Abstract
In this paper, we state and prove Wardowski type fixed point theorems in metric space by using a modified generalized F-contraction maps. These theorems extend other well-known fundamental metrical fixed point theorems in the literature (Dung and Hang in Vietnam J. Math. 43:743-753, 2015 and Piri and Kumam in Fixed Point Theory Appl. 2014:210, 2014, etc.). Examples are provided to support the usability of our results.
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1 Introduction and preliminaries
One of the most well-known results in generalizations of the Banach contraction principle is the Wardowski fixed point theorem [3]. Before providing the Wardowski fixed point theorem, we recall that a self-map T on a metric space \((X,d)\) is said to be an F-contraction if there exist \(F\in\mathcal{F}\) and \(\tau\in (0, \infty)\) such that
where \(\mathcal{F}\) is the family of all functions \(F :(0,\infty) \to\mathbb{R} \) such that
-
(F1)
F is strictly increasing, i.e. for all \(x,y\in\mathbb {R}_{+}\) such that \(x < y\), \(F (x) < F (y)\);
-
(F2)
for each sequence \(\{\alpha_{n}\}_{n=1}^{\infty}\) of positive numbers, \(\lim_{n\to\infty}\alpha_{n}=0\) if and only if \(\lim_{n\to\infty}F(\alpha_{n})=-\infty\);
-
(F3)
there exists \(k\in(0,1)\) such that \(\lim_{\alpha\to 0^{+}}\alpha^{k}F(\alpha)=0\).
Obviously every F-contraction is necessarily continuous. The Wardowski fixed point theorem is given by the following theorem.
Theorem 1.1
[3]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be an F-contraction. Then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Later, Wardowski and Van Dung [4] have introduced the notion of an F-weak contraction and prove a fixed point theorem for F-weak contractions, which generalizes some results known from the literature. They introduced the concept of an F-weak contraction as follows.
Definition 1.2
Let \((X,d)\) be a metric space. A mapping \(T:X\rightarrow X\) is said to be an F-weak contraction on \((X,d)\) if there exist \(F\in\mathcal{F}\) and \(\tau>0\) such that, for all \(x,y\in X\),
where
By using the notion of F-weak contraction, Wardowski and Van Dung [4] have proved a fixed point theorem which generalizes the result of Wardowski as follows.
Theorem 1.3
[4]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be an F-weak contraction. If T or F is continuous, then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Recently, by adding values \(d(T^{2}x, x)\), \(d(T^{2}x, T x)\), \(d(T^{2}x, y)\), \(d(T^{2}x,T y)\) to (2), Dung and Hang [1] introduced the notion of a modified generalized F-contraction and proved a fixed point theorem for such maps. They generalized an F-weak contraction to a generalized F-contraction as follows.
Definition 1.4
Let \((X,d)\) be a metric space. A mapping \(T:X\rightarrow X\) is said to be a generalized F-contraction on \((X,d)\) if there exist \(F\in\mathcal{F}\) and \(\tau>0\) such that
where
By using the notion of a generalized F-contraction, Dung and Hang have proved the following fixed point theorem, which generalizes the result of Wardowski and Van Dung [4].
Theorem 1.5
[1]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be a generalized F-contraction. If T or F is continuous, then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Very recently, Piri and Kumam [2] described a large class of functions by replacing the condition (F3) in the definition of F-contraction introduced by Wardowski with the following one:
- (F3′):
-
F is continuous on \((0,\infty)\).
They denote by \(\mathfrak{F}\) the family of all functions \(F:\mathbb{R}_{+}\rightarrow \mathbb{R}\) which satisfy conditions (F1), (F2), and (F3′). Under this new set-up, Piri and Kumam proved some Wardowski and Suzuki type fixed point results in metric spaces as follows.
Theorem 1.6
[2]
Let T be a self-mapping of a complete metric space X into itself. Suppose there exist \(F\in\mathfrak{F}\) and \(\tau>0\) such that
Then T has a unique fixed point \(x^{*}\in X\) and for every \(x_{0}\in X\) the sequence \(\{T^{n}x_{0}\}_{n=1}^{\infty}\) converges to \(x^{*}\).
Theorem 1.7
[2]
Let T be a self-mapping of a complete metric space X into itself. Suppose there exist \(F\in\mathfrak{F}\) and \(\tau>0\) such that
Then T has a unique fixed point \(x^{*}\in X\) and for every \(x_{0}\in X\) the sequence \(\{T^{n}x_{0}\}_{n=1}^{\infty}\) converges to \(x^{*}\).
The aim of this paper is to introduce the modified generalized F-contractions, by combining the ideas of Dung and Hang [1], Piri and Kumam [2], Wardowski [3] and Wardowski and Van Dung [4] and give some fixed point result for these type mappings on complete metric space.
2 Main results
Let \(\mathfrak{F}_{G}\) denote the family of all functions \(F:\mathbb{R}_{+}\rightarrow\mathbb{R}\) which satisfy conditions (F1) and (F3′) and \(\mathcal{F}_{G}\) denote the family of all functions \(F:\mathbb{R}_{+}\rightarrow\mathbb{R}\) which satisfy conditions (F1) and (F3).
Definition 2.1
Let \((X,d)\) be a metric space and \(T:X\rightarrow X\) be a mapping. T is said to be modified generalized F-contraction of type (A) if there exist \(F\in\mathfrak{F}_{G}\) and \(\tau>0\) such that
where
Remark 2.2
Note that \(\mathfrak{F}\subseteq\mathfrak{F}_{W}\). Since, for \(\beta\in(0,\infty)\), the function \(F(\alpha)=\frac{-1}{\alpha+\beta}\) satisfies the conditions (F1) and (F3′) but it does not satisfy (F2), we have \(\mathfrak{F}\subsetneq\mathfrak{F}_{W}\).
Definition 2.3
Let \((X,d)\) be a metric space and \(T:X\rightarrow X\) be a mapping. T is said to be modified generalized F-contraction of type (B) if there exist \(F\in\mathcal{F}_{G}\) and \(\tau>0\) such that
Remark 2.4
Note that \(\mathcal{F}\subseteq\mathcal{F}_{W}\). Since, for \(\beta\in (0,\infty)\), the function \(F(\alpha)=\ln(\alpha+\beta)\) satisfies the conditions (F1) and (F3) but it does not satisfy (F2), we have \(\mathcal{F}\subsetneq\mathcal{F}_{W}\).
Remark 2.5
-
(1)
Every F-contraction is a modified generalized F-contraction.
-
(2)
Let T be a modified generalized F-contraction. From (3) for all \(x,y\in X\) with \(Tx\neq Ty\), we have
$$\begin{aligned} F\bigl(d(Tx,Ty)\bigr) < &\tau+F\bigl(d(Tx,Ty)\bigr) \\ \leq& F\biggl(\max \biggl\{ d(x,y),\frac{d(x,Ty)+d(y,Tx)}{2},\frac {d(T^{2}x,x)+d(T^{2}x,Ty)}{2}, \\ &d \bigl(T^{2}x,Tx\bigr),d\bigl(T^{2}x,y\bigr),d\bigl(T^{2}x,Ty\bigr)+d(x,Tx), \\ & d(Tx,y)+d(y,Ty)\biggr\} \biggr). \end{aligned}$$Then, by (F1), we get
$$\begin{aligned} d(Tx,Ty) < &\max \biggl\{ d(x,y),\frac{d(x,Ty)+d(y,Tx)}{2},\frac {d(T^{2}x,x)+d(T^{2}x,Ty)}{2},d \bigl(T^{2}x,Tx\bigr), \\ &d\bigl(T^{2}x,y\bigr),d\bigl(T^{2}x,Ty\bigr)+d(x,Tx), d(Tx,y)+d(y,Ty)\biggr\} , \end{aligned}$$for all \(x,y\in X\), \(Tx\neq Ty\).
The following examples show that the inverse implication of Remark 2.5(1) does not hold.
Example 2.6
Let \(X=[0,2]\) and define a metric d on X by \(d(x,y)=\mid x- y\mid\) and let \(T: X\to X\) be given by
Obviously, \((X,d)\) is complete metric space. Since T is not continuous, T is not an F-contraction. For \(x\in[0,2)\) and \(y=2\), we have
and
Therefore
So, by choosing \(F(\alpha)=\ln(\alpha)\) and \(\tau=\ln\frac{1}{5}\) we see that T is modified generalized F-contraction of type (A) and type (B).
Example 2.7
Let \(X=\{-2,-1,0,1,2\}\) and define a metric d on X by
Then \((X,d)\) is a complete metric space. Let \(T : X \to X\) be defined by
First observe that
Now we consider the following cases:
Case 1. Let \(x\in\{-2,-1,0\}\wedge y=1\), then
Case 2. Let \(x\in\{-2,-1,0\}\wedge y=2\), then
Case 3. Let \(x=1\wedge y=2\), then
In Case 1, we have
This proves that for all \(F\in\mathcal{F}\cup\mathfrak{F}\), T is not an F-weak contraction and generalized F-contraction. Since every F-contraction is an F-weak contraction and a generalized F-contraction, T is not an F-contraction. However, we see that
Hence, by choosing \(F(\alpha)=\ln(\alpha)\) and \(\tau=\ln\frac{1}{2}\) we see that T is modified generalized F-contraction of type (A) and type (B).
Theorem 2.8
Let \((X,d)\) be a complete metric space and \(T:X\rightarrow X\) be a modified generalized F-contraction of type (A). Then T has a unique fixed point \(x^{*}\in X\) and for every \(x_{0}\in X\) the sequence \(\{T^{n}x_{0}\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Proof
Let \(x_{0}\in X\). Put \(x_{n+1}=T^{n}x_{0}\) for all \(n\in\mathbb{N}\). If, there exists \(n\in\mathbb{N}\) such that \(x_{n+1}=x_{n}\), then \(Tx_{n}=x_{n}\). That is, \(x_{n}\) is a fixed point of T. Now, we suppose that \(x_{n+1}\neq x_{n}\) for all \(n\in\mathbb{N}\). Then \(d(x_{n+1}, x_{n})>0\) for all \(n\in\mathbb{N}\). It follows from (3) that, for all \(n\in\mathbb{N}\),
If there exists \(n\in\mathbb{N}\) such that \(\max\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}=d(x_{n},x_{n+1})\) then (4) becomes
Since \(\tau>0\), we get a contradiction. Therefore
Thus, from (4), we have
It follows from (5) and (F1) that
Therefore \(\{d(x_{n+1},x_{n})\}_{n\in\mathbb{N}}\) is a nonnegative decreasing sequence of real numbers, and hence
Now, we claim that \(\gamma=0\). Arguing by contradiction, we assume that \(\gamma>0\). Since \(\{d(x_{n+1},x_{n})\}_{n\in\mathbb{N}}\) is a nonnegative decreasing sequence, for every \(n \in\mathbb{N}\), we have
From (6) and (F1), we get
for all \(n \in\mathbb{N}\). Since \(F(\gamma)\in\mathbb{R}\) and \(\lim_{n\rightarrow\infty}[F(d(x_{0},x_{1}))-n \tau]=-\infty\), there exists \(n_{1}\in\mathbb{N}\) such that
It follows from (7) and (8) that
It is a contradiction. Therefore, we have
As in the proof of Theorem 2.1 in [2], we can prove that \(\{x_{n}\}_{n=1}^{\infty}\) is a Cauchy sequence. So by completeness of \((X,d)\), \(\{x_{n}\}_{n=1}^{\infty}\) converges to some point \(x^{*}\) in X. Therefore,
Finally, we will show that \(x^{*}=Tx^{*}\). We only have the following two cases:
-
(I)
\(\forall n\in\mathbb{N}\), \(\exists i_{n}\in\mathbb {N}\), \(i_{n}> i_{n-1}\), \(i_{0}=1\) and \(x_{i_{n}+1} =Tx^{*}\),
-
(II)
\(\exists n_{3}\in\mathbb{N}\), \(\forall n\geq n_{3}\), \(d(Tx_{n},Tx^{*})>0\).
In the first case, we have
In the second case from the assumption of Theorem 2.8, for all \(n\geq n_{3}\), we have
From (F3′), (10), and taking the limit as \(n\rightarrow\infty\) in (11), we obtain
This is a contradiction. Hence, \(x^{*}=Tx^{*}\). Now, let us to show that T has at most one fixed point. Indeed, if \(x^{*},y^{*}\in X\) are two distinct fixed points of T, that is, \(Tx^{*}=x^{*}\neq y^{*}=Ty^{*}\), then
It follows from (3) that
which is a contradiction. Therefore, the fixed point is unique. □
Theorem 2.9
Let \((X,d)\) be a complete metric space and \(T:X\rightarrow X\) be a continuous modified generalized F-contraction of type (B). Then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Proof
By using a similar method to that used in the proof of Theorem 2.8, we have
and
As in the proof of Theorem 2.1 in [3], we can prove that \(\{x_{n}\}_{n=1}^{\infty}\) is a Cauchy sequence. So, by completeness of \((X,d)\), \(\{x_{n}\}_{n=1}^{\infty}\) converges to some point \(x^{*}\in X\). Since T is continuous, we have
Again by using similar method as used in the proof of Theorem 2.8, we can prove that \(x^{*}\) is the unique fixed point of T. □
3 Some applications
Theorem 3.1
[2]
Let T be a self-mapping of a complete metric space X into itself. Suppose there exist \(F\in\mathfrak{F}\) and \(\tau>0\) such that
Then T has a unique fixed point \(x^{*}\in X\) and for every \(x_{0}\in X\) the sequence \(\{T^{n}x_{0}\}_{n=1}^{\infty}\) converges to \(x^{*}\).
Proof
Since
from (F1) and Theorem 2.8 the proof is complete. □
Theorem 3.2
[3]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be an F-contraction. Then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Proof
Since
So from (F1) and Theorem 2.9 the proof is complete. □
Theorem 3.3
[4]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be an F-weak contraction. If T or F is continuous, then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Proof
Since
if F is continuous, from (F1) and Theorem 2.8 the proof is complete. If T is continuous, from (F1) and Theorem 2.9 the proof is complete. □
Theorem 3.4
[1]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be a generalized F-contraction. If T or F is continuous, then T has a unique fixed point \(x^{*}\in X\) and for every \(x\in X\) the sequence \(\{T^{n}x\}_{n\in\mathbb{N}}\) converges to \(x^{*}\).
Proof
Since
if F is continuous, from (F1) and Theorem 2.8 the proof is complete. If T is continuous, from (F1) and Theorem 2.9 the proof is complete. □
Theorem 3.5
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be a function with the following property:
where α, β, and γ are nonnegative and satisfy \(\alpha+\beta+\gamma<1\). Then T has a unique fixed point.
Proof
From (12), we have
Then if \(d(Tx,Ty)>0\), we have
Therefore by taking \(F(\alpha)=\ln(\alpha)\) and \(\tau=\ln\frac{1}{\alpha+\beta+\gamma}\) in Theorem 2.8 or in Theorem 2.9 the proof is complete. □
Remark 3.6
Our theorems are extensions of the above theorems in the following aspects:
-
(1)
Theorem 2.8 gives all consequences of Theorem 2.1 of [2] without assumption (F2) used in its proof.
-
(2)
Theorem 2.9 gives all consequences of Theorem 2.1 of [3] without assumption (F2) used in its proof.
-
(3)
If in Theorem 3 of [1] F is continuous, Theorem 2.8 gives all consequences of Theorem 3 of [1] without assumptions (F2) and (F3) used in its proof.
-
(4)
If in Theorem 3 of [1] T is continuous, Theorem 2.9 gives all consequences of Theorem 3 of [1] without assumption (F2) used in its proof.
-
(5)
Because every F-weak contraction is a generalized F-contraction, (3) and (4) are also true for Theorem 2.4 of [4].
References
Dung, NV, Hang, VL: A fixed point theorem for generalized F-contractions on complete metric spaces. Vietnam J. Math. 43, 743-753 (2015)
Piri, H, Kumam, P: Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210 (2014). doi:10.1186/1687-1812-2014-210
Wardowski, D: Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)
Wardowski, D, Van Dung, N: Fixed points of F-weak contractions on complete metric spaces. Demonstr. Math. 1, 146-155 (2014)
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Piri, H., Kumam, P. Wardowski type fixed point theorems in complete metric spaces. Fixed Point Theory Appl 2016, 45 (2016). https://doi.org/10.1186/s13663-016-0529-0
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DOI: https://doi.org/10.1186/s13663-016-0529-0