EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 135-143 ISSN 1307-5543 – ejpam.com Published by New York Business Global Quadruple g-best Proximity Point for New Contraction in Complete Metric Space Savita Rathee1, Monika Swami1,∗ 1 Department of Mathematics, Maharshi Dayanand University, Rohtak, India Abstract. The aim of this manuscript is to propose a contraction to pursue the existence of g-best proximity point results. The finding of this manuscript generalize and unify the results of Rohen and Mlaiki by using the new contraction with P-property and prove the existence and uniqueness of quadruple best proximity point alongwith an example. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Best proximity point, quadruple best proximity point, metric space, contraction 1. Introduction Fixed point theory is a flourished theory due to its functioning in physics, computer science, engineering etc. As always it is not possible to find fixed point for every self- contractive mappings, then there is possibility of existence of a point with minimum dis- tance between the point and its image. This point is known as best proximity point which was introduced by Fan [8] and extended by Basha [5] and many more researchers. In 1987, Guo and Lakshmikantham [10], introduced coupled fixed point and proved its related fixed point theorems under appropriate conditions. After that, Lakshmikantham and Ciric in [13] extend these results by defining the g-monotone property. The results of [10] leads to the development of tripled fixed point by Berinde and Borcut [7]. In [7], they proved the existence and uniqueness of the introduced tripled fixed point for non-linear mappings in Partially ordered complete metric space and later on many results exists between coupled and tripled fixed points on different spaces under different contractions. In 2012, tripled fixed point was extended to quadruple fixed point by Karapinar and Luong [11] in complete metric space. Motivated from [18], Rohen and Maliki [17] gave the notion of tripled best proximity points theorem graced with P-property and the developed contraction. See references [15], [2], [9], [14] for further research in coupled best proximity point results. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4171 Email addresses: dr.savitarathee@gmail.com (S. Rathee), monikaswami06@gmail.com (M. Swami) http://www.ejpam.com 135 © 2022 EJPAM All rights reserved. S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 136 Rohen and Maliki [17] and Karapinar and Luong [11], motivated us, in the direction to precede the quadruple best proximity point. We propose the quadruple best proximity point results with P-property and the newly introduce contraction. Also, examples are supplied in favour of our results. 2. Preliminaries Definition 1. [3] Let (X, d) be metric space, Q and R be two non-empty subset of X. Define d(Q,R) = inf{d(x, y) : x ∈ Q, y ∈ R}, Q0 = {x ∈ Q : there exists some y ∈ R such that d(x, y) = d(Q,R)}, R0 = {y ∈ R : there exists some x ∈ Q such that d(x, y) = d(Q,R)}. In 2011, Basha [4] proved sufficient conditions when Q0 and R0 are non-empty. Definition 2. [6] Let (X, d) be metric space and Q ≠ ϕ,R ≠ ϕ are subsets of X. Let G : Q → R be a mapping. Then x ∈ Q is said to be best proximity point if and only if d(x,Gx) = d(Q,R). Definition 3. [7] Let G : X ×X ×X → X. An element (x, y, z) is said to be tripled fixed point of G if G(x, y, z) = x,G(y, x, z) = y and G(z, y, x) = z. Definition 4. [11] Let G : X×X×X×X → X. An element (x, y, z, t) is said to be quadru- ple fixed point of G if G(x, y, z, t) = x,G(y, x, z, t) = y,G(z, y, x, t) = z and G(t, y, z, x) = t. Definition 5. [16] Let (Q,R) be non-empty pair of subsets of mertic space (X, d) with Q0 ̸= ϕ, then the pair (Q,R) has P-property if and only if{ d(x1, y1) = d(Q,R) d(x2, y2) = d(Q,R) =⇒ d(x1, x2) = d(y1, y2), where x1, x2 ∈ Q and y1, y2 ∈ R. Definition 6. [12] Let (Q,R) be non-empty pair of subsets of mertic space (X, d) . Con- sider g : Q → Q and G : Q → R be mappings then a point x ∈ Q is a best proximity g- point of the pair (g,G) if d(gx,Gx) = d(Q,R). Definition 7. [1] Let Ψ represent the family of functions ψ such that ψ : [0,∞) → [0,∞) which satisfy (i) ψ(x) = 0 if and only if x = 0. (ii) ψ(x) is continuous and non-decreasing. S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 137 Let Θ signify the collection of functions of type θ : [0,∞)8 → [0,∞) such that θ(x, y, z, t, a, b, c, u) = min{x, y, z, t, a, b, c, u} for all x, y, z, t, a, b, c, u ∈ [0,∞). Definition 8. [17] Let (X, d) be a complete metric space and Q ̸= ϕ and R ̸= ϕ are closed subsets. An element (x, y, z) ∈ X × X × X is said to be a tripled best proximity point of G : X × X × X → X if x, z ∈ Q and y ∈ R such that d(x,G(x, y, z)) = d(Q,R), d(y,G(y, x, y)) = d(Q,R) and d(z,G(z, y, x)) = d(Q,R). 3. Main Results Definition 9. Let (X, d) be a complete metric space and (Q,R) be a pair of non-empty subset of X such that Q0 is non-empty. Consider g : X → X and G : X4 → X be two mappings, then (x, y, z, t) is said to be quadruple g-best proximity point of G and g if d(gx,G(x, y, z, t)) = d(Q,R), d(gy,G(y, x, z, t)) = d(Q,R), d(gz,G(z, y, x, t)) = d(Q,R) and d(gt,G(t, y, z, x)) = d(Q,R) for all x, z ∈ Q and y, t ∈ R. If g = I (Identity mapping) then (x, y, z, t) is said to be quadruple best proximity point of G if d(x,G(x, y, z, t)) = d(Q,R), d(y,G(y, x, z, t)) = d(Q,R), d(z,G(z, y, x, t)) = d(Q,R) and d(t, G(t, y, z, x)) = d(Q,R) for all x, z ∈ Q and y, t ∈ R. Theorem 1. Let Q and R be non-empty subset of complete metric space (X, d) such that Q0 and R0 are non-empty and g : X → X is an isometry such that Q0 ⊆ g(Q0) and R0 ⊆ g(R0), let G : X4 → X be continuous mapping and ψ, ζ ∈ Ψ and θ ∈ Θ, satisfies the preceeding conditions: (i) For every x, y, z, t, a, b, c, u ∈ X ψ(d(gx, ga)) =ψ(d(G(x, y, z, t), G(a, b, c, u))) ≤ψ{max(d(x, a), d(y, b), d(z, c), d(t, u))} − ζ{max(d(x, a), d(y, b), d(z, c), d(t, u))} + θ[d(ga,G(x, y, z, t))− d(Q,R), d(gb,G(y, x, z, t))− d(Q,R), d(gc,G(z, y, x, t))− d(Q,R), d(gu,G(t, y, z, x))− d(Q,R), d(gx,G(x, y, z, t))− d(Q,R), d(gy,G(y, x, z, t))− d(Q,R), d(gz,G(z, y, x, t))− d(Q,R), d(gt,G(t, y, z, x))− d(Q,R)] (1) (ii) G(Q0,R0,Q0,R0) ⊆ R0 (iii) G(R0,Q0,R0,Q0) ⊆ Q0 (iv) Pair (Q,R) has P-property then (a, a, a, a) is the unique quadruple g-best proximity point of the pair (g,G). Proof. Choose x0, z0 ∈ Q0 and y0, t0 ∈ R0. Since G(x0, y0, z0, t0), G(z0, y0, x0, t0) ∈ R0 and G(y0, x0, t0, z0), G(t0, z0, y0, x0) ∈ Q0, there exist x1, z1 ∈ Q and y1, t1 ∈ R such that d(gx1, G(x0, y0, z0, t0)) = d(gy1, G(y0, x0, z0, t0)) = d(gz1, G(z0, y0, x0, t0)) = S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 138 d(gt1, G(t0, z0, y0, x0)) = d(Q,R). Continuing like this, we get a sequence of {gxn}, {gzn} ∈ Q and {gyn}, {gtn} ∈ R such that d(gxn+1, G(xn, yn, zn, tn)) = d(Q,R) d(gyn+1, G(yn, xn, tn, zn)) = d(Q,R) d(gzn+1, G(zn, yn, xn, tn)) = d(Q,R) d(gtn+1, G(tn, zn, yn, xn)) = d(Q,R) for all n ∈ IN ∪ {0} (2) If d(gxn, gxn+1) = d(gyn, gyn+1) = d(gzn, gzn+1) = d(gtn, gtn+1) = 0 for all n ∈ IN ∪ {0} then nothing to prove. Suppose d(gxn, gxn+1) > 0 or d(gyn, gyn+1) > 0 or d(gzn, gzn+1) > 0 or d(gtn, gtn+1) > 0. From(1), P-property, and d(gxn+1, G(xn, yn, zn, tn)) = d(Q,R), d(gxn, G(xn−1, yn−1, zn−1, tn−1)) = d(Q,R), we have d(gxn, gxn+1) = d(G(xn−1, yn−1, zn−1, tn−1), G(xn, yn, zn, tn)) ψ(d(gxn, gxn+1)) = ψ(d(G(xn−1, yn−1, zn−1, tn−1), G(xn, yn, zn, tn))) ≤ψ{max(d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn))} − ζ{max(d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn))} + θ[d(gxn, G(xn−1, yn−1, zn−1, tn−1)− d(Q,R), d(gyn, G(yn−1, xn−1, tn−1, zn−1)− d(Q,R), d(gzn, G(zn−1, yn−1, xn−1, tn−1)− d(Q,R), d(gtn, G(tn−1, zn−1, yn−1, xn−1)− d(Q,R), d(gxn−1, G(xn−1, yn−1, zn−1, tn−1)− d(Q,R), d(gyn−1, G(yn−1, xn−1, tn−1, zn−1)− d(Q,R), d(gzn−1, G(zn−1, yn−1, xn−1, tn−1)− d(Q,R), d(gtn−1, G(tn−1, zn−1, yn−1, xn−1)− d(Q,R)] = ψ{max(d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn))} − ζ{max(d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn))} (3) Similarly for d(gyn+1, G(yn, xn, tn, zn)) = d(Q,R), d(gyn, G(yn−1, xn−1, tn−1, zn−1)) = d(Q,R), d(gzn+1, G(zn, yn, xn, tn)) = d(Q,R), d(gzn, G(zn−1, yn−1, xn−1, tn−1)) = d(Q,R) and d(gtn+1, G(tn, zn, yn, xn)) = d(Q,R), d(gtn, G(tn−1, zn−1, yn−1, xn−1)) = d(Q,R), we have ψ(d(gyn, gyn+1)) ≤ψ{max(d(yn−1, yn), d(xn−1, xn), d(zn−1, zn), d(tn−1, tn))} − ζ{max(d(yn−1, yn), d(xn−1, xn), d(zn−1, zn), d(tn−1, tn))} (4) ψ(d(gzn, gzn+1)) ≤ψ{max(d(zn−1, zn), d(yn−1, yn), d(xn−1, xn), d(tn−1, tn))} − ζ{max(d(zn−1, zn), d(yn−1, yn), d(xn−1, xn), d(tn−1, tn))} (5) ψ(d(gtn, gtn+1)) ≤ψ{max(d(tn−1, tn), d(yn−1, yn), d(zn−1, zn), d(xn−1, xn))} − ζ{max(d(tn−1, tn), d(yn−1, yn), d(zn−1, zn), d(xn−1, xn))} (6) From (3), (4), (5) and (6), we obtain ψ[max{d(gxn, gxn+1),d(gyn, gyn+1), d(gzn, gzn+1), d(gtn, gtn+1)}] S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 139 ≤ ψ[max{d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn)}] − ζ[max{d(xn−1, xn), d(yn−1, yn), d(zn−1, zn), d(tn−1, tn)}] = ψ[max{d(gxn−1, gxn), d(gyn−1, gyn), d(gzn−1, gzn), d(gtn−1, gtn)}] − ζ[max{d(gxn−1, gxn), d(gyn−1, gyn), d(gzn−1, gzn), d(gtn−1, gtn)}] (7) ≤ ψ[max{d(gxn−1, gxn), d(gyn−1, gyn), d(gzn−1, gzn), d(gtn−1, gtn)}] As ψ is continuous function, therefore, max{d(gxn, gxn+1),d(gyn, gyn+1), d(gzn, gzn+1), d(gtn, gtn+1)} ≤ max{d(gxn−1, gxn), d(gyn−1, gyn), d(gzn−1, gzn), d(gtn−1, gtn)} implies {d(gxn, gxn+1), d(gyn, gyn+1), d(gzn, gzn+1), d(gtn, gtn+1)} is a non-increasing se- quence of positive real number, it must converge to a positive real number, say τ =⇒ lim n→∞ {d(gxn, gxn+1),d(gyn, gyn+1), d(gzn, gzn+1), d(gtn, gtn+1)} = τ. Taking limit on both side in (7), we have ψ(τ) ≤ ψ(τ)− ζ(τ) =⇒ ζ(τ) = 0 τ = 0. Hence, lim n→∞ d(gxn, gxn+1) = lim n→∞ d(gyn, gyn+1) = lim n→∞ d(gzn, gzn+1) = lim n→∞ d(gtn, gtn+1) = 0 Now, we prove that {gxn}, {gyn}, {gzn} and {gtn} are Cauchy sequences, i.e. max{d(gxn(ι), gxm(ι), d(gyn(ι), gym(ι)), d(gzn(ι), gzm(ι)), d(gtn(ι), tm(ι))} < ϵ ∀m(ι) > n(ι) > ι. Let if possible sequences are not Cauchy then there exists an ϵ > 0 such that for all ι > 0 there are m(ι) > n(ι) > ι which satisfies the conditions max{d(gxn(ι), gxm(ι), d(gyn(ι), gym(ι)), d(gzn(ι), gzm(ι)), d(gtn(ι), tm(ι))} ≥ ϵ and max{d(gxn(ι)−1, gxm(ι), d(gyn(ι)−1, gym(ι)), d(gzn(ι)−1, gzm(ι)), d(gtn(ι)−1, tm(ι))} < ϵ. Then, we have ϵ ≤ d(gxn(ι), gxm(ι)) ≤ d(gxn(ι), gxn(ι)−1) + d(gxn(ι)−1, gxm(ι) ≤ d(gxn(ι), gxn(ι)−1) + ϵ This gives us ϵ ≤ d(gxn(ι), gxn(ι)−1) + ϵ For ι→ ∞, we have lim ι→∞ d(gxn(ι), gxm(ι)) = ϵ (8) Also, from triangular inequality, we find d(gxn(ι)−1, gxm(ι)−1) ≤ d(gxn(ι)−1, gxm(ι)) + d(gxm(ι), gxm(ι)−1) ≤ ϵ Hence, d(gxn(ι)−1, gxm(ι)−1) ≤ ϵ. (9) Since d(gxn(ι), G(xn(ι)−1, yn(ι)−1, zn(ι)−1, tn(ι)−1)) = d(Q,R) S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 140 and d(gxm(ι), G(xm(ι)−1, ym(ι)−1, zm(ι)−1, tm(ι)−1)) = d(Q,R). From P-property, we have d(gxn(ι), gxm(ι)) = d(G(xn(ι)−1, yn(ι)−1, zn(ι)−1, tn(ι)−1), G(xm(ι)−1, ym(ι)−1, zm(ι)−1, tm(ι)−1)) Now from (1) and using the continuity of ψ, we obtain ψ(d(gxn(ι),gxm(ι))) =ψ(d(G(xn(ι)−1, yn(ι)−1, zn(ι)−1, tn(ι)−1), G(xm(ι)−1, ym(ι)−1, zm(ι)−1, tm(ι)−1))) ≤ ψ[max{d(xn(ι)−1, xm(ι)−1), d(yn(ι)−1, ym(ι)−1), d(zn(ι)−1, zm(ι)−1), d(tn(ι)−1, tm(ι)−1)}]− ζ[max{d(xn(ι)−1, xm(ι)−1), d(yn(ι)−1, ym(ι)−1), d(zn(ι)−1, zm(ι)−1), d(tn(ι)−1, tm(ι)−1)}] + θ[d(gxm(ι)−1, G(xn(ι)−1, yn(ι)−1, zn(ι)−1, tn(ι)−1)), d(gym(ι)−1, G(yn(ι)−1, xn(ι)−1, tn(ι)−1, zn(ι)−1)), d(gzm(ι)−1, G(zn(ι)−1, yn(ι)−1, xn(ι)−1, tn(ι)−1)), d(gtm(ι)−1, G(tn(ι)−1, zn(ι)−1, yn(ι)−1, xn(ι)−1)), d(gxn(ι)−1, G(xn(ι)−1, yn(ι)−1, zn(ι)−1, tn(ι)−1)), d(gyn(ι)−1, G(yn(ι)−1, xn(ι)−1, tn(ι)−1, zn(ι)−1)), d(gzn(ι)−1, G(zn(ι)−1, yn(ι)−1, xn(ι)−1, tn(ι)−1)), d(gtn(ι)−1, G(tn(ι)−1, zn(ι)−1, yn(ι)−1, xn(ι)−1))] = ψ[max{d(xn(ι)−1, xm(ι)−1), d(yn(ι)−1, ym(ι)−1), d(zn(ι)−1, zm(ι)−1), d(tn(ι)−1, tm(ι)−1)}]− ζ[max{d(xn(ι)−1, xm(ι)−1), d(yn(ι)−1, ym(ι)−1), d(zn(ι)−1, zm(ι)−1), d(tn(ι)−1, tm(ι)−1)}]. Similary, from the same techinque, we obtain ψ[max{d(gxn(ι), gxm(ι)), d(gyn(ι), gym(ι)), d(gzn(ι), gzm(ι)), d(gtn(ι), gtm(ι))}] ≤ ψ[max{d(gxn(ι)−1, gxm(ι)−1), d(gyn(ι)−1, gym(ι)−1), d(gzn(ι)−1, gzm(ι)−1), d(gtn(ι)−1, gtm(ι)−1)}]− ζ[max{d(gxn(ι)−1, gxm(ι)−1), d(gyn(ι)−1, gym(ι)−1), d(gzn(ι)−1, gzm(ι)−1), d(gtn(ι)−1, gtm(ι)−1)}]. Now, from (8) and (9), we get ψ(ϵ) ≤ ψ(ϵ)− ζ(ϵ) ζ(ϵ) = 0 =⇒ ϵ = 0. Thus, for ι tends to infinity, it gives us lim ι→∞ {d(gxn(ι), gxm(ι)), d(gyn(ι), gym(ι)), d(gzn(ι), gzm(ι)), d(gtn(ι), gtm(ι))} = 0, which is contradiction to our suppostion that ϵ > 0. Hence, {gxn}, {gzn} are Cauchy sequences in Q and {gyn}, {gtn} in R. Since (X, d) is complete metric space, then there exist, a, b, c, u ∈ X such that lim n→∞ gxn = a, lim n→∞ gyn = b, lim n→∞ gzn = c and lim n→∞ gtn = u. As Q,R are closed subset of X, then a, c ∈ Q and b, u ∈ R. Since G is continuous, then lim n→∞ d(gxn, G(xn, yn, zn, tn)) = d(Q,R) =⇒ d(ga,G(a, b, c, u)) = d(Q,R). S. Rathee, M. Swami, / Eur. J. Pure Appl. Math, 15 (1) (2022), 135-143 141 Similarly, d(gb,G(b, a, c, u)) = d(Q,R), d(gc,G(c, b, a, u)) = d(Q,R) and d(gu,G(u, b, c, a)) = d(Q,R). Thus, (a, b, c, u) is quadruple g-best proximity point of the pair (g,G). Now, we show that ga = gb = gc = gu. Again from P-property, g-isometry and condition (1), we calculate d(ga, gc) = d(G(a, b, c, u), G(c, b, a, u)) ψ(d(ga, gc)) = ψ(d(G(a, b, c, u), G(c, b, a, u))) ≤ ψ(d(a, c)) = ψ(d(ga, gc)) =⇒ a = c. Therefore, ga = gb = gc = gu. To prove the uniqueness of quadruple g-best proximity point , consider q as another point. Now d(ga, gq) = d(G(a, a, a, a), G(q, q, q, q)) ψ(d(ga, gq)) = ψ(d(G(a, a, a, a), G(q, q, q, q))) ≤ ψ(d(a, q)) = ψ(d(ga, gq)) =⇒ a = q. Hence, the result. Theorem 2. Let Q and R be non-empty subset of complete metric space (X, d) such that Q0 and R0 are non-empty and g : X → X is an isometry such that Q0 ⊆ g(Q0) and R0 ⊆ g(R0), let G : X4 → X be continuous mapping and ψ, ζ ∈ Ψ and θ ∈ Θ, satisfies the preceeding conditions: (i) For every x, y, z, t, a, b, c, u ∈ X ψ(d(gx, ga)) =ψ(d(G(x, y, z, t), G(a, b, c, u))) ≤ψ{max(d(x, a), d(y, b), d(z, c), d(t, u))} − ζ{max(d(x, a), d(y, b), d(z, c), d(t, u))} + θ[d(ga,G(x, y, z, t))− d(Q,R), d(gb,G(y, x, z, t))− d(Q,R), d(gc,G(z, y, x, t))− d(Q,R), d(gu,G(t, y, z, x))− d(Q,R), d(gx,G(x, y, z, t))− d(Q,R), d(gy,G(y, x, z, t))− d(Q,R), d(gz,G(z, y, x, t))− d(Q,R), d(gt,G(t, y, z, x))− d(Q,R)] (10) (ii) G(Q0,Q0,Q0,Q0) ⊆ R0 (iii) G(R0,R0,R0,R0) ⊆ Q0 (iv) Pair (Q,R) has P-property then (a, a, a, a) is the unique quadruple g-best proximity point of the pair (g,G). Proof. Consider x0, y0, z0, t0 ∈ Q0 thenG(x0, y0, z0, t0), G(y0, x0, z0, t0), G(z0, y0, x0, t0) and G(t0, y0, z0, x0) ∈ R0. Then by the same process as in theorem (1), we obtain (a, a, a, a) as the unique quadruple g-best proximity point of the pair (g,G). REFERENCES 142 Corollary 1. Let Q be non-empty subset of complete metric space (X, d) such that Q0 is non-empty and g : X → X be mapping such that Q ⊆ g(Q) and R ⊆ g(R), let G : X4 → X be continuous mapping and ψ, ζ ∈ Ψ and θ ∈ Θ, satisfies the preceeding conditions: (i) For every x, y, z, t, a, b, c, u ∈ X ψ(d(gx, ga)) =ψ(d(G(x, y, z, t), G(a, b, c, u))) ≤ψ{max(d(x, a), d(y, b), d(z, c), d(t, u))} − ζ{max(d(x, a), d(y, b), d(z, c), d(t, u))} + θ[d(ga,G(x, y, z, t))− d(Q,R), d(gb,G(y, x, z, t))− d(Q,R), d(gc,G(z, y, x, t))− d(Q,R), d(gu,G(t, y, z, x))− d(Q,R), d(gx,G(x, y, z, t))− d(Q,R), d(gy,G(y, x, z, t))− d(Q,R), d(gz,G(z, y, x, t))− d(Q,R), d(gt,G(t, y, z, x))− d(Q,R)] (11) (ii) G(Q,Q,Q,Q) ⊆ Q (iii) g is an isometry then (a, a, a, a) is the unique quadruple g-fixed point of the pair (g,G). Proof. By taking Q = R in Theorem (1), we have the desired result. Example 1. Consider X = [1, 5] with d(x, y) = ∥x − y∥for all x, y ∈ X. Let Q = [2, 3] and R = [3, 4] be subsets of X and G : X4 → X, g : Q → Q both are continuous mappings given by G(x, y, z, t) = 1 2(x − y + z + t) and g(x) = x respectively for all x, y, z, t ∈ X. Consider θ : [0,∞)8 → [0,∞) defined by θ(x, y, z, t, a, b, c, u) = min{x, y, z, t, a, b, c, u} and ψ, ζ : [0,∞) → [0,∞) are given by ψ(q) = 1 2q, ζ(q) = 1 3q. Here Q0 = {3} and R0 = {3} with d(Q,R) = 0. Taking x0, z0 ∈ Q0 and y0, t0 ∈ R0, then G(Q0,R0,Q0,R0) ⊆ R0, G(R0,Q0,R0,Q0) ⊆ Q0. Also, the remaining conditions of the theorem are satisfied. Hence by the theorem (1), (3, 3, 3, 3) is the quadruple g-best proximity point of g and G. Acknowledgements Special thanks to CSIR to fund PhD through file number 09/382(0187)/2017-EMR-1 References [1] A A Aserkar and M P Gandhi. Quadruple fixed point theorem for four mappings. Gen. Math. Notes, 25(2):95–109, 2014. 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