Fixed points for cyclic weakly Chatterjea type contractions in ordered b-metric spaces and applications to integral equations

In this paper, we introduce the notion of a cyclic (ψ,φ,A,B)-weakly Chatterjea type contraction and subsequently establish some fixed point and common fixed point results for this class of mappings in partially ordered b-metric spaces. The presented theorems extend some known fixed point results in the literature. Furthermore, an example as well as an application to integral equations are given here to illustrate the usability of our obtained results.


Introduction and Preliminaries
The celebrated Banach contraction principle is a fundamental piece in several branches of functional analysis as well as in many applications.Due to its relevance, generalizations of Banach's fixed point theorem have been investigated heavily by many authors and there have been many theorems dealing with mappings satisfying various types of contractive inequalities.Following this trend, Chatterjea [9] introduced the concept of C-contraction as follows.
Definition 1.1.( [9]) Let (X, d) be a metric space.A mapping f : X → X is said to be a C-contraction if there exists α ∈ (0, 1  2 ) such that d( f x, f y) ≤ α ( d(x, f y) + d(y, f x) ) holds for all x, y ∈ X.
In this interesting paper, Chatterjea [9] proved that every C-contraction f on a complete metric space has a unique fixed point.In 2009, Choudhury [10] introduced the concept of weakly C-contractive mapping as a generalization of C-contractive mapping.
Moreover, Choudhury in [10] proved the existence of a unique fixed point for such mappings on complete metric spaces.Some of contractive conditions are based on functions called control functions which alter the distance between two points in a metric space.Such functions were introduced by Khan et al. [21] as follows.
In 2004, Ran and Reurings [25] introduced a new concept of a metric space endowed with a partial ordering and proved a fixed point theorem which generalizes Banach contraction principle.They applied this result to solve some linear and nonlinear matrix equations.After that, Nieto and Rodŕiguez-López [22] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions.Recently, many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and obtained many fixed point results in such spaces.
In [27], Shatanawi, by considering an altering distance function, established some fixed point theorems for a nonlinear weakly C-contraction type mapping in the framework of ordered metric spaces.
Theorem 1.1.( [27]) Let (X, ≼) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space.Let f : X → X be a nondecreasing mapping such that for all comparable x, y ∈ X, where ψ is an altering distance function and φ : [0, +∞) × [0, +∞) → [0, +∞) is a continuous function such that φ(x, y) = 0 if and only if x = y = 0. Assume that f is continuous, or the space (X, ≼, d) is regular.If there exists x 0 ∈ X such that x 0 ≼ f x 0 , then f has a fixed point.
Here, the ordered metric space (X, ≼, d) is called regular if for any nondecreasing sequence {x n } in X such that x n → x ∈ X, as n → +∞, one has x n ≼ x for all n ∈ N.
The concept of metric spaces has been generalized in many directions.The notion of a b-metric space was introduced by Bakhtin in [4], and later extensively used by Czerwik in [11,12].After that, several interesting results about the existence of fixed points for single-valued and multi-valued operators in b-metric (sometimes also called metric type) spaces have been obtained.For further works in this direction, we refer to [2,3,5,6,7,8,14,19,23,26,30].Khmasi and Hussain [20] and Hussain and Shah [17] discussed KKM mappings and related results in b-metric and cone b-metric spaces.
For the sake of convenience, we briefly recall some concepts and notations which will be needed in the sequel.
Definition 1.4.( [12]) Let X be a (nonempty) set and s ≥ 1 be a given real number.A function d : X × X → [0, +∞) is a b-metric if, for all x, y, z ∈ X, the following conditions hold: In this case, the pair (X, d) is called a b-metric space.
It should be noted that, the class of b-metric spaces is effectively larger than the class of metric spaces, since every metric is a b-metric with s = 1.
The following example shows that in general a b-metric space does not necessarily need to be a metric space (see also [30, p. 264]).
Example 1.1.Let (X, d) be a metric space and ρ(x, y) = (d(x, y)) p , where p > 1 is a real number.Then ρ is a b-metric with s = 2 p−1 .However, if (X, d) is a metric space, then (X, ρ) is not necessarily a metric space.For example, if X = R is the set of real numbers and d(x, y) = |x − y| is the usual Euclidean metric, then ρ(x, y) = (x − y) 2 is a b-metric on R with s = 2.But is not a metric on R.
Definition 1.5.Let X be a nonempty set.Then (X, ≼, d) is called a partially ordered b-metric space if and only if d is a b-metric on a partially ordered set (X, ≼).
The notions of b-convergent and b-Cauchy sequences, as well as of b-complete b-metric spaces are introduced in an obvious way (see, e.g., [5,6]).
Note that a b-metric is not always a continuous function of its variables (see, e.g., [15,Example 2]), whereas an ordinary metric is.
Since in general a b-metric is not continuous, we need the following simple lemma about the b-convergent sequences in the proof of our main results.
Lemma 1.1.( [2]) Let (X, d) be a b-metric space with parameter s ≥ 1, and suppose that the sequences {x n } and {y n } are b-convergent to x and y, respectively.Then, we have In particular, if x = y, then lim n→+∞ d(x n , y n ) = 0.Moreover, for each z ∈ X, we have One of the interesting generalizations of the Banach contraction principle was obtained by Kirk et al. [16].They introduced the following notion of cyclic representation.
Definition 1.6.( [16]) Let A and B be nonempty subsets of a metric space (X, d) and T : The following interesting theorem for a cyclic map was given in [16].
Theorem 1.2.( [16]) Let A and B be nonempty closed subsets of a complete metric space (X, d) and suppose that T : for all x ∈ A and y ∈ B, where λ ∈ [0, 1).Then T has a unique fixed point in A ∩ B.
They named such contractive conditions, cyclical contractive conditions.The mappings satisfying the conditions of Theorem 1.2 are called cyclic contractions.Notice that although a contraction is continuous, cyclic contractions need not be.This is one of the important gains of this approach.After the remarkable paper of Kirk et al. [16], several authors proved many results in fixed point theory for cyclic mappings, satisfying various (nonlinear) contractive conditions.The readers interested in this topic are referred to [1,15,18,24,28,29].
In this work, we introduce the concept of a cyclic (ψ, φ, A, B)-weakly Chatterjea type contraction and then derive fixed point and common fixed point theorems for these cyclic contractions in the framework of partially ordered b-complete b-metric spaces.Our main results extend and generalize several well known comparable results in the existing literature.Moreover, we present an example as well as an application to integral equations in order to illustrate the effectiveness of the obtained results.

Main Results
From now on, we assume that Definition 2.1.Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1. Suppose that f , g : X → X be two self-mappings and A and B be nonempty closed subsets of X.The pair ( f , g) is called a cyclic (ψ, φ, A, B)-weakly Chatterjea type contraction if (ii) there exist the functions ψ ∈ Ψ and φ ∈ Φ such that for arbitrary comparable elements x, y ∈ X with x ∈ A and y ∈ B, we have ) Let (X, ≼) be a partially ordered set and suppose that A and B be closed subsets of X with X = A ∪ B. Let f , g : X → X be two self-mappings.The pair ( f , g) is said to be (A, B)-weakly increasing if f x ≼ g f x for all x ∈ A and gy ≼ f gy for all y ∈ B.
Theorem 2.1.Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1 and suppose that A and B be closed subsets of X.Let f , g : X → X be two (A, B)-weakly increasing mappings with respect to ≼. Assume that (a) the pair ( f , g) is a cyclic (ψ, φ, A, B)-weakly Chatterjea type contraction, (b) f or g is continuous.
Then f and g have a common fixed point u ∈ A ∩ B.
Proof.We prove that u ∈ A ∩ B is a fixed point of f if and only if u is a fixed point of g.Suppose that u is a fixed point of f , that is, f u = u.As u ≼ u and u ∈ A ∩ B, by applying the inequality (2.1), we have since ψ is nondecreasing.This yields that d(u, gu) = 0 by our assumptions about φ.Therefore, gu = u.Similarly, we can show that if u is a fixed point of g, then u is a fixed point of f .Let x 0 ∈ A and let Continuing this process, we can construct a sequence {x n } in X such that x 2n+1 = f x 2n and x 2n+2 = gx 2n+1 with x 2n ∈ A and x 2n+1 ∈ B. Since f and g are (A, B)-weakly increasing, we have International Scientific Publications and Consulting Services If x 2n = x 2n+1 for some n ∈ N, then x 2n = f x 2n .Thus, x 2n is a fixed point of f .By the first part of proof, we conclude that x 2n is also a fixed point of g.Similarly, 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 of proof, we conclude that x 2n+1 is also a fixed point of f .Therefore, we assume that x n ̸ = x n+1 for all n ∈ N. Now, we complete the proof in the following steps.
Step 1.We will show that As x 2n and x 2n+1 are comparable and x 2n ∈ A and x 2n+1 ∈ B, by applying the inequality (2.1), we have Therefore, by using the triangular inequality and the properties of ψ and φ, we get ) .
Since ψ is nondecreasing, we have Similarly, we can show that From (2.4) and (2.5), we get that {d(x n , x n+1 ) : n ∈ N ∪ {0}} is a monotone decreasing sequence of nonnegative real numbers.Then there exists r ≥ 0 such that From the above argument, we have By taking limit as n → +∞ in the above inequality, we obtain lim n→+∞ d(x 2n , x 2n+2 ) = s 2 (s + 1)r.Now, letting n → +∞ in (2.3) and using the continuity of ψ and the properties of φ, we obtain Therefore, φ ( s 2 (s + 1)r, 0 ) = 0 which by our assumptions as regards φ implies that r = 0.
Step 2. We will prove that {x n } is a b-Cauchy sequence in X.To do this, it is sufficient to show that the subsequence {x 2n } is a b-Cauchy sequence.Assume on the contrary that {x 2n } is not a b-Cauchy sequence.Then there exists ε > 0 for which we can find two subsequences {x 2m k } and {x 2n k } of {x n } such that n k is the smallest index for which This means that From (2.6) and (2.7) and by using the triangular inequality, we have By taking the upper limit as k → +∞ in the above inequality and using (2.2), we get Further, from and thanks to (2.2) and (2.8), we get On the other hand, from and by using (2.2) and (2.8), we obtain lim sup Consequently, we have Again, by using the triangular inequality, we have Passing to the upper limit as k → +∞ in the above inequality and taking into account (2.2), we get lim sup From (2.6) and by using the triangular inequality again, we have By taking the upper limit as k → +∞ in the above inequality and thanks to (2.2), we get .
Thus, we have By using the property of φ, we have lim inf which contradicts (2.11) and (2.12).So, we deduce that {x n } is a b-Cauchy sequence.
Step 3. Existence of a common fixed point for f and g.Since {x n } is a b-Cauchy sequence in X which is a b-complete b-metric space, it follows that there exists u ∈ X such that lim n→+∞ x n = u and Now, without any loss of generality, we may assume that f is continuous.By using the triangular inequality, we have Now, by taking the upper limit as n → +∞ in the above inequality and using the continuity of f , we get Thus, we have f u = u.Hence, u is a fixed point of f and since A and B are closed subsets of X, it follows that u ∈ A ∩ B. By the first part of proof, we conclude that u is also a fixed point of g.
The assumption of continuity of one of the mappings f or g in Theorem 2.1 can be replaced by another condition, which is often used in similar situations.Namely, we shall use the notion of a regular partially ordered b-metric space, which is defined analogously to the case of the standard metric.
Theorem 2.2.Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1 and suppose that A and B be closed subsets of X.Let f , g : X → X be two (A, B)-weakly increasing mappings with respect to ≼. Assume that (a) the pair ( f , g) is a cyclic (ψ, φ, A, B)-weakly Chatterjea type contraction, (b) the space (X, ≼, d) is regular.
Then f and g have a common fixed point.
Proof.Following similar arguments as those given in the proof of Theorem 2.1, we construct an increasing sequence {x n } in X such that lim n→+∞ x n = u for some u ∈ X.As A and B are closed subsets of X, we have u ∈ A ∩ B. By using the given assumption on X, we have x n ≼ u for all n ∈ N. Now, we show that f u = gu = u.Putting x = x 2n and y = u International Scientific Publications and Consulting Services in (2.1) and by taking the upper limit as n → +∞ and using Lemma 1.1 and the properties of ψ and φ, we obtain This yields that lim inf n→+∞ d(x 2n , gu) = 0 by our assumptions about φ.By applying the triangular inequality, we have Now, letting n → +∞ in the above inequality, we conclude that d(u, gu) = 0. Thus, we have u = gu.Hence, u is a fixed point of g.On the other hand, similar to the first part of the proof of Theorem 2.1, we can show that f u = u.Therefore, u is a common fixed point of f and g.
By setting A = B = X in Theorems 2.1 and 2.2, we get the following result.
Corollary 2.1.Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1 and suppose that f , g : X → X be two weakly increasing mappings with respect to ≼. Assume that (a) there exist two functions ψ ∈ Ψ and φ ∈ Φ such that for all comparable elements x, y ∈ X, we have Then f and g have a common fixed point.
By taking ψ(t) = t and φ(t where k ∈ [0, 1 s+1 ) in Theorems 2.1 and 2.2, we obtain immediately the following result.By putting f = g in Theorems 2.1 and 2.2, we get the following result.
Corollary 2.3.Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1 and suppose that f : X → X be a nondecreasing mapping with respect to ≼. Assume that (a) X = A ∪ B is a cyclic representation of X with respect to f , that is, f A ⊆ B and f B ⊆ A, (b) there exist the functions ψ ∈ Ψ and φ ∈ Φ such that for all comparable elements x, y ∈ X with x ∈ A and y ∈ B, we have If there exists x 0 ∈ X such that x 0 ≼ f x 0 , then f has a fixed point.Now, in order to support the usability of our main results, we present the following example.
Thus, we have the following: (1) (X, ≼, d) is a partially ordered b-complete b-metric space.
(2) A ∪ B is a cyclic representation of X with respect to the pair ( f , g).
(4) f and g are continuous.
(5) The pair ( f , g) is a cyclic (ψ, φ, A, B)-weakly Chatterjea type contraction, that is, for all comparable elements x, y ∈ X with x ∈ A and y ∈ B, we have Proof.The proof of (1) is clear.Since f A = [0, 1  6 ] ⊆ B and gB = [0, 1  14 ] ⊆ A, it follows that X = A ∪ B is a cyclic representation of X with respect to the pair ( f , g).So (2) holds.To prove (3), given x ∈ A. Since x we have f x ≼ g f x for all x ∈ A. Similarly, one can show that gx ≼ f gx for all x ∈ B. Thus, The pair ( f , g) is (A, B)weakly increasing.It is easy to see that f and g are continuous.So (4) holds.
To prove (5), given two comparable elements x, y ∈ X with x ∈ A and y ∈ B. Then the following cases are possible.
Hence, we have  International Scientific Publications and Consulting Services

Corollary 2 . 2 . 2 (
Let (X, ≼, d) be a partially ordered b-complete b-metric space with parameter s ≥ 1 and suppose that A and B be closed subsets of X.Let f , g : X → X be two (A, B)-weakly increasing mappings with respect to ≼. Assume that (a) X = A ∪ B is a cyclic representation of X with respect to the pair ( f , g), (b) there exists k ∈ [0, 1 s+1 ) such that for all comparable elements x, y ∈ X with x ∈ A and ∈ B, we haved( f x, gy) ≤ k s d(x, gy) + d(y, f x) ) , (c) f or g is continuous, or (c ′ ) the space (X, ≼, d) is regular.Then f and g have a common fixed point u ∈ A ∩ B.