Common fıxed points of almost generalized (ψ, ϕ)-contractive mappings in ordered metric spaces

In this article, we introduce the notion of almost generalized (ψ, ϕ)-contractive mappings in ordered metric spaces and we establish some fixed and common fixed point results in ordered complete metric spaces. Our results generalize several well-known comparable results in the literature. Finally, an example and an application are given in order to support the useability of our results.

Mathematics Subject Classifications 2000: 54H25; 47H10; 54E50

A fundamental principle in computer science is iteration. Iterative techniques are used to find roots of equations, solutions of linear and nonlinear systems of equations, and solution of differential equations. So the attraction of the fixed point iteration is understandable to a large number of mathematicians.

The Banach contraction principle (see [1]) is a very popular tool for solving problems in nonlinear analysis. Some authors generalized this interesting theorem in different ways (see for example [2-20]).

Berinde [21-24] initiated the concept of almost contractions and studied many interesting fixed point theorems for a Ćirić strong almost contraction. So, let us recall the following definition.

Definition 1.1. [21]A single valued mapping f : X × X is called a Ćirić strong almost contraction if there exist a constant α ∈ [0, 1) and some L ≥ 0 such that

for all x, y ∈ X, where

Babu [25] introduced the class of mappings which satisfy "condition (B)".

Definition 1.2. [25]Let (X, d) be a metric space. A mapping f : X → X is said to satisfy "condition (B)" if there exist a constant δ ∈ (0, 1) and some L ≥ 0 such that

for all x, y ∈ X.

Moreover, Babu proved in [25] the existence of fixed point theorem for such mappings on complete metric spaces.

Ćirić et al. [26] introduced the concept of almost generalized contractive condition and they proved some existing results.

Definition 1.3. [26]Let (X,≼) be a partially ordered set. Two mappings f, g : X → X are said to be strictly weakly increasing if fx ≺ gfx and gx ≺ fgx, for all x ∈ X.

Definition 1.4. [26]Let f and g be two self mappings on a metric space (X, d). Then they are said to satisfy almost generalized contractive condition if there exist a constant δ ∈ (0, 1) and some L ≥ 0 such that

for all x, y ∈ X.

Then Ćirić et al. [26] proved the following theorems.

Theorem 1.1. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric space (X, d) is complete. Let f : X → X be a strictly increasing continuous mapping with respect to ≼. Suppose that there exist a constant δ ∈ [0, 1) and some L ≥ 0 such that

for all comparable x, y ∈ X, where

If there exists x₀ ∈ X such that x₀ ≼ fx₀, then f has a fixed point in X.

Theorem 1.2. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric space (X, d) is complete. Let f, g : X → X be two strictly weakly increasing mappings which satisfy (1) with respect to ≼, for all comparable elements x, y ∈ X. If either f or g is continuous, then f and g have a common fixed point in X.

Khan et al. [27] introduced the concept of altering distance function as follows.

Definition 1.5. [27]The function ϕ : [0, +∞) → [0, +∞) is called an altering distance function, if the following properties are satisfied:

(1) ϕ is continuous and non-decreasing.

(2) ϕ(t) = 0 if and only if t = 0.

Many authors studied fixed point theorems which are based on altering distance functions, see for example [27-36].

In this article, we introduce the notion of almost generalized (ψ, ϕ)-contractive mapping and we establish some results in complete ordered metric spaces, where ψ and ϕ are altering distance functions. Our results generalize Theorems 1.1 and 1.2.

In this section, we define the notion of almost generalized (ψ, ϕ)-contractive mapping, then we present and prove our new results. In particular, we generalize Theorems 2.1, 2.2 and 2.3 of Ćirić et al. [26].

Let (X, ≼, d) be an ordered metric space and let f : X → X be a mapping. Set

and

Now, we introduce the following definition.

Definition 2.1. Let (X, d) be a metric space and let ψ and ϕ be altering distance functions. We say that a mapping f : X → X is an almost generalized (ψ, ϕ)-contractive mapping if there exists L ≥ 0 such that

for all comparable x, y ∈ X.

Throughout this article, the mappings ψ and ϕ denote altering distance functions. Now, let us prove our first result.

Theorem 2.1. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric (X, d) is complete. Let f : X → X be a non-decreasing continuous mapping with respect to ≼. Suppose that f is an almost generalized (ψ, ϕ)-contractive mapping. If there exists x₀ ∈ X such that x₀ ≼ fx₀, then f has a fixed point.

Proof. Let x₀ ∈ X. Then, we define a sequences (x_n) in X such that x_n+1 = fx_n. Since x₀ ≼ fx₀ = x₁ and f is non-decreasing, we have x₁ = fx₀ ≼ x₂ = fx₁. Again, as x₁ ≼ x₂ and f is non-decreasing, we have x₂ = fx₁ ≼ x₃ = fx₂. By induction, we show that

If x_n = x_n+1 for some n ∈ ℕ, then x_n = fx_n and hence x_n is a fixed point of f. So, we may assume that x_n ≠ x_n+1 for all n ∈ ℕ. By (2), we have

where

and

From (3)-(5) and the properties of ψ and ϕ, we get

If

then by (6) we have

which gives a contradiction. Thus,

Therefore (6) becomes

Since ψ is a non-decreasing mapping, we have {d(x_n, x_n+1): n ∈ ℕ ∪ {0}} is a non-increasing sequence of positive numbers. So, there exists r ≥ 0 such that

Letting n → +∞ in (7), we get

Therefore ϕ(r) = 0, and hence r = 0. Thus, we have

Next, we show that (x_n) is a Cauchy sequence in X. Suppose to the contrary; that is, (x_n) is not a Cauchy sequence. Then there exists ε > 0 for which we can find two subsequences (x_m(i)) and (x_n(i)) of (x_n) such that n(i) is the smallest index for which

This means that

From (9), (10) and the triangular inequality, we get

Using (8) and letting i → +∞, we get

From (2), we have

where

and

Letting i → +∞ in (13) and (14) then using (8) and (11), we get

and

Letting i → +∞ in (12) then using (11), (15) and (16) we have

which gives a contradiction. Thus (x_n+1 = fx_n) is a Cauchy sequence in X. As X is a complete space, there exists u ∈ X such that

Now, suppose that f is continuous, then fx_n → fu. By the uniqueness of limit, we have fu = u. Thus u is a fixed point of f. □

Notice that the continuity of f in Theorem 2.1 is not necessary and can be dropped.

Theorem 2.2. Under the same hypotheses of Theorem 2.1 and without assuming the continuity of f, assume that whenever (x_n) is a non-decreasing sequence in X such that x_n → x ∈ X implies x_n ≼ x, for all n ∈ ℕ. Then f has a fixed point in X.

Proof. Following similar arguments to those given in Theorem 2.1, we construct an increasing sequence (x_n) in X such that x_n → u for some u ∈ X. Using the assumption of X, we have x_n ≼ u for all n ∈ ℕ. Now, we show that fu = u. By (2), we have

where

and

Letting n → +∞ in (18) and (19) then we get M(x_n, u) → d(u, fu) and N(x_n, u) → 0. Again when n → +∞ in (17) then we get

Therefore, d(u, fu) = 0. Thus u = fu and hence u is a fixed point of f. □

Corollary 2.1. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric (X, d) is complete. Let f : X → X be a non-decreasing continuous mapping with respect to ≼. Suppose that there exist k ∈ [0, 1) and L ≥ 0 such that

for all comparable x, y ∈ X. If there exists x₀ ∈ X such that x₀ ≤ fx₀, then f has a fixed point.

Proof. Follows from Theorem (2.1) by taking ψ(t) = t and ϕ(t) = (1-k)t for all t ∈ [0, +∞) and noticing that f is an almost generalized (ψ, ϕ)-contractive mapping. □

The continuity of f in Corollary 2.1 is not necessary and can be dropped.

Corollary 2.2. Under the hypotheses of Corollary 2.1 and without assuming the continuity of f, assume that whenever (x_n) is a non-decreasing sequence in X such that x_n → x ∈ X implies x_n ≼ x for all n ∈ ℕ. Then f has a fixed point in X.

Proof. Follows from Theorem (2.2) by taking ψ(t) = t and ϕ(t) = (1 -k)t for all t ∈ [0, +∞). □

Now, let (X, ≼, d) be an ordered metric space and let f, g : X → X be two mappings. Set

and

Then we introduce the following definition.

Definition 2.2. Let (X, ≼) be a partially ordered set having a metric d, and let ψ and ϕ be altering distance functions. We say that a mapping f : X → X is an almost generalized (ψ, ϕ)-contractive mapping with respect to a mapping g : X → X if there exists L ≥ 0 such that

for all comparable x, y ∈ X.

Definition 2.3. Let (X, ≼) be a partially ordered set. Then two mappings f, g : X → X are said to be weakly increasing if fx ≼ gfx and gx ≼ fgx, for all X ∈ X.

Note that every strictly weakly increasing mapping is weakly increasing.

Theorem 2.3. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric (X, d) is complete, and let f, g : X → X be two weakly increasing mappings with respect to ≼. Suppose that f is an almost generalized (ψ, ϕ)-contractive mapping with respect to g. If either f or g is continuous, then f and g have a common fixed point.

Proof. Let us divide the proof into two parts:

First part: We prove that u is a fixed point of f if and only if u is a fixed point of g. Now, suppose that u is a fixed point of f, then fu = u. As u ≼ u, by (20), we have

Thus we have ϕ(d(u, gu)) = 0. Therefore d(u, gu) = 0 and hence gu = u. Similarly, we show that if u is a fixed point of g, then u is a fixed point of f.

Second part (construction a sequence by iterative technique):

Let x₀ ∈ X. We construct a sequence (x_n) in X such that x_2n+1 = fx_2n and x_2n+2 = gx_2n+1, for all non-negative integers, i.e. n ∈ ℕ ∪ {0}. As f and g are weakly increasing with respect ≼, we obtain the following:

If x_2n = x_2n+1 for some n ∈ ℕ, then x_2n = fx_2n. Thus x_2n is a fixed point of f. By the first part, we conclude that x_2n is also a fixed point of g.

If x_2n+1 = x_2n+2 for some n ∈ ℕ, then x_2n+1 = gx_2n+1. Thus x_2n+1 is a fixed point of g. By the first part, we conclude that x_2n+1 is also a fixed point of f.

Therefore, we may assume that x_n ≠ x_n+1 for all n ∈ ℕ. Now we complete the proof in the following steps:

Step 1. We will prove that

As x_2n+1 and x_2n+2 are comparable, by (20), we have

where

and

So, we have

If

then (21) becomes

which gives a contradiction. So

and hence (21) becomes

Similarly, we show that

By (22) and (23), we get that {d(x_n, x_n+1); n ∈ ℕ} is a non-increasing sequence of positive numbers. Hence there is r ≥ 0 such that

Letting n →+∞ in (22), we get

which implies that ϕ(r) = 0 and hence r = 0.

So, we have

Step 2. We will prove that (x_n) is a Cauchy sequence. It is sufficient to show that (x_2n) is a Cauchy sequence. Suppose to the contrary; that is, (x_2n) is not a Cauchy sequence. Then there exists ε > 0 for which we can find two subsequences of positive integers (x_2m(i)) and (x_2n(i)) such that n(i) is the smallest index for which

This means that

From (25), (26) and the triangular inequality, we get

By letting i → +∞ in the above inequality and using (24), we have

Moreover,

Using (24), (27) and letting i → +∞, we get

Since x_2n(i) and x_2m(i)+1 are comparable, so by (20) we have

where

and

By letting i → +∞ in (30) and (31), we get

Now, letting i → +∞ in (29) we get

So ϕ(ε) = 0 and then ε = 0, which is a contradiction. Hence (x_n) is a Cauchy sequence in X.

Step 3. (A common fixed point)

As (x_n) is a Cauchy sequence in X which is a complete space, there exists u ∈ X such that x_n → u as n → +∞. Without loss of generality, we may assume that f is continuous. Since x_2n → u as n → +∞, we have x_2n+1 = fx_2n → fu as n → +∞. By the uniqueness of limit we get fu = u. Thus u is a fixed point of f. By the first part, we conclude that u is also a fixed point of g. □

The continuity of one of the functions f or g in Theorem 2.3 is not necessary and can be dropped.

Theorem 2.4. Under the hypotheses of Theorem 2.3 and without assuming the continuity of one of the functions f or g, assume that whenever (x_n) is a non-decreasing sequence in X such that x_n → x ∈ X implies x_n ≼ x, for all n ∈ ℕ. Then f and g have a common fixed point in X.

Proof. Following similar arguments to those given in Theorem 2.3, we construct an increasing sequence (x_n) in X such that x_n → u for some u ∈ X. Using the assumption on X, we have x_n ≼ u for all n ∈ ℕ. Now, we show that fu = gu = u. By (2), we have

where

and

Letting n → +∞ in (33) and (34) then we get M(x_2n, u) → d(u, gu) and N(x_2n, u) → 0 as n → +∞. Again when n → +∞ in (32), we get

Therefore, d(u, gu) = 0 thus u = gu and then u is a fixed point of f. Similarly, we may show that fu = u. Hence u is a common fixed point of f and g. □

Then we have the following consequence results.

Corollary 2.3. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that the metric (X, d) is complete. Let f, g : X → X be two strictly weakly increasing mappings with respect to ≼. Suppose that there exist k ∈ [0, 1) and L ≥ 0 such that

for all comparable x, y ∈ X. If either f or g is continuous, then f and g have a common fixed point.

Proof. Define ψ, ϕ : [0, +∞) → [0, +∞) by ψ(t) = t and ϕ(t) = (1-k)t. Then f is an almost generalized (ψ, ϕ)-contractive mapping with respect to g. The proof follows from Theorem 2.3. □

The continuity of one of the functions f or g in Corollary 2.3 is not necessary and can be dropped.

Corollary 2.4. Under the hypotheses of Corollary 2.3 and without assuming the continuity of one of the functions f or g, assume that whenever (x_n) is a non-decreasing sequence in × such that x_n → X ∈ X implies x_n ≼ x, for all n ∈ ℕ. Then f and g have a common fixed point in X.

Proof. Follows from Theorem 2.4. □

Now, in order to support the useability of our results, let us introduce the following example.

Example 2.1. Let × = {0, 1, 2, ...}. Define the function f, g : X → X by

and

Let d : X × X → ℝ⁺ be given by

Define ψ, ϕ: [0, +∞) → [0, +∞) by ψ(t) = t² and ϕ(t) = t. Define a relation ≼ on X by × ≼ y iff y ≤ x. Then we have the following:

(1) (X, ≼) is a partially ordered set having the metric d, where the metric space (X, d) is complete.

(2) f and g are weakly increasing mappings with respect to ≼.

(3) f is continuous.

(4) f is an almost generalized (ψ, ϕ)-contractive mapping with respect to g, that is,

for x, y ∈ X with × ≼ y and L ≥ 0, where

and

Proof. The proof of (1) is clear. To prove (2), let x ∈ X. If x ∈ {0, 1, 2}, then fgx = 0 ≤ gx = 0 and gfx = 0 ≤ fx. So gx ≼ fgx and fx ≼ gfx. While, if x ≥ 3, then fgx = x - 3 ≤ x - 2 = gx and gfx = x - 3 ≤ x - 1 = fx. So gx ≼ fgx and fx ≼ gfx. Hence f and g are weakly increasing mappings with respect to ≼. To prove that f is continuous, let (x_n) be a sequence in X such that x_n → x ∈ X. By the definition of the metric d, there exists k ∈ ℕ such that x_n = x for all n ≥ k. So fx_n = fx for all n ≥ k. Hence fx_n → fx that is, f is continuous. To prove (4), given x, y ∈ X with x ≼ y. So y ≤ x. Thus, we have the following cases:

Case 1. x ∈ {0, 1} and y ∈ {0, 1}. Then d(fx, fy) = 0 and hence

for each L ≥ 0.

Case 2. x, y ≥ 2 and x = y. Then d(fx, fy) = 0 and hence

for each L ≥ 0.

Case 3. x > y ≥ 2.

If x = y + 1, then

and

Since

we have

for each L ≥ 0.

If x > y + 1, then

and

As

we have

for each L ≥ 0. By combining all cases together, we conclude that f is an almost generalized (ψ, ϕ)-contractive mapping with respect to g. Thus f, g, ψ and ϕ satisfy all the hypotheses of Theorem 2.3 and hence f and g have a common fixed point. Indeed, 0 is the unique common fixed point of f and g. □

Let Φ denote the set of functions ϕ : [0, +∞) → [0, +∞) satisfying the following hypotheses:

(1) Every ϕ ∈ Φ is a Lebesgue integrable function on each compact subset of [0, +∞),

(2) For any ϕ ∈ Φ and ε > 0,

It is an easy matter, to check that the mapping ψ : [0, +∞) → [0, +∞) defined by

is an altering distance function. Therefore, we have the following results.

Corollary 3.1. Let (X, ≼) be a partially ordered set having a metric d, such that the metric space (X, d) is complete. Let f : X → X be a non-decreasing continuous mapping with respect to ≼. Suppose that there exist k ∈ [0, 1) and L ≥ 0 such that

for all comparable x, y ∈ X. If there exists x₀ ∈ X such that x₀ ≼ fx₀, then f has a fixed point.

Proof. Follows from Corollary 2.1 by taking □

Corollary 3.2. Let (X, ≼) be a partially ordered set having a metric d, such that the metric space (X, d) is complete. Let f, g : X → X be two weakly increasing mappings with respect to ≼. Suppose that there exist k ∈ [0, 1) and L ≥ 0 such that

for all comparable x, y ∈ X. If there exists x₀ ∈ X such that x₀ ≼ fx₀, then f has a fixed point.

Proof. Follows from Corollary 2.3 by taking □

Finally, let us finish this article by noticing the following remarks:

Remark 1. Theorem 2.1 of [26]is a special case of Corollary 2.1

Remark 2. Theorem 2.2 of [26]is a special case of Corollary 2.2

Remark 3. Theorem 2.3, Corollaries 2.4 and 2.6 of [26]are special cases of Corollary 2.3

The authors declare that they have no competing interests.

All the authors contributed equally. All authors read and approved the final manuscript.

The authors thank the editor and the referees for their useful comments and suggestions.

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