Some Coupled Fixed Point Theorems on Quasi-Partial b-metric Spaces

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1 International Journal of Mathematical Analysis Vol. 9, 2015, no. 6, HIKARI Ltd, Some Coupled Fixed Point Theorems on Quasi-Partial b-metric Spaces Anuradha Gupta Department of Mathematics, Delhi College of Arts and Commerce, University of Delhi, India Pragati Gautam Department of Mathematics, Kamala Nehru College (University of Delhi), August Kranti Marg, Delhi , India Copyright c 2014 Anuradha Gupta and Pragati Gautam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we introduce the notion of quasi-partial b-metric space and then study coupled fixed point results in a quasi-partial b-metric space. Some examples are also given in support of the obtained results. Mathematics Subject Classification: 47H10, 54H25 Keywords: partial metric space; quasi-partial metric space; b-metric space; quasi-partial b-metric space; coupled fixed point 1 Introduction A generalization of metric space can be obtained as a partial metric space defined by Matthews [1] in 1994 by replacing the condition d(x, x) = 0 with the condition d(x, x) d(x, y) for all x, y in the definition of the metric. Further

2 294 Anuradha Gupta and Pragati Gautam Czerwik [2] introduced the concept of b-metric space as another generalization of the concept of metric space. Several authors have focused on fixed point theorems for metric space, partial metric space and partial b-metric space. In the year 2012, Karapinar [3] introduced the concept of quasi-partial metric space and studied some fixed point theorems on these spaces whereas Shukla [4] explained partial b-metric spaces. Later on Shatanawi [5] studied some coupled fixed point theorems on these spaces. The aim of this paper is to examine some topological structure of these spaces. In this context we introduce the concept of quasi-partial b-metric space and using it we prove coupled fixed point theorems supported by some examples. 2 Preliminaries and Definitions We begin the section with some basic definitions and concepts. Definition 2.1 ([3]). A quasi-partial metric on a non-empty set X is a function q : X X R + which satisfies: (QPM 1 ) If q(x, x) = q(x, y) = q(y, y), then x = y, (QPM 2 ) q(x, x) q(x, y) (QPm 3 ) q(x, x) q(y, x),and (QPM 4 ) q(x, y) + q(z, z) q(q, z) + q(z, y) for all x, y X. A quasi-partial metric space is a pair (X, q) such that X is a non-empty set and q is a quasi-partial metric on X. Then d q (x, y) = q(x, y) + q(y, x) q(x, x) q(y, y) is a metric on X. Lemma 2.2 ([3]). For a quasi-partial metric q on X, p q (x, y) = 1 [q(x, y) + q(y, x)], 2 x, y X is a partial metric on X Lemma 2.3 ([3]). Let (X, q) be a quasi-partial metric space, let (X, p q ) be the corresponding partial metric space, and let (X, d pq ) be the corresponding metric space. Then the sequence {x n } is Cauchy in (X, q) iff the sequence {x n } is Cauchy in (X, p q ) iff the sequence {x n } is Cauchy in (X, d pq ). Lemma 2.4 ([3]). Let (X, q) be a quasi-partial metric space, let (X, p q ) be the corresponding partial metric space, and let (X, d pq ) be the corresponding metric space. Then (X, q) is complete iff (X, p q ) is complete iff (X, d pq ) is complete.

3 Some coupled fixed point theorems on quasi-partial b-metric spaces 295 Moreover, lim d p q (x, x n ) = 0 p q (x, x) = lim p q (x, x n ) = lim p q(x n, x m ) n,m q(x, x) = lim q(x, x n ) = lim q(x n, x m ) n,m = lim q(x n, x) = lim q(x m, x n ). n,m Bhaskar and Lakshmikantham [6] introduced the concept of coupled fixed point for a metric space. Later, Lakshmikantham and Ćirić [7] introduced the following notion of a coupled coincidence point of mappings on a metric space. Definition 2.5 ([6]). Let X be a non-empty set. An element (x, y) X X, is a coupled fixed point of the mapping F : X X X if F (x, y) = x and F (y, x) = y. Definition 2.6 ([7]). An element (x, y)inx 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. The concept of w-compatible mappings was given by Abbas et al. [8] which is defined as: Definition 2.7 ([8]). Let X be a non-empty set. The mappings F : X X X and g : X X are w-compatible if gf (x, y) = F (gx, gy) whenever gx = F (x, y) and gy = F (y, x). We will now introduce the concept of quasi-partial b-metric space. 3 Quasi-partial b-metric space Definition 3.1. A quasi-partial b-metric on a non-empty set X is a mapping qp b : X X R + such that for some real number s 1 and all x, y, z X (QPb 1 ) qp b (x, x) = qp b (x, y) = qp b (y, y) x = y, (QPb 2 ) qp b (x, x) qp b (x, y), (QPb 3 ) qp b (x, x) qp b (y, x), and (QPb 4 ) qp b (x, y) s[qp b (x, z) + qp b (z, y)] qp b (z, z). A quasi-partial b-metric space is a pair (X, qp b ) such that X is a non-empty set and (X, qp b ) is a quasi-partial b-metric on X. The number s is called the coefficient of (X, qp b ). Let qp b be a quasi-partial b-metric on the set X. Then d qpb (x, y) = qp b (x, y) + qp b (y, x) qp b (x, x) qp b (y, y) is a b-metric on X.

4 296 Anuradha Gupta and Pragati Gautam Lemma 3.2. Every quasi-partial metric space is a quasi-partial b-metric space. But the converse need not be true. Example 3.3. Let X = [0, 1]. Define qp b (x, y) = x y + x. Here qp b (x, y) = qp b (x, y) = qp b (y, y) x = y as x = x y + x = y gives x = y. Again, qp b (x, y) qp b (x, y) as x x y +x and similarly, qp b (x, x) qp p (y, x) as x y x + y for 0 < x < y. Also qp b (x, y) + qp b (z, z) s[qp b (x, z) + qp b (z, y)] as x y + x + z s[ x z + x + z y + z] for all s 1. It can be observed that x y + x + z = x z + z y + x + z x z + z y + x + z. So, (X, qp b ) is a quasi-partial b-metric space with s 1. Definition 3.4. Let (X, qp b ) be a Quasi-partial b-metric. Then (i) a sequence {x n } X converges to x X if and only if qp b (x, x) = lim qp b (x, x n ) = lim qp b (x n, x). (ii) a sequence {x n } X is called a Cauchy sequence if and only if lim qp b(x n, x m ) and lim qp b(x m, x n ) exist (and are finite) n,m n,m (iii) the quasi partial b-metric space (X, qp b ) is said to be complete if every Cauchy sequence {x n } X converges with respect to τ qpb to a point x X such that qp b (x, x) = lim n,m qp b (x m, x n ) = lim n,m qp b (x n, x m ). (iv) a mapping f : X X is said to be continuous at x 0 X if, for every ɛ > 0, there exist δ > 0 such that f(b(x 0, δ)) B(f(x 0 ), ɛ). Lemma 3.5. Let (X, qp b ) be a Quasi partial b-metric space. Then the following hold (A) If qp b (x, y) = 0 then x = y. (B) If x y, then qp b (x, y) > 0 and qp b (y, x) > 0. Proof. is similar as for the case of quasi-partial metric space [2]. Shatanawi [5] studied coupled fixed point theorems in quasi-partial metric space. Motivated by his work we have studied some coupled fixed point theorems in quasi partial b-metric space.

5 Some coupled fixed point theorems on quasi-partial b-metric spaces The Main Results Theorem 4.1. Let (X, qp b ) be a quasi-partial b-metric space, g : X X and F : X X X be two mappings. Suppose that there exist k 1, k 2, k 3 in [0, 1) with k 1 + k 2 + k 3 < 1 and k 3 < 1 where s 1 such that the condition s qp b (F (x, y), F (x, y )) + qp b (F (y, x), F (y, x )) k 1 (qp b (gx, gx ) + qp b (gy, gy )) + k 2 (qp b (gx, F (x, y)) + qp b (gy, F (y, x))) + k 3 (qp b (gx, F (x, y )) + qp b (gy, F (y, x ))) (4.1) holds for all x, y, x, y X. Also, suppose the following hypotheses: (1) F (X X) g(x). (2) g(x) is a complete subspace of X with respect to the quasi partial b-metric qp b. Then the mappings F and g have a coupled coincidence point (u, v) satisfying gu = F (u, v) = F (v, u) = gv. Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u, u). Proof. Let x 0, y 0 X. Since F (X X) g(x), we put gx 1 = F (x 0, y 0 ) and gy 1 = F (y 0, x 0 ). Again, since F (X X) g(x), we put gx 2 = F (x 1, y 1 ) and gy 2 = F (y 1, x 1 ). Continuing this process, we can construct two sequences {gx n } and {gy n } in X such that gx n = F (x n 1, y n 1 ), for all n N and gy n = F (y n 1, x n 1 ), for all n N. Let n N. Then by inequality (4.1), we get qp b (gx n, gx n+1 ) + qp b (gy n, gy n+1 ) = qp b (F (x n 1, y n 1 ), F (x n, y n )) + qp b (F (y n 1, x n 1 ), F (y n, x n )) k 1 (qp b (gx n 1, gx n ) + qp b (gy n 1, gy n )) + k 2 ((qp b (gx n 1, F (x n 1, y n 1 )) + qp b (gy n 1, F (y n 1, x n 1 ))) + k 3 ((qp b (gx n, F (x n, y n )) + qp b (gy n, F (y n, x n ))) = k 1 (qp b (gx n 1, gx n ) + qp b (gy n 1, gy n )) + k 2 (qp b (gx n 1, gx n ) + qp b (gy n 1, gy n )) + k 3 (qp b (gx n, gx n+1 ) + qp b (gy n, gy n+1 )). (4.2) From (4.2), we have qp b (gx n, gx n+1 ) + qp b (gy n, gy n+1 ) k 1 + k 2 1 k 3 (qp b (gx n 1, gx n ) + qp b (gy n 1, gy n )). (4.3)

6 298 Anuradha Gupta and Pragati Gautam Put k = k 1+k 2 1 k 3. Then k < 1. Repeating (4.3) n times, we get qp b (gx n, gx n+1 ) + qp b (gy n, gy 1+1 ) k n (qp b (gx 0, gx 1 ) + qp b (gy 0, gy 1 )). Let m and n be natural numbers with m > n, then using (QP b 4 ), we get qp b (gx n, gx n+2 ) s{qp b (gx n, gx n+1 ) + qp b (gx n+1, gx n+2 )} qp b (gx n+1, gx n+1 ) s{qp b (gx n, gx n+1 ) + qp b (gx n+1, gx n+2 )}. qp b (gx n, gx n+3 ) s{qp b (gx n, gx n+2 ) + qp b (gx n+2, gx n+3 )} qp b (gx n+2, gx n+2 ) On generalization, we get Similarly, s(s(qp b (gx n, gx n+1 ) + qp b (gx n+1, gx n+2 )) + sqp b (gx n+2, gx n+3 ) s 2 qp b (gx n, gx n+1 ) + s 2 qp b (gx n+1, gx n+2 ) + sqp b (gx n+2, gx n+3 ). qp b (gx n, gx m ) s m n 1 {qp b (gx n, gx n+1 ) + qp b (gx n+1, gx n+2 )} = = = qp b (gy n, gy m ) + s m n 2 {qp b (gx n+2, gx n+3 )} + + s{qp b (gx, gx m )} +1 On adding (4.4) and (4.5), we get s m i {qp b (gx i, gx i+1 )} + s m n 1 {qp b (gx n, gx n+1 )} s m i {qp b (gx i, gx i+1 )} + s m n 1 (qp b {(gx n, gx n+1 )} s m n {qp b (gx n, gx n+1 )} qp b (gx n, gx m ) + qp b (gy n, gy m ) s m i {qp b (gx i, gx i+1 )} s m n {(qp b (gx n, gx n+1 )} s m i {qp b (gx i, gx i+1 )} (4.4) s m i {qp b (gy i, gy i+1 )}. (4.5) ( 1 1 ) s s m i {qp b (gx i, gx i+1 ) + qp b (gy i, gy i+1 )} (4.6) s m i k i {qp b (gx 0, gx 1 ) + qp b (gy 0, gy 1 )}

7 Some coupled fixed point theorems on quasi-partial b-metric spaces 299 ( ) i k s m {qp b (gx 0, gx 1 ) + qp b (gy 0, gy 1 )} s ( ) i k s m {qp b (gx 0, gx 1 ) + qp b (gy 0, gy 1 )} (4.7) s On letting n in (4.7); holding m fixed, we get But Thus, lim [qp b(gx n, gx m ) + qp b (gy n, gy m )] 0. lim [qp b(gx n, gx m ) + qp b (gy n, gy m )] 0. lim qp b(gx n, gx m ) = lim qp b (gy n, gy m ) = 0. Now letting m + lim qp b(gx n, gx m ) = lim qp b(gy n, gy m ) = 0. (4.8) n,m n,m Similarly, we can show that lim qp b(gx m, gx n ) = lim qp b(gy m, gy n ) = 0. (4.9) n,m + n,m + Thus the sequences {gx n } and {gy n } are Cauchy in (gx, qp b ). Since (gx, qp b ) is complete, there are u and v in X such that gx n gu and gy n gv with respect to τ qpb, that is, and qp b (gu, gu) = qp b (gv, gv) = From (4.8) and (4.9), we have qp b (gu, gu) = lim qp b(gu, gx n ) = lim qp b(gx n, gu) n + n + = lim qp b(gx m, gx n ) = lim qp b(gx n, gx m ) n,m + n,m + lim qp b(gv, gy n ) = lim qp b(gy n, gv) n + n + = lim qp b(gy m, gy n ) = lim qp b(gy n, gy m ). n,m + n,m + lim qp b(gx n, gx m ) = 0 (4.10) n,m +

8 300 Anuradha Gupta and Pragati Gautam and qp b (gv, gv) = For n N, we get qp b (gx n+1, F (u, v)) lim qp b(gy n, gy m ) = 0. (4.11) n,m + s[qp b (gx n+1, gu) + qp b (gu, F (u, v))] qp b (gu, gu) s[qp b (gx n+1, gu) + qp b (gu, F (u, v))] s[qp b (gx n+1, gu) + s{qp b (gu, gx n+1 ) + qp b (gx n+1, F (u, v))} qp b (gx n+1, gx n+1 )] s[qp b (gx n+1, gu)] + s 2 [qp b (gu, gx n+1 )] + s 2 [qp b (gx n+1, F (u, v))]. On letting n + in the above inequalities and using (4.10) and (4.11), we have Also since, lim qp b(gx n+1, F (u, v)) s[qp b (gu, F (u, v)]. n + s[qp b (gx n+1, gu) + qp b (gu, F (u, v))] s[qp b (gx n+1, gu)] + s 2 [qp b (gu, gx n+1 )] + s 2 [qp b (qx n+1, F (u, v))]. On taking n +, we get or Similarly, qp b (gu, F (u, v)) s lim qp b (gx n+1, F (u, v)) 1 s qp b(gu, F (u, v)) lim qp b (gx n+1, F (u, v)). (4.12) 1 s qp b(gv, F (v, u)) lim qp b (gy n+1, F (v, u)). (4.13) We show that gu = F (u, v) and gv = F (v, u). For n N, we have qp b (gx n+1, F (u, v)) + qp b (gy n+1, F (v, u)) = qp b (F (x n, y n ), F (u, v)) + qp b (F (y n, x n ), F (v, u)) k 1 (qp b (gx n, gu) + qp b (gy n, gv)) + k 2 (qp b (gx n, F (x n, y n ) + qp b (gy n, F (y n, x n )) + k 3 (qp b (gu, F (u, v)) + qp b (gv, F (v, u)) = k 1 (qp b (gx n, gu) + qp b (gy n, gv)) + k 2 (qp b (gx n, gx n+1 ) + qp b (gy n, gy n+1 )) + k 3 (qp b (gu, F (u, v)) + qp b (gv, F (v, u))).

9 Some coupled fixed point theorems on quasi-partial b-metric spaces 301 On letting n in above inequalities and using (4.12) and (4.13), we have lim (qp b(gx n+1, F (u, v)) + qp b (gy n+1, F (v, u))) n + k 3 (qp b (gu, F (u, v)) + qp b (gv, F (v, u))) 1 s (qp b(gu, F (u, v)) + qp b (gv, F (v, u))) k 3 (qp b (gu, F (u, v)) + qp b (gv, F (v, u))) ( ) 1 {qp b (gu, F (u, v)) + qp b (gv, F (v, u))} s k 3 0. Since k 3 < 1 s, therefore {qp b(gu, F (u, v)) + qp b (gv, F (v, u))} 0. But {qp b (gu, F (u, v)) + qp b (gv, F (v, u))} 0 qp b (gu, F (u, v)) + qp b (gv, F (v, u)) = 0 qp b (gu, F (u, v)) = qp b (gv, F (v, u)) = 0. By Lemma 3.5, we get gu = F (u, v) and gv = F (v, u). Next we will show that gu = gv. Now from (4.1) we have qp b (gu, gv) + qp b (gv, gu) = qp b (F (u, v), F (v, u)) + qp b (F (v, u), F (u, v)) k 1 (qp b (gu, gv) + qp b (gv, gu)) + k 2 (qp b (gu, F (u, v)) + qp b (gv, F (v, u))) + k 3 (qp b (gv, F (v, u)) + qp b (gu, F (u, v))) = k 1 (qp b (gu, gv) + qp b (gv, gu)) + k 2 (qp b (gu, gu) + qp b (gv, gv)) + k 3 (qp b (gv, gv) + qp b (gu, gu)). Using (4.10) and (4.11), we get qp b (gu, gv) + qp b (gv, gu) k 1 (qp b (gu, gv) + qp b (gv, gu)). Since k 1 < 1, we have qp b (gu, gv) = qp b (gv, gu) = 0. By Lemma 3.5, we get that gu = gv. Finally, assume that g and F are w-compatible. Let u 1 = gu and v 1 = gv. Then and gu 1 = ggu = g(f (u, v)) = F (gu, gv) = F (u 1, v 1 ) (4.14) gv 1 = ggv = g(f (v, u)) = F (gv, gu) = F (v 1, u 1 ). (4.15)

10 302 Anuradha Gupta and Pragati Gautam From (4.14) and (4.15) we can show that qp b (gu 1, gu 1 ) = qp b (gv 1, gv 1 ). We claim that gu 1 = gu and gv 1 = gv. From (4.1), we have qp b (gu 1, gu) + qp b (gv 1, gv) = qp b (F (u 1, v 1 ), F (u, v)) + qp b (F (v 1, u 1 ), F (v, u)) k 1 (qp b (gu 1, gu) + qp b (gv 1, gv)) + k 2 (qp b (gu 1, F (u 1, v 1 )) + qp b (gv 1, F (v 1, u 1 ))) + k 3 (qp b (gu, F (u, v)) + qp b (gv, F (v, u))) = k 1 (qp b (gu 1, gu) + qp b (gv 1, gv)) + k 2 (qp b (gu 1, gu 1 ) + qp b (gv 1, gv 1 )) + k 3 (qp b (gu, gu)) + qp b (gv, gv)) = k 1 (qp b (gu 1, gu) + qp b (gv 1, gv)). (1 k 1 )(qp b (gu 1, gu) + qp b (gv 1, gv)) 0. Since k 1 < 1 we have Also, (qp b (gu 1, gu) + qp b (gv 1, gv)) 0. qp b (gu 1, gu) 0 and qp b (gv 1, gv) 0. qp b (gu 1, gu) + qp b (gv 1, gv) 0. Combining the two we conclude that qp b (gu 1, gu) + qp b (gv 1, gv) = 0. Hence qp b (gu 1, gu) = qp b (gv 1, gv) = 0. By Lemma 3.5, we get gu 1 = gu and gv 1 = gv. Therefore, u 1 = gu 1 and v 1 = gv 1. Again, since gu = gv, we get u 1 = v 1. Hence F and g have a unique common coupled fixed point of the form (u, u). Example 4.2. On the set X = [0, 1], define qp b : X X R +, qp b (x, y) = x y + x. Also, define F : X X X, F (x, y) = x + y and g : X X by g(x) = 10x. Then

11 Some coupled fixed point theorems on quasi-partial b-metric spaces 303 (1) (g(x), qp b ) is a complete quasi-partial b-metric space. (2) F (X X) g(x). (3) For any x, y, x, y X, we have qp b (F (x, y), F (x, y )) + qp b (F (y, x), F (y, x )) 1 4 (qp b(gx, gx ) + qp b (gy, gy )). To prove (1) we proceed by observing that qp b (x, y) = x y + x is a quasipartial b-metric with s = 1. Hence a quasi-partial metric. By Lemma 2.4, (g(x), qp b ) is complete if and only if (X, d qpb ) is complete. Here, and p qpb (x, y) = 1 2 [qp b(x, y) + qp b (y, x)] = x y + x + y 2 d pqpb (x, y) = 2p qpb (x, y) p qpb (x, x) p qpb (y, y) = 2 x y + x + y x y = 2 x y. Clearly (g(x), d pqpb ) is a complete metric space being a compact space. Now we prove (2). Let F (x, y) be an arbitrary element of F (X X). We need to show F (x, y) g(x) = {g(x) : x X} = {10x : x [0, 1]} = [0, 10]. Since x, y X = [0, 1], therefore 0 x + y 2, F (x, y) = x + y [0, 2] [0, 10] = g(x). Hence F (X X) g(x). Now we prove (3). Here we have qp b (F (x, y), F (x, y )) + qp b (F (y, x), F (y, x )) = qp b (x + y, x + y ) + qp b (y + x, y + x ) = x + y x y + x + y + y + x y x + y + x x x + y y + x + y + y y + x x + y + x = 2 x x + 2 y y + 2(x + y) = 2( x x + y y + (x + y)) 10 4 ( x x + y y + (x + y)) = 1 4 ( 10x 10x + 10y 10y + (10x + 10y))

12 304 Anuradha Gupta and Pragati Gautam = 1 4 ( 10x 10x + 10x + 10y 10y + 10y) = 1 4 (qp b(gx, gx ) + qp b (gy, gy )). Thus F and g satisfy all the hypotheses of Theorem 4.1. So F and g have a unique coupled coincidence point (u, v) = (0, 0) satisfying gu = F (u, v) = F (v, u) = gv. Corollary 4.3. Let (X, p) be a quasi-partial b-metric space, g : X X and F : X X X be two mappings. Suppose that there exist a, b, c, d, e, f in [0, 1) with a + b + c + d + e + f < 1 and e + f < 1 such that s qp b (F (x, y), F (x, y )) aqp b (gx, gx ) + bqp b (gy, gy ) + cqp b (gx, F (x, y)) + dqp b (gy, F (y, x)) + eqp b (gx, F (x, y )) + fqp b (gy, F (y, x )) (4.16) holds for all x, y, x, y X. Also, suppose the following hypotheses: (1) F (X X) g(x). (2) g(x) is a complete subspace of X with respect to the quasi-partial b-metric qp b. Then F and g have a coupled coincidence point (u, v) satisfying gu = F (u, v) = F (v, u) = gv. Moreover, if F and g are w-compatible, then F and g have a unique common fixed point of the form (u, u). Proof. Given x, y, x, y X. From (4.16), we have qp b (F (x, y), F (x, y )) and aqp b (gx, gx ) + bqp b (gy, gy ) + cqp b (gx, F (x, y)) + dqp b (fy, F (y, x)) + eqp b (gx, F (x, y )) + fqp b (gy, F (y, x )) (4.17) qp b (F (x, y), F (x, y )) aqp b (gy, gy ) + bqp b (gx, gx ) + cqp b (gy, F (y, x)) + dqp b (gx, F (y, x)) + eqp b (gy, F (y, x )) + fqp b (gx, F (x, y )) (4.18)

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