C-class functions with new approach on coincidence point results for generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\psi ,\varphi )$$\end{document}(ψ,φ)-weakly contractions in ordered b-metric spaces

In this paper, by using the C-class functions and a new approach we present some coincidence point results for four mappings satisfying generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\psi ,\phi )$$\end{document}(ψ,ϕ)-weakly contractive condition in the setting of ordered b-metric spaces. Also, an application and example are given to support our results.

a complete metric space. If f , g : X → X are generalized ϕ-weak contractive mappings, then there exists a unique point u ∈ X such that u = fu = gu.
The concept of b-metric space was introduced by Czerwik in Czerwik (1998). Since then, several papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in b-metric spaces (see also Akkouchi 2011;Boriceanu 2009a, b;Boriceanu et al. 2010;Bota et al. 2011;Hussain et al. 2012;Hussain and Shah 2011;Olatinwo 2008;Mustafa 2014;Pacurar 2010;Ansari et al. 2014).
Definition 8 (Jungck 1996) Let f , g : X → X be given self-mappings on X. The pair (f, g) is said to be weakly compatible if f and g commute at their coincidence points (i.e., fgx = gfx, whenever fx = gx).
Definition 9 Let (X, ) be a partially ordered set and d be a metric on X. We say that (X, d, ) is regular if the following conditions hold: (1) If a non-decreasing sequence x n → x, then x n x for all n.
(2) If a non-increasing sequence y n → y, then y n y for all n.
In Nashine and Samet (2011), established some coincidence point and common fixed point theorems for mappings satisfying a generalized weakly contractive condition in an ordered complete metric space by considering a pair of altering distance functions (ψ, ϕ) . In fact, they proved the following theorem.
Theorem 11 (Nashine and Samet 2011 Theorem 2.4.) Let (X, ) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Let T , R : X → X be given mappings satisfying for every pair (x, y) ∈ X × X such that Rx and Ry are comparable, where ψ and ϕ are altering distance functions. We suppose the following hypotheses: (i) T and R are continuous, (ii) TX ⊆ RX, (iii) T is weakly increasing with respect to R, (iv) the pair (T, R) is compatible.
Then, T and R have a coincidence point, that is, there exists u ∈ X such that Ru = Tu.
Further, they showed that by replacing the continuity hypotheses on T and R with the regularity of (X, d, ) and omitting the compatibility of the pair (T, R), the above theorem is still valid (see, Theorem 2.6 of Nashine and Samet 2011).
Also, in Shatanawi and Samet (2011), Shatanawi and Samet studied common fixed point and coincidence point for three self mappings T, S and R satisfying (ψ, ϕ)-weakly contractive condition in an ordered metric space (X, d), where S and T are weakly increasing with respect to R and ψ, ϕ are altering distance functions. Their result generalize Theorem 11. ψ(d(Tx, Ty)) ≤ ψ(d(Rx, Ry)) − ϕ(d(Rx, Ry)), Analogous to the work in Nashine and Samet (2011), Shatanawi and Samet proved the above result by replacing the continuity hypotheses of T, S and R with the regularity of X and omitting the compatibility of the pair (T, R) and (S, R) (See, Theorem 2.2 of Shatanawi and Samet 2011).
Consistent with Czerwik (1998),  and Singh and Prasad (2008), the following definitions and results will be needed in the sequel.
Definition 12 (Czerwik 1998) Let X be a (nonempty) set and s ≥ 1 be a given real number. A function d : X × X → R + is a b-metric iff, for all x, y, z ∈ X, the following conditions are satisfied: The pair (X, d) is called a b-metric space. Note that, the class of b-metric spaces is effectively larger than the class of metric spaces, since a b-metric is a metric, when s = 1.
The following example shows that in general a b-metric need not necessarily be a metric (see, also, Singh and Prasad 2008, p. 264).
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 not a metric on R. Definition 14 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, ).
Definition 15 (Boriceanu et al. 2010) Let (X, d) be a b-metric space. Then a sequence {x n } in X is called b-convergent if and only if there exists x ∈ X such that d(x n , x) → 0, as n → +∞. In this case, we write lim n→∞ x n = x. (i) A b-convergent sequence has a unique limit.
(ii) Each b-convergent sequence is b-Cauchy. (iii) In general, a b-metric need not be continuous.
In 2014 Ansari (2014) introduced the concept of C-class functions which cover a large class of contractive conditions. Definition 21 (Ansari 2014) A mapping F : [0, ∞) 2 → R is called a C-class function if it is continuous and satisfies following axioms: (1) F (r, t) ≤ r; (2) F (r, t) = r implies that either r = 0 or t = 0; for all r, t ∈ [0, ∞).
We denote a C-class functions as C.
Motivated by the works in Nashine and Samet (2011), Jamal (2015), In this paper, by using the C-class functions and a new approach, we present some coincidence point results for four mappings satisfying generalized (ψ, φ) -weakly contractive condition in the setting of ordered b-metric spaces where ψ is altering distance function and ϕ is Ultra-altering distance function. Also, an application and example are given to support our results.
Theorem 24 Let (X, , d) be an ordered complete b-metric space (with parametr s > 1). Let f , g, T , h : X → X be four mappings such that f (X) ⊆ T (X) and g(X) ⊆ h(X). Suppose that for every x, y ∈ X with comparable elements hx, Ty, there exists N (x, y) such that where ψ is altering distance function and ϕ is Ultra altering distance function, a > 1 and F is a C-class function such that F is increasing with respect to first variable and decreasing with respect to second variable. Let f, g, T and h are continuous, the pairs (f, h) and (g, T) are compatible and the pairs (f, g) and (g, f) are partially weakly increasing with respect to T and h , respectively. Then, the pairs (f, h) and (g, T) have a coincidence point w in X.

Moreover, if Rw and Sw are comparable, then w is a coincidence point of f, g, T and h.
Proof Let x 0 ∈ X be an arbitrary point. Since f (X) ⊆ T (X) and g(X) ⊆ h(X), one can find x 1 , x 2 ∈ X such that fx 0 = Tx 1 and gx 1 = hx 2 .
Continuing this process, we construct a sequence {w n } defined by: and for all n ≥ 0.
Since, x 1 ∈ T −1 (fx 0 ) and x 2 ∈ h −1 (gx 1 ), and the pairs (g, f) and (f, g) are partially weakly increasing with respect to T and h, respectively, we have, Repeating this process, we obtain w 2n+1 � w 2n+2 for all n ≥ 0. The proof will be done in three steps.
Step I We will show that lim k→∞ d(w k , w k+1 ) = 0.
From definition of F, and condition (2) we get that Thus, from the monotonocity increasing of ψ we have for all k ≥ 0 Analogously, in all cases, we see that {d(w k , w k+1 )} is a non-increasing sequence of nonnegative real numbers. Therefore, there is an r ≥ 0 such that We know that, Taking the limit as n → ∞ in above and (9), we have which implies that, Step II Using 10 and Lemma (23) we get {w n } is a b-Cauchy sequence in X.
Step III In this step we prove that f, g, T and h have a coincidence point. Since {w n } is a b-Cauchy sequence in the complete b-metric space X, there exists w ∈ X such that and Hence, Moreover, from lim n→∞ d(fx 2n , w) = 0, lim n→∞ d(hx 2n , w) = 0 and the continuity of h and f, we obtain, By the triangle inequality, we have, Taking the limit as n → ∞ in (18), we obtain that which yields that fw = hw, that is w is a coincidence point of f and h.

Corollary 25
Let (X, , d) be an ordered complete b-metric space (with parametr s > 1 ). Let f , g, T , h : X → X be four mappings such that f (X) ⊆ T (X) and g(X) ⊆ h(X In the following theorem, we replace the compatibility of the pairs (f, h) and (g, T) by weak compatibility of the pairs and we omit the continuity assumption of f, g, T and h and Theorem 26 Let (X, , d) be a regular partially ordered b-metric space (with parametr s > 1), f , g, T , h : X → X be four mappings such that f (X) ⊆ T (X) and g(X) ⊆ h(X) and TX and hX are complete subsets of X. Suppose that for comparable elements hx, Ty ∈ X, we have, where ψ is altering distance function and ϕ is Ultra altering distance function and a>1 and F is C-class function such that F is increasing with respect to first varaible. Then, the pairs (f, h) and (g, T) have a coincidence point w in X provided that the pairs (f, h) and (g, T) are weakly compatible and the pairs (f, g) and (g, f) are partially weakly increasing with respect to T and h, respectively. Moreover, if Tw and hw are comparable, then w ∈ X is a coincidence point of f, g, T and h.
Proof Following to the construction of the sequence w n in the proof of Theorem (24), there exists w ∈ X such that ψ d(fw, gw) ≤ ψ s a d(fw, gw) Since T(X) is complete and {w 2n+1 } ⊆ T (X), this implies that w ∈ T (X). Hence, there exists u ∈ X such that w = Tu and Similarly, there exists v ∈ X such that w = Tu = hv and We prove that v is a coincidence point of f and h.
As f and h are weakly compatible, we have fw = fhv = hfv = hw. Thus, w is a coincidence point of f and h.
Similarly it can be shown that w is a coincidence point of the pair (g, T). The rest of the proof can be done using similar arguments as in Theorem 24.
Taking h = T in Theorem 24, we obtain the following result.

Corollary 27
Let (X, , d) be a partially ordered complete b-metric space (with parametr s > 1) and f , g, T : X → X be three mappings such that f (X) ∪ g(X) ⊆ T (X) and T is continuous. Suppose that for every x, y ∈ X with comparable elements Tx, Ty, we have, where ψ is altering distance function, ϕ is Ultra altering distance function, a >1 and F is C-class function such that F is increasing with respect to first variable. Then, f, g and T have a coincidence point in X provided that the pair (f, g) is weakly increasing with respect to T and either, a. the pair (f, T) is compatible and f is continuous, or, b. the pair (g, T) is compatible and g is continuous.
Taking T = h and f = g in Theorem 24, we obtain the following coincidence point result.

Corollary 28
Let (X, , d) be a partially ordered complete b-metric space (with parameter s > 1) and f , T : X → X be two mappings such that f (X) ⊆ R(X). Suppose that for every x, y ∈ X for which Tx, Ty are comparable, we have, where, ψ is altering distance function, ϕ is Ultra altering distance function, a >1 and F a is C-class function such that F is increasing with respect to first variable. Then, the pair (f, T) has a coincidence point in X provided that f and T are continuous, the pair (f, T) is compatible and f is weakly increasing with respect to T.
Example 29 Let F (r, t) = r 1+t , X = [0, ∞) and d on X be given by d(x, y) = x − y 2 , for all x, y ∈ X. We define an ordering " " on X as follows: Define self-maps f, g, h and T on X by To prove that (f, g) is partially weakly increasing with respect to T, let x, y ∈ X be such that y ∈ T −1 fx, that is, Ty = fx. By the definition of f and T, we have ln 1 + x = exp(7y) − 1 and y = ln(1+ln(1+x)) 7 . , Therefore, fx gy. Hence (f, g) is partially weakly increasing with respect to T.
Furthermore, fX = gX = hX = TX = [0, ∞) and the pairs (f, h) and (g, T) are compatible. Indeed, let {x n } is a sequence in X such that lim n→∞ d(t, fx n ) = lim n→∞ d(t, hx n ) = 0, for some t ∈ X. Therefore, we have, Continuity of ln x and exp(21x) − 1 on X implies that, and the uniqueness of the limit gives that exp(t) + 1 = ln t+1 21 . But, So, we have t = 0. Since f and h are continuous, we have Define ψ, ϕ : [0, ∞) → [0, ∞) as ψ(t) = 441 256 t and ϕ(t) = 313 128 for all t ∈ [0, ∞). Using the mean value theorem for the functions ln(1 + z) and exp(z) on the intervals [x, y 3 ] ⊂ X and [21x, 7y] ⊂ X, respectively, we have, Thus, (2) is satisfied for all x, y ∈ X with a = 7 and N (x, y) = d(hx, Ty). Therefore, all the conditions of Theorem 24 are satisfied. Moreover, 0 is a coincidence point of f, g, T and h.
Corollary 30 Let (X, , d) be a regular partially ordered b-metric space (with parametr s > 1), f , g, T : X → X be three mappings such that f (X) ⊆ T (X) and g(X) ⊆ T (X) and TX is a complete subset of X. Suppose that for comparable elements Tx, Ty ∈ X, we have, lim n→∞ d(fhx n , hfx n ) = lim n→∞ fhx n − hfx n 2 = 0.
where where ψ is altering distance function, ϕ is Ultra altering distance function, a >1 and F is C-class function such that F is increasing with respect to first variable. Then, the pairs (f, T) and (g, T) have a coincidence point w in X provided that the pair (f, g) is weakly increasing with respect to T.
Corollary 31 Let (X, , d) be a regular partially ordered b-metric space (with parameter s > 1), f , T : X → X be two mappings such that f (X) ⊆ T (X) and TX is a complete subset of X. Suppose that for comparable elements Tx, Ty ∈ X, we have, where, ψ is altering distance function, ϕ is Ultra altering distance function, a >1 and F is C-class function such that F is increasing with respect to first variable. Then, the pair (f, T) have a coincidence point w in X provided that f is weakly increasing with respect to T.
Taking T = h = I X (the identity mapping on X) in Theorems 24 and 26, we obtain the following common fixed point result.
Corollary 32 Let (X, , d) be a partially ordered complete b-metric space(with parametr s > 1). Let f , g : X → X be two mappings. Suppose that for every comparable elements x, y ∈ X, where, ψ is altering distance function, ϕ is Ultra altering distance function, a >1 and F is C-class function such that F is increasing with respect to first variable. Then, the pair (f, g) have a common fixed point w in X provided that the pair (f, g) is weakly increasing and either, a. f or g is continuous, or, b. X is regular.
Define mappings f , g : X → X by Proof Clearly from condition (4), the mappings f, g are weakly increasing with respect to . Let X and f, g be as defined above. For all x, y ∈ X define the b-metric on X by Clearly that (X, d) is a complete b-metric space with constant (s = 2). Moreover, in Nieto and Rodaiguez-Loez (2007) it is proved that (X, ) is regular.