Single-valued signals, multi-valued signals, and fixed point of contractive signals

1. Introduction

Many subfields of pure and applied mathematics rely heavily on the theory of multi-valued mappings for their work. This is because it may be used, for example, in actual, thorough analysis, and also problems involving best management. The significance of this hypothesis has increased over time. There are now a great number of publications in the academic literature that are devoted to the examination of both theoretical and real-world issues that are associated with multi-valued mappings. In point of fact, among the many different ways that were used to build fixed-point theory, it provides the foundation for this fascinating theory. This is frequently because fixed-point theorems tend to have a constructive character, particularly in the metric branch of the field (see [18]). Therefore, Nadler [18] was the first author to create the fixed-point theorem, which linked the idea of contraction with multi-valued mappings. Subsequently, several authors explored the findings of and presented their generalizations and applications (see [1,2,5,17,46]). However, there are a few different ways in which the concept of metric space has been altered. So it can better address much more broad circumstances, such as those that arise in computer science and elsewhere (see [8]). For further information, see [8,16]; in this study, we use the term “b-metric space”, which stands for a metric space where a loosened triangle inequality holds. Numerous investigations were carried out using this method, with varying results when applied to a position in the b-metric scale [10,12,14]. In 2012, Samet et al. [13] introduced the concept of contractive mapping and established sufficient requirements for the existence of fixed points in this class of mappings. In this study, we enhance [20] by looking at convergence points for contractions with multiple values on 2-metric spaces. Fixed-point results offer a wonderful set of circumstances for the study of mathematical analysi. These results can be used to approximate the solutions of linear and nonlinear differential and integral equations. Because it is a peculiar mix of geometry, topology, and analysis, the theory of fixed point has emerged as an effective and vital instrument for the study of nonlinear processes [29]. This is because the theory itself is a strange combination. This theory [21], which has been around for a while, is an important component of both pure and applied mathematics. The normally fixed-point theory has proven to be helpful in a wide variety of fields, including game theory, biology, engineering, nonlinear programming, economics, and the theory of differential equations [9,30,40,45]. For the past 20–30 years, numerous mathematicians have found success in the field of analysis known as fixed-point theory. The study of the existence and uniqueness of fixed-point mappings has received a significant amount of attention in recent years as a result of the widespread applications [34] of the Banach contraction principle [43]. Numerous academics devoted their time and energy to developing this theory, and as a result, fresh extensions of Banach and Nadler’s theorems were proposed in a variety of areas. The concept of Hausdorff distance is crucial to the study of a variety of subfields within computer science and mathematics [3,46], including the theory of optimization, image processing, and fractals, amongst others. The Hausdorff metric is a crucial and extremely important notion. It can be defined as the greatest of all the distances that separate a point in one set from the point in the other set that is closest to it. This includes the development of a method for researching the fixed-point theory of set-valued mappings in spaces that have the structure of generalized metrics. In b-metric space, fixed-point and common fixed-point results for single-valued as well as multi-valued mappings have been researched (see [4,6,7,22,23,24,25,26,27,31,33,41,44]). Recently published research works centered the viewpoints more or less identical regarding coincidence and fixed points of multi-valued F-contraction in generalized metric spaces with application (see [37]). Several phenomenological research works are sought to have very identical reflections on the concept of fixed points of Suzuki-type generalized multi-valued (f, $\theta$, L) almost contraction (see). The motivating factor in such marvelous studies led me to the point that the common fixed-point theory is more generalized as a particular case is necessary to be studied closely (see [15,19,28,35,36,38,39]).

Let $(F,\rho )$ be a two-dimensional space.

Let $P,Q,R$ be three non-empty subsets of F then we define the following relations:

Let us suppose that CB$(F)$ be the collection of all convex and finite subsets that are not empty F. Regarding geometry, this area is a 2-metric one. L is known as Gahler 2-metric. Now, we extend a result of [20].

Theorem 1. If $(F,\rho )$ have the properties of a full 2-metric and $h:F\to \mathrm{\text{CB}}(F)$ be satisfiable by multi-valued signals

$\delta ,\mu \ge 0$ and $2\delta +\mu -1<0$ then the signal has a fixed point where $p,q,w\in F$.

Proof. Let us consider that $c_0$ be a point in $F_1$ and $c_1\in h(c_0)$.

When $L(h(x_0),h(x_1),\{w\})=0$ for all $w\in F$.

Consequently $h(c_0)=h(c_1)$ (L is the portion of the set of all sets that is a Gahler 2-metric F).

Hence, $c_1\in h(c_1)$.

If we put $\delta =\mu =0$ then we get our required result.

Again, let $\mu +2\delta >0$ and $L(h(w),h(c_1),\{w\}).$

Put

Let $t=\frac{L(h(c_0),h(c_1),\{w\})}{g}$.

Then $t>L(h(c_0),h(c),\{w\})$.

Thus, by hypothesis, we have

meaning that $t>\rho (c_1,c_2,w)$.

Now,

where $b=\frac{\delta +\mu }{g-\delta }$ and $b\in (0,1)$.

For pairing $c_1,c_2$, we obtain

(i)	$L(h(c_1),h(c_2),\{w\})=0$,
(ii)	$L(h(c_1),h(c_2),\{w\})>0$.

For $L(h(c_1),h(c_2),\{w\})=0$, then there exists $c_2$ such that $c_2\in h(c_2)$.

For $L(h(c_1),h(c_2),\{w\})>0$, then there exists a point $c_3$ of $h(c_2)$ such that $\rho (c_2,c_3,w)\le b\rho (c_1,c_2,w)$ and in the same way, if $L(h(c_j),h(c_{j+1}),\{w\})>0$, then there exists a point $c_{j+2}\in h(c_{j+1})$ which satisfies $\rho (c_{j+1},c_{j+2},w)\le b\rho (c_j,c_{j+1},w)$.

Now, we next show that

if $n=0,1$ then In the event that this thesis holds true for $2\le n\le \vee -1$.

i.e. $\rho (c_0,c_1,c_{\vee -1})=0$,

hence by mathematical induction $\rho (c_0,c_1,c_n)=0\forall n\ge 0$ for $x>n$, $\Rightarrow \{c_x\}$ is a Cauchy sequence.

Thus, by definition, when we say that F is all subsets are closed and bounded, then find the upper bound on $\{c_n\}$.

Let a be the limit point of $\{c_n\}$, then

since $1-\delta >0$, then $d(a,h(a),w)=0$ for all $w\in F$, thus $a\in h(a)$. □

Theorem 2. If $(F,\rho )$ be an open 2-metric space with no boundaries, and be a set of related signals with values ranging from F to BN (F). If we choose a monotonic series of positive numbers, then BN (F) is the collection of all finite subsets of that are not empty F. $\delta$ and ${\delta }_1$ in order that everyone benefits $p,q,w\in F$ and $e,f$ with $e\ne f$,

where ${\delta }_1+2\delta -1<0$ then the monotonic sequence of multi-valued signals $\{h_m\}$ possesses a singular shared fixed point.

Proof. Let $\sqrt{{\delta }_1+2\delta }>0$ and $g-1<0$.

Now, we define a sequence $\{{\epsilon }_m\}$ of single-valued signal ${\epsilon }_m:F\to F$ such that ${\epsilon }_m(p)$ is a point q in $h_m(p)$ which satisfy $\rho (p,q,w)=\rho (p,{\epsilon }_e^{n_e}(p),w)\ge gL(p,h_e^{n_e}(p),w)$.

For ${\epsilon }_m$,

thus by hypothesis of [20], $\{{\epsilon }_m\}$ possesses a single fixed point a, ${\epsilon }_m(a)=a$.

Here’s a,

$\rho (a,{\epsilon }_m(a),w)\ge gL(a,{\epsilon }_m(a),w)$ and $\rho (a,{\epsilon }_m(a),w)=0$

Let $r\in {\epsilon }_m(r)$ and $L(r,{\epsilon }_m(r),w)>0$ which is false.

Thus ${\epsilon }_m(r)=\{r\}$.

Let $K({\epsilon }_m(r),{\epsilon }_m(a),\{w\})$

$\Rightarrow {\epsilon }_m(r)$ has a single, unchanging center. □

Theorem 3. If (F, $\rho$) is a complete bounded 2-metric space and $h_x$ ($x\ge$1) is a family of multi-valued signals from F into F and also, we consider a monotonic sequence of non-negative numbers $\{n_r\}$ and $\delta$ and ${\delta }_1$ so that for all p, q, $w\in F$ and e, f with e and f are not equal, $\rho (h_e^{n_e}$(p), $h_f^{n_f}(q)$, $w)\ge \delta$  [(p, $h_e^{n_e}(p)$,$w)+\rho$ (q,$h_f^{n_f}(q)$, $w)$] + ${\delta }_1\rho$ (p, q, w) where ${\delta }_1+2\delta -1<0$ then the monotonic sequence of multi-valued signals $\{h_m\}$ will have a unique common fixed point.

Similar proof as Theorem 2.

Theorem 4. If (F, $\rho )$ is a complete bounded 2-metric space and if ${\zeta }_1$ and ${\zeta }_2$ fromF into bounded multi-valued mappings so that forp, q, w are elements in non-empty setF, there exist positive numbers a, b, c, d, and u  where $a+b+c+d+u$ – 1 is negative and ($a-b$) ($c-u)$ is non-negative

then the both multi-valued signals have unique common fixed point.

Proof. We consider a non-negative number $1>\frac{1}{{(2a+4v+s)}^{\frac{-1}{2}}}>0$.

Then by hypothesis, there exists a single-valued signal $\mu :F\to F$ such that $\mu (p)$ is a point q in $\sigma (p)$ and it satisfies the following property:

then Hence by hypothesis of theorem $\rho$ (p, $\pi (q),q)$ = 0 and $\rho (q,\pi (p),p)$ = 0 and we get a point t such that $\mu (t)=t$.

Consequently $t\in \sigma (t)$

Let $\theta \in \sigma (\theta )$, K({$\theta$},$\sigma (\theta ),w).$

Then d ($\sigma (q),\sigma (q),w)\le K$ ({q},$\sigma (q),w),$ which gives a contradiction.

Thus $\sigma (\theta )=\{\theta \}$.

Also let $d(\sigma (\theta ),\sigma (t),w)\le {\delta }_1$$[K(\theta ),\sigma (p),w)+K(\{t\},\sigma (\theta ),w)+s\rho (t,\theta ,w))\le (2{\delta }_1+s)$$\rho (\theta ,t,w)$.

Consequently $t=\theta \mathrm{\text{ i.e. }}\sigma$ has a unique fixed point. □

2. Conclusion

Techniques based on fixed points are instruments that are both incredibly helpful and appealing. This theory has the potential to be useful in many different areas of nonlinear functional analysis, including functional inclusions, optimization theory, fractal graphics, discrete dynamics for set-valued operators, and more. The DP and CFP theorems for single-valued mappings that fulfill Ciric type contractions have been generalized and proven within the context of the sb-MS. Additionally, in these spaces, two FP theorems for multi-valued mappings with Nadler’s type contractions have been established and shown. These theorems have been established and demonstrated. These sweeping generalizations might prove valuable in research and applications in the future.

Acknowledgments

The authors would like to express gratitude to Department of Technical Education, Uttar Pradesh, India and Department of Mathematics, MIT Campus, T.U., Janakpurdham, Nepal.

ORCID

Binay Kumar Pandey  https://orcid.org/0000-0002-4041-1213