# Fixed Points for Multivalued Weighted Mean Contractions in a Symmetric Generalized Metric Space

## Abstract

**:**

## 1. Introduction

## 2. Main Results

- $P\left(X\right)$—the set of all subsets of X, which are nonempty;
- ${P}_{c}\left(X\right)$—the set of all compact subsets of X, which are nonempty;
- P
_{cl}(X)—the set of all nonempty closed subsets of X.

- $D:P\left(X\right)\times P\left(X\right)\to \left[0,\infty \right),D\left(Z,Y\right)\u2254inf\{d\left(x,y\right):x\in Z,y\in Y\},Z,Y\in P\left(X\right)$—the gap functional;
- $H:P\left(X\right)\times P\left(X\right)\to \left[0,\infty \right),H\left(Z,Y\right)\u2254max\{su{p}_{x\in Z}in{f}_{y\in Y}d\left(x,y\right),su{p}_{y\in Y}in{f}_{x\in Z}d\left(x,y\right)\}$—the Pompeiu–Hausdorff functional.

**Definition 1**

**.**The functional $F:X\to R$ is known as regular-global-inf (RGI) in $x\in X$ if and only if $F\left(x\right)>infF$ implies that there is a $p>infF$, such that $D\left(x,{L}_{p}\right)>0$. The application F is called RGI in X if it is RGI in all $x\in X$.

**Proposition 1**

**.**

**Proposition 2**

**.**Let $\left(X,d\right)$ be a complete metric space, and $F:X\to \left[0,\infty \right)$ be an RGI function in X. If $\underset{p\downarrow infF}{lim}{\alpha}_{K}\left({L}_{p}\right)=0$, we obtain that the set of the global minimum points of F is nonempty and compact.

**Definition 2**

**.**Let X be a nonempty set. We consider the vector space of vectors with positive real components ${R}_{+}^{m},$ equipped with the usual component-wise partial order. The operator $d:X\times X\to {R}_{+}^{m}$, for which the usual axioms of the metric take place, is called a generalized metric in the sense of Perov.

_{i}denotes the Pompeiu–Hausdorff metric on ${P}_{c}\left(X\right)$ generated by d

_{i}, $i\in \{1,2\dots ,m\}$, then we denote by $H:{P}_{c}\left(X\right)\times {P}_{c}\left(X\right)\to {R}_{+}^{m}$, $H\u2254\left({H}_{1},{H}_{2},\dots ,{H}_{m}\right)$ the vector-valued Pompeiu–Hausdorff metric.

**Definition 3**

**Definition 4**

**.**Let $A\in {M}_{m,m}\left({R}_{+}\right)$ be a matrix convergent to zero. It is said that any A multivalued operator $T:Y\subset X$→${P}_{c}\left(X\right)$ is a multivalued left A-contraction in the sense of Nadler, if the following inequality takes place:

**Definition 5.**

**Definition 6.**

**Definition 7.**

**Definition 8**

- (i)
- $d\left(x,y\right)=0$ if and only if $x=y$;
- (ii)
- $d\left(x,y\right)=d\left(y,x\right)$.

**Definition 9**

**.**A vector $d\u2254\left({d}_{1},{d}_{2},\dots ,{d}_{m}\right),$ for which each element ${d}_{i}$ is a nonnegative real-valued function on $X\times X$, such that

- (i)
- ${d}_{i}\left(x,y\right)=0$ if and only if $x=y$ for all $i\in \{1,2\dots ,n\}$;
- (ii)
- ${d}_{i}\left(x,y\right)={d}_{i}\left(y,x\right)$ for all $i\in \{1,2\dots ,n\}$

**Theorem 1.**

**Proof.**

- (i)
- inf F = 0;
- (ii)
- $\underset{p\downarrow 0}{\mathrm{lim}}{\alpha}_{K}\left({L}_{p}\right)=0;$
- (iii)
- F is RGI in X.

**Corollary 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Corollary 2.**

**Proof.**

## 3. Applications to the Inclusion Systems

**Corollary 3.**

**Proof.**

**Corollary 4.**

**Proof.**

**Example 1.**

- restart:
- rsx:=- x(t)+x(t)*y(t);
- rsy:=- y(t)+3*y(t)*z(t)-x(t)*y(t);
- rsz:= 3*z(t)-z(t)ˆ2-y(t)*z(t);

- sol:=dsolve({diff(x(t),t)=rsx,diff(y(t),t)=rsy,diff(z(t),t)=rsz,x(0)=0,y(0)=0,z(0)=2},{x(t),y(t),z(t)},
- type=numeric,output=listprocedure);
- zsol:=subs(sol,z(t)); zsol(2);
- plot(zsol,0..30,color=red);

- sol:=dsolve({diff(x(t),t)=rsx,diff(y(t),t)=rsy,diff(z(t),t)=rsz,x(0)=0,y(0)=1,z(0)=2},{x(t),y(t),z(t)},
- type=numeric,output=listprocedure);
- ysol:=subs(sol,y(t));zsol:=subs(sol,z(t));
- plot([ysol,zsol],0..30,color=[red,blue]);

- sol:=dsolve({diff(x(t),t)=rsx,diff(y(t),t)=rsy,diff(z(t),t)=rsz,x(0)=1,y(0)=1,z(0)=2},{x(t),y(t),z(t)},
- type=numeric,output=listprocedure);
- xsol:=subs(sol,x(t));
- ysol:=subs(sol,y(t));
- zsol:=subs(sol,z(t));
- plot([xsol,ysol,zsol],0..30,color=[red,blue,green]);

**Particular Case of Corollary 3.**

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sebeşul de Sus village (in German, Ober-Schewesch). (Source: https://ro.wikipedia.org/wiki/Sebe%C8%99u_de_Sus,_Sibiu#/media/Fi%C8%99ier:Ulita_din_Sebesu_de_Sus.jpg).

**Figure 3.**Red frog, European bullhead, and trout that live in the mountain river. (Sources: http://herpetolife.ro/broasca-rosie-de-munte-rana-temporaria/, https://ro.wikipedia.org/wiki/Zglăvoacă, and http://www.zooland.ro/pastravul-de-munte-salmo-trutta-fario-2226).

**Figure 4.**Equilibrium state for the volume of the frog population (represented in red) in the absence of European bullheads and trout (figure created by the author using Maple software).

**Figure 5.**The equilibrium state of the European bullhead population (blue graph) and the frog population (red graph) in the absence of trout (figure created by the author using Maple software).

**Figure 6.**The equilibrium state of the European bullhead (blue graph) and frog population (red graph) in the presence of trout (green graph) (figure created by the author using Maple software).

**Figure 7.**Analytical relations between the volumes of frog, European bullhead, and trout populations. (figure created by the author using Maple software).

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**MDPI and ACS Style**

Bucur, A.
Fixed Points for Multivalued Weighted Mean Contractions in a Symmetric Generalized Metric Space. *Symmetry* **2020**, *12*, 134.
https://doi.org/10.3390/sym12010134

**AMA Style**

Bucur A.
Fixed Points for Multivalued Weighted Mean Contractions in a Symmetric Generalized Metric Space. *Symmetry*. 2020; 12(1):134.
https://doi.org/10.3390/sym12010134

**Chicago/Turabian Style**

Bucur, Amelia.
2020. "Fixed Points for Multivalued Weighted Mean Contractions in a Symmetric Generalized Metric Space" *Symmetry* 12, no. 1: 134.
https://doi.org/10.3390/sym12010134