Ratio Mathematica Volume 47, 2023 Families of mappings satisfying a mixed implicit relation Rajesh Kumar* Sanjay Kumar† Abstract A fixed point for a suitable map or operator is identical to the presence of a solution to a theoretical or real-world problem. As a result, fixed points are crucial in many fields of mathematics, science and engi- neering. The purpose of this paper is to prove unique common fixed point theorems for families of weakly compatible mappings. Given mappings satisfy common limit range property and a mixed implicit relation. Our results generalize, extend and improve the results of Im- dad (2013) and Popa (2018). We provide an application for integral type contraction condition. An example is also mentioned to check the authenticity of our results. Keywords: common fixed point, weakly compatible mappings, mixed implicit relation, almost altering distance, common limit range prop- erty. 2020 AMS subject classifications: 47H10, 54H25. 1 *Department of Mathematics, Institute of Higher Learning, BPS Mahila Vishwavidyalaya, Khanpur Kalan, Sonipat-131305, Haryana (India); rkdubaldhania@gmail.com. †Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technol- ogy, Murthal, Sonepat-131039, Haryana (India); sanjaymudgal2004@yahoo.com 1Received on August 26, 2022. Accepted on February 10, 2023. Published on March 5, 2023. DOI:10.23755/rm.v41i0.839. ISSN: 1592-7415. eISSN: 2282-8214. 219 R. Kumar and S. Kumar 1 Introduction and Preliminaries Fixed point theory is an important tool of modern mathematics as it helps to find a unique fixed point of multi-valued and single-valued mappings by restrict- ing the condition of the domain of the function. It also helps to find the results of many differential as well as integral equations which can further be used to solve many industrial based problems. The most popular tool in fixed point the- ory is Banach contraction [6] principle which states that every contraction map- ping on a complete metric space has a unique fixed point. Various authors have extended and generalized this principle in various directions. In 1976, Jungck [9] used the concept of commuting maps to prove a common fixed point theo- rem. Several authors have investigated various concepts of minimal commuting maps. A pair of self-mappings (P,Q) on a metric space (X ,d) is said to be compatible [10] if limn→∞ d(PQun,QPun) = 0, whenever {un} is a sequence in X such that limn→∞ Pun = limn→∞ Qun = u for some u ∈ X . A pair of self-mappings (P,Q) on a metric space (X ,d) is called weakly compatible [11] if P and Q commute at their points of coincidence. Pant ([13], [14], [15]) ini- tiated the study of common fixed points for non-compatible mappings. Further, Aamri and El-Moutawakil [1] introduced (E.A) property as a generalization of non-compatible mappings. A pair of self-mappings (P,Q) on a metric space (X ,d) is said to satisfy (E.A) property [1] if there exists a sequence {un} in X such that limn→∞ Pun = limn→∞ Qun = u , for some u ∈ X . In 2011, Sintunavarat and Kumam [22] introduced the concept of common limit range property. Definition 1.1. [22] A pair of self-mappings (P,Q) on a metric space (X ,d) is said to satisfy the common limit range property with respect to Q, denoted by CLRQ, if there exists a sequence {un} in X such that lim n→∞ Pun = lim n→∞ Qun = u, for some u ∈ Q(X). Thus one can note that the mappings P and Q satisfying property (E.A) along with the closedness of the subspace Q(X) always have CLRQ property with re- spect to Q. In 2013, Imdad et al. [8] extended the notion of common limit range property for pairs of self mappings. Definition 1.2. [8] Two pairs of self-mappings (P,Q) and (S,T ) on a metric space (X ,d) are said to satisfy common limit range property with respect to Q and T , denoted by (CLR)(Q,T ), if there exists two sequences {un} and {vn} in X such that lim n→∞ Pun = lim n→∞ Qun = lim n→∞ Svn = lim n→∞ T vn = t, 220 Families of mappings satisfying a mixed implicit relation where t ∈ Q(X) ∩ T (X). In 2018, Popa et al. [17] introduced a new type of limit range property as follows. Definition 1.3. [17] Let P,Q, T be self mappings of a metric space (X ,d). The pair (P,Q) is said to satisfy common limit range property with respect to T if there exists a sequence {un} in X such that limn→∞ Pun = limn→∞ Qun = u, for some u ∈ Q(X) ∩ T (X). Now we extend the definition 1.3 for families of mappings. Definition 1.4. Let Q1,Q2,...,Q2n and P be self mappings of a metric space (X ,d). The pair (P,Q1Q3...Q2n−1) is said to satisfy common limit range prop- erty with respect to Q2Q4...Q2n if there exists a sequence {un} in X such that limn→∞ Pun = limn→∞ Q1Q3...Q2n−1un = u, for some u ∈ Q1Q3...Q2n−1(X)∩ Q2Q4...Q2n(X). Remark 1.1. [17] Let P,Q, S and T be self mappings of a metric space (X ,d). If the pairs (P,Q) and (S,T ) satisfy the common limit range property with respect to Q and T , then (P,Q) satisfy the limit range property with respect to T , but the converse does not hold. Boyd and Wong [5] introduced ϕ contraction condition and generalized the Banach contraction principle using this contraction. A self mapping P on a complete metric space (X ,d) is said to satisfy ϕ contraction if d(Pα,Pβ) ≤ ϕ(d(α,β)), for all α,β ∈ X , where ϕ : [0,∞) → [0,∞) is an upper semi con- tinuous function from right such that 0 ≤ ϕ(t) < t for all t > 0. The theorems of existence of fixed points for self mappings in Hilbert spaces satisfying ϕ-weak contraction were studied by Alber and Guerre-Delabriere [3]. Further Rhoades [21] extended this concept in complete metric space. Some fixed point results are proved in [7], [8] and in other papers for mappings with common limit range property satisfying (ϕ,ψ)-weak contractive conditions. The following theorem is proved in [8]. Theorem 1.1. [8] Let P , Q, S and T be self mappings of a metric space (X ,d) satisfying ψ(d(Px,Qy)) ≤ ψ(m(x,y)) − ϕ(m(x,y)), for all x,y ∈ X and for some ϕ, ψ, where m(x,y) = max{d(Sx,Ty),d(Sx,Px),d(Ty,Qy),d(Sx,Qy),d(Ty,Px)} and ϕ, ψ : [0,∞) → [0,∞) such that ϕ is a lower semi-continuous function and ϕ−1(0) = 0 and ψ is a non-decreasing continuous function with ψ−1(0) = 0. If the pairs (P,Q) and (S,T) satisfy the (CLR)(S,T) property and are weakly compatible, then P , Q, S and T have a unique common fixed point. 221 R. Kumar and S. Kumar Definition 1.5. [12] An altering distance is a function ψ : [0,∞) → [0,∞) satisfying: (ψ1): ψ is increasing and continuous, (ψ2): ψ(t) = 0 if and only if t = 0. Definition 1.6. [18] A function ψ : [0,∞) → [0,∞) is an almost altering distance if it satisfies: (ψ′1): ψ is continuous, (ψ′2): ψ(t) = 0 if and only if t = 0. Example 1.1. Define a function ψ : [0,∞) → [0,∞) by ψ(t) = { 2t, t ∈ [0,1] 1 1+t , t ∈ (1,∞). Here we note that every altering distance is an almost altering distance, but converse is not true. Various authors have unified several common fixed point theorems by using im- plicit functions. In 2008, Ali and Imdad [2] introduced a new class of implicit functions. Definition 1.7. [2] Let F be the family of lower semi-continuous functions F : R6+ → R which are satisfying: (F1) for all u > 0, F(u,0,u,0,0,u) > 0; (F2) for all u > 0, F(u,0,0,u,u,0) > 0; (F3) for all u > 0, F(u,u,0,0,u,u) > 0; Definition 1.8. [17] Let FD be the set of all lower semi-continuous functions F : R6+ → R which are satisfying: (F1D) for all u > 0, F(u,0,u,0,0,u) ≥ 0; (F2D) for all u > 0, F(u,0,0,u,u,0) ≥ 0; (F3D) for all u > 0, F(u,u,0,0,u,u) ≥ 0; Now we provide some examples in support of definition 1.8. 1. Let F(u1, ...,u6) = u1 − tmax{u2,u3,u4,u5,u6}, where t ∈ [0,1]. 2. Let F(u1, ...,u6) = u1 − tmax{u2,u3,u4, u5+u63 }, where t ∈ [0,1]. 3. Let F(u1, ...,u6) = u1 − α max{u2,u3,u4} − β(u5 + u6), where α, β ≥ 0 and α + 2β < 1. 4. Let F(u1, ...,u6) = u1 − α max{u2,u3,u4, 12(u5 + u6), u3u4 1+u2 , u5u6 1+u1 }, where α ∈ [0,1). 222 Families of mappings satisfying a mixed implicit relation 5. Let F(u1, ...,u6) = u1 − max{cu2,cu3,cu4,au5 + bu6)}, where c > 0, a,b ≥ 0 and a + b + c ≤ 1. Definition 1.9. [17] Let GD be the set of all lower semi-continuous functions G : R5+ → R such that G(t1, ..., t5) > 0 if one of t1, ..., t5 > 0. The following functions belong to the set GD. 1. G(t1, ..., t5) = max{t1, ..., t5}. 2. G(t1, ..., t5) = max{t1, t2+t32 , t4+t5 2 }. 3. G(t1, ..., t5) = t21 + t 2 2 + t 2 3 + t 2 4 + t 2 5. 4. G(t1, ..., t5) = 1 t1+t2+t3+t4+t5 . Definition 1.10. [17] A function ϕ(u1, ...,u6, t1, ..., t5) = F(u1, ...,u6)+G(t1, ..., t5) is called a mixed implicit relation. The aim of this paper is to prove general fixed point theorems for families of weakly compatible mappings with common limit range property satisfying a mixed implicit relation. Our results generalize, extend and improve the results of Popa [17] and Imdad [8]. 2 Main Results In 2018, Popa et al. [17] proved the following theorem. Theorem 2.1. [17] Let (X ,d) be a metric space and P,Q,S and T be four self mappings on X satisfying F(ψ(d(Px,Qy)),ψ(d(Sx,Ty)),ψ(d(Sx,Px)),ψ(d(Ty,Qy)),ψ(d(Sx,Qy)), ψ(d(Ty,Px))) + G(ψ(d(Sx,Ty)),ψ(d(Sx,Px)),ψ(d(Ty,Qy)),ψ(d(Sx,Qy)), ψ(d(Ty,Px))) ≤ 0, for all x,y ∈ X , for some F ∈ FD, G ∈ GD and ψ is an almost altering distance. If the pairs (P,S) and (Q,T) are weakly compatible and (P,S) and T satisfy (CLR)(P,S)T property, then P,Q,S and T have a unique common fixed point. Now we extend the Theorem 2.1 for any even number of weakly compatible mappings. 223 R. Kumar and S. Kumar Theorem 2.2. Let Q1,Q2, ...,Q2n,P0 and P1 be self mappings on a metric space (X ,d), satisfying the following conditions: (C1) Q2(Q4...Q2n) = (Q4...Q2n)Q2, Q2Q4(Q6...Q2n) = (Q6...Q2n)Q2Q4, ... Q2...Q2n−2(Q2n) = (Q2n)Q2...Q2n−2, P1(Q4...Q2n) = (Q4...Q2n)P1, P1(Q6...Q2n) = (Q6...Q2n)P1, ... P1Q2n = Q2nP1, Q1(Q3...Q2n−1) = (Q3...Q2n−1)Q1, Q1Q3(Q5...Q2n−1) = (Q5...Q2n−1)Q1Q3, ... Q1...Q2n−3(Q2n−1) = (Q2n−1)Q1...Q2n−3, P0(Q3...Q2n−1) = (Q3...Q2n−1)P0, P0(Q5...Q2n−1) = (Q5...Q2n−1)P0, ... P0Q2n−1 = Q2n−1P0, (C2) the pairs (P0,Q1...Q2n−1) and (P1,Q2...Q2n) are weakly compatible and (P0,Q1....Q2n−1) and Q2...Q2n satisfy (CLR)(P0,Q1...Q2n−1)Q2...Q2n property, (C3) F(ψ(d(P0x,P1y)),ψ(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)),ψ(d(Q1Q3...Q2n−1x,P0x)), ψ(d(Q2Q4...Q2ny,P1y)),ψ(d(Q1Q3...Q2n−1x,P1y)),ψ(d(Q2Q4...Q2ny,P0x))) + G(ψ(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)),ψ(d(Q1Q3...Q2n−1x,P0x)), ψ(d(Q2Q4...Q2ny,P1y)),ψ(d(Q1Q3...Q2n−1x,P1y),ψ(d(Q2Q4...Q2ny,P0x))) ≤ 0, for all x,y ∈ X , some F ∈ FD, G ∈ GD and ψ is an almost altering distance. Then Q1, Q2,...,Q2n, P0 and P1 have a unique common fixed point in X . Proof. Let Q′1 = Q1Q3...Q2n−1 and Q′2 = Q2Q4...Q2n. Since (P0,Q′1) and Q′2 satisfy (CLR)(P0,Q′1)Q′2 property, there exists a sequence {un} in X such that lim n→∞ P0un = lim n→∞ Q′1un = lim n→∞ Q1Q3...Q2n−1un = z, where z ∈ Q′1(X) ∩ Q′2(X) = Q1...Q2n−1(X) ∩ Q2...Q2n(X). Since z ∈ Q2Q4...Q2n(X), there exists u ∈ X such that z = Q2Q4...Q2nu. Using 224 Families of mappings satisfying a mixed implicit relation (C3) with x = un and y = u, we get F(ψ(d(P0un,P1u)),ψ(d(Q1Q3...Q2n−1un,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1un,P0un)), ψ(d(Q2Q4...Q2nu,P1u)),ψ(d(Q1Q3...Q2n−1un,P1u)),ψ(d(Q2Q4...Q2nu,P0un))) + G(ψ(d(Q1Q3...Q2n−1un,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1un,P0un)), ψ(d(Q2Q4...Q2nu,P1u)),ψ(d(Q1Q3...Q2n−1un,P1u),ψ(d(Q2Q4...Q2nu,P0un))) ≤ 0. Taking limits as n → ∞, we have F(ψ(d(z,P1u)),0,0,ψ(d(z,P1u)),ψ(d(z,P1u)),0) + G(0,0,ψ(d(z,P1u)), ψ(d(z,P1u)),0) ≤ 0. If d(z,P1u) > 0, then G(0,0,ψ(d(z,P1u)),ψ(z,P1u),0) > 0, which implies that F(ψ(d(z,P1u)),0,0,ψ(d(z,P1u)),ψ(d(z,P1u),0) < 0, a contradiction of (F2D). Hence d(z,P1u) = 0 i.e., z = P1u = Q2Q4...Q2nu. Since (P1,Q2Q4...Q2n) is weakly compatible, we have P1z = P1Q2Q4...Q2nu = Q2Q4...Q2nP1u = Q2Q4...Q2nz. Since z ∈ Q1Q3...Q2n−1(X), which implies z = Q1Q3...Q2n−1v for some v ∈ X . On putting x = v and y = u in (C3), we have F(ψ(d(P0v,P1u)),ψ(d(Q1Q3...Q2n−1v,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1v,P0v)), ψ(d(Q2Q4...Q2nu,P1u)),ψ(d(Q1Q3...Q2n−1v,P1u)),ψ(d(Q2Q4...Q2nu,P0v))) + G(ψ(d(Q1Q3...Q2n−1v,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1v,P0v)), ψ(d(Q2Q4...Q2nu,P1u)),ψ(d(Q1Q3...Q2n−1v,P1u),ψ(d(Q2Q4...Q2nu,P0v))) ≤ 0. On simplification, we get F(ψ(d(P0v,z)),0,ψ(d(P0v,z)),0,0,ψ(d(P0v,z))) + G(0,ψ(d(P0v,z)),0,0, ψ(d(P0v,z))) ≤ 0. If d(P0v,z) > 0, then G(0,ψ(d(P0v,z)),0,0,ψ(d(P0v,z))) > 0. 225 R. Kumar and S. Kumar Therefore, we obtain F(ψ(d(P0v,z)),0,ψ(d(P0v,z))0,0,ψ(d(P0v,z)) < 0, a contradiction of (F1D). Hence d(P0v,z) = 0, which implies that z = P0v = Q1Q3...Q2n−1v. Since (P0,Q1Q3...Q2n−1) is weakly compatible, we get P0z = P0Q1Q3...Q2n−1v = Q1Q3...Q2n−1P0v = Q1Q3...Q2n−1z. Now, we prove that z = P1z. On putting x = v and y = z in (C3), we get F(ψ(d(P0v,P1z)),ψ(d(Q1Q3...Q2n−1v,Q2Q4...Q2nz)),ψ(d(Q1Q3...Q2n−1v,P0v)), ψ(d(Q2Q4...Q2nz,P1z)),ψ(d(Q1Q3...Q2n−1v,P1z)),ψ(d(Q2Q4...Q2nz,P0v))) + G(ψ(d(Q1Q3...Q2n−1v,Q2Q4...Q2nz)),ψ(d(Q1Q3...Q2n−1v,P0v)), ψ(d(Q2Q4...Q2nz,P1z)),ψ(d(Q1Q3...Q2n−1v,P1z),ψ(d(Q2Q4...Q2nz,P0v))) ≤ 0, which implies that F(ψ(d(z,P1z)),ψ(d(z,P1z)),0,0,ψ(d(z,P1z)),ψ(d(z,P1z))) + G(ψ(d(z,P1z)),0,0, ψ(d(P1z,z)),ψ(d(z,P1z))) ≤ 0. If d(z,P1z) > 0, then G(ψ(d(z,P1z)),0,0,ψ(d(P1z,z)),ψ(d(z,P1z))) > 0. Thus from above, we get F(ψ(d(z,P1z)),ψ(d(z,P1z)),0,0,ψ(d(z,P1z)),ψ(d(z,P1z))) < 0, a contradiction of (F3D). Hence d(z,P1v) = 0 i.e., P1z = z and hence P1z = Q2Q4...Q2nz = z. Further on putting x = y = z in (C3), we get F(ψ(d(P0z,P1z)),ψ(d(Q1Q3...Q2n−1z,Q2Q4...Q2nz)),ψ(d(Q1Q3...Q2n−1z,P0z)), ψ(d(Q2Q4...Q2nz,P1z)),ψ(d(Q1Q3...Q2n−1z,P1z)),ψ(d(Q2Q4...Q2nz,P0z))) + G(ψ(d(Q1Q3...Q2n−1z,Q2Q4...Q2nz)),ψ(d(Q1Q3...Q2n−1z,P0z)), ψ(d(Q2Q4...Q2nz,P1z)),ψ(d(Q1Q3...Q2n−1z,P1z),ψ(d(Q2Q4...Q2nz,P0z))) ≤ 0. On simplification, we have F(ψ(d(P0z,z)),ψ(d(P0z,z)),0,0,ψ(d(P0z,z)),ψ(d(P0z,z))) + G(ψ(d(P0z,z)),0, 0,ψ(d(P0z,z)),ψ(d(P0z,z))) ≤ 0. 226 Families of mappings satisfying a mixed implicit relation If d(P0,z) > 0, then G(ψ(d(P0z,z)),0, ,0,ψ(d(P0z,z)),ψ(d(P0z,z))) > 0, which implies that F(ψ(d(P0z,z)),ψ(d(P0z,z)),0,0,ψ(d(P0z,z)),ψ(d(P0z,z))) < 0, a contradiction of (F3D). Hence d(P0z,z) = 0 i.e., P0z = z and hence P0z = Q1Q3...Q2n−1z = z. On putting x = z and y = Q4...Q2nz in (C3) and using (C1), Q′1 = Q1Q3...Q2n−1 and Q′2 = Q2Q4...Q2n, we get F(ψ(d(P0z,P1Q4...Q2nz)),ψ(d(Q′1z,Q ′ 2Q4...Q2nz)),ψ(d(Q ′ 1z,P0z)), ψ(d(Q′2Q4...Q2nz,P1Q4...Q2nz)),ψ(d(Q ′ 1z,P1Q4...Q2nz)),ψ(d(Q ′ 2Q4...Q2nz,P0z))) + G(ψ(d(Q′1z,Q ′ 2Q4...Q2nz)),ψ(d(Q ′ 1z,P0z)),ψ(d(Q ′ 2Q4...Q2nz,P1Q4...Q2nz)), ψ(d(Q′1z,P1Q4...Q2nz)),ψ(d(Q ′ 2Q4...Q2nz,P0z))) ≤ 0. From this we get F(ψ(d(z,Q4...Q2nz)),ψ(d(z,Q4...Q2nz)),0,0,ψ(d(z,Q4...Q2nz)),ψ(d(Q4...Q2nz,z))) + G(ψ(d(z,Q4...Q2nz)),0,0,ψ(d(z,Q4...Q2nz)),ψ(d(Q4...Q2nz,z))) ≤ 0. If (d(z,Q4...Q2nz)) > 0 then G(ψ(d(z,Q4...Q2nz)),0,0,ψ(d(z,Q4...Q2nz)),ψ(d(Q4...Q2nz,z))) > 0. Therefore, we have F(ψ(d(z,Q4...Q2nz)),ψ(d(z,Q4...Q2nz)),0,0,ψ(d(z,Q4...Q2nz)), ψ(d(Q4...Q2nz,z))) < 0, a contradiction to (F3D). Hence d(z,Q4...Q2nz)) = 0 i.e., Q4...Q2nz = z. Hence Q2Q4...Q2nz = Q2z = z. Continuing like this, we have P1z = Q2z = Q4z = ... = Q2n = z. (1) On putting x = Q3...Q2n−1z and y = z in (C3) and using (C1), Q′1 = Q1Q3...Q2n−1 and Q′2 = Q2Q4...Q2n, we get F(ψ(d(P0Q3...Q2n−1z,P1z)),ψ(d(Q′1Q3...Q2n−1z,Q ′ 2z)), ψ(d(Q′1Q3...Q2n−1z,P0Q3...Q2n−1z)),ψ(d(Q ′ 2z,P1z)), ψ(d(Q′1Q3...Q2n−1z,P1z)),ψ(d(Q ′ 2z,P0Q3...Q2n−1z))) + G(ψ(d(Q′1Q3...Q2n−1z,Q ′ 2z)),ψ(d(Q ′ 1Q3...Q2n−1z,P0Q3...Q2n−1z)), ψ(d(Q′2z,P1z)),ψ(d(Q ′ 1Q3...Q2n−1z,P1z)),ψ(d(Q ′ 2z,P0Q3...Q2n−1z))) ≤ 0, 227 R. Kumar and S. Kumar which implies that F(ψ(d(Q3...Q2n−1z,z)),ψ(d(Q3...Q2n−1z,z)),0,0,ψ(d(Q3...Q2n−1z,z)), ψ(d(z,Q3...Q2n−1z))) + G(d(ψ(d(Q3...Q2n−1z,z)),0,0,ψ(d(Q3...Q2n−1z,z)), ψ(d(z,Q3...Q2n−1z))) ≤ 0. If d(z,Q3...Q2n−1z) > 0 then G(d(ψ(d(Q3...Q2n−1z,z)),0,0,ψ(d(Q3...Q2n−1z,z)),ψ(d(z,Q3...Q2n−1z))) > 0. Thus from above, we obtain F(ψ(d(Q3...Q2n−1z,z)),ψ(d(Q3...Q2n−1z,z)),0,0,ψ(d(Q3...Q2n−1z,z)), ψ(d(z,Q3...Q2n−1z))) < 0, a contradiction to (F3D). Hence d(z,Q3...Q2n−1z) = 0 i.e., Q3...Q2n−1z = z. Hence Q1Q3...Q2n−1z = Q1z = z. Continuing like this, we have P0z = Q1z = Q3z = ... = Q2n−1 = z. (2) Hence from (1) and (2), we have P0z = P1 = Q1z = Q2z = Q3z = ... = Q2n−1 = Q2nz = z. Therefore, z is a common fixed point of the given self mappings. Uniqueness. Let w be another fixed point of the given mappings. Then P0w = P1w = Q1w = Q2w = Q3w = ... = Q2nw = w. Suppose that z ̸= w. Putting x = z and y = w in condition (C3), we have F(ψ(d(z,w)),ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z)) + G(ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z))) ≤ 0. If d(z,w) > 0, then G(ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z))) > 0. Therefore, we obtain F(ψ(d(z,w)),ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z)) < 0, a contradiction of (F3D). Hence z = w. Therefore, z is a unique common fixed point of the given mappings. Now we prove a theorem for families of mappings. 228 Families of mappings satisfying a mixed implicit relation Theorem 2.3. Let {Sα}α∈J and {Qi} 2p i=1 be two families of self-mappings on a metric space (X ,d). Suppose that there exists a fixed β ∈ J such that: (C4) Q2(Q4...Q2n) = (Q4...Q2n)Q2, Q2Q4(Q6...Q2n) = (Q6...Q2n)Q2Q4, ... Q2...Q2n−2(Q2n) = (Q2n)Q2...Q2n−2, Sβ(Q4...S2n) = (S4...S2n)Sβ, Sβ(Q6...Q2n) = (Q6...Q2n)Sβ, ... SβQ2n = Q2nSβ, Q1(Q3...Q2n−1) = (Q3...Q2n−1)Q1, Q1Q3(Q5...Q2n−1) = (Q5...Q2n−1)Q1Q3, ... Q1...Q2n−3(Q2n−1) = (Q2n−1)Q1...Q2n−3, Sα(Q3...Q2n−1) = (Q3...Q2n−1)Sα, Sα(Q5...Q2n−1) = (Q5...Q2n−1)Sα, ... SαS2n−1 = S2n−1Sα, (C5) the pairs (Sα,Q1...Q2n−1) and (Sβ,Q2...Q2n) are weakly compatible and (Sα,Q1...Q2n−1) and Q2...Q2n satisfy (CLR)(Sα,Q1...Q2n−1)Q2...Q2n property, (C6) F(ψ(d(Sαx,Sβy)),ψ(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)),ψ(d(Q1Q3...Q2n−1x,Sαx)), ψ(d(Q2Q4...Q2ny,Sβy)),ψ(d(Q1Q3...Q2n−1x,Sβy)),ψ(d(Q2Q4...Q2ny,Sαx))) + G(ψ(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)),ψ(d(Q1Q3...Q2n−1x,Sαx)), ψ(d(Q2Q4...Q2ny,Sβy)),ψ(d(Q1Q3...Q2n−1x,Sβy)),ψ(d(Q2Q4...Q2ny,Sαx))) ≤ 0, for all x,y ∈ X and some F ∈ FD, G ∈ GD and ψ is an almost altering distance. Then all Sα and Qi have a unique common fixed point in X . Proof. Let Sα0 be a fixed element in {Sα}α∈J . By Theorem 2.2 with P0 = Sα and P1 = Sα0 it follows that there exists some u ∈ X such that Sαu = Sα0u = Q1Q3...Q2n−1u = Q2Q4...Q2nu = u. Let β ∈ J be arbitrary. Then from (C6), we get F(ψ(d(Sαu,Sβu)),ψ(d(Q1Q3...Q2n−1u,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1u,Sαu)), ψ(d(Q2Q4...Q2nu,Sβu)),ψ(d(Q1Q3...Q2n−1u,Sβu)),ψ(d(Q2Q4...Q2nu,Sαu))) + G(ψ(d(Q1Q3...Q2n−1u,Q2Q4...Q2nu)),ψ(d(Q1Q3...Q2n−1u,Sαu)), ψ(d(Q2Q4...Q2nu,Sβu)),ψ(d(Q1Q3...Q2n−1u,Sβu)),ψ(d(Q2Q4...Q2nu,Sαu))) ≤ 0. 229 R. Kumar and S. Kumar Hence F(ψ(d(u,Sβu)),ψ(d(u,u)),ψ(d(u,u)),ψ(d(u,Sβu)),ψ(d(u,Sβu)),ψ(d(u,u))) +G(ψ(d(u,u)),ψ(d(u,u)),ψ(d(u,Sβu)),ψ(d(u,Sβu)),ψ(d(u,u))) ≤ 0, i.e., F(ψ(d(u,Sβu)),0,0,ψ(d(u,Sβu)),ψ(d(u,Sβu)),0) +G(0,0,ψ(d(u,Sβu)),ψ(d(u,Sβu)),0) ≤ 0. If d(u,Sβu) > 0, we get G(0,0,ψ(d(u,Sβu)),ψ(d(u,Sβu)),0) > 0, which implies that F(ψ(d(u,Sβu)),0,0,ψ(d(u,Sβu)),ψ(d(u,Sβu)),0) < 0, a contradiction by (F2D) and hence ψ(d(u,Sβu)) = 0 i.e., Sβu = u for each β ∈ J. Uniqueness follows easily. If we take ψ(t) = t in Theorem 2.2, we get Theorem 2.4. Let Q1,Q2, ...,Q2n,P0 and P1 be self mappings on a metric space (X ,d), satisfying conditions (C1), (C2) and the following condition: (C7) F((d(P0x,P1y)),(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)),(d(Q1Q3...Q2n−1x,P0x)), (d(Q2Q4...Q2ny,P1y)),(d(Q1Q3...Q2n−1x,P1y)),(d(Q2Q4...Q2ny,P0x))) + G(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny),d(Q1Q3...Q2n−1x,P0x),d(Q2Q4...Q2ny,P1y), d(Q1Q3...Q2n−1x,P1y),d(Q2Q4...Q2ny,P0x)) ≤ 0, for all x,y ∈ X , some F ∈ FD, G ∈ GD and ψ is an almost altering distance. Then Q1, Q2,...,Q2n, P0 and P1 have a unique common fixed point in X . If we take ψ(t) = t in Theorem 2.3, we get Theorem 2.5. Let {Sα}α∈J and {Qi} 2p i=1 be two families of self-mappings on a metric space (X ,d). Suppose that there exists a fixed β ∈ J such that conditions (C4) and (C5) are satisfied. Moreover, (C8) F(d(Sαx,Sβy),d(Q1Q3...Q2n−1x,Q2Q4...Q2ny),d(Q1Q3...Q2n−1x,Sαx), (d(Q2Q4...Q2ny,Sβy),d(Q1Q3...Q2n−1x,Sβy),d(Q2Q4...Q2ny,Sαx)) + G(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny),d(Q1Q3...Q2n−1x,Sαx), d(Q2Q4...Q2ny,Sβy),d(Q1Q3...Q2n−1x,Sβy),d(Q2Q4...Q2ny,Sαx)) ≤ 0, for all x,y ∈ X and some F ∈ FD, G ∈ GD and ψ is an almost altering distance. Then all Sα and Qi have a unique common fixed point in X . 230 Families of mappings satisfying a mixed implicit relation Remark 2.1. (i). Let ψ and ϕ be as in Theorem 1.1. Then F(u1, ...,u6) = ψ(u1) − ψ(m(x,y)) and G(v1, ...,v5) = ϕ(m(x,y)). Then F(u,0,u,0,0,u) = F(u,0,0,u,u,0) = F(u,u,0,0,u,u) = 0 and G(v1, ...,v5) = ϕ(max{v1, ...,v5}) > 0, if one of v1, ...,v5 > 0. Hence F ∈ FD and G ∈ GD. Then by Theorem 2.4, we get a generalization and extension of Theorem 1.1 for any even number of weakly compatible mappings. Similarly, Theorem 2.5 is a generalization and extension of Theorem 1.1 for fam- ilies of weakly compatible mappings. (ii). Theorems 2.2 and 2.3 are extension of Theorem 2.1 for any even number of weakly compatible mappings and families of weakly compatible mappings respec- tively. Now we give an example in support of our theorems. Example 2.1. Let X = [0,1] and d be usual metric on X . Define Sα(x) = x4 1 + x4 for each α ∈ J and all x ∈ X , Qi(x) = x n √ 4 for each i ∈ {1,2, ...,2n} and all x ∈ X . Then Q2Q4...Q2nx = x4, Q1Q3...Q2n−1x = x4. The pairs (Sα,Q1...Q2n−1) and (Sβ,Q2...S2n) are weakly compatible.. Define implicit function F such that Let F(u1, ...,u6) = u1 − 9 10 max{u2,u3,u4,u5,u6}. and G(t1, ..., t5) = 1 100(t1 + t2 + t3 + t4 + t5) . Then F ∈ FD and G ∈ GD. Thus all the conditions of Theorems 2.2 (for α = 0,1) and 2.3 are satisfied for ψ(t) = t and 0 is the unique common fixed point of the mappings. 231 R. Kumar and S. Kumar 3 Application In 2002, Branciari [4] obtained Banach contraction principle for mappings satisfying an integral type contraction condition. In the same way, we analyze Theorem 2.3 for mappings satisfying integral type contraction condition. Lemma 3.1. [19] Let r : [0,∞) → [0,∞) is a Lebesgue measurable mapping which is summable on each compact subset of [0,∞) such that ∫ ∞ 0 r(t)dt > 0, for ϵ > 0. Then ψ(t) = ∫ t 0 r(x)dx is an almost altering distance. Theorem 3.1. Let {Sα}α∈J and {Qi} 2p i=1 be two families of self-mappings on a metric space (X ,d). Suppose that there exists a fixed β ∈ J such that conditions (C4) and (C5) are satisfied. Moreover, (C12) F (∫ d(Sαx,Sβy) 0 r(t)dt, ∫ d(Q1Q3...Q2n−1x,Q2Q4...Q2ny) 0 r(t)dt, ∫ d(Q1Q3...Q2n−1x,Sαx) 0 r(t)dt, ∫ d(Q2Q4...Q2ny,Sβy) 0 r(t)dt, ∫ d(Q1Q3...Q2n−1x,Sβy) 0 r(t)dt, ∫ d(Q2Q4...Q2ny,Sαx) 0 r(t)dt ) + G (∫ d(Q1Q3...Q2n−1x,Q2Q4...Q2ny) 0 r(t)dt, ∫ d(Q1Q3...Q2n−1x,Sαx) 0 r(t)dt, ∫ d(Q2Q4...Q2ny,Sβy) 0 r(t)dt, ∫ d(Q1Q3...Q2n−1x,Sβy) 0 r(t)dt, ∫ d(Q2Q4...Q2ny,Sαx) 0 r(t)dt ) ≤ 0, for all x,y ∈ X and some F ∈ FD and G ∈ GD. Then all Sα and Qi have a unique common fixed point in X . Proof. Let ψ(t) be as in Lemma 3.1. Then ψ(d(Sαx,Sβy)) = ∫ d(Sαx,Sβy) 0 r(t)dt,ψ(d(Q1Q3...Q2n−1x,Sαx) = ∫ d(Q1Q3...Q2n−1x,Sαx) 0 ψ(d(Q1Q3...Q2n−1x,Q2Q4...Q2ny)) = ∫ d(Q1Q3...Q2n−1x,Q2Q4...Q2ny) 0 r(t)dt, ψ(d(Q2Q4...Q2ny,Sβy)) = ∫ d(Q2Q4...Q2ny,Sβy) 0 r(t)dt, ψ(d(Q1Q3...Q2n−1x,Sβy)) = ∫ d(Q1Q3...Q2n−1x,Sβy) 0 r(t)dt, ψ(d(Q2Q4...Q2ny,Sαx)) = ∫ d(Q2Q4...Q2ny,Sαx) 0 r(t)dt. 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