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Fixed point results for $(\alpha \psi ,\beta \phi )$contractive mappings for a generalized altering distance
 Maher Berzig^{1} and
 Erdal Karapınar^{2}Email author
https://doi.org/10.1186/168718122013205
© Berzig and Karapınar; licensee Springer 2013
 Received: 24 April 2013
 Accepted: 23 July 2013
 Published: 29 July 2013
Abstract
In this manuscript, we extend the concept of altering distance, and we introduce a new notion of $(\alpha \psi ,\beta \phi )$contractive mappings. We prove the existence and uniqueness of a fixed point for such mapping in the context of complete metric space. The presented theorems of this paper generalize, extend and improve some remarkable existing results in the literature. We also present several applications and consequences of our results.
Keywords
 Binary Relation
 Fixed Point Theorem
 Cauchy Sequence
 Contractive Mapping
 Monotone Property
1 Introduction and preliminaries
Fixed point theory is one of the core research areas in nonlinear functional analysis since it has a broad range of application potential in various fields such as engineering, economics, computer science, and many others. Banach contraction mapping principle [1] is considered to be the initial and fundamental result in this direction. Fixed point theory and hence the Banach contraction mapping principle have evidently attracted many prominent mathematicians due to their wide application potential. Consequently, the number of publications in this theory increases rapidly; we refer the reader to [2–19].
In this paper, by introducing a new notion of $(\alpha \psi ,\beta \phi )$contractive mappings, we aim to establish a more general result to collect/combine a number of existing results in the literature.
We start by recalling the notion of altering distance function introduced by Khan et al. [12] as follows.
Definition 1.1 A function $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is called an altering distance function if the following properties are satisfied:

$\psi (t)$ is continuous and nondecreasing.

$\psi (t)=0$ if and only if $t=0$.
Now, we present a definition, which will be useful later.
In the sequel, let ℕ denote the set of all nonnegative integers, let ℝ denote the set of all real numbers.
Example 1.1 Let $X=\mathbb{R}$ and a function $T:X\to X$ defined as $Tx={e}^{x}$.
Define the first binary relation ${\mathcal{R}}_{1}$ by $x{\mathcal{R}}_{1}y$ if and only if $\alpha (x,y)\le 1$, and define the second binary relation by $x{\mathcal{R}}_{2}y$ if and only if $\beta (x,y)\ge 1$. Then, we easily obtain that T is simultaneously ${\mathcal{R}}_{1}$preserving and ${\mathcal{R}}_{2}$preserving.
The following remark is a consequence of the previous definition.
 (i)
If ℛ is transitive, then it is Ntransitive for all $N\in \mathbb{N}$.
 (ii)
If ℛ is Ntransitive, then it is kNtransitive for all $k\in \mathbb{N}$.
2 Main results
Before we start the introduction of the concept $(\alpha \psi ,\beta \phi )$contractive mappings, we introduce the notion of a pair of generalized altering distance as follows:
 (a1)
ψ is continuous;
 (a2)
ψ is nondecreasing;
 (a3)
${lim}_{n\to \mathrm{\infty}}\phi ({t}_{n})=0\u27f9{lim}_{n\to \mathrm{\infty}}{t}_{n}=0$.
The condition (a3) was introduced by Popescu in [15] and Moradi and Farajzadeh in [14].
where $\alpha ,\beta :X\times X\to [0,+\mathrm{\infty})$.
In the sequel, the binary relations ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$ are defined as following.
Now we are ready to state our first main result.
 (i)
${\mathcal{R}}_{i}$ is Ntransitive for $i=1,2$;
 (ii)
T is ${\mathcal{R}}_{i}$preserving for $i=1,2$;
 (iii)
there exists ${x}_{0}\in X$ such that ${x}_{0}{\mathcal{R}}_{i}T{x}_{0}$ for $i=1,2$;
 (iv)
T is continuous.
Then, T has a fixed point, that is, there exists ${x}^{\ast}\in X$ such that $T{x}^{\ast}={x}^{\ast}$.
From the continuity of T, it follows that ${x}_{n+1}=T{x}_{n}\to T{x}^{\ast}$ as $n\to \mathrm{\infty}$. Due to the uniqueness of the limit, we derive that $T{x}^{\ast}={x}^{\ast}$, that is, ${x}^{\ast}$ is a fixed point of T. □
Theorem 2.2 In Theorem 2.1, if we replace the continuity of T by the $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$regularity of $(X,d)$, then the conclusion of Theorem 2.1 holds.
Letting $k\to \mathrm{\infty}$ in the above inequality, we get $d({x}^{\ast},T{x}^{\ast})=0$, that is, ${x}^{\ast}=T{x}^{\ast}$. □
Theorem 2.3 Adding to the hypotheses of Theorem 2.1 (respectively, Theorem 2.2) that X is $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$directed, we obtain uniqueness of the fixed point of T.
Using (23) and (24), the uniqueness of the limit gives us ${x}^{\ast}={y}^{\ast}$. □
3 Some corollaries
In this section, we derive new results from the previous theorems.
3.1 Coupled fixed point results in complete metric spaces
Let us recall the definition of a coupled fixed point introduced by Guo and Lakshmikantham in [5].
Definition 3.1 (Guo and Lakshmikantham [5])
where $\alpha ,\beta :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty})$.
In this section, we define two binary relations ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ as follows.
We have the following result.
 (i)
${\mathcal{S}}_{i}$ is Ntransitive for $i=1,2$ ($N>0$);
 (ii)For all $(x,y),(u,v)\in X\times X$, we have$(x,y){\mathcal{S}}_{i}(u,v)\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}(F(x,y),F(y,x)){\mathcal{S}}_{i}(F(u,v),F(v,u))\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.25em}{0ex}}i=1,2;$
 (iii)There exists $({x}_{0},{y}_{0})\in X\times X$ such that$({x}_{0},{y}_{0}){\mathcal{S}}_{i}(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})),\phantom{\rule{2em}{0ex}}(F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})){\mathcal{S}}_{i}({y}_{0},{x}_{0})\phantom{\rule{1em}{0ex}}\mathit{\text{for}}\phantom{\rule{0.25em}{0ex}}i=1,2;$
 (iv)
F is continuous, or $(X\times X,d)$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$biregular.
Then, F has a coupled fixed point $({x}^{\ast},{y}^{\ast})\in X\times X$. Moreover, if $X\times X$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$bidirected, then we have the uniqueness of the coupled fixed point.
and $T:Y\to Y$ is given by (25). We shall prove that T is $(a\psi ,b\phi )$contractive mapping.
Then our claim holds.
Let $\xi =({\xi}_{1},{\xi}_{2}),\eta =({\eta}_{1},{\eta}_{2})\in Y$ such that $a(\xi ,\eta )\le 1$ and $b(\xi ,\eta )\ge 1$. Using condition (ii), we obtain immediately that $a(T\xi ,T\eta )\le 1$ and $b(T\xi ,T\eta )\ge 1$. Then T is ${\mathcal{R}}_{j}$preserving for $j=1,2$. Moreover, from condition (iii), we know that there exists $({x}_{0},{y}_{0})\in Y$ such that $({x}_{0},{y}_{0}){\mathcal{R}}_{j}T({x}_{0},{y}_{0})$ for $j=1,2$. If F is continuous, then T also is continuous. Then all the hypotheses of Theorem 2.1 are satisfied. If $(X\times X,d)$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$biregular, then we easily have that $(X\times X,d)$ is $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$regular. Hence, Theorem 2.2 yields the result. We deduce the existence of a fixed point of T that gives us from (25) the existence of a coupled fixed point of F. Now, since $X\times X$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$bidirected, one can easily derive that $X\times X$ is $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$directed by regarding Lemma 3.1 and Definition 3.5. Finally, by using Theorem 2.3, we obtain the uniqueness of the fixed point of T, that is, the uniqueness of the coupled fixed point of F. □
3.2 Fixed point results on metric spaces endowed with Ntransitive binary relation
In [18], Samet and Turinci established fixed point results for contractive mappings on metric spaces, endowed with an amorphous arbitrary binary relation. Very recently, this work has been extended by Berzig in [2] to study the coincidence and common fixed points.
In this section, we establish a fixed point theorem on metric space endowed with Ntransitive binary relation .
 (i)
 (ii)
 (iii)
there exists ${x}_{0}\in X$ such that ${x}_{0}\mathcal{S}T{x}_{0}$;
 (iv)
Then T has a fixed point. Moreover, if X is directed, we have the uniqueness of the fixed point.
Next, by using (28), (29) and Definition 2.3, the conclusion follows directly from Theorems 2.1, 2.2 and 2.3. □
3.3 Fixed point results for cyclic contractive mappings
In [13], Kirk et al. have generalized the Banach contraction principle. They obtained a new fixed point results for cyclic contractive mappings.
Theorem 3.1 (Kirk et al. [13])
 (i)
$T({A}_{i})\subseteq {A}_{i+1}$ for all $i\in \{1,\dots ,N\}$ with ${A}_{N+1}:={A}_{1}$;
 (ii)there exists $k\in (0,1)$ such that$d(Tx,Ty)\le kd(x,y).$
Then T has a unique fixed point in ${\bigcap}_{i=1}^{N}{A}_{i}$.
Let us define the binary relations ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$.
Now, based on Theorem 2.2, we will derive a more general result for cyclic mappings.
 (i)
$T({A}_{i})\subseteq {A}_{i+1}$ for all $i\in \{1,\dots ,N\}$ with ${A}_{N+1}:={A}_{1}$;
 (ii)there exist two altering distance functions ψ and φ such that$\psi (d(Tx,Ty))\le \psi (d(x,y))\phi (d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}(x,y)\in {A}_{i}\times {A}_{i+1}\phantom{\rule{0.25em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}i\in \{1,\dots ,N\}.$
Then T has a unique fixed point in ${\bigcap}_{i=1}^{N}{A}_{i}$.
Proof Let $Y:={\bigcup}_{i=1}^{N}{A}_{i}$. For all $i\in \{1,\dots ,N\}$, we have by assumption that each ${A}_{i}$ is nonempty closed subset of the complete metric space X, which implies that $(Y,d)$ is complete.
Hence, Definition 2.3 is equivalent to Definition 3.6.
which implies that $({x}_{0},{x}_{N+1})\in {A}_{i}\times {A}_{i+1}\subseteq Y$. Hence, we obtain $\alpha ({x}_{0},{x}_{N+1})\le 1$ and $\beta ({x}_{0},{x}_{N+1})\ge 1$, that is, ${x}_{0}{\mathcal{R}}_{1}{x}_{N+1}$ and ${x}_{0}{\mathcal{R}}_{2}{x}_{N+1}$, which implies that ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$ are Ntransitive.
for all $x,y\in Y$. Thus, T is $(\alpha \psi ,\beta \phi )$contractive mapping.
We claim next that T is ${\mathcal{R}}_{1}$preserving and ${\mathcal{R}}_{2}$preserving. Indeed, let $x,y\in Y$ such that $x{\mathcal{R}}_{1}y$ and $x{\mathcal{R}}_{2}y$, that is, $\alpha (x,y)\le 1$ and $\beta (x,y)\ge 1$; hence, there exists $i\in \{1,\dots ,N\}$ such that $x\in {A}_{i}$, $y\in {A}_{i+1}$. Thus, $(Tx,Ty)\in {A}_{i+1}\times {A}_{i+2}\subseteq \mathrm{\Gamma}$, then $\alpha (Tx,Ty)\le 1$ and $\beta (Tx,Ty)\ge 1$, that is, $Tx{\mathcal{R}}_{1}Ty$ and $Tx{\mathcal{R}}_{2}Ty$. Hence, our claim holds.
Also, from (i), for any ${x}_{0}\in {A}_{i}$ for all $i\in \{1,\dots ,N\}$, we have $({x}_{0},T{x}_{0})\in {A}_{i}\times {A}_{i+1}$, which implies that $\alpha ({x}_{0},T{x}_{0})\le 1$ and $\beta ({x}_{0},T{x}_{0})\ge 1$, that is, ${x}_{0}{\mathcal{R}}_{1}T{x}_{0}$ and ${x}_{0}{\mathcal{R}}_{2}T{x}_{0}$.
hence $\alpha ({x}_{n(k)},x)\le 1$ and $\beta ({x}_{n(k)},x)\ge 1$ for all k, that is, ${x}_{n(k)}{\mathcal{R}}_{1}x$ and ${x}_{n(k)}{\mathcal{R}}_{2}x$, which proves our claim.
Hence, all the hypotheses of Theorem 2.2 are satisfied on $(Y,d)$, and we deduce that T has a fixed point ${x}^{\ast}$ in Y. Since ${x}^{\ast}\in {A}_{i}$ for some $i\in \{1,\dots ,N\}$ and ${x}^{\ast}=T{x}^{\ast}\in {A}_{i+1}$ for all $i\in \{1,\dots ,N\}$, then ${x}^{\ast}\in {\bigcap}_{i=1}^{N}{A}_{i}$.
Moreover, it is easy to check that X is $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$directed. Indeed, let $x,y\in Y$ with $x\in {A}_{i}$, $y\in {A}_{j}$, $i,j\in \{1,\dots ,N\}$. For $z={x}^{\ast}\in Y$, we have $((\alpha (x,z)\le 1)\wedge (\alpha (y,z)\le 1))$ and $((\beta (x,z)\ge 1)\wedge (\beta (y,z)\ge 1))$. Thus, X is $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$directed.
Finally, the uniqueness follows by Theorem 2.3. □
4 Related fixed point theorems
In this section, we show that many existing results in the literature can be deduced from our results.
4.1 Classical fixed point results
Corollary 4.1 (Dutta and Choudhury [4])
where ψ and φ are altering distance functions. Then T has a unique fixed point.
Proof Let $\alpha ,\beta :X\times X\to [0,+\mathrm{\infty})$ be the mapping defined by $\alpha (x,y)=\beta (x,y)=1$ for all $x,y\in X$. Then T is $(\alpha \psi ,\beta \phi )$contractive mappings. It is easy to show that all the hypotheses of Theorems 2.1 and 2.2 are satisfied. Consequently, T has a unique fixed point. □
Corollary 4.2 (Rhoades [17])
where φ is an altering distance functions. Then T has a unique fixed point.
Proof Following the lines of the proof of Corollary 4.1, by taking $\psi (t)=t$, we get the desired result. □
4.2 Fixed point results in partially ordered metric spaces
We start by defining the binary relations ${\mathcal{R}}_{i}$ for $i=1,2$ and the concept of ≤directed.
 1.We define two binary relations ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$ on X by$x,y\in X:x{\mathcal{R}}_{i}y\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}x\le y\phantom{\rule{1em}{0ex}}\text{for}i=1,2.$
 2.
We say that X is ≤directed if for all $x,y\in X$ there exists a $z\in X$ such that $x\le z$ and $y\le z$.
Corollary 4.3 (Harjani and Sadarangani [8])
 (i)
T is a nondecreasing mapping;
 (ii)
there exists ${x}_{0}\in X$ with ${x}_{0}\le T{x}_{0}$;
 (iii)
T is continuous or,
(iii′) for every sequence $\{{x}_{n}\}$ in X such that ${x}_{n}\to x\in X$, and $\{{x}_{n}\}$ is a nondecreasing sequence, there exists a subsequence $\{{x}_{n(k)}\}$ such that ${x}_{n(k)}\le x$ for all $k\ge 0$.
Then T has a fixed point. Moreover, if X is ≤directed, we have the uniqueness of the fixed point.
In case $x\nleqq y$, the functions α and β are well defined, because the altering functions $\phi (d(x,y))$ and $\psi (d(x,y))$ are null only, and only if $d(x,y)=0$, that is, $x=y$ which is not the case.
We can verify easily that ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{2}$ are 1transitive.
hence, our claim holds.
Thus T is ${\mathcal{R}}_{i}$preserving for $i=1,2$. Now, if condition (iii) is satisfied, that is, T is continuous, the existence of a fixed point follows from Theorem 2.1. Suppose now, that the condition (iii′) is satisfied, and let $\{{x}_{n}\}$ be a nondecreasing sequence in X, that is, $\alpha ({x}_{n},{x}_{n+1})\le 1$ and $\beta ({x}_{n},{x}_{n+1})\ge 1$ for all n. Suppose also that ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$. From (iii′), there exists a subsequence $\{{x}_{n(k)}\}$ such that ${x}_{n(k)}\le x$ for all k. This implies from the definition of α and β that $\alpha ({x}_{n(k)},x)\le 1$ and $\beta ({x}_{n(k)},x)\ge 1$ for all k, which implies ${x}_{n(k)}{\mathcal{R}}_{i}x$ for $i=1,2$ and for all k. In this case, the existence of a fixed point follows from Theorem 2.2.
To show the uniqueness, suppose that X is ≤directed, that is, for all $x,y\in X$ there exists a $z\in X$ such that $x\le z$ and $y\le z$, which implies from the definition of α and β that $(\alpha (x,z)\le 1)\wedge (\alpha (y,z)\le 1)$ and $(\beta (x,z)\ge 1)\wedge (\beta (y,z)\ge 1)$. Hence, Theorem 2.3 gives us the uniqueness of this fixed point. □
4.3 Coupled fixed point theorems
Next, in order to prove a coupled fixed point results in partially ordered set, we need to define an order relation on the set $X\times X$.
Corollary 4.4 (Harjani et al. [6])
 (i)
F is a mixed monotone mapping;
 (ii)
there exists ${x}_{0},{y}_{0}\in X$ with $({x}_{0},{y}_{0})\u2aaf(F{x}_{0},F{y}_{0})$;
 (iii)
F is continuous or $(X\times X,d)$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$biregular.
Then F has a coupled fixed point $({x}^{\ast},{y}^{\ast})\in X\times X$. Moreover, if $X\times X$ is $({\mathcal{S}}_{1},{\mathcal{S}}_{2})$bidirected, we have the uniqueness of the fixed point.
Proof The conclusions then follows directly from Corollary 3.1. □
4.4 Fixed point results for cyclic contractive mappings
In this section, we will derive from our results the fixed point theorem of Karapınar and Sadarangani [11] for cyclic weak $(\phi \psi )$contractive mappings.
Definition 4.3 (Păcurar and Rus [16])
 1.
${X}_{i}$, $i=1,\dots ,m$ are nonempty sets;
 2.
$T({X}_{1})\subset {X}_{2},\dots ,T({X}_{m1})\subset {X}_{m}$, $T({X}_{m})\subset {X}_{1}$.
Definition 4.4 (Karapınar and Sadarangani [10])
 1.
${\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of Y with respect to T, and
 2.$\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is an altering distance function such that$\psi (d(Tx,Ty))\le \psi (d(x,y))\phi (d(x,y))$(30)
for any $x\in {A}_{i}$, $y\in {A}_{i+1}$, where ${A}_{m+1}:={A}_{1}$.
Corollary 4.5 (Karapınar and Sadarangani [10])
Let $(X,d)$ be a complete metric space, let m be a positive integer, let ${A}_{1},{A}_{2},\dots ,{A}_{m}$ be closed nonempty subsets of X and let $Y={\bigcup}_{i=1}^{m}{A}_{i}$. Suppose that $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is an altering distance function, and T is a cyclic weak φcontraction, where $Y={\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of Y with respect to T. Then, T has a unique fixed point $z\in {\bigcap}_{i=1}^{m}{A}_{i}$.
Proof The proof follows immediately from Corollary 3.3. □
Declarations
Acknowledgements
The authors thank Professor MirceaDan Rus for his remarkable comments, suggestion and ideas that helped to improve this paper.
Authors’ Affiliations
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