Common Fixed Points for Multivalued Mappings

Transcription

1 Advaces i Applied Mathematical Bioscieces. ISSN Volume 5, Number (04), pp. 9-5 Iteratioal Research Publicatio House Commo Fixed Poits for Multivalued Mappigs Lata Vyas* ad Shoyeb Ali Sayyed** * Lecture, Dept. of Mathematics, Lakshmi Narai College of Techology, Idore (M. P.) Idia. ** Pricipal, Royal College of Techology, Idore (M. P.) Idia. ABSTRACT I this paper we have exteded the result of Sayyed [0]. The result is a geeralized cocept of commutig ad compatible mappigs uder some coditios ad correspodig result of Beg ad Azam [], Falset et. al [], Jugck [3, 4], Kaeko [5] Nadler [7], Reich [8], Yadav et. al [3], Wag ad Sog [] ad may others. Key words ad Phrases: Hausdorff metric, Multivalued mappigs, compatible mappig, complete metric space ad coicidece poit. AMS (00) subject classificatios: primary 54H5; Secodary 47H0 INTRODUCTION Baach obtaied a fixed poit theorem for cotractio mappig, appearace of the celebrated baach cotractio priciple, several geeralizatios of this theorem i the settig of poit mappigs have bee obtaied. Nadler [7] was the first to exted Baach cotractio priciple to multivalued cotractig mappig. Rhoades [9] gave a complete ad compariso of various defiitios of cotractio mappig ad also survey of the subject. I this directio sayyed et. al [], Lateef et. al. [6] proved a commo fixed poit theorem for multivalued ad compatible maps. The purpose of this paper to further demostrate the effectiveess of the compatible map cocept as a mea of multivalued ad sigle valued maps satisfyig a cotractive type coditio.. PRELIMINARIES Let (X, d) be a metric space ad let CB (X) deote the family of all o-empty bouded closed subsets of X. For A, B CB (X), let H (A, B) deote the distace betwee A ad B i Hausdorff metric, that is

2 0 Lata Vyas ad Shoyeb Ali Sayyed H (A, B) = if E AB Where E AB = {ε > 0: A N (ε, B), B N (ε, A) } N (ε, A) = {x : d (x, A) < ε}. A poit x is said to be a fixed poit of a sigle valued mappig f : X X (multivalued mappig T : X CB (X)) provided x = fx (x Tx). The poit x is called coicidece poit of f ad T, if fx Tx. If each elemet of X is a coicidece poit of f ad T, the f is called a selectio of T. Let T : X CB (X) be a mappig, the C = { f : X X : TX fx ad ( x X ) ( ftx = Tfx)}. Tad f are said to be commutig mappigs if for each x X, ftx ( ) = ftx= Tfx= Tfx ( ). Lemma. : {Beg [, Lemma. ] }. Let S, T be two multivalued mappigs of X ito CB (X). Let x0, x X. The for each y T ( x ) oe has dysx (, ) HTx (, Sx ). 0 0 Theorem. : Let S, T be two mappigs from a complete metric space X ito CB (X) ad let f C S C T be cotiuous mappig. Suppose that for all x, y X, [ HSxTy (, )] α [ dfxsxdfyty (, ) (, ) dfxtydfysx (, ) (, )] β [ d( fx, Sx) d( fy, Sx) d( fy, Ty) d( fx, Ty )] γ [ dfxsx (, ) dfyty (, )] HSxTy (, )... () Where α, β, γ 0 ad 0 α β γ <. The there exists a commo coicidece poit of f ad T ad f ad S. Proof: Defie M = αβγ β γ T. Let x0 be a arbitrary, but fixed elemet of X. We shall costruct two sequeces { x } ad { y } as follows. Let x X be such that y = fx Sx 0, usig the defiitio of Hausdorff metric ad fact that Tx fx, we may choose x X such that y = fx Txad dy (, y) = dfx (, fx) HSx ( 0, Tx ) ( αβ γ). Sice SX ( ) fx ( ), we may choose x X such that 3 y3 = fx3 Sxad ( α βγ) d( y, y3) = d( fx, fx3) H( Tx, Sx ). β γ By iductio, we produce two sequece of poits of X such that y = fx Sx k, y = fx Tx (),

3 Commo Fixed Poits for Multivalued Mappigs Where k is ay positive iteger. Further more dy (, y ) = dfx (, fx ) ( αβγ) HSx (, Tx ) ( β γ) d( y, y ) = d( fx, fx ) k k 3 3 ( αβγ) HTx (, Sx ) ( β γ) Hece [ dfx (, fx )] <α[ dfx (, Sx )] dfx (, Tx ) k k dfx ( k, Tx ) dfx (, Sx k)] β[ dfx ( k, Sxk) dfx (, Sx k) dfx (, Tx ) dfx ( k, Tx )] γ [d (fx k, Sx k ) d (fx, Tx )] d (fx, fx ) ( αβγ) k ( β γ) d( fx, fx ) < ( αβ γ ) d( fx, fx ) ( β γ) d( fx, fx )] k ( αβγ) ( β γ) k ( αβγ ) ( αβγ) dfx (, fx ) dfx (, fx ) ( β γ) ( β γ) k Therefore, dfx (, fx ) Mdfx (, fx ) M k Similarly, ( α βγ) dfx (, fx ) HdTx (, Sx ) ( β γ) k k k k k Therefore, dfx (, fx ) Mdfx (, fx ) M k k k It further implies that dy (, y ) Mdy (, y) M (, ) ( ) M d y y M M d( fx, fx ) ( ) M k

4 Lata Vyas ad Shoyeb Ali Sayyed for p, we have dy (, y ) dy (, y ) dy (, y )... dy (, y ) p 3 p p { M d( fx, fx) M } M d fx fx M p p M d fx fx p M p p i i M d( fx, fx) im i= i= { (, ) ( ) }... { (, ) ( ) } It follows that the sequece { y } is Cauchy sequece. Hece there exists z i X such that y z. Therefore fx z ad fx z. From (), we have f x = ffx fsx Sfx, k k Ad f x = ffx ftx Tfx. Now usig lemma. [ dfzsz (, )] [ dfzfx (, ) dfx (, Sz )] [ dfzfx (, ) HTfx (, Sz )] = [ dfzfx (, )] HTfx (, Szdfzfx ) (, ) [ HTfx (, Sz )] [ dfzfx (, )] HTfx (, Szdfzfx ) (, ) α [ dfzszdfx (, ) (, Tfx ) dfztfx (, ) dfx (, Sz )] β[ dfzszdfx (, ) (, Sz ) dfx (, Tfx ) dfztfx (, )] γ [ dfzsz (, ) dfx (, Tfx )] HTfx (, Sz ) [ dfzfx (, )] HTfx (, Szdfzfx ) (, ) α [ d( fz, Sz) d( f x, f x ) d( fz, f x ) dfx (, Sz )] β[ dfzszdfx (, ) (, Sz ) d( f x, f x ) d( fz, f x )] γ [ d( fz, Sz) d( f x, f x )] d( f x, Sz ) Sice f is cotiuous, by lettig K, we obtai [ dfzsz (, )] ( βγ)[ dfzsz (, )]

5 Commo Fixed Poits for Multivalued Mappigs 3 or dfzsz (, ) ( βγ) dfzsz (, ). Thus fz Sz. similarly, [ dfztz (, )] [ dfzfx (, ) dfx (, Tz )] [ dfzfx (, ) HSfx (, Tz )] βdfzsz [ (, )] k Therefore fz Tz. Hece Z is a coicidece poit of f ad S ad f ad T. Corollary. 3: Let S, T be cotiuous mappigs from a complete metric space X ito CB (X) ad f C C be a cotiuous mappig. Assume that () is satisfied. If S T f( z) Sz Tzimplies lim f z = t, the t is a commo fixed poit of S, T ad f. Proof: Clearly, fx Sz implies that f z fsz Sfz. Therefore follows that t St. Similarly t Tt. Moreover. = lim ft f f z = lim f z = t. f z Sf Z. If Hece t is a commo fixed poit of f, S ad T. I the followig theorem the cotiuity of f ad its commutativity with S ad T are ot required. Theorem. 4: Let S, T be two mappigs from a metric space X ito CB (X) ad let f : X X be a mappig such that f (X) is complete, T( X) f( X) ad SX ( ) fx ( ). Suppose that () is satisfied, the there exists a commo coicidece poit of f ad T ad f ad S. Proof: As i the proof of theorem. we costruct the Cauchy sequece y = fx X. By our hypothesis it follows that there exists a poit u i X such that y z = fu. Now usig Lemma., we have [ d( fu, Tu)] [ d( fu, fx ) d( fx, Tu )] [ dfufx (, ) HSx ( k, Tu )] [ dfufx (, )] HSx ( k, Tudfufx ) (, ) [ HSx ( k, Tu )] [ d( fu, fx )] d( fu, fx ) H( Sxk, Tu ) α [ dfx ( k, Sxk) dfutu (, ) dfx ( k, Tu ) dfusx (, )] k

6 4 Lata Vyas ad Shoyeb Ali Sayyed β [ dfx (, Sx ) dfusx (, ) dfutu (, ) k k k dfx (, Tu)] γ [ dfx (, Sx ) dfutu (, )] HSx (, Tu ) k k k k [ d( fu, fx )] d( fu, fx ) d( fx, Tu ) α [ dfx (, fx ) dfutu (, ) dfx (, Tu ) k k dfufx (, )] β [ d( fx, fx ) d( fu, fx ) d( fu, Tu ) k dfx (, Tu)] γ [ dfx (, fx ) dfutu (, )] dfx (, Tu ). k k Lettig k, we obtai [ dfutu (, )] ( βγ)[ dfutu (, )] or dfutu (, ) ( βγ) dfutu (, ) Hece fu Tu. Similarly, [ d( fu, Su)] [ d( fu, fx ) d( fx, Su)] dfufx (, ) HTx (, Su)] ( βγ)[ dfusu (, )] Hece fu Su. Example: Let Sx ( ) = x ad T( x) = 3 xwith x = R. Sx ( ) Tx ( ) = x 3 x 0 if ad oly if x ad lim ST( x ) TS( x ) = lim6 x = 0 if x Thus S ad T are compatible but ot weakly commutig pair. Referece [] Beg ad A. Azam, commo fixed poits for commutig ad compatible maps, Discussioes Mathemticae Differetial iclusios 6 (996), -35. [] J. G. Falset, E. L. Fuster ad E. M. Galvez, fixed poit theory for multivalued geeralized o expasive mappigs, Appl. Aal. Discrete Math. 6 (0), [3] G. Jugck, Commutig mappig ad fixed poits, Amer. Math. Mothly 83 (976), 6-63.

7 Commo Fixed Poits for Multivalued Mappigs 5 [4] G. Jugck, Commo fixed poits for commutig ad compatible maps o compacta, Proc. Amer. Math. Soc. 03 (3) (988), [5] H. Kaeko, Sigle valued ad multivalued f-cotractio, Boll. U. M. I. 4A (985), [6] D. Lateef, S. A. Sayyed ad A. Bhattacharyya, Commom fixed poit for multivalued ad compatible maps, Ultra Scietist, Vol. () M, , (009) [7] S. B. Nadler, Jr. Multivalued cotractio mappigs, pacific J. Math. 30 (969), [8] S. Reich, Some remarks cocerig cotractio mappigs, Caad Math. Bull. 4 (97), -4. [9] B. E. Rhoades, A compariso of various defiitios of cotractive mappigs, Tras, Amer. Math. Soc. 6 (977), [0] S. A. Sayyed, Some results o commo fixed poit for multivalued ad compatible maps, Ultra Egieer, Vol. (), (0), 9-94 [] S. A. Sayyed, F. Sayyed ad V. H. Badshah, Fixed poit theorem ad multivalued mappigs, Acta Ciecia Idica, Vol. XXXVIIIM, No. (00), [] Q. Wag, M. Sog, Commom fixed poit theorems of multivalued maps i ultrametric spaces, Applied Mathematics, 03, 4, [3] H. Yadav, S. A. Sayyed ad V. H. Babshah, Fixed poit theorem for multivalued mappigs satisfyig fuctioal iequality, Orietal Joural Of Computer Sciece ad Techology, Vol. 4 () -3, 0.

8 6 Lata Vyas ad Shoyeb Ali Sayyed