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Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 17 (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.
1 Introduction and preliminaries
Throughout this article, by ℝ+, ℝ we denote the sets of all nonnegative real numbers and all real numbers, respectively, while ℕ is the set of all natural numbers. Let (X, d) be a metric space, D be a subset of X and f: D → X be a map. We say f is contractive if there exists α ∈ [0,1) such that for all x, y ∈ D,
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: X → X on a metric space is called Φ-contraction if there exists an upper semi-continuous function Φ: [0, ∞) → [0, ∞) such that
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 ∈ ℕ and f: X → X an operator. Then is called a cyclic representation of X with respect to f if
-
(1)
A i , i = 1, 2,..., m are nonempty subsets of X;
-
(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 ∈ ℕ, A1, A2,..., A m , closed nonempty subsets of X and . Suppose that f satisfies the following condition.
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 .
Recently, the fixed theorems for an operator f: X → X 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., [6–10]). 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 ∈ ℕ, A1, A2,...,A m closed nonempty subsets of X and . An operator f: X → X is called a cyclic weaker φ-contraction if
-
(1)
is a cyclic representation of X with respect to f;
-
(2)
there exists a continuous, non-decreasing function φ: [0, ∞) → [0, ∞) with φ(t) > 0 for t ∈ (0, ∞) and φ(0) = 0 such that
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 ∈ ℕ, A1, A2,..., A m closed nonempty subsets of X and . Suppose that f is a cyclic weaker φ-contraction. Then, f has a fixed point .
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 η ≤ t < 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 η ≤ t < η + δ, there exists n0 ∈ ℕ such that .
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, ∞), {ϕn(t)}n∈ℕis decreasing;
-
(ϕ3) for t n ∈ [0, ∞), we have that
-
(a)
if limn→∞t n = γ > 0, then limn→∞ϕ (t n ) < γ, and
-
(b)
if limn→∞t n = 0, then limn→∞ϕ (t n ) = 0.
-
(a)
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 ∈ ℕ, A1, A2,..., A m nonempty subsets of X and . An operator f: X → X is called a cyclic weaker ϕ ○ φ-contraction if
-
(i)
is a cyclic representation of X with respect to f;
-
(ii)
for any x ∈ A i , y ∈ Ai+1, i = 1, 2,..., m,
where Am+1= A1.
Theorem 3 Let (X, d) be a complete metric space, m ∈ ℕ, A1, A2, ..., A m nonempty subsets of X and . Let f: X → X be a cyclic weaker ϕ ○ φ-contraction. Then, f has a unique fixed point .
Proof. Given x0 and let xn+1= fx n = fn+1x0, for n ∈ ℕ∪{0}. If there exists n0 ∈ ℕ ∪ {0} such that , then we finished the proof. Suppose that xn+1≠ x n for any n ∈ ℕ ∪ {0}. Notice that, for any n > 0, there exists i n ∈ {1,2,...,m} such that and . Since f: X → X is a cyclic weaker ϕ ○ φ-contraction, we have that for all n ∈ ℕ
and so
Since {ϕn(φ(d(x0, x 1)))}n∈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 η ≤ φ(d(x0, x1)) < δ + η, there exists n0 ∈ ℕ such that . Since limn→∞ϕn(φ(d(x0, x1))) = η, there exists p0 ∈ ℕ such that η ≤ ϕp(φ(d(x0, x1)) < δ + η, for all p ≥ p0. Thus, we conclude that . So we get a contradiction. Therefore limn→∞ϕn(φ(d(x0, x1))) = 0, that is,
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) ∈ ℕ such that for all p, q ≥ n0(ε),
We shall prove (*) by contradiction. Suppose that (*) is false. Then there exists some ε > 0 such that for all n ∈ ℕ , there are p n , q n ∈ ℕ with p n > q n ≥ n satisfying:
-
(i)
, and
-
(ii)
p n is the smallest number greater than q n such that the condition (i) holds.
Since
hence we conclude . Since φ is subadditive and nondecreasing, we conclude
and so
Letting n → ∞, we also have
Thus, there exists i, 0 ≤ i ≤ m - 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,
Letting n → ∞. Then by, we have
a contradiction. Therefore , by the condition (φ3), we also have . The case i ≠ 0 is similar. Thus, {x n } is a Cauchy sequence. Since X is complete, there exists 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 , and also we conclude that . Since
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 . Since f is a cyclic weaker ϕ ○ φ-contraction, we have
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 = ℝ3 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 ∈ ℝ}, B = {(0,y,0):y ∈ ℝ},C = {(0,0, z): z ∈ ℝ} be three subsets of X. Define f: A ∪ B ∪ C → A ∪ B ∪ C by
We define φ: [0, ∞) → [0, ∞) by
and φ: [0, ∞) → [0, ∞) by
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, ∞), {ϕn(t)}n∈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 ∈ ℕ, A1, A2,..., A m nonempty subsets of X and . An operator f: X → X is called a cyclic weaker (ϕ,φ)-contraction if
-
(i)
is a cyclic representation of X with respect to f;
-
(ii)
for any x ∈ A i , y ∈ Ai+1, i = 1, 2,..., m,
where Am+ 1= A1.
Theorem 4 Let (X, d) be a complete metric space, m ∈ ℕ, A1, A2,..., A m nonempty subsets of X and . Let f: X → X be a cyclic weaker (ϕ, φ)-contraction. Then f has a unique fixed point .
Proof. Given x0 and let xn+1= fx n = fn+1x0, for n ∈ ℕ ∪{0}. If there exists n ∈ ℕ ∪{0} such that , then we finished the proof. Suppose that xn+ 1≠ x n for any n ∈ ℕ ∪ {0}. Notice that, for any n > 0, there exists i n ∈ {1,2,...,m} such that and . Since f: X → X is a cyclic weaker (ϕ, φ)-contraction, we have that n ∈ℕ
and so
Since {ϕn(d(x0, x1))}n∈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 η ≤ d(x0, x1) < δ + η, there exists n0 ∈ ℕ such that . Since limn→∞, ϕn(d(x0, x1)) = η, there exists p0 ∈ ℕ such that η ≤ ϕp(d(x0, x1)) < δ + η, for all p ≥ p0. Thus, we conclude that . So we get a contradiction. Therefore limn→∞ϕn(d(x0, x1)) = 0, that is,
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 ∈ ℕ such that if p, q ≥ n 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 ∈ℕ, there are p n , q n ∈ ℕ with p n > q n ≥ n with p n - q n = 1 mod m satisfying
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 . Therefore and
Letting n → ∞ , we obtain that
On the other hand, we can conclude that
Letting n → ∞, we obtain that
Since and lie in different adjacently labeled sets A i and Ai+1for certain 1 ≤ i ≤ m, by using the fact that f is a cyclic weaker (ϕ, φ)-contraction, we have
Letting n → ∞, by using the condition ϕ3 of the function ϕ, we obtain that
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 ∈ ℕ such that if p, q ≥ n1 with p - q = 1 mod m, then
Since limn→∞d(x n , xn+1) = 0, there exists n2 ∈ ℕsuch that
for any n ≥ n2.
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
Thus, {x n } is a Cauchy sequence. Since X is complete, there exists such that limn→∞x n = ν. Since 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 of {x n } with and also all converge to ν. Since
Letting k → ∞ , we have
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 . Since f is a cyclic weaker (ϕ, φ)-contraction, we have
and we can conclude that
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: X → X by for all x ∈ X, and let ϕ, φ: [0,∞) → [0, ∞) be . Then f is a cyclic weaker (ϕ, φ)-contraction and 0 is the unique fixed point.
Example 3 Let X = ℝ+ with the metric d:X × X → ℝ+ given by
Let A1 = A2 = ... = A m = ℝ+. Define f: X → X by
and let ϕ, φ: [0, ∞) → [0,∞) be and
Then f is a cyclic weaker (ϕ, φ)-contraction and 0 is the unique fixed point.
Example 4 Let X = ℝ3 and we define d: X × X → [0, ∞) by
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 ∪ C → A ∪ B ∪ C by
We define φ: [0, ∞) → [0,∞) by
and φ: [0, ∞) → [0,∞) by
Then f is a cyclic weaker (ϕ, φ)-contraction and (0,0,0) is the unique fixed point.
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The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the article.
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Chen, CM. Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory Appl 2012, 17 (2012). https://doi.org/10.1186/1687-1812-2012-17
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DOI: https://doi.org/10.1186/1687-1812-2012-17
Keywords
- fixed point theory
- weaker Meir-Keeler function
- cyclic weaker ϕ ○ φ-contraction
- cyclic weaker (ϕ, φ)-contraction