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Fixed point theorems of generalized cyclic orbital MeirKeeler contractions
Fixed Point Theory and Applications volume 2013, Article number: 91 (2013)
Abstract
In this paper, we introduce two class of generalized cyclic orbital MeirKeeler contractions and we study the existence and uniqueness of fixed points for these mappings. Our results in this paper extend and generalize several existing fixedpoint theorems in the literature.
MSC:47H10, 54C60, 54H25, 55M20.
1 Introduction and preliminaries
Throughout this paper, by {\mathbb{R}}^{+}, we denote the set of all nonnegative numbers, while ℕ is the set of all natural numbers. It is well known and easy to prove that if (X,d) is a complete metric space, and if f:X\to X is continuous and f satisfies
then f has a fixed point in X. Using the above conclusion, Kirk, Srinivasan and Veeramani [1] proved the following fixedpoint theorem.
Theorem 1 [1]
Let A and B be two nonempty closed subsets of a complete metric space (X,d), and suppose f:A\cup B\to A\cup B satisfies

(i)
f(A)\subset B and f(B)\subset A,

(ii)
d(fx,fy)\le k\cdot d(x,y) for all x\in A, y\in B and k\in (0,1).
Then A\cap B is nonempty and f has a unique fixed point in A\cap B.
The following definitions and results will be needed in the sequel. Let A and B be two nonempty subsets of a metric space (X,d). A mapping f:A\cup B\to A\cup B is called a cyclic map if f(A)\subseteq B and f(B)\subseteq A. In 2010, Karpagam and Agrawal [2] introduced the notion of cyclic orbital contraction, and obtained a unique fixed point theorem for such a map.
Definition 1 [2]
Let A and B be nonempty subsets of a metric space (X,d), f:A\cup B\to A\cup B be a cyclic map such that for some x\in A, there exists a {\kappa}_{x}\in (0,1) such that
Then f is called a cyclic orbital contraction.
Theorem 2 [2]
Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let f:A\cup B\to A\cup B be a cyclic orbital contraction. Then f has a fixed point in A\cap B.
Further, many results dealing with cyclic contractions have appeared in the literature (see, e.g., [3–16]).
In 2012, Chen [17] introduced the below notion of cyclic orbital stronger MeirKeeler contraction, and obtained a unique fixedpoint theorem for such class of mappings.
Definition 2 [17]
Let (X,d) be a metric space. We call \psi :{\mathbb{R}}^{+}\to [0,1) a stronger MeirKeeler type mapping in X if the mapping ψ satisfies the following condition:
Definition 3 [17]
Let A and B be nonempty subsets of a metric space (X,d). Suppose f:A\cup B\to A\cup B is a cyclic map such that for some x\in A, there exists a stronger MeirKeeler type mapping {\psi}_{x}:{\mathbb{R}}^{+}\to [0,1) in X such that
for all n\in \mathbb{N} and y\in A. Then f is called a cyclic orbital stronger MeirKeeler {\psi}_{x}contraction.
Clearly, if f:A\cup B\to A\cup B is a cyclic orbital contraction, then f is a cyclic orbital stronger MeirKeeler {\psi}_{x}contraction, where {\psi}_{x}(t)={k}_{x} for all t\in {\mathbb{R}}^{+}.
Theorem 3 [17]
Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let {\psi}_{x}:{\mathbb{R}}^{+}\to [0,1) be a stronger MeirKeeler type mapping in X. Suppose f:A\cup B\to A\cup B is a cyclic orbital stronger MeirKeeler {\psi}_{x}contraction. Then A\cap B is nonempty and f has a unique fixed point in A\cap B.
Chen [17] also introduced the below notion of cyclic orbital weaker MeirKeeler contraction, and obtained a unique fixedpoint theorem for such class of mappings.
Definition 4 [17]
Let (X,d) be a metric space, and \psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}. Then ψ is called a weaker MeirKeeler type mapping in X, if the mapping ψ satisfies the following condition:
Definition 5 [17]
Let (X,d) be a metric space. We call f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} a ψmapping in X if the function f satisfies the following conditions:
({\psi}_{1}) f is a weaker MeirKeeler type mapping in X with f(0)=0;
({\psi}_{2})

(a)
if {lim}_{n\to \mathrm{\infty}}{t}_{n}=\gamma >0, then {lim}_{n\to \mathrm{\infty}}f({t}_{n})\le \gamma, and

(b)
if {lim}_{n\to \mathrm{\infty}}{t}_{n}=0, then {lim}_{n\to \mathrm{\infty}}f({t}_{n})=0;
({\psi}_{3}) {\{{f}^{n}(t)\}}_{n\in \mathbb{N}} is decreasing, for each t\in {\mathbb{R}}^{+}\mathrm{\setminus}\{0\}.
Definition 6 [17]
Let A and B be nonempty subsets of a metric space (X,d). Suppose f:A\cup B\to A\cup B is a cyclic map such that for some x\in A, there exists a ψmapping {\psi}_{x}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} in X such that
for all n\in \mathbb{N} and y\in A. Then f is called a cyclic orbital weaker MeirKeeler {\psi}_{x}contraction.
Theorem 4 [17]
Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let {\psi}_{x}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} be a ψmapping in X. Suppose f:A\cup B\to A\cup B is a cyclic orbital weaker MeirKeeler {\psi}_{x}contraction. Then A\cap B is nonempty and f has a unique fixed point in A\cap B.
2 Fixedpoint theorems (I)
In this section, we will introduce the class of generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction and we study the existence and uniqueness of fixed points for such mappings. Our results in this section extend and generalize several existing fixedpoint theorems in the literature, including Theorem 2 and Theorem 3.
In the sequel, we denote by Θ the class of functions \phi :{{\mathbb{R}}^{+}}^{5}\to {\mathbb{R}}^{+} satisfying the following conditions:
({\phi}_{1}) φ is a strictly increasing, continuous function in each coordinate;
({\phi}_{2}) for all t>0, \phi (t,t,t,0,2t)<t, \phi (t,t,t,2t,0)<t, \phi (t,0,0,t,t)<t, \phi (0,0,t,t,0)<t, and \phi (0,0,0,0,0)=0.
Example 1 Let \phi :{{\mathbb{R}}^{+}}^{5}\to {\mathbb{R}}^{+} denote
Then φ satisfies the above conditions ({\phi}_{1}) and ({\phi}_{2}).
We now denote the below notion of generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction.
Definition 7 Let A and B be nonempty subsets of a metric space (X,d). Suppose f:A\cup B\to A\cup B is a cyclic map such that for some x\in A, there exist a stronger MeirKeeler type mapping {\psi}_{x}:{\mathbb{R}}^{+}\to [0,1) in X and \phi \in \mathrm{\Theta} such that
where
for all n\in \mathbb{N} and y\in A. Then f is called a generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction.
Our main result is the following.
Theorem 5 Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let {\psi}_{x}:{\mathbb{R}}^{+}\to [0,1) be a stronger MeirKeeler type mapping in X and \phi \in \mathrm{\Theta}. Suppose f:A\cup B\to A\cup B is a generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction. Then A\cap B is nonempty and f has a unique fixed point in A\cap B.
Proof Since f:A\cup B\to A\cup B is a generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction and for x\in A, we have {f}^{2n}x\in A. Put y={f}^{2n}x, for n\in \mathbb{N}. Then we have that for each n\in \mathbb{N}
and by the conditions of the function φ, we get
and
Similarly, we put y={f}^{2n}x and for each n\in \mathbb{N}
and by the conditions of the function φ, we get
and
Using inequalities (2.1) and (2.2), we deduce that \{d({f}^{n}x,{f}^{n+1}x)\} is a decreasing sequence and hence it is convergent. Let {lim}_{n\to \mathrm{\infty}}d({f}^{n}x,{f}^{n+1}x)=\eta. Then there exists {\kappa}_{0}\in \mathbb{N} and \delta >0 such that for all n\ge {\kappa}_{0},
Taking into account the above inequality and the definition of stronger MeirKeeler type mapping {\psi}_{x} in X, corresponding to η use, there exists {\gamma}_{\eta}\in [0,1) such that
Put {n}_{0}=[\frac{{\kappa}_{0}+3}{2}], where [\frac{{\kappa}_{0}+3}{2}] is the integer part of \frac{{\kappa}_{0}+3}{2}. It follows from (2.1), (2.2) and (2.3) that we deduce that for all n\ge {n}_{0},
and
It follows from (2.4) and (2.5) that for each n\in \mathbb{N}\cup \{0\}
Since {\gamma}_{\eta}<1, we get
For m,n\in \mathbb{N} with m>n, we have
and hence d({f}^{n}x,{f}^{m}x)\to 0, since 0<{\gamma}_{\eta}<1. So, \{{f}^{n}x\} is a Cauchy sequence. Since (X,d) is a complete metric space, A and B are closed, \{{f}^{n}x\}\subset A\cup B, there exists \nu \in A\cup B such that {lim}_{n\to \mathrm{\infty}}{f}^{n}x=\nu. Now \{{f}^{2n}x\} is a sequence in A and \{{f}^{2n+1}x\} is a sequence in B, and also both converge to ν. Since A and B are closed, \nu \in A\cap B, and so A\cap B is nonempty. Next, we want to show that ν is a fixed point of f. Suppose that ν is not a fixed point of f. Then d(\nu ,f\nu )>0. Since {lim}_{n\to \mathrm{\infty}}d({f}^{2n1}x,\nu )=0 and
where
we obtain that
This leads to a contradiction. So, d(\nu ,f\nu )=0, that is, ν is a fixed point of f.
Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have \nu ,\mu \in A\cap B. Since f is a generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction, we have
and
where
It follows from (2.7), (2.8) and the condition ({\phi}_{2}) of the mapping φ that
This leads to a contradiction. Therefore, \nu =\mu, and so ν is the unique fixed point of f. □
We give the following example to illustrate Theorem 5.
Example 2 Let A=B=X={\mathbb{R}}^{+} and we define d:X\times X\to {\mathbb{R}}^{+} by
and let f:X\to X denote
We next define {\psi}_{x}:{\mathbb{R}}^{+}\to [0,1) by
and let \phi :{{\mathbb{R}}^{+}}^{5}\to {\mathbb{R}}^{+} denote
Then f is a generalized cyclic orbital stronger MeirKeeler ({\psi}_{x},\phi )contraction and 0 is the unique fixed point.
3 Fixedpoint theorems (II)
In this section, we will introduce the class of generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction and we study the existence and uniqueness of fixed points for such mappings.
In the sequel, we denote by Φ the class of functions \varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} satisfying the following conditions:
({\varphi}_{1}) ϕ is lower semicontinuous, and
({\varphi}_{2}) \varphi (0)=0 if and only if t=0.
Definition 8 Let A and B be nonempty subsets of a metric space (X,d). Suppose f:A\cup B\to A\cup B is a cyclic map such that for some x\in A, there exist a ψmapping {\psi}_{x}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} in X and \varphi \in \mathrm{\Phi} such that
Then f is called a generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction.
Our second main result is the following.
Theorem 6 Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let {\psi}_{x}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} be a ψmapping in X and \varphi \in \mathrm{\Phi}. Suppose f:A\cup B\to A\cup B is a generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction. Then A\cap B is nonempty and f has a unique fixed point in A\cap B.
Proof Since f:A\cup B\to A\cup B is a generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction and for x\in X, there exist a ψmapping {\psi}_{x}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} in X and \varphi \in \mathrm{\Phi} such that (3.1) is satisfied. Put y={f}^{2n}x for all n\in \mathbb{N}. Then we have that for each n\in \mathbb{N}
and
Generally, we have that for each n\in \mathbb{N}
and so we conclude that for each n\in \mathbb{N}
Since {\{{\psi}_{x}^{n}(d(x,fx))\}}_{n\in \mathbb{N}} is decreasing, it must converge to some \eta \ge 0. We claim that \eta =0. On the contrary, assume that \eta >0. Then by the definition of weaker MeirKeeler type mapping {\psi}_{x} in X, there exists \delta >0 such that for x,y\in X with \eta \le d(x,y)<\delta +\eta, there exists {n}_{0}\in \mathbb{N} such that {\psi}_{x}^{{n}_{0}}(d(x,y))<\eta. Since {lim}_{n\to \mathrm{\infty}}{\psi}_{x}^{n}(d(x,fx))=\eta, there exists {m}_{0}\in \mathbb{N} such that \eta \le {\psi}_{x}^{m}(d(x,fx))<\delta +\eta, for all m\ge {m}_{0}. Thus, we conclude that {\psi}_{x}^{{m}_{0}+{n}_{0}}(d({x}_{0},{x}_{1}))<\eta, and we get a contradiction. So, {lim}_{n\to \mathrm{\infty}}{\psi}_{x}^{n}(d(x,fx))=0, that is,
We now claim that \{{f}^{n}x\} is a Cauchy sequence. It is sufficient to show that \{{f}^{2n}x\} is a Cauchy sequence. Suppose \{{f}^{2n}x\} is not Cauchy. Then there exists \epsilon >0 such that for all k\in \mathbb{N}, there are {m}_{k},{n}_{k}\in \mathbb{N} with {m}_{k}>{n}_{k}\ge k satisfying:

(i)
d({f}^{2{m}_{k}}x,{f}^{2{n}_{k}})\ge \epsilon, and

(ii)
{m}_{k} is the smallest number greater than {n}_{k} such that the condition (i) holds.
Using (3.2), we have
Let k\to \mathrm{\infty}, we get
On the other hand, applying (3.1) with y={f}^{2{n}_{k}}x for all k\in \mathbb{N}, we get
Since for each k\in \mathbb{N}
and
taking k\to \mathrm{\infty} and using the inequalities (3.3), (3.5) and (3.6), we get
and
Taking into account the inequalities (3.4), (3.7) and (3.8), and by the definitions of the functions ϕ and {\psi}_{x} , we get
which implies that \epsilon =0. Thus, \{{f}^{n}x\} is a Cauchy sequence.
Since (X,d) is a complete metric space, A and B are closed, \{{f}^{n}x\}\subset A\cup B, there exists \nu \in A\cup B such that {lim}_{n\to \mathrm{\infty}}{f}^{n}x=\nu. Now \{{f}^{2n}x\} is a sequence in A and \{{f}^{2n+1}x\} is a sequence in B, and also both converge to ν. Since A and B are closed, \nu \in A\cap B, and so A\cap B is nonempty. On the other hand, since {lim}_{n\to \mathrm{\infty}}d({f}^{2n1}x,\nu )=0 and
taking n\to \mathrm{\infty},we obtain that
and hence d(\nu ,f\nu )=0, that is, ν is a fixed point of f.
Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have \nu ,\mu \in A\cap B. Since f is a generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction, we have
Letting n\to \mathrm{\infty}, and by the definitions of the functions ϕ and {\psi}_{x}, we obtain that
which implies that d(\nu ,\mu )=0. Therefore, \nu =\mu, and so ν is the unique fixed point of f. □
We give the following example to illustrate Theorem 6.
Example 3 Let A=B=X={\mathbb{R}}^{+} and we define d:X\times X\to {\mathbb{R}}^{+} by
Define f:X\to X by
and define \psi ,\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} by
Then f is a generalized cyclic orbital weaker MeirKeeler ({\psi}_{x},\varphi )contraction and 0 is the unique fixed point.
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Acknowledgements
This research was supported by the National Science Council of the Republic of China. The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.
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Chen, CM. Fixed point theorems of generalized cyclic orbital MeirKeeler contractions. Fixed Point Theory Appl 2013, 91 (2013). https://doi.org/10.1186/16871812201391
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DOI: https://doi.org/10.1186/16871812201391
Keywords
 fixedpoint theorem
 cyclic map
 generalized cyclic orbital MeirKeeler contraction