@ Appl. Gen. Topol. 15, no. 1 (2014), 65-84doi:10.4995/agt.2014.1823 c© AGT, UPV, 2014 Unified common fixed point theorems under weak reciprocal continuity or without continuity Zoran Kadelburg a, Mohammad Imdad b and Sunny Chauhan c a University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Ser- bia. (kadelbur@matf.bg.ac.rs) b Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India. (mhimdad@yahoo.co.in) c Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor 246 701, Uttar Pradesh, India. (sun.gkv@gmail.com) Abstract The purpose of this paper is two fold. Firstly, using the notion of weak reciprocal continuity due to Pant et al. [Weak reciprocal conti- nuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(1), 181–190 (2011)], we prove unified common fixed point theorems for various variants of compatible and R-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. Secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable sub- spaces. The realized improvements in our results are also substantiated using appropriate examples. 2010 MSC: Primary 47H10; Secondary 54H25. Keywords: Metric space; compatible mappings; R-weakly commuting map- pings; weak reciprocal continuity; coincidentally commuting; im- plicit relation. 1. Introduction and preliminaries Indeed, Banach Contraction Principle is a fundamental and fruitful result of metric fixed point theory. After this classical result, several authors have Received 1 November 2013 – Accepted 22 January 2014 http://dx.doi.org/10.4995/agt.2014.1823 Z. Kadelburg, M. Imdad and S. Chauhan contributed to the vigorous development of metric fixed point theory in dif- ferent ways (e.g. [4, 7, 10, 11, 12, 17, 30]). As patterned in Jungck [13], a result on the existence of common fixed points generally involves conditions on commutativity, continuity and contraction along with a suitable condition on the containment of range of one mapping into the range of other. Hence, one is always required to improve one or more of these conditions to prove a new fixed point theorem. Jungck [13] obtained a common fixed point theorem for a pair of commuting mappings. In 1982, Sessa [29] formulated the notion of weak commutativity and established a common fixed point theorem for such pairs. Jungck [14] gen- eralized the notion of weakly commuting mappings by introducing the concept of compatible mappings. Thereafter, Pant [20] defined R-weak commutativity and proved common fixed point results for R-weakly commuting mappings in metric spaces. The study of common fixed points of non-compatible mappings is equally natural which was indeed noted in Pant [22]. In 1997, Pathak et al. [24] intro- duced the notions of R-weakly commuting mappings of types (Ag) and (Af ) and utilize the same to prove common fixed point theorems in metric spaces. In 1998, Pant [21] introduced the notion of reciprocal continuity and utilize the same to prove results on common fixed points which also remains a point of discontinuity of the involved mappings. Recently, Pant et al. [23] improved the notion of reciprocal continuity by introducing weak reciprocal continuity and observed that weak reciprocal continuity is applicable to compatible as well as to non-compatible mappings. In fact, they proved the following result. Theorem 1.1. Let f and g be weakly reciprocally continuous self-mappings of a complete metric space (X,d) such that (i) f(X) ⊆ g(X); (ii) d(fx,fy) ≤ ad(gx,gy) + bd(fx,gx) + cd(fy,gy), for some a,b,c ≥ 0 with a + b + c < 1 and for all x,y ∈ X. If f and g are either compatible or R-weakly commuting of type (Ag) or R- weakly commuting of type (Af ) then f and g have a unique common fixed point. The notions and definitions utilized thus far and also to be utilized in the sequel are presented in the following multitude of definitions: Definition 1.2. Let f,g : X → X be two self-mappings of a metric space (X,d). Then the pair (f,g) is said to be (1) commuting if fgx = gfx, for all x ∈ X, (2) weakly commuting [29] if d(fgx,gfx) ≤ d(fx,gx), for all x ∈ X, (3) R-weakly commuting [20] if there exists some real number R > 0 such that d(fgx,gfx) ≤ Rd(fx,gx) for all x ∈ X, (4) pointwise R-weakly commuting [20] if given x ∈ X there exists some real number R > 0 such that d(fgx,gfx) ≤ Rd(fx,gx), c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 66 Unified common fixed point theorems... (5) R-weakly commuting of type (Ag) [24] if there exists some real number R > 0 such that d(ffx,gfx) ≤ Rd(fx,gx) for all x ∈ X, (6) R-weakly commuting of type (Af ) [24] if there exists some real number R > 0 such that d(fgx,ggx) ≤ Rd(fx,gx) for all x ∈ X, (7) compatible [14] if lim n→∞ d(fgxn,gfxn) = 0 for each sequence {xn} in X such that lim n→∞ fxn = lim n→∞ gxn, (8) g-compatible [25] if lim n→∞ d(ffxn,gfxn) = 0 for each sequence {xn} in X such that lim n→∞ fxn = lim n→∞ gxn, (9) f-compatible [25] if lim n→∞ d(fgxn,ggxn) = 0 for each sequence {xn} in X such that lim n→∞ fxn = lim n→∞ gxn, (10) non-compatible [22] if there exists a sequence {xn} in X such that lim n→∞ fxn = lim n→∞ gxn but lim n→∞ d(fgxn,gfxn) is either nonzero or nonex- istent, (11) reciprocally continuous [21] if lim n→∞ fgxn = ft and lim n→∞ gfxn = gt whenever {xn} is a sequence such that lim n→∞ fxn = lim n→∞ gxn = t for some t ∈ X, (12) weakly reciprocally continuous [23] if lim n→∞ fgxn = ft or lim n→∞ gfxn = gt whenever {xn} is a sequence such that lim n→∞ fxn = lim n→∞ gxn = t for some t ∈ X. (13) coincidentally commuting [5] (or weakly compatible [15]) if they merely commute at their coincidence points. In the foregoing definitions, it is assumed that there is at least one sequence in the underlying space meeting the prescribed requirements. Definition 1.3 ([8]). Two families of self mappings {fi}mi=1 and {gk}nk=1 are said to be pairwise commuting if (1) fifj = fjfi for all i,j ∈ {1,2, . . . ,m}, (2) gkgl = glgk for all k,l ∈ {1,2, . . . ,n}, (3) figk = gkfi for all i ∈ {1,2, . . . ,m} and k ∈ {1,2, . . . ,n}. For more details on systematic comparisons and illustrations of earlier de- scribed notions, we refer to Murthy [19], Singh and Tomar [31], Pant et al. [23] and Kadelburg et al. [16]. In this paper, utilizing the notion of implicit relation due to Popa et al. [27], we prove common fixed point results for variants of compatible (g-compatible and f-compatible) and R-weakly commuting (R-weakly commuting of type (Ag) and R-weakly commuting of type (Af )) mappings, employing an implicit relation which is a slightly refined form of an implicit function due to Popa [26]. Further, we prove a more general result for coincidentally commuting mappings without any requirement of weak reciprocal continuity. Some related results are also derived besides furnishing illustrative examples which exhibit the superiority of our results over some of the known ones. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 67 Z. Kadelburg, M. Imdad and S. Chauhan 2. Implicit relations Implicit functions are often used very effectively to cover various contraction conditions in one go rather than proving a separate theorem for each contrac- tion condition. The first ever attempt to coin an implicit relation can be traced back to Popa [26]. In 2008, Ali and Imdad [1] introduced a new class of im- plicit functions which covers several classes of contractions conditions. Since then, many mathematicians utilized implicit relations with various properties to prove a number of fixed point theorems (e.g. [3, 27, 28]). In order to de- scribe the implicit function we intend to use here, let Φ be the family of lower semi-continuous functions φ : R6+ → R satisfying the following conditions: (φ1): φ is non-increasing in variables t5 and t6, (φ2): there exists h ∈ (0,1) such that for u,v ≥ 0, (φ2a): φ(u,v,v,u,u + v,0) ≤ 0 or (φ2b): φ(u,v,u,v,0,u + v) ≤ 0 implies u ≤ hv, (φ3): φ(u,u,0,0,u,u) > 0, for all u > 0. The following examples of functions φ ∈ Φ appeared in Popa [26] and Imdad and Ali [7]. Example 2.1. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3, t4, 1 2 (t5 + t6) } , where k ∈ (0,1). Example 2.2. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − t1(αt2 + βt3 + γt4) − ηt5t6, where α > 0, β,γ,η ≥ 0, α + β + γ < 1 and α + η < 1. Example 2.3. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 3 1 − αt21t2 − βt1t3t4 − γt25t6 − ηt5t26, where α > 0, β,γ,η ≥ 0, α + β < 1 and α + γ + η < 1. Example 2.4. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 3 1 − k t23t 2 4 + t 2 5t 2 6 1 + t2 + t3 + t4 , where k ∈ (0,1). Example 2.5. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − αt22 − β t5t6 1 + t23 + t 2 4 , where α > 0, β ≥ 0 and α + β < 1. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 68 Unified common fixed point theorems... Example 2.6. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − α max{t22, t23, t24} − β max{t3t5, t4t6} −γt5t6 where α > 0, β,γ ≥ 0, α + 2β < 1 and α + γ < 1. Example 2.7. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3, t4, t5 2 , t6 2 } , where k ∈ (0,1). Example 2.8. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − k max { t2, t3 + t4 2 , t5 + t6 2 } , where k ∈ (0,1). Example 2.9. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − (αt2 + βt3 + γt4 + ηt5 + λt6), where α + β + γ + η + λ < 1. Example 2.10. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − k 2 max {t2, t3, t4, t5, t6} , where k ∈ (0,1). Example 2.11. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − [αt2 + βt3 + γt4 + η(t5 + t6)], where α + β + γ + 2η < 1. Since verifications of requirements (φ1), (φ2) and (φ3) for Examples 2.1–2.11 are straightforward, the details are not included. Here, one may notice that some other well known contraction conditions can also be deduced as particular cases of implicit relation of Popa [26]. In order to strengthen this viewpoint, we include some further examples and utilize them to demonstrate how this implicit relation can cover several other known contraction conditions and is also good enough to yield further natural contraction conditions as well. Example 2.12. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) =    t1 − a1 t23 + t 2 4 t3 + t4 − a2t2 − a3(t5 + t6), if t3 + t4 6= 0; t1, if t3 + t4 = 0, where ai ≥ 0 with at least one ai non-zero and a1 + a2 + 2a3 < 1. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 69 Z. Kadelburg, M. Imdad and S. Chauhan Example 2.13. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) =    t1 − a1t2 − a2t3t4 + a3t5t6 t3 + t4 , if t3 + t4 6= 0; t1, if t3 + t4 = 0, where a1,a2,a3 ≥ 0 such that 1 < 2a1 + a2 < 2. Example 2.14. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − a1 [ a2 max { t2, t3, t4, 1 2 (t5 + t6) } +(1 − a2) [ max { t22, t3t4, t5t6, t3t6 2 , t4t5 2 }] 1 2 ] , where a1 ∈ (0,1) and 0 ≤ a2 ≤ 1. Example 2.15. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − a1 max { t22, t 2 3, t 2 4 } − a2 max { t3t5 2 , t4t6 2 } −a3t5t6, where a1,a2,a3 ≥ 0 and a1 + a2 + a3 < 1. Very recently, Popa et al. [27] proved several fixed point theorems satisfying suitable implicit relations from which Husain and Sehgal [6] type contraction conditions can be deduced. A slight modification in condition (φ1) is used as follows: (φ′1): φ is decreasing in variables t2, . . . , t6. Hereafter, let φ : R6+ → R be a continuous function which satisfies the conditions φ′1, φ2 and φ3 and let Φ ′ be the family of such functions φ. In this paper, we employ such implicit relations to prove our results. But before we proceed further, let us furnish some examples to highlight the utility of the modifications instrumental herein. Example 2.16. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − ψ ( max{t2, t3, t4, 1 2 (t5 + t6)} ) , where ψ : R+ → R+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. Example 2.17. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t1 − ψ (t2, t3, t4, t5, t6) , where ψ : R5+ → R+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 70 Unified common fixed point theorems... Example 2.18. Define φ : R6+ → R as φ(t1, t2, t3, t4, t5, t6) = t 2 1 − ψ ( t22, t3t4, t5t6, t3t6, t4t5 ) , where ψ : R5+ → R+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. Clearly, apart from these examples, there are many other functions which meet the requirements (φ′1), (φ2) and (φ3). 3. Results under weak reciprocal continuity In the following proposition, we notice that under the prescribed setting, the set of common fixed point of the involved mappings is always singleton provided such points exist. Proposition 3.1. Let (X,d) be a metric space and let f,g : X → X be two mappings satisfying (3.1) φ(d(fx,fy),d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)) ≤ 0, for all x,y ∈ X where φ enjoys the property (φ3). Then f and g have at most one common fixed point in X. Proof. Let, on the contrary, the mappings f and g have two common fixed points w and w′ such that w 6= w′. On using (3.1), we have φ(d(fw,fw′),d(gw,gw′),d(fw,gw),d(fw′,gw′),d(gw,fw′),d(gw′,fw)) ≤ 0, or φ(d(w,w′),d(w,w′),d(w,w),d(w′,w′),d(w,w′),d(w′,w)) ≤ 0, so that φ(d(w,w′),d(w,w′),0,0,d(w,w′),d(w′,w)) ≤ 0, which contradicts (φ3), yielding thereby w = w ′. � Let f and g be two mappings from a metric space (X,d) into itself satisfying the following condition: (3.2) f(X) ⊆ g(X). For an arbitrary point x0 ∈ X there exists a point x1 ∈ X such that fx0 = gx1. Continuing in this way, one can inductively define a sequence {yn} in X such that (3.3) yn = fxn = gxn+1, for all n = 0,1,2, . . . . Lemma 3.2. Let f and g be two mappings from a metric space (X,d) into itself which satisfy conditions (3.1) (for some φ ∈ Φ′) and (3.2), and let {yn} be defined by (3.3). Then {yn} is a Cauchy sequence. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 71 Z. Kadelburg, M. Imdad and S. Chauhan Proof. On using (3.1) with x = xn and y = xn+1, we have φ ( d(fxn,fxn+1),d(gxn,gxn+1),d(fxn,gxn), d(fxn+1,gxn+1),d(gxn,fxn+1),d(gxn+1,fxn) ) ≤ 0, or φ ( d(yn,yn+1),d(yn−1,yn),d(yn,yn−1), d(yn+1,yn),d(yn−1,yn+1),d(yn,yn) ) ≤ 0, or φ ( d(yn,yn+1),d(yn−1,yn),d(yn−1,yn), d(yn,yn+1),d(yn−1,yn) + d(yn,yn+1),0 ) ≤ 0, implying thereby d(yn,yn+1) ≤ hd(yn−1,yn) (due to (φ2a)) where h ∈ (0,1). In general, for all n = 0,1,2, . . ., we have d(yn,yn+1) ≤ hd(yn−1,yn). Again using (3.1) with x = xn+2 and y = xn+1, we get φ ( d(fxn+2,fxn+1),d(gxn+2,gxn+1),d(fxn+2,gxn+2), d(fxn+1,gxn+1),d(gxn+2,fxn+1),d(gxn+1,fxn+2) ) ≤ 0, or φ ( d(yn+2,yn+1),d(yn+1,yn),d(yn+2,yn+1), d(yn+1,yn),d(yn+1,yn+1),d(yn,yn+2) ) ≤ 0, or φ ( d(yn+1,yn+2),d(yn,yn+1),d(yn+1,yn+2), d(yn,yn+1),0,d(yn,yn+1) + d(yn+1,yn+2) ) ≤ 0. In view of (φ2b), we have d(yn+1,yn+2) ≤ hd(yn,yn+1) where h ∈ (0,1). For all n = 0,1,2, . . . we get d(yn+1,yn+2) ≤ hd(yn,yn+1). Moreover, for every integer p > 0, we obtain d(yn,yn+p) ≤ d(yn,yn+1) + d(yn+1,yn+2) + · · · + d(yn+p−1,yn+p) ≤ (1 + h + · · · + hp−1)d(yn,yn+1) ≤ ( 1 1 − h ) hnd(y0,y1), i.e., d(yn,yn+p) → 0 as n → ∞. Therefore {yn} is a Cauchy sequence in X. � Utilizing the recently introduced notion of weak reciprocal continuity due to Pant et al. [23], we prove a common fixed point theorem in a complete metric space for variants of compatible (g-compatible and f-compatible) and R-weakly commuting (R-weakly commuting of type (Ag) and R-weakly commuting of type (Af )) mappings via an implicit function studied by Popa [27] which is a slightly refined version of some known implicit relations. Some related results are also derived besides furnishing illustrative examples which establish the superiority of our results over some of the known results especially the ones contained in Pant et al. [23]. Theorem 3.3. Let f and g be weakly reciprocally continuous self-mappings of a complete metric space (X,d) satisfying conditions (3.1) (for some φ ∈ Φ′) and (3.2). If the mappings f and g are either compatible or g-compatible or f-compatible or R-weakly commuting or R-weakly commuting of type (Ag) or R-weakly commuting of type (Af ), then f and g have a unique common fixed point in X. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 72 Unified common fixed point theorems... Proof. Let {yn} be a sequence defined by (3.3). By Lemma 3.2, {yn} is a Cauchy sequence in X. Since X is complete, there exists a point z in X such that lim n→∞ yn = lim n→∞ fxn = lim n→∞ gxn+1 = z. Case I: Suppose that the mappings f and g are compatible. The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) Let gfxn → gz as n → ∞. The compatibility of f and g yields lim n→∞ d(fgxn,gfxn) = 0, hence fgxn → gz as n → ∞. By (3.2), this yields fgxn+1 = ffxn → gz as n → ∞. On using (3.1) with x = z and y = fxn, we get φ(d(fz,ffxn),d(gz,gfxn),d(fz,gz),d(ffxn,gfxn),d(gz,ffxn),d(gfxn,fz)) ≤ 0. Taking the limit as n → ∞, we have φ(d(fz,gz),d(gz,gz),d(fz,gz),d(gz,gz),d(gz,gz),d(gz,fz)) ≤ 0, or, equivalently, φ(d(fz,gz),0,d(fz,gz),0,0,d(gz,fz)) ≤ 0, implying thereby d(fz,gz) = 0 (due to (φ2b)). Hence fz = gz which shows that z is a coincidence point of the mappings f and g. Also compatibility of f and g implies commutativity at a coincidence point. Hence ffz = fgz = gfz = ggz. Now we assert that fz is a common fixed point of f and g. To accomplish this, using (3.1) with x = z and y = fz, we have φ(d(fz,ffz),d(gz,gfz),d(fz,gz),d(ffz,gfz),d(gz,ffz),d(gfz,fz)) ≤ 0. and so φ(d(fz,ffz),d(fz,ffz),0,0,d(fz,ffz),d(ffz,fz)) ≤ 0, yielding thereby d(fz,ffz) = 0 (due to (φ3)). Therefore, fz = ffz = gfz which shows that fz is a common fixed point of the mappings f and g. (ii) Assume next that fgxn → fz as n → ∞. By virtue of (3.2), we have fz = gt for some t ∈ X hence fgxn → gt as n → ∞. The compatibility of the mappings f and g implies lim n→∞ d(fgxn,gfxn) = 0, hence gfxn → gt as n → ∞. Again by (3.2), fgxn+1 = ffxn → gt as n → ∞. On using (3.1) with x = t and y = fxn, we get φ(d(ft,ffxn),d(gt,gfxn),d(ft,gt),d(ffxn,gfxn),d(gt,ffxn),d(gfxn,ft)) ≤ 0. Taking the limit as n → ∞, we have φ(d(ft,gt),0,d(ft,gt),0,0,d(gt,ft)) ≤ 0, which implies d(ft,gt) = 0 (due to (φ2b)). Hence ft = gt which shows that t is a coincidence point of the mappings f and g. Also, compatibility of f and g implies commutativity at a coincidence point. Hence fft = fgt = gft = ggt. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 73 Z. Kadelburg, M. Imdad and S. Chauhan We show that ft is a common fixed point of f and g. To assert this, using (3.1) with x = t and y = ft, we have φ(d(ft,fft),d(gt,gft),d(ft,gt),d(fft,gft),d(gt,fft),d(gft,ft)) ≤ 0, or, equivalently, φ(d(ft,fft),d(ft,fft),0,0,d(ft,fft),d(fft,ft)) ≤ 0, implying thereby d(ft,fft) = 0 (due to (φ3)). Therefore, ft = fft = gft which shows that ft is a common fixed point of the mappings f and g. Case II: Let the mappings f and g be g-compatible. The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) Suppose that gfxn → gz as n → ∞. Then g-compatibility of f and g yields lim n→∞ d(ffxn,gfxn) = 0, hence ffxn → gz as n → ∞. On using (3.1) with x = z and y = fxn, φ(d(fz,ffxn),d(gz,gfxn),d(fz,gz),d(ffxn,gfxn),d(gz,ffxn),d(gfxn,fz)) ≤ 0. Taking the limit as n → ∞, we get fz = gz (due to (φ2b)) which shows that z is a coincidence point of the mappings f and g. Since g-compatibility of f and g implies commutativity at a coincidence point, we have ffz = fgz = gfz = ggz. Now, we assert that fz is a common fixed point of f and g. On using (3.1) with x = z and y = fz, we have φ(d(fz,ffz),d(gz,gfz),d(fz,gz),d(ffz,gfz),d(gz,ffz),d(gfz,fz)) ≤ 0, implying thereby fz = ffz = gfz (due to (φ3)). Hence fz is a common fixed point of the mappings f and g. (ii) Assume that fgxn → fz as n → ∞. By (3.2), we have fz = gt for some t ∈ X. Thus fgxn → gt as n → ∞. Also (3.2) implies fgxn+1 = ffxn → gt as n → ∞. The g-compatibility of the mappings f and g implies lim n→∞ d(ffxn,gfxn) = 0, hence gfxn → gt as n → ∞. On using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (due to (φ2b)). Hence t is a coincidence point of the mappings f and g. Also, g- compatibility of f and g implies commutativity at a coincidence point. Hence fft = fgt = gft = ggt. We assert that ft is a common fixed point of f and g. On using (3.1) with x = t, y = ft we obtain ft = fft = gft (due to (φ3)). Therefore, ft is a common fixed point of the mappings f and g. Case III: Suppose that the pair (f,g) is f-compatible. The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) Let us consider gfxn → gz as n → ∞. Then f-compatibility of f and g implies lim n→∞ d(fgxn,ggxn) = 0. By (3.2), we get gfxn = ggxn+1 → gz as n → ∞ and so fgxn+1 = ffxn → gz as n → ∞. Putting x = z and y = fxn in (3.1) and taking the limit as n → ∞, we get fz = gz (in view of (φ2b)). Hence z is a coincidence point of the mappings f and g. Since f- compatibility of f and g implies commutativity at a coincidence point, we have ffz = fgz = gfz = ggz. Now we show that fz is a common fixed point of f c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 74 Unified common fixed point theorems... and g. Putting x = z and y = fz in (3.1), we have fz = ffz = gfz (due to (φ3)). Thus fz is a common fixed point of the mappings f and g. (ii) Assume that fgxn → fz as n → ∞. By (3.2), we have fz = gt for some t ∈ X. Thus fgxn → gt as n → ∞. Then f-compatibility of the mappings f and g implies lim n→∞ d(fgxn,ggxn) = 0, hence ggxn → gt as n → ∞. By (3.2), we get fgxn+1 = ffxn → gt as n → ∞ and so gfxn = ggxn+1 → gt as n → ∞. On using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we obtain ft = gt (in view of (φ2b)). Thus t is a coincidence point of the mappings f and g. Also, f-compatibility of f and g implies commutativity at a coincidence point. Hence fft = fgt = gft = ggt. By putting x = t, y = ft in (3.1), we get ft = fft = gft (due to (φ3)). Hence ft is a common fixed point of the mappings f and g. Case IV: Let the pair (f,g) be R-weakly commuting. The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) First we assume that gfxn → gz as n → ∞. Then R-weak commutativity of the pair (f,g) implies d(fgxn,gfxn) ≤ Rd(fxn,gxn). Taking the limit as n → ∞, we have fgxn → gz as n → ∞. By (3.2), we get fgxn+1 = ffxn → gz as n → ∞. On using (3.1) with x = z, y = fxn and taking the limit as n → ∞, we get fz = gz (in view of (φ2b)), hence z is a coincidence point of the mappings f and g. It is obvious that R-weakly commuting mappings commute at their coincidence points, i.e., d(fgz,gfz) ≤ Rd(fz,gz), implying fgz = gfz and so ffz = fgz = gfz = ggz. Now we assert that fz is a common fixed point of f and g. On using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)) and fz is a common fixed point of the mappings f and g. (ii) Suppose that fgxn → fz as n → ∞. By (3.2), we have fz = gt for some t ∈ X. Thus fgxn → gt as n → ∞. Then R-weak commutativity of the pair (f,g), i.e., d(fgxn,gfxn) ≤ Rd(fxn,gxn) implies gfxn → gt as n → ∞. By (3.2), we get fgxn+1 = ffxn → gt as n → ∞. On using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)) which shows that t is a coincidence point of the mappings f and g. Also, R-weak commutativity of the pair (f,g), d(fgt,gft) ≤ Rd(ft,gt) implies fft = fgt = gft = ggt. By putting x = t, y = ft in (3.1), we can easily obtain ft = fft = gft (due to (φ3)). Hence ft is a common fixed point of the mappings f and g. Case V: Assume that the pair (f,g) is R-weakly commuting of type (Ag). The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) Let us assume that gfxn → gz as n → ∞. Then R-weak commutativity of type (Ag) of the pair (f,g) implies d(ffxn,gfxn) ≤ Rd(fxn,gxn). On letting n → ∞, we have ffxn → gz as n → ∞. On using (3.1) with x = z, y = fxn and taking the limit as n → ∞, we have fz = gz (in view of (φ2b)) which shows that z is a coincidence point of the mappings f and g. R-weak commutativity of type (Ag), d(ffz,gfz) ≤ Rd(fz,gz) yields ffz = gfz and so ffz = fgz = gfz = ggz. Now we show that fz is a common fixed point of c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 75 Z. Kadelburg, M. Imdad and S. Chauhan f and g. On using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)) and fz is a common fixed point of the mappings f and g. (ii) Suppose that fgxn → fz as n → ∞. In view of (3.2), fz = gt for some t ∈ X. Thus fgxn → gt as n → ∞. By (3.2), fgxn+1 = ffxn → gt as n → ∞. Then R-weak commutativity of type (Ag) of the pair (f,g), d(ffxn,gfxn) ≤ Rd(fxn,gxn) implies gfxn → gt as n → ∞. On using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)), hence t is a coincidence point of the mappings f and g. Also, R-weak commutativity of type (Ag) of the pair (f,g), d(fft,gft) ≤ Rd(ft,gt) implies fft = fgt = gft = ggt. Again using (3.1) with x = t, y = ft, we get ft = fft = gft (due to (φ3)). Thus ft is a common fixed point of the mappings f and g. Case VI: Finally, let us consider that the pair (f,g) be R-weakly commuting of type (Af ). The weak reciprocal continuity of f and g implies that fgxn → fz or gfxn → gz as n → ∞. (i) Suppose that gfxn → gz as n → ∞. By (3.2), we get gfxn = ggxn+1 → gz as n → ∞. Then R-weak commutativity of type (Af ) of the pair (f,g) implies d(fgxn,ggxn) ≤ Rd(fxn,gxn). This yields fgxn → gz as n → ∞. In view of (3.2), fgxn+1 = ffxn → gz as n → ∞. Putting x = z and y = fxn in (3.1) with limit as n → ∞, we get fz = gz (in view of (φ2b)). Thus z is a coincidence point of the mappings f and g. Since R-weak commutativity of type (Af ), d(fgz,ggz) ≤ Rd(fz,gz) yields fgz = ggz and so ffz = fgz = gfz = ggz. Now we assert that fz is a common fixed point of f and g. On using (3.1) x = z, y = fz, we have fz = ffz = gfz (due to (φ3)). Hence fz is a common fixed point of the mappings f and g. (ii) Assume that fgxn → fz as n → ∞. In view of (3.2), fz = gt for some t ∈ X. Thus fgxn → gt as n → ∞. Then R-weak commutativity of type (Af ) of the pair (f,g), d(fgxn,ggxn) ≤ Rd(fxn,gxn) implies ggxn → gt as n → ∞. By (3.2), we get gfxn = ggxn+1 → gt and fgxn+1 = ffxn → gt as n → ∞. On using (3.1) with x = t, y = fxn and taking the limit as n → ∞, we get ft = gt (in view of (φ2b)) which shows that t is a coincidence point of the mappings f and g. Also, R-weak commutativity of type (Af ) of the pair (f,g), d(fgt,ggt) ≤ Rd(ft,gt) implies fft = fgt = gft = ggt. Again using (3.1) with x = t, y = ft, we get ft = fft = gft (due to (φ3)). Thus ft is a common fixed point of the mappings f and g. In view of Proposition 3.1, z is the unique common fixed point of the map- pings f and g. � Obviously, Theorem 1.1 follows as a special case of the obtained result (using φ(t1, t2, t3, t4, t5, t6) = t1 − at2 − bt3 − ct4, with a,b,c ≥ 0, a + b + c < 1). We state some other corollaries. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 76 Unified common fixed point theorems... Corollary 3.4. The conclusions of Proposition 3.1, Lemma 3.2 and Theorem 3.3 remain true if condition (3.1) is replaced by one of the following contraction conditions (each inequality is supposed to hold for all x,y ∈ X). (3.4) d(fx,fy) ≤ k max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] } , where k ∈ (0,1). d2(fx,fy) ≤ d(fx,fy)[αd(gx,gy) + βd(fx,gx) + γd(fy,gy)](3.5) +ηd(gx,fy)d(gy,fx), where α > 0, β,γ,η ≥ 0, α + β + γ < 1 and α + η < 1. d3(fx,fy) ≤ αd2(fx,fy)d(gx,gy) + βd(fx,fy)d(fx,gx)d(fy,gy)(3.6) + γd(gx,fy)2d(gy,fx) + ηd(gx,fy)d(gy,fx)2, where α > 0, β,γ,η ≥ 0, α + β < 1 and α + γ + η < 1. (3.7) d3(fx,fy) ≤ kd 2(fx,gx)d2(fy,gy) + d2(gx,fy)d2(gy,fx) 1 + d(gx,gy) + d(fx,gx) + d(fy,gy) , where k ∈ (0,1). (3.8) d2(fx,fy) ≤ αd2(gx,gy) + β d(gx,fy)d(gy,fx) 1 + d2(fx,gx) + d2(fy,gy) , where α > 0, β ≥ 0 and α + β < 1. d2(fx,fy) ≤ αmax { d2(gx,gy),d2(fx,gx),d2(fy,gy) } (3.9) + β max{d(fx,gx)d(gx,fy),d(fy,gy)d(gy,fx)} + γd(gx,fy)d(gy,fx), where α > 0, β,γ ≥ 0, α + 2β < 1 and α + γ < 1. (3.10) d(fx,fy) ≤ k max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 d(gx,fy), 1 2 d(gy,fx) } , where k ∈ (0,1). (3.11) d(fx,fy) ≤ k max { d(gx,gy), 1 2 [d(fx,gx) + d(fy,gy)], 1 2 [d(gx,fy) + d(gy,fx)] } , where k ∈ (0,1). (3.12) d(fx,fy) ≤ αd(gx,gy) + βd(fx,gx) + γd(fy,gy) + ηd(gx,fy) + λd(gy,fx), where α + β + γ + η + λ < 1. (3.13) d(fx,fy) ≤ k 2 max {d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)} , c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 77 Z. Kadelburg, M. Imdad and S. Chauhan where k ∈ (0,1). (3.14) d(fx,fy) ≤ αd(gx,gy) + βd(fx,gx) + γd(fy,gy) + η[d(gx,fy) + d(gy,fx)], where α + β + γ + 2η < 1. (3.15) d(fx,fy) ≤      a1 d 2(fx,gx)+d2(fy,gy) d(fx,gx)+d(fy,gy) + a2d(gx,gy) +a3[d(gx,fy) + d(gy,fx)], if d(fx,gx) + d(fy,gy) 6= 0; 0, if d(fx,gx) + d(fy,gy) = 0, where ai ≥ 0 with at least one ai non-zero and a1 + a2 + 2a3 < 1. (3.16) d(fx,fy) ≤      a1d(gx,gy) + a2d(fx,gx)d(fy,gy)+a3d(gx,fy)d(gy,fx) d(fx,gx)+d(fy,gy) , if d(fx,gx) + d(fy,gy) 6= 0; 0, if d(fx,gx) + d(fy,gy) = 0, d(fx,fy) ≤ a1[a2 max{d(gx,gy),d(fx,gx),d(fy,gy),(d(gx,fy) + d(gy,fx))/2} (3.17) + (1 − a2)(max{d2(gx,gy),d(fx,gx)d(fy,gy),d(gx,fy)d(gy,fx), (d(fx,gx)d(gy,fx))/2,(d(fy,gy)d(gx,fy))/2}) 12 ], where a1 ∈ (0,1) and 0 ≤ a2 ≤ 1. d2(fx,fy) ≤ a1 max{d2(gx,gy),d2(fx,gx),d2(fy,gy)} (3.18) + a2 max { 1 2 [d(fx,gx)d(gx,fy)], 1 2 [d(fy,gy)d(gy,fx)] } + a3d(gx,fy)d(gy,fx), where a1,a2,a3 ≥ 0 and a1 + a2 + a3 < 1. (3.19) d(fx,fy) ≤ ψ ( max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] }) , where ψ : R+ → R+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. (3.20) d(fx,fy) ≤ ψ (max {d(gx,gy),d(fx,gx),d(fy,gy),d(gx,fy),d(gy,fx)}) , where ψ : R5+ → R+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. (3.21) d2(fx,fy) ≤ ψ ( max { d2(gx,gy),d(fx,gx)d(fy,gy),d(gx,fy)d(gy,fx), d(fx,gx)d(gy,fx),d(fy,gy)d(gx,fy) }) , c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 78 Unified common fixed point theorems... where ψ : R5+ → R+ is an upper semi-continuous and non-decreasing function in each coordinate variable such that ψ(t,t,at,bt,ct) < t for each t > 0 and a,b,c ≥ 0 with a + b + c ≤ 3. Proof. The proof of each case (3.4)–(3.21) easily follows from Theorem 3.3 in view of Examples 2.1–2.18. � The following example demonstrates the genuineness of Theorem 1.1 over Theorem 3.3. Example 3.5. Let X = [1,5.5] (with the standard metric d) and consider the mappings f,g : X → X given by fx =      1, for x = 1, 3, for 1 < x ≤ 2.5, 1, for 2.5 < x ≤ 5.5; gx =          1, for x = 1, 4, for 1 < x ≤ 2.5, 2x+1 6 , for 2.5 < x < 5.5, 3, for x = 5.5. Firstly, we show that Theorem 1.1 does not work in the context of this example. To accomplish this, we show that the contraction condition (ii) of Theorem 1.1 is not satisfied. To accomplish this, take 1 < x ≤ 2.5 and 2.5 < y ≤ 5.5 so that d(fx,fy) = 2, and ad(gx,gy) + bd(fx,gx) + cd(fy,gy) = a ( 4 − 2y + 1 6 ) + b + c ( 2y + 1 6 − 1 ) . The right-hand side of the last equality is equal to 2a + b + c for y = 5.5 and tends to 3a + b for y → 2.5. Hence, in order that conditon (ii) of Theorem 1.1 holds, one must simultaneously require a,b,c ≥ 0, a + b + c < 1, 2a + b + c ≥ 2 and 3a + b ≥ 2 which is indeed impossible. Next, we show that all the conditions of Theorem 3.3 are fulfilled with the function φ taken from Example 2.9. First of all, the mappings f and g are R-weakly commuting of type (Ag) ( the condition d(ffx,gfx) ≤ Rd(fx,gx) can be easily checked). Also, f(X) = {1,3} ⊆ [1,2) ∪ {3,4} = g(X). In order to show that f and g are weakly reciprocally continuous, suppose that {xn} is a sequence in X such that fxn → t and gxn → t yielding thereby t = 1. Then, we are left with two possibilities: either xn = 1 for almost each n, or xn = 2.5 + αn where αn → +0 as n → ∞. In the first case, obviously, fgxn → 1 = f1 and gfxn → 1 = g1 as n → ∞. In the second case gxn = 1 + 1 3 αn and fgxn = 3 6= f1, but fxn = 1 and gfxn = 1 = g1. Hence, f and g are weakly reciprocally continuous which is not reciprocally continuous as well as non-compatible. Finally, we check the contractive condition (3.12) of Corollary 3.4 with η = 2 3 , λ = 2 9 and α = β = γ = 0 (note that α + β + γ + η + λ < 1) so that (3.22) d(fx,fy) ≤ 2 3 d(gx,fy) + 2 9 d(fx,gy). Without loss of generality, we can assume that x ≤ y. Consider the following cases: 1◦ x = y = 1; 2◦ x = 1, 1 < y ≤ 2.5; 3◦ x = 1, y > 2.5; 4◦ 1 < c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 79 Z. Kadelburg, M. Imdad and S. Chauhan x,y ≤ 2.5; 5◦ 1 < x ≤ 2.5, y > 2.5; 6◦ x,y > 2.5. The cases 1◦, 3◦, 4◦ and 6◦ are trivial as d(fx,fy) = 0. In the Case 2◦ we have d(fx,fy) = 2 and 2 3 d(gx,fy) + 2 9 d(fx,gy) = 2 3 · 2 + 2 9 · 3 = 2, while in Case 5◦ we have d(fx,fy) = 2 and 2 3 d(gx,fy) + 2 9 d(fx,gy) ≥ 2 3 · 3 + 2 9 · 0 = 2. Therefore, condition (3.22) is satisfied. Thus all the conditions of Theorem 3.3 are satisfied and the mappings f and g have a unique common fixed point at z = 1. Remark 3.6. Notice that the preceding example confirms the importance of weak reciprocal continuity instead of reciprocal continuity when the given pair of mappings is not even compatible. In the following proposition, we notice that a non-compatible reciprocally continuous pair of self-mappings cannot have a common fixed point under certain contractive assumptions (see also [23, Remark 1]). Proposition 3.7. Let (X,d) be a metric space and let f,g : X → X satisfy the following assumptions: (1) f and g are noncompatible. (2) f and g are reciprocally continuous. (3) f and g satisfy the condition (3.1) for some φ ∈ Φ′. Then f and g cannot have a common fixed point. Proof. Suppose, to the contrary, that there is a point z ∈ X such that (3.23) fz = gz = z. By the assumption (1), there exists a sequence {xn} in X such that lim n→∞ fxn = lim n→∞ gxn = t, but lim n→∞ d(fgxn,gfxn) is non-zero or non-existent. The assump- tion (2) implies that fgxn → ft and gfxn → gt as n → ∞, which means that ft 6= gt. Now, put x = z and y = xn in the condition (3.1) to obtain φ ( d(fz,fxn),d(gz,gxn),d(fz,gz), d(fxn,gxn),d(gz,fxn),d(fz,gxn) ) ≤ 0, or φ ( d(z,fxn),d(z,gxn),d(z,z), d(fxn,gxn),d(z,fxn),d(z,gxn) ) ≤ 0. On taking the limit as n → ∞, we get that φ ( d(z,t),d(z,t),0,0,d(z,t),d(z,t) ) ≤ 0, which implies that d(z,t) = 0 (by (φ3)) so that z = t, which is a contradiction to (3.23) as ft 6= gt. � 4. Results without continuity A critical examination of the recent literature reveals the fact that more natural results can be proved if we replace the completeness of the entire space by completeness of suitable subspaces (e.g. Imdad et al. [12]) under relatively tighter conditions. Concretely speaking, a common fixed point theorem under c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 80 Unified common fixed point theorems... entire space completeness condition requires five conditions while the similar results can be proved under four conditions if the completeness of suitable subspaces are assumed. On the lines of the results proved in Imdad et al. [12], we state and prove our next theorem which is an improvement over Theorem 3.3 in the following three respects: (1) to relax the weak reciprocal continuity of the involved mappings, (2) to reduce the commutativity requirements of the mappings to coinci- dence points, (3) to replace the completeness of the whole space by the completeness of any one of the underlying subspaces. Theorem 4.1. Let f and g be two self-mappings of a metric space (X,d) satisfying conditions (3.1) (for some φ ∈ Φ′) and (3.2). If one of f(X) and g(X) is a complete subspace of X, then f and g have a coincidence point. Moreover, f and g have a unique common fixed point provided the pair (f,g) is coincidentally commuting. Proof. Let {yn} be a sequence defined by (3.3). By Lemma 3.2, {yn} is a Cauchy sequence in X. Now assume that g(X) is a complete subspace of X, then the subsequence fx2n+1 = gx2n which is contained in g(X) must has a limit z ∈ g(X). As {yn} is a Cauchy sequence containing a convergent subse- quence {y2n+1}, therefore {yn} also converges implying thereby the convergence of the subsequence {y2n}, i.e. lim n→∞ fx2n = lim n→∞ gx2n+1 = z. Let u ∈ g−1(z) such that gu = z. By using inequality (3.1) with x = x2n, y = u, we have φ ( d(fx2n,fu),d(gx2n,gu),d(fx2n,gx2n), d(fu,gu),d(gx2n,fu),d(gu,fx2n) ) ≤ 0, which on making n → ∞, gives rise φ(d(z,fu),d(z,z),d(z,z),d(fu,z),d(z,fu),d(z,z)) ≤ 0, φ(d(z,fu),0,0,d(fu,z),d(z,fu) + 0,0) ≤ 0, implying thereby d(z,fu) = 0 (due to (φ2a)). Hence z = fu = Su which shows that u is a coincidence point of the mappings f and g. If one assumes that f(X) is a complete subspace of X, then z ∈ f(X) ⊆ g(X). Similarly, we can easily show that f and g have a coincidence point. Since f and g are weakly compatible and fu = gu = z, then fgu = gfu which implies fz = gz. By using inequality (3.1) with x = z, y = u, we have φ(d(fz,fu),d(gz,gu),d(fz,gz),d(fu,gu),d(gz,fu),d(gu,fz)) ≤ 0, or, equivalently, φ(d(fz,z),d(gz,z),0,0,d(z,fu),d(z,fz)) ≤ 0, c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 81 Z. Kadelburg, M. Imdad and S. Chauhan a contradiction to (φ3) if d(z,fz) > 0. Thus z = fz = gz which shows that z is a common fixed point of the mappings f and g. In view of Proposition 3.1, z is the unique common fixed point of the mappings f and g. � Remark 4.2. The conclusion of Theorem 4.1 remains valid if we replace condi- tion (3.1) by anyone of the contraction conditions (3.4)-(3.21) shown in Corol- lary 3.4 (each inequality is supposed to hold for all x,y ∈ X). Example 4.3. Let X = [0,5] (with the standard metric d) and consider the mappings f,g : X → X given by f(x) = { 0, if x = 0; 1, if 0 < x ≤ 5. f(x) =    0, if x = 0; 5, if 0 < x < 5; 1, if x = 5. Notice that the mappings f and g are discontinuous at ‘0’ which is also their common fixed point. Also the pair (f,g) is coincidentally commuting with f(X) = {0,1} ⊆ {0,1,5} = g(X). Define φ : R6+ → R as (4.1) φ(t1, t2, t3, t4, t5, t6) = t1 − ψ ( max{t2, t3, t4, 1 2 (t5 + t6)} ) , where ψ : R+ → R+ is an increasing upper semi-continuous function with ψ(0) = 0 and ψ(t) < t for each t > 0. In view of (4.1), condition (3.1) implies (for all x,y ∈ X) (4.2) d(fx,fy) ≤ ψ ( max { d(gx,gy),d(fx,gx),d(fy,gy), 1 2 [d(gx,fy) + d(gy,fx)] }) . By a routine calculation one can verify that implicit contraction condition (4.2) is satisfied if we choose ψ(s) = √ s. Thus, all the conditions of Theorem 4.1 are satisfied. Notice that 0 is the unique common fixed point of the involved mappings f and g. As an application of Theorem 4.1, we have the following interesting result involving two finite families of self mappings. Corollary 4.4. Let {fi}pi=1 and {gj} q j=1 be two finite families of self-mappings of a metric space (X,d) with f = f1f2 . . .fp and g = g1g2 . . .gq satisfying conditions (3.1) (for some φ ∈ Φ′) and (3.2). Then (f,g) has a point of coincidence. Moreover, {fi}pi=1 and {gj} q j=1 have a unique common fixed point if the fami- lies ({fi},{gj}) commutes pairwise wherein i ∈ {1,2, . . . ,p} and j ∈ {1,2, . . . ,q}. Proof. The proof of this theorem can be completed on the lines of Theorem 2.2 of Imdad et al. [12]. � By setting f1 = f2 = . . . = fp = f and g1 = g2 = . . . = gq = g in Corollary 4.4, one deduces the following corollary which is a slight but partial generalization of Theorem 4.1 as the commutativity requirements are slightly stronger as compared to Theorem 4.1. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 82 Unified common fixed point theorems... Corollary 4.5. Let f and g be two self-mappings of a metric space (X,d) satisfying (4.3) fp(X) ⊆ gq(X), (4.4) φ(d(fpx,fpy),d(gqx,gqy),d(fpx,gqx),d(fpy,gqy),d(gqx,fpy),d(gqy,fpx)) ≤ 0, for all x,y ∈ X, some φ ∈ Φ′ and p,q are fixed positive integers. If one of fp(X) and gq(X) is a complete subspace of X, then the pair (fp,gq) has a coincidence point. Moreover, the mappings f and g have a unique common fixed point provided that the pair (f,g) commutes. Conclusion Theorem 3.3 extends a result of Pant et al. [23, Theorem 1] to various variants of compatible and R-weakly commuting mappings which is substan- tiated well by an appropriate example. Proposition 3.7 is designed to exhibit the importance of weak reciprocal continuity over reciprocal continuity. The- orem 4.1 scores a superiority over all those results wherein the continuity of the mappings and completeness of the whole spaces are assumed for the exis- tence of coincidence point. 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