 0.0013\\ 0.0017& 0.0016& 0.0013& 1.0020\end{array}\right].$$

The graphical view of convergence and the surface plot of $\widehat{\mathcal{K}}$ are shown in Figure 3 and Figure 4, respectively.

Figure 3. Iteration vs. error graph.

Figure 4. Solution surface plot.

Example 10.

Consider the matrices ${\mathcal{D}}_1,{\mathcal{D}}_2,\mathcal{G}\in {\mathcal{C}}^{4\times 4}$ as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_1& =\left[\begin{array}{cccc}0.0024+0.0389i& 0.0022-0.0134i& 0.0046-0.0075i& 0.0058+0.0314i\\ 0.0084-0.0124i& 0.0023+0.0044i& 0.0064+0.0065i& 0.0043-0.0174i\\ 0.0086-0.0061i& 0.0054+0.0330i& 0.0092+0.0374i& 0.0088+0.0215i\\ 0.0096-0.0186i& 0.0076+0.0129i& 0.0016-0.0126i& 0.0039+0.0387i\end{array}\right],\hfill \\ \hfill {\mathcal{D}}_2& =\left[\begin{array}{cccc}0.0194+0.0080i& -0.0067+0.0077i& -0.0037+0.0051i& 0.0157+0.0348i\\ -0.0062+0.0141i& 0.0022+0.0180i& 0.0032+0.0322i& -0.0087+0.0274i\\ -0.0030+0.0087i& 0.0165+0.0273i& 0.0187+0.0030i& 0.0108+0.0205i\\ -0.0093+0.0001i& 0.0064+0.0187i& -0.0063+0.0049i& 0.0194+0.0246i\end{array}\right],\hfill \\ \hfill \mathcal{G}& =\left[\begin{array}{cccc}1.0020+0.0000i& 0.0011-0.0008i& 0.0009+0.0003i& 0.0011+0.0004i\\ 0.0011+0.0008i& 1.0024+0.0000i& 0.0013+0.0013i& 0.0011+0.0005i\\ 0.0009-0.0003i& 0.0013-0.0013i& 1.0020+0.0000i& 0.0013-0.0002i\\ 0.0011-0.0004i& 0.0011-0.0005i& 0.0013+0.0002i& 1.0015+0.0000i\end{array}\right].\hfill \end{array}$$

We began with the following initializations to demonstrate the convergence of the sequence $\left\{{\mathcal{K}}_n\right\}$, which was specified in (29) as follows:

$$\begin{array}{c}{\mathcal{K}}_0=1.0e^{-03}\times \\ \left[\begin{array}{cccc}0.2797+0.0000i& -0.1056+0.0476i& -0.0156+0.0116i& 0.0487+0.0235i\\ -0.1056-0.0476i& 0.0653-0.0000i& 0.0270-0.0532i& -0.0342-0.0104i\\ -0.0156-0.0116i& 0.0270+0.0532i& 0.3252+0.0000i& 0.1075+0.0201i\\ 0.0487-0.0235i& -0.0342+0.0104i& 0.1075-0.0201i& 0.2339-0.0000i\end{array}\right],\\ {\mathcal{L}}_0=\left[\begin{array}{cccc}0.0000+2.0000i& 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+0.0000i\\ 0.0000+0.0000i& 0.0000+2.0000i& 0.0000+0.0000i& 0.0000+0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+2.0000i& 0.0000+0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+2.0000i\end{array}\right],\\ {\mathcal{J}}_0=\left[\begin{array}{cccc}1.0041+0.0000i& 0.0022-0.0016i& 0.0019+0.0006i& 0.0023+0.0007i\\ 0.0022+0.0016i& 1.0049+0.0000i& 0.0026+0.0026i& 0.0021+0.0009i\\ 0.0019-0.0006i& 0.0026-0.0026i& 1.0040+0.0000i& 0.0025-0.0004i\\ 0.0023-0.0007i& 0.0021-0.0009i& 0.0025+0.0004i& 1.0031+0.0000i\end{array}\right],\end{array}$$

where ${\mathcal{K}}_0,{\mathcal{L}}_0,{\mathcal{J}}_0\in \mathcal{P}\left(n\right)$. Table 1 shows the details of the iteration, where the PDS is given by $\widehat{\mathcal{K}}$, and the convergence graph is shown in Figure 5.

$$\widehat{\mathcal{K}}=\left[\begin{array}{cccc}1.0052+0.0000i& 0.0005-0.0006i& 0.0012+0.0002i& 0.0029+0.0009i\\ 0.0005+0.0006i& 1.0059+0.0000i& 0.0037+0.0008i& 0.0039+0.0003i\\ 0.0012-0.0002i& 0.0037-0.0008i& 1.0053+0.0000i& 0.0026+0.0001i\\ 0.0029-0.0009i& 0.0039-0.0003i& 0.0026-0.0001i& 1.0088+0.0000i\end{array}\right].$$

Table 1. Three initialization based analysis.

Int. Mat.	$\mathit{\varrho }\left(\mathcal{K}\right)$	(a,b)	Dim.	Iter. No.	CPU	Error	Min. (Eig.)
${\mathcal{K}}_0$	${\mathcal{K}}^2$	(0.93,0.1)	4	6	0.019362	$0.005480$	1.0034 + 0.0000i
${\mathcal{L}}_0$	${\mathcal{L}}^2$	(0.93,0.1)	4	6	0.014226	$0.003246$	1.0034 + 0.0000i
${\mathcal{J}}_0$	${\mathcal{J}}^2$	(0.93,0.1)	4	5	0.013140	$0.003165$	1.0034 + 0.0000i

Figure 5. Iteration vs. error graph.

10. Conclusions

In this work, a new concept of controlled extended Branciari quasi-b -metric spaces and a ${\mathcal{G}}_q$-implicit-type mapping were introduced. Further, we derived some new fixed-point, periodic-point, right and left Ulam-Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results under the new space settings. We applied the results to find the solutions to the Riesz-Caputo fractional differential equations with integral anti-periodic boundary values and nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples.