COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES

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1 Iteratioal Joural of Egieerig Cotemporary Mathematics ad Scieces Vol. No. 1 (Jauary-Jue 016) ISSN: COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES N. CHANDRA M. C. ARYA AND MAHESH C. JOSHI Abstract. Recetly Suzuki i 008 obtaied a geeralizatio of Baach cotractio priciple. Subsequetly a umber of ew fixed/commo fixed poit theorems for mappigs i metric spaces/partial metric spaces have bee established by may authors. I this paper we obtai a commo fixed poit theorem for multivalued maps i partial metric spaces which geeralizes some well kow results ad also exteds may results i the settigs of partial metric. 1. Itroductio I 1994 Mathews[1] itroduced the otio of partial metric spaces(pms) as a part of deotatioal sematics of data for etworks ad proved the Baach cotractio priciple i partial metric cotext for the applicatios i program verificatio. Defiitio 1.1. [1] A partial metric o a o empty set X is a fuctio p : X X R + such that for all x y z X: (a) x = y p(x x) = p(x y) = p(y y). (b) p(x x) p(x y). (c) p(x y) = p(y x). (d) p(x y) p(x z) + p(z y) p(z z). The the pair (X p) is said to be partial metric space. If p(x y) = 0 the (a) ad (b) imply that x = y. But coverse is ot true i geeral. A obvious example of the partial metric space is (R + p) where partial metric p is defied as p(x y) = max x y. If p is a partial metric o X the the mappig p s : X X R + defied by p s (x y) = p(x y) p(x x) p(y y) is a metric o X. Let (X p) be a partial metric space. The a sequece x i X is called: (i) Coverget to a poit x X iff lim + p(x x) = p(x x); (ii) Cauchy sequece wheever lim m + p(x m x ) exists ad fiite. A partial metric space (X p) is said to be complete if every Cauchy sequece x i X is coverget with respect to τ p. Furthermore lim p(x m x ) = lim p(x x ) = p(x x). m Mathematics Subject Classificatio. 47H10 54H5. Key words ad phrases. Partial metric space Multivalued map Commo fixed poit. 1 1

2 N. CHANDRA M. C. ARYA AND MAHESH C. JOSHI Lemma 1.. [1] Let (X p) be a partial metric space the (i) A sequece x i X is a Cauchy sequece i (X p) iff it is Cauchy sequece i metric space (x p s ). (ii) A partial metric space (X p) is complete if ad oly if the metric space (X p s ) is complete. After the itroductio of above cocept by Mathews[1] various geeralizatios of cotractio priciple are obtaied by researchers i partial metric spaces(see for istace [] [3] [6] ad refereces therei). Most of the cotractive coditios used i provig existece of fixed poits i partial metric spaces (PMS) are extesios of well kow cotractive coditios used to ivestigate the existece of fixed poits for maps i metric spaces (see [16]). Nadler [13] was the first who exteded the Baach cotractio cocept for multivalued mappigs ad proved the remarkable result for multivalued cotractios i metric spaces. Afterwards there appears may geeralizatios of Nadler s result (see for istace [5] [8] [9] [10] [18] ad refereces therei). Recetly Ayadi et al.[4] itroduced the partial Hausdorff metric showig that the Nadler s fixed poit theorem ca be geeralized to the partial metric spaces also. Let (X p) be a partial metric space ad CB p (X)(resp. CL p (X)) be the collectio of o-empty closed ad bouded (resp. closed) subsets of Xrespectively. The Hausdorff (resp. Geeralized Hausdorff) metric H p is defied by H p (A B) = max sup x A p(x B) sup p(y A) y B for every A B CB p (X)(resp. CL p (X)) where p(x A) = if y A p(x y). For a o-empty subset A of a partial metric space (X p) a A if ad oly if p(a A) = p(a a) where A deotes the closure of A with respect to partial metric p. Note that A is closed i (X p) if ad oly if A = A. Throughout this paper for x y X we follow the followig otatios where f g S ad T are mappigs to be defied specifically i a particular cotext: M(Sx T y) = max p(x y) M 1 (Sx T y) = max p(fx gy) p(x Sx) + p(y T y) p(fx Sx) + p(gy T y) p(x T y) + p(y Sx) p(fx T y) + p(gy Sx) Suzuki[19] i 008 itroduced a ew type of mappig ad proved a good geeralizatio of Baach cotractio priciple. The may results appeared i the literature o Suzuki type cotractio coditios for the existece of fixed poits for siglevalued as well as multivalued mappigs i metric spaces (see [5] [7] [11] [17] ad refereces therei). Further these results have bee exteded i the settig of partial metric spaces by may authors (see [1][] [14] [15] ad refereces therei). Recetly Rao et al.[15] itroduced the ew coditio (W.C.C.) ad obtaied the Suzuki type fixed poit theorems for a geeralized multivalued mappigs o partial Hausdorff metric spaces.

3 CFPTMMPMS 3 Defiitio 1.3. [15] Let (X p) be a partial metric space with f g : X X ad S : X CB p (X). The the triplet (f g; S) is said to satisfy coditio (W.C.C.) if p(fx gy) p(y Sx) for all x y X. Theorem 1.4. [15] Let (X p) be a complete partial metric space ad S T : X CB p (X) ad f g : X X. Assume that there exists 0 r < 1 such that for every x y X φ(r) mip(fx Sx) p(gy T y) p(fx gy) H p (Sx T y) r M 1 (Sx T y) (1.1) where x X Sx g(x) ad x X T x f(x) ad φ : [0 1) (0 1] defied as 1 if 0 r < 1 φ(r) = (1 r) if 1 r < 1. (1.) If the triplet (f g; S) or the triplet (f g; T ) satisfy the coditio(w.c.c.) the f g S ad T have a uique commo fixed poit i X. Here we remark that a commo fixed poit for the mappigs exists eve if the coditios of the above theorem are ot satisfied. To prove our claim we give followig couter example where the coditios (1.1) ad (W.C.C) of the above theorem are ot satisfied but there is a commo fixed poit for the maps. Example 1.5. Let X = 0 1 be edowed with the partial metric p : X X R + defied by p(0 0) = p(1 1) = 0 p( ) = 1 3 p(0 1) = p(1 0) = 1 4 p(0 ) = p( 0) = 5 p(1 ) = p( 1) = Defie the mappigs f g : X X as idetity maps ad S T : X CB p (X) by 0 if x if x 0 1 Sx = ad T x = 0 1 if x = 1 if x =. I this example if we take x = 1 y = 1 the p(fx gy) = 0 H p (Sx T y) = 0 mip(fx T x) p(gy T y) = 1 4 ad M(Sx T y) = 1 4. It is clear that φ(r)mip(fx T x) p(gy T y) > p(fx gy) but H p (Sx T y) r M(Sx T y). Ad also if we take x = 1 y = 0 the p(fx gy) = 1 4 p(y Sx) = 0 ad p(y T x) = 0. It meas that coditio (W.C.C.) is also ot satisfied but 0 is the commo fixed poit of S ad T. Now we give our mai result which is more geeral for the existece of commo fixed poits of mappigs i partial metric spaces. We use the followig lemma essetially due to Nadler [13] i case of metric spaces ad also true for the partial metric cotext as i [4]. 3

4 4 N. CHANDRA M. C. ARYA AND MAHESH C. JOSHI Lemma 1.6. Let A B CL p (X) ad a A the for ay ɛ > 0 there exists a poit b B such that p(a b) H p (A B) + ɛ.. Mai Results Theorem.1. Let X be a complete partial metric space ad let S ad T be maps from X to CL p (X). If there exists r [0 1) such that for all x y X mip(x Sx) p(y T y) (1 + r)p(x y) implies H p (Sx T y) rm(sx T y). (.1) The there exists a elemet z X such that z Sz T z. Proof. Here we take M(Sx T y) > 0. Otherwise if M(Sx T y) = 0 the x = y is a commo fixed poit of S ad T. Let β = r + ɛ where ɛ > 0. Let x 0 X ad x 1 T x 0 by lemma 1.6 there exists x Sx 1 such that p(x x 1 ) H p (Sx 1 T x 0 ) + M(Sx 1 T x 0 ). Similarly there exists x 3 T x such that p(x 3 x ) H p (T x Sx 1 ) + ɛm(t x Sx 1 ). Cotiuig i this maer we fid a sequece x i X such that x +1 T x ad x + Sx +1 ad p(x +1 x ) H p (T x Sx 1 ) + ɛm(t x Sx 1 ) p(x + x +1 ) H p (Sx +1 T x ) + M(Sx +1 T x ). Now we show that for ay N p(x +1 x ) β p(x 1 x ). (.) Suppose p(x 1 Sx 1 ) p(x T x ) the mip(x 1 Sx 1 ) p(x T x ) p(x 1 Sx 1 ) (1 + r)p(x 1 x ). 4

5 CFPTMMPMS 5 By (.1) we get p(x +1 x ) H p (T x Sx 1 ) rm(sx 1 T x ) rm(sx 1 T x ) + ɛ M(Sx 1 T x ) = β M(Sx 1 T x ) = β max p(x 1 x ) p(x 1 Sx 1 ) + p(x T x ) p(x 1 T x ) + p(x Sx 1 ) β max p(x 1 x ) p(x 1 x ) + p(x x +1 ) p(x 1 x +1 ) + p(x x ) β maxp(x 1 x ) p(x x +1 ) β p(x 1 x ) which proves (.). If p(x T x ) p(x 1 Sx 1 ) the mip(x 1 Sx 1 ) p(x T x ) = p(x 1 Sx 1 ) Now from (.1) we have p(x +1 x ) H p (T x Sx 1 ) p(x 1 x ) (1 + r)p(x 1 x ). M(Sx 1 T x ) + ɛm(sx 1 T x ) = β M(Sx 1 T x ) = β max p(x 1 x ) p(x 1 Sx 1 ) + p(x T x ) p(x 1 T x ) + p(x T x 1 ) β max p(x 1 x ) p(x 1 x ) + p(x x +1 ) p(x 1 x +1 ) + p(x x ) β maxp(x 1 x ) p(x x +1 ) β p(x 1 x ). This yields (.). I a aalogous maer we ca show that p(x + x +1 ) βp(x +1 x ). (.3) Now we coclude from (.) ad (.3) that for ay N p(x x +1 ) β p(x x 1 ) p(x x +1 ) β p(x 1 x 0 ). 5

6 6 N. CHANDRA M. C. ARYA AND MAHESH C. JOSHI Thus for all m N with m > we get p(x m x ) p(x x +1 ) + p(x +1 x + ) p(x m 1 x m ) β p(x 1 x 0 ) + β +1 p(x 1 x 0 ) β m+ 1 p(x 1 x 0 ) ( ) ( ) 1 β β m β p(x 1 x 0 ) p(x 1 x 0 ). 1 β 1 β It implies lim m + p(x m x ) = 0 ad so x is a Cauchy sequece. Sice X is complete x coverges to some poit z X i.e. lim + p(x z) = p(z z). Furthermore lim p(x m x ) = lim p(x z) = p(z z) = 0. m + + Sice x z there exists 0 N such that As i ([14] 913) Therefore p(x z) 1 3 p(z y) for y z ad all 0. (1 + r) 1 p(x 1 Sx 1 ) p(x 1 Sx 1 ) p(x 1 x ) p(x 1 z) + p(z x ) 3 p(y z) = p(y z) 1 p(y z) 3 p(y z) p(x 1 z) p(x 1 y). p(x 1 Sx +1 ) (1 + r)p(x 1 y). (.4) Now either p(x 1 Sx 1 ) p(y T y) or p(y T y) p(x 1 Sx 1 ). I either case by (.4) ad (.1) we have p(x T y) H p (Sx 1 T y) rm(sx 1 T y) r max p(x 1 y) p(x 1 Sx 1 ) + p(y T y) p(x 1 T y) + p(y Sx 1 ) Makig we get p(z z) + p(y T y) p(z T y) + p(y z) p(z T y) r max p(z y) p(z T y) + p(y z) r max p(y z) It is clear from (.5) that Now we show that H p (Sz T z) r max p(z T y) r max p(y z) p(z y) p(z T y) + p(y z). (.5) p(z T y) r p(z y). (.6) p(z Sz) + p(y T y) p(z T y) + p(y Sz). (.7). 6

7 CFPTMMPMS 7 Suppose that y z the for every N there exists z T y such that So by (.6) we obtai Hece p(z z ) p(z T y) + 1 p(y z). p(z T y) p(y z ) p(y z) + p(z z ) p(y z) + p(z T y) + 1 p(y z) p(y z) + r p(y z) + 1 p(y z) = (1 + r + 1 )p(y z). p(z T y) (1 + r) p(y z). (.8) Now either p(z Sz) p(y T y) or p(y T y) p(z Sz). So i either case by (.8) ad the assumptio we have H p (Sz T y) r M(Sz T y) which is (.7). Now takig y = x i (.7) we get P (Sz x +1 ) H p (Sz T x ) Takig the limit as we have p(sz z) r p(sz z) p(sz z) = 0 = p(z z) z Sz = Sz. r max r max p(zsz)+p(xt x) p(zt x)+p(xsz) p(z x ) p(z x ) p(zsz)+p(xx+1) p(zx+1)+p(xsz) With similar argumets we ca show that z T z. Hece z Sz T z. Now we show that the Example 1.5 satisfies the coditios (.1) of the Theorem.1 with r = 10 1 for all x y X. Note that Sx ad T x are closed for all x X uder the give partial metric p. (i) If x = y = 0 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 0 ad M(Sx T y) = 0. (ii) If x = 0 y = 1 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 0 ad M(Sx T y) = 1 4. (iii) If x = 0 y = the H p (Sx T y) = 1 4 mip(x Sx) p(y T y) = 0 ad M(Sx T y) = (iv) If x = 1 y = 0 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 0 ad M(Sx T y) = 1 4. (v) If x = 1 y = 1 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 1 4 ad M(Sx T y) =

8 8 N. CHANDRA M. C. ARYA AND MAHESH C. JOSHI (vi) If x = 1 y = the H p (Sx T y) = 1 4 mip(x Sx) p(y T y) = 1 4 ad M(Sx T y) = (vii) If x = y = 0 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 0 ad M(Sx T y) = 5. (viii) If x = y = 1 the H p (Sx T y) = 0 mip(x Sx) p(y T y) = 1 4 ad M(Sx T y) = (ix) If x = y = the H p (Sx T y) = 1 4 mip(x Sx) p(y T y) = 5 ad M(Sx T y) = Thus for all x y X with r = 10 1 we get mip(x T x) p(y T y) (1 + r)p(x y) implies H p (Sx T y) rm(sx T y). Evidetly 0 S0 T 0. Here we remark that our result i.e. Theorem.1 is also geeralizatio of the result of R. Kamal et al. ([11] Theorem.) i partial metric cotext. Now if we take S ad T as sigle valued mappigs of X we get followig result which is geeralizatio of ([14] Theorem ) ad extesio of ([11] Corollory.3). Theorem.. Let X be a complete partial metric space ad S T : X X. Assume there exists r [0 1) such that for every x y X mi p(x Sx) p(y T y) (1 + r)p(x y) implies d(sx T y) r M(Sx T y). The S ad T have a uique commo fixed poit. Proof. It ca be proved easily by takig S ad T as sigle valued maps i Theorem.1. Uiqueess of the commo fixed poit is obvious. Takig S = T i Theorem.1 we get followig Corollaries which are geeralizatios of results of [18] i the settigs of partial metric. Corollary.3. Let X be a complete partial metric space ad T : X CL(X). Assume there exists r [0 1) such that for every x y X p(x T x) (1 + r)p(x y) implies H p (T x T y) r M(T x T y). The there exists z X such that z T z. Corollary.4. Let X be a complete partial metric space ad T : X X. Assume there exists r [0 1) such that for every x y X p(x T x) (1 + r)p(x y) implies p(t x T y) r M(T x T y). The T has a uique fixed poit. Refereces 1. M. Abbas B. Ali C. Vetro: A Suzuki type fixed poit theorem for a geeralized multivalued mappig o partial metric spaces Topol. Appl. 160(013) I. Altu F. Sola H. Simsek: Geeralized cotractio o partial metric spaces Topol. Appl. 157 (010) H. Aydi M. Abbas C. Vetro: Commo fixed poits for multivalued geeralized cotractio o partial metric spaces Revista de la Real Academia de Ciecias exacts Fisicas y Naturales A (013)

9 CFPTMMPMS 9 4. H. Aydi M. Abbas C. Vetro: Partial Hausdorff metric ad Nadler s fixed poit theorem o partial metric spacestopol. Appl. 159 (01) R. K. Bose: Some Suzuki type fixed poit theorems for multivalued mappigs ad applicatios It. J. Pure ad Appl. Math. 9(4)(014) L. Ciric B. Samet H. Aydi C. Vetro: Commo fixed poits of geeralized cotractio o partial metric spaces ad a applicatio Appl. Math. Comp. 18 (011) D. Doric ad R. Lazovic: Some Suzuki-type fixed poit theorem for geeralized multivalued mappigs ad applicatios Fixed Poit Theory ad Appl :40 doi: L. S. Dube S. P. Sigh: O multivalued cotractio mappigs Bull. Math. Soc. Set. Math. R. S. Roumaie 14 (1979) T. Hu: fixed poit theorems for multivalued mappigs Caad. Math. bull. 3(1980) S. Itoh W. Takahashi: Sigle valued mappigs Multivalued mappigs ad fixed poit theorems J. Math. Aal. Appl. 59 (1977) R. Kamal R. Chugh S. L. Sigh ad S. N. Mishra: New commo fixed poit theorems for multivalued maps Appl. Ge. Topol. 15()(014) S. G. Matthews Partial metric topology: Proceedigs of the 8th Summer Coferece o Geeral Topology ad Applicatios i: A. New York Acad. Sci. 78 (1994) S. B. Nadler: Multivalued cotractio mappigs Pacific J. Math. 30 (1969) D. Paesao P. Vetro: Suzuki s type characterizatio of completeess for partial metric spaces ad fixed poits for partially ordered metric spaces Topology Appl. 159(3)(01) K. P. R. Rao K. R. K. Rao M. Imdad: A Suzuki type uique commo fixed poit theorem for two pairs of hybrid maps uder a ew coditio i partial metric spaces Demostratio Mathematica 49(016) B. E. Rhoades: A compariso of various defiitios of cotractive mappigs Tras. Amer. Math. Soc. 6 (1977) S. L. Sigh S. N. Mishra R. Chugh ad R. Kamal: Geeral Commo fixed poit theorms ad applicatios J. Applied Mathematics Vol. 01 Article ID S. L. Sigh ad S. N. Mishra: Coicidece theorems for certai classes of hybrid cotractios Fixed Poit Theory Appl. 010(010) Article ID pp. 19. T. Suzuki: A geeralized Baach cotractio priciple that characterizes metric completeess Proc. Amer. Math. Soc. 136(008) Departmet of Mathematics D. S. B. Campus Kumau Uiversity Naiital Idia address: cavee39@gmail.commcarya1986@gmail.com ad mcjoshi69@gmail.com 9