SQU Journal for Science, 2015, 19(2), 55-61 © 2014 Sultan Qaboos University 55 A Unique Common Coupled Fixed Point Theorem for Four Maps in Partial b-Metric- Like Spaces Mohammad S. Khan 1 *, Konduru P.R. Rao 2 and Kandipalli V.S. Parvathi 3 1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, PC 123, Al-Khod, Muscat, Sultanate of Oman. 2 Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar -522 510, A.P., India. 3 Department of Applied Mathematics,Krishna University-M.R.Appa Row P.G.Center, Nuzvid-521 201, Andhra Pradesh, India. *Email: mohammad@squ.edu.om. ABSTRACT: We prove the existence of a unique common coupled fixed point theorem for four mappings satisfying a general contractive condition on partial b-metric-like spaces. We also give an example to illustrate our main theorem. Our theorem generalizes and improves the theorem of [1]. Keywords: b-Metric-like space; Coupled fixed point; w-Compatible maps. ب الجزئي الفراغ شبه فيالربعة اقترانات العامة المزدوجة الثابتة النقطة نظرية كانذورو ب.ر.راو و كانذيبالي ف.س. برافاثي ، محمذ سعيذ خان تحقق شزوط االنقباض العامة على شبو الفزاغ الجزئي ألربعة إقتزانات نظزية وجود نقطة ثابتة مزدوجة عامة بإثبات في ىذه الورقة قمنا ملخص: .[1]نظزيتنا ىي تعميم و تحديث للنظزية المثبتة في .مثاال لتوضيح نظزيتنا الزئيسيةأحضزنا ب. و قد W.المتوافقة توافق الدوال و مزدوجة ثابتة نقطة ، الجزئي ب الفزاغ شبو : كلمات مفتاحية 1. Introduction and Preliminaries he concept of b-metric space was introduced by Czerwik [2] as follows: Definition 1.1 [2]: A b-metric on a non-empty set X is a function d : X × X → [0, ∞) such that for all x, y, z ε X and a constant k ≥ 1 the following three conditions hold true: (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x), (iii) d(x, y) ≤ k [d(x, z) + d(z, y)] . The triad (X, d, k) is called a b-metric space. Alghamdi et al. [3] introduced the concept of b-metric -like spaces and proved some fixed point theorems for a single map. Definition 1.2 [3]: A b-metric-like on a non-empty set X is a function d : X × X → [0, ∞) such that for all x, y, z ε X and a constant k ≥ 1 the following three conditions hold true: (i) d(x, y) = 0 implies x = y, (ii) d(x, y) = d(y, x), (iii) d(x, y) ≤ k[d(x, z) + d(z, y)] . The triad (X, d, k) is called a b-metric-like space. Mathews [4] introduced the concept of a partial metric space as follows: Definition 1.3 [4]: A mapping p : X × X → [0, ∞), where X is a non-empty set, is said to be a partial metric on X if for any x, y, z ε X the following are satisfied: (i) x = y if and only if p(x, x) = p(x, y) = p(y, y), (ii) p(x, x) ≤ p(x, y) , (iii) p(x, y) = p(y, x), T mailto:mohammad@squ.edu.om MOHAMMAD S. KHAN ET AL. 56 (iv) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z). The pair (X, p) is called a partial metric space. Now we give the following definition by combining the Definitions 1.2 and 1.3. Definition 1.4: A partial b-metric-like on a non-empty set X is a function p : X × X → [0, ∞) such that for all x, y, z ε X and a constant k ≥ 1 the following are satisfied: (p1) p(x, y) = 0 implies x = y, (p2) p(x, x) ≤ p(x, y), p(y, y) ≤ p(x, y), (p3) p(x, y) = p(y, x), (p4) p(x, y) ≤ k[p(x, z) + p(z, y) − p(z, z)]. The triad (X, p, k) is called a partial b-metric-like space. Definition 1.5: Let (X, p, k) be a partial b-metric-like space and let {xn} be a sequence in X and x ε X. The sequence {xn} is said to be convergent to x if n lim p(xn, x) = p(x, x). Definition 1.6: Let (X, p, k) be a partial b-metric-like space. (i) A sequence {xn} in (X, p, k) is said to be a Cauchy sequence if mn, lim p(xn, xm) exists and is finite . (ii) A partial b-metric-like space (X, p, k) is said to be complete if every Cauchy sequence {xn} in X converges to a point x ε X so that mn, lim p(xn, xm) = p(x, x) = n lim p(xn, x). One can prove easily the following remark. Remark 1.7: Let (X, p, k) be a partial b-metric-like space and {xn} be a sequence in X such that n lim p(xn, x) = 0. Then (i) x is unique , (ii) k 1 p(x, y) ≤ n lim p(xn, y) ≤ k p(x, y) for all y ε X , (iii) p(xn, x0) ≤ kp(x0, x1) + k 2 p(x1, x2) + ··· + k n−1 p(xn−2, xn−1) + k n p(xn−1, xn) whenever  kx n k 0 ε X. Let (X, p, k) be a partial b-metric-like space and F, G : X × X and f, g : X → X. For x, y, u, v ε X, we denote x,y u,v p(fx, gu), p(fy, gv), p(fx, F (x, y)), p(fy, F (y, x)), p(gu, G(u, v)), p(gv, G(v, u)), 1 min [p(fx, G(u, v)) + p(gu, F (x, y))], 2k 1 [p(fy, G(v, u)) + p(gv, F (y, x))] 2k .M                    and , , p(fx, F (x, y)), p(fy, F (y, x)), p(gu, G(u, v)), p(gv, G(v, u)), 1 1 max p(fx, G(u, v)), p(gu, F (x, y)), k k 1 1 p(fy, G(v, u)), p(gv, F (y, x)) k k . x y u vm                    Recently Bhaskar and Lakshmikantham [5] introduced the concept of coupled fixed point and discussed some problems of the uniqueness of a coupled fixed point and applied their results to the problems of the existence and uniqueness of a solution for the periodic boundary value problems. Later Lakshmikantham and Ciric [6] proved some coupled coincidence and coupled common fixed point results in partially ordered metric spaces. A UNIQUE COMMON COUPLED FIXED POINT THEOREM 57 Definition 1.8 [6] An element (x, y) ε X × X is called (i) a coupled coincident point of mappings F : X×X → X and g: X → X if gx = F (x, y) and gy = F (y, x). (ii) a common coupled fixed point of mappings F : X × X → X and g : X → X if x = gx = F (x, y) and y = gy = F (y, x) . Definition 1.9 [7] The mappings F : X × X → X and g : X → X are called w-compatible if g(F (x, y)) = F (gx, gy) and g(F (y, x)) = F (gy, gx), whenever gx = F (x, y) and gy = F (y, x). Recently, Abbas et al. [8] proved a common fixed point theorem for two maps of Jungck type satisfying generalized condition (B) in metric spaces (See Theorem 2.2, [8]). As a generalization of Theorem 2.2 of [8], Kaewcharoen et al. [1] obtained a common fixed point theorem for four maps satisfying a generalized condition in partial metric spaces. In this paper, we obtain the existence of a unique common coupled fixed point theorem for four mappings satisfying a general contractive condition on partial b-metric-like spaces. We also give an example to illustrate our main theorem. Our theorem generalizes and improves the theorems of [1] and [8]. 2. Main Result Theorem 2.1: Let (X, p, k) be a complete partial b-metric-like space, F, G : X × X → X and f, g : X → X be mappings satisfying (2.1.1) F (X × X)  g(X), G(X × X)  f(X), (2.1.2) k p(F(x,y),G(u,v)) ≤ δ , , x y u vM + L , , x y u vm for all x, y, u, v ε X, where δ > 0 and L ≥ 0, k l < 1, where l = max         L L   , 1 , (2.1.3) f(X) or g(X) is closed, (2.1.4) the pairs (F, f), and (G, g) are w-compatible. Then F, G, f and g have a unique common coupled fixed point. Proof. Let (x0, y0) ε X × X. Since F (X × X)  g(X), there exist x1, y1 ε X such that gx1 = F (x0, y0) and gy1 = F (y0, x0). Since G (X × X)  f(X), there exist x2, y2 ε X such that fx2 = G(x1, y1) and fy2 = G(y1, x1). Continuing this process, we construct sequences {xn} and {yn} in X such that gx2n+1 = F (x2n, y2n) = z2n, gy2n+1 = F (y2n, x2n) = w2n, fx2n+2 = G(x2n+1, y2n+1) = z2n+1, fy2n+2 = F (y2n+1, x2n+1) = w2n+1, n = 0, 1, 2, 3, ··· Now consider p(z2n, z2n+1) ≤ k p(F (x2n, y2n), G(x2n+1, y2n+1)) ≤ 2 2 2 1 2 1 2 2 2 1 2 1 , , , , n n n n n n n n x y x y x y x y LM m     (1) where 2 2 2 1 2 1 , , n n n n x y x yM   2 1 2n 2 1 2n 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 2 1 2n+1 2n 2n 2 1 2n+1 2n 2n p(z , z ), p(w , w ), p(z , z ), p(w , w ), p(z ,z ), p(w ,w ), 1 max [p(z , z ) + p(z , z )], 2k 1 [p(w , w ) + p(w , w )] 2k n - n - n - n - n - n -                    ≤ max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      from k ≥ 1 and from (p4) MOHAMMAD S. KHAN ET AL. 58 2 2 2 1 2 1 , , n n n n x y x ym   2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 2 1 2n+1 2n 2n 2 1 2n+1 2n 2n p(z , z ), p(w , w ), p(z ,z ), p(w ,w ), 1 1 min p(z , z ) , p(z , z ), k k 1 1 p(w , w ), p(w , w ) k k n - n - n - n -                    2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 2 1 2n 2 2n+1 2n 2n 2 1 2n 2 2n+1 2n 2n p(z , z ), p(w , w ),p(z ,z ), min p(w ,w ), p(z , z ) p(z , z ) , p(z , z ),p(w , w ) p(w , w ), p(w , w ) n - n - n - n n - n              = min {p(z2n, z2n), p(w2n, w2n)}, from (p2) ≤ max {p(z2n, z2n), p(w2n, w2n)} ≤ max {p(z2n, z2n-1), p(w2n, w2n-1)}, from (p2). Thus p(z2n, z2n+1) ≤ δ max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      + L max{p(z2n, z2n-1), p(w2n, w2n-1)}. Similarly p(w2n,w2n+1) ≤ δ max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      + L max{p(z2n, z2n-1), p(w2n, w2n-1)}. Thus max 2 2n 1 2n 2n+1 p(z , z ), p(w ,w ) n       ≤ δ max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      + L max 2 2n 1 2n 2n-1 p(z , z ), p(w ,w ) n       (2) If max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      ≤ max 2 2n 1 2n 2n+1 p(z , z ), p(w ,w ) n       , then from (2) max {p(z2n, z2n+1), p(w2n, w2n+1)} ≤ 1 L  max { p(z2n−1, z2n), p(w2n−1, w2n)}. If max 2 1 2n 2 1 2n 2n 2n+1 2n 2n+1 p(z , z ), p(w , w ), p(z ,z ), p(w ,w ) n - n -      ≤ max 2 1 2n 2n-1 2n p(z , z ), p(w ,w ) n       , then from (2) max{ p(z2n, z2n+1), p(w2n, w2n+1)} ≤ (δ + L) max{ p(z2n−1, z2n), p(w2n−1, w2n)}. Hence max {p(z2n, z2n+1), p(w2n, w2n+1) } ≤ l max {p(z2n−1, z2n), p(w2n−1, w2n)} where l = max  , 1 L L    < 1. Similarly we can show that max{p(z2n−1, z2n), p(w2n−1, w2n) } ≤ l max { p(z2n−2, z2n−1), p(w2n−2, w2n−1)}. Hence Max {p(zn, zn+1), p(wn, wn+1)} ≤ l max { p(zn−1, zn), p(wn−1, wn)} , n = 1, 2, 3, ··· Thus max{ p(zn, zn+1), p(wn, wn+1) } ≤ l n max{ p(z0, z1), p(w0, w1)}. (3) From (3), it follows that lim n p(zn, zn+1) = 0 = lim n p(wn, wn+1). (4) For m > n, consider max {p(zn, zm), p(wn, wm)} ≤ max 2 m-n-1 m-n-1 n n 1 n 1 n 2 m-2 m-1 m-1 m 2 m-n-1 m-n-1 n n 1 n 1 n 2 m-2 m-1 m-1 m k p(z ,z ) k p(z ,z ) ... k p(z ,z ) k p(z ,z ), k p(w ,w ) k p(w ,w ) ... k p(w ,w ) k p(w ,w )                     ≤ k max {p(zn, zn+1), p(wn, wn+1)} + k 2 max {p(zn+1, zn+2), p(wn+1, wn+2)} + .... + k m−n−1 max {p(zm−2, zm−1), p(wm−2, wm−1)} + k m−n−1 max {p(zm−1, zm), p(wm−1, wm)} A UNIQUE COMMON COUPLED FIXED POINT THEOREM 59 ≤  112112 ....   mnmmnmnn lklklkkl max 0 1 0 1 p(z , z ), p(w ,w )       =  1222221 ....1   nmnmnmnmn lklklkklkl max 0 1 0 1 p(z , z ), p(w ,w )       ≤  1122221 ....1   nmnmnmnmn lklklkklkl max 0 1 0 1 p(z , z ), p(w ,w )       ≤ 1 n kl kl max 0 1 0 1 p(z , z ), p(w ,w )       , since kl < 1. Hence , iml n m p(zn, zm) = 0 = , lim n m p(wn, wm). (5) Thus {zn} and {wn} are Cauchy in (X, p, k). Since X is complete, the sequences {zn} and {wn} converge to some α and β in X respectively such that , iml n m p(zn, zm) = p(α, α) and , iml n m p(wn, wm) = p(β, β). Also lim n p(zn, α) = p(α, α) and lim n p(wn, β) = p(β, β). Now from (5), we have p(α, α) = 0 = p(β, β). (6) Suppose f(X) is closed. Since fx2n+2 = z2n+1 → α and fy2n+2 = w2n+1 → β, it follows that α = fu and β = fv for some u, v ε X. Consider p(α, F (u, v)) ≤ kp(α, z2n+1) + kp(F (u, v), G(x2n+1, y2n+1)) ≤ kp(α, z2n+1) + δ 2 1 2 1 , ,n n u v x yM   + L 2 1 2 1 , ,n n u v x ym   2 1 2 1 , ,n n u v x yM   2n 2n 2n 2n+1 2n 2n+1 2n+1 2n 2n+1 2n p(fu, z ), p(fv, w ), p(fu, F(u, v)), p(fv, F(v, u)), p(z ,z ), p(w ,w ), 1 max [p(fu, z ) + p(z , F(u, v))], 2k 1 [p(fv, w ) + p(w , F(v, u)] 2k                    0, 0, p( , F(u, v)), p( , F(v, u)), 0, 0, 1 max [0 + p( , F(u, v))], 2k 1 [0 + p( , F(v, u)] 2k                        , from (4) and Remark 1.7 (ii) = max { p(α, F (u, v)), p(β, F (v, u))}. Also 2 1 2 1 , ,n n u v x ym   → 0. Thus p(α, F (u, v)) ≤ δ max {p(α, F (u, v)), p(β, F (v, u))} . Similarly we can show that p(β, F (u, v)) ≤ δ max {p(α, F (u, v)), p(β, F (v, u))} . Hence max {p(α, F (u, v)), p(β, F (v, u))} ≤ δ max {p(α, F (u, v)), p(β, F (v, u))}, which in turn yields that α = F (u, v) and β = F (v, u). Thus fu = α = F (u, v) and fv = β = F (v, u). Since the pair (F, f) is w- compatible, we have fα = F (α, β) and fβ = F (β, α). (7) Since α = F (u, v) ε F (X × X)  g(X), there exists r ε X such that α = gr. Since β = F (v, u) ε F (X × X)  g(X), there exists t ε X such that β = gt. MOHAMMAD S. KHAN ET AL. 60 Now p(α, G(r, t)) ≤ s p(F (u, v), G(r, t)) ≤ δ , , u v r tM + L , , u v r tm , , u v r tM p(fu, gr), p(fv, gt), p(fu, F(u, v)), p(fv, F(v, u)), p(gr, G(r, t)), p(gt, G(t,r)), 1 max [p(fu, G(r, t)) + p(gr, F(u, v))], 2k 1 [p(fv, G(t, r)) + p(gt, F(v, u)] 2k                    0, 0, 0, 0 p( , G(r, t)), p( , G(t,r)), 1 max [p( , G(r, t)) + p( , F(u, v))], 2k 1 [p( , G(t, r)) + p( , F(v, u)] 2k                          = max{ p(α, G(r, t)), p(β, G (t, r))}. , , u v r tm = 0. Thus p(α, G(r, t)) ≤ δ max {p(α, G(r, t)), p(β, G(t, r))} . Similarly we can show that p(β, G(t, r)) ≤ δ max {p(α, G(r, t)), p(β, G(t, r))} . Hence max {p(α, G(r, t)), p(β, G(t, r))} ≤ δ max {p(α, G(r, t)), p(β, G(t, r))} which in turn yields that gr = α = G(r, t) and gt = β = G(t, r). Since the pair (G, g) is w-compatible, we have gα = G(α, β) and gβ = G(β, α). Now consider p(fα, α) ≤ k p(F (α, β), G(r, t)) ≤ δ , ,r tM   + L , ,r tm   , ,r tM   p(f , gr), p(f , gt), p(f , F( , )), p(f , F( , )), p(gr, G(r, t)), p(gt, G(t,r)), 1 max [p(f , G(r, t)) + p(gr, F( , ))], 2k 1 [p(f , G(t, r)) + p(gt, F( , ))] 2k                                  p(f , ), p(f , ), 0, 0, 0, 0, 1 max [p(f , ) + p( , f ) ], 2k 1 [p(f , ) + p( , f )] 2k                               . = max { p(fα, α), p(fβ, β)}. Also , ,r tm   = 0. Thus p(fα,α) ≤ δ max {p(fα, α), p(fβ, β)}. Similarly we can show that p(fβ, β) ≤ δ max {p(fα, α), p(fβ, β)} . Hence max {p(fα, α), p(fβ, β)} ≤ δ max {p(fα, α), p(fβ, β)} which in turn yields that fα = α and fβ = β. Similarly we can show that gα = α and gβ = β. Thus A UNIQUE COMMON COUPLED FIXED POINT THEOREM 61 F (α, β) = fα = α = gα = G(α, β) and F (β, α) = fβ = β = gβ = G(β, α). Hence (α, β) is a common coupled fixed point of F, G, f and g. Uniqueness of this common coupled fixed point follows easily from (2.1.2). Now, we give an example to illustrate our main Theorem 2.1. Example 2.2 Let X = [0, 1] and p(x, y) = max{x 2 , y 2 }. Then (X, p, k) is a complete partial b-metric-like space with k = 2. Define F, G : X × X → X and f, g : X → X as F(x, y) = 0, G(x, y) = 4 x , fx = 2 x and gx = x. Then k p(F(x, y), G(u, v)) = 2 max 2 0, 16 u      = 2 8 u , p(gu, G(u, v)) = max 2 2 , 16 u u       = u 2 . Thus p(F (x, y), G(u, v)) = 1 8 u 2 = 1 8 p(gu, G(u, v)) ≤ 1 8 , , x y u vM + 0 , , x y u vm . Here δ = 1 8 , L = 0, k l = 1 4 < 1. Clearly (2.1.1), (2.1.3) and (2.1.4) are satisfied and (0, 0) is the unique common coupled fixed point of F, G, f and g. Theorem 2.1 is a generalization and improvement of the following: Theorem 2.3 (Theorem 2.1, [1]): Let (X, p) be a complete partial metric space. Suppose that f, g, F, G : X → X satisfying the following conditions (2.3.1) f(X)  g(X) and F (X)  G(X), (2.3.2) there exist δ > 0 and L ≥ 0 with δ + 2L < 1 such that p(Fx, fy) ≤ δ M(x, y) + L min{p(gx, Fx), p(Gy, fy), p(gx, fy), Gy, Fx)} for all x, y ε X, where M(x, y) = max{p(gx, Gy), p(gx, Fx), p(Gy, fy), 1 2 [p(gx, fy) + p(Gy, Fx)]}, (2.3.3)f(X) or g(X) is closed and (2.3.4) the pairs (f, G) and (g, F ) are w-compatible. Then f, g, F and G have a unique common fixed point in X. Acknowledgement The authors are thankful to the referees for their valuable suggestions in improving the manuscript. References 1. Kaewcharoen, A. and Yuying, T. Unique common fixed point theorems on partial metric spaces. J. Nonlinear Sciences and Applications, 2014, 7, 90-101. 2. Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. 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On common fixed points of weakly compatible mappings satisfying generalized condition(B). Filomat, 2011, 25(2), 9-19. Received 24 August 2014 Accepted 20 December 2014