Some Valid Generalizations of Boyd and Wong Inequality and (ψ,φ)-Weak Contraction in Partially Ordered b−Metric Spaces

Department of Mathematics, University of Malakand, Chakdara, Pakistan Department Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Department of Mathematics, GMR Institute of Technology, Rajam 532 127, Andhra Pradesh, India


Introduction
Banach contraction principle [1] is one of the most important result in analysis. It is widely considered as the source of fixed point theory. Banach principle has been modified by many researchers by changing either the contractive condition or the underlying spaces. Boyd and Wong [2] modified the contraction condition in metric spaces (X, d) with control function d(ft, fb) ≤ ψ(d (t, b)), for all t, b in X.
(1) e ϕ-weak contraction condition was initiated by Alber and Guerre-Delabriere in the Hilbert spaces, see [3]. Rhoudes, in [4], showed that every ϕ-weak contractive mapping in metric spaces has a unique fixed point. e generalized ϕ-weak contractive condition in complete metric spaces was introduced by Zhang and Song in [5]. In [6], Dutta and Choudhury studied unique fixed point results with the help of (ψ, ϕ)-weak contraction in metric spaces. Doric, in [7], studied the result of [6] with generalized (ψ, ϕ)-weak contraction in metric spaces. e concept of fixed point results for two pairs of mappings was introduced by Jungck in [8], for commuted mappings. Jungck in [8,9] states this new assumption, with compatibility and weak compatibility of mappings. Some further work for generalized (ψ, ϕ)-weak contraction in metric spaces can be found in [6,10,11]. Ran and Reurring, in [12], introduced partially ordered metric spaces. A b-metric space with partial ordering is called partially ordered b-metric space. Recently, (ψ, ϕ)-weak contraction in partially ordered b-metric spaces gain the attention of many researcher. In this direction, fixed point and coincidence point results are discussed by many authors. For more details of fixed point and coincidence point results and their applications, comparison of different contraction conditions, and related results in b-metric spaces, we refer the reader to  along with the references mentioned therein. To show that a sequence is Cauchy in metric-type spaces, Jovanovic et al. [41] proved Lemma 1. As b-metric spaces have discontinuous structure, therefore, in this manuscript, we give a generalization of Boyd and Wong [2] inequality and (ψ, ϕ)-weak contraction, to establish coincidence and fixed point results.

Preliminaries
Definition 1 (see [42]). Let (X, ≺ ) be a partially ordered set, and assume that the map d: X × X ⟶ R + satisfies the following conditions, For all b, t, r ∈ X and s ≥ 1: en, (X, d, ≺ ) is called partially ordered b-metric space.
Definition 2 (see [9]). e pair of mappings (f, g) is compatible in the metric space (X, d) if and only if lim m⟶∞ d fgn m , gfn m � 0, whenever n m in sequence such that Definition 3 (see [8]). In the metric space (X, d), the pair of mappings f, g: X ⟶ X is weakly compatible if (f, g) is commutative at the point of coincidence (i.e., gfn � fgn whenever fn � gn).
Definition 4 (see [43]). e pair of mappings f, g: X ⟶ X defined on a partially ordered set X is called (a) Weakly increasing if fn ≺ g(fn) and gn ≺ f(gn), for all n ∈ X (b) Partially weakly increasing if ∀n ∈ X, fn ≺ g(fn) Definition 5 (see [44,45]). Let f, g, h: X ⟶ X be three mappings on partially ordered set (X, ≺ ) such that fn ≺ hn and gn ≺ hn. en, pair (f, g) is said to be (i) Weakly increasing with respect to h if and only if for all n ∈ X, fn ≺ gw, for all w ∈ h − 1 (fn) and gn ≺ fw, and for all w ∈ h − 1 (gn) (ii) Partially weakly increasing w.r.t h if and only if fn ≺ gw for all w ∈ h − 1 (fn).
Theorem 1 (see [45]). Let self-mappings f, g be continuous mappings in partially ordered complete metric space (X, ≺ , d) with f(X)⊆g(X). Assume that the pair (f, g) is weakly compatible, whereas f is partially weakly increasing with respect to g such that gb and gt are comparable elements and satisfying the following condition: where ϕ, ψ: [0, ∞) ⟶ [0, ∞) are altering distance functions. en, mappings f, g have coincidence point.
Theorem 2 (see [46]). Let f, g: X ⟶ X be mappings on a partially ordered complete b-metric space (X, ≺ , d). Assume that f, g is partially weakly increasing and satisfying the following condition: where ϕ, ψ: [0, ∞) ⟶ [0, ∞) are altering distance and lower semicontinuous functions, respectively, with ϕ(0) � 0 and where s ≥ 1. Furthermore, assume that either f or g is continuous mapping or space X is regular. en, f, g have a common fixed point.
Theorem 3 (see [19]). "Let (X, ≺ , d) be partially ordered complete b-metric space and f, g, h, and k: X ⟶ X be continuous mappings such that f(X) ⊆ h(X) and g(X) ⊆ k(X). Suppose for comparable elements kb and ht, the following condition hold Theorem 4 (see [18]). "Let f, g, h, k: X ⟶ X be continuous mappings on a partially ordered complete b-metric space (X, ≺, d) with f(X) ⊆ h(X) and g(X) ⊆ k(X). Assume that compatible pairs (f, h) and (g, k) and comparable elements kt and hb satisfy the following condition:  [20]). Let the mappings f, g: X ⟶ X be defined on a partially ordered complete b-metric space (X, ≺ , d) and f be g-weakly isotone increasing. Let ψ: [0, ∞) ⟶ [0, ∞) be a function such that ψ(n) < n for all n > 0 and ψ(0) � 0 satisfy the condition: where s ≥ 1. If the mappings f or g are continuous in X. en, (f, g) has a common fixed point.

Main Results
We begin with the following result.
International Journal of Mathematics and Mathematical Sciences erefore, Using triangle inequality and assumption (19), we have Since ψ(n) ≤ n for all n > 0, thus, from (17), one has is is again contradiction. Hence, for all cases, we conclude that Step II: in this step, we have to prove that the sequence b m is b-Cauchy by using Lemma 1, for all three cases.
From (17), one can write As ψ(n) ≤ n for n > 0, so Taking K � 1/s ϵ , we have (17), one has Since ψ(n) ≤ n, for n > 0, therefore us, Let K � 1/s ϵ . en, Using triangle inequality and (26), it is reduced to case (2). erefore, (2) holds for three cases. Similarly, if Ft 2m+1 � b 2m+1 and Gt 2m+2 � b 2m+2 , one can easily show that Hence, Define en, from (17), Since ψ(n) ≤ n for all n > 0 and ψ(0) � 0, therefore, is is only Similarly, one can show for the remaining two cases. By the same process, if Consequently, the subsequences will also converge to b ∈ X: Step III: now, we show that b is a coincidence point of Taking limit m ⟶ ∞ and using (41) and (42) (11), one can write where is is possible if Ab � Bb. Hence, Ab � Bb � Fb � Gb. On the same process, from the other two cases, we can show that Ab � Bb � Fb � Gb.
In the following, the result condition of continuity and compatibility is relaxed for mappings.
where ϵ > 1 and ψ: Proof. Following the lines of the proof of eorem 6, one can easily show that there exists a sequence b m which converges to some b ∈ X. erefore, If b 2m+1 ⊆F(X) and b 2m+2 ⊆G(X) but F(X) and G(X) are b-closed subsets of X, then there must be some x, k ∈ X, from which b � Fx and b � Gk. Hence, from the construction of the sequence given in eorem 6, we have Now, we prove that Ab � Gb. From regularity of X, we have Ft 2m+1 ≺ Gk and Bt 2m+1 ≺ Gk.
us, from (49), one can write

Remark 2. As
In the remaining part of this manuscript, we discuss coincidence point of two compatible pairs of mappings with generalized (ψ, ϕ)-weak contractive condition.
roughout the rest of this paper, we consider the following, ϵ > 1, and for all t, b in X, we define  A, B) and (B, A) are partially weakly increasing with respect to F and G, respectively, and satisfy the following condition: Proof. Let t 0 ∈ X. Since A(X)⊆F(X) and B(X)⊆G(X), so there exist t 1 , t 2 ∈ X such that At 0 � Ft 1 and Bt 1 � Gt 2 . Construct the sequence b m as follows: Since the pairs (A, B) and (B, A) are partially weakly increasing with respect to F and G, therefore, Repeating the above process, we can write We discuss the proof in three steps.
Step I: first, we prove that where Here, we discuss three possible cases of M(t 2m , t 2m− 1 ).
As ψ is nondecreasing, therefore which contradicts assumption (69). us, is is only possible if ϕ (d(b 2m , b 2m+1 )) � 0. However, (76) Using triangle inequality and (69), we have en, International Journal of Mathematics and Mathematical Sciences erefore, in all three cases, we concluded that Step II: in this step, we will show that the sequence b m is b-Cauchy sequence by using Lemma 1 for all three cases.
en, by using triangle inequality and (79) it will be converted to case (2). erefore, (2) holds for all cases. Similarly, if Ft 2m+1 � b 2m+1 and Gt 2m+2 � b 2m+2 , again we have Hence, (89) en, from (64), From the above, we can obtain Since ψ is nondecreasing, therefore erefore, the sequence b k is constant sequence for k ≥ k 0 . erefore, (2) also holds for constant sequence b k . us, by Lemma 1, b m is a b-Cauchy sequence. From the completeness of X one can say that b-Cauchy sequence b m converges to some b in X.
Step III: now, we show that b is a coincidence point of A and F: where   F and G, respectively, whereas (A, G) and (B, F) are weakly compatible and satisfy the following condition: (M(t, b)), ∀t, b ∈ X.
(104) en, these four mappings have a coincidence point. If Gb and Fb are comparable then Ab � Bb � Fb � Gb.
Proof. As eorem 8, one can easily construct a sequence b m which converges to some b ∈ X. us, If b 2m+1 ⊆F(X), b 2m+2 ⊆G(X). Since F(X) and G(X) are b-closed subsets X, therefore, there exist some x, k ∈ X, such that b � Fx and b � Gk. Hence, Now, we prove that Ab � Gb. Since (X, ≺ , d) is regular, therefore Hence, (104) implies that    M(t, b)) − ϕ (M(t, b)), ∀t, b ∈ X, either A and F are continuous or space X is regular. en, A, F have a coincidence point. If we put F � G � I (identity mapping) in eorem 8, then the following corollary is obtained with Corollary 5. Let A, B: X ⟶ X be mappings on (X, ≺ , d) such that the pair (A, B) is partially weakly increasing and satisfies the following condition: either A and B are continuous or space X is regular. en, A, B have a common fixed point.

Example 2.
Let t ≺ b if and only if b ≤ t, for all t, b in X, is partially ordering on X � [0, ∞). Define b-metric on X by d(t, b) � |t − b| 2 , with s � 2. Suppose A, B, F, and G be continuous mappings defined by First, we check the compatibility of (A, F). Let t a be sequence which converges to some a ∈ X. en,