Fixed points of almost quadratic Geraghty contractions and property(p) in partially ordered metric spaces

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1 Journal of Advanced Research in Applied Mathematics Online ISSN: Vol. 7, Issue., 2015, pp. 1-9 doi: /jaram Fixed points of almost quadratic Geraghty contractions and property(p) in partially ordered metric spaces G.V.R. Babu 1,, K.K.M. Sarma 1, P.H. Krishna 2 1 Department of Mathematics, Andhra University, Visakhapatnam , India. 2 Department of Mathematics, Viswanadha Institute of Technology and Management, Visakhapatnam , India. 8 Abstract. We introduce almost quadratic Geraghty contraction maps and prove the existence of fixed points in partially ordered metric spaces. Further, we study property(p) of these contractions. Examples are provided in support of our results Keywords: Geraghty contraction; Quadratic contraction; Almost quadratic contraction; Property(P); Strongly comparable. 11 Mathematics Subject Classification 2010: 47H10, 54H Introduction Contraction principle has been extended and generalized by many researchers. One among those generalizations is due to Geraghty. In 1973, Geraghty [7] introduced an extension of the contraction in which the contraction constant was replaced by a function having some specific properties. On the other hand, the notion of fixed points in partially ordered sets was introduced by Brondsted [6]. In 2004, Banach contraction principle has been extended and generalized to partially ordered metric spaces by Ran and Reurings [16]. For more works on the existence of fixed points in ordered metric spaces, we refer [1, 5, 11 16]. In 2007, Jeong and Rhoades [9, 10] defined property(p) on metric spaces. Definition 1.1. [9] A selfmap T : X X is said to have the property(p) whenever F (T ) = F (T n ) for all n 1, where F (T ) is the set of all fixed points of T. Correspondence to: G.V.R. Babu, Department of Mathematics, Andhra University, Visakhapatnam , India. gvr babu@hotmail.com. Received: 20 February 2015, accepted: 9 July c 2015 Institute of Advanced Scientific Research

2 2 Fixed points of almost quadratic Geraghty contractions and property(p) Definition 1.2. [8] Let (X, ) be a partially ordered set. We define X = {(x, y) X X : x y or y x}. We say that a selfmap T : X X has the strongly comparable property whenever (T n 1 y, T n y) X for all n 1 and m 2 where y F (T m ). Example 1.1. Let X = R, we define partial order on R by a b if and only if a b, where { a b means b = ka for some positive integer k. We define T : X X by 1, if x is rational T (x) = 1, if x is irrational. Then F (T ) = { 1} = F (T m ), m 2 and (T n 1 ( 1), T n ( 1)) = ( 1, 1) X. Therefore T has the strongly comparable property. Example 1.2. Let X = {1, 2, 3, 4}, we define T : X X by T 1 = 1, T 2 = 3, T 3 = 2 and T 4 = 4. Then F (T ) = {1, 4} and F (T 2 ) = {1, 2, 3, 4} = X. We define X = {(3, 4)}. i.e., X = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 4)}. For y = 2 F (T 2 ), we consider (T n 1 y, T n y) = (y, T y) = (2, 3) for n = 1 and (2, 3) does not belong to X. Therefore T does not satisfy the strongly comparable property. We use the following notation throughout this paper. (X, d) denotes a metric space and we write it as X. Let T : X X be a selfmap of X and F (T ) denotes the set of all fixed points of T. We denote S = {β : [0, ) [0, 1)/ β(t n ) 1 t n 0}. Definition 1.3. [7] A selfmap T : X X is said to be a Geraghty contraction if there exists β S such that d(t x, T y) β(d(x, y))d(x, y) for all x, y X. (1.1) Theorem 1.4. (Geraghty [7]) Let (X, d) be a complete metric space. Let T : X X be a selfmap. Assume that there exists β S such that d(t x, T y) β(d(x, y))d(x, y) for all x, y X. (1.2) Then T has a unique fixed point z X and, for any choice of the initial point x 0 X, the sequence {x n } defined by x n = T x n 1 for each n 1 converges to the point z. As a generalization of contraction maps, Berinde [3] introduced weak contractions as follows. Definition 1.5. Let (X, d) be a metric space. A map T : X X is called a weak contraction if there exist a constant δ (0, 1) and L 0 such that d(t x, T y) δ d(x, y)+l d(y, T x) for all x, y X. (1.3) Remark 1.6. If T is a contraction with contraction constant α then T is a weak contraction with δ = α and L = 0. But its converse need not be true due to the following example. 57 Example 1.3. [3] Let X = [0, 1] with the usual metric. We define T : X X by T x = 2 3 for x [0, 1) and T 1 = 0. Then T satisfies (1.3) with δ and L δ so that 59 T is a weak contraction. Here we observe that T is not continuous on [0, 1] and hence T 60 is not a contraction.

3 G.V.R. Babu, K.K.M. Sarma and P.H. Krishna Here we note that Berinde [4] renamed weak contractions as almost contractions. Definition 1.7. Let X be a nonempty set and T be a selfmap on X. Let x X, we define the orbit of x w.r.t. T by O T (x) = {T n (x)/n = 0, 1, 2,...}. Here T 0 = I, I is the identity map of X. Definition 1.8. A mapping T : X X is said to be orbitally continuous at a point z in X with respect to x in X, if for any sequence {x n } O T (x) with x n z as n implies T x n T z as n. Remark 1.9. Every continuous map T : X X is orbitally continuous at each point of X but every orbitally continuous map need not be continuous. In this paper, (X,, d) denotes a partially ordered metric space, where (X, ) is a partially ordered set, and d is a metric on (X, ). If X is complete with respect to the metric d then we call (X,, d), a partially ordered complete metric space. The following theorems are established by V. Ghorbanian, Sh Rezapour, N Shahzad [8] in partially ordered complete metric spaces. Theorem [8] Let (X, d, ) be a partially ordered complete metric space. Let T be a selfmap on X. Assume that there exists λ (0, 1) such that min{d 2 (T x, T y), d(x, y)d(t x, T y), d 2 (y, T y)} min{d 2 (x, T x), d(y, T y)d(x, T y), d 2 (y, T x)} λ d(x, T x)d(y, T y) for all x, y X. If there exists x 0 X such that (T n 1 x 0, T n x 0 ) X for all n 1 and T is orbitally continuous at x 0, then T has a fixed point. Moreover, if T has the strongly comparable property, then T has the property(p ). In the following, we introduce almost quadratic Geraghty contraction maps. Definition Let (X, d, ) be a partially ordered metric space. T be a selfmap of X. We say that T is an almost quadratic Geraghty contraction if there exist β S and L 0 such that d 2 (T x, T y) β(m(x, y))m(x, y) + Ld(x, T x).d(y, T x) (1.4) where m(x, y) = max{d(x, T x).d(y, T y), d(x, T y).d(y, T x)} for all x, y X with x y. Example 1.4. Let X = {1, 2, 3, 4} be endowed with the usual metric. We define partial order on X by : {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 3)} and we define T : X X by T 1 = 4, T 2 = 3, { T 3 = 3 and T 4 = 4. 0 if t = 0 We define β : [0, ) [0, 1) by β(t) = 1 1+t if t > 0. Then β S and T is an almost quadratic Geraghty contraction. The following lemma is useful in proving our main results.

4 4 Fixed points of almost quadratic Geraghty contractions and property(p) Lemma [2] Let (X, d) be a metric space. Let {x n } be a sequence in X such that d(x n+1, x n ) 0 as n. If {x n } is not a Cauchy sequence then there exist an ϵ > 0 and sequences of positive integers {m(k)} and {n(k)} with m(k) > n(k) > k and (i) (iii) lim d(x m(k) 1, x n(k)+1 ) = ϵ, (ii) lim d(x m(k), x n(k) ) = ϵ lim d(x m(k) 1, x n(k) ) = ϵ and (iv) lim d(x m(k), x n(k)+1 ) = ϵ. In Section 2, we prove the existence of fixed points of almost quadratic Geraghty contraction maps in partially ordered metric spaces. Further we study property(p) of these contractions. 2 Main results Theorem 2.1. Let (X, d, ) be a partially ordered complete metric space. Let T be a selfmap of X. Suppose that T is an almost quadratic Geraghty contraction. Suppose that there exists x 0 X such that (T n 1 x 0, T n x 0 ) X for all n 1. Then the sequence {T n x 0 } is Cauchy in X. Let lim T n x 0 = u, u X. If T is orbitally continuous at u w.r.t. x 0, then T has a fixed point. Proof. Let x 0 X be as in the hypothesis. We define x 1 = T x 0 and x n+1 = T x n for n = 0, 1, 2, 3,.... (2.1) If x n = x n+1 for some n, then x n = T x n and hence x n is a fixed point. Now, without loss of generality, we suppose that x n x n+1 for all n = 0, 1, 2, 3,.... i.e., d(x n, x n+1 ) > 0. Since, T is an almost quadratic Geraghty contraction, by using the inequality (1.4) d 2 (x n+1, x n+2 ) = d 2 (T x n, T x n+1 ) β(m(x n, x n+1 ))m(x n, x n+1 ) + Ld(x n, T x n ).d(x n+1, T x n ) where m(x n, x n+1 ) = max{d(x n, T x n ).d(x n+1, T x n+1 ), d(x n, T x n+1 ).d(x n+1, T x n )} = max{d(x n, x n+1 ).d(x n+1, x n+2 ), d(x n, x n+2 ).d(x n+1, x n+1 )} = max{d(x n, x n+1 ).d(x n+1, x n+2 ), d(x n, x n+2 ).0} = d(x n, x n+1 ).d(x n+1, x n+2 ) and d(x n, T x n ).d(x n+1, T x n ) = d(x n, x n+1 ).d(x n+1, x n+1 ) = 0. Therefore d 2 (x n+1, x n+2 ) β(d(x n, x n+1 ).d(x n+1, x n+2 ))d(x n, x n+1 ).d(x n+1, x n+2 ). (2.2) Since d(x n, x n+1 ) > 0, d(x n+1, x n+2 ) > 0 and β S, it follows that d 2 (x n+1, x n+2 ) < d(x n, x n+1 ).d(x n+1, x n+2 ) and hence d(x n+1, x n+2 ) < d(x n, x n+1 ). Thus the sequence {d(x n, x n+1 )} is a decreasing sequence of non-negative real numbers. So lim d(x n, x n+1 )) exists and it is r (say). i.e., lim d(x n, x n+1 ) = r 0. (2.3)

5 G.V.R. Babu, K.K.M. Sarma and P.H. Krishna We now show that r = 0. (2.4) Suppose r > 0, then from (2.2), we have d 2 (x n+1, x n+2 ) β(d(x n, x n+1 ).d(x n+1, x n+2 ))d(x n, x n+1 ).d(x n+1, x n+2 ) d(x n+1, x n+2 ) β(d(x n, x n+1 ).d(x n+1, x n+2 )).d(x n, x n+1 ) d(x n+1,x n+2 ) d(x n,x n+1 ) β(d(x n, x n+1 ).d(x n+1, x n+2 )) < 1 for all n 1. On letting n, we get 1 lim β(d(x n, x n+1 ).d(x n+1, x n+2 )) 1 so that β(d(x n, x n+1 ).d(x n+1, x n+2 )) 1 as n. Since β S, it follows that d(x n, x n+1 ).d(x n+1, x n+2 ) 0 as n so that r 2 = 0 which implies that r = 0, and (2.4) holds. Therefore lim d(x n, x n+1 ) = 0. Now, we show that {x n } is Cauchy. Suppose that {x n } is not a Cauchy sequence. Then, by Lemma 1.12, there exists an ϵ > 0 for which we can find sequences of positive integers {m(k)} and {n(k)} with m(k) > n(k) > k and d(x n(k), x m(k) ) ϵ and d(x m(k) 1, x n(k) ) < ϵ. The following identities are satisfied. (i) (iii) lim d(x n(k), x m(k) ) = ϵ, (ii) lim d(x m(k) 1, x n(k) ) = ϵ, lim d(x m(k), x n(k)+1 ) = ϵ, and (iv) lim d(x n(k), x m(k)+1 ) = ϵ. we now consider d 2 (x n(k)+1, x m(k)+1 ) = d 2 (T x n(k), T x m(k) ) β(m(x n(k), x m(k) ))m(x n(k), x m(k) ) + L.d(x n(k), T x n(k) ).d(x m(k), T x n(k) ), where m(x n(k), x m(k) ) = max{d(x n(k), T x n(k) ).d(x m(k), T x m(k) ), d(x n(k), T x m(k) ).d(x m(k), T x n(k) )} = max{d(x n(k), x n(k)+1 ).d(x m(k), x m(k)+1 ), d(x n(k), x m(k)+1 ).d(x m(k), x n(k)+1 )}. On using the above (iii) and (iv) identities, we get lim m(x n(k), x m(k) ) = lim max{d(x n(k), x n(k)+1 ).d(x m(k), x m(k)+1 ), d(x n(k), x m(k)+1 ).d(x m(k), x n(k)+1 )} = max{0.0, ϵ.ϵ}= ϵ 2 and lim d(x n(k), x n(k)+1 ).d(x m(k), x n(k)+1 ) = 0.ϵ = 0. From (2.2), on letting k, we have lim d2 (x n(k)+1, x m(k)+1 ) lim β(max{d(x n(k), x n(k)+1 ).d(x m(k), x m(k)+1 ), d(x n(k), x m(k)+1 ).d(x m(k), x n(k)+1 )} lim m(x n(k), x m(k) ) + L. lim d(x n(k), T x n(k) ).d(x m(k), T x n(k) ) ϵ 2 lim β(m(x n(k), x m(k) ).ϵ 2 + L.0 1 = ϵ2 lim ϵ β(m(x n(k), x 2 m(k) ) 1, so that lim β(m(x n(k), x m(k) ) = 1. Since β S, it follows that m(x n(k), x m(k) ) 0 as k. i.e., max{0, ϵ.ϵ} 0 implies that ϵ 2 = 0 so that ϵ = 0, a contradiction.

6 6 Fixed points of almost quadratic Geraghty contractions and property(p) Therefore {x n } is a Cauchy sequence. Since (X, d) is complete, there exists u X such that lim x n = u. Since T is orbitally continuous at u w.r.t. x 0 it follows that x n+1 = T x n T u so that T u = u. Hence u is a fixed point of T. Theorem 2.2. In addition to the hypotheses of Theorem 2.1, if T has the strongly comparable property, then T has the property(p ). Proof. By Theorem 2.1, T has a fixed point. We now show that F (T n ) F (T ) for all n 2. Let n 2 and v F (T n ). If d(v, T v) = 0 then T v = v so that v is a fixed point of T. Suppose that d(v, T v) > 0. Since T has the strongly comparable property, for i {1, 2, 3,..., n} we have (T n i v, T n i+1 v) X. Now by taking x = T n i v and y = T n i+1 v in the inequality (1.4), we have d 2 (T n i+1 v, T n i+2 v) β(m(t n i v, T n i+1 v)).m(t n i v, T n i+1 v) + L.d(T n i v, T n i+1 v).d(t n i+1 v, T n i+1 v) (2.5) where m(t n i v, T n i+1 v) = max{d(t n i v, T n i+1 v).d(t n i+1 v, T n i+2 v), d(t n i v, T n i+2 v).d(t n i+1 v, T n i+1 v)} = d(t n i v, T n i+1 v).d(t n i+1 v, T n i+2 v) From (2.5), we have d 2 (T n i+1 v, T n i+2 v) β(d(t n i v, T n i+1 v).d(t n i+1 v, T n i+2 v)). d(t n i v, T n i+1 v).d(t n i+1 v, T n i+2 v). If d(t n i+1 v, T n i+2 v) = 0 then T n i+1 v = T n i+2 v. If i = 1, we have T n v = T n+1 v = T (T n v), and since v F (T n ) it implies that v = T v. Now, we consider the case d(t n i+1 v, T n i+2 v) > 0 for all i {1, 2, 3,..., n}. On using the inequality (2.5) we have d 2 (T n i+1 v, T n i+2 v) < d(t n i v, T n i+1 v).d(t n i+1 v, T n i+2 v), which implies that d(t n i+1 v, T n i+2 v) < d(t n i v, T n i+1 v). Similarly it can be shown that d(t n i v, T n i+1 v) < d(t n i 1 v, T n i v) < d(t n i 2 v, T n i 1 v) <... < d(v, T v). Hence for i {1, 2, 3,..., n}, we have d(t n i v, T n i+1 v) < d(t n i 1 v, T n i v) < d(t n i 2 v, T n i 1 v) <... < d(v, T v). In particular for i = n, we have d(v, T v) < d(v, T v), a contradiction. Therefore d(v, T v) = 0 so that T v = v. Thus F (T n ) F (T ) for all n 2. Hence T has property(p) and this completes the proof of the theorem. The following is an example in support of Theorem 2.2.

7 G.V.R. Babu, K.K.M. Sarma and P.H. Krishna Example Let X = {1, 2, 3, 4} be endowed with the usual metric. 212 We define partial order on X by : {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (1, 4), (2, 4), (3, 4)} 213 and X := {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (2, 4), (4, 2)(3, 4), (4, 3)}. 214 We define T : X X by T 1 = 4, T 2 = 1, T 3 = 3 and T 4 = 3. 0 if t = We define β : [0, ) [0, 1) by β(t) = 1+t if t (0, 2) 3 4 if t then β S and T satisfies the inequality (1.4) with L = We verify it in the following cases. 219 Case (i) : (x, y) = (1, 3). 220 In this case d 2 (T 1, T 3) = 1, m(1, 3) = 2, d(1, T 1).d(3, T 1) = 3, and d 2 (T 1, T 3) = L.3 = β(m(1, 3)).m(1, 3) + L.d(1, T 1).d(3, T 1) 222 holds with L Case (ii) : (x, y) = (3, 1). 224 In this case d 2 (T 3, T 1) = 1, m(3, 1) = 2, d(3, T 3).d(1, T 3) = 0, and d 2 (T 3, T 1) = L.0 = β(m(3, 1)).m(3, 1) + L.d(3, T 3).d(1, T 3) 226 holds with L Case (iii) : (x, y) = (1, 4). 228 In this case d 2 (T 1, T 4) = 1, m(1, 4) = 3, d(1, T 1).d(4, T 1) = 0, and d 2 (T 1, T 4) = L.0 = β(m(1, 4)).m(1, 4) + L.d(1, T 1).d(4, T 1) 230 holds with L Case (iv) : (x, y) = (4, 1). 232 In this case d 2 (T 4, T 1) = 1, m(4, 1) = 3, d(4, T 4).d(1, T 4) = 2, and d 2 (T 4, T 1) = L.2 = β(m(4, 1)).m(4, 1) + L.d(4, T 4).d(1, T 4) 234 holds with L Case (v) : (x, y) = (2, 4). 236 In this case d 2 (T 2, T 4) = 4, m(2, 4) = 2, d(2, T 2).d(4, T 2) = 3, and d 2 (T 2, T 4) = L.3 = β(m(2, 4)).m(2, 4) + L.d(2, T 2).d(4, T 2) 238 holds with L = Case (vi) : (x, y) = (4, 2). 240 In this case d 2 (T 4, T 2) = 4, m(4, 2) = 2, d(4, T 4).d(2, T 4) = 1, and d 2 (T 4, T 2) = L.1 = β(m(4, 2)).m(4, 2) + L.d(4, T 4).d(2, T 4) 242 holds with L = In the remaining cases inequality (1.4) holds trivially. 244 Therefore T is an almost quadratic Geraghty contraction with L = Clearly T is orbitally continuous at z = 3 with respect to x 0 = Therefore T satisfies all the hypotheses of Theorem 2.1 and T has a fixed point 3. Since y = 3 F (T m ) for m 2, we have (T n 1 3, T n 3) = (3, 3) X. Therefore T has the strongly comparable property. Therefore T satisfies all the hypotheses of Theorem 2.2 also T has the property(p). We observe that the inequality (1.4) fails to hold when L = 0, from Case (v) and Case (vi), and it shows the importance of L in the inequality (1.4).

8 8 Fixed points of almost quadratic Geraghty contractions and property(p) Definition 2.3. Let (X, d, ) be a partially ordered complete metric space. T be a selfmap of X. We say that T is an almost quadratic contraction if there exist λ [0, 1) and L 0 such that d 2 (T x, T y) λ. m(x, y) +Ld(x, T x).d(y, T x) (2.6) where m(x, y) = max{d(x, T x).d(y, T y), d(x, T y).d(y, T x)} for all x, y X with x y. As a consequence of Theorem 2.2, we obtain the following corollary when T is an almost quadratic contraction. Corollary 2.4. : Let (X, d, ) be a partially ordered complete metric space. Let T be a selfmap of X. Suppose that T ia an almost quadratic contraction. Suppose that there exist x 0 X such that (T n 1 x 0, T n x 0 ) X for all n 1 and T is orbitally continuous at x 0, then T has a fixed point. Moreover, if T has the strongly comparable property, then T has the property(p ). References [1] A. Amini-Harindi, H. Emami. A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Analysis, 2010, 72(5): , doi: /j.na [2] G.V.R. Babu, P.D. Sailaja. A fixed point theorem of generalized weak contractive maps in orbitally complete metric spaces. Thai Journal of Mathematics, 2011, 9(1): [3] V. Berinde. Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Analysis forum, 2004, 9(1): [4] V. Berinde. General contractive fixed point theorems for Ciric-type almost contractions in metric spaces. Carpathian J. of Math., 2008, 24(2): [5] T.G. Bhaskar, V. Lakshmikantham. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis, 2006, 65: [6] A. Brondsted. Common fixed points and partial orders. Proc. Amer. Math. Soc., 1979, 77(3): [7] M.A. Geraghty. On contractive maps. Proc. of Amer. Math. Soc., 1973, 40: , doi: /S [8] V. Ghorbanian, Sh. Rezapour, N. Shahzad. Some ordered fixed point results and the property (P). Computers and Mathematics with Applications, 2012, 63: [9] S.G. Jeong, B.E. Rhoades. Maps for which F (T ) = F (T n ). Fixed Point Theory and Applications, Nova Sci., New York, 2007, 6: [10] S.G. Jeong, B.E. Rhoades. More maps for which F (T ) = F (T n ). Demonstratio Mathematica, 2007, 40(3): [11] Z. Kadelburg, M. Pavlovic, S. Radenovic. Common fixed point theorems for ordered contractions and quasi contractions in ordered cone metric spaces. Computers and Mathematic with Applications, 2010, 59: [12] J. Harjani, B. Lopez, K. Sadarangani. Fixed point theorems for weakly contractive mappings in ordered metric spaces. Computers & Mathematics with Applications, 2011, 61: , doi: /j.camwa [13] J. Harjani, K. Sadarangani. Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Analysis, 2009, 71: , doi: /j.na [14] J. Harjani, K. Sadarangani. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Analysis, 2010, 72:

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