II. EXPANSION MAPPINGS WITH FIXED POINTS

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1 Geeralizatio Of Selfmaps Ad Cotractio Mappig Priciple I D-Metric Space. U.P. DOLHARE Asso. Prof. ad Head,Departmet of Mathematics,D.S.M. College Jitur ,Dist. Parbhai (M.S.) Idia ABSTRACT Large umber of fixed poit results for selfmappigs satisfyig various types of cotractive iequalities which i [ 5, 9,12]. I this paper, theorems o selfmaps ad some fixed poit theorems are proved i D-metric spaces. The Geeralizatio of cotractio mappig priciple i D-metric space which iclude some fixed poit results i U.P.Dolhare [1], B.C. Dhage, U.P.Dolhare ad Adria Petrusel [ 3 ],Rhoades [ 12 ] i D-metric spaces as special cases. Key words: D-Metric space, fixed poit etc.,(2000) Mathematics subject classificatio:47h10, 54H25 I. INTRODUCTION The study of fixed poits of Selfmappigs satisfyig cotractive coditios which is oe of the research activity. Metric fixed poit theory used to Baach fixed poit theorem (1922). Fixed poit theorems are applied i various fields of sciece. The theory started with the geeralizatio of the Baach fixed poit theorem usig cotractio ad o expasive mappigs Pat [15], Fisher B [ 6], Rhoades B.E.[8], Cric [12]. worked 0 these mappig. Dhage, Dolhare U.P. ad Adria Petrusel [2] used o-selfmaps to obtai some fixed poit theorems i metric space. M.S.Kha[9] proved some iterestig results for multivalued cotiuous mappigs. We also geeralized the results of Chtterjee s.[14] to expasive selfmaps ad o-selfmaps. II. EXPANSION MAPPINGS WITH FIXED POINTS Rhoades proved the follig theorem: Theorem 2.1 :If f is a selfmap of complete metric space (x,d), f is oto, ad there is a costat α > 1 such that d( f x, f y ) αd(x,y), for all x, y X (2.1) the f has a uique fixed poit. Pat.R,P.[16] geeralizig this theorem by cosidered the followig coditio. d( f x, f y ) > mi{d(x,y), d(x, f x ), d(y, f y )} for all x, y X, x y. (2.2) He proved each cotiuous selfmap f of a compact metric space satisfyig the above coditio has a fixed poit. Dhage,Dolhare u.p ad Ntouyas s.k.[4] proved some iterestig results for multivalued cotiuous mappigs. We ivestigate the fixed poits of cotiuous mappigs, expasive mappigs for o-selfmaps ad obtai some results. M.S.Kha [9] used the square root coditio for h < 1 to obtai a commo fixed poit of two cotiuous self mappigs S ad T d( S x,t y ) h{ d( x,s x ). d( y, T y ) } ½ for h < 1 (2.3) he used the followig expasive coditio to obtai a idetity mappig All rights Reserved Page 17

2 d ( T x, T y ) { d( x, T x ).d( y, T y ) } ½ for all x, y X if we replace the cotractive coditio (2.3) with the similar expasive coditio : d( S x,t z ) h{ d( x, S x ). d( z, T z ) } ½ for h>1 (2.4) This coditio which is useful for to fid the fixed poit. III. D-CONTRACTION MAPPING PRINCIPLE: Oe of the most fudametal fixed poit theorems for Cotractio mappigs i complete metric space is proved by Polish Mathematicia Stefa Baach i year 1922 which as follows. Theorem 3.1 : Let f be a self mappig of a complete metric space X satisfyig x, y X ad 0 d( f x,f y ) α d( x, y ) for all α < 1 the f has uique fixed poit x* ad the sequece {f 1 ( x) } X of the successive iteratios of f 1 ( x) X coverges to x* Similarly the fudametal fixed poit theorems for cotractio as will as cotractive mappigs i D- Metric spaces are proved by Dolhare ad Dhage [5] which as follows. Theorem 3.2 : Let f be self mappig of a complete ad bouded D- metric space X satisfyig ρ ( f x,f y, f z ) α ρ ( x, y, z ) for all x, y, z X ad 0 α < 1 the f has a uique fixed poit x* ad the sequece {f x} X of the successive iteratios f x X coverges to x*. Also Caccioppoli [10] proved the followig fixed poit theorem i D-Metric spaces. Theorem 3.4 : Let f be a selfmap of a complete metric space X satisfyig d(f x, f y ) a d ( x,y ) for all x, y X where a > 0, N (3.1) ad are idepedet of x ad y if =1 a < the f has a uique fixed poit. Dhage,Dolhare ad Ntouyas S.K. [4] give the followig form of well kow D-Cotractio mappig priciple. Theorem 3.5 : Let f be a selfmaps of f-orbitally complete D-metric space X satisfyig ρ( f x, f y, f z ) λ ρ( x, y, z ) for all x, y, z X, where 0 λ < 1 the f has a uique fixed poit. IV. GENERALIZATION OF CONTRACTION MAPPING PRINCIPLE IN D-METRIC SPACE : The study of the results cocerig oliear self mappigs satisfyig a special type of cotractio coditio o a D-metric space are obtaied. The study of the oliear mappig f o a D-metric space X ito itself satisfyig the cotractive coditio of the form ρ( f x, f y, f z ) [max{ρ(x, y, z), ρ(x, f x, f y ), ρ(x, f x, f z ), ρ(y, f y, f x ), ρ ( y, f y, f z ), ρ( y, f z, f y ), ρ (z, f z, f x ),)}] for all x, y, z X where : R + R + is a cotiuous o decreasig fuctio satisfyig (t) < t, t>0 ad =1 ⁿ t < for each t [0, ) ad the existece of a uique fixed poit is proved uder the coditio of completeess ad boudedess of All rights Reserved Page 18

3 I 1922 Baach first proved his well-kow cotractio mappig priciple for the selfmappigs i ordiary metric spaces satisfyig certai cotractio coditio.i 1968 Kaa give a ew ter to Baach fixed poit theorem ad discovered a ew class of cotractio mappigs. Baach cotractio mappig priciple are several extesio ad geeralizatio of the above cotractio priciple i literature see Rhoades [11], Ciric[12]. Theorem 4.1 : Ciric[12] : Let f be a selfmap of a complete metric space X d( f x, f y ) C max {d (x, y), d(x, f x ), d(y, f y ), d(x, f y ), d(y, f x )} for all x, y X, where 0 C 1 the f has a uique fixed poit. Rhodes [11] Characterized geeral Cotractio mappigs by the iequality d(f x, f y ) C max{ d(x, y), d(x, f x ), d(y, f y ), [d(x, f y ) + d(y, f x ) 2]} the f has uique for all x, y X, where 0 C 1 the f has uique fixed poit. Dolhare [1] ad Dhage ad Rhoades[13] shows that fixed poit theorem i D-metric space are atural geeralizatio of the fixed poit theorems i metric spaces. V. CONTRACTION MAPPINGS IN D-METRIC SPACES : The basic cotractio mappig priciple i D-Metric spaces developed by Dhage [3] as follows Defiitio : Let X be a D-metric space ad f: X X. Let O f (x) deote a f orbit of f at a poit x X defied by O f (x) = { x, f x, f 2 x,------} D-metric space is called f-orbitally complete if every D-Cauchy sequece i O f (x) coverges to a poit i X Similarly X is called f-orbitally bouded if (O f (x)) < for each x X Theorem 5.1 [ Basic cotractio mappig priciple] Let X be a D-metric space ad Let f: X x X X a mappig satisfyig ρ(f x, f y, f z ) λ(max{ρ(x, y, z), ρ(x, f x, z), ρ(x, f y, z), ρ(y, f x, z), ρ(f x, f y, f z )}) x, y, z X, where 0 λ 1 further if X is f-orbitally bouded ad f-orbitally complete, the f has a uique fixed poit. Lemma 5.1 : If Φ the (0) = 0 for each N ad lim ⁿ t = 0 for all t > 0 By geeralizig cotractio mappig priciple Dolhare [7] we prove our mai result VI. MAIN RESULT The atural geeralizatio of cotractio mappig priciple i D-metric is as follows. Theorem 6.1 : Let X be a D-metric space ad let f :X X be a mappig satisfyig ρ(f x, f y, f z ) max{ ρ(x, y, z), ρ(x, f x, f y ), ρ(x, f x, f z ), ρ(y, f y, f z ), ρ(y, f y, f x ), ρ(z, f z, f x ), ρ (z, f z, f y )}. (6.1) Proof: Let x X be arbitrary ad defie {x } X by x 0 = x, x -1 = fx, 0 if x r = x -1 for some r N, the U = x r is a fixed poit of f.therefore we assume that x x -1 for each N. We show that {x } is D-Cauchy. By usig the coditio x = x 0, y = x 1 ad z = x m-1, m > 1 i (6.1) we obtai ρ(x 1, x 2, x m ) = ρ (fx 0, fx 1, fx m-1 ) (max{ ρ(x 0,x 1,x m-1 ), ρ(x 0,fx 0,fx 1 ), ρ(x 0,fx 0,fx m-1 ), ρ(x 1,fx 1,fx m-1 ), ρ(x 1,fx 1,fx 0 ), ρ(x m- 1,fx m-1,fx 0 ), ρ(x m-1,fx m-1,fx 1 )}) = (max(ρ(x 0, x 1, x m-1 ), ρ(x 0, x 1, x 1 ), ρ(x 0, x 1, x m ),ρ (x 1, x 2, x m ), ρ(x 1, x 2, x 1 ), ρ(x m-1, x m, x 1 ), ρ (x m-1, x m, x 2 All rights Reserved Page 19

4 maxρ(xa, xb, xc) 0 a + 1 (k) Agai gettig x = x 1, y = y 2, z = x m-1, m > 2 i 6.1 yields ρ(x 2,x 3,x m ) = ρ( fx 1, fx 2, fx m-1 ) (max { ρ(x 1,x 2,x m-1 ), ρ(x 1,x 2,x 3 ), ρ(x 1,x 2,x m ) ρ(x 2,x 3,x m ),ρ(x 2,x 3,x 2 ), ρ (x m-1,x m,x 2 ), ρ(x m-1,x m,x 3 )}) max ϱ xa, xb, xc 2 0 a 3 2 (k) By iductio ρ (x,x -1,x m ) max ϱ xa, xb, xc 0 a + 1 (k) for all m > N Now a applicatio of theorem 5.1 yield that {x } is D-Cauchy X beig a complete D-metric space there is a poit u X such that lim x = u. We show that u is a fixed poit of f Now ( u,u,f) = lim ρ(x +1,x +1,f u ) = lim (fx, fx, f u ) lim (max{ρ(x, x, u), ρ(x, x -1, x -1 ), ρ(x, x -1, f u ), ρ(x, x -1, f u ), ρ (x, x -1, x -1 ), ρ (u, f u, x -1 ), ρ (u, f u, x -1 )}) = (max { 0,0, ρ(u,u,f u )}) = (ρ (u,u,f)) Which is possible oly whe u = f u thus f has uique fixed poit Refereces [1] Dolhare U.P.: Noliear mappig ad fixed poits theorems i D-Metric Spaces; Ph.D. Thesis S.R.T.M. Uiversity, Naded, Idia, Dec [2] Dolhare U,.P., Dhage B.C., Adria Petrusel: Some commo fixed poit theorems or sequeces of Noself Multivatlued operatios i Metrically covex D-Metric Spaces. Maths, Fixed poit Theory. Iteratioal Joural, Romaia, Volue. 4. No. (2003), [3] Dhage B.C:A study of some fixed poit theorems,ph.d. Thesis Marth. Uiv. A bad Idia1984 [4] Dhage B.C.,Dolhare u. p. ad Ntouyas S.K: Existece Theorems for Noliear fuctioal Equatios i Baach Algebras Commuicatios o Applied oliear Aalysis. A Great America Joural Volume 10. (2003). Number 4, (59-69) America. [5] Dolhare U.P., Dhage B.C., Lokesh V. ad Giiswamy: Max. mi. priciple ad cotractio mappigs i IR Maths. Bull. Calcutta Math. Soc Volume 3 (2) All rights Reserved Page 20

5 [6] Fisher.B. Commo fixed poit mappigs.idia J.Math.20(2)(1978) [7] Dolhare U.P.:Some fixed poit Theorems i D-metric spaces Published i Natioal Cof. O Noliear Aalysis applicatio March 24(2005)28. [8] Rhoades B.E.: A fixed poit theorems for geeralized metric spaces. Iter. J. Math. ad Sci.9(3)(1996) [9] M.S. kha: Commo fixed poit theorems for multi valued mappigs. Pacific J. Math. 95 (1981): [10] Caccioppoli:Utheorem a geerale bull existece di-elemets uiti i uatras formazroe fuctioal Ahi Acad. Na Licei 6 (11): (1930), [11] Rhoades: A compariso of various defiitio of cotractio mappig Tras. Amer. Math.Soc. 226 (1977): [12] Cric LJ. B : A geeralizatio of Baach Cotractio Priciple Proc, Amer. Math, Soc.45(1974), [13] Dhage ad Rhoades: A fixed poit theorem for geeralized metric space. Iteret J. Math ad Sci. 9 (3) (1996): [14] Chatterjee S. : Fixed poit theorems, Red. Acad. Bulgare Sci. 25 (1972): [15] Pat R.P. ad Ta. : Commo fixed poit theorems for Cotractive maps. Math. Aal. Appl. 226 (1998): All rights Reserved Page 21