COMMON FIXED POINT THEOREMS VIA w-distance

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1 Bulleti of Mathematical Aalysis ad Applicatios ISSN: , URL: Volume 3 Issue 3, Pages COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA KUMAR MOHANTA Abstract. We prove some commo fixed poit theorems with the help of the otio of w-distace i a metric space. Our results will improve ad supplemet some results of [1] ad [6]. 1. Itroductio Metric fixed poit theory is playig a icreasig role i mathematics because of its wide rage of applicatios i applied mathematics ad scieces. Over the past two decades a cosiderable amout of research work for the developmet of fixed poit theory have executed by several authors. I 1996, Kada et.al.[1] itroduced the cocept of w-distace i a metric space ad studied some fixed poit theorems. The aim of this paper is to obtai some commo fixed poit results by usig the otio of w-distace i a metric space. Our results geeralize some results of [1] ad [6]. Now, we recall some basic defiitios which will be eeded i the sequel. Throughout this paper we deote by N the set of all positive itegers. Defiitio 1.1. [1] Let (X, d) be a metric space. The a fuctio p : X X [0, ) is called a w-distace o X if the followig coditios are satisfied: (i) p(x, z) p(x, y) + p(y, z) for ay x, y, z X; (ii) for ay x X, p(x,.) : X [0, ) is lower semicotiuous; (iii) for ay ϵ > 0, there exists δ > 0 such that p(z, x) δ ad p(z, y) δ imply d(x, y) ϵ. Example 1.1. [5] If X = { 1 : N} {0}. For each x, y X, d(x, y) = x + y if x y ad d(x, y) = 0 if x = y is a metric o X. Moreover, by defiig p(x, y) = y, p is a w-distace o (X, d) Mathematics Subject Classificatio. 54H25, 47H10. Key words ad phrases. w-distace, expasive mappig, commo fixed poit i a metric space. c 2011 Uiversiteti i Prishtiës, Prishtië, Kosovë. Submitted Jue 18, Published August 2,

2 COMMON FIXED POINT THEOREMS VIA w-distance 183 Defiitio 1.2. Let (X, d) be a metric space. A mappig T : X X is said to be expasive if there exists a real costat c > 1 satisfyig d(t (x), T (y)) cd(x, y) for all x, y X. 2. Mai Results Before presetig our results we recall the followig lemma due to O. Kada et. al.[1] that will play a crucial role i this sectio. Lemma 2.1. Let (X, d) be a metric space ad let p be a w-distace o X. Let (x ) ad (y ) be sequeces i X, let (α ) ad (β ) be sequeces i [0, ) covergig to 0, ad let x, y, z X. The the followig hold: (i) If p(x, y) α ad p(x, z) β for ay N, the y = z. I particular, if p(x, y) = 0 ad p(x, z) = 0, the y = z; (ii) if p(x, y ) α ad p(x, z) β for ay N, the (y ) coverges to z; (iii) if p(x, x m ) α for ay, m N with m >, the (x ) is a Cauchy sequece; (iv) if p(y, x ) α for ay N, the (x ) is a Cauchy sequece. Theorem 2.1. Let p be a w-distace o a complete metric space (X, d). Let T 1, T 2 be mappigs from X ito itself. Suppose that there exists r [0, 1) such that max { p(t 1 (x), T 2 T 1 (x)), p(t 2 (x), T 1 T 2 (x))} r mi { p(x, T 1 (x)), p(x, T 2 (x))} (2.1) for every x X ad that if {p(x, y) + mi {p(x, T 1 (x)), p(x, T 2 (x))} : x X} > 0 (2.2) for every y X with y is ot a commo fixed poit of T 1 ad T 2. The there exists z X such that z = T 1 (z) = T 2 (z). Moreover, if v = T 1 (v) = T 2 (v), the p(v, v) = 0. Proof. Let u 0 X be arbitrary ad defie a sequece (u ) by The if N is odd, we have u = T 1 (u 1 ), if is odd p(u, u +1 ) = p(t 1 (u 1 ), T 2 (u )) = T 2 (u 1 ), if is eve. = p(t 1 (u 1 ), T 2 T 1 (u 1 )) max {p(t 1 (u 1 ), T 2 T 1 (u 1 )), p(t 2 (u 1 ), T 1 T 2 (u 1 ))} r mi {p(u 1, T 1 (u 1 )), p(u 1, T 2 (u 1 ))}, by (2.1) r p(u 1, T 1 (u 1 )) = r p(u 1, u ).

3 184 SUSHANTA KUMAR MOHANTA If is eve, the by (2.1), we have p(u, u +1 ) = p(t 2 (u 1 ), T 1 (u )) = p(t 2 (u 1 ), T 1 T 2 (u 1 )) max {p(t 2 (u 1 ), T 1 T 2 (u 1 )), p(t 1 (u 1 ), T 2 T 1 (u 1 ))} r mi {p(u 1, T 2 (u 1 )), p(u 1, T 1 (u 1 ))} r p(u 1, T 2 (u 1 )) = r p(u 1, u ). Thus for ay positive iteger, it must be the case that By repeated applicatio of (2.3), we obtai So, if m >, the p(u, u +1 ) r p(u 1, u ). (2.3) p(u, u +1 ) r p(u 0, u 1 ). p(u, u m ) p(u, u +1 ) + p(u +1, u +2 ) + + p(u m 1, u m ) [ r + r r m 1] p(u 0, u 1 ) r 1 r p(u 0, u 1 ). By Lemma 2.1(iii), (u ) is a Cauchy sequece i X. Sice X is complete, (u ) coverges to some poit z X. Let N be fixed. The sice (u m ) coverges to z ad p(u,.) is lower semicotiuous, we have p(u, z) lim if p(u, u m ) r m 1 r p(u 0, u 1 ). Assume that z is ot a commo fixed poit of T 1 ad T 2. The by hypothesis 0 < if { p(x, z) + mi { p(x, T 1 (x)), p(x, T 2 (x))} : x X} if p(u, z) + mi { p(u, T 1 (u )), p(u, T 2 (u ))} : N} { } r if 1 r p(u 0, u 1 ) + p(u, u +1 ) : N { } r if 1 r p(u 0, u 1 ) + r p(u 0, u 1 ) : N = 0 which is a cotradictio. Therefore, z = T 1 (z) = T 2 (z). If v = T 1 (v) = T 2 (v) for some v X, the p(v, v) = max { p(t 1 (v), T 2 T 1 (v)), p(t 2 (v), T 1 T 2 (v))} r mi { p(v, T 1 (v)), p(v, T 2 (v))} = r mi { p(v, v), p(v, v)} = r p(v, v) which gives that, p(v, v) = 0.

4 COMMON FIXED POINT THEOREMS VIA w-distance 185 The followig Corollary is the result [1, T heorem 4]. Corollary 2.1. Let X be a complete metric space, let p be a w-distace o X ad let T be a mappig from X ito itself. Suppose that there exists r [0, 1) such that for every x X ad that p(t (x), T 2 (x)) r p(x, T (x)) if {p(x, y) + p(x, T (x)) : x X} > 0 for every y X with y T (y). The there exists z X such that z = T (z). Moreover, if v = T (v), the p(v, v) = 0. Proof. Takig T 1 = T 2 = T i Theorem 2.1, the coclusio of the Corollary follows. So Corollary 2.1 ca be treated as a special case of Theorem 2.1. We ow supplemet Theorem 2.1 by examiatio of coditios (2.1) ad (2.2) i respect of their idepedece. We furish Examples 2.1 ad 2.2 below to show that these two coditios are idepedet i the sese that Theorem 2.1 shall fall through by droppig oe i favour of the other. Example 2.1. Take X = { 0 } { 1 2 : 1 }, which is a complete metric space with usual metric d of reals. Defie T : X X by T (0) = 1 2 ad T ( ) 1 2 = for 1. Clearly, T has got o fixed poit i X. It is easy to check that d(t (x), T 2 (x)) 1 2 d(x, T (x)) for all x X. Thus coditio (2.1) holds for T 1 = T 2 = T. However, T (y) y for all y X ad so if {d(x, y) + d(x, T (x)) : x, y X with y T (y)} = if {d(x, y) + d(x, T (x)) : x, y X} = 0. Thus, coditio (2.2) is ot satisfied for T 1 = T 2 = T. We ote that Theorem 2.1 is ivalid without coditio (2.2). Example 2.2. Take X = [2, ) {0, 1}, which is a complete metric space with usual metric d of reals. Defie T : X X where T (x) = 0, for x (X \ {0}) = 1, for x = 0. Clearly, T possesses o fixed poit i X. Now, if {d(x, y) + d(x, T (x)) : x, y X with y T (y)} = if {d(x, y) + d(x, T (x)) : x, y X} > 0. Thus, coditio (2.2) is satisfied for T 1 = T 2 = T. But, for x = 0, we fid that d(t (x), T 2 (x)) = 1 > r d(x, T (x)) for ay r [0, 1). So, coditio (2.1) does ot hold for T 1 = T 2 = T. I this case we observe that Theorem 2.1 does ot work without coditio (2.1). Note :I examples above we treat d as a w-distace o X i referece to Theorem 2.1.

5 186 SUSHANTA KUMAR MOHANTA Theorem 2.2. Let p be a w-distace o a complete metric space (X, d). Let T 1, T 2 be mappigs from X oto itself. Suppose that there exists r > 1 such that mi { p(t 2 T 1 (x), T 1 (x)), p(t 1 T 2 (x), T 2 (x))} r max { p(t 1 (x), x), p(t 2 (x), x)} (2.4) for every x X ad that if {p(x, y) + mi { p(t 1 (x), x), p(t 2 (x), x)} : x X} > 0 (2.5) for every y X with y is ot a commo fixed poit of T 1 ad T 2. The there exists z X such that z = T 1 (z) = T 2 (z). Moreover, if v = T 1 (v) = T 2 (v), the p(v, v) = 0. Proof. Let u 0 X be arbitrary. Sice T 1 is oto, there is a elemet u 1 satisfyig u 1 T1 1 (u 0 ). Sice T 2 is also oto, there is a elemet u 2 satisfyig u 2 T2 1 (u 1 ). Proceedig i the same way, we ca fid u 2+1 T1 1 (u 2 ) ad u 2+2 T2 1 (u 2+1 ) for = 1, 2, 3,. Therefore, u 2 = T 1 (u 2+1 ) ad u 2+1 = T 2 (u 2+2 ) for = 0, 1, 2,. If = 2m, the usig (2.4) p(u 1, u ) = p(u 2m 1, u 2m ) = p(t 2 (u 2m ), T 1 (u 2m+1 )) = p(t 2 T 1 (u 2m+1 ), T 1 (u 2m+1 )) mi {p(t 2 T 1 (u 2m+1 ), T 1 (u 2m+1 )), p(t 1 T 2 (u 2m+1 ), T 2 (u 2m+1 ))} r max {p(t 1 (u 2m+1 ), u 2m+1 ), p(t 2 (u 2m+1 ), u 2m+1 )} r p(t 1 (u 2m+1 ), u 2m+1 ) = r p(u 2m, u 2m+1 ) = r p(u, u +1 ). If = 2m + 1, the by (2.4), we have p(u 1, u ) = p(u 2m, u 2m+1 ) = p(t 1 (u 2m+1 ), T 2 (u 2m+2 )) = p(t 1 T 2 (u 2m+2 ), T 2 (u 2m+2 )) mi {p(t 2 T 1 (u 2m+2 ), T 1 (u 2m+2 )), p(t 1 T 2 (u 2m+2 ), T 2 (u 2m+2 ))} r max {p(t 1 (u 2m+2 ), u 2m+2 ), p(t 2 (u 2m+2 ), u 2m+2 )} r p(t 2 (u 2m+2 ), u 2m+2 ) = r p(u 2m+1, u 2m+2 ) = r p(u, u +1 ). Thus for ay positive iteger, it must be the case that which implies that, p(u 1, u ) r p(u, u +1 ) p(u, u +1 ) 1 r p(u 1, u ) Let α = 1 r, the 0 < α < 1 sice r > 1. ( ) 1 p(u 0, u 1 ). (2.6) r

6 COMMON FIXED POINT THEOREMS VIA w-distance 187 Now, (2.6) becomes So, if m >, the p(u, u +1 ) α p(u 0, u 1 ). p(u, u m ) p(u, u +1 ) + p(u +1, u +2 ) + + p(u m 1, u m ) [ α + α α m 1] p(u 0, u 1 ) α 1 α p(u 0, u 1 ). By Lemma 2.1(iii), (u ) is a Cauchy sequece i X. Sice X is complete, (u ) coverges to some poit z X. Let N be fixed. The sice (u m ) coverges to z ad p(u,.) is lower semicotiuous, we have p(u, z) lim if p(u, u m ) α m 1 α p(u 0, u 1 ). Assume that z is ot a commo fixed poit of T 1 ad T 2. The by hypothesis 0 < if { p(x, z) + mi { p(t 1 (x), x), p(t 2 (x), x)} : x X} if p(u, z) + mi { p(t 1 (u ), u ), p(t 2 (u ), u )} : N} { } α if 1 α p(u 0, u 1 ) + p(u 1, u ) : N { } α if 1 α p(u 0, u 1 ) + α 1 p(u 0, u 1 ) : N = 0 which is a cotradictio. Therefore, z = T 1 (z) = T 2 (z). If v = T 1 (v) = T 2 (v) for some v X, the p(v, v) = mi { p(t 2 T 1 (v), T 1 (v)), p(t 1 T 2 (v), T 2 (v))} r max { p(t 1 (v), v), p(t 2 (v), v)} = r max { p(v, v), p(v, v)} = r p(v, v) which gives that, p(v, v) = 0. Corollary 2.2. Let p be a w-distace o a complete metric space (X, d) ad let T : X X be a oto mappig. Suppose that there exists r > 1 such that for every x X ad that p(t 2 (x), T (x)) rp(t (x), x) (2.7) if{p(x, y) + p(t (x), x) : x X} > 0 (2.8) for every y X with y T (y). The T has a fixed poit i X. Moreover, if v = T (v), the p(v, v) = 0. Proof. Takig T 1 = T 2 = T i Theorem 2.2, we have the desired result.

7 188 SUSHANTA KUMAR MOHANTA The followig Corollary is the result [6, T heorem 4]. Corollary 2.3. Let (X, d) be a complete metric space ad T be a mappig of X ito itself. If there is a real umber r with r > 1 satisfyig d(t 2 (x), T (x)) rd(t (x), x) for each x X, ad T is oto cotiuous, the T has a fixed poit. Proof. We treat d as a w-distace o X. The d satisfies coditio (2.7) of Corollary 2.2. Assume that there exists y X with y T (y) ad if{d(x, y) + d(t (x), x) : x X} = 0. The there exists a sequece (x ) such that lim {d(x, y) + d(t (x ), x )} = 0. So, we have d(x, y) 0 ad d(t (x ), x ) 0 as. Now, d(t (x ), y) d(t (x ), x ) + d(x, y) 0 as. Sice T is cotiuous, we have ( T (y) = T lim x This is a cotradictio. Hece if y T (y), the ) = lim T (x ) = y. if{d(x, y) + d(t (x), x) : x X} > 0, which is coditio (2.8) of Corollary 2.2. By Corollary 2.2, there exists z X such that z = T (z). As a applicatio of Corollary 2.2, we have the followig result [6, T heorem 3]. Corollary 2.4. Let (X, d) be a complete metric space ad T be a mappig of X ito itself. If there is a real umber r with r > 1 satisfyig d(t (x), T (y)) r mi{d(x, T (x)), d(t (y), y), d(x, y)} (2.9) for ay x, y X, ad T is oto cotiuous, the T has a fixed poit. Proof. We treat d as a w-distace o X. Replacig y by T (x) i (2.9), we obtai d(t (x), T 2 (x)) r mi{d(x, T (x)), d(t 2 (x), T (x)), d(x, T (x))} (2.10) for all x X. Without loss of geerality, we may assume that T (x) T 2 (x). For, otherwise, T has a fixed poit. Sice r > 1, it follows from (2.10) that d(t 2 (x), T (x)) rd(t (x), x) for every x X. By the argumet similar to that used i Corollary 2.3, we ca prove that, if y T (y), the if{d(x, y) + d(t (x), x) : x X} > 0, which is coditio (2.8) of Corollary 2.2. So, Corollary 2.2 applies to obtai a fixed poit of T.

8 COMMON FIXED POINT THEOREMS VIA w-distance 189 Remark 2.1. The class of mappigs satisfyig coditio (2.9) is strictly larger tha the class of all expasive mappigs. For, if T : X X is expasive, the there exists r > 1 such that d(t (x), T (y)) r d(x, y) r mi {d(x, T (x)), d(t (y), y), d(x, y)} for all x, y X. O the otherhad, the idetity mappig satisfies coditio (2.9) but it is ot expasive. Refereces [1] Osamu Kada, Tomoari Suzuki ad Wataru Takahashi, Nocovex miimizatio theorems ad fixed poit theorems i complete metric spaces, Math. Japoica 44(2), 1996, [2] H. Lakzia ad F. Arabyai, Some fixed poit theorems i coe metric spaces with w-distace, It. Joural of Math. Aalysis, 3(22), 2009, [3] J. Caristi, Fixed poit theorems for mappigs satisfyig iwardess coditios, Tras. Amer. Math. Soc. 215(1976), [4] Sushata Kumar Mohata, A fixed poit theorem via geeralized w-distace, Bulleti of Mathematical Aalysis ad Applicatios, 3(2), 2011, [5] Sushata Kumar Mohata, Geeralized w-distace ad a fixed poit theorem, It. J. Cotemp. Math. Scieces, 6(18), 2011, [6] Shag Zhi Wag, Bo Yu Li, Zhi Mi Gao ad Kiyoshi Iseki, Some fixed poit theorems o expasio mappigs, Math. Japoica, 29(4), 1984, Departmet of Mathematics, West Begal State Uiversity, Barasat, 24 Pargaas (North), West Begal, Kolkata , Idia. address: smwbes@yahoo.i