Some coupled common fixed point theorems for a pair of ... - Core

Journal Nonlinear Analysis and Application 2013 (2013) 1-6

Available online at www.ispacs.com/jnaa Volume 2013, Year 2013 Article ID jnaa-00174, 6 Pages doi:10.5899/2013/jnaa-00174 Research Article

Some coupled common fixed point theorems for a pair of mappings satisfying a contractive condition of rational type Sumit Chandok 1∗, Mohammad Saeed Khan2 , K.P.R. Rao3 (1)Department of Mathematics, Khalsa College of Engineering Technology (Punjab Technical University), Amritsar-143001, India. (2)Department of Mathematics and Statistics, College of Science, Sultan Qaboos University Al-Khod, Sultanate of Oman. (3)Department of Mathematics, Acharya Nagarjuna University, India. c Sumit Chandok, Mohammad Saeed Khan and K.P.R. Rao. This is an open access article distributed under the Creative Commons Copyright 2013 ⃝ Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract The purpose of this paper is to establish some coupled coincidence point theorems for a pair of mappings having a mixed g-monotone property satisfying a contractive condition of rational type in the framework of partially ordered metric spaces. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend several well-known results in the literature. Keywords: Coupled fixed point, mixed g-monotone property, Ordered metric spaces.

1 Introduction Fixed point theory is one of the famous and traditional theories in mathematics and has a large number of applications. The Banach contraction mapping is one of the pivotal results of analysis. It is very popular tool for solving existence problems in many different fields of mathematics. There are a lot of generalizations of the Banach contraction principle in the literature. Ran and Reurings [13] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and Rod´riguez-L´opez [12] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [2] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results on a first order differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results in cone metric spaces, partially ordered metric spaces and others (see [1]-[14]). The purpose of this paper is to establish some coupled coincidence point results in partially ordered metric spaces for a pair of mappings having mixed g-monotone property satisfying a contractive condition of rational type. Also, we present a result on the existence and uniqueness of coupled common fixed points. Definition 1.1. Let (X, d) be a metric space and F : X × X → X and g : X → X, F and g is said to commute if F(gx, gy) = g(F(x, y)), for all x, y ∈ X. ∗ Corresponding

author. Email address: [email protected]

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Definition 1.2. Let (X, ≼) be a partially ordered set and F : X → X. The mapping F is said to be non-decreasing if for x, y ∈ X, x ≼ y implies F(x) ≼ F(y) and non-increasing if for x, y ∈ X, x ≼ y implies F(x) ≽ F(y). Definition 1.3. Let (X, ≼) be a partially ordered set and F : X × X → X and g : X → X. The mapping F is said to have the mixed g-monotone property if F(x, y) is monotone g-non-decreasing in x and monotone g-non-increasing in y, that is, for any x, y ∈ X, x1 , x2 ∈ X, gx1 ≼ gx2 ⇒ F(x1 , y) ≼ F(x2 , y), and y1 , y2 ∈ X, gy1 ≼ gy2 ⇒ F(x, y1 ) ≽ F(x, y2 ). If g =identity mapping in Definition 1.3, then the mapping F is said to have the mixed monotone property. Definition 1.4. An element (x, y) ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F(x, y) = gx, and F(y, x) = gy. If g =identity mapping in Definition 1.4, then (x, y) ∈ X × X is called a coupled fixed point. 2 Main Results 2.1 Coupled common fixed point theorems In this section, we prove some coupled common fixed point theorems in the context of ordered metric spaces. Theorem 2.1. Let (X, ≼) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose that F : X × X → X and g : X → X are self mappings on X such that F has the mixed g-monotone property on X such that there exists two elements x0 , y0 ∈ X with g(x0 ) ≼ F(x0 , y0 ) and g(y0 ) ≽ F(y0 , x0 ). Suppose that there exists α ∈ [0, 1) such that d(gx, F(x, y))d(gu, F(u, v)) , d(gx, gu) d(gx, F(u, v))d(gu, F(x, y)) d(gy, F(y, x))d(gv, F(v, u)) , , d(gx, gu) d(gy, gv) d(gy, F(v, u))d(gv, F(y, x)) }, d(gy, gv)

d(F(x, y), F(u, v)) ≤ α max{d(gx, gu), d(gy, gv),

(2.1)

satisfies for all x, y, u, v ∈ X, gx ̸= gu, gy ̸= gv with gx ≽ gu and gy ≼ gv. Further suppose that F is continuous, F(X × X) ⊆ g(X), g is continuous non-decreasing and commutes with F. Then there exist x, y ∈ X such that either gx = F(x, y) or gy = F(y, x) or gx = F(x, y) and gy = F(y, x) i.e F and g have a coupled coincidence point (x, y) ∈ X × X. Proof. Let x0 , y0 ∈ X be such that gx0 ≼ F(x0 , y0 ) and gy0 ≽ F(y0 , x0 ). Since F(X × X) ⊆ g(X), we can construct sequences {xn } and {yn } in X such that

We claim that for all n ≥ 0,

gxn+1 = F(xn , yn ) and gyn+1 = F(yn , xn ), ∀n ≥ 0.

(2.2)

gxn ≼ gxn+1 ,

(2.3)

gyn ≽ gyn+1 .

(2.4)

and We shall use the mathematical induction. Let n = 0. Since gx0 ≼ F(x0 , y0 ) and gy0 ≽ F(y0 , x0 ), in view of gx1 = F(x0 , y0 ) and gy1 = F(y0 , x0 ), we have gx0 ≼ gx1 and gy0 ≽ gy1 , that is, (2.3) and (2.4) hold for n = 0. Suppose that (2.3) and (2.4) hold for some n > 0. As F has the mixed g-monotone property and gxn ≼ gxn+1 and gyn ≽ gyn+1 , from (2.2), we get gxn+1 = F(xn , yn ) ≼ F(xn+1 , yn ) ≼ F(xn+1 , yn+1 ) = gxn+2 , (2.5)

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and gyn+1 = F(yn , xn ) ≽ F(yn+1 , xn ) ≽ F(yn+1 , xn+1 ) = gyn+2 .

(2.6)

Now from (2.5) and (2.6), we obtain that gxn+1 ≼ gxn+2 and gyn+1 ≽ gyn+2 . Thus by the mathematical induction, we conclude that (2.3) and (2.4) hold for all n ≥ 0. Therefore gx0 ≼ gx1 ≼ gx2 ≼ . . . ≼ gxn ≼ gxn+1 ≼ . . . ,

(2.7)

gy0 ≽ gy1 ≽ gy2 ≽ . . . ≽ gyn ≽ gyn+1 ≽ . . . .

(2.8)

and Since gxn ≽ gxn−1 and gyn ≼ gyn−1 , from (2.1) and (2.2), we have d(gxn+1 , gxn ) = d(F(xn , yn ), F(xn−1 , yn−1 )) d(gxn , F(xn , yn ))d(gxn−1 , F(xn−1 , yn−1 )) , d(gxn , gxn−1 ) d(gxn , F(xn−1 , yn−1 ))d(gxn−1 , F(xn , yn )) , d(gxn , gxn−1 ) d(gyn , F(yn , xn ))d(gyn−1 , F(yn−1 , xn−1 )) d(gyn , F(yn−1 , xn−1 ))d(gyn−1 , F(yn , xn )) , } d(gyn , gyn−1 ) d(gyn , gyn−1 ) d(gxn , gxn+1 )d(gxn−1 , gxn ) = α max{d(gxn , gxn−1 ), d(gyn , gyn−1 ), , d(gxn , gxn−1 ) d(gxn , gxn )d(gxn−1 , gxn+1 ) d(gyn , gyn+1 )d(gyn−1 , gyn ) d(gyn , gyn )d(gyn−1 , gyn+1 ) , , } d(gxn , gxn−1 ) d(gyn , gyn−1 ) d(gyn , gyn−1 ) = α max{d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gxn , gxn+1 ), d(gyn , gyn+1 )}, ≤ α max{d(gxn , gxn−1 ), d(gyn , gyn−1 ),

which implies that d(gxn+1 , gxn ) ≤ α max{d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gxn , gxn+1 ), d(gyn , gyn+1 )}. Similarly, we have d(gyn+1 , gyn ) ≤ α max{d(gxn , gxn−1 ), d(gyn , gyn−1 ), d(gxn , gxn+1 ), d(gyn+1 , gyn )}. Set {ρn := max{d(gxn+1 , gxn ), d(gyn+1 , gyn )}}. Hence max{d(gxn+1 , gxn ), d(gyn+1 , gyn )} ≤ α max{d(gxn , gxn−1 ), d(gyn , gyn−1 )} = αρn−1 . By induction we get that max{d(gxn+1 , gxn ), d(gyn+1 , gyn )} ≤ α n ρ0 . It easily follows that for each m, n ∈ N, m < n, we have d(gxm , gxn ) ≤

km ρ0 , 1−k

d(gym , gyn ) ≤

km ρ0 . 1−k

and

Therefore, {gxn } and {gyn } are Cauchy sequences. Since X is a complete metric space, there is (x, y) ∈ X ×X such that gxn → x and gyn → y. Since g is continuous, g(gxn ) → gx and g(gyn ) → gy. As F is continuous. Then F(gxn , gyn ) → F(x, y) and F(gyn , gxn ) → F(y, x). As, F commutes with g, we have F(gxn , gyn ) = gF(xn , yn ) = g(gxn+1 ) → gx and F(gyn , gxn ) = gF(yn , xn ) = g(gyn+1 ) → gy. By the uniqueness of the limit, we get gx = F(x, y) and gy = F(y, x). Thus F and g have a coupled coincidence point.

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Now, we shall prove the existence and uniqueness of a coupled common fixed point. Note that, if (X, ≼) is a partially ordered set, then we endow the product space X × X with the following partial order relation: for (x, y), (u, v) ∈ X × X, (u, v) ≼ (x, y) ⇔ x ≼ u, y ≽ v. Theorem 2.2. In addition to hypotheses of Theorem 2.1, suppose that for every (x, y), (z,t) ∈ X × X, there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable to (F(x, y), F(y, x)) and (F(z,t), F(t, z)). Then F and g have a unique coupled common fixed point, that is, there exist x, y ∈ X such that either gx = F(x, y) or gy = F(y, x) or (x, y) is the unique coupled common fixed point of F and g. Proof. From Theorem 2.1, the set of coupled coincidence points of F and g is non-empty. Suppose that (x, y) and (z,t) are coupled coincidence points of F and g, that is, gx = F(x, y), gy = F(y, x), gz = F(z,t) and gt = F(t, z). We shall show that gx = gz and gy = gt. By the assumption, there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable with (F(x, y), F(y, x)) and (F(z,t), F(t, z)). Put u0 = u, v0 = v and choose u1 , v1 ∈ X so that gu1 = F(u0 , v0 ) and gv1 = F(v0 , u0 ). Then similarly as in the proof of Theorem 2.1, we can inductively define sequences {gun }, {gvn } as gun+1 = F(un , vn ) and gvn+1 = F(vn , un ) for all n. Further, set x0 = x, y0 = y, z0 = z, t0 = t and on the same way define the sequences {gxn }, {gyn }, and {gzn }, {gtn }. Then as in Theorem 2.1, we can show that gxn → gx = F(x, y), gyn → gy = F(y, x), gzn → gz = F(z,t), gtn → gt = F(t, z), for all n ≥ 1. Since (F(x, y), F(y, x)) = (gx1 , gy1 ) = (gx, gy) and (F(u, v), F(v, u)) = (gu1 , gv1 ) are comparable, then gx ≽ gu1 and gy ≼ gv1 . Now, we shall show that (gx, gy) and (gun , gvn ) are comparable, that is, gx ≽ gun and gy ≼ gvn for all n. Suppose that it holds for some n ≥ 0, then by the mixed g-monotone property of F, we have gun+1 = F(un , vn ) ≼ F(x, y) = gx and gvn+1 = F(vn , un ) ≽ F(y, x) = gy. Hence gx ≽ gun and gy ≼ gvn hold for all n. Thus from (2.1), we have d(gx, gun+1 ) = d(F(x, y), F(un , vn )) d(gx, F(x, y))d(gun , F(un , vn )) , d(gx, gun ) d(gx, F(un , vn ))d(gun , F(x, y)) d(gy, F(y, x))d(gvn , F(vn , un )) , , d(gx, gun ) d(gy, gvn ) d(gy, F(vn , un ))d(gvn , F(y, x)) } d(gy, gvn ) = α max{d(gx, gun ), d(gy, gvn ), d(gx, gun+1 ), d(gy, gvn+1 )}. ≤ α max{d(gx, gun ), d(gy, gvn ),

(2.9)

Similarly, we can prove that d(gy, gvn+1 ) ≤ α max{d(gx, gun ), d(gy, gvn ), d(gx, gun+1 ), d(gy, gvn+1 )}. Hence max{d(gx, gun+1 ), d(gy, gvn+1 )}

≤ α max{d(gx, gun ), d(gy, gvn )}

max{d(gx, gun+1 ), d(gy, gvn+1 )}

≤ α n max{d(gx, gu1 ), d(gy, gv1 )}

and by induction

On taking limit, n → ∞, we get limn→∞ d(gx, gun+1 ) = 0 and limn→∞ d(gy, gvn+1 ) = 0. Similarly, we can prove that limn→∞ d(gz, gun+1 ) = 0 = limn→∞ d(gt, gvn+1 ). Finally, we have d(gx, gz) ≤ d(gx, gun )+ d(gun , gz) and d(gy, gt) ≤ d(gy, gvn ) + d(gvn , gt). Taking n → ∞ in these inequalities, we get d(gx, gz) = 0 = d(gy, gt), that is gx = gz and gy = gt. Denote gx = p and gy = q. we have that gp = g(gx) = g(F(x, y)) and gq = g(gy) = g(F(y, x)). By the definition of sequences {xn } and {yn }, we have gxn = F(x, y) = F(xn−1 , yn−1 ) and gyn = F(y, x) = F(yn−1 , xn−1 ), and so gxn → F(x, y), F(xn−1 , yn−1 ) → F(x, y) and gyn → F(y, x), F(yn−1 , xn−1 ) → F(y, x). Compatibility of F and g implies that limn→∞ d(g(F(xn , yn )), F(gxn , gyn )) → 0 i.e. gF(x, y) = F(gx, gy). Then gp = F(p, q)

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and similarly, gq = F(q, p). Thus (p, q) is a coupled coincidence point. Thus, it follows gp = gx and gq = gy, that is, gp = p and gq = q. Hence p = gp = F(p, q) and q = gq = F(q, p). Therefore, (p, q) is a coupled common fixed point of F and g. To prove the uniqueness, assume that (r, s) is another coupled common fixed point. Then as above, we have r = gr = gp = p and s = gs = gq = q. Hence we get the result. Theorem 2.3. In addition to hypotheses of Theorem 2.1, if gx0 and gy0 are comparable. Then F and g have a coupled coincidence point, that is, there exist x, y ∈ X such that either gx = F(x, y) or gy = F(y, x) or gx = F(x, y) = F(y, x) = gy. Proof. By Theorem 2.1, we can construct two sequences {gxn } and {gyn } in X such that gxn → gx and gyn → gy, where (x, y) is a coincidence point of F and g. Suppose gx0 ≼ gy0 . We shall show that gxn ≼ gyn , where gxn = F(xn−1 , yn−1 ), gyn = F(yn−1 , xn−1 ), for all n. Suppose it holds for some n ≥ 0. Then by mixed g-monotone property of F, we have gxn+1 = F(xn , yn ) ≼ F(yn , xn ) = gyn+1 . From (2.1), we have d(gxn+1 , gyn+1 ) =

d(F(xn , yn ), F(yn , xn )) d(gxn , F(xn , yn ))d(gyn , F(yn , xn )) , d(gxn , gyn ) d(gxn , F(yn , xn ))d(gyn , F(xn , yn )) d(gyn , F(yn , xn ))d(gxn , F(xn , yn )) , , d(gxn , gyn ) d(gyn , gxn ) d(gyn , F(xn , yn ))d(gxn , F(yn , xn )) }. d(gyn , gxn )

≤ α max{d(gxn , gyn ), d(gyn , gxn ),

On taking n → ∞, we obtain d(gy, gx) ≤ α d(gy, gx). Since α < 1, d(gy, gx) = 0. Hence F(x, y) = gx = gy = F(y, x). A similar arguments can be used if gy0 ≼ gx0 . x Example 2.1. Let X = R and d(x, y) = |x − y|. Define F(x, y) = x−y 8 and gx = 2 . Relation is ordinary ≤. All the conditions of Theorem 2.1 are satisfied including the contractive condition

d(F(x, y), F(u, v)) ≤ ≤

1 max{d(gx, gu), d(gy, gv)} 2 1 d(gx, F(x, y))d(gu, F(u, v)) max{d(gx, gu), d(gy, gv), , 2 d(gx, gu) d(gx, F(u, v))d(gu, F(x, y)) d(gy, F(y, x))d(gv, F(v, u)) , , d(gx, gu) d(gy, gv) d(gy, F(v, u))d(gv, F(y, x)) }. d(gy, gv)

Clearly, (0, 0) is a coupled coincidence point of F and g. References [1] R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008) 1-8. [2] T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393. http://dx.doi.org/10.1016/j.na.2005.10.017 [3] S. Chandok, Some common fixed point theorems for generalized f -weakly contractive mappings, J. Appl. Math. Informatics 29 (2011), 257-265.

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[4] S. Chandok, Some common fixed point theorems for generalized nonlinear contractive mappings, Computers and Mathematics with Applications 62 (2011), 3692-3699. http://dx.doi.org/10.1016/j.camwa.2011.09.009 [5] S. Chandok, Common fixed points, invariant approximation and generalized weak contractions, Internat. J. Math. Math. Sci. Vol. 2012 (2012) Article ID 102980. http://dx.doi.org/10.1155/2012/102980 [6] S. Chandok, Common fixed points for generalized nonlinear contractive mappings in metric spaces, Mat. Vesnik, 65 (2013) 29-34. [7] S. Chandok, Some common fixed point results for generalized weak contractive mappings in partially ordered metric spaces, J. Nonlinear Anal. Opt. (in press). [8] S. Chandok, J. K. Kim, Fixed point theorem in ordered metric spaces for generalized contractions mappings satisfying rational type expressions, J. Nonlinear Functional Anal. Appl, 17 (2012) 301-306. [9] S. Chandok, Y.J. Cho, Coupled common fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces, communicated. [10] A. Kaewkhao, W. Sintunavarat, P. Kumam, Common fixed point theorems of C-distance on cone metric spaces, Journal of Nonlinear Analysis and Application, Volume 2012 (2012) , 1-11. http://dx.doi.org/10.5899/2012/jnaa-00137 [11] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially orderd metric spaces, Nonlinear Anl. 70(2009), 4341-4349. http://dx.doi.org/10.1016/j.na.2008.09.020 [12] J.J. Nieto, R.R. L´opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(2005), 223-239. http://dx.doi.org/10.1007/s11083-005-9018-5 [13] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (5) (2004), 1435-1443. http://dx.doi.org/10.1090/S0002-9939-03-07220-4 [14] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler Contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. http://dx.doi.org/10.1016/j.na.2010.02.026

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