Fixed point theorems for contraction mappings in modular metric spaces

In this article, we study and prove the new existence theorems of fixed points for contraction mappings in modular metric spaces.

AMS: 47H09; 47H10.

Let (X, d) be a metric space. A mapping T : X → X is a contraction if

for all x, y ∈ X, where 0 ≤ k <1. The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 [1]. Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [2-10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11-13] and others. Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14-18] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4,19-26].

In 2008, Chistyakov [27] introduced the notion of modular metric spaces generated by F-modular and develop the theory of this spaces, on the same idea he was defined the notion of a modular on an arbitrary set and develop the theory of metric spaces generated by modular such that called the modular metric spaces in 2010 [28].

In this article, we study and prove the existence of fixed point theorems for contraction mappings in modular metric spaces.

We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14,15,27-29] for more details).

Definition 2.1. Let X be a vector space over ℝ (or ℂ). A functional ρ : X → [0, ∞] is called a modular if for arbitrary x and y, elements of X satisfies the following three conditions :

(A.1) ρ(x) = 0 if and only if x = 0;

(A.2) ρ(αx) = ρ(x) for all scalar α with |α| = 1;

(A.3) ρ(αx + βy) ≤ ρ(x) + ρ(y), whenever α, β ≥ 0 and α + β = 1.

If we replace (A.3) by

(A.4) ρ(αx + βy) ≤ α^s ρ(x) + β^s ρ(y), for α, β ≥ 0, α^s + β^s = 1 with an s ∈ (0, 1], then the modular ρ is called s-convex modular, and if s = 1, ρ is called a convex modular.

If ρ is modular in X, then the set defined by

is called a modular space. X_ρ is a vector subspace of X it can be equipped with an F-norm defined by setting

In addition, if ρ is convex, then the modular space X_ρ coincides with

and the functional x ρ * = inf { λ > 0 : ρ x λ ≤ 1 } is an ordinary norm on X ρ * which is equivalence to x ρ (see [16]).

Let X be a nonempty set, λ ∈ (0, ∞) and due to the disparity of the arguments, function w : (0, ∞) × X × X → [0, ∞] will be written as w_λ(x, y) = w(λ, x, y) for all λ >0 and x, y ∈ X.

Definition 2.2. [[28], Definition 2.1] Let X be a nonempty set. A function w : (0, ∞) × X × X → [0, ∞] is said to be a metric modular on X if satisfying, for all x, y, z ∈ X the following condition holds:

(i) w_λ(x, y) = 0 for all λ >0 if and only if x = y;

(ii) w_λ(x, y) = w_λ(y, x) for all λ >0;

(iii) w_{λ + μ} (x, y) ≤ w_λ(x, z) + w_μ(z, y) for all λ, μ >0.

If instead of (i), we have only the condition

(i') w_λ(x, x) = 0 for all λ >0, then w is said to be a (metric) pseudomodular on X.

The main property of a (pseudo) modular w on a set X is a following: given x, y ∈ X, the function 0 < λ ↦ w_λ(x, y) ∈ [0, ∞] is a nonincreasing on (0, ∞).

In fact, if 0 < μ < λ, then (iii), (i') and (ii) imply

It follows that at each point λ >0 the right limit w λ + 0 ( x , y ) : = lim ε → + 0 w λ + ε ( x , y ) and the left limit w λ - 0 ( x , y ) : = lim ε → + 0 w λ - ε ( x , y ) exists in [0, ∞] and the following two inequalities hold :

Definition 2.3. [[28], Definition 3.3] A function w : (0, ∞) × X × X → [0, ∞] is said to be a convex (metric) modular on X if it is satisfies the conditions (i) and (ii) from Definition 2.2 as well as this condition holds;

(iv) w λ + μ ( x , y ) = λ λ + μ w λ ( x , z ) + μ λ + μ w μ ( z , y ) f o r a l l λ , μ > 0 a n d x , y , z ∈ X .

If instead of (i), we have only the condition (i') from Definition 2.2, then w is called a convex(metric) pseudomodular on X.

From [27,28], we know that, if x₀ ∈ X, the set X w = { x ∈ X : lim λ → ∞ w λ ( x , x 0 ) = 0 } is a metric space, called a modular space, whose metric is given by d w ∘ ( x , y ) = inf { λ > 0 : w λ ( x , y ) ≤ λ } for all x, y ∈ X_w. Moreover, if w is convex, the modular set X_w is equal to X w * = { x ∈ X : ∃ λ = λ ( x ) > 0 such that w_λ(x, x₀) <∞} and metrizable by d w * ( x , y ) = inf { λ > 0 : w λ ( x , y ) ≤ 1 } for all x , y ∈ X w * . We know that (see [[28], Theorem 3.11]) if X is a real linear space, ρ : X → [0, ∞] and

then ρ is modular (convex modular) on X in the sense of (A.1)-(A.4) if and only if w is metric modular (convex metric modular, respectively) on X. On the other hand, if w satisfy the following two conditions (i) w_λ(μx, 0) = w_λ/μ (x, 0) for all λ, μ >0 and x ∈ X, (ii) w_λ(x + z, y + z) = w_λ(x, y) for all λ >0 and x, y, z ∈ X, if we set ρ(x) = w₁(x, 0) with (2.6) holds, where x ∈ X, then

(i) X_ρ = X_w is a linear subspace of X and the functional x ρ = d w ∘ ( x , 0 ) , x ∈ X_ρ, is an F-norm on X_ρ;

(ii) if w is convex, X ρ * ≡ X w * ( 0 ) = X ρ is a linear subspace of X and the functional x ρ = d w * ( x , 0 ) , x ∈ X ρ * , is an norm on X ρ * .

Similar assertions hold if replace the word modular by pseudomodular. If w is metric modular in X, we called the set X_w is modular metric space.

By the idea of property in metric spaces and modular spaces, we defined the following:

Definition 2.4. Let X_w be a modular metric space.

(1) The sequence (x_n)_n∈ℕ in X_w is said to be convergent to x ∈ X_w if w_λ(x_n, x) → 0, as n → ∞ for all λ > 0.

(2) The sequence (x_n) _n∈ℕ in X_w is said to be Cauchy if w_λ (x_m, x_n) → 0, as m, n → ∞ for all λ >0.

(3) A subset C of X_w is said to be closed if the limit of a convergent sequence of C always belong to C.

(4) A subset C of X_w is said to be complete if any Cauchy sequence in C is a convergent sequence and its limit is in C.

(5) A subset C of X_w is said to be bounded if for all λ >0 δ_w(C) = sup{w_λ(x, y); x, y ∈ C} <∞.

In this section, we prove the existence of fixed points theorems for contraction mapping in modular metric spaces.

Definition 3.1. Let w be a metric modular on X and X_w be a modular metric space induced by w and T : X_w → X_w be an arbitrary mapping. A mapping T is called a contraction if for each x, y ∈ X_w and for all λ >0 there exists 0 ≤ k <1 such that

Theorem 3.2. Let w be a metric modular on X and X_w be a modular metric space induced by w. If X_w is a complete modular metric space and T : X_w → X_w is a contraction mapping, then T has a unique fixed point in X_w. Moreover, for any x ∈ X_w, iterative sequence {Tⁿx} converges to the fixed point.

Proof. Let x₀ ba an arbitrary point in X_w and we write x₁ = Tx₀, x₂ = Tx₁ = T²x₀, and in general, x_n = Tx_n-1 = Tⁿx₀ for all n ∈ ℕ. Then,

for all λ >0 and for each n ∈ ℕ. Therefore, lim n → ∞ w λ ( x n + 1 , x n ) = 0 for all λ >0. So for each λ >0, we have for all ∊ > 0 there exists n₀ ∈ ℕ such that w_λ(x_n, x_n+1) < ∊ for all n ∈ ℕ with n ≥ n₀. Without loss of generality, suppose m, n ∈ ℕ and m > n. Observe that, for λ m - n > 0 , there exists n_λ/(m-n) ∈ ℕ such that

for all n ≥ n_λ/(m-n). Now, we have

for all m, n ≥ n_λ/(m-n). This implies {x_n}_n∈ℕ is a Cauchy sequence. By the completeness of X_w, there exists a point x ∈ X_w such that x_n → × as n → ∞.

By the notion of metric modular w and the contraction of T, we get

for all λ >0 and for each n ∈ ℕ. Taking n → ∞ in (3.2) implies that w_λ(Tx, x) = 0 for all λ >0 and thus Tx = x. Hence, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z is another fixed point of T. We see that

for all λ >0. Since 0 ≤ k <1, we get w_λ(x, z) = 0 for all λ >0 this implies that x = z. Therefore, x is a unique fixed point of T and the proof is complete.   □

Theorem 3.3. Let w be a metric modular on X and X_w be a modular metric space induced by w. If X_w is a complete modular metric space and T : X_w → X_w is a contraction mapping. Suppose x* ∈ X_w is a fixed point of T, {ε_n} is a sequence of positive numbers for which lim n → ∞ ε n = 0 , and {y_n} ⊆ X_w satisfies

for all λ >0. Then, lim n → ∞ y n = x * .

Proof. For each m ∈ ℕ, we observe that

for all λ >0. Thus, we get

Next, we claimed that lim m → ∞ w λ ( y m + 1 , x * ) = 0 for all λ >0.

Now let ε >0. Since lim n → ∞ ε n = 0 , there exists N ∈ ℕ such that for m ≥ N, ε_m ≤ ε. Thus,

Taking limit as m → ∞ in (3.5), we have

Since x₀ is a fixed point of T and using result of Theorem 3.2, we get the sequence {Tⁿx} converge to x*. This implies that

for all λ >0. From (3.4), (3.6) and (3.7), we have

for all λ >0 which implies that lim n → ∞ y n = x * .   □

Theorem 3.4. Let w be a metric modular on X and X_w be a modular metric space induced by w. If X_w is a complete modular metric space and T : X_w → X_w is a mapping, which T^N is a contraction mapping for some positive integer N. Then, T has a unique fixed point in X_w.

Proof. By Theorem 3.2 , T^N has a unique fixed point u ∈ X_w. From T^N(T_u) = T^N+1u = T(T^Nu) = Tu, so Tu is a fixed point of T^N. By the uniqueness of fixed point of T^N, we have Tu = u. Thus, u is a fixed point of T. Since fixed point of T is also fixed point of T^N, we can conclude that T has a unique fixed point in X_w.   □

Theorem 3.5. Let w be metric modular on X, X_w be a complete modular metric space induced by w and for x* ∈ X_w we define

If T : B_w(x*, γ) → X_w is a contraction mapping with

for all λ >0, where 0 ≤ k <1. Then, T has a unique fixed point in B_w(x*, γ).

Proof. By Theorem 3.2 , we only prove that B_w(x*, γ) is complete and Tx ∈ B_w(x*, γ), for all x ∈ B_w(x*, γ). Suppose that {x_n} is a Cauchy sequence in B_w(x*, γ), also {x_n} is a Cauchy sequence in X_w. Since X_w is complete, there exists x ∈ X_w such that

for all λ >0. Since for each n ∈ ℕ, x_n ∈ B_w(x*, γ), using the property of metric modular, we get

for all λ >0. It follows the inequalities (3.10) and (3.11), we have w_λ(x*, x) ≤ γ which implies that x ∈ B_w(x*, γ). Therefore, {x_n} is convergent sequence in B_w(x*, γ) and also B_w(x*, γ) is complete.

Next, we prove that Tx ∈ B_w(x*, γ) for all x ∈ B_w(x*, γ). Let x ∈ B_w(x*, γ). From the inequalities (3.9), using the contraction of T and the notion of metric modular, we have

Therefore, Tx ∈ B_w(x*, γ) and the proof is complete.

Theorem 3.6. Let w be a metric modular on X, X_w be a complete modular metric space induced by w and T : X_w → X_w. If

for all x, y ∈ X_w and for all λ >0, where k ∈ [ 0 , 1 2 ) , then T has a unique fixed point in X_w. Moreover, for any x ∈ X_w, iterative sequence {Tⁿx} converges to the fixed point.

Proof. Let x₀ be an arbitrary point in X_w and we write x₁ = Tx₀, x₂ = Tx₁ = T²x₀, and in general, x_n = Tx_n-1 = Tⁿx₀ for all n ∈ ℕ. If T x n 0 - 1 = T x n 0 for some n₀ ∈ ℕ, then T x n 0 = x n 0 . Thus, x n 0 is a fixed point of T. Suppose that Tx_n-1 ≠ Tx_n for all n ∈ ℕ. For k ∈ [ 0 , 1 2 ) , we have

for all λ >0 and for all n ∈ ℕ. Hence,

for all λ >0 and for all n ∈ ℕ. Put β : = k 1 - k , since k ∈ [ 0 , 1 2 ) , we get β ∈ [0, 1) and hence

for all λ >0 and for all n ∈ ℕ. Similar to the proof of Theorem 3.2, we can conclude that {x_n} is a Cauchy sequence and by the completeness of X_w there exists a point x ∈ X_w such that x_n → x as n → ∞. By the property of metric modular and the inequality (3.12), we have

for all λ >0 and for all n ∈ ℕ. Taking n → ∞ in the inequality (3.16), we obtained that

Since k ∈ [ 0 , 1 2 ) , we have Tx = x. Thus, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z be another fixed point of T. We note that

for all λ >0, which implies that x = z. Therefore, x is a unique fixed point of T.   □

Now, we shall give a validate example of Theorem 3.2 .

Example 3.7. Let X = {(a, 0) ∈ ℝ²|0 ≤ a ≤ 1} ∪ {(0, b) ∈ ℝ²|0 ≤ b ≤ 1}.

Defined the mapping w : (0, ∞) × X × X → [0, ∞] by

and

We note that if we take λ → ∞, then we see that X = X_w and also X_w is a complete modular metric space. We let a mapping T : X_w → X_w is define by

and

Simple computations show that

for all (a₁, b₁), (a₂, b₂) ∈ X_w. Thus, T is a contraction mapping with constant k = 3 4 . Therefore, T has a unique fixed point that is (0, 0) ∈ X_w.

On the Euclidean metric d on X_w, we see that

for all k ∈ [0, 1). Thus, T is not a contraction mapping and then the Banach contraction mapping cannot be applied to this example.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The authors thank the referee for comments and suggestions on this manuscript. The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0029/2553). The second author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for the Ph.D. Program. The third author was supported by the Commission on Higher Education and the Thailand Research Fund (Grant No.MRG5380044). Moreover, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC Project No. 54000267).

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