Common fixed point theorems for nonlinear contractions in a Menger space

Abstract

The main purpose of this paper is to introduce a new class of Jungck-type contraction and to present some common fixed point theorems for this mapping. Several examples are given to show that our result is a proper extension of many known results.

MSC:47H10.

1 Introduction

Probabilistic metric space has been introduced and studied in 1942 by Menger in USA [1], and since then the theory of probabilistic metric spaces has developed in many directions [2–6]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situation when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. Such a probabilistic generalization of metric spaces appears to be well adapted for the investigation of physiological thresholds and physical quantities, particularly in connection with both string and E-infinity which were introduced and studied by a well-known scientific hero El Naschie [7–9].

It is observed by many authors that the contraction condition in a metric space may be exactly translated into a probabilistic metric space endowed with min norms. Sehgal and Bharucha-Reid [10] obtained a generalization of the Banach contraction principle on a complete Menger space, which is a milestone in developing fixed point theorems in a Menger space.

Jungck’s fixed point theorem [11] has many applications in nonlinear analysis. This theorem is extended by several authors; see [12–16] and the references therein.

In this paper, we introduce a new class of Jungck-type contraction and present some common fixed point theorems for this mapping. Several examples are given to show that our result is a proper extension of many known results.

2 Preliminaries

Throughout this paper we denote by N the set of all positive integers, by Q the set of all rational numbers, by $Z^{+}$ the set of all nonnegative integers, by R the set of all real numbers and by $R^{+}$ the set of all nonnegative real numbers. We shall recall some definitions and lemmas related to a Menger space.

Definition 2.1 A mapping $F:R\to R^{+}$ is called a distribution if it is nondecreasing left continuous with $inf\{F(t):t\in R\}=0$ and $sup\{F(t):t\in R\}=1$. We shall denote by L the set of all distribution functions. The specific distribution function $H:R\to R^{+}$ is defined by

$$H(t)=\{\begin{array}{cc}0,\hfill & t\le 0,\hfill \\ 1,\hfill & t>0.\hfill \end{array}$$

Definition 2.2 ([13])

Probabilistic metric space (PM-space) is an ordered pair $(X,F)$, where X is an abstract set of elements and $F:X\times X\to L$ is defined by $(p,q)\to F_{p,q}$, where $\{F_{p,q}:p,q\in X\}\subseteq L$, where the functions $F_{p,q}$ satisfy the following:

• (a) $F_{p,q}(x)=1$ for all $x>0$ if and only if $p=q$;

• (b) $F_{p,q}(0)=0$;

• (c) $F_{p,q}=F_{q,p}$;

• (d) $F_{p,q}(x)=1$ and $F_{q,r}(y)=1$, then $F_{p,r}(x+y)=1$.

Definition 2.3 A mapping $t:[0,1]\times [0,1]\to [0,1]$ is called a t-norm if

• (e) $t(0,0)=0$ and $t(a,1)=a$ for all $a\in [0,1]$;

• (f) $t(a,b)=t(b,a)$ for all $a,b\in [0,1]$;

• (g) $t(a,b)\le t(c,d)$ for all $a,b,c,d\in [0,1]$ with $a\le c$ and $b\le d$;

• (h) $t(t(a,b),c)=t(a,t(b,c))$ for all $a,b,c\in [0,1]$.

Definition 2.4 A Menger space is a triplet $(X,F,t)$, where $(X,F)$ is a PM-space and t is a t-norm such that for all $p,q,r\in X$ and all $x,y\ge 0$,

$$F_{p,r}(x+y)\ge t(F_{p,q}(x),F_{q,r}(y)).$$

Definition 2.5 ([13])

Let $(X,F,t)$ be a Menger space and $f:X\to X$.

• (1) A sequence $\{p_n\}$ in X is said to converge to a point p in X (written as $p_n\to p$) if for every $\epsilon >0$ and $\lambda >0$, there exists a positive integer $M(\epsilon ,\lambda )$ such that $F_{p_n,p}(\epsilon )>1-\lambda$ for all $n\ge M(\epsilon ,\lambda )$.

• (2) A sequence $\{p_n\}$ in X is said to be Cauchy if for each $\epsilon >0$ and $\lambda >0$, there is a positive integer $M(\epsilon ,\lambda )$ such that $F_{p_n,p_m}(\epsilon )\ge 1-\lambda$ for all $n,m\in N$ with $n,m\ge M(\epsilon ,\lambda )$.

• (3) A Menger space $(X,F,t)$ is said to be complete if every Cauchy sequence in X converges to a point of it.

• (4) f is said to be continuous at a point p in X if for every sequence $\{p_n\}$ in X, which converges to p, the sequence $\{f(p_n)\}$ in X converges to $f(p)$.

• (5) f is said to be continuous on X if f is continuous at every point in X.

Definition 2.6 ([4])

A t-norm t is said to be of H-type if a family of functions ${\{t^n(a)\}}_{n=1}^{\mathrm{\infty }}$ is equicontinuous at $a=1$, that is, for any $\epsilon \in (0,1)$, there exists $\delta \in (0,1)$ such that $a>1-\delta$ and $n\ge 1$ imply $t^n(a)>1-\epsilon$. The t-norm $t=min$ is a trivial example of a t-norm of H-type, but there are t-norms of H-type with $t\text{-norm}\ne min$ (see, e.g., Hadzic [5]).

From Definition 2.1-Definition 2.5, we can prove easily the following lemmas.

Lemma 2.7 ([10])

If $(X,d)$ is a metric, then the metric induces a mapping $X\times X\to L$, defined by $F_{p,q}(x)=H(x-d(p,q))$, $p,q\in X$ and $x\in R$. Further, if the t-norm $t:[0,1]\times [0,1]\to [0,1]$ is defined by $t(a,b)=min\{a,b\}$ for all $a,b\in [0,1]$, then $(X,F,t)$ is a Menger space. It is complete if $(X,d)$ is complete.

Lemma 2.8 In a Menger space $(X,F,t)$, if $t(x,x)\ge x$ for all $x\in [0,1]$, then $t(a,b)=min\{a,b\}$ for all $a,b\in [0,1]$.

3 Jungck-type fixed point theorems

In 1976, Jungck proved the following theorem.

Theorem A (Jungck [11], 1976)

Let f be a continuous mapping of a complete metric space $(X,d)$ into itself and let $g:X\to X$ be a map that satisfies the following conditions:

for all $x,y\in X$ and for some $0<k<1$. Then f and g have a unique common fixed point.

Definition 3.1 Let $(X,F,t)$ be a Menger space with $t(x,x)\ge x$ for all $x\in [0,1]$ and let $f,g:X\to X$ be two self-mappings of X. We will say that f and g are Jungck-type generalized contraction if

for all $p,q\in X$ and $x>0$, where $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a mapping such that $\phi (x)<x$ for all $x>0$, and for all $p,q\in X$ and $x\in R$, $F_{p,q}(x)$ is the same as in Definition 2.2.

Remark 3.2

• (1) It is clear that (∗1) implies (∗2) if $F_{p,q}(x)=H(x-d(p,q))$ for all $p,q\in X$, $x\in R$, and $\phi (x)=kx$ for all $x\in R^{+}$, where $0<k<1$.

• (2) In Example 3.10, we shall show that the condition (∗2) is satisfied, but the condition (∗1) is not satisfied.

Definition 3.3 Let $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a mapping such that $\phi (x)<x$ for all $x>0$. We say that φ is the U-generalized contraction if

for all $x>0$ and $r\in Z^{+}$.

Lemma 3.4 Let $k\in (0,1)$ be as in (c) of Theorem  A and let $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be defined by

$$\phi (x)=\{\begin{array}{cc}(\frac{1+k}{2})x+(\frac{1-k}{2})x^2,\hfill & 0\le x\le k,\hfill \\ (\frac{1}{2}+k-\frac{k^2}{2})x,\hfill & k<x.\hfill \end{array}$$

Then φ is an U-generalized contraction.

Proof It follows from hypotheses that for all $x>0$,

Now we shall show that condition (∗3) is satisfied. Since

$$(x-\phi (x)){\left(\frac{\phi (x)}{x}\right)}^r\le k\text{for all }x\in (0,k]\text{ and }r\in Z^{+},$$

there are three cases which need to be considered.

Case 1. Let $x\in (0,k]$ and $r\in Z^{+}$. Then, since

$$(1-\frac{\phi (x)}{x}){\left(\frac{\phi (x)}{x}\right)}^r\le 1,$$

(∗3) is satisfied.

Case 2. Let $x\in (k,\mathrm{\infty })$ and $r\in Z^{+}$ with $(x-\phi (x)){(\frac{\phi (x)}{x})}^r\le k$. Then, since

$$\left(\frac{1+k}{2}\right)+\left(\frac{1-k}{2}\right)(x-\phi (x)){\left(\frac{\phi (x)}{x}\right)}^r\le \frac{1}{2}+k-\frac{k^2}{2},$$

(∗3) is satisfied.

Case 3. Let $x\in (k,\mathrm{\infty })$ and $r\in Z^{+}$ with $k<(x-\phi (x)){(\frac{\phi (x)}{x})}^r$. Then, since

$$\frac{1}{2}+k-\frac{k^2}{2}=\frac{\phi (x)}{x},$$

(∗3) is satisfied. From (∗4), Case 1, Case 2 and Case 3, φ is U-generalized contraction. □

The following example shows that f and g do not have a common fixed point even though f, g and φ satisfy (∗2) and (∗3).

Example 3.5 Let $k\in (0,1)$ and $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be as in Lemma 3.4. Let $f,g:R\to R$ be defined by $f(x)=x+1$ and $g(x)=\frac{k}{2}x$. Define $F_{p,q}:R\to R^{+}$ by

$$F_{p,q}(x)=H(x-|p-q|)\text{for all }p,q\in R\text{ and }x\in R,$$

where $F_{p,g}$ and H are the same as in Definition 2.1 and Definition 2.2. Let $t:[0,1]\times [0,1]\to [0,1]$ be defined by $t(a,b)=min\{a,b\}$ for all $a,b\in [0,1]$. Then, by Lemma 3.4 and simple calculations, (∗2) and (∗3) are satisfied. But f and g do not have a common fixed point.

Remark 3.6 It follows from Example 3.5 that f and g must satisfy (∗2) and (∗3), and other conditions additionally in order to have a common fixed point of f and g.

The following is Jungck-type common fixed point theorem which is a generalization of Jungck’s common fixed point theorem [11].

Theorem 3.7 Let $(X,F,t)$ be a complete Menger space with continuous t-norm and $t(x,x)\ge x$ for all $x\in [0,1]$, let f be a continuous self-mapping on X and let $\phi :R^{+}\to R^{+}$ be a mapping that satisfies the following conditions:

• (i) $g(X)\subseteq f(X)$;

• (ii) g commutes with f;

• (iii) f, g and φ satisfy (∗2) and (∗3);

• (iv) φ is a strictly increasing and bijective;

• (v) ${lim}_{n\to \mathrm{\infty }}{\phi }^{-n}(x)=\mathrm{\infty }$ for each $x>0$, where ${\phi }^{-n}$ is n-times repeated composition of ${\phi }^{-1}$ with itself.

Then f and g have a unique common fixed point.

Proof

$$\begin{array}{r}\text{It is easy to see that the self-mapping }g\text{ on }X\text{ in Theorem 3.7 is}\\ \text{continuous on }X.\end{array}$$

(3.1)

Let $x_0\in X$. By (i), there exists a sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ in X such that

$$f(x_n)=g(x_{n-1})\text{for all }n\in N.$$

(3.2)

From (iii) and (3.2), we have

$$F_{g(x_{n-1}),g(x_n)}(\phi (x))\ge F_{f(x_{n-1}),f(x_n)}(x)\text{for all }n\in N\text{ and }x>0.$$

(3.3)

By virtue of (iv), (3.2) and (3.3), we obtain

$$F_{f(x_n),f(x_{n+1})}(x)=F_{g(x_{n-1}),g(x_n)}(x)\ge F_{f(x_{n-1}),f(x_n)}({\phi }^{-1}(x))$$

(3.4)

for all $n\in N$ and $x>0$. In view of (3.4), we have

$$F_{f(x_n),f(x_{n-1})}(x)\ge F_{f(x_0),f(x_1)}({\phi }^{-n}(x))$$

(3.5)

for all $n\in N$ and $x>0$. By repeated application of (3.5), we have

$$F_{f(x_{n+j}),f(x_{n+1+j})}(x)\ge F_{f(x_n),f(x_{n+1})}({\phi }^{-j}(x))$$

(3.6)

for all $n,j\in N$ and $x>0$. From (iii), we have

$$0<\frac{\phi (x)}{x}<1\text{for all }x>0.$$

(3.7)

On account of (3.7), we obtain that

$$\sum _{k=0}^{\mathrm{\infty }}{\left[\frac{\phi (x)}{x}\right]}^k=\frac{1}{1-(\frac{\phi (x)}{x})}\text{for all }x>0.$$

(3.8)

In terms of (3.8), we get that

$$x=(x-\phi (x))\sum _{k=o}^{\mathrm{\infty }}{\left(\frac{\phi (x)}{x}\right)}^k\text{for all }x>0.$$

(3.9)

Now we shall show that $\{f(x_n)\}$ is a Cauchy sequence.

$$\text{Let }n,m\in N\text{ be such that }n<m.$$

(3.10)

From (iii), (iv), (3.5)-(3.10) and Definition 2.4, we deduce that

$$\begin{array}{r}{F}_{f(x_n),f(x_m)}(x)\\ =F_{f(x_n),f(x_m)}\left((x-\phi (x))\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{\phi (x)}{x}\right)}^k\right)\\ \ge F_{f(x_n),f(x_m)}\left((x-\phi (x))\sum _{k=0}^{m-n-1}{\left(\frac{\phi (x)}{x}\right)}^k\right)\\ \ge min\{F_{f(x_n),f(x_{n+1})}\left((x-\phi (x))\right),\\ F_{f(x_{n+1}),f(x_{n+2})}\left((x-\phi (x))\left(\frac{\phi (x)}{x}\right)\right),\dots ,\\ F_{f(x_{m-1}),f(x_m)}\left((x-\phi (x)){\left(\frac{\phi (x)}{x}\right)}^{m-n-1}\right)\}\\ \ge min\{F_{f(x_n),f(x_{n+1})}\left((x-\phi (x))\right),\\ F_{f(x_n),f(x_{n+1})}\left({\phi }^{-1}\left((x-\phi (x))\left(\frac{\phi (x)}{x}\right)\right)\right),\dots ,\\ F_{f(x_n),f(x_{n+1})}\left({\phi }^{-(m-n-1)}\left((x-\phi (x)){\left(\frac{\phi (x)}{x}\right)}^{m-n-1}\right)\right)\}\\ \ge F_{f(x_n),f(x_{n+1})}\left((x-\phi (x))\right)\\ \ge F_{f(x_0),f(x_1)}\left({\phi }^{-n}(x-\phi (x))\right)\end{array}$$

(3.11)

for all $x>0$ and $n,m\in N$ with $n<m$. In terms of (iii), (v) and Definition 2.2, we have

$$\underset{n\to \mathrm{\infty }}{lim}{F}_{f(x_0),f(x_1)}\left({\phi }^{-n}(x-\phi (x))\right)=1\text{for all }x>0.$$

(3.12)

By (3.11), (3.12) and Definition 2.2, $\{f(x_n)\}$ is a Cauchy sequence in X. Since X is complete and $\{f(x_n)\}$ is a Cauchy sequence in X, there exists $z\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}f(x_n)=z.$$

(3.13)

On account of (3.2) and (3.13), we have

$$\underset{n\to \mathrm{\infty }}{lim}g(x_n)=z.$$

(3.14)

By (ii), (3.1), (3.13), (3.14) and hypotheses,

$$\begin{array}{r}f(g(x_n))=g(f(x_n))\text{for all }n\in N,\\ \underset{n\to \mathrm{\infty }}{lim}f(g(x_n))=f(z)\end{array}$$

(3.15)

and

$$\underset{n\to \mathrm{\infty }}{lim}g(f(x_n))=g(z).$$

From (3.15), we get that

$$f(z)=g(z).$$

(3.16)

In view of (ii), (3.16) and (∗2), we have

$$\begin{array}{rl}{F}_{g(z),g(g(z))}(\phi (x))& \ge F_{f(z),f(g(z))}(x)\ge F_{g(z),g(f(z))}(x)\\ \ge F_{g(z),g(g(z))}(x)\end{array}$$

(3.17)

for all $x>0$.

By (iv) and (3.17),

$$F_{g(z),g(g(z))}(x)\ge F_{g(z),g(g(z))}({\phi }^{-n}(x))$$

(3.18)

for all $n\in N$ and $x>0$. Due to (v), (3.18), Definition 2.1 and Definition 2.2, we get that

$$g(z)=g(g(z)).$$

(3.19)

From (ii), (3.16) and (3.19), we have

$$g(z)=g(g(z))=g(f(z))=f(g(z)).$$

(3.20)

By (3.20), $g(z)$ is a common fixed point of f and g. To prove the uniqueness of a common fixed point of f and g, let u and w be common fixed points of f and g. Then $f(u)=g(u)=u$ and $f(w)=g(w)=w$. Putting $p=u$ and $q=w$ in (∗2), we get

$$F_{g(u),g(w)}(\phi (x))=F_{u,w}(\phi (x))\ge F_{f(u),f(w)}(x)=F_{u,w}(x)$$

(3.21)

for all $x>0$, which gives $u=w$. Thus $g(z)$ is a unique common fixed point of f and g. □

Now we give an example to support Theorem 3.7.

Example 3.8 Let $X=R$ be the set of reals with the usual metric and let $f,g:X\to X$ and $\phi :R^{+}\to R^{+}$ be mappings defined as follows:

$$f(x)=3x,g(x)=2x\text{and}\phi (x)=\{\begin{array}{cc}\frac{2}{3}x+\frac{1}{3}{x}^2,\hfill & 0\le x\le \frac{2}{3},\hfill \\ \frac{8}{9}x,\hfill & \frac{2}{3}<x.\hfill \end{array}$$

(3.22)

Let the mappings $F_{p,q}$, H and t be as in Example 3.5. Then from Lemma 2.7, $(X,F,t)$ is a complete Menger space. By the same method as in Lemma 3.4 and simple calculations, the conditions of Theorem 3.7 are satisfied. Thus f and g have a unique common fixed point 0.

From Theorem 3.7, we have the following corollary.

Corollary 3.9 Let $(X,F,t)$ be a complete Menger space with continuous t-norm and $t(x,x)\ge x$ for all $x\in [0,1]$. Let $f,g:`X\to X$ be maps that satisfy the following conditions:

• (a) $g(X)\subseteq f(X)$;

• (b) f is continuous;

• (c) g commutes with f;

• (d) $F_{g(p),g(q)}(kx)\ge F_{f(p),f(q)}(x)$ for all $p,q\in X$, $x>0$ and for some $0<k<1$. Then f and g have a unique common fixed point.

Proof Let $\phi :R^{+}\to R^{+}$ be defined by

$$\phi (x)=kx,0<k<1.$$

(3.23)

From (b) and (d), we deduce that g is continuous. Thus, by (3.23), the same method as in Lemma 3.4 and simple calculations, the conditions of Theorem 3.7 are satisfied. Therefore f and g have a unique common fixed point.

In the next example, we shall show that all the conditions of Theorem 3.7 are satisfied, but condition (d) in Corollary 3.9 and condition (∗1) in Theorem A are not satisfied. □

Example 3.10 Let $k\in (0,1)$ be as in (c) of Theorem A and let $X=R$ be the set of reals with usual metric. Suppose that $f,g:X\to X$ and $\phi :R^{+}\to R^{+}$ are mappings defined as follows:

$$f(x)=kx,g(x)=\left(\frac{k+k^2}{2}\right)x$$

and

$$\phi (x)=\{\begin{array}{cc}(\frac{1+k}{2})x+(\frac{1-k}{2})x^2,\hfill & 0\le x\le k,\hfill \\ (\frac{1}{2}+k-\frac{k^2}{2})x,\hfill & k<x.\hfill \end{array}$$

Let the mappings $F_{p,q}$, H and t be the same as in Example 3.5. Then, from Lemma 2.7 and Lemma 3.4, $(X,F,t)$ is a complete Menger space and φ satisfies (∗3). Since

$$|g(p)-g(g)|\le \phi \left(|f(p)-f(g)|\right)$$

for all $p,q\in X$, we deduce that

$$F_{g(p),g(q)}(\phi (x))\ge F_{f(p),f(q)}(x)$$

for all $p,q\in X$ and $x>0$, which implies (∗2). By simple calculations, conditions (i), (ii), (iv) and (v) of Theorem 3.7 are satisfied. Thus all the conditions of Theorem 3.7 are satisfied. Hence f and g have a unique common fixed point 0. By hypotheses, there exist $p_1=-k\in R$, $q_1=0\in R$ and $x_1=\frac{3}{4}{k}^2+\frac{1}{4}k>0$ such that

$$F_{g(p_1),g(q_1)}(k{x}_1)<F_{f(p_1),f(q_1)}(x_1),$$

which implies that condition (d) of Corollary 3.9 is not satisfied. By hypotheses, there exist $p_2=-k^2\in R$ and $q_2=0\in R$ such that

$$|g(p_2)-g(q_2)|>k|f(p_2)-f(q_2)|,$$

which implies that condition (∗1) in Theorem A is not satisfied. Therefore Theorem 3.7 is a proper extension of Theorem A and Corollary 3.9.

A natural question arises from Example 3.5.

Question Would Theorem 3.7 remain true if (i)-(v) in Theorem 3.7 were substituted by some suitable conditions?