Generalization of Contraction Principle on G-Metric Spaces
|
|
- Reynard Turner
- 5 years ago
- Views:
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
Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfying Generalized Contractive Condition of Integral Type
Iteratioal Refereed Joural of Egieerig ad Sciece (IRJES ISSN (Olie 239-83X (Prit 239-82 Volume 2 Issue 4(April 23 PP.22-28 Uique Commo Fixed Poit Theorem for Three Pairs of Weakly Compatible Mappigs Satisfyig
More informationSome Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions
ISSN(Olie): 319-8753 ISSN (Prit): 347-671 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) Some Commo Fixed Poit Theorems i Coe Rectagular Metric
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationInternational Journal of Mathematical Archive-7(6), 2016, Available online through ISSN
Iteratioal Joural of Mathematical Archive-7(6, 06, 04-0 Available olie through www.ijma.ifo ISSN 9 5046 COMMON FIED POINT THEOREM FOR FOUR WEAKLY COMPATIBLE SELFMAPS OF A COMPLETE G METRIC SPACE J. NIRANJAN
More informationFixed Point Theorems for Expansive Mappings in G-metric Spaces
Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
More informationII. EXPANSION MAPPINGS WITH FIXED POINTS
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 -431509,Dist. Parbhai (M.S.) Idia ABSTRACT Large umber
More informationOn the Variations of Some Well Known Fixed Point Theorem in Metric Spaces
Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow
More informationCOMMON FIXED POINT THEOREMS VIA w-distance
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA
More informationStatistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function
Applied Mathematics, 0,, 398-40 doi:0.436/am.0.4048 Published Olie April 0 (http://www.scirp.org/oural/am) Statistically Coverget Double Sequece Spaces i -Normed Spaces Defied by Orlic Fuctio Abstract
More informationCOMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS
PK ISSN 0022-2941; CODEN JNSMAC Vol. 49, No.1 & 2 (April & October 2009) PP 33-47 COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS *M. A. KHAN, *SUMITRA AND ** R. CHUGH *Departmet
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More informationA Common Fixed Point Theorem Using Compatible Mappings of Type (A-1)
Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: 79-087X (P), 79-0888(olie) Published o 7 September 07 www.researchmathsci.org DOI: http://dx.doi.org/0.457/apam.v4a8 Aals of A Commo Fixed
More informationOn common fixed point theorems for weakly compatible mappings in Menger space
Available olie at www.pelagiaresearchlibrary.com Advaces i Applied Sciece Research, 2016, 7(5): 46-53 ISSN: 0976-8610 CODEN (USA): AASRFC O commo fixed poit theorems for weakly compatible mappigs i Meger
More informationCommon Fixed Points for Multivalued Mappings
Advaces i Applied Mathematical Bioscieces. ISSN 48-9983 Volume 5, Number (04), pp. 9-5 Iteratioal Research Publicatio House http://www.irphouse.com Commo Fixed Poits for Multivalued Mappigs Lata Vyas*
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationOn Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings
Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad
More informationCOMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES
Iteratioal Joural of Egieerig Cotemporary Mathematics ad Scieces Vol. No. 1 (Jauary-Jue 016) ISSN: 50-3099 COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES N. CHANDRA M. C. ARYA
More informationA Common Fixed Point Theorem in Intuitionistic Fuzzy. Metric Space by Using Sub-Compatible Maps
It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 55, 2699-2707 A Commo Fixed Poit Theorem i Ituitioistic Fuzzy Metric Space by Usig Sub-Compatible Maps Saurabh Maro*, H. Bouharjera** ad Shivdeep Sigh***
More informationConvergence of Random SP Iterative Scheme
Applied Mathematical Scieces, Vol. 7, 2013, o. 46, 2283-2293 HIKARI Ltd, www.m-hikari.com Covergece of Radom SP Iterative Scheme 1 Reu Chugh, 2 Satish Narwal ad 3 Vivek Kumar 1,2,3 Departmet of Mathematics,
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationA COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE MAPPINGS
Volume 2 No. 8 August 2014 Joural of Global Research i Mathematical Archives RESEARCH PAPER Available olie at http://www.jgrma.ifo A COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE
More informationOn n-collinear elements and Riesz theorem
Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (206), 3066 3073 Research Article O -colliear elemets ad Riesz theorem Wasfi Shataawi a, Mihai Postolache b, a Departmet of Mathematics, Hashemite
More informationCOMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES
I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES Ail Kumar Dube, Madhubala Kasar, Ravi
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationGeneralized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces
Turkish Joural of Aalysis a Number Theory, 08, Vol. 6, No., 43-48 Available olie at http://pubs.sciepub.com/tjat/6// Sciece a Eucatio Publishig DOI:0.69/tjat-6-- Geeralize Dyamic Process for Geeralize
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationStrong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types
It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationCOMMON FIXED POINT THEOREM FOR FINITE NUMBER OF WEAKLY COMPATIBLE MAPPINGS IN QUASI-GAUGE SPACE
IJRRAS 19 (3) Jue 2014 www.arpapress.com/volumes/vol19issue3/ijrras_19_3_05.pdf COMMON FIXED POINT THEOREM FOR FINITE NUMBER OF WEAKLY COMPATIBLE MAPPINGS IN QUASI-GAUGE SPACE Arihat Jai 1, V. K. Gupta
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationCommon Fixed Point Theorem for Expansive Maps in. Menger Spaces through Compatibility
Iteratioal Mathematical Forum 5 00 o 63 347-358 Commo Fixed Poit Theorem for Expasive Maps i Meger Spaces through Compatibility R K Gujetiya V K Gupta M S Chauha 3 ad Omprakash Sikhwal 4 Departmet of Mathematics
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationSome Fixed Point Theorems in Generating Polish Space of Quasi Metric Family
Global ad Stochastic Aalysis Special Issue: 25th Iteratioal Coferece of Forum for Iterdiscipliary Mathematics Some Fied Poit Theorems i Geeratig Polish Space of Quasi Metric Family Arju Kumar Mehra ad
More informationIterative Method For Approximating a Common Fixed Point of Infinite Family of Strictly Pseudo Contractive Mappings in Real Hilbert Spaces
Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 89-4966 Volume 2, Number 2 (207), pp. 293-303 Research Idia Publicatios http://www.ripublicatio.com Iterative Method For Approimatig a Commo
More informationAPPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS
Joural of Mathematical Iequalities Volume 6, Number 3 0, 46 47 doi:0.753/jmi-06-43 APPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS HARK-MAHN KIM, JURI LEE AND EUNYOUNG SON Commuicated by J. Pečarić
More informationCOMMON FIXED POINT THEOREM USING CONTROL FUNCTION AND PROPERTY (CLR G ) IN FUZZY METRIC SPACES
Iteratioal Joural of Physics ad Mathematical Scieces ISSN: 2277-2111 (Olie) A Ope Access, Olie Iteratioal Joural Available at http://wwwcibtechorg/jpmshtm 2014 Vol 4 (2) April-Jue, pp 68-73/Asati et al
More informationEquivalent Banach Operator Ideal Norms 1
It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity
More informationResearch Article Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces
Iteratioal Scholarly Research Network ISRN Mathematical Aalysis Volume 2011, Article ID 576108, 13 pages doi:10.5402/2011/576108 Research Article Covergece Theorems for Fiite Family of Multivalued Maps
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More informationJournal of Applied Research and Technology ISSN: Centro de Ciencias Aplicadas y Desarrollo Tecnológico.
Joural of Applied Research ad Techology ISSN: 665-64 jart@aleph.cistrum.uam.mx Cetro de Ciecias Aplicadas y Desarrollo Tecológico México Shahi, Priya; Kaur, Jatiderdeep; Bhatia, S. S. Commo Fixed Poits
More informationON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS
Vol. 9, No., pp. 3449-3453, Jue 015 Olie ISSN: 190-3853; Prit ISSN: 1715-9997 Available olie at www.cjpas.et ON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS Awar A. Al-Badareh Departmet
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationA constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference
MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at http://mpra.ub.ui-mueche.de/55889/
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 3, ISSN:
Available olie at http://scik.org J. Math. Comput. Sci. 2 (202, No. 3, 656-672 ISSN: 927-5307 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS L. HORVÁTH,
More informationReal Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim
Real Aalysis Fall 004 Take Home Test SOLUTIONS. Use the defiitio of a limit to show that (a) lim si = 0 (b) Proof. Let ε > 0 be give. Defie N >, where N is a positive iteger. The for ε > N, si 0 < si
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES
J. Noliear Var. Aal. (207), No. 2, pp. 20-22 Available olie at http://jva.biemdas.com ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES SHIH-SEN CHANG,, LIN WANG 2, YUNHE ZHAO 2 Ceter
More informationMulti parameter proximal point algorithms
Multi parameter proximal poit algorithms Ogaeditse A. Boikayo a,b,, Gheorghe Moroşau a a Departmet of Mathematics ad its Applicatios Cetral Europea Uiversity Nador u. 9, H-1051 Budapest, Hugary b Departmet
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationSome Approximate Fixed Point Theorems
It. Joural of Math. Aalysis, Vol. 3, 009, o. 5, 03-0 Some Approximate Fixed Poit Theorems Bhagwati Prasad, Bai Sigh ad Ritu Sahi Departmet of Mathematics Jaypee Istitute of Iformatio Techology Uiversity
More informationHomework 4. x n x X = f(x n x) +
Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.
More informationCommon Fixed Point Theorems for Four Weakly Compatible Self- Mappings in Fuzzy Metric Space Using (JCLR) Property
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 3 Ver. I (May - Ju. 05), PP 4-50 www.iosrjourals.org Commo Fixed Poit Theorems for Four Weakly Compatible Self- Mappigs
More informationNotes #3 Sequences Limit Theorems Monotone and Subsequences Bolzano-WeierstraßTheorem Limsup & Liminf of Sequences Cauchy Sequences and Completeness
Notes #3 Sequeces Limit Theorems Mootoe ad Subsequeces Bolzao-WeierstraßTheorem Limsup & Limif of Sequeces Cauchy Sequeces ad Completeess This sectio of otes focuses o some of the basics of sequeces of
More informationA FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE. Abdolrahman Razani (Received September 2004)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 (2006), 109 114 A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE Abdolrahma Razai (Received September 2004) Abstract. I this article, a fixed
More informationf n (x) f m (x) < ɛ/3 for all x A. By continuity of f n and f m we can find δ > 0 such that d(x, x 0 ) < δ implies that
Lecture 15 We have see that a sequece of cotiuous fuctios which is uiformly coverget produces a limit fuctio which is also cotiuous. We shall stregthe this result ow. Theorem 1 Let f : X R or (C) be a
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationA General Iterative Scheme for Variational Inequality Problems and Fixed Point Problems
A Geeral Iterative Scheme for Variatioal Iequality Problems ad Fixed Poit Problems Wicha Khogtham Abstract We itroduce a geeral iterative scheme for fidig a commo of the set solutios of variatioal iequality
More informationSome iterative algorithms for k-strictly pseudo-contractive mappings in a CAT (0) space
Some iterative algorithms for k-strictly pseudo-cotractive mappigs i a CAT 0) space AYNUR ŞAHİN Sakarya Uiversity Departmet of Mathematics Sakarya, 54187 TURKEY ayuce@sakarya.edu.tr METİN BAŞARIR Sakarya
More informationResearch Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property
Discrete Dyamics i Nature ad Society Volume 2011, Article ID 360583, 6 pages doi:10.1155/2011/360583 Research Article A Note o Ergodicity of Systems with the Asymptotic Average Shadowig Property Risog
More informationINTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010
Fixed Poits theorem i Fuzzy Metric Space for weakly Compatible Maps satisfyig Itegral type Iequality Maish Kumar Mishra 1, Priyaka Sharma 2, Ojha D.B 3 1 Research Scholar, Departmet of Mathematics, Sighaia
More informationLocal Approximation Properties for certain King type Operators
Filomat 27:1 (2013, 173 181 DOI 102298/FIL1301173O Published by Faculty of Scieces ad athematics, Uiversity of Niš, Serbia Available at: http://wwwpmfiacrs/filomat Local Approimatio Properties for certai
More informationUniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations
Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie
More information2.4.2 A Theorem About Absolutely Convergent Series
0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationOn Some New Entire Sequence Spaces
J. Aa. Num. Theor. 2, No. 2, 69-76 (2014) 69 Joural of Aalysis & Number Theory A Iteratioal Joural http://dx.doi.org/10.12785/jat/020208 O Some New Etire Sequece Spaces Kuldip Raj 1, ad Ayha Esi 2, 1 School
More informationFuzzy n-normed Space and Fuzzy n-inner Product Space
Global Joural o Pure ad Applied Matheatics. ISSN 0973-768 Volue 3, Nuber 9 (07), pp. 4795-48 Research Idia Publicatios http://www.ripublicatio.co Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationKorovkin type approximation theorems for weighted αβ-statistical convergence
Bull. Math. Sci. (205) 5:59 69 DOI 0.007/s3373-05-0065-y Korovki type approximatio theorems for weighted αβ-statistical covergece Vata Karakaya Ali Karaisa Received: 3 October 204 / Revised: 3 December
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationINVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R + )
Electroic Joural of Mathematical Aalysis ad Applicatios, Vol. 3(2) July 2015, pp. 92-99. ISSN: 2090-729(olie) http://fcag-egypt.com/jourals/ejmaa/ INVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R +
More informationSequences. A Sequence is a list of numbers written in order.
Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,
More informationWeakly Connected Closed Geodetic Numbers of Graphs
Iteratioal Joural of Mathematical Aalysis Vol 10, 016, o 6, 57-70 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ijma01651193 Weakly Coected Closed Geodetic Numbers of Graphs Rachel M Pataga 1, Imelda
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationAn elementary proof that almost all real numbers are normal
Acta Uiv. Sapietiae, Mathematica, 2, (200 99 0 A elemetary proof that almost all real umbers are ormal Ferdiád Filip Departmet of Mathematics, Faculty of Educatio, J. Selye Uiversity, Rolícej šoly 59,
More information} is said to be a Cauchy sequence provided the following condition is true.
Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r
More informationResearch Article Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function
Hidawi Publishig Corporatio Abstract ad Applied Aalysis, Article ID 88020, 5 pages http://dx.doi.org/0.55/204/88020 Research Article Ivariat Statistical Covergece of Sequeces of Sets with respect to a
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationBanach Lattices and the Weak Fixed Point Property
Baach Lattices ad the Weak Fixed Poit Property Tim Dalby ad Brailey Sims School of Mathematics, Statistics ad Computer Sciece, The Uiversity of New Eglad, NSW 2351, Australia, e-mail: timd@turig.ue.edu.au
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationResearch Article Fixed Points and Stability of an Additive Functional Equation of n-apollonius Type in C -Algebras
Abstract ad Applied Aalysis Volume 008, Article ID 67618, 13 pages doi:10.1155/008/67618 Research Article Fixed Poits ad Stability of a Additive Fuctioal Equatio of -Apolloius Type i C -Algebras Fridou
More informationInternational Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN
Iteratioal Joural of Mathematical Archive-3(4,, Page: 544-553 Available olie through www.ima.ifo ISSN 9 546 INEQUALITIES CONCERNING THE B-OPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet
More informationResearch Article Generalized Vector-Valued Sequence Spaces Defined by Modulus Functions
Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 00, Article ID 45789, 7 pages doi:0.55/00/45789 Research Article Geeralized Vector-Valued Sequece Spaces Defied by Modulus Fuctios
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More information1. Introduction. g(x) = a 2 + a k cos kx (1.1) g(x) = lim. S n (x).
Georgia Mathematical Joural Volume 11 (2004, Number 1, 99 104 INTEGRABILITY AND L 1 -CONVERGENCE OF MODIFIED SINE SUMS KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Abstract. New modified sie sums are itroduced
More informationSeveral properties of new ellipsoids
Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationMetric Space Properties
Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All
More informationJournal of Mathematical Analysis and Applications 250, doi: jmaa , available online at http:
Joural of Mathematical Aalysis ad Applicatios 5, 886 doi:6jmaa766, available olie at http:wwwidealibrarycom o Fuctioal Equalities ad Some Mea Values Shoshaa Abramovich Departmet of Mathematics, Uiersity
More informationResearch Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More information