INTUITIONISTIC RANDOM STABILITY OF A QUADRATIC FUNCTIONAL EQUATION ORIGINATING FROM THE SUM OF THE MEDIANS OF A TRIANGLE

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1 Iteratioal Joural of Differece Equatios ISSN Volume 1 Number 1 (017) pp Research Idia Publicatios INTUITIONISTIC RANDOM STABIITY OF A QUADRATIC FUNCTIONA EQUATION ORIGINATING FROM THE SUM OF THE MEDIANS OF A TRIANGE M. Arukumar 1 ad S. Karthikeya * 1 Departmet of Mathematics Govermet Arts College Tiruvaamalai TamilNadu Idia. aaru00@yahoo.co.i * Departmet of Mathematics R.M.K. Egieerig College Kavaraipettai Tamil Nadu Idia. ABSTRACT I this paper the authors proved the ituitioistic radom stability of quadratic fuctioal equatio x y y z z x 3 f z f x f y f x y f y z f z x 4 origiatig from the sum of the medias of a triagle usig direct ad fixed poit methods. 010 Mathematics Subject Classificatio. : 39B5 3B7 3B8. Key words ad phrases: Quadratic fuctioal equatio Geeralized Ulam-Hyers stability ituitioistic radom ormed space fixed poit method. 1. Itroductio The stability of fuctioal equatios had bee first raised by S.M. Ulam [46]. I 1941 D. H. Hyers [4] remarked a positive aswer to the questio of Ulam regards to Baach spaces. I 1950 T. Aoki [3] was cosidered as the secod author to hadle this problem for additive mappigs.

2 56 M. Arukumar ad S. Karthikeya Evetually Th.M. Rassias [39] succeeded i extedig the result of Hyers' Theorem by weakeig the coditio for the Cauchy differece cotrolled by p p x y ; [01) p to be ubouded. While cosiderig the ifluece of Ulam Hyers ad Rassias o the developmet of stability problems of fuctioal equatios the stability pheomeo that was proved by Th.M. Rassias is called Hyers-Ulam- Rassias stability oe ca refer [ 19 5]. I 198 J.M. Rassias [35] followed the iovative approach of the Th.M. Rassias theorem [39] i which he replaced the factor p p x y by p q x y for p q with p q 1. The comprehesive state of above results were obtaied by P. Gavruta [1] i 1994 by replacig the ubouded Cauchy differece by a geeral cotrol fuctio ( xy ) i the spirit of Rassias approach. I 008 a special case of Gavruta's theorem for the ubouded Cauchy differece was procured by Ravi et al. [40] i view of the summatio of both the sum ad the product of two p orms i the sprit of Rassias approach. I 003 V. Radu [11] proposed a ew method successively developed i [ ] to obtaiig the existece of the exact solutios ad the error estimatios based o the fixed poit alterative. The theory of radom ormed spaces (RN-spaces) is importat as a geeralizatio of the determiistic result of liear ormed spaces ad also i the study of radom operator equatios. The RN-spaces may also provide us with the appropriate tools to study the geometry of uclear physics ad have importat applicatio i quatum particle physics. Recetly J.M. Rassias et al. [36] ivestigated the ituitioistic radom stability of the quartic fuctioal equatio ad C. Park et al. [33] preseted the Hyers-Ulam stability of the additive-quadratic fuctioal equatio i ituitioistic radom ormed space. Very recetly Joh M. Rassias M. Arukumar ad S. Karthikeya [38] ivestigated the ituitioistic radom stability of a quadratic reciprocal fuctioal equatio f ( x y) f ( x y) usig direct ad fixed poit methods. f ( x) f ( y) 5 f ( x) 5 f ( y) 8 f ( x) f ( y) f ( x) f ( y) 5 f ( x) f ( y) (1.1) I 015 Joh M. Rassias M. Arukumar ad S. Karthikeya [37] proved the solutio i vector space ad the geeralized Ulam-Hyers stability of the terary

3 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio 57 quadratic homomorphisms ad terary quadratic derivatios betwee fuzzy terary Baach algebras associated to the quadratic fuctioal equatio x y y z z x 3 f z f x f y f x y f y z f z x 4 (1.) origiatig from the sum of the medias of a triagle by usig direct ad fixed poit methods. A applicatio of this fuctioal equatio is also studied. I this paper we prove the ituitioistic radom stability of a quadratic fuctioal equatio (1.) origiatig from the sum of the medias of a triagle usig direct ad fixed poit methods. Prelimiaries of Ituitioistic Radom Normed Spaces I this sectio usig the idea of ituitioistic radom ormed spaces itroduced by Chag et al. [16] we defie the otio of ituitioistic radom ormed spaces as i [ ]. Defiitio.1 A measure distributio fuctio is a fuctio : [01] which is left cotiuous o-decreasig o if t ( t) 1 ad suf t ( t) 1. We will deote by D the family of all measure distributio fuctios ad by H a special elemet of D defied by 0 if t 0 Ht () 1 if t 0. (.1) If X is a oempty set the : X D is called a probabilistic measure o X ad ( x) is deoted by x. Defiitio.. A o-measure distributio fuctio is a fuctio : [01] which is right cotiuous o-decreasig o if t ( t) 1 ad suf t ( t) 1. We will deote by B the family of all o-measure distributio fuctios ad by G a special elemet of B defied by 1 if t 0 Gt () 0 if t 0. (.)

4 58 M. Arukumar ad S. Karthikeya If X is a oempty set the : X D is called a probabilistic o-measure o X ad ( x) is deoted by x. emma.3. [8 0] Cosider the set * ad the order relatio * defied by: * ( x x ) : ( x x ) [01] ad x x ( x x ) ( y y ) x y x y ( x x ) ( y y ) *. 1 * The * * is a complete lattice. Defiitio.4. [8] A ituitioistic fuzzy set A i a uiversal set U is a object for all u U ( u) 01 ad ( u) 01 A ( A( u) A( u)) u U are called the membership degree ad the o-membership degree respectively of u i A ad furthermore they satisfy A(u) + A(u) 1. orm We deote its uits by 0 * (01) ad 1 * (01). Classically a triagular T A o [01] is defied as a icreasig commutative associative mappig T :[01] [01] satisfyig T(1 x) 1 x x for all x [01]. A triagular co-orm S is defied as a icreasig commutative associative mappig S :[01] [01] satisfyig S(0 x) 0x x for all x [01]. Usig the lattice * * these defiitios ca be straightforwardly exteded. Defiitio.5. [8] A triagular orm ( t orm) o * is a mappig the followig coditios: A T : (*) * satisfyig * 3 4 * * * ( i) * T ( x1 ) x (boudary coditios); ( ii) ( x y) * T( x y) T( y x) (commutativity); ( iii) ( x y z) * T x T( y z) T T( x y) z (associativity); ( iv) ( x x ' y y ') * x x ' ad y y ' T( x y) T( x ' y ') (mootoically). If * Tis a Abelia topological mooid with uit 1 * the * is said to be a * cotiuous t orm.

5 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio 59 Defiitio.6. [8] A cotiuous t orms T o * is said to be cotiuous t represetable if there exist a cotiuous t orm ad a cotiuous t coorm o [01] such that for all For example x ( x x ) y ( y y ) * T( x y) ( x y x y ) T( a b) a b mi a b ad M( a b) mi a b max a b for all a ( a1 a) b ( b1 b ) * are cotiuous t represetable. Now we defie a sequece Defiitio.7. [45] T recursively by 1 T T ad (1) ( 1) ( 1) (1) ( ) ( 1) ( i T x x T T x x x x ) *. A egator o * is ay decreasig mappig N : * * satisfyig N:(0 ) ad N(1 ) 0. If N( N( x)) x for all x * the N is called a * * * ivolutive egator. A egator o [01] is a decreasig mappig N :[01] [01] satisfyig Pµ (0) 1 ad Pµ (1) 0. Ns deotes the stadard egator o [01] defied by N ( x) 1 x [01]. Defiitio.8. [45] s et ad be measure ad o- measure distributios fuctios from X 0 to [01] such that ( t) ( t) 1 for all x X ad all t 0. The triple X P T x x is said to be a ituitioistic radom ormed space (briefly IRN-space) if X is a vector space T is a cotiuous t represetable ad Pµ is a mappig X 0 * satisfyig the followig coditios: for all x y X ad ts 0 ( IRN1) P ( x0) 0 ; µ * ( IRN ) P ( x t) 1 if ad oly if x 0; µ * t ( IRN3) Pµ ( x t) Pµ x for all 0; ( IRN 4) P ( x y t s) T P ( x t) P ( y s). µ * µ µ

6 60 M. Arukumar ad S. Karthikeya I this case Pµ is called a ituitioistic radom orm. Here P x t t t. ( ) x( ) x( ). Example.9. [45] et x. be a ormed space. et T( a b) ( a1 b1 mi ( a b 1)) for all a ( a a ) b ( b b ) * ad be measure ad o-measure distributios 1 1 fuctios defied by t x P. ( x t) x( t) x( t) t R. t t t x The ( X P T) is a IRN-sapce. Defiitio.10. [45] A sequece x i a IRN-space ( X P T) is called a Cauchy sequece if for ay 0 ad t 0 there exists 0 N that P ( x x t) * N ( ) m where N s is the stadard egator.. m s 0 Defiitio.11. [45] The sequece x is said to be coverget to a poit x X such. (deoted by P x x) if P. ( x x t) 1 * as for every t 0. Defiitio.1. [45] A IFN-space ( X P T) is said to be complete if every Cauchy sequece i X is coverget to a poit x X. Now we use the followig otatio for a give mappig f : X Y for all x y z X. x y y z z x ( x y z) f z f x f y 3 f x y f y z f z x 4

7 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio STABIITY RESUTS: DIRECT METHOD I this sectio the authors preset the geeralized Ulam-Hyers stability of the fuctioal equatio (1.) i IRN- space usig direct method. Hereafter throughout this paper let us cosider X be a liear space ad M be a complete ituitioistic radom ormed space. ( Y P ) Theorem 3.1. et f : X Y be a mappig with f (0) 0 for which there are 3 : X D ( x y z) x y z x y z is deoted by x y z ad ( x y z) is deoted by x y z furthur ( t) ( t) is deoted by P ( x y z t) with the property ( ) * (3.1) P x y z t P x y z t for all x y z X ad all t > 0. If i i i ( i ) * Ti 1 P x x x t = 1 (3.) ad lim P x x x t = 1 * (3.3) for all x X ad all t >0the there exists a uique quadratic mappig : satisfies the iequality 1 ( ) ( ) * i i i i Q X P f x Q x t P x x x t (3.4) for all x X ad all t >0. Proof. Replacig x y z by x x x i (3.1) we get Y f ( x) t P f ( x) * P x x x t (3.5) for all x X ad all t >0. Replacig x by x ad usig ( IRN 3) i the above equatio we have 1 f ( x) f ( x) t ( 1) ( 1) * P P x x x t (3.6) for all x X ad all r >0which implies that 1 f ( x) f ( x) t 1 ( 1) 1 * P P x x x t (3.7) holds for all x X ad all t >0. As 1... by the triagular iequality it follows

8 6 M. Arukumar ad S. Karthikeya i1 i 1 f ( x) 1 f ( x) f ( x) 1 P f ( x) t T i 0 P t ( i1) i i1 i0 * T P x x x t i i i i1 i0 (3.8) for all x X ad all t >0. I order to prove the covergece of the sequece f( x) replacig x by m x i (3.8) we obtai m m f ( x) f ( x) 1 ( ) * 1 im im im i m P t T m m i P x x x t (3.9) for all x X ad all t >0 ad all m 0. Sice the right had-side of the iequality teds f( x) to 1 * as m teds to ifiity the sequece we may defie the mappig Q : X Y by is a Cauchy sequece. Therefore f( x) P Q( x) t 1 * as t 0 for all x X. Now we prove that Q satisfies (1.). Replacig x y z by x y z i (3.1) we get P 1 x y z t P x y z t * (3.10) for all x y z X ad all t > 0. defiitio of Qx ( ) we see that Q satisfies (1.) for all ettig i the above iequality ad usig the x y z X. Fially to prove the uiqueess of the quadratic fuctio Qx ( ) subject to (3.4) let us assume that there exists aother quadratic fuctio Rx ( ) which satisfies (3.4). Obviously we have for all x X ad. Hece it follows from (3.4) that R x R x ( ) ( ) ( ) ( ) * ( ) ( ) P Q x R x t P Q x R x t * T P Q( x) f ( x) t P f ( x) R( x) t Q( x) Q( x) i i i i1 i i i i1 * T Ti 1 P x x x t Ti 1 P x x x t (3.11) for all x X ad all t >0. By lettig i (3.11) we fid that Q R. This completes the proof. ad

9 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio 63 From Theorem 3.1 we obtai the followig corollary cocerig the Hyers-Ulam- Rassias ad JMRassias stabilities for the fuctioal equatio (1.). Corollary 3. Suppose that a fuctio f : X Y satisfies the iequality P P t s s s P x y z t s ; ( x y z) t * s s s P x y z t s ; 3 s s s 3s 3s 3s P x y z x y z t s 3 (3.1) for all all x y z X ad all t >0 a uique quadratic mappig Q : X Y such that where s are costats with >0. The there exists P 3 t 3 s P x t s P f ( x) Q( x) t * 3s P x t 3s 4 3s P x t 3s (3.13) for all x X ad all t >0. 4. STABIITY RESUTS: FIXED POINT METHOD I this sectio usig the fixed poit alterative approach we prove the geeralized Ulam - Hyers stability of the fuctioal equatio (1.) i ituitioistic radom ormed spaces. Now we will recall the fudametal results i fixed poit theory. Theorem 4.1 (Baach Cotractio Priciple) et ( d) is a o-archimedea geeralized complete metric space ad cosider a mappig <1. The T : which is strictly cotractive mappig that is (A1). d( Tx Ty) d( x y) for all x y T for some (ipschitz costat) (i) The mappig T has oe ad oly fixed poit x = T( x );

10 64 M. Arukumar ad S. Karthikeya (ii)the fixed poit for each give elemet (A). lim = T x x for ay startig poit x ; (iii) Oe has the followig estimatio iequalities: x is globally attractive that is 1 1 (A3). d( T x x ) d( T x T x) 0 x; 1 1 (A4). d ( x x ) d( x x ) x.. 1 Theorem 4. [13](The alterative of fixed poit) Suppose that for a complete geeralized metric space (A d ) ad a strictly cotractive mappig T : A A with ipschitz costat. The for each give elemet x A either ( B1) d( T x T 1 x) = 0 or (B ) there exists a atural umber 0 such that: (i) d( T x T 1 x) < for all 0 ; (ii) The sequece ( T x) is coverget to a fixed poit y of T (iii) 0 y is the uique fixed poit of T i the set Y ={ y A : d( T x y) < }; 1 (iv) d( y y) d( y Ty) 1 for all y Y. Usig above fixed poit theorems to prove the stability results we defie the ad is the set such that followig costat if i = 0 i = 1 if i =1 = g g : X Y g(0) = 0. Theorem 4.3. et f : X Y be a mappig for which there exist a fuctio 3 :X D with the coditio i i i i 1 * Ti 1 P x x x t = 1 (4.1)

11 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio 65 ad lim P x x x t = 1 * (4.) ad satisfyig the fuctioal iequality ( ) * > 0. (4.3) P x y z t P x y z t x y z X t If there exists such that the fuctio x x x x ( x) = (4.4) has the property P ( ) = i x r P ( x) t x X t > 0 i (4.5) the there exists uique quadratic fuctio Q : X Y satisfyig the fuctioal equatio (1.) ad 1i P f ( x) Q( x) t * P ( x) t x X t > 0. 1 (4.6) Proof. et d be a geeral metric o such that * d( g h) = if K (0 ) P g( x) h( x) t P K ( x) t x X t 0. It is easy to see that d is complete. Defie : by g( x) = g( ) i x for all x X. Now for all gh we have * d g h K P g x h x t P K ( x) t x X 1 1 P g i x h i x t * P ' K ( i x) i t x X i i 1 1 K P g i x h i x t * P ' ( ) i x t x X i i i * d Tg Th K P Tg( x) Th( x) t P ' K ( x) t x X. 1 i (4.7) This gives d Tg Th d g h for all gh i.e. T is a strictly cotractive mappig of with ipschitz costat. Replacig ( x y z ) by ( x x x) i (4.3) we get * P f ( x) f ( x) t P x x x t x X t > 0. (4.8)

12 66 M. Arukumar ad S. Karthikeya Usig ( IFN 3) i (4.5) we arrive f( x) P f x t P x x x t x X t ( ) * > 0. (4.9) With the help of (4.5) whe i =0 it follows from (4.8) that f( x) P f ( x) t * P x t x X t > d f f = i. (4.10) Replacig x by x i (4.8) ad usig (IRN3) we obtai x x x x P f x f t * P t x X t > 0. (4.11) With the help of (4.5) whe i =1 it follows from (4.11) that x P f x f t * P ( x) t x X t > d f f 1 = i. (4.1) The from (4.10) ad (4.1) we ca coclude 1 d f f i <. Now from the fixed poit alterative i both cases it follows that there exists a fixed poit Q of i such that 1 lim P f i x Q( x) t 1 * x X t 0. i Replacig ( x y z ) by i x i y i z i (4.3) we arrive * i i i i i i i i (4.13) P x y z r P x y z t x y z X t > 0. (4.14) By proceedig the same procedure as i the Theorem 3.1 we ca prove the fuctio Q : X Y satisfies the fuctioal equatio (1.). By fixed poit alterative sice Q is uique fixed poit of i the set such that = f d f Q < ( ) ( ) * ( ) > 0. (4.15) P f x Q x t P K x t x X t

13 Ituitioistic Radom Stability Of A Quadratic Fuctioal Equatio 67 Agai usig the fixed poit alterative we obtai Hece we have 1i 1 d f Q d f f d f Q i P f ( x) Q( x) t * P ( x) t x X t > 0. 1 This completes the proof of the theorem. (4.16) From Theorem 4.3 we obtai the followig corollary cocerig the stability for the fuctioal equatio (1.). Corollary 4.4 Suppose that a fuctio f : X Y satisfies the iequality P P t s s s P x y z t s ; ( x y z) t * s s s P x y z t s ; 3 s s s 3s 3s 3s P x y z x y z t s 3 (4.17) for all all x y z X ad all t >0 a uique quadratic mappig Q : X Y such that where s are costats with >0. The there exists P 3 t 3 s P x t s P f ( x) Q( x) t * 3s P x t 3s 4 3s P x t 3s (4.18) for all x X ad all t >0. REFERENCES [1] Abasalt Bodaghi ad Sag Og Kim Approximatio o the quadratic reciprocal fuctioal equatio Joural of Fuctio Spaces

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