Some Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions

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1 ISSN(Olie): ISSN (Prit): 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 Space uder T Kaa ad T Reich Cotractive Coditios M. Ragamma 1, P. Mallikarju Reddy * Professor, Departmet of Mathematics, Osmaia Uiversity, Hyderabad, Telagaa, Idia 1 Lecturer i Mathematics, Govt. Polytechic for wome (Mi.), Badagpet, R.R. District, Telagaa, Idia ABSTRACT: The purpose of this paper is to establish some commo fixed poit theorems for two self mappigs which satisfy T- Kaa ad T- Reich cotractive coditios i coe rectagular metric space. KEYWORDS: coe rectagular metric space, commo fixed poit theorem, coicidece poit, cotractive coditio. I. INTRODUCTION Recetly, Huag ad Zhag [5] itroduced the otio of coe metric space. They have replaced real umber system by a ordered Baach space ad established some fixed poit theorems for cotractive type mappigs i a ormal coe metric space. The study of fixed poit theorems i such spaces is followed by some other mathematicias; see [1], [5], [8], [11], [14]. I 9, Azam, Arshad ad Beg [] exteded the otio of coe metric spaces by replacig the triagular iequality by a rectagular iequality ad they proved Baach cotractio Priciple i a complete ormal coe rectagular metric space. Several authors proved some fixed poit theorems i such spaces see; [6], [9], [1], [1], [15]. I 9, Jleli, Samet [6] exteded the Kaa s fixed poit theorem i a complete ormal coe rectagular metric space. I 1, R. A. Rashwa ad S. M. Saleh [1] exteded Baach cotractio priciple i coe rectagular metric space with two self mappigs ad proved commo fixed poit theorem for T- cotractive coditio i coe rectagular metric space. I 13, Malhotra et al. [1] geeralized the result of Azam et al. [] i ordered coe rectagular metric space ad proved some fixed poit results for ordered Reich type cotractios. I this paper, we prove some commo fixed poit theorems for two self mappigs which satisfy T Kaa ad T Reich cotractive coditios i coe rectagular metric space. Our results geeralize ad exted the results of M. Jleli et al. [6] ad Malhotra et al. [1] o coe rectagular metric spaces. II. PRELIMINARIES First, we recall some stadard defiitios ad other results that will be eeded i the sequel..1. Defiitio [5]: A subset P of a real Baach space E is called a coe if it has followig properties: (1) P is o empty, closed ad P ; () a, b R, a, b, x, y P implies ax by P ; (3) P P. For a give coe P E, we ca defie a partial orderig o E with respect to P by x y y x P. We shall write x y if x y ad x y, deotes the iterior of P. while x << y will stads for y x it ( P), if ad oly if where it P Copyright to IJIRSET DOI: /IJIRSET

2 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio).. Defiitio [5]: The coe P is called ormal if there is a umber k > 1 such that for all x, y X x y implies x k y. The least positive umber k satisfyig the above coditio is called the ormal costat of P..3. Defiitio [5]: Let X is a o empty set, E is a real Baach space ad P is a coe i E with it P ad is a partial orderig with respect to P. Suppose the mappig d : XX E satisfies: (1) () d x, y d y, x, for all x, y X ; (3) d x, y, for all x, y X ad d x, y if ad oly if x y; d x, y d x, z d z, y, for all x, y, z X. The d is called a coe metric o X ad (X, d) is called a coe metric space..4. Defiitio []: Let X is a o empty set, E is a real Baach space ad P is a coe i E with it P ad is a partial orderig with respect to P. Suppose that the mappig d : XX E satisfies: (1) d x, y, for all x, y X ad d x, y if ad oly if x y; () d x, y d y, x, for all x, y X ; (3) d x, y d x, w d w, z d z, y, for all x, y X ad for all distict poits w, z X x, y [rectagular property]. The d is called a coe rectagular metric o X ad (X, d) is called a coe rectagular metric space. x be a sequece i (X, d) ad x X. If for.5. Defiitio []: Let (X, d) be a coe rectagular metric space. Let every c E, with c N, there is such that for all x coverges to x ad x is the limit of x. d x, x c, the We deote this by lim x x or x x, as. x is said to be coverget,.6. Defiitio []: Let (X, d) be a coe rectagular metric space. Let x be a sequece i (X, d). If for every c E, with c there is N such that for all m,, d x, x c, the x m is called a Cauchy sequece i (X, d)..7. Defiitio []: Let (X, d) be a coe rectagular metric space. If every Cauchy sequece is coverget i (X, d), the (X, d) is called a complete coe rectagular metric space..8. Lemma []: Let (X, d) be a coe rectagular metric space ad P be a ormal coe with ormal costat k, let x be a sequece i X. The x coverges to x if ad oly if d( x, x), as..9. Lemma []: Let (X, d) be a coe rectagular metric space, P be a ormal coe with ormal costat k. Let x be a sequece i X. The x is a Cauchy sequece if ad oly if d x, x, as, m. m.1. Defiitio [3]: A self map T: X X o a metric space (X, d) is called Baach cotractio if for all x, y X, there exists λ [, 1) such that, d(tx, Ty) λ d(x, y)..11. Defiitio [7]: A self map T: X X o a metric space (X, d) is called Kaa cotractio if for all x, y X, there exists λ [, 1/) such that, d(tx, Ty) λ [d(x, Tx) + d(y, Ty)]..1. Defiitio [4]: Let T ad S be two self mappig of a metric space (X, d). The self mappig S of X is said to be a T cotractio if for all x, y X there exists a real umber λ < 1 such that d(tsx, TSy) λ d(tx, Ty)..13. Defiitio [13]: A self map T: X X o a metric space (X, d) is called Reich cotractio if for all x, y X, there exists λ, μ, δ [, 1) with λ + μ + δ < 1 such that, d(tx, Ty) λd(x, y) + μ d(x, Tx) + δ d(y, Ty). Copyright to IJIRSET DOI: /IJIRSET

3 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) III. MAIN RESULTS First we give defiitios of T- Kaa cotractive ad T-Reich cotractive mappigs o coe rectagular metric spaces which are based o the ideas of Morales ad Rojas [11] Defiitio: Let (X, d) be a coe rectagular metric space ad T, S: X X two fuctios: (i) A mappig S is said to be T Kaa cotractio, if there is λ [, ½) such that, d TSx, TSy d Tx, TSx d Ty, TSy, for all x, y X. (ii) A mappig S is said to be T Reich cotractio, if there is λ 1, λ, λ 3 with λ 1 + λ + λ 3 < 1 such that, d TSx, TSy d Tx, TSx d Ty, TSy d Tx, Ty, for all x, y X. 3.. Theorem: Let (X, d) be a coe rectagular metric space, P be a ormal coe with ormal costat k ad let the mappigs S ad T: X X satisfy the followig: d TSx, TSy d Tx, TSx d Ty, TSy, (1) for all x, y X, where λ [, ½). Suppose T is oe to oe ad T(X) is a complete subspace of X, the the mappig S has a uique fixed poit i X. Moreover, if S ad T are commutig at the fixed poit of S, the S ad T have a uique commo fixed poit i X. Proof: Let x be a arbitrary poit i X. Defie a sequece {x } i X such that x +1 = Sx, for all =,1,,.... If x m = x m+1, for some m N, the x m = Sx m. That is, S has a fixed poit x m i X. Assume x x +1, for all N. The from (1) it follows that, d( Tx, Tx ) d( TSx, TSx ) 1 1,, 1 1,, d Tx TSx d Tx TSx d Tx Tx d Tx Tx 1 1 which implies that, d( Tx, Tx ) d( Tx, Tx ), for all =,1,, Thus, d ( Tx, Tx ) d ( Tx, Tx ) d ( Tx, Tx ) d ( Tx, Tx ), () For all, where 1. 1 From (1), () ad usig the facts λ μ ad λ < ½ < 1, we get, d( Tx, Tx ) d( TSx, TSx ) 1 1 d Tx, TSx d Tx, TSx d Tx, Tx d Tx, Tx d Tx, Tx 1 d Tx, Tx d Tx, Tx 1 d Tx, Tx dtx Tx (1 ),, for all (3) For the sequece {Tx } we cosider d(tx, Tx +p ) i two cases. If p is odd say m +1, for m 1, the by usig rectagular iequality ad () we get, d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) m1 m1 m m m1 m1 Copyright to IJIRSET DOI: /IJIRSET

4 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) m m1 m1 m m1 m d( Tx, Tx )... d( Tx, Tx ) d( Tx, Tx ) m m3 1 1 d( Tx, Tx ) d( Tx, Tx )... d( Tx, Tx ) d( Tx, Tx ) 1 1 m1 m m m1 1 m1 m d( Tx, Tx ) d( Tx, Tx )... d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) Hece, d ( Tx, Tx ) d ( Tx, Tx ), for all 1, m 1. m1 1 If p is eve say m, for m 1, the by usig rectagular iequality, (), (3) ad the fact that μ <1 we get, d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) m m m1 m1 m m d( Tx, Tx ) d( Tx, Tx )... d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) m1 m m m d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx )... d( Tx, Tx ) d ( Tx, Tx ) m m1 m1 m 3 (1 ) d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx ) Hece, d( Tx, Tx ) d ( Tx, Tx ), for all 1, m 1. m 1 Thus, combiig the above two cases we have, d ( Tx, Tx ) d ( Tx, Tx ), for all N, p N. p 1 Sice, P is a ormal coe with ormal costat k ad 1, we have k d ( Tx, Tx ) d ( Tx, Tx ), as, p 1 i. e., d( Tx, Tx ), as, p N. p m m1... d( Tx, Tx ) d( Tx, Tx ) Therefore, {Tx } is a Cauchy sequece i X. Sice, T(X) is a complete subspace of X, the there exists a poit z i T(X) such that limtx limtsx z (5) 1 Also, we ca fid x X such that z = Tx We shall show that TSx = z. Usig Rectagular property ad (1) we get, Which implies that, d( z, TSx) d( z, Tx ) d( Tx, TSx ) d( TSx, TSx) 1 d( z, Tx ) d( Tx, Tx ) d Tx, TSx d Tx, TSx d( z, Tx ) d( Tx, Tx ) d Tx, Tx d z, TSx 1 1 (4) Copyright to IJIRSET DOI: /IJIRSET

5 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) 1 1 d( z, TSx) d( z, Tx ) d( Tx, Tx ), 1 for all Sice, P is a ormal coe with ormal costat k, usig (4) ad (5) we get, k 1 d( z, TSx) d( z, Tx ) k d( Tx, Tx ), as, i. e., d ( z, TSx) Therefore, TSx = Tx = z. Sice T is oe to oe, x = Sx. Hece, x is a fixed poit of S i X. Now, we prove the uiqueess of the fixed poit of S. Let y be aother fixed poit of S, that is y = Sy, the, d( Tx, Ty) d( TSx, TSy) d Tx, TSx d Ty, TSy d Tx, Tx d Ty, Ty. Hece, Tx = Ty. Sice T is oe to oe, we coclude that x = y. Sice, S ad T are commutig at the fixed poit of S, TSx = STx = Tx. Therefore Tx is a fixed poit of S. Sice S has a uique fixed poit, Tx = x. Hece Tx = Sx = x, x is a uique commo fixed poit of S ad T i X Corollary [6]: Let (X, d) be a complete coe rectagular metric space, P be a ormal coe with ormal costat k. Suppose the mappig T : X X satisfies the cotractive coditio: d Tx, Ty d Tx, x d Ty, y, for all x, y X where λ [, ½). The T has a uique fixed poit i X. Proof: If we put S = I, i Theorem (3.), where I is the idetity mappig i X, the we obtai the corollary Theorem: Let (X, d) be a coe rectagular metric space, P be a ormal coe with ormal costat k ad let the mappigs S ad T: X X satisfy the followig: d TSx, TSy d Tx, TSx d Ty, TSy d Tx, Ty, (6) for all x, y X ad λ 1, λ, λ 3 with λ 1 + λ + λ 3 < 1. Suppose T is oe to oe ad T(X) is a complete subspace of X, the the mappig S has a uique fixed poit i X. Moreover, if S ad T are commutig at the fixed poit of S, the S ad T have a uique commo fixed poit i X. Proof: Let x be a arbitrary poit i X. Defie a sequece {x } i X such that x +1 = Sx, for all =,1,,.... If x m = x m +1, for some m N, the x m = Sx m. That is, S has a fixed poit x m i X. Assume x x +1, for all N. The from (6) it follows that, d( Tx, Tx ) d( TSx, TSx ) 1 1 which implies that, d Tx Tx d Tx 1 Tx,,,, d Tx, TSx d Tx, TSx d Tx, Tx, d Tx Tx d Tx Tx d Tx Tx (, ) (, ), 1 1 for all =,1,,.... Thus, d( Tx, Tx ) d( Tx, Tx ) d ( Tx, Tx ) d ( Tx, Tx ), (7) for all, where 1. 1 Usig the same argumet i the proof of theorem (3.) to prove that {Tx } is a Cauchy sequece i X. Sice, T(X) is a complete subspace of X, the there exists a poit z i T(X) such that limtx lim TSx z. 1 Cosequetly, we ca fid x X such that z = Tx Copyright to IJIRSET DOI: /IJIRSET

6 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) Now cosider, d( z, TSx) d( z, Tx ) d( Tx, TSx ) d( TSx, TSx) d( z, Tx ) d( Tx, Tx ) d Tx, TSx d Tx, TSx d Tx, Tx d( z, Tx ) d( Tx, Tx ) d Tx, Tx d z, TSx d Tx, z Which implies that, d ( z, TSx) d ( z, Tx ) d ( Tx, Tx ), for all 1. Sice, P is a ormal coe with ormal costat k, usig (7) ad (8) we get, d( z, TSx) k d( z, Tx ) k d ( Tx, Tx ), as, i. e., d ( z, TSx) Therefore, TSx = Tx = z. Sice T is oe to oe, x = Sx. Hece x is a fixed poit of S. Now, we prove the uiqueess of the fixed poit of S. Let y be aother fixed poit of S, that is y = Sy, the, d( Tx, Ty) d( TSx, TSy) d Tx, TSx d Ty, TSy d( Tx, Ty) d Tx, Tx d Ty, Ty d( Tx, Ty) d( Tx, Ty) 3 Sice, λ 3 < 1, d(tx, Ty) = Ɵ which implies that Tx = Ty. Sice T is oe to oe, we coclude that x = y. Sice, S ad T are commutig at the fixed poit of S, TSx = STx = Tx. Therefore Tx is a fixed poit of S. Sice S has a uique fixed poit, Tx = x. Hece Tx = Sx = x, x is uique commo fixed poit of S ad T i X. IV. CONCLUSION I this article we have proved that the existece ad uiqueess of commo fixed poit theorems for T-Kaa ad T- Reich cotractios i coe rectagular metric spaces. We ote that the results of this paper geeralize the results of M. Jleli et al. [6] ad Malhotra et al. [1] o coe rectagular metric spaces. REFERENCES [1] Abbas. M ad Jugck. G, Commo fixed poit results for o commutig mappigs without cotiuity i coe metric spaces, Joural of Mathematical Aalysis ad Applicatios, 341, 416 4, 8. [] Azam. A, Arshad. M ad Beg. I, Baach cotractio priciple o coe rectagular metric spaces, Appl. Aal. Discrete Math., 3, 36 41, 9. [3] Baach. S, Sur les op eratios das les esembles abstraits et leur applicatio aux equatios it egrales, Fudameta Mathematicae 3(1), , 19. [4] Beiravad. A, Moradi. S, Omid. M ad Pazadeh. H, Two fixed poit theorem for special mappig, arxiv: v1[math.FA], 9. [5] Huag. L. G, Zhag. X, Coe metric spaces ad fixed poit theorems of cotractive mappigs, J. Math. Aal. Appl., 33, , 7. [6] Jleli. M, Samet. B, The Kaa fixed poit theorem i a coe rectagular metric space, J. Noliear Sci. Appl.,, , 9. [7] Kaa. R, Some results o fixed poit, Bull. Calcutta Math. Soc. 6, 71-76, [8] Malhotra. S. K, Shukla. S ad Se. R, Some coicidece ad commo fixed poit theorems i coe metric spaces, Bulleti of Mathematical Aalysis ad Applicatios 4(), 64-71, 1. [9] Malhotra. S. K, Sharma. J. B ad Shukla. S, g-weak cotractio i ordered coe rectagular metric spaces, The Scietific World Joural, 13a. [1] Malhotra. S. K, Shukla. S ad Se. R, Some fixed poit theorems for ordered reich type cotractios i coe rectagular metric spaces, Acta Mathematica Uiversitatis Comeiaae (), , 13b. Copyright to IJIRSET DOI: /IJIRSET

7 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) [11] Morales. J. R ad Rojas. E, Coe metric spaces ad fixed poit theorems of T-Kaa cotractive mappigs, It. Joural of Math. Aalysis, 4(4), , 1. [1] Rashwa. R. A, Saleh. S. M, Some Fixed Poit Theorems i Coe Rectagular Metric Spaces, Mathematica Aetera, Vol., o. 6, , 1. [13] Reich. S, Some remarks cocerig cotractio mappigs, Caad. Nth. Bull. 14, 11-14,1971. [14] Rezapour. S, Hamlbarai. R, Some otes o paper Coe metric spaces ad fixed poit theorems of cotractive mappigs, J. Math. Aal. Appl., 345 (), , 8. [15] Satish Shukla, Reich Type Cotractios o Coe Rectagular Metric Spaces Edowed with a Graph, Theory ad Applicatios of Mathematics & Computer Sciece 4 (1), 14 5, 14. Copyright to IJIRSET DOI: /IJIRSET