Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
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A common random fixed point theorem for rational ineqality in hilbert space using integral type mappings

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A common random fixed point theorem for rational ineqality in hilbert space using integral type mappings

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications160A Common Random Fixed Point Theorem for Rational Ineqalityin Hilbert Space using Integral Type MappingsRAMAKANT BHARDWAJAssociate Professor in Truba Institute of Engineering & Information Technology, Bhopal (M.P)Mr. PIYUSH M. PATEL*Research Scholar of CMJ University, Shillong (Meghalaya), pmpatel551986@gmail.comdrrkbhardwaj100@gmail.com*Corresponding authorABSTRACT: The object of this paper is to obtain a common random fixed point theorem for four continuousrandom operators defined on a non empty closed subset of a separable Hilbert space for integral type mapping.Key wards: common fixed point, rational expression, hilbert space random variable1. INTRODUCTION AND PRELIMINARIESImpact of fixed point theory in different branches of mathematics and its applications is immense. The firstresult on fixed points for Contractive type mapping was the much celebrated Banach’s contraction principle by S.Banach [9] in 1922. In the general setting of complete metric space, this theorem runs as the follows,Theorem 1.1 (Banach’s contraction principle) Let (X,d)be a complete metric space, c (0,1)∈ and f: X→X bea mapping such that for each x, y ∈X, d (fx,fy) cd x,y≤ Then f has a unique fixed point a ∈X, such that foreach x ∈ X, lim nn f x a→∞ = .After the classical result, Kannan [7] gave a subsequently new contractive mapping to prove the fixed pointtheorem. Since then a number of mathematicians have been worked on fixed point theory dealing with mappingssatisfying various type of contractive conditions.In 2002, A. Branciari [1] analyzed the existence of fixed point for mapping f defined on a complete metricspace (X,d) satisfying a general contractive condition of integral type.Theorem 1.2 (Branciari) Let (X,d) be a complete metric space, c (0,1)∈ and let f: X→X be a mapping suchthat for each x, y ∈ X,:[0, ) [0, )ξ +∞ → +∞ is a LesbesgueWhereintegrable mapping which is summable on each compact subset of[0, )+∞ , non negative, and such that for eachϵ > o, ( )0t dtξ∈∫ , then f has a unique fixed point a ∈ X such that for each x∈X, limnf x an =→∞ . After thepaper of Branciari, a lot of a research works have been carried out on generalizing contractive conditions ofintegral type for different contractive mappings satisfying various known properties. A fine work has been doneby Rhoades [2] extending the result of Branciari by replacing the condition [1.2] by the followingThe aim of thispaper is togeneralize some mixed type of contractive conditions to the mapping and then a pair of mappings satisfyinggeneral contractive mappings such as Kannan type [7], Chatrterjee type [8], Zamfirescu type [11], etc.In recent years, the study of random fixed points has attracted much attention. Some of the recent literatures inrandom fixed point may be noted in Rhoades [3], and Binayak S. Choudhary [4].In this paper, we construct asequence of measurable functions and consider its convergence to the common unique random fixed point offour continuous random operators defined on a non-empty closed subset of a separable Hilbert space. For thepurpose of obtaining the random fixed point of the four continuous random operators. We have used a rationalinequality (from B. Fisher [5] and S.S. Pagey [10]) and the parallelogram law. Throughout this paper, (Ω, Σ)denotes a measurable space consisting of a set Ω and sigma algebra Σ of subsets of Ω, H stands for a separableHilbert space and C is a nonempty closed subset of H.d (fx,fy) ( , )( ) ( )0 0d x yt dt t dtξ ξ≤∫ ∫0 0( , ) ( , )max ( , ), ( , ), ( , ),( , ) 2( ) ( )d x fy d y fxd x y d x fx d y fyd fx fyt dt t dtξ ξ     +≤∫ ∫
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications161Definition 1.3. A function :F CΩ → is said to be measurable if1( )f B C−∩ ∈∑ for every Borel subset Bof H.Definition 1.4. A function :F C CΩ× → is said to be a random operator if (., ) :F x CΩ → is measurablefor every x C∈Definition 1.5. A measurable :g CΩ → is said to be a random fixed point of the random operator:F C CΩ× → if ( , ( )) ( )F t g t g t= for all t ∈ΩDefinition 1.6. A random operator :F C CΩ× → is said to be continuous if for fixed t ∈Ω ,( ,.) :F t C C→ is continuous.Condition (A). Four mappings E, F, T, S: C → C, where C is a non-empty closed subset of a Hilbert space H, issaid to satisfy condition (A) ifES = SE, FT = TF, E (H) ⊂ T (H) and F (H) ⊆ S (H) --------------------------- (1)0 00002 2 222 22 2 2212 22( ) ( )1( )2( )3( )42 2with , 0 and , , , 01 2 3 4Sx Ex Ty Fy Ex TyE Fx y Sx Ty Ex TySx Ex Ex Ty Ty FySx Ex Ex Ty T Fy ySx Ex Ty FySx Tyt dt t dtt dtt dtt dtSx Ty Sx Ty Ex Tyξ α ξα ξα ξα ξα α α α   − − + −− − + −− + − + −+ − ⋅ − ⋅ −− + −−≤+++≠ − + − ≠ ≥−∫ ∫∫∫∫(2)− − − − − − − − − − − − − − − − − − − − −where ξ ∶ℛ+→ ℛ+is a lebesgue- integrable mapping which is summable on each compact subset of ℛ+, non negative, andsuch that for each foreach ϵ > o ,( )0t d tξ∈∫. )3(21121321432−−−−−−−−−−−−−−−−−−−−−++−++pαααααα
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications1622. MAIN RESULTSTheorem 2.1. Let C be a non-empty closed subset of a separable Hilbert space H . Let , ,E F S and T befour continuous random operators defined on C such that for, t ∈Ω ( ,.), ( ,.), ( ,.), ( ,.) :E t F t T t S t C C→satisfy condition (A).Then , ,E F S and T have unique common random fixed point.Proof: Let the function :g CΩ → be arbitrary measurable function. By (1), there exists a function1 :g CΩ → such that 1 0( , ( )) ( , ( ))T t g t E t g t= fort ∈Ω and for this function 1 :g CΩ → , we can chooseanother function 2 :g CΩ → such that 1 2( , ( ) ( , ( ))F t g t S t g t= for t ∈Ω , and so on. Inductively, we candefine a sequence of functions fort ∈Ω , { }( )ny t such that2 2 1 22 1 2 1 2 1( ) ( , ( )) ( , ( )) an d( ) ( , ( )) ( , ( ))fo r 1, 2 , 3 ........(4 )n n nn n ny t T t g t E t g ty t S t g t F t g tt n++ + += == =∈ Ω =− − − − − − − − − − − − − − − −From condition (A), we have for, t ∈Ω0 002 2 2( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( ))2 2 2 1 2 1 2 2 12( , ( )) ( , ( )) ( , ( )) ( ,2 2 1 2 22 2( ) ( ) ( , ( )) ( , ( ))2 2 1 2 2 1( ) ( )( )1S t g t E t g t T t g t F t g t E t g t T t g tn n n n n nS t g t T t g t E t g t T t gn n n ny t y t E t g t F t g tn n n nd ddξ λ λ ξ λ λα ξ λ λ   − − + −+ + +− + −+− −+ +=≤∫ ∫002( ))12 2 2( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( ))2 2 2 2 1 2 1 2 11 ( , ( )) ( , ( )) . ( , ( )) ( , ( )) . ( , ( )) ( , ( ))2 2 2 2 1 2 1 2 1( , ( )2 2( )2( )3tS t g t E t g t E t g t T t g t T t g t F t g tn n n n n nS t g t E t g t E t g t T t g t T t g t F t g tn n n n n nS t g tnddα ξ λ λα ξ λ λ+− + − + −+ + ++ − − −+ + ++++∫∫02) ( , ( )) ( , ( )) ( , ( ))2 1 2 1 2 12( , ( )) ( , ( ))2 2 1by(2)( )4E t g t T t g t F t g tn n nS t g t T t g tn ndα ξ λ λ− + −+ + +− ++∫∫
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications163002 2 2( ) ( ) ( ) ( ) ( ) ( )2 1 2 2 2 1 2 22 2( ) ( ) ( ) ( )2 1 2 2 22 2 2( ) ( ) ( ) ( ) ( ) ( )2 1 2 2 2 2 2 12 21 ( ) ( ) . ( ) ( ) . ( ) (2 1 2 2 2 2 2 112( )( )y t y t y t y t y t y tn n n n n ny t y t y t y tn n n ny t y t y t y t y t y tn n n n n ny t y t y t y t y t yn n n n n nddααξ λ λξ λ λ   − − + −− +− + −−− + − + −− ++ − − −− +=+∫000 002)2 2( ) ( ) ( ) ( )2 1 2 2 2 12 2( ) ( ) ( ) ( )2 1 2 2 2 12 2( ) ( ) ( ) ( )2 2 1 2 1 22( ) ( )2 2 132( )( ) by (4)4( ) ( ) ( ) ( )1 2 3 2 3 4( )ty t y t y t y tn n n ny t y t y t y tn n n ny t y t y t y tn n n ny t y tn nddd ddααξ λ λα ξ λ λα α α ξ λ λ α α α ξ λ λξ λ λ− + −+ +− + −+ +− −+ −− ++≤+= + + + + +⇒∫∫∫∫ ∫∫ 00 02( ) ( )2 1 2( ) ( ) ( ) ( )2 2 1 2 1 23 41 1 2 3122 3 41 1 2 3( )( ) ( )y t y tn ny t y t y t y tn n n ndd dα αα α αα α αα α αξ λ λξ λ λ ξ λ λ−−− −+ −+ +− + ++ +⇒ ≤− + +      ∫∫ ∫
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications164In integralNow, we shall prove that for t ∈Ω , ( )ny t is a Cauchy sequence. For this for Common random fixed pointtheorem every positive integer p, we have0 0( ) ( ) ( ) ( )2 2 1 2 1 2( ) ( )1212 3 4where by (3)1 21 2 3y t y t y t y tn n n nKd dKξ λ λ ξ λ λα α αα α α− −+ −≤+ +=− + +   ∫ ∫p0 0( ) ( ) ( ) ( )1 0 1for(5)( ) ( )y t y t y t y tn nK td dξ λ λ ξ λ λ− −+′′⇒ ≤ ∈Ω−−−−−−−−−−−−−−−−−−−−−−−−−−−−∫ ∫0 000( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( ) ( )1 1 2 2 1( ) ( ) ( ) ( ) ... ( ) ( )1 1 2 1( ) ( )0 111...21( ) ( )( )( )y t y t y t y t y t y t y t y t y tn n p n n pn n n n n py t y t y t y t y t y tn n pn n n n py t y tn pn nk k knk k kd dddξ λ λ ξ λ λξ λ λξ λ λ− − + − + + + −+ ++ + + + + −− + − + + − ++ + + + −−=≤+ −+≤ + + += + +  ∫ ∫∫∫( )000( ) ( )0 1( ) ( )0 1( ) ( )1...10 for(6)( )( )( )y t y ty t y ty t y tn n ppknkkLim tndddξ λ λξ λ λξ λ λ−−− +−+ +≤−⇒ = ∈Ω→∞−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−  ∫∫∫
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications165From equation (6), it follows that for, { }( )ny t is a Cauchy sequence and hence is convergent in closed subset Cof Hilbert space H.For t ∈Ω , letAgain, closeness of C gives that g is a function from C to C. and consequently the subsequences{ }2( , ( ))nE t g t,{ }2 1( , ( ))nF t g t+ { }2 1( , ( ))nT t g t+and{S(t,g2n+2(t))}of{ }( )ny tfor t ∈Ω , also converges to the( )y t .........(*) and continuity of E,F, T and S gives(From (1)) ------------------------------- (8)Existence of random fixed point: Consider for t ∈Ω{ }0( )lim ( ) (7)( )y tny tnt dtξ = − − − − − − − − − − − − − − − − − − − − − −→∞ ∫0 0 0 00 0 0 00( , ( , ( ))) ( , ( )) ( , ( , ( ))) ( , ( ))( , ( , ( ))) ( , ( )) ( , ( , ( ))) ( , ( ))( , ( ))and( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ),,E t S t g t E t y t S t E t g t S t y tn nF t T t g t F t y t T t F t g t T t y tn nE t y td ddd d d dd d d dd dλ λλξ λ λ ξ λ λ ξ λ λ ξ λ λξ λ λ ξ λ λ ξ λ λ ξ λ λξ λ λ ξ λ λ→ →→ →=∫ ∫ ∫ ∫∫ ∫ ∫ ∫∫ 0 0 0( , ( )), ( , ( )) ( , ( )), ( ) ( ) forS t y t F t y t T t y td d tξ λ λ ξ λ λ= ∈Ω∫ ∫ ∫
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications166000 002( , ( )) ( ))2( , ( )) ( ) ( ) ( ))2 1 2 12 2( , ( )) ( , ( )) ( ) ( ))2 1 2 122( , ( )) ( , ( )) ( , ( )) ( , ( )) (2 1 2 12 1( )( )2 ( ) 2 ( ) by(4)( )E t y t y tE t y t y t y t y tn nE t y t F t g t y t y tn nS t y t E t y t T t g t F t g t E tn nddd ddαξ λ λξ λ λξ λ λ ξ λ λξ λ λ−− + −+ +− −+ +− − ++ +≤== +∫∫∫ ∫02, ( )) ( , ( ))2 12 2( , ( )) ( , ( )) ( , ( )) ( , ( ))2 1 2 12 22( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( ))2 1 2 1 2 11 ( , ( )) ( , ( )) . ( , ( )) ( , 22 2 ( )y t T t g tnS t y t T t g t E t y t T t g tn nS t y t E t y t E t y t T t g t T t g t F t g tn n nS t y t E t y t E t y t T t g dα ξ λ λ   − +− + −+ +− + − + −+ + ++ − −+∫0( )) . ( , ( )) ( , ( ))1 2 1 2 12( , ( )) ( , ( )) ( , ( )) ( , ( ))2 1 2 1 2 12 3 ( )t T t g t F t g tn n nS t y t E t g t T t g t F t g tn n ndα ξ λ λ−+ + +− + −+ + ++∫∫0 02 2( , ( )) ( , ( )) ( ) ( )2 1 2 12 2 ( )4 ( )S t y t T t g t y t y tn nddα ξ λ λξ λ λ− −+ ++ +∫ ∫Therefore for t ∈Ω( )0 0002 2( , ( )) ( )) ( , ( )) ( ))2( , ( )) ( ))2( , ( )) ( ))2 41 2 04( ) ( )( )1( ) 0 as 42E t y t y t E t y t y tE t y t y tE t y t y td dddααξ λ λ ξ λ λξ λ λξ λ λ α− −−−≤⇒ − ≤⇒ =∫ ∫∫∫ p
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications167Common random fixed point theorem:0 0( , ( )) ( )( ) ( )E t y t y td dξ λ λ ξ λ λ=∫ ∫ for t ∈Ω ----------------------------------(9)From (8) and (9) we have for all0 0 0( , ( )) ( ) ( , ( ))( ) ( ) ( )E t y t y t S t y td d dξ λ λ ξ λ λ ξ λ λ= =∫ ∫ ∫ ------------------------------ (10)In an exactly similar way, we can prove that for all0 0 0( , ( )) ( ) ( , ( ))( ) ( ) ( )F t y t y t T t y td d dξ λ λ ξ λ λ ξ λ λ= =∫ ∫ ∫ -------------------------------- (11)Again, if A : Ω×C → C is a continuous random operator on a nonempty closed subset C of a separable Hilbertspace H, then for any measurable function f : Ω → C, the function h(t) = A(t, f(t)) is also measurable [4].It follows from the construction of { }( )ny t for t ∈Ω , (by (4)) and above consideration that { }( )ny t is asequence of measurable function. From (7), it follows that ( )y t for t ∈Ω , is also measurable function. This factalong with (10) and (11) shows that g: Ω → C is a common random fixed point of E, F, S and T.Uniqueness: Let h: Ω → C be another random fixed point common toE, F, T and S, that is, for t ∈Ω ,( , ( )) ( ), ( , ( )) ( ), ( , ( )) ( ) ( , ( )) ( )(12)F t h t h t F t h t h t T t h t h t and S t h t h t= = = =− − − − − − − − − − − − − − − − −Then for t ∈Ωt ∈Ωt ∈Ω
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications1682( ) ( )0002 2 2( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( ))2 2( , ( )) ( , ( )) ( , ( )) ( , ( ))2 2( , ( )) ( , ( )) ( , ( )) ( , ( )) ( , ( ))2( )( )1( )g t h tS t g t E t g t T t h t F t h t E t g t T t h tS t g t T t h t E t g t T t h tS t g t E t g t E t g t T t h t T t h t Fdddαξ λ λα ξ λ λξ λ λ−   − − + −− + −− + − + −+≤∫∫22( , ( )) ( , ( )) ( , ( )) ( , ( ))0( , ( )) ( , ( ))02( , ( ))1 ( , ( )) ( , ( )) . ( , ( )) ( , ( )) . ( , ( )) ( , ( ))4( )3( )S t g t E t g t T t h t F t h tS t g t T t h tt h tS t g t E t g t E t g t T t h t T t h t F t h tddαα ξ λ λξ λ λ− + −−+ − − −++∫∫∫2 222( ) ( ) ( ) ( )0( ) ( )0( ) ( )0( ) ( )0 0( ) ( ) by (12)41 ( ) 041( ) 0 (as )42( ) ( ) for t( ) ( ) for t( - )g t h t g t h tg t h tg t h tg t h td dddd dg t h tξ λ λ α ξ λ λα ξ λ λξ λ λ αξ λ λ ξ λ λ− −−−⇒ ≤⇒ ≤⇒⇒ ∈ Ω⇒ = ∈ Ω==∫ ∫∫∫∫ ∫p
  10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications169This completes the proof of the theorem (2.1).ACKNOWLEDGEMENTS: The authors are thankful to Prof. B.E.Rhoades [Indiana University, Bloomington,USA] S.S.Pagey and Neeraj malviya for providing us necessary Literature of fixed point theory.References:[1] A. Branciari, A fixed point theorem for mappings satisfying a general con- tractive condition of integraltype, Int.J.Math.Math.Sci, 29(2002), no.9, 531 - 536.[2] B.E Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integraltype, International Journal of Mathematics and Mathematical Sciences, 63, (2003), 4007 - 4013.[3] B.E. Rhoades, Iteration to obtain random solutions and fixed points of oper-ators in uniformly convex Banach spaces, Soochow Journal of mathematics ,27(4) (2001), 401 – 404[4] Binayak S. Choudhary, A common unique fixed point theorem for two ran- dom operators in Hilbert space,IJMMS 32(3)(2002), 177 - 182.[5] B. Fisher, Common fixed point and constant mapping satisfying a rational inequality, Math. Sem. Kobe Univ,6(1978), 29 - 35.[6] C.J. Himmelberg, Measurable relations, Fund Math, 87 (1975), 53 - 72.[7] R. Kannan, Some results on fixed points, Bull.Calcutta Math. Soc. 60(1968), 71-76[8] S.K.Chatterjea, Fixedpoint theorems, C.R.Acad.Bulgare Sci. 25(1972), 727- 730.[9] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrals, Fund.Math.3, (1922)133181 (French).[10] S.S. Pagey, Shalu Srivastava and Smita Nair, Common fixed point theorem for rational inequality in a quasi2-metric space, Jour. Pure Math., 22(2005), 99 - 104.[11] T.Zamfrescu, Fixed point theorems in metric spaces, Arch.Math.(Basel) 23(1972), 292-298.
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