Open Access

Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces

Fixed Point Theory and Applications20122012:17

https://doi.org/10.1186/1687-1812-2012-17

Received: 14 November 2011

Accepted: 16 February 2012

Published: 16 February 2012

Abstract

In this article, we introduce the notions of cyclic weaker ϕ φ-contractions and cyclic weaker (ϕ, φ)-contractions in complete metric spaces and prove two theorems which assure the existence and uniqueness of a fixed point for these two types of contractions. Our results generalize or improve many recent fixed point theorems in the literature.

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

Keywords

fixed point theoryweaker Meir-Keeler functioncyclic weaker ϕ φ-contractioncyclic weaker (ϕ, φ)-contraction

1 Introduction and preliminaries

Throughout this article, by +, we denote the sets of all nonnegative real numbers and all real numbers, respectively, while is the set of all natural numbers. Let (X, d) be a metric space, D be a subset of X and f: DX be a map. We say f is contractive if there exists α [0,1) such that for all x, y D,
d ( f x , f y ) α d ( x , y ) .
The well-known Banach's fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. In 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping f: XX on a metric space is called Φ-contraction if there exists an upper semi-continuous function Φ: [0, ∞) → [0, ∞) such that
d ( f x , f y ) Φ ( d ( x , y ) ) for all x , y X .

Generalization of the above Banach contraction principle has been a heavily investigated branch research. (see, e.g., [3, 4]). In 2003, Kirk et al. [5] introduced the following notion of cyclic representation.

Definition 1 [5] Let X be a nonempty set, m and f: XX an operator. Then X = i = 1 m A i is called a cyclic representation of X with respect to f if

  1. (1)

    A i , i = 1, 2,..., m are nonempty subsets of X;

     
  2. (2)

    f (A1) A2, f (A2) A3,..., f (Am-1) A m , f (A m ) A1.

     

Kirk et al. [5] also proved the below theorem.

Theorem 1 [5] Let (X, d) be a complete metric space, m , A1, A2,..., A m , closed nonempty subsets of X and X = i = 1 m A i . Suppose that f satisfies the following condition.
d ( f x , f y ) ψ ( d ( x , y ) ) , f o r a l l x A i , y A i + 1 , i { 1 , 2 , . . . , m } ,

where ψ: [0, ∞) → [0, ∞) is upper semi-continuous from the right and 0 ≤ ψ(t) < t for t > 0. Then, f has a fixed point z i = 1 n A i .

Recently, the fixed theorems for an operator f: XX that defined on a metric space X with a cyclic representation of X with respect to f had appeared in the literature. (see, e.g., [610]). In 2010, Pǎcurar and Rus [7] introduced the following notion of cyclic weaker φ-contraction.

Definition 2 [7] Let (X, d) be a metric space, m , A1, A2,...,A m closed nonempty subsets of X and X = i = 1 m A i . An operator f: XX is called a cyclic weaker φ-contraction if

  1. (1)

    X = i = 1 m A i is a cyclic representation of X with respect to f;

     
  2. (2)
    there exists a continuous, non-decreasing function φ: [0, ∞) → [0, ∞) with φ(t) > 0 for t (0, ∞) and φ(0) = 0 such that
    d ( f x , f y ) d ( x , y ) - φ ( d ( x , y ) ) ,
     

for any x A i , y Ai+1, i = 1,2,...,m where Am+1= A1.

And, Pǎcurar and Rus [7] proved the below theorem.

Theorem 2 [7] Let (X, d) be a complete metric space, m , A1, A2,..., A m closed nonempty subsets of X and X = i = 1 m A i . Suppose that f is a cyclic weaker φ-contraction. Then, f has a fixed point z i = 1 n A i .

In this article, we also recall the notion of Meir-Keeler function (see [11]). A function ϕ: [0, ∞) → [0, ∞) is said to be a Meir-Keeler function if for each η > 0, there exists δ > 0 such that for t [0, ∞) with ηt < η + δ, we have ϕ (t) < η. We now introduce the notion of weaker Meir-Keeler function ϕ: [0, ∞) → [0,∞), as follows:

Definition 3 We call ϕ: [0, ∞) → [0, ∞) a weaker Meir-Keeler function if for each η > 0, there exists δ > 0 such that for t [0, ∞) with ηt < η + δ, there exists n0 such that ϕ n 0 ( t ) < η .

2 Fixed point theory for the cyclic weaker ϕ φ-contractions

The main purpose of this section is to present a generalization of Theorem 1. In the section, we let ϕ: [0, ∞) → [0, ∞) be a weaker Meir-Keeler function satisfying the following conditions:

  • (ϕ1) ϕ(t) > 0 for t > 0 and ϕ (0) = 0;

  • (ϕ2) for all t (0, ∞), {ϕ n (t)}nis decreasing;

  • (ϕ3) for t n [0, ∞), we have that
    1. (a)

      if limn→∞t n = γ > 0, then limn→∞ϕ (t n ) < γ, and

       
    2. (b)

      if limn→∞t n = 0, then limn→∞ϕ (t n ) = 0.

       

And, let φ: [0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying

  • (φ1) φ(t) > 0 for t > 0 and φ(0) = 0;

  • (φ2) φ is subadditive, that is, for every μ1, μ2 [0, ∞), φ( μ1 + μ2) ≤ φ(μ1) + φ(μ2);

  • (φ3) for all t (0, ∞), limn→∞t n = 0 if and only if limn→∞φ(t n ) = 0.

We state the notion of cyclic weaker ϕ φ-contraction, as follows:

Definition 4 Let (X, d) be a metric space, m , A1, A2,..., A m nonempty subsets of X and X = i = 1 m A i . An operator f: XX is called a cyclic weaker ϕ φ-contraction if

  1. (i)

    X = i = 1 m A i is a cyclic representation of X with respect to f;

     
  2. (ii)
    for any x A i , y Ai+1, i = 1, 2,..., m,
    φ ( d ( f x , f y ) ) ϕ ( φ ( d ( x , y ) ) ) ,
     

where Am+1= A1.

Theorem 3 Let (X, d) be a complete metric space, m , A1, A2, ..., A m nonempty subsets of X and X = i = 1 m A i . Let f: XX be a cyclic weaker ϕ φ-contraction. Then, f has a unique fixed point z i = 1 m A i .

Proof. Given x0 and let xn+1= fx n = fn+1x0, for n {0}. If there exists n0 {0} such that x n 0 + 1 = x n 0 , then we finished the proof. Suppose that xn+1x n for any n {0}. Notice that, for any n > 0, there exists i n {1,2,...,m} such that x n - 1 A i n and x n A i n + 1 . Since f: XX is a cyclic weaker ϕ φ-contraction, we have that for all n
φ ( d ( x n , x n + 1 ) ) = φ ( d ( f x n - 1 , f x n ) ) ϕ ( φ ( d ( x n - 1 , x n ) ) ) ,
and so
φ ( d ( x n , x n + 1 ) ) ϕ ( φ ( d ( x n 1 , x n ) ) ) ϕ ( ϕ ( φ ( d ( x n 2 , x n 1 ) ) ) = ϕ 2 ( φ ( ( d ( x n 2 , x n 1 ) ) ) ϕ n ( φ ( d ( x 0 , x 1 ) ) ) .
Since {ϕ n (φ(d(x0, x 1)))}nis decreasing, it must converge to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then by the definition of weaker Meir-Keeler function ϕ, there exists δ > 0 such that for x0, x1 X with ηφ(d(x0, x1)) < δ + η, there exists n0 such that ϕ n 0 ( φ ( d ( x 0 , x 1 ) ) ) < η . Since limn→∞ϕ n (φ(d(x0, x1))) = η, there exists p0 such that ηϕ p (φ(d(x0, x1)) < δ + η, for all pp0. Thus, we conclude that ϕ p 0 + n 0 ( φ ( d ( x 0 , x 1 ) ) ) < η . So we get a contradiction. Therefore limn→∞ϕ n (φ(d(x0, x1))) = 0, that is,
lim n φ ( d ( x n , x n + 1 ) ) = 0 .

Next, we claim that {x n } is a Cauchy sequence. We claim that the following result holds:

Claim: for each ε > 0, there is n0(ε) such that for all p, qn0(ε),
φ ( d ( x p , x q ) ) < ε , ( * )
We shall prove (*) by contradiction. Suppose that (*) is false. Then there exists some ε > 0 such that for all n , there are p n , q n with p n > q n n satisfying:
  1. (i)

    φ ( d ( x p n , x q n ) ) ε , and

     
  2. (ii)

    p n is the smallest number greater than q n such that the condition (i) holds.

     
Since
ε φ ( d ( x p n , x q n ) ) φ ( d ( x p n , x p n - 1 ) + d ( x p n - 1 , x q n ) ) φ ( d ( x p n , x p n - 1 ) ) + φ ( d ( x p n - 1 , x q n ) ) φ ( d ( x p n , x p n - 1 ) ) + ε ,
hence we conclude lim p φ ( d ( x p n , x q n ) ) = ε . Since φ is subadditive and nondecreasing, we conclude
φ ( d ( x p n , x q n ) ) φ ( d ( x p n , x q n + 1 ) + d ( x p n + 1 , x q n ) ) φ ( d ( x p n , x q n + 1 ) ) + φ ( d ( x p n + 1 , x q n ) ) ,
and so
φ ( d ( x p n , x q n ) ) - φ ( d ( x p n , x p n + 1 ) ) φ ( d ( x p n + 1 , x q n ) ) φ ( d ( x p n , x p n + 1 ) + d ( x p n , x q n ) ) φ ( d ( x p n , x p n + 1 ) ) + φ ( d ( x p n , x q n ) ) .
Letting n → ∞, we also have
lim n φ ( d ( x p n + 1 , x q n ) ) = ε .
Thus, there exists i, 0 ≤ im - 1 such that p n - q n + i = 1 mod m for infinitely many n. If i = 0, then we have that for such n,
ε φ ( d ( x p n , x q n ) ) φ ( d ( x p n , x p n + 1 ) + d ( x p n + 1 , x q n + 1 ) + d ( x q n + 1 , x q n ) ) φ ( d ( x p n , x p n + 1 ) ) + φ ( d ( x p n + 1 , x q n + 1 ) ) + φ ( d ( x q n + 1 , x q n ) ) = φ ( d ( x p n , x p n + 1 ) ) + φ ( d ( f x p n , f x q n ) ) + φ ( d ( x q n + 1 , x q n ) ) φ ( d ( x p n , x p n + 1 ) ) + ϕ ( φ ( d ( x p n , x q n ) ) ) + φ ( d ( x q n + 1 , x q n ) ) .
Letting n → ∞. Then by, we have
ε 0 + lim n ϕ ( φ ( d ( x p n , x q n ) ) ) + 0 < ε ,
a contradiction. Therefore lim n φ ( d ( x p n , x q n ) ) = 0 , by the condition (φ3), we also have lim n d ( x p n , x q n ) = 0 . The case i ≠ 0 is similar. Thus, {x n } is a Cauchy sequence. Since X is complete, there exists ν i = 1 m A i such that limn→∞x n = ν. Now for all i = 0, 1, 2,..., m - 1, {fx mn-i } is a sequence in A i and also all converge to ν. Since A i is clsoed for all i = 1, 2,..., m, we conclude ν i = 1 m A i , and also we conclude that i = 1 m A i ϕ . Since
φ ( d ( ν , f ν ) ) = lim n φ ( d ( f x m n , f ν ) ) lim n ϕ ( φ ( d ( f x m n - 1 , ν ) ) ) = 0 ,

hence φ(d(ν, fν)) = 0, that is, d(ν, fν) = 0, ν is a fixed point of f.

Finally, to prove the uniqueness of the fixed point, let μ be another fixed point of f. By the cyclic character of f, we have μ , ν i = 1 n A i . Since f is a cyclic weaker ϕ φ-contraction, we have
φ ( d ( ν , μ ) ) = φ ( d ( ν , f μ ) ) = lim n φ ( d ( f x m n , f μ ) ) lim n ϕ ( φ ( d ( f x m n - 1 , μ ) ) ) < φ ( d ( ν , μ ) ) ,

and this is a contradiction unless φ(d(ν, μ)) = 0, that is, μ = ν. Thus ν is a unique fixed point of f.

Example 1 Let X = 3 and we define d: X × X → [0,∞) by d(x,y) = |x1-y1 |+| x2-y2 |+| x3-y3|, for x = (x1, x2, x3), y = (y1, y2, y3) X, and let A = {(x, 0,0):x }, B = {(0,y,0):y },C = {(0,0, z): z } be three subsets of X. Define f: A B CA B C by
f ( ( x , 0 , 0 ) ) = 0 , 1 4 x , 0 ; f o r a l l x ; f ( ( 0 , y , 0 ) ) = 0 , 0 , 1 4 y ; f o r a l l y ; f ( ( 0 , 0 , z ) ) = 1 4 z , 0 , 0 ; f o r a l l z .
We define φ: [0, ∞) → [0, ∞) by
ϕ ( t ) = 1 3 t f o r t [ 0 , ) ,
and φ: [0, ∞) → [0, ∞) by
φ ( t ) = 1 2 t f o r t [ 0 , ) .

Then f is a cyclic weaker ϕ φ-contraction and (0, 0, 0) is the unique fixed point.

3 Fixed point theory for the cyclic weaker (ϕ, φ-contractions

The main purpose of this section is to present a generalization of Theorem 2. In the section, we let ϕ: [0, ∞) → [0, ∞) be a weaker Meir-Keeler function satisfying the following conditions:

  • (ϕ1) ϕ (t) > 0 for t > 0 and ϕ(0) = 0;

  • (ϕ2) for all t (0, ∞), {ϕ n (t)}nis decreasing;

  • (ϕ3) for t n [0, ∞), if limn→∞t n = γ, then limn→∞ϕ(t n ) ≤ γ.

And, let φ: [0, ∞) → [0, ∞) be a non-decreasing and continuous function satisfying φ(t) > 0 for t > 0 and φ(0) = 0.

We now state the notion of cyclic weaker (ϕ, φ)-contraction, as follows:

Definition 5 Let (X, d) be a metric space, m , A1, A2,..., A m nonempty subsets of X and X = i = 1 m A i . An operator f: XX is called a cyclic weaker (ϕ,φ)-contraction if

  1. (i)

    X = i = 1 m A i is a cyclic representation of X with respect to f;

     
  2. (ii)
    for any x A i , y Ai+1, i = 1, 2,..., m,
    d ( f x , f y ) ϕ ( d ( x , y ) ) - φ ( d ( x , y ) ) ,
     

where Am+ 1= A1.

Theorem 4 Let (X, d) be a complete metric space, m , A1, A2,..., A m nonempty subsets of X and X = i = 1 m A i . Let f: XX be a cyclic weaker (ϕ, φ)-contraction. Then f has a unique fixed point z i = 1 m A i .

Proof. Given x0 and let xn+1= fx n = fn+1x0, for n {0}. If there exists n {0} such that x n 0 + 1 = x n 0 , then we finished the proof. Suppose that xn+ 1x n for any n {0}. Notice that, for any n > 0, there exists i n {1,2,...,m} such that x n - 1 A i n and x n A i n + 1 . Since f: XX is a cyclic weaker (ϕ, φ)-contraction, we have that n
d ( x n , x n + 1 ) = d ( f x n - 1 , f x n ) ϕ ( d ( x n - 1 , x n ) ) - φ ( d ( x n - 1 , x n ) ) ϕ ( d ( x n - 1 , x n ) ) ,
and so
d ( x n , x n + 1 ) ϕ ( d ( x n 1 , x n ) ) ϕ ( ϕ ( d ( x n 2 , x n 1 ) ) = ϕ 2 ( d ( x n 2 , x n 1 ) ) ϕ n ( d ( x 0 , x 1 ) ) .
Since {ϕ n (d(x0, x1))}nis decreasing, it must converge to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then by the definition of weaker Meir-Keeler function ϕ, there exists δ > 0 such that for x0, x1 X with ηd(x0, x1) < δ + η, there exists n0 such that ϕ n 0 ( d ( x 0 , x 1 ) ) < η . Since limn→∞, ϕ n (d(x0, x1)) = η, there exists p0 such that ηϕ p (d(x0, x1)) < δ + η, for all pp0. Thus, we conclude that ϕ p 0 + n 0 ( d ( x 0 , x 1 ) ) < η . So we get a contradiction. Therefore limn→∞ϕ n (d(x0, x1)) = 0, that is,
lim n d ( x n , x n + 1 ) = 0 .

Next, we claim that {x n } is a Cauchy sequence. We claim that the following result holds:

Claim: For every ε > 0, there exists n such that if p, qn with p-q = 1 mod m, then d(x p , x q ) < ε.

Suppose the above statement is false. Then there exists ϵ > 0 such that for any n , there are p n , q n with p n > q n n with p n - q n = 1 mod m satisfying
d ( x q n , x p n ) ε .
Now, we let n > 2m. Then corresponding to q n n use, we can choose p n in such a way, that it is the smallest integer with p n > q n n satisfying p n - q n = 1 mod m and d ( x q n , x p n ) ε . Therefore d ( x q n , x p n - m ) ε and
ε d ( x q n , x p n ) d ( x q n , x p n - m ) + i = 1 m d ( x p n - i , x p n - i + 1 ) < ε + i = 1 m d ( x p n - i , x p n - i + 1 ) .
Letting n → ∞ , we obtain that
lim n d ( x q n , x p n ) = ε .
On the other hand, we can conclude that
ε d ( x q n , x p n ) d ( x q n , x q n + 1 ) + d ( x q n + 1 , x p n + 1 ) + d ( x x p n + 1 , p n ) d ( x q n , x q n + 1 ) + d ( x q n + 1 , x q n ) + d ( x q n , x p n ) + d ( x p n , x p n + 1 ) + d ( x x p n + 1 , p n ) .
Letting n → ∞, we obtain that
lim n d ( x q n + 1 , x p n + 1 ) = ε .
Since x q n and x p n lie in different adjacently labeled sets A i and Ai+1for certain 1 ≤ im, by using the fact that f is a cyclic weaker (ϕ, φ)-contraction, we have
d ( x q n + 1 , x p n + 1 ) = d ( f x q n , f x p n ) ϕ ( d ( x q n , x p n ) ) - φ ( d ( x q n , x p n ) ) .
Letting n → ∞, by using the condition ϕ3 of the function ϕ, we obtain that
ε ε - φ ( ε ) ,

and consequently, φ (ϵ) = 0. By the definition of the function φ, we get ϵ = 0 which is contraction. Therefore, our claim is proved.

In the sequel, we shall show that {x n } is a Cauchy sequence. Let ε > 0 be given. By our claim, there exists n1 such that if p, qn1 with p - q = 1 mod m, then
d ( x p , x q ) ε 2 .
Since limn→∞d(x n , xn+1) = 0, there exists n2 such that
d ( x n , x n + 1 ) ε 2 m ,

for any nn2.

Let p, q ≥ max{n 1, n2} and p > q. Then there exists k {1, 2,..., m} such that p -q = k mod m. Therefore, p - q + j = 1 mod m for j = m - k + 1, and so we have
d ( x q , x p ) d ( x q , x p + j ) + d ( x p + j , x p + j - 1 ) + + d ( x p + 1 , x p ) ε 2 + j × ε 2 m ε 2 + m × ε 2 m = ε .
Thus, {x n } is a Cauchy sequence. Since X is complete, there exists ν i = 1 m A i such that limn→∞x n = ν. Since X = i = 1 m A i is a cyclic representation of X with respect to f, the sequence {x n } has infinite terms in each A i for i {1,2,...,m}. Now for all i = 1,2,...,m, we may take a subsequence { x n k } of {x n } with x n k A i - 1 and also all converge to ν. Since
d ( x n k + 1 , f ν ) = d ( f x n k , f ν ) ϕ ( d ( x n k , ν ) ) - φ ( d ( x n k , ν ) ) ϕ ( d ( x n k , ν ) ) .
Letting k → ∞ , we have
d ( ν , f ν ) 0 ,

and so ν = fν.

Finally, to prove the uniqueness of the fixed point, let μ be the another fixed point of f. By the cyclic character of f, we have μ , ν i = 1 n A i . Since f is a cyclic weaker (ϕ, φ)-contraction, we have
d ( ν , μ ) = d ( ν , f μ ) = lim n d ( x n k + 1 , f μ ) = lim n d ( f x n k , f μ ) lim n [ ϕ ( d ( x n k , μ ) ) - φ ( d ( x n k , μ ) ) ] d ( ν , μ ) - φ ( d ( ν , μ ) ) ,
and we can conclude that
φ ( d ( ν , μ ) ) = 0 .

So we have μ = ν. We complete the proof.

Example 2 Let X = [-1,1] with the usual metric. Suppose that A1 = [-1,0] = A3 and A2 = [0,1] = A4. Define f: XX by f ( x ) = - x 6 for all x X, and let ϕ, φ: [0,∞) → [0, ∞) be ϕ ( t ) = 1 2 , φ ( t ) = t 4 . Then f is a cyclic weaker (ϕ, φ)-contraction and 0 is the unique fixed point.

Example 3 Let X = + with the metric d:X × X+ given by
d ( x , y ) = max { x , y } , f o r x , y X .
Let A1 = A2 = ... = A m = +. Define f: XX by
f ( x ) = x 2 77 f o r x X ,
and let ϕ, φ: [0, ∞) → [0,∞) be φ ( t ) = t 3 2 ( t + 2 ) and
ϕ ( t ) = 2 t 3 3 t + 8 , i f t 1 ; t 2 2 , i f t < 1 .

Then f is a cyclic weaker (ϕ, φ)-contraction and 0 is the unique fixed point.

Example 4 Let X = 3 and we define d: X × X → [0, ∞) by
d ( x , y ) = max x 1 - y 1 , x 2 - y 2 , x 3 - y 3 ,

for x = (x1,x2,x3), y = (y1, y2, y3) X, and let A = {(x,0,0): x [0,1]}, B = {(0,y,0): y [0,1]}, C = {(0,0, z): z [0,1]} be three subsets of X.

Define f: A B CA B C by
f ( ( x , 0 , 0 ) ) = 0 , 1 8 x 2 , 0 ; f o r a l l x [ 0 , 1 ] ; f ( ( 0 , y , 0 ) ) = 0 , 0 , 1 8 y 2 ; f o r a l l y [ 0 , 1 ] ; f ( ( 0 , 0 , z ) ) = 1 8 z 2 , 0 , 0 ; f o r a l l z [ 0 , 1 ] .
We define φ: [0, ∞) → [0,∞) by
ϕ ( t ) = t 2 t + 1 f o r t [ 0 , ) ,
and φ: [0, ∞) → [0,∞) by
φ ( t ) = t 2 t + 2 f o r t [ 0 , ) .

Then f is a cyclic weaker (ϕ, φ)-contraction and (0,0,0) is the unique fixed point.

Declarations

Acknowledgements

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the article.

Authors’ Affiliations

(1)
Department of Applied Mathematics, National Hsinchu University of Education

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© Chen; licensee Springer. 2012

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