Fixed Point Results for Frum-Ketkov Type Contractions in b-Metric Spaces

Abstract

The purpose of this paper is to present some fixed point results for Frum-Ketkov type operators in complete b-metric spaces.

1. Introduction and Preliminaries

In [1], Frum-Ketkov obtained a fixed point theorem, which was later generalized by Nussbaum [2] and Buley [3]. Later, Park and Kim [4] obtained other forms of the Frum-Ketkov theorem. Recently, Petrusel, Rus and Serban [5] gave sufficient conditions ensuring that a Frum-Ketkov operator is a weakly Picard operator and studied also some generalized Frum-Ketkov operators, see also [6].

The purpose of this paper is to obtain similar results for generalized Frum-Ketkov operators in the context of b-metric spaces.

We start by recalling the definition of Frum-Ketkov operators and some notions given in [5].

Let $\left(M,d\right)$ be a metric space. We denote by $P\left(M\right)$ the family of all nonempty subsets of M, by $P_{cl}\left(M\right)$ the family of all nonempty closed subsets of M and by $P_{cp}\left(M\right)$ the family of all nonempty compact subsets of M.

The $\omega$-limit set of $x\in M$ under the self-mapping f is defined as

$${\omega }_f\left(x\right)=\bigcap _{n=0}^{+\infty }\overline{\left\{f^k\left(x\right):k\ge n\right\}},$$

where $f^k$ is the iterate of order k of f.

Remark 1.

Ref. [5] ${\omega }_f\left(x\right)=\left\{x^{*}\in M:there exists n_k such that f^{n_k}\left(x\right)\to x^{*}\right\}.$

Definition 1.

Ref. [5] Let $\left(M,d\right)$ be a metric space. A self-mapping $f:M\to M$ is called:

1. 

l-contraction if $l\in \left(0,1\right)$ and $d\left(f\right(x),f(y\left)\right)\le ld(x,y)$, for every $x,y\in M$;

2. 

Contractive if $d\left(f\right(x),f(y\left)\right)<d(x,y)$, for every $x,y\in M$ with $x\ne y$;

3. 

Nonexpansive if $d\left(f\right(x),f(y\left)\right)\le d(x,y)$, for every $x,y\in M$;

4. 

Quasinonexpansive if $F_f\ne ⌀$ and, if $x*\in F_f$ then $d(f\left(x\right),x^{*})\le d(x,x^{*})$, for every $x\in M$, where $F_f$ is the set of fixed point of the mapping f;

5. 

Asymptotical regular in a point $x\in M$, if $d\left(f^n\left(x\right),f^{n+1}\left(x\right)\right)\to 0$, as $n\to +\infty .$

Definition 2.

Ref. [7] Let $X\in P_{cl}\left(M\right)$ and $f:X\to X$. f is called weakly Picard operator (WPO) if the sequence of successive approximation ${\left\{f^k\left(x\right)\right\}}_{n\in \mathbb{N}}$ converges for all $x\in X$ and its limit (which in general depends on x) is a fixed point of f. If f is a WPO with a unique fixed point, then f is called Picard operator (PO).

Definition 3.

Ref. [5] Let $\left(M,d\right)$ be a metric space, $X\in P_{cl}\left(M\right)$ and $K\in P_{cp}\left(M\right)$. A continuous operator $f:X\to X$ is said to be a Frum-Ketkov $\left(l,K\right)$-operator if $l\in \left(0,1\right)$ and

$$d\left(f\left(x\right),K\right)\le ld\left(x,K\right), for every x\in X,$$

where

$$d(x,K)=inf\left\{d\right(x,z):z\in K\}.$$

In what follows, we recollect the definition of b-metric that was considered by several authors, including Bakhtin [8] and Czerwik [9].

Definition 4.

Let M be a nonempty set and let $s\ge 1$ be a given real number. A functional $d:M\times M\to [0,+\infty )$ is said to be a b-metric with constant s, if

1. 

d is symmetric, that is, $d(x,y)=d(y,x)$ for all $x,y$,

2. 

d is self-distance, that is, $d(x,y)=0$ if and only if $x=y$,

3. 

d provides s-weighted triangle inequality, that is

$$d(x,z)\le s\left[d\right(x,y)+d(y,z\left)\right], for all x,y,z\in M.$$

In this case the triple $(M,d,s)$ is called a b-metric space with constant $s\ge 1$.

It is evident that the notions of b-metric and standard metric coincide in case of $s=1$. For more details on b-metric spaces see, e.g., [10,11,12] and corresponding references therein.

Example 1.

Let $M=[0,+\infty )$ and $d:M\times M\to [0,+\infty )$ such that $d\left(x,y\right)={\left|x-y\right|}^p,p>1.$ It’s easy to see that d is a b-metric with $s=2^{p-1}$, but is not a metric.

Definition 5.

A mapping $\phi :[0,+\infty )\to [0,+\infty )$ is called a comparison function if it is increasing and ${\phi }^n\left(t\right)\to 0$, as $n\to +\infty$, for any $t\in [0,+\infty )$.

Lemma 1.

Ref. [11] If $\phi :[0,+\infty )\to [0,+\infty )$ is a comparison function, then:

1. 

Each iterate ${\phi }^k$ of φ, $k\ge 1$, is also a comparison function;

2. 

φ is continuous at 0;

3. 

$\phi \left(t\right)<t$, for any $t>0$.

Definition 6.

A function $\phi :[0,+\infty )\to [0,+\infty )$ is said to be a $c$-comparison function if

1. 

φ is increasing;

2. 

There exists $k_0\in \mathbb{N}$, $a\in (0,1)$ and a convergent series of nonnegative terms ${\displaystyle \sum _{k=1}^{+\infty }}{v}_k$ such that ${\phi }^{k+1}\left(t\right)\le a{\phi }^k\left(t\right)+v_k$, for $k\ge k_0$ and any $t\in [0,+\infty ).$

In order to give some fixed point results to the class of b-metric spaces, the notion of $c$-comparison function was extended to b-comparison function by V. Berinde [12].

Definition 7.

Ref. [12] Let $s\ge 1$ be a real number. A mapping $\phi :[0,+\infty )\to [0,+\infty )$ is called a b-comparison function if the following conditions are fulfilled

1. 

φ is monotone increasing;

2. 

There exist $k_0\in \mathbb{N}$, $a\in (0,1)$ and a convergent series of nonnegative terms ${\displaystyle \sum _{k=1}^{+\infty }}{v}_k$ such that $s^{k+1}{\phi }^{k+1}\left(t\right)\le a{s}^k{\phi }^k\left(t\right)+v_k$, for $k\ge k_0$ and any $t\in [0,+\infty ).$

The following lemma is very important in the proof of our results.

Lemma 2.

Ref. [12] If $\phi :[0,+\infty )\to [0,+\infty )$ is a b-comparison function, then we have the following conclusions:

1. 

The series ${\displaystyle \sum _{k=0}^{+\infty }}{s}^k{\phi }^k\left(t\right)$ converges for any $t\in \left[0,+\infty \right)$;

2. 

The function $S_b:[0,+\infty )\to [0,+\infty )$ defined by $S_b\left(t\right)={\displaystyle \sum _{k=0}^{+\infty }}{s}^k{\phi }^k\left(t\right),t\in [0,+\infty )$, is increasing and continuous at 0.

Remark 2.

Due to the Lemma 1.2, any b-comparison function is a comparison function.

2. Frum-Ketkov Operators in $\mathit{b}$-Metric Spaces

Definition 8.

Let $(M,d)$ be a b-metric space with constant $s\ge 1$, $X\in P_{cl}\left(M\right)$ and $K\in P_{cp}\left(M\right)$. A continuous function $f:X\to X$ is said to be a Frum-Ketkov $\left(\phi ,K\right)$-operator if there exists $\phi :[0,+\infty )\to [0,+\infty )$ a b-comparison function such that

$$d\left(f\left(x\right),K\right)\le \phi \left(d\left(x,K\right)\right), for every x\in X.$$

Example 2.

Let $M=\left[0,+\infty \right),d:M\times M\to \left[0,+\infty \right),d\left(x,y\right)={\left(x-y\right)}^2,s=2$. From Example 1.1. we have that $\left(M,d\right)$ is a b-metric space. Let $X=\left[0,1\right],K=\left\{0\right\},$ $f:X\to X,$ $f\left(x\right)=\frac{x}{x+2},\phi :[0,+\infty )\to [0,+\infty ),\phi \left(t\right)=\frac{t}{t+4}$. f is Frum-Ketkov operator.

Theorem 1.

Let $(M,d)$ be a b-metric space with constant $s\ge 1$, $X\in P_{cl}\left(M\right)$, $K\in P_{cp}\left(M\right)$ and $f:X\to X$ a Frum-Ketkov $\left(\phi ,K\right)$-operator. Then the following conclusion hold:

(i) 

${\omega }_f\left(x\right)\ne ⌀$ and ${\omega }_f\left(x\right)\subset X\cap K$, for every $x\in X$;

(ii) 

$F_f\subset X\cap K$;

(iii) 

$f\left(X\cap K\right)\subset X\cap K$;

(iv) 

If f is asymptotically regular, then ${\varpi }_f\left(x\right)\subset F_f$, for every $x\in X$. If, in addition, f is quasinonexpansive, then f is WPO.

Proof. (i) Let $x\in X$ arbitrary. Because $K\in P_{cp}\left(M\right)$, there exists $\left(y_n\right)$ such that $d\left(f\left(x\right),K\right)=d\left(f\left(x\right),y_n\right)$

$$\begin{array}{c}d\left(f\left(x\right),y_n\right)\le \phi \left(d\left(x,y_n\right)\right)\hfill \\ d\left(f^2\left(x\right),y_n\right)\le \phi \left(d\left(f\left(x\right),y_n\right)\right)\le {\phi }^2\left(d\left(x,y_n\right)\right)\hfill \end{array}$$

Inductively, we obtain

$$d\left(f^n\left(x\right),y_n\right)\le {\phi }^n\left(d\left(x,y_n\right)\right)\to 0,asn\to +\infty .$$

Hence, $d\left(f^n\left(x\right),y_n\right)\to 0,asn\to +\infty .$

As $K\in P_{cp}\left(M\right)$, there exists a subsequence $\left(y_{n_k}\right)$ of $\left(y_n\right)$, such that $y_{n_k}\to y^{*}\left(x\right)\in K$, $n_k\to +\infty .$

Since $d\left(f^n\left(x\right),y_n\right)\to 0$, then $d\left(f^{n_k}\left(x\right),y^{*}\left(x\right)\right)\to 0$ and hence $f^{n_k}\left(x\right)\to y^{*}\left(x\right),n_k\to +\infty ,$ and thus $y^{*}\left(x\right)\in {\omega }_f\left(x\right).$

In this way ${\omega }_f\left(x\right)\ne ⌀$ and ${\omega }_f\left(x\right)\subset X\cap K$, for every $x\in X.$

(ii) Let $x\in F_f$. Suppose $d\left(x,K\right)\ne 0.$

$$d\left(x,K\right)=d\left(f\left(x\right),K\right)\le \phi \left(d\left(x,K\right)\right)<d\left(x,K\right),$$

which is a contradiction.

Hence, $d\left(x,K\right)=0$ which implies $x\in K$ and thus $F_f\subset X\cap K.$

(iii) Let $x\in X\cap K$

$$d\left(f\left(x\right),K\right)\le \phi \left(d\left(x,K\right)\right)=\phi \left(0\right)=0.$$

Hence, $f\left(x\right)\in K.$

(iv) From (i) we have that ${\omega }_f\left(x\right)\ne ⌀$, for every $x\in X$. Let $x^{*}\left(x\right)\in$ ${\omega }_f\left(x\right)$. There exists $n_k$ such that $f^{n_k}\left(x\right)\to x^{*}\left(x\right)$ as $n_k\to +\infty .$

$$\begin{array}{cc}\hfill d\left(x^{*},f\left(x^{*}\right)\right)& \le sd\left(x^{*},f^{n_k}\left(x^{*}\right)\right)+sd\left(f^{n_k}\left(x^{*}\right),f\left(x^{*}\right)\right)\hfill \\ \hfill & \le sd\left(x^{*},f^{n_k}\left(x^{*}\right)\right)+s^{2}d\left(f^{n_k}\left(x^{*}\right),f^{n_k+1}\left(x^{*}\right)\right)+s^{2}d\left(f^{n_k+1}\left(x^{*}\right),f\left(x^{*}\right)\right)\hfill \end{array}$$

(1)

From (i) and (iii) since $x^{*}\left(x\right)\in$ ${\omega }_f\left(x\right)$ we have that

$$d\left(f^2\left(x^{*}\right),f\left(x^{*}\right)\right)\le \phi \left(d\left(x^{*},f\left(x^{*}\right)\right)\right).$$

Inductively, we obtain

$$d\left(f^{n_k}\left(x^{*}\right),f^{n_k+1}\left(x^{*}\right)\right)\le {\phi }^{n_k}\left(d\left(x^{*},f\left(x^{*}\right)\right)\right).$$

Now, if in (1) we consider $n_k\to +\infty$, then we obtain $d\left(x^{*},f\left(x^{*}\right)\right)$, which implies that $x^{*}\in F_f$ and thus ${\varpi }_f\left(x\right)\subset F_f.$

Consider now that, in addition, f is quasinonexpansive and let $x\in X$ and $f^{n_k}\left(x\right)\to y^{*}\left(x\right),n_k\to +\infty$ (see (i)). Because f is asymptotically regular, $y^{*}\left(x\right)\in F_f.$

$$\begin{array}{c}d\left(f\left(x\right),y^{*}\right)\le \phi \left(d\left(x,y^{*}\right)\right)\hfill \\ d\left(f^2\left(x\right),y^{*}\right)\le \phi \left(d\left(f\left(x\right),y^{*}\right)\right)<d\left(f\left(x\right),y^{*}\right).\hfill \end{array}$$

Hence the sequence $\left(d\left(f^n\left(x\right),y^{*}\right)\right)$ is decreasing and since $\left(d\left(f^{n_k}\left(x\right),y^{*}\right)\right)\to 0$ as $n_k\to +\infty$, we obtain $d\left(f^n\left(x\right),y^{*}\right)\to 0$ as $n\to +\infty$ and thus f is WPO. □

3. Conclusions

Frum-Ketkov type contractions are an interesting topic that has been overlooked and has not attracted anyone’s attention for many years. The very attractive recent publication of Petrusel–Rus–Serban [5] is the one that brought this shadowy concept to light. In this paper, we consider the Frum-Ketkov type contractions in the framework of b-metric space. For this reason, this paper should be considered as an initial paper that opens a new trend in metric fixed point theory.