Generalization of Contraction Principle on G-Metric Spaces

Transcription

1 Global Joural of Pure ad Applied Mathematics. ISSN Volume 14, Number ), pp Research Idia Publicatios Geeralizatio of Cotractio Priciple o G-Metric Spaces G Sudhaamsh Moha Reddy Departmet of Mathematics, Faculty of Sciece ad Techology ICFAI Foudatio for Higher Educatio, Hyderabad , Idia Abstract I this paper, we prove certai fixed poit theorems of G -metric spaces usig the geeralized cotractio priciple o G -metric spaces. Key words: G-metric space, Cauchy sequece. Mathematics Subject Classificatio: 54H25, 47H10 1. INTRODUCTION AND PRELIMINARIES: Metric fixed poit theory is a importat Mathematical disciplie because of its applicatios i areas such as variatio ad liear iequalities, optimizatio ad approximatio theor etc..,. The geeralizatio of metric spaces were proposed by Gahler [5] called 2-metric spaces) ad Dhage [2, 3, 4] called D-metric spaces). Hsiao [7] showed that every cotractive defiitio, with x T x0, every orbit is liearly depedet, thus givig fixed poit theorem i such spaces. However HA et. al. [6] have poited out that the results obtaied by Gahler for his 2-metric spaces are idepedet, rather tha the geeralizatios of correspodig results i metric spaces. While Mustafa ad Sims [8] have proved that the Dhage s otio of D-metric space is fudametally icorrect ad most of the results claimed by Dhage ad others are ivalid. Mustafa ad Sims [8] i 2003 have itroduced a more appropriate ad robust otio of geeralized metric spaces as follows: 1.1 Defiitio: See [8]) Let X be a o empty set ad let G : X X X [0, ) be a fuctio satisfyig the followig axioms:

2 1160 G Sudhaamsh Moha Reddy G1) G 0 if x y z. G2) 0 G y), for all y X with x y G3) G y) G for all z X with y z G4) G G ) for all z X, where is a permutatio of the set x, z Symmetry i all three variables) G5) G G a, a) G a, for all z, a X Rectagular iequality) The the fuctio G is called a geeralized metric or more specifically a G - metric o X. The pair X, G) is called a G -metric space. 1.2 Defiitio: Let, G) x be a sequece of poits of X, we say that x is G coverget to x if for every give 0, there exist X be a G -metric space ad let N N set of all atural umbers) such that G x, x, x ) for all m, N. We deote it as 1.3 Defiitio: Lim G x, x ) 0 m, m m Let, G) x i X is called G cauchy if for every give 0, there exists N N such that, X be a metric space, a sequece G x, x, x ) for all, m, l N, that is if, m l 1.4 Defiitio: Lim G x, x, x ) 0, m, l m l A G-metric space X, G) is said to be G -complete or a complete G -metric space) if every G -Cauchy sequece i X, G) is G -coverget to some poit i X, G). 1.5 Defiitio: A G -metric space X, G) is said to be symmetric if G y) G x) for all y X 1.6 Defiitio: A fuctio :[0, ) [0, ) is called a Alterig distace fuctio, if the followig properties are satisfied. i) 0) 0 ii) is cotiuous ad mootoically o-decreasig.

3 Geeralizatio of Cotractio Priciple o G-Metric Spaces Defiitio: Let X, G) be a G -metric space ad let T : X X be a mappig. T is called a cotractio of X if 1.7.1) G T T T k G z ) for all z X 1.18 Defiitio: A mappigt : X X, where X, G) is a G -metric space, is said to be weakly cotractive if G T T T G G ) for all z X, where :[0, ) [0, ) is a cotiuous ad o decreasig fuctio such that t) 0 if ad oly if t Defiitio: Let T be a self map of a complete G -metric space X, G) with o empty fixed poit set F T ) set of all fixed poits of T ). The we say that T satisfies property P if F T) F T ) for all N. 2. MAIN THEOREM Very recetly i 2008 P.N. Dutta et al [4] have obtaied fixed poit theorem of metric spaces usig the cocept of geeralizatio of cotractio priciple. Here we state the theorem proved by Dutta et al [4] Theorem: Let X, G) be a complete matrix space ad T : X X be a self mappig satisfyig the iequality ) d T Ty)) d y)) d y)), for all y X,where, : 0, 0, are both cotiuous ad mootoe o decreasig fuctios with t) 0 t) if ad oly if t 0. The T has uique fixed poit. I this paper, we have establish ad geeralized the above theorem for G -metric spaces Theorem: Let X, G) be a complete G -metric space ad Let T : X X be a self mappig satisfyig the iequality ) G T T T ) G ) G ) for all z X where, : 0, 0, are both cotiuous ad mootoe o decreasig fuctios with t) 0 t) if ad oly if t 0. The T has uique fixed poit.

4 1162 G Sudhaamsh Moha Reddy Proof: Let x X 0. We costruct the sequece x by x Tx 1,2,3,... choosig x x 1, y x, z x i obtai.), we obtai, 2.2.2) G x, x, x )) = G Tx, Tx, Tx )) Which implies G x, x, x ) ) - G x, x, x ) 1 1 G x, x, x )) G x, x, x ) ) sice G x, x, x ) 0. 1 Now, usig the mootoe property of, we get ) G x, x, x ) x, x, x ) G 1 1 This shows that the sequece { G x, x 1, x 1) } is mootoe decreasig ad bouded below by 0 i the complete G -metric space X, G). Hece there exist r 0 such that G x, x, x ) r as Now, Lettig r 0 Hece, 2.2.4) G x, x, x ) 0 as. i 2.2.2), we get r) r) r) it holds oly whe Now, we prove that x is ot a Cauchy sequece the there exist some 0 for which we ca fid the sub sequeces x m k, x of x with k such that 2.2.5) G x, x, x ) Further, correspodig to m k we ca choose smallest iteger satisfyig 2.2.5). The, 2.2.6) G x, x, x ), ow, we have G x, x, x ) G x, x, x ) + x, x, x ) Takig < + x, x, x ) G 1 G 1 k o both the sides ad usig 2.2.4), we have 2.2.7) Lt G x, x, x ) = k for k i such a way that it is the

5 Geeralizatio of Cotractio Priciple o G-Metric Spaces 1163 Agai, 2.2.8) G x, x, x ) G x, x, x ) + G x, x, x ) m k + x, x, x ) G 1 m k G x, x, x ) G x, x, x ) + G x, x, x ) Lettig We get, 1 m k + G x, x, x ) k i the above two iequalities ad usig 2.2.4) ad 2.2.7) ) Lt G x, x, x ) = k Choosig x xm, y x, k z x k i 2.2.2) ad usig 2.2.5) We obtai Takig G x, x, x )) G x, x, x )) - G x, x, x )) k o both the sides. ) ) ) Which is a cotradictio if 0 Therefore, 0. This shows that x is a Cauchy sequece i complete G -metric space X, G) ad hece is coverget to some u X. That is, ) x coverget to x say) as. Now we claim that u is a fixed poit of T Cosider x x 1, y z u i 2.2.1). We obtai ) G x, T Tu)) G x 1, u)) - G x 1, u)) Lettig ad usig ), we get G T Tu)) G u)) - G u)) This is G T Tu)) 0) - 0) = 0

6 1164 G Sudhaamsh Moha Reddy Hece G T Tu) = 0 which gives Tu u. To prove the uiquecess of the fixed poit, Let us suppose that u1, are two fixed poits of T. That is T u1) u1, T ) Takig x u1, y, z i 2.2.1). This is This gives, G Tu1, T, T )) G u1,, )) - G u, u, )) G u, u, )) G u, u, )) - G u, u, )) G u, u, )) 0, it holds oly whe G u, u, ) = 0 That is u1. Showig T has uique fixed poit Corollary: Let X, G) be a complete G -metric space, T : X X be a self mappig which satisfyig the followig iequality ) G T T T) k G ) for all z X where 0 k : 0, 0, is a cotiuous ad mootoe o decreasig fuctios with t) 0 if ad oly if t 0. The T has uique fixed poit. Proof of the corollary follows by takig t) 1 k) t) i theorem Corollary: Let T : X X be a weakly cotractive mappig of a complete G -metric space X, G), the T has uique fixed poit. Proof: Give T is weakly cotractive mappig that is G T T T G G ) for all z X where : 0, 0, is a cotiuous ad o decreasig fuctios. Takig t) t i theorem 2.3 corollary follows.

7 Geeralizatio of Cotractio Priciple o G-Metric Spaces 1165 REFERENCES [1] Circic. Lj. B,. "A geeralizatio of Baach's cotractio priciple," Proceedigs of the America Mathematical Societ Vol. 45, pp , [2] Dhage, B.C., "Geeralized metric space ad mappig with fixed poits", Bulleti of the Calcutta Mathematical Societ Vol. 84, pp , [3] Dhage. B.C., "Geeralized metric spaces ad topological structure I", Aalete Stiitifice ale Uiversitatii." Al. I. Cuza" dia Iasi. Serie Nova, Mathametical, Vol. 46, o. 1, pp. 3 24, [4] Dutta. P.N. ad Choudhaur Biayak S, "A geeralizatio of cotractio priciple i metric spaces," Joural of Fixed Poit Theory ad Applicatios, Vol [5] Gahler. S, "2 metriche Raume udihre topologische struktur," Mathematische Nachrichte. Vol. 26, o. 1-4, pp , [6] Ha. R.S., Cho. Y.J. ad White A, "Strictly Covex ad Strictly 2 covex 2 ormed spaces," Mathematica Japaica, vol. 33, o. 3, pp , [7] Hsio. C.R., "A property of cotractive type mappigs i 2-metric spaces," Iaabha, Vol. 16, pp , [8] Mustafa. Z ad Sims. B, "Some remarks cocerig D-metric spaces," i proceedigs of the Iteratioal Coferece o Fixed Poit Theorey ad Applicatios, pp , Valecica, Spai, July [9] Mustafa. Z, A ew structure for geeralized metric spaces with applicatios to fixed poit theor Ph.D. thesis, the Uiversity of New Castle, Callagha, Australia, [10] Mustafa. Z ad Sims. B, "A ew approach to geeralized metric spaces," Joural of Noliear ad Covex Aalysis, Vol. 7, No. 2, pp , [11] Rashwa. R. A. ad A.M. Sadeek, "A commo fixed poit theorem i complete metric spaces," Southwest Joural of Pure ad Applied Mathematics, Vol. 01, 1996, pp

8 1166 G Sudhaamsh Moha Reddy