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Fixed point theorems of generalized cyclic orbital Meir-Keeler contractions

Abstract

In this paper, we introduce two class of generalized cyclic orbital Meir-Keeler contractions and we study the existence and uniqueness of fixed points for these mappings. Our results in this paper extend and generalize several existing fixed-point theorems in the literature.

MSC:47H10, 54C60, 54H25, 55M20.

1 Introduction and preliminaries

Throughout this paper, by R + , we denote the set of all non-negative numbers, while 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:XX is continuous and f satisfies

d ( f x , f 2 x ) kd(x,fx),for all xX and k(0,1),

then f has a fixed point in X. Using the above conclusion, Kirk, Srinivasan and Veeramani [1] proved the following fixed-point theorem.

Theorem 1 [1]

Let A and B be two nonempty closed subsets of a complete metric space (X,d), and suppose f:ABAB satisfies

  1. (i)

    f(A)B and f(B)A,

  2. (ii)

    d(fx,fy)kd(x,y) for all xA, yB and k(0,1).

Then AB is nonempty and f has a unique fixed point in AB.

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:ABAB is called a cyclic map if f(A)B and f(B)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:ABAB be a cyclic map such that for some xA, there exists a κ x (0,1) such that

d ( f 2 n x , f y ) κ x d ( f 2 n 1 x , y ) ,nN,yA.

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:ABAB be a cyclic orbital contraction. Then f has a fixed point in AB.

Further, many results dealing with cyclic contractions have appeared in the literature (see, e.g., [316]).

In 2012, Chen [17] introduced the below notion of cyclic orbital stronger Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

Definition 2 [17]

Let (X,d) be a metric space. We call ψ: R + [0,1) a stronger Meir-Keeler type mapping in X if the mapping ψ satisfies the following condition:

η>0,δ>0, γ η [0,1),x,yX ( η d ( x , y ) < δ + η ψ ( d ( x , y ) ) < γ η ) .

Definition 3 [17]

Let A and B be nonempty subsets of a metric space (X,d). Suppose f:ABAB is a cyclic map such that for some xA, there exists a stronger Meir-Keeler type mapping ψ x : R + [0,1) in X such that

d ( f 2 n x , f y ) ψ x ( d ( f 2 n 1 x , y ) ) d ( f 2 n 1 x , y ) ,

for all nN and yA. Then f is called a cyclic orbital stronger Meir-Keeler ψ x -contraction.

Clearly, if f:ABAB is a cyclic orbital contraction, then f is a cyclic orbital stronger Meir-Keeler ψ x -contraction, where ψ x (t)= k x for all t R + .

Theorem 3 [17]

Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let ψ x : R + [0,1) be a stronger Meir-Keeler type mapping in X. Suppose f:ABAB is a cyclic orbital stronger Meir-Keeler ψ x -contraction. Then AB is nonempty and f has a unique fixed point in AB.

Chen [17] also introduced the below notion of cyclic orbital weaker Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

Definition 4 [17]

Let (X,d) be a metric space, and ψ: R + R + . Then ψ is called a weaker Meir-Keeler type mapping in X, if the mapping ψ satisfies the following condition:

η>0,δ>0,x,yX ( η d ( x , y ) < δ + η n 0 N ψ n 0 ( d ( x , y ) ) < η ) .

Definition 5 [17]

Let (X,d) be a metric space. We call f: R + R + a ψ-mapping in X if the function f satisfies the following conditions:

( ψ 1 ) f is a weaker Meir-Keeler type mapping in X with f(0)=0;

( ψ 2 )

  1. (a)

    if lim n t n =γ>0, then lim n f( t n )γ, and

  2. (b)

    if lim n t n =0, then lim n f( t n )=0;

( ψ 3 ) { f n ( t ) } n N is decreasing, for each t R + {0}.

Definition 6 [17]

Let A and B be nonempty subsets of a metric space (X,d). Suppose f:ABAB is a cyclic map such that for some xA, there exists a ψ-mapping ψ x : R + R + in X such that

d ( f 2 n x , f y ) ψ x ( d ( f 2 n 1 x , y ) ) ,

for all nN and yA. Then f is called a cyclic orbital weaker Meir-Keeler ψ x -contraction.

Theorem 4 [17]

Let A and B be two nonempty closed subsets of a complete metric space (X,d), and let ψ x : R + R + be a ψ-mapping in X. Suppose f:ABAB is a cyclic orbital weaker Meir-Keeler ψ x -contraction. Then AB is nonempty and f has a unique fixed point in AB.

2 Fixed-point theorems (I)

In this section, we will introduce the class of generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction and we study the existence and uniqueness of fixed points for such mappings. Our results in this section extend and generalize several existing fixed-point theorems in the literature, including Theorem 2 and Theorem 3.

In the sequel, we denote by Θ the class of functions φ: R + 5 R + satisfying the following conditions:

( φ 1 ) φ is a strictly increasing, continuous function in each coordinate;

( φ 2 ) for all t>0, φ(t,t,t,0,2t)<t, φ(t,t,t,2t,0)<t, φ(t,0,0,t,t)<t, φ(0,0,t,t,0)<t, and φ(0,0,0,0,0)=0.

Example 1 Let φ: R + 5 R + denote

φ( t 1 , t 2 , t 3 , t 4 , t 5 )= 2 3 max { t 1 , t 2 , t 3 , 1 2 t 4 , 1 2 t 5 } .

Then φ satisfies the above conditions ( φ 1 ) and ( φ 2 ).

We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction.

Definition 7 Let A and B be nonempty subsets of a metric space (X,d). Suppose f:ABAB is a cyclic map such that for some xA, there exist a stronger Meir-Keeler type mapping ψ x : R + [0,1) in X and φΘ such that

d ( f 2 n x , f y ) ψ x ( d ( f 2 n 1 x , y ) ) θ,

where

θ=φ ( d ( f 2 n 1 x , y ) , d ( f 2 n 1 x , f 2 n x ) , d ( f y , y ) , d ( f 2 n 1 x , f y ) , d ( f 2 n x , y ) )

for all nN and yA. Then f is called a generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-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 ψ x : R + [0,1) be a stronger Meir-Keeler type mapping in X and φΘ. Suppose f:ABAB is a generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction. Then AB is nonempty and f has a unique fixed point in AB.

Proof Since f:ABAB is a generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction and for xA, we have f 2 n xA. Put y= f 2 n x, for nN. Then we have that for each nN

and by the conditions of the function φ, we get

θ<d ( f 2 n 1 x , f 2 n x ) ,

and

d ( f 2 n x , f 2 n + 1 x ) < ψ x ( d ( f 2 n 1 x , f 2 n x ) ) d ( f 2 n 1 x , f 2 n x ) d ( f 2 n 1 x , f 2 n x ) .
(2.1)

Similarly, we put y= f 2 n x and for each nN

and by the conditions of the function φ, we get

θ<d ( f 2 n x , f 2 n + 1 x ) ,

and

d ( f 2 n + 1 x , f 2 n + 2 x ) < ψ x ( d ( f 2 n + 1 x , f 2 n x ) ) d ( f 2 n x , f 2 n + 1 x ) d ( f 2 n x , f 2 n + 1 x ) .
(2.2)

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 d( f n x, f n + 1 x)=η. Then there exists κ 0 N and δ>0 such that for all n κ 0 ,

ηd ( f n x , f n + 1 x ) <η+δ.

Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping ψ x in X, corresponding to η use, there exists γ η [0,1) such that

ψ x ( d ( f n x , f n + 1 x ) ) < γ η for all n κ 0 .
(2.3)

Put n 0 =[ κ 0 + 3 2 ], where [ κ 0 + 3 2 ] is the integer part of κ 0 + 3 2 . It follows from (2.1), (2.2) and (2.3) that we deduce that for all n n 0 ,

d ( f 2 n x , f 2 n + 1 x ) < ψ x ( d ( f 2 n 1 x , f 2 n x ) ) d ( f 2 n 1 x , f 2 n x ) < γ η d ( f 2 n 1 x , f 2 n x ) ,
(2.4)

and

d ( f 2 n + 1 x , f 2 n + 2 x ) < ψ x ( d ( f 2 n + 1 x , f 2 n x ) ) d ( f 2 n x , f 2 n + 1 x ) < γ η d ( f 2 n x , f 2 n + 1 x ) .
(2.5)

It follows from (2.4) and (2.5) that for each nN{0}

d ( f 2 n 0 + n x , f 2 n 0 + n + 1 x ) < γ η n d ( f 2 n 0 1 x , f 2 n 0 x ) .
(2.6)

Since γ η <1, we get

lim n d ( f 2 n 0 + n x , f 2 n 0 + n + 1 x ) =0.

For m,nN with m>n, we have

d ( f 2 n 0 + n x , f 2 n 0 + m x ) i = n m 1 d ( f 2 n 0 + i x , f 2 n 0 + i + 1 x ) < γ η m 1 1 γ η d ( f 2 n 0 x , f 2 n 0 + 1 x ) ,

and hence d( f n x, f m x)0, since 0< γ η <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}AB, there exists νAB such that lim n f n x=ν. Now { f 2 n x} is a sequence in A and { f 2 n + 1 x} is a sequence in B, and also both converge to ν. Since A and B are closed, νAB, and so AB 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(ν,fν)>0. Since lim n d( f 2 n 1 x,ν)=0 and

d ( f 2 n x , f ν ) ψ x ( d ( f 2 n 1 x , ν ) ) θ,

where

θ=φ ( d ( f 2 n 1 x , ν ) , d ( f 2 n 1 x , f 2 n x ) , d ( f ν , ν ) , d ( f 2 n 1 x , f ν ) , d ( f 2 n x , ν ) ) ,

we obtain that

d ( ν , f ν ) = lim n d ( f 2 n x , f ν ) γ η φ ( d ( ν , ν ) , d ( ν , ν ) , d ( f ν , ν ) , d ( ν , f ν ) , d ( ν , ν ) ) φ ( 0 , 0 , d ( ν , f ν ) , d ( ν , f ν ) , 0 ) < d ( ν , f ν ) .

This leads to a contradiction. So, d(ν,fν)=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 ν,μAB. Since f is a generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction, we have

d(ν,μ)=d(ν,fμ)= lim n d ( f 2 n x , f μ ) ,
(2.7)

and

d ( f 2 n x , f μ ) ψ x ( d ( f 2 n 1 x , μ ) ) θ< γ η θ,
(2.8)

where

θ=φ ( d ( f 2 n 1 x , μ ) , d ( f 2 n 1 x , f 2 n x ) , d ( f μ , μ ) , d ( f 2 n 1 x , f μ ) , d ( f 2 n x , μ ) ) .

It follows from (2.7), (2.8) and the condition ( φ 2 ) of the mapping φ that

d ( ν , μ ) < γ η φ ( d ( ν , μ ) , d ( ν , ν ) , d ( f μ , μ ) , d ( ν , f μ ) , d ( ν , μ ) ) φ ( d ( ν , μ ) , 0 , 0 , d ( ν , μ ) , d ( ν , μ ) ) < d ( ν , μ ) .

This leads to a contradiction. Therefore, ν=μ, 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= R + and we define d:X×X R + by

d(x,y)=|xy|,for x,yX,

and let f:XX denote

f(x)={ 0 , if  0 x < 1 ; 1 16 , if  x 1 .

We next define ψ x : R + [0,1) by

ψ x (t)={ 1 3 , if  0 t 1 ; t t + 1 , if  t > 1 ,

and let φ: R + 5 R + denote

φ( t 1 , t 2 , t 3 , t 4 , t 5 )= 1 2 max { t 1 , t 2 , t 3 , 1 2 t 4 , 1 2 t 5 } .

Then f is a generalized cyclic orbital stronger Meir-Keeler ( ψ x ,φ)-contraction and 0 is the unique fixed point.

3 Fixed-point theorems (II)

In this section, we will introduce the class of generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-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 ϕ: R + R + satisfying the following conditions:

( ϕ 1 ) ϕ is lower semi-continuous, and

( ϕ 2 ) ϕ(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:ABAB is a cyclic map such that for some xA, there exist a ψ-mapping ψ x : R + R + in X and ϕΦ such that

d ( f 2 n x , f y ) ψ x ( d ( f 2 n 1 x , y ) ) ϕ ( d ( f 2 n 1 x , y ) ) ,nN,yA.
(3.1)

Then f is called a generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-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 ψ x : R + R + be a ψ-mapping in X and ϕΦ. Suppose f:ABAB is a generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-contraction. Then AB is nonempty and f has a unique fixed point in AB.

Proof Since f:ABAB is a generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-contraction and for xX, there exist a ψ-mapping ψ x : R + R + in X and ϕΦ such that (3.1) is satisfied. Put y= f 2 n x for all nN. Then we have that for each nN

d ( f 2 n x , f 2 n + 1 x ) ψ x ( d ( f 2 n 1 x , f 2 n x ) ) ϕ ( d ( f 2 n 1 x , f 2 n x ) ) ψ x ( d ( f 2 n 1 x , f 2 n x ) ) ,

and

d ( f 2 n + 1 x , f 2 n + 2 x ) = d ( f 2 n + 2 x , f 2 n + 1 x ) ψ x ( d ( f 2 n + 1 x , f 2 n x ) ) ϕ ( d ( f 2 n + 1 x , f 2 n x ) ) ψ x ( d ( f 2 n + 1 x , f 2 n x ) ) .

Generally, we have that for each nN

d ( f n x , f n + 1 x ) ψ x ( d ( f n 1 x , f n x ) ) ,

and so we conclude that for each nN

d ( f n x , f n + 1 x ) ψ x ( d ( f n 1 x , f n x ) ) ψ x 2 ( d ( f n 2 x , f n 1 x ) ) ψ x n ( d ( x , f x ) ) .

Since { ψ x n ( d ( x , f x ) ) } n N is decreasing, it must converge to some η0. We claim that η=0. On the contrary, assume that η>0. Then by the definition of weaker Meir-Keeler type mapping ψ x in X, there exists δ>0 such that for x,yX with ηd(x,y)<δ+η, there exists n 0 N such that ψ x n 0 (d(x,y))<η. Since lim n ψ x n (d(x,fx))=η, there exists m 0 N such that η ψ x m (d(x,fx))<δ+η, for all m m 0 . Thus, we conclude that ψ x m 0 + n 0 (d( x 0 , x 1 ))<η, and we get a contradiction. So, lim n ψ x n (d(x,fx))=0, that is,

lim n d ( f n x , f n + 1 x ) =0.
(3.2)

We now claim that { f n x} is a Cauchy sequence. It is sufficient to show that { f 2 n x} is a Cauchy sequence. Suppose { f 2 n x} is not Cauchy. Then there exists ε>0 such that for all kN, there are m k , n k N with m k > n k k satisfying:

  1. (i)

    d( f 2 m k x, f 2 n k )ε, and

  2. (ii)

    m k is the smallest number greater than n k such that the condition (i) holds.

Using (3.2), we have

ε d ( f 2 m k x , f 2 n k ) d ( f 2 m k x , f 2 m k 1 ) + d ( f 2 m k 1 x , f 2 m k 2 ) + d ( f 2 m k 2 x , f 2 n k ) d ( f 2 m k x , f 2 m k 1 ) + d ( f 2 m k 1 x , f 2 m k 2 ) + ε .

Let k, we get

lim n d ( f 2 m k x , f 2 n k ) =ε.
(3.3)

On the other hand, applying (3.1) with y= f 2 n k x for all kN, we get

d ( f 2 m k x , f 2 n k + 1 ) ψ x ( d ( f 2 m k 1 x , f 2 n k ) ) ϕ ( d ( f 2 m k 1 x , f 2 n k ) ) .
(3.4)

Since for each kN

d ( f 2 m k x , f 2 n k + 1 ) d ( f 2 m k x , f 2 n k ) +d ( f 2 n k x , f 2 n k + 1 ) ,
(3.5)

and

d ( f 2 m k 1 x , f 2 n k ) d ( f 2 m k 1 x , f 2 m k ) +d ( f 2 m k x , f 2 n k ) ,
(3.6)

taking k and using the inequalities (3.3), (3.5) and (3.6), we get

lim n d ( f 2 m k x , f 2 n k + 1 ) =ε,
(3.7)

and

lim n d ( f 2 m k 1 x , f 2 n k ) =ε.
(3.8)

Taking into account the inequalities (3.4), (3.7) and (3.8), and by the definitions of the functions ϕ and ψ x , we get

ε = lim n d ( f 2 m k x , f 2 n k + 1 ) lim n ψ x ( d ( f 2 m k 1 x , f 2 n k ) ) lim n ϕ ( d ( f 2 m k 1 x , f 2 n k ) ) ε ϕ ( ε ) ,

which implies that ε=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}AB, there exists νAB such that lim n f n x=ν. Now { f 2 n x} is a sequence in A and { f 2 n + 1 x} is a sequence in B, and also both converge to ν. Since A and B are closed, νAB, and so AB is nonempty. On the other hand, since lim n d( f 2 n 1 x,ν)=0 and

d ( f 2 n x , f ν ) ψ x ( d ( f 2 n 1 x , ν ) ) ϕ ( d ( f 2 n 1 x , ν ) ) ,

taking n,we obtain that

d(ν,fν)0ϕ ( d ( ν , ν ) ) =0,

and hence d(ν,fν)=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 ν,μAB. Since f is a generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-contraction, we have

d ( f 2 n x , f μ ) ψ x ( d ( f 2 n 1 x , μ ) ) ϕ ( d ( f 2 n 1 x , μ ) ) .

Letting n, and by the definitions of the functions ϕ and ψ x , we obtain that

d(ν,μ)=d(ν,fμ)= lim n d ( f 2 n x , f μ ) d(ν,μ)ϕ ( d ( ν , μ ) ) ,

which implies that d(ν,μ)=0. Therefore, ν=μ, 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= R + and we define d:X×X R + by

d(x,y)=|xy|,for x,yX.

Define f:XX by

f(x)={ 0 , if  0 x < 1 ; 1 16 , if  x 1

and define ψ,ϕ: R + R + by

ψ x (t)= 1 3 tandϕ(t)= 1 6 tfor t R + .

Then f is a generalized cyclic orbital weaker Meir-Keeler ( ψ x ,ϕ)-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 Meir-Keeler contractions. Fixed Point Theory Appl 2013, 91 (2013). https://doi.org/10.1186/1687-1812-2013-91

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Keywords

  • fixed-point theorem
  • cyclic map
  • generalized cyclic orbital Meir-Keeler contraction