Weakly compatible mappings with respect to a generalized c-distance and common fixed point results

Abstract In this paper, we consider weakly compatible mappings with respect to a generalized -distance in cone -metric spaces and obtain new common fixed-point theorems. Our results provide a more general statement, since we need not to nor the continuity of mappings and nor the normality of cone. In particular, we refer to the results of M. Abbas and G. Jungck [Common fixed point results for non-commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008) 416–420]. Some corollaries and examples are presented to support the main result proved herein.


Introduction
In 1976, Jungck (Jungck, 1976) proved a common fixed point theorem for two commuting mappings. This theorem has many applications but it requires the continuity of one of the two mappings. Then, Sessa (Sessa, 1982) defined the concept of weakly commuting to obtain common fixed point for a pair of mappings. Jungck generalized the idea of commuting mappings, first to compatible mappings (Jungck, 1988) and then to weakly compatible mappings (Jungck, 1996). In the sequel, Jungck and Rhoades (Jungck & Rhoades, 2006) proved some fixed and common fixedpoint theorems for noncommuting and compatible mappings in metric spaces (also, see  and references therein).

Reza Babaei
ABOUT THE AUTHOR Hamidreza Rahimi is Professor of mathematics at Central Tehran Branch, Islamic Azad University, Tehran, Iran. He received his M.S from Sharif University in 1993 in non-commutative algebra and he received his Ph.D. degree in mathematics from Science and Research Branch, IAU, in 2003 in Harmonic Analysis. His research interest includes Harmonic Analysis on semigroups and groups, and fixed point theory. He is editor-inchief of the Journal of Linear and Topological Algebra.

PUBLIC INTEREST STATEMENT
Fixed point theory is an important and useful tool for different branches of both mathematical analysis and nonlinear analysis. Accordingly, from when Banach introduced his famous principle in 1929, fixed-point theory and its application in various metrics and different distances have been developed by other scholars. One of these spaces and distances is generalized $c$distance in cone $b$-metric spaces introduced by Bao et al.
Ordered normed spaces and cones have many applications in applied mathematics. In particular, the usage of ordered normed spaces in functional analysis date back to 1940s. It seems that Kurepa (Kurepa, 1934) was the first to use ordered normed spaces as the codomain of a metric. Later on, such metric spaces appeared occasionally under names K-metric spaces, abstract metric spaces, and generalized metric spaces (see (Zabrejko, 1997)). In 2007, Huang and Zhang (Huang & Zhang, 2007) reintroduced such spaces under the name of cone metric spaces by substituting an ordered normed space for the real numbers. On the other hands, a new type of spaces which they called b-metric spaces are defined by Bakhtin (Bakhtin, 1989) and Czerwik (Czerwik, 1993). In the sequel, analogously with definition of a b-metric space and a cone metric space, Cvetković et al. (Ćvetković et al., 2011) and Hussain and Shah (Hussain & Shah, 2011) defined cone b-metric spaces.
In 1996, Kada et al. (Kada et al., 1996) introduced the concept of w-distance in metric spaces, where nonconvex minimization problems were treated. After that, Cho et al. (Cho et al., 2011) defined the concept of c-distance which is a cone version of the w-distance and proved some fixed-point theorems under c-distance in cone metric spaces (also, see (Fallahi et al., 2018;Rahimi & Soleimani Rad, 2014;). In 2014, Hussain et al. (Hussain et al., 2014) introduced the concept of wt-distance on a b-metric space. In the sequel, Bao et al. (Bao et al., 2015) defined generalized cdistance in cone b-metric spaces and obtained several fixed-point results in ordered cone b-metric spaces (also, see (Fadail & Bin Ahmad, 2015;Soleimani Rad et al., 2019)).
Definition 1.1 ( (Deimling, 1985;Huang & Zhang, 2007)). Let E be a real Banach space and P a subset of E. Then P is called a cone if and only if ðaÞ P is closed, non-empty and P�fθg; ðbÞ a; b 2 R; a; b � 0; x; y 2 P implies ax þ by 2 P and ðcÞ if x 2 P and À x 2 P, then x ¼ θ.
Given a cone P � E, a partial ordering � with respect to P is defined by x � y , y À x 2 P. We shall write x � y to mean x � y and x�y. Also, we write x � y if and only if y À x 2 int P (where int P is the interior of P). If int P�;, the cone P is called solid. A cone P is called normal if there exists a number K > 0 such that θ � x � y implies that k x k� K k y k for all x; y 2 E. Definition 1.2 ((Ćvetković et al., 2011;Hussain & Shah, 2011)). Let X be a nonempty set, E be a real Banach space equipped with the partial ordering � with respect to the cone P � E and θ be the zero vector of E. Suppose that a mapping d : X � X ! E satisfies the following conditions: (d 1 ) θ � dðx; yÞ for all x; y 2 X and dðx; yÞ ¼ θ if and only if x ¼ y; (d 2 ) dðx; yÞ ¼ dðy; xÞ for all x; y 2 X; (d 3 ) dðx; zÞ � s½dðx; yÞ þ dðy; zÞ� for all x; y; z 2 X.
Then, d is called a cone b-metric and ðX; dÞ is called a cone b-metric space (or cone metric type space).
Obviously, for s ¼ 1, the cone b-metric space is a cone metric space. Moreover, if X is any nonempty set, E ¼ R and P ¼ ½0; 1Þ, then cone b-metric on X is a b-metric on X. For notions such as convergent and Cauchy sequences, completeness, continuity, and etc in cone b-metric spaces, we refer to (Ćvetković et al., 2011;Hussain & Shah, 2011). Also, we use of the following properties for all u; v; w; c 2 E when the cone P may be non-normal.
(p 1 ) If u � v and v � w, then u � w.
(p 4 ) Let a n ! θ in E, θ � a n and θ � c. Then, there exists a positive integer n 0 such that a n � c for each n > n 0 . Definition 1.3 ((Bao et al., 2015)). Let ðX; dÞ be a cone b-metric space with parameter s � 1. A mapping q : X � X ! E is said to be a generalized c-distance on X if for any x; y; z 2 X; the following properties are satisfied: (q 1 ) θ � qðx; yÞ; (q 2 ) qðx; zÞ � s½qðx; yÞ þ qðy; zÞ�; (q 3 ) q is b-lower semi-continuous in its second variable i.e., if qðx; y n Þ � u for all n � 1 and for some u ¼ u x , then qðx; yÞ � su, where fy n g is a sequence in X which converges to y 2 X; (q 4 ) for any c 2 int P, there exists e 2 E with θ � e such that qðz; xÞ � e and qðz; yÞ � e imply that dðx; yÞ � c. (Cho et al., 2011). Also, set s ¼ 1, E ¼ R and P ¼ ½0; 1Þ in the above definition. Then, we obtain the definition of w-distance (Kada et al., 1996) (for more details, see ). Moreover, for any generalized c-distance q, qðx; yÞ ¼ θ is not necessarily equivalent to x ¼ y for all x; y 2 X and qðx; yÞ ¼ qðy; xÞ does not necessarily hold for all x; y 2 X.
and consider the non-normal cone P ¼ fx 2 E : xðtÞ � 0 for all t 2 ½0; 1�g. Also, let X ¼ ½0; 1Þ and define a mapping d : Lemma 1.5 ((Soleimani Rad et al., 2019)). Let ðX; dÞ be a cone b-metric space and q be a generalized c-distance on X. Let fx n g and fy n g be sequences in X, fu n g and fv n g be two convergent sequences in P. For any x; y; z 2 X, (qp 1 ) if for all n 2 N, qðx n ; yÞ � u n and qðx n ; zÞ � v n , then y ¼ z. In particular, if qðx; yÞ ¼ θ and qðx; zÞ ¼ θ, then y ¼ z; (qp 2 ) if for all n 2 N, qðx n ; y n Þ � u n and qðx n ; zÞ � v n , then fy n g converges to z; (qp 3 ) if for m; n 2 N, with m > n; we have qðx n ; x m Þ � u n , then fx n g is a Cauchy sequence in X; (qp 4 ) if for all n 2 N, qðy; x n Þ � u n then fx n g is a Cauchy sequence in X. Definition 1.6 ( (Jungck & Rhoades, 2006)). Let f and g be two self-mappings defined on a set X. If fw ¼ gw ¼ z for some w 2 X, then w is called a coincidence point of f and g, and z is called a point of coincidence of f and g. Also, the mappings f and g are said to be weakly compatible if they commute at every coincidence point; that is, if fgw ¼ gfw for all coincidence points w. Lemma 1.7 ((Abbas & Jungck, 2008)). Let f and g be weakly compatible self-mappings on a set X. If f and g have a unique point of coincidence z ¼ fw ¼ gw, then z is the unique common fixed point of f and g.

Main results
Our main result is the following theorem. We prove a common fixed point theorem under the concept of a generalized c-distance on cone b-metric spaces without assumption of normality for a cone.
Now, by a similar procedure in Lemma 1.7, we can prove w is a unique common fixed point as follows. Since f and g are weakly compatible and w ¼ fz ¼ gz, we obtain fw ¼ fgz ¼ gfz ¼ gw; that is, fw ¼ gw is a point of coincidence f and g. But w is unique point of coincidence of f and g. Thus, w ¼ fw ¼ gw. Also, if z ¼ fz ¼ gz, then z is a point of coincidence of f and g. Therefore, by uniqueness, z ¼ w; i.e., w is a unique common fixed point of f and g.
The following corollary can be obtained as consequences of Theorem 2.6 which are the extension of some results of Abbas and Jungck (Abbas & Jungck, 2008) under the concept of a generalized c-distance in cone b-metric spaces over a solid cone and by applying control function instead of constant coefficient. Corollary 2.3. Let ðX; dÞ be a cone b-metric space over a solid cone P with given real number s � 1. Also, let q be a generalized c-distance and f ; g : X ! X be two mappings with f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X. Suppose that there exists mapping α : X ! ½0; 1Þ such that αðfxÞ � αðgxÞ for all x 2 X and qðfx; fyÞ � αðgxÞqðgx; gyÞ for all x; y 2 X, where sαðxÞ < 1. Then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.
In Theorem 2.1 and Corollary 2.3, set s ¼ 1. Then, we obtain the following result in the framework of cone metric spaces associated with a c-distance.
Corollary 2.5. Let ðX; dÞ be a cone metric space over a solid cone P. Also, let q be a c-distance and f ; g : X ! X be two mappings with f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X. Suppose that there exists mapping α : X ! ½0; 1Þ such that αðfxÞ � αðgxÞ for all x 2 X and qðfx; fyÞ � αðgxÞqðgx; gyÞ for all x; y 2 X where αðxÞ < 1. Then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.

□
The following corollaries can be obtained as consequences of Theorem 2.6 which are the extension of some results of Abbas and Jungck (Abbas & Jungck, 2008), and Shi and Xu (Shi & Xu, 2013) under the concept of a generalized c-distance in cone b-metric spaces over a solid cone. These are same Corollary 4.4 and Corollary 4.5 of Fadail and Bin Ahmad (Fadail & Bin Ahmad, 2015).
Corollary 2.7. Let ðX; dÞ be a cone b-metric space over a solid cone P with given real number s � 1 and q be a generalized c-distance. Suppose that there exist two mappings f ; g : X ! X such that qðfx; fyÞ � αqðgx; gyÞ for all x; y 2 X, where α 2 ½0; 1 s Þ. If f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X, then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.
Corollary 2.9. Let ðX; dÞ be a cone b-metric space over a solid cone P with given real number s � 1 and q be a generalized c-distance. Suppose that there exist two mappings f ; g : X ! X such that qðfx; fyÞ � δqðgx; fxÞ þ γqðgy; fyÞ for all x; y 2 X, where δ and γ are nonnegative coefficients with sδ þ γ < 1. If f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X, then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.
In Theorem 2.6 and its corollaries, set s ¼ 1. Then, we obtain the following results in the framework of cone metric spaces associated with a c-distance. These are same Theorem 3.1, Corollary 3.1 and Corollary 3.2 of Fadail et al. (Fadail et al., 2013).
The following corollaries can be obtained as consequences of Theorem 2.10 which are the extension of some results of Abbas and Jungck (Abbas & Jungck, 2008) under the concept of a c-distance.
Corollary 2.11. Let ðX; dÞ be a cone metric space over a solid cone P and q be a c-distance. Suppose that there exist two mappings f ; g : X ! X such that qðfx; fyÞ � αqðgx; gyÞ for all x; y 2 X, where α 2 ½0; 1Þ. If f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X, then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.
Corollary 2.12. Let ðX; dÞ be a cone metric space over a solid cone P and q be a c-distance. Suppose that there exist two mappings f ; g : X ! X such that qðfx; fyÞ � δqðgx; fxÞ þ γqðgy; fyÞ for all x; y 2 X, where δ and γ are nonnegative coefficients with δ þ γ < 1. If f ðXÞ � gðXÞ and gðXÞ be a complete subspace of X, then f and g have a coincidence point z 2 X. Moreover, if w ¼ gz ¼ fz, then qðw; wÞ ¼ θ. Also, if f and g are weakly compatible, then f and g have a unique common fixed point.

Conclusion and suggestion
Here, we considered the concept of weakly compatible mappings with respect to a generalized cdistance in cone b-metric spaces and proved several fixed-point theorems. Our results are significant, since (1) the class of generalized c-distance in cone b-metric spaces is bigger than of the class of usual c-distance in cone metric spaces. Hence, the authors can prove their results with respect to a c-distance without complete and repetitive proof (by considering s ¼ 1 in generalized c-distance).
(2) the class of generalized c-distance in cone b-metric spaces is bigger than of the class of usual wt-distance in b-metric spaces. Hence, the authors can prove their results with respect to a wt-distance without complete and repetitive proof (by considering E ¼ R and P ¼ ½0; þ1Þ in generalized c-distance).
(3) we need not to nor the continuity of mapping and nor the normality of cone in the procedure the proof of main results.
To continue this paper, the readers can consider some former researches from 2007 until now and can obtain new results with respect to this distance with its application.