() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 2, 2012 pp. 193-206 Tripled coincidence and fixed point results in partial metric spaces Hassen Aydi and Mujahid Abbas Abstract In this paper, we introduce the concept of W -compatiblity of mappings F : X × X × X → X and g : X → X and based on this notion, we obtain tripled coincidence and common tripled fixed point results in the setting of partial metric spaces. The presented results generalize and extend several well known comparable results in the existing literature. We also provide an example to support our results. 2010 MSC: 54H25, 47H10. Keywords: W -compatible mappings, tripled coincidence point, common tripled fixed point, partial metric space. 1. Introduction Matthews [19, 20] introduced the notion of a partial metric space which is a generalization of usual metric spaces in which the distance of a point to itself is no longer necessarily zero. A partial metric space (see [19, 20]) is a pair (X, p) such that X is a (non- empty) set and p : X × X → R+ (where R+ denotes the set of all non negative real numbers) satisfies (p1) p(x, y) = p(y, x) (symmetry) (p2) If p(x, x) = p(x, y) = p(y, y) then x = y (equality) (p3) p(x, x) ≤ p(x, y) (small self-distances) (p4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangularity) for all x, y, z ∈ X. In this case we say that p is a partial metric on X. Each partial metric p on X generates a T0 topology τp on X with a base of the family open of p-balls {Bp(x, ε) : x ∈ X, ε > 0}, where Bp(x, ε) = {y ∈ 194 H. Aydi and M. Abbas X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0. Similarly, a closed p-ball is defined as Bp[x, ε] = {y ∈ X : p(x, y) ≤ p(x, x) + ε}. Definition 1.1 ([19, 20]). (i) A sequence {xn} in a partial metric space (X, p) is called Cauchy if lim n,m→∞ p(xn, xm) exists (and finite), (ii) A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn} in X converges, with respect to τp, to a point x ∈ X such that p(x, x) = lim n,m→∞ p(xn, xm). Notice that for a partial metric p on X, the function ps : X ×X → R+ given by (1.1) ps(x, y) = 2p(x, y) − p(x, x) − p(y, y) is a (usual) metric on X. It is well known and easy to see that (1.2) lim n→∞ ps(x, xn) = 0 ⇔ p(x, x) = lim n→∞ p(x, xn) = lim n,m→∞ p(xn, xm). Lemma 1.2 ([19, 20]). (A) A sequence {xn} is Cauchy in a partial metric space (X, p) if and only if {xn} is Cauchy in the metric space (X, p s). (B) A partial metric space (X, p) is complete if and only if the metric space (X, ps) is complete. For simplicity, we denote from now on X × X · · · X × X︸ ︷︷ ︸ k terms by Xk where k ∈ N and X a non-empty set. We start by recalling some definitions where X is a non-empty set. Definition 1.3 (Bhashkar and Lakshmikantham [13]). An element (x, y) ∈ X2 is called a coupled fixed point of a mapping F : X2 → X if x = F(x, y) and y = F(y, x). Definition 1.4 (Lakshmikantham and Ćirić [18]). An element (x, y) ∈ X2 is called (i) a coupled coincidence point of mappings F : X2 → X and g : X → X if gx = F(x, y) and gy = F(y, x), and (gx, gy) is called coupled point of coincidence; (ii) a common coupled fixed point of mappings F : X2 → X and g : X → X if x = gx = F(x, y) and y = gy = F(y, x). Note that if g is the identity mapping, then Definition 1.4 reduces to Defi- nition 1.3. In 2011, Samet and Vetro [21] introduced a fixed point of order N ≥ 3. In particular, for N = 3 we have following definition. Definition 1.5 (Samet and Vetro [21]). An element (x, y, z) ∈ X3 is called a tripled fixed point of a given mapping F : X3 → X if x = F(x, y, z), y = F(y, z, x) and z = F(z, x, y). Tripled coincidence and fixed point results 195 Note that, Berinde and Borcut [12] defined differently the notion of a tripled fixed point in the case of ordered sets in order to keep true the mixed monotone property. For more details, see [12]. Now, we give the following definitions. Definition 1.6. An element (x, y, z) ∈ X3 is called (i) a tripled coincidence point of mappings F : X3 → X and g : X → X if gx = F(x, y, z), gy = F(y, x, z) and gz = F(z, x, y). In this case (gx, gy, gz) is called tripled point of coincidence; (ii) a common tripled fixed point of mappings F : X3 → X and g : X → X if x = gx = F(x, y, z), y = gy = F(y, z, x) and z = gz = F(z, x, y). Fixed point theorems on partial metric spaces have received a lot of attention in the last years (see, for instance, [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 22, 23, 24] and their references). Abbas et al. [1] introduced the concept of w-compatible mappings and obtained coupled coincidence point and coupled point of coincidence for mappings satisfying a contractive condition in cone metric spaces. Very recently, Aydi et al. [11] introduced the concepts of w̃- compatible mappings and generalized the results in [1]. The aim of this paper is to introduce the concepts of W-compatible map- pings. Based on this notion, tripled coincidence point and common tripled fixed point for mappings F : X × X × X → X and g : X → X are obtained in partial metric space. The presented theorems generalize and extend several well known comparable results in the literature. An example is also given in support of our results. Definition 1.7 (Abbas, Khan and Radenović [1]). The mappings F : X2 → X and g : X → X are called w-compatible if g(F(x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x). Definition 1.8. Mappings F : X3 → X and g : X → X are called W- compatible if F(gx, gy, gz) = g(F(x, y, z)) whenever F(x, y, z) = gx, F(y, z, x) = gy and F(z, y, x) = gz. 2. Main results We present our first result as follows. Theorem 2.1. Let (X, p) be a partial metric space and F : X3 −→ X and g : X → X be mappings such that F(X3) ⊆ g(X) and g(X) is a complete subspace of X. Suppose that for any x, y, z, u, v, w ∈ X, the following condition p(F(x, y, z), F(u, v, w)) ≤ a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(F(u, v, w), gu) + a5p(F(v, w, u), gv) + a6p(F(w, u, v), gw) + a7p(F(u, v, w), gx) + a8p(F(v, w, u), gy) + a9p(F(w, u, v), gz) + a10p(F(x, y, z), gu) + a11p(F(y, z, x), gv) + a12p(F(z, x, y), gw) + a13p(gx, gu) + a14p(gy, gv) + a15p(gz, gw), 196 H. Aydi and M. Abbas holds, where ai, i = 1, · · · , 15 are nonnegative real numbers. If 9∑ i=7 ai + 15∑ i=1 ai < 1 or 12∑ i=10 ai + 15∑ i=1 ai < 1, then F and g have a tripled coincidence point. Proof. Let x0, y0 and z0 be three arbitrary points in X. By given assumptions, there exists (x1, y1, z1) such that F(x0, y0, z0) = gx1, F(y0, z0, x0) = gy1 and F(z0, x0, y0) = gz1. Continuing this process, we construct three sequences {xn}, {yn} and {zn} in X such that (2.1) F(xn, yn, zn) = gxn+1, F(yn, zn, xn) = gyn+1 and F(zn, xn, yn) = gzn+1 ∀ n ∈ N. Denote δn = p(gxn+1, gxn) + p(gyn+1, gyn) + p(gzn+1, gzn). We claim that (2.2) δn+1 ≤ κδn ∀ n ∈ N, where κ ∈ [0, 1) will be chosen conveniently. First, taking (x, y, z) = (xn, yn, zn) and (u, v, w) = (xn+1, yn+1, zn+1) in the considered contractive condition and using (2.1), we have p(gxn+1, gxn+2) = p(F(xn, yn, zn), F(xn+1, yn+1, zn+1)) ≤ a1p(F(xn, yn, zn), gxn) + a2p(F(yn, zn, xn), gyn) + a3p(F(zn, xn, yn), gzn) + a4p(F(xn+1, yn+1, zn+1), gxn+1) + a5p(F(yn+1, zn+1, xn+1), gyn+1) + a6p(F(zn+1, xn+1, yn+1), gzn+1) + a7p(F(xn+1, yn+1, zn+1), gxn) + a8p(F(yn+1, zn+1, xn+1), gyn) + a9p(F(zn+1, xn+1, yn+1), gzn) + a10p(F(xn, yn, zn), gxn+1) + a11p(F(yn, zn, xn), gyn+1) + a12p(F(zn, xn, yn), gzn+1) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1) = a1p(gxn+1, gxn) + a2p(gyn+1, gyn) + a3p(gzn+1, gzn) + a4p(gxn+2, gxn+1) + a5p(gyn+2, gyn+1) + a6p(gzn+2, gzn+1) + a7p(gxn+2, gxn) + a8p(gyn+2, gyn) + a9p(gzn+2, gzn) + a10p(gxn+1, gxn+1) + a11p(gyn+1, gyn+1) + a12p(gzn+1, gzn+1) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1). Then, using (p2) and the triangular inequality (which holds even for partial metrics), one can write for any n ∈ N (1 − a4 − a7 − a10)p(gxn+2, gxn+1) ≤ (a1 + a7 + a13)p(gxn+1, gxn) + (a2 + a8 + a14)p(gyn, gyn+1) +(a3 + a9 + a15)p(gzn, gzn+1) + (a5 + a8 + a11)p(gyn+2, gyn+1) + (a6 + a9 + a12)p(gzn+2, gzn+1). (2.3) Tripled coincidence and fixed point results 197 Similarly, following similar arguments to those given above, we obtain (1 − a4 − a7 − a10)p(gyn+2, gyn+1) ≤ (a1 + a7 + a13)p(gyn+1, gyn) + (a2 + a8 + a14)p(gzn, gzn+1) +(a3 + a9 + a15)p(gxn, gxn+1) + (a5 + a8 + a11)p(gzn+2, gzn+1) + (a6 + a9 + a12)p(gxn+2, gxn+1), (2.4) and (1 − a4 − a7 − a10)p(gzn+2, gzn+1) � (a1 + a7 + a13)p(gzn+1, gzn) + (a2 + a8 + a14)p(gxn, gxn+1) +(a3 + a9 + a15)p(gyn, gyn+1) + (a5 + a8 + a11)p(gxn+2, gxn+1) + (a6 + a9 + a12)p(gyn+2, gyn+1). (2.5) Adding (2.3) to (2.5) we have (2.6) (1−a4−a5−a6−a7−a8−a9−a10−a11−a12)δn+1 ≤ (a1+a2+a3+a7+a8+a9+a13+a14+a15)δn, that is (2.7) δn+1 ≤ κ1 δn ∀ n ∈ N, where κ1 = a1 + a2 + a3 + a7 + a8 + a9 + a13 + a14 + a15 1 − 12∑ i=4 ai . As 9∑ i=7 ai + 15∑ i=1 ai < 1, so 0 ≤ κ1 < 1. Hence (2.2) holds for κ = κ1. On the other hand, we have p(gxn+2, gxn+1) = p(F(xn+1, yn+1, zn+1), F(xn, yn, zn)) ≤ a1p(F(xn+1, yn+1, zn+1), gxn+1) + a2p(F(yn+1, zn+1, xn+1), gyn+1) + a3p(F(zn+1, xn+1, yn+1), gzn+1) + a4p(F(xn, yn, zn), gxn) + a5p(F(yn, zn, xn), gyn) + a6p(F(zn, xn, yn), gzn) + a7p(F(xn, yn, zn), gxn+1) + a8p(F(yn, zn, xn), gyn+1) + a9p(F(zn, xn, yn), gzn+1) + a10p(F(xn+1, yn+1, zn+1), gxn) + a11p(F(yn+1, zn+1, xn+1), gyn) + a12p(F(zn+1, xn+1, yn+1), gzn) + a13p(gxn+1, gxn) + a14p(gyn+1, gyn) + a15p(gzn+1, gzn) = a1p(gxn+2, gxn+1) + a2p(gyn+2, gyn+1) + a3p(gzn+2, gzn+1) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gxn+1) + a8p(gyn+1, gyn+1) + a9p(gzn+1, gzn+1) + a10p(gxn+2, gxn) + a11p(gyn+2, gyn) + a12p(gzn+2, gzn) + a13p(gxn, gxn+1) + a14p(gyn, gyn+1) + a15p(gzn, gzn+1). Thus, using again (p2) and the triangular inequality (1 − a1 − a7 − a10)p(gxn+2, gxn+1) ≤ (a4 + a10 + a13)p(gxn+1, gxn) + (a5 + a11 + a14)p(gyn, gyn+1) +(a6 + a12 + a15)p(gzn, gzn+1) + (a2 + a8 + a11)p(gyn+2, gyn+1) + (a3 + a9 + a12)p(gzn+2, gzn+1). (2.8) 198 H. Aydi and M. Abbas Similarly, (1 − a1 − a7 − a10)p(gyn+2, gyn+1) ≤ (a4 + a10 + a13)p(gyn+1, gyn) + (a5 + a11 + a14)p(gzn, gzn+1) +(a6 + a12 + a15)p(gxn, gxn+1) + (a2 + a8 + a11)p(gzn+2, gzn+1) + (a3 + a9 + a12)p(gxn+2, gxn+1) (2.9) and (1 − a1 − a7 − a10)p(gzn+2, gzn+1) ≤ (a4 + a10 + a13)p(gzn+1, gzn) + (a5 + a11 + a14)p(gxn, gxn+1) +(a6 + a12 + a15)p(gyn, gyn+1) + (a2 + a8 + a11)p(gxn+2, gxn+1) + (a3 + a9 + a12)p(gyn+2, gyn+1). (2.10) Adding (2.8) to (2.10), we obtain that (2.11) (1−a1−a2−a3−a7−a8−a9−a10−a11−a12)δn+1 ≤ (a4+a5+a6+a10+a11+a12+a13+a14+a15)δn. From (2.11), one can write δn+1 ≤ κ2δn ∀ n ∈ N where κ2 = a4 + a5 + a6 + a10 + a11 + a12 + a13 + a14 + a15 1 − a1 − a2 − a3 − ∑12 i=7 ai . Since 12∑ i=10 ai + 15∑ i=1 ai < 1, so 0 ≤ κ2 < 1. Thus, (2.2) holds for κ = κ2. By (2.2), we have (2.12) δn ≤ κδn−1 ≤ · · · ≤ κ nδ0. If δ0 = 0, we get p(gx0, gx1)+p(gy0, gy1) = p(gz0, gz1) = 0, that is, gx0 = gx1, gy0 = gy1 and gz0 = gz1. Therefore, from (2.1) we have F(x0, y0, z0) = gx1 = gx0, F(y0, z0, x0) = gy1 = gy0 and F(z0, x0, y0) = gz1 = gz0, that is, (x0, y0, z0) is a tripled coincidence point of F and g. Now, assume that δ0 6= 0. If m > n, we have p(gxm, gxn) ≤ p(gxm, gxm−1) + p(gxm−1, gxm−2) + · · · + p(gxn+1, gxn), p(gym, gyn) ≤ p(gym, gym−1) + p(gym−1, gym−2) + · · · + p(gyn+1, gyn), and p(gzm, gzn) ≤ p(gzm, gzm−1) + p(gzm−1, gzm−2) + · · · + p(gzn+1, gzn). Adding above inequalities and using (2.12), we obtain (for m > n) p(gxm, gxn) + p(gym, gyn) + p(gzm, gzn) ≤ δm−1 + δm−2 + · · · + δn ≤ (κm−1 + κm−1 + · · · + κn)δ0 ≤ κn 1 − κ δ0 → 0 since κ ∈ [0, 1). This implies that (2.13) lim n,m→∞ p(gxm, gxn) = lim n,m→∞ p(gym, gyn) = lim n,m→∞ p(gzm, gzn) = 0. Tripled coincidence and fixed point results 199 We deduce that {gxn}, {gyn} and {gzn} are Cauchy sequences in (g(X), p) which is complete, then by Lemma 1.2, {gxn}, {gyn} and {gzn} are Cauchy sequences in the metric subspace (g(X), ps). Since is also (g(X), ps) complete, so that there exist x, y, z ∈ X such that (2.14) lim n→∞ ps(gxn, gx) = lim n→∞ ps(gyn, gy) = lim n→∞ ps(gzn, gz) = 0. Again, by Lemma 1.2 and (2.13), we get that (2.15) p(gx, gx) = lim n→∞ p(gxn, gx) = lim n,m→∞ p(gxm, gxn) = 0, (2.16) p(gy, gy) = lim n→∞ p(gyn, gy) = lim n,m→∞ p(gym, gyn) = 0, and (2.17) p(gz, gz) = lim n→∞ p(gzn, gz) = lim n,m→∞ p(gzm, gzn) = 0. Now, we prove that F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz. Note that p(F(x, y, z), gx) ≤ p(F(x, y, z), F(xn, yn, zn) + p(F(xn, yn, zn), gx) = p(F(x, y, z), F(xn, yn, zn) + p(gxn+1, gx).(2.18) On the other hand, applying the given contractive condition, we obtain p(F(x, y, z), F(xn, yn, zn)) ≤ a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(F(xn, yn, zn), gxn) + a5p(F(yn, zn, xn), gyn) + a6p(F(zn, xn, yn), gzn) + a7p(F(xn, yn, zn), gx) + a8p(F(yn, zn, xn), gy) + a9p(F(zn, xn, yn), gz) + a10p(F(x, y, z), gxn) + a11p(F(y, z, x), gyn) + a12p(F(z, x, y), gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn) = a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + a10p(F(x, y, z), gxn) + a11p(F(y, z, x), gyn) + a12p(F(z, x, y), gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn). Combining above inequality with (2.18) and using a triangular inequality, we have p(F(x, y, z), gx) � a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + a7p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + a10p(F(x, y, z), gx) + a10p(gx, gxn) + a11p(F(y, z, x), gy) + a11p(gy, gyn) + a12p(F(z, x, y), gz) + a12p(gz, gzn) + a13p(gx, gxn) + a14p(gy, gyn) + a15p(gz, gzn) + p(gxn+1, gx). 200 H. Aydi and M. Abbas Therefore, (1 − a1 − a10)p(F(x, y, z), gx) − (a2 + a11)p(F(y, z, x), gy) − (a3 + a12)p(F(z, x, y), gz) ≤ a4p(gxn+1, gxn) + a5p(gyn+1, gyn) + a6p(gzn+1, gzn) + (1 + a7)p(gxn+1, gx) + a8p(gyn+1, gy) + a9p(gzn+1, gz) + (a10 + a13)p(gx, gxn) + (a11 + a14)p(gy, gyn) + (a12 + a15)p(gz, gzn). (2.19) Similarly, we obtain (1 − a1 − a10)p(F(y, z, x), gy) − (a2 + a11)p(F(z, x, y), gz) − (a3 + a12)p(F(x, y, z), gx) ≤ a4p(gyn+1, gyn) + a5p(gzn+1, gzn) + a6p(gxn+1, gxn) + (1 + a7)p(gyn+1, gy) + a8p(gzn+1, gz) + a9p(gxn+1, gx) + (a10 + a13)p(gy, gyn) + (a11 + a14)p(gz, gzn) + (a12 + a15)p(gx, gxn), (2.20) and (1 − a1 − a10)p(F(z, x, y), gz) − (a2 + a11)p(F(x, y, z), gx) − (a3 + a12)p(F(y, z, x), gy) ≤ a4p(gzn+1, gzn) + a5p(gxn+1, gxn) + a6p(gyn+1, gyn) + (1 + a7)p(gzn+1, gz) + a8p(gxn+1, gx) + a9p(gyn+1, gy) + (a10 + a13)p(gz, gzn) + (a11 + a14)p(gx, gxn) + (a12 + a15)p(gy, gyn). (2.21) Letting in (2.19)-(2.21) and using (2.15)-(2.17), we get that (1−a1−a2−a3−a10−a11−a12)[p(F(x, y, z), gx)+p(F(y, z, x), gy)+p(F(z, x, y), gz)] = 0. It follows that p(F(x, y, z), gx) = p(F(y, z, x), gy) = p(F(z, x, y), gz) = 0, that is F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz. � As consequences of Theorem 2.1, we give the following corollaries. Corollary 2.2. Let (X, p) be a partial metric space. Let F : X3 −→ X and g : X → X be mappings such that F(X3) ⊆ g(X) and g(X) is a complete subspace of X. Suppose that for any x, y, z, u, v, w ∈ X p(F(x, y, z), F(u, v, w)) ≤ α1[p(F(x, y, z), gx) + p(F(y, z, x), gy) + p(F(z, x, y), gz)] + α2[p(F(u, v, w), gu) + p(F(v, w, u), gv) + p(F(w, u, v), gw)] + α3[p(F(u, v, w), gx) + p(F(v, w, u), gy) + p(F(w, u, v), gz)] + α4[p(F(x, y, z), gu) + p(F(y, z, x), gv) + p(F(z, x, y), gw)] + α5[p(gx, gu) + p(gy, gv) + p(gz, gw)], where αi, i = 1, · · · , 5 are nonnegative real numbers. If α3 + 5∑ i=1 αi < 1/3 or α4 + 5∑ i=1 αi < 1/3, then F and g have a tripled coincidence. Tripled coincidence and fixed point results 201 Proof. Take a1 = a2 = a3 = α1, a4 = a5 = a6 = α2, a7 = a8 = a9 = α3, a10 = a11 = a12 = α4 and a13 = a14 = a15 = α5 in Theorem 2.1 with α3 + 5∑ i=1 αi < 1/3 or α4 + 5∑ i=1 αi < 1/3. The result follows. � Corollary 2.3. Let (X, p) be a partial metric space. Let F̃ : X2 −→ X and g : X → X be mappings satisfying F̃(X2) ⊆ g(X), (g(X), p) is a complete subspace of X and for any x, y, u, v ∈ X, p(F̃(x, y), F̃(u, v)) ≤ a1p(F̃(x, y), gx) + a2p(F̃(u, v), gu) + a3p(F̃(u, v), gx) a4p(F̃(x, y), gu) + a5p(gx, gu) + a6p(gy, gv),(2.22) where ai, i = 1, · · · , 6 are nonnegative real numbers such that a3 + 6∑ i=1 ai < 1 or a4 + 6∑ i=1 ai < 1. Then F̃ and g have a coupled coincidence point (x, y) ∈ X 2, that is, F̃(x, y) = gx and F̃(y, x) = gy. Proof. Consider the mappings F : X3 → X defined by F(x, y, z) = F̃(x, y) for all x, y, z ∈ X. Then, the contractive condition (2.22) implies that, for all x, y, z, u, v, w ∈ X p(F(x, y, z), F(u, v, w)) ≤ a1p(F(x, y, z), gx) + a2p(F(u, v, w), gu) + a3p(F(x, y, z), gu) + a4p(F(u, v, w), gx) + a5p(gx, gu) + a6p(gy, gv). Then F and g satisfy the contractive condition of Theorem 2.1. Clearly other conditions of Theorem 2.1 are also satisfied as F̃(X2) ⊆ g(X) and g(X) is a complete subspace of X. Therefore, from Theorem 2.1, F and g have a tripled fixed point (x, y, z) ∈ X3 such that F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz, that is, F̃(x, y) = gx and F̃(y, x) = gy. This makes end to the proof. � Now, we are ready to state and prove a result of common tripled fixed point. Theorem 2.4. Let F : X3 → X and g : X → X be two mappings which satisfy all the conditions of Theorem 2.1. If F and g are W-compatible, then F and g have a unique common tripled fixed point. Moreover, common tripled fixed point of F and g is of the form (u, u, u) for some u ∈ X. Proof. First, we’ll show that the tripled point of coincidence is unique. Suppose that (x, y, z) and (x∗, y∗, z∗) ∈ X3 with    gx = F(x, y, z) gy = F(y, z, x) gz = F(z, x, y), and    gx∗ = F(x∗, y∗, z∗) gy∗ = F(y∗, z∗, x∗) gz∗ = F(z∗, x∗, y∗). 202 H. Aydi and M. Abbas Using contractive condition in Theorem 2.1 and (p2), we obtain p(gx, gx∗) = p(F(x, y, z), F(x∗, y∗, z∗)) ≤ a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(F(x ∗, y∗, z∗), gx∗) + a5p(F(y ∗, z∗, x∗), gy∗) + a6p(F(z ∗, x∗, y∗), gz∗) + a7p(F(x ∗, y∗, z∗), gx) + a8p(F(y ∗, z∗, x∗), gy) + a9p(F(z ∗, x∗, y∗), gz) + a10p(F(x, y, z), gx ∗) + a11p(F(y, z, x), gy ∗) + a12p(F(z, x, y), gz ∗) + a13p(gx, gx ∗) + a14p(gy, gy ∗) + a15p(gz, gz ∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gx ∗, gx) + (a2 + a5 + a8 + a11 + a14)p(gy ∗, gy) + (a3 + a6 + a9 + a12 + a15)p(gz ∗, gz). Similarly, we have p(gy, gy∗) = p(F(y, z, x), F(y∗, z∗, x∗)) ≤ (a1 + a4 + a7 + a10 + a13)p(gy ∗, gy) + (a2 + a5 + a8 + a11 + a14)p(gz ∗, gz) + (a3 + a6 + a9 + a12 + a15)p(gx ∗, gx), and p(gz, gz∗) = p(F(z, x, y), F(z∗, x∗, y∗)) ≤ (a1 + a4 + a7 + a10 + a13)p(gz ∗, gz) + (a2 + a5 + a8 + a11 + a14)p(gx ∗, gx) + (a3 + a6 + a9 + a12 + a15)p(gy ∗, gy). Adding above three inequalities, we get p(gx, gx∗)+p(gy, gy∗)+p(gz, gz∗) ≤ ( 15∑ i=1 ai)[p(gx, gx ∗)+p(gy, gy∗)+p(gz, gz∗)]. Since 15∑ i=1 ai < 1, we obtain p(gx, gx∗) + p(gy, gy∗) + p(gz, gz∗) = 0, which implies that (2.23) gx = gx∗, gy = gy∗ and gz = gz∗, which implies uniqueness of the tripled point of coincidence of F and g, that is, (gx, gy, gz). Note that p(gx, gy∗) = p(F(x, y, z), F(y∗, z∗, x∗)) ≤ a1p(F(x, y, z), gx) + a2p(F(y, z, x), gy) + a3p(F(z, x, y), gz) + a4p(F(y ∗, z∗, x∗), gy∗) + a5p(F(z ∗, x∗, y∗), gz∗) + a6p(F(x ∗, y∗, z∗), gx∗) + a7p(F(y ∗, z∗, x∗), gx) + a8p(F(z ∗, x∗, y∗), gy) + a9p(F(x ∗, y∗, z∗), gz) + a10p(F(x, y, z), gy ∗) + a11p(F(y, z, x), gz ∗) + a12p(F(z, x, y), gx ∗) + a13p(gx, gy ∗) + a14p(gy, gz ∗) + a15p(gz, gx ∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gy ∗, gx) + (a2 + a5 + a8 + a11 + a14)p(gz ∗, gy) + (a3 + a6 + a9 + a12 + a15)p(gx ∗, gz). Similarly p(gy, gz∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gz ∗, gy) + (a2 + a5 + a8 + a11 + a14)p(gx ∗, gz) + (a3 + a6 + a9 + a12 + a15)p(gy ∗, gx), Tripled coincidence and fixed point results 203 and p(gz, gx∗) ≤ (a1 + a4 + a7 + a10 + a13)p(gx ∗, gz) + (a2 + a5 + a8 + a11 + a14)p(gy ∗, gx) + (a3 + a6 + a9 + a12 + a15)p(gz ∗, gy). Adding the above inequalities, we obtain p(gx, gy∗)+p(gy, gz∗)+p(gz, gx∗) ≤ ( 15∑ i=1 ai)(p(gx, gy ∗)+p(gy, gz∗)+p(gz, gx∗)), which again yields that (2.24) gx = gy∗, gy = gz∗ and gz = gx∗. In view of (2.23) and (2.24), one can assert that (2.25) gx = gy = gz. That is, the unique tripled point of coincidence of F and g is (gx, gy, gz). Now, let u = gx, then we have u = gx = F(x, y, z) = gy = F(y, z, x) = gz = F(z, x, y). Since F and g are W-compatible, we have F(gx, gy, gz) = g(F(x, y, z)), which due to (2.25) gives that F(u, u, u) = gu. Consequently, (u, u, u) is a tripled coincidence point of F and g, and so (gu, gu, gu) is a tripled point of coincidence of F and g, and by its uniqueness, we get gu = gx. Thus, we obtain u = gx = gu = F(u, u, u). Hence, (u, u, u) is the unique common tripled fixed point of F and g. This completes the proof. � Corollary 2.5. Let (X, p) be a cone partial metric space. Let F̃ : X2 −→ X and g : X → X be mappings satisfying F̃(X2) ⊆ g(X), (g(X), p) is a complete subspace of X and for any x, y, u, v ∈ X, p(F̃(x, y), F̃(u, v)) ≤ a1p(F̃(x, y), gx) + a2p(F̃(u, v), gu) + a3p(F̃(u, v), gx) + a4p(F̃(x, y), gu) + a5p(gx, gu) + a6p(gy, gv), where ai, i = 1, · · · , 6 are nonnegative real numbers such that a3 + 6∑ i=1 ai < 1 or a4 + 6∑ i=1 ai < 1. If F̃ and g are w-compatible, then F̃ and g have a unique common coupled fixed point. Moreover, the common fixed point of F̃ and g is of the form (u, u) for some u ∈ X. 204 H. Aydi and M. Abbas Proof. Consider the mappings F : X3 → X defined by F(x, y, z) = F̃(x, y) for all x, y, z ∈ X. From the proof of Corollary 2.3 and the result given by Theorem 2.4, we have only to show that F and g are W-compatible. Let (x, y, z) ∈ X3 such that F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz. From the definition of F , we get F̃(x, y) = gx and F̃(y, x) = gy. Since F̃ and g are w-compatible, this implies that (2.26) g(F̃(x, y)) = F̃(gx, gy). Using (2.26), we have F(gx, gy, gz) = F̃(gx, gy) = g(F̃(x, y)) = g(F(x, y, z)). Thus, we proved that F and g are W-compatible mappings, and the desired result follows immediately from Theorem 2.4. � Remark 2.6. • Theorem 2.1 of Aydi [5] is a particular case of Corollary 2.5 by taking a1 = a1 = a3 = a4 = 0 and g = IX, the identity on X. • Theorem 2.4 of Aydi [5] is a particular case of Corollary 2.5 by taking a3 = a4 = a5 = a6 = 0 and g = IX, the identity on X. • Theorem 2.5 of Aydi [5] is a particular case of Corollary 2.5 by taking a1 = a2 = a5 = a6 = 0 and g = IX, the identity on X. • Corollary 2.2 extends Theorem 2.9 of Samet and Vetro [21] to partial metric spaces (corresponding to the case N = 3). • Theorem 2.4 extends Theorem 2.10 of Samet and Vetro [21] to partial metric spaces (case N = 3). • Theorem 2.4 extends Theorem 2.11 of Samet and Vetro [21] to partial metric spaces ( case N = 3). Similar to the Corollaries 2.3 and 2.5, by considering F(x, y, z) = fx for all x, y, z ∈ X where f : X → X, we may state the following consequence of Theorem 2.4. Corollary 2.7. Let (X, p) be a partial metric space and f, g : X → X be mappings such that p(fx, fu) ≤ a1p(fx, gx) + a2p(fu, gu) + a3p(fu, gx) + a4p(fx, gu) + a5p(gu, gx)(2.27) for all x, u ∈ X, where ai ∈ [0, 1), i = 1, · · · , 5 and a3 + 5∑ i=1 ai < 1 or a4 + 5∑ i=1 ai < 1. Suppose that f and g are weakly compatible, f(X) ⊆ g(X) and g(X) is a complete subspace of X. Then the mappings f and g have a unique common fixed point. Now, we give an example to illustrate our obtained results. Tripled coincidence and fixed point results 205 Example 2.8. Let X = R+ endowed with the partial metric metric p(x, y) = max(x, y) for all x, y ∈ X. Define the mappings F : X3 → X and g : X → X by gx = x 2 and F(x, y, z) = 2x + 3y + 4z 72 . We will check that all the hypotheses of Theorem 2.1 are satisfied. Note that F(X3) ⊆ g(X) with g(X) is complete in X. Now, for all x, y, z, u, v, w ∈ X, we have p(F(x, y, z), F(u, v, w)) = max(F(x, y, z), F(u, v, w)) ≤ max( 2x + 3y + 4z 72 , 2u + 3v + 4w 72 ) ≤ 1 4 [max{ x 2 , u 2 } + max{ y 2 , v 2 } + max{ z 2 , w 2 }] = 1 4 p(gx, gu) + 1 4 p(gy, gv) + 1 4 p(gz, gw). Then, the contractive condition is satisfied with ai = 0 for all i = 1, · · · , 12 and a13 = a14 = a15 = 1/4. All conditions of Theorem 2.1 are satisfied. Consequently, (x, y, z) is a tripled coincidence point of F and g if and only if x = y = z = 0. This implies that F and g are W-compatible. Applying our Theorem 2.4, we obtain the existence and uniqueness of a common tripled fixed point of F and g. In this example, (0, 0, 0) is the unique common tripled fixed point. Acknowledgements. The authors are grateful to the reviewers and the editor for their useful comments. References [1] M. Abbas, M. Ali Khan and S. Radenović, Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. 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Aydi (hassen.aydi@isima.rnu.tn) Université de Sousse, Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Route GP1-4011, H. Sousse, Tunisie. M. Abbas (mujahid@lums.edu.pk) Department of Mathematics, Lahore University of Management Sciences, 54792- Lahore, Pakistan. Tripled coincidence and fixed point results in partial metric spaces. By H. Aydi and M. Abbas