On a fixed point theorems for multivalued maps in b-metric space. Department of Mathematics, College of Science, University of Basrah,Iraq

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1 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 O a fixed poit theorems for multivalued maps i -metric space AMAL M. HASHM DUAA L.BAQAIR Departmet of Mathematics, College of Sciece, Uiversity of Basrah,Iraq Astract: amalmhashim@yahoo.com I this paper, we prove the existece of the fixed poits for ciric's cotractive coditio comig with Beride coditio the class of so called ciric strog almost cotractio i -metric spaces. The ciric strog almost cotractio appear to e oe of the most geeral metrical coditio for which the set of fixed poits is ot a sigleto. Our results exted ad uify a multitude of fixed poit theorems for multivalued maps. Mathematics Suject Classificatios (000): 54H5, 47H0. Keywords: fixed poit, -metric space, multivalued map 6

2 Hashm, A.M. & Baqair, D.L. -Itroductio: The cocept of -metric space was itroduced y Czerwik((998), sice the several papers deal with fixed poit theory for siglevalued ad multivalued operators i -metric see Sigh et al.(005),boriceau (009) ad Hashim (0). Our purpose is to show that some well-kow fixed poit theorems are valid i -metric spaces. Let ( X, d) e a metric space ad CL( X ) deotes the class of all oempty O a fixed poit theorems for multivalued closed suset of X ad CB ( X ) deotes the class of all oempty closed ouded suset of X. For A, B, C we cosider, if, ;, the distace d x B d x y y B etwee x ad B. For ay A, B CL( X ), defie a fuctio CL X H : CL X 0, H A, B max {sup d x, B,sup d ( y, A)}, x A y B the, H is said to e the geeralize Housdorffmetric o CL( X ) iduced y the metric d, see for istace Czerwik S. Let (, ) Theorem (.):Czerwik(998) X d e a complete -metric space. If T : X CL( X ), satisfies the iequality H(Tx,Ty) d(x,y), x,y X, where 0, The (i) for every x X, there exist a sequece o { x } X ad u X such that x Tx, 0,,,3...,ad lim x u, (ii) the poitu is a fixed poit of T,. i e u Tu. Theorem (.): Beride et al.(007) Let ( X, d ) e a complete metric space, T : X CB ( X ) e a multivalued map ad L 0. Assume that H Tx Ty d x y d x y L d y Tx for all x, y X, where α is a fuctio,,,,, from [0, ) ito [0, ) satisfyig lim sup( s) st FT ( ) where FT ( ) poit of T. for all t [0, ). The, the set of fixed As kow, a mappig : R Ris called a compariso fuctio if it is icreasig ad t 0,, for ay t R 7

3 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 Lemma (.3):Rus(00) If : R R is a compariso fuctio, the; ) each iterate φ k of φ, k, is also a compariso fuctio: ) is cotiuous at zero; 3) t t, for ayt 0. Let T : X CL( X ) ad cosider the followig coditio for all x, y X : ) H Tx, Ty d x, y Ld y, Tx, where 0 ) H Tx Ty, M ( x, y ), L ad 0,,,,,,,,,, wherem x y d x y d x Tx d y Ty d x Ty d y Tx 3) H Tx Ty M x y Ld y Tx,,,, 4) H Tx, Ty M x, y Ld y, Tx. Where M x y d x y d x Tx d y Ty d x Ty d y Tx, {,,,,,,,, }, where is -compariso fuctio with z, Remark (.4): z 0 0,. Coditio () is a very geeral cotractive coditio that allows the operator T to have more tha oe uique fixed poit ad it icludes may cotractive coditios from Rhoade's classificatio Rhoade(977) ad it is called a multivalued weak (almost) cotractio, see Beride et al.(007).coditio () ad () are idepedets. Coditio (3) itroduced y Beride(009) ad y replacig the term d x, y i () y M ( x, y ) we get cotractive coditio (). It is ovious that coditio (4) is more geeral 8

4 Hashm, A.M. & Baqair, D.L. cotractive coditio tha (),() ad (3).. Prelimiaries Cosistet with Hashim(0) ad Czerwik(998) we use the followig otatios ad defiitios. ) d x, y 0 iff x y, ) d x y, d ( x, y ), 3) d x z d x y d y z (, ) [,, ]. O a fixed poit theorems for multivalued Defiitio (.): Czerwik(998). Let X e a oempty set ad a give real umer. Afuctio d : X X R (oegative real umers) is called a -metric space provided that, for all x, y, z X, The pair ( X, d) is called a -metric space. We remark that a metric space is evidetly a -metric space. However, The followig example show -metric o X ot e a metric o X. Czerwik [] has show a -metric o X eed ot e a metric space o X. Example (.):Czerwik(998). Let X x x x 3 {,, } ad d : X X R such that,,,, ad d x, x 0, d x, x d x, x d x x a d x x d x x for k k a k,,,3...,. The, d ( x, x k ) [ d x, x i d x i, x k ],, k, i,,3. The ( X, d ) is a -metric space. Ad if a, the ordiary triagle iequalitydoes ot hold. Defiitio (.3):Boriceau (009) Let ( X, d ) e a -metric space. The a sequece { } x N i X is called 9

5 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 a) Cauchy if ad oly if for every 0 there exists () N such that for each, m ( ) we have d ( x, x ). m ) Coverget if ad oly if there exists x X such that for all 0 there exists () N such that for all () we have d ( x, x ). c) The -metric space is complete if every Cauchy sequece coverges. Lemma (.4):Czerwik(998). For ay A, B, C CL( X ). (i) d x B d x y,,, for ay y B. (ii) d A B H A B,,, (iii) d x B H A, B,, x A (iii) H A C s H A B H B C,,,, (iv) d x A sd x y sd y A,,,, x, y X. Lemma (.5)]Czerwik(998).: Let ( X, d ) e a -metric space ad A, B CL ( X ). The for each 0ad for all B there exists a A such that,, d a H A B. Lemma (.6): Sigh et al. (008). Let ( X, d ) e a -metric space ad { y } is a sequece i X such that d y, y qd y, y, 0,,, The the sequece i X { y } is a Cauchy sequece i X, provided that sq where q (0,) ad s Lemma.7: Pacurar(00) Ay -compariso fuctio is a compariso fuctio. 30

6 Hashm, A.M. & Baqair, D.L. O a fixed poit theorems for multivalued 3. Mai Results Theorem (3.): Let ( X, d ) e a complete -metric space with costat ad T : X CL ( X ) a multivalued operator. Suppose that there exists a cotiuous : R R 0 0, ad for all x, y iequality if M ( x, y ) 0. Where z with z, X. We have H Tx, Ty M x, y Ld y, Tx, with strict M x y Max d x y d x Tx d y Ty d x Ty d y Tx,,,,,,,,, fixed poit. Proof: Let x 0 implies that Fix ( T ) Let ( 0, ) 0 X ad x Tx 0. H Tx 0, Tx 0. The, Tx 0 Tx. If H Tx Tx. Sice H Tx Tx M x x Ld x Tx may choose 0 with 0 0. The T has a ; that is x T, which (, ) (,,, y Lemma (.5) we H Tx, Tx M x, x Ld ( x, Tx ) ( M x, x, Next we choose x Tx such that d x, x H Tx, Tx M x, x max{ d x, x, There is 4 cases, M x, x d ( x, x ). The, ) If 0 0 d x, x d x, x, (3.) 0 d x 0, Tx 0, d x, Tx, d x 0, Tx d x, Tx 0 } 3

7 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 ) If M x x d x Tx,,, the we have, d x, x H Tx, Tx M x, x Ld x, Tx d x, x d x, x, ) If M x x d x Tx,,, the, 0 d x, x d x, Tx Ld x, Tx d x, Tx d x, x, 0 4) If M x 0, x d x 0, Tx d x, Tx 0 d x 0, Tx, the, d ( x, x ) [ d x 0, Tx ] d ( x 0, Tx ) d ( x 0, x ) [ d x 0, x d x, x,. Hece d x, x d x, x 0 Thus (3.)is true i all cases, d ( x, x ) d ( x 0, x ) d ( x 0, x ). Next we assume M x x choose δ > 0 with H Tx, Tx ( M x, x Ld x, Tx (, ) 0 Let x 3 Tx such that, d x, x 3 H Tx, Tx ( M ( x, x ). If M x, x 0, the Fix ( T ) 3 If M ( x, x 3) 0. The, we will show that d ( x, x ) ( d ( x, x ) d ( x, x ), (3.) 3 0 Now if M x x d x x,,, the (3.) is true. If M x x d x Tx,,, the we have d x, x d x, Tx Ld x, Tx d x, x, thus (3.) is true. 3 3 M x, x d ( x, Tx )), the If d ( x, x ) ( d ( x, Tx ) d ( x, Tx ) d ( x, x ), which is a cotractio

8 Hashm, A.M. & Baqair, D.L. O a fixed poit theorems for multivalued,,, the, we have If M x x d x Tx d x, x 3,, (, 3),, 3, d x Tx d x Tx d x x d x x d x x Thusd x x d x x,,, which is a cotractio. Thus (3.) is true. 3 Hece i all cases (3.) is true. By a iductively procedure we will otai x Tx, d x, x M x, x, The argumet aove guaratees that for 3,4, with d ( x, x ) ( d ( x, x ) ( d ( x, x ). 0 Next we will prove that {x } is a Cauchy sequece p p p p d x, x d x, x d x, x d x, x, p p p d x, x d x, x d x, x d x, x, Which ca also e writte as d x, 0, 0, p p x p d x x d x x d x 0, x j j d x 0, x 0, as. J Thus { x } is Cauchy sequece i the complete -metric space( X, d ).So there is x * such that x * lim x. X 33

9 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 I the followig we prove that x is a fixed poit of Ti.e. x Tx. * * * * * * * (, ) [,, ] [,, ] [, * * * * M x, x Ld x, Tx d x, x M x, x. d x Tx d x x d x Tx d x x H Tx Tx d x x For, d x Tx d x Tx d x Tx d x Tx d x Tx * * * * * * * * * * (, ) (max{0,0,,, (, },,, * * Which implies that d x, Tx 0, so that x Tx Tx, ( T is the closure of T ). Remark (3.): () If L 0 i coditio the we otai theorem (3.) of Boriceau(009). () If z z i coditio (4) the we otai corollary(3.) of Sigh et al.(008) with Y X ad f I. (3) Theorem (3.) geeralize the mai results ofpacurar (00),Pacurar (03)ad Sigh et al.(0). Refereces Beride M. ad Beride V., (007) Ogeeral class of multivalued weakly Picard mappig, J. Math. Aal, 36, Beride,V., (009)Some remarks o a fixed poit theorem for Ciric-type almost 34 cotractio, Carpathia J-Math 5,No.,57-6. Boriceau M., (009)Strict fixed poit theorem for multivalued operators i - metric spaces Iteratioal Joural of Moder Mathematics 4(), -7. Czerwik S., (998) Noliear set valued cotractio i -metric spaces.attisem. Mat.Uiv. Modea, 46, Hashim A.M., (0)Staility of iterative procedures for hyrid maps i - metric space. Basrah Joural of sciece(a) 9(), Kir.M, Kiziltuc.H, (03): O some well-kow fixed poit theorems i - metric space, Turkish Joural of aalysis ad umer theory.,3-6.

10 Hashm, A.M. & Baqair, D.L. Pacurar,M., (00) A fixed poit result for -cotractio o -metric spaces without the oudedees assumptio, Fasciculi Mathematici,No.43, Rhoade,B.E, ( 977) A compariso of various defiitios of cotractive mappigs, Tras, Amer, Math.Soc.6, Rus,I.A.,(00) Geeralized Cotractios ad Applicatios, Cluj uiversity press Cluj- Napoca. Sigh S.L., Bhatgar C, Mishra S.N, (005) Staility of iterative procedure for O a fixed poit theorems for multivalued multivalued maps i metric spaces, Demostratio. Math, 37, Sigh S.L., Czerwik S., Krol Krzysztof, ad Sigh Aa., (008)Coicideces ad fixed poits of hyrid cotractios, Tamsui Oxford Joural of Mathematical scieces,4(4), Sigh, A ad Alam, A, (0)Zamfirescu maps ad it's staility o geeralized space.iteratioal Joural of Egieerig ad Techology (IJEST) Vol. 4 No.ol,

11 Basrah Joural of Sciece (A) Vol.33(),6-36, 05 حول مبرهنات النقطة الصامدة للدوال المتعددة القيم في الفضاء المتري امل محمدهاشم البطاط دعاء لفتة باقر قسم الرياضيات- كلية العلوم جامعة البصرة الخالصة في هذا البحث برهن على وجود النقاط الصامده باستخدام شرط Ciricوشرط Berideمعا في الفضاء المتري. ان شرطCiricالمسمى باالنكماش القوي تقريبا هو من اكثر الشروط تعميما والذي تكون فية النقاط الصامدة ليست وحيدة. في هذاالبحث توحيد وتوسيع بعض مبرهنات النقطة الصامدة للدوال متعددة القيم. 36