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1 Malaya Joural of Maemak, Vol. 6, No., 6-76, 08 hps://do.org/0.667/mjm060/00 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods Maa J. Rassas, M. Arukumar, P. Agla * Absrac I hs paper, he auhors roduce ad esablsh he geeral soluo ad geeralzed Ulam- Hyers sably of a r ype dmesoal Quadrac-Cubc fucoal equao [ f (r x r x )] # r v r r ()uv r r ()u r () u v ( f (() x () x )) u0 v0 r r r r r r r r f (x ) f (x ) r r r r r r r r f (x ) f (x ) " where r, r R {0}, ( 0,, ) ad s a posve eger Radom ormed spaces. Keywords Quadrac fucoal equao, Cubc fucoal equao, Mxed fucoal equao, Geeralzed Ulam - Hyers sably,fxed po, Radom ormed spaces. AMS Subjec Classfcao 9B5, B7, B8. Deparme of Sascal Scece, Uversy College odo, -9 Torrgo Place, 0, odo, WCE 7HB, UK. Deparme of Mahemacs, Goverme Ars College, Truvaamala , TamlNadu, Ida. Deparme of Mahemacs, Jeppaar Isue of Techology, Srperumbudur, Chea , Taml Nadu, Ida. *Correspodg auhor: maa@sas.ucl.ac.uk; aaru00@gmal.co.; aglram@gmal.com Arcle Hsory: Receved November 07; December 0 December 07 Coes Iroduco Geeral Soluo Prelmares of Radom Normed Spaces Sably Resuls : Drec Mehod Sably Resuls: Fxed po Mehod c 07 MJM. Refereces Iroduco I 90, Ulam [0] a he Uversy of Wscos proposed he followg sably problem: A basc queso he heory of fucoal equaos s as follows: whe s rue ha a fuco, whch approxmaely sasfes a fucoal equao, mus be close o a

2 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 6/76 exac soluo of he equao? I 9, D. H. Hyers [] gave a affrmave aswer o he queso of S.M. Ulam for Baach spaces. I 950, T. Aok [] was he secod auhor o rea hs problem for addve mappgs. I 978, Th.M. Rassas [] succeeded exedg Hyers Theorem by weakeg he codo for he Cauchy dfferece corolled by ( x p y p ), p [0, ), o be ubouded. I 98, J.M. Rassas [] replaced he facor x p y p by x p y q for p, q R. A geeralzao of all he above sably resuls was obaed by P. Gavrua [8] 99 by replacg he ubouded Cauchy dfferece by a geeral corol fuco ϕ(x, y). I 008, a specal case of Gavrua s heorem for he ubouded Cauchy dfferece was obaed by K.Rav eal., [6] by cosderg he summao of boh he sum ad he produc of wo p orms. These ermologes are also appled o he folder of oher fucoal equaos ad has bee exesvely vesgaed by a umber of auhors ad here are couless remarkable resuls perag o hs problem ogeher wh mxed ype fucoal equaos (see[,, 7, 9, 5]) ad refereces ced here. The fucoal equao oal equaos f (x ky) f (x ky) k f (x y) k f (x y) (k ) f (kx) k (k ) (k k k ) 8 f (y) f (x) f (y) (k ) (.5) f (y) f (y) for fxed egers k wh k 6 where f (y) 0, ±, Radom Normed Spaces. Recely M. Arukumar ad P. Agla [] roduced ad vesgaed he soluo ad sably of geeralzed UlamHyers Sably of a r ype -dmesoal Addve Quadrac fucoal equao rj j f r x f (x ) () f (x ) j r r j < j () p0 pq p q f [() x () x j ] q0 (.6) f (x y) f (x y) f (x) f (y) (.) s relaed o a symmerc b-addve mappg (see[, 9]). I s aural ha hs equao s called a quadrac fucoal equao. I parcular, every soluo of he quadrac equao (.) s sad o be a quadrac mappg. K.W.Ju ad H.M.Km [5] cosdered he followg fucoal equao where r ad are posve egers wh quas bea ormed spaces. I hs paper, he auhors esablsh he geeral soluo ad geeralzed Ulam- Hyers sably of a r ype dmesoal Quadrac-Cubc fucoal equao [ f (r x r x )] f (xy) f (xy) f (xy) f (xy) f (x) (.) whch s called a cubc fucoal equao ad every soluo of he cubc fucoal equao s sad o be a cubc mappg. G.H.Km, H.Y.Sh [0] roduced ad vesgaed he geeralzed Ulam-Hyers Sably of he followg more geeralzed cubc fucoal equao f (rx sy) f (rx sy) rs f (x y) rs f (x y) r(r s ) f (x) r r ()uv ()u r r u0 v0 r r ()v 6 f (x y) 6 f (x y) f (y) f (x y) ( f (()u x ()v x )) r r r r r r r r r r r r r r r r (.) where r 6 ±, 0, s are real umbers. I 00, I.S.Chag ad Y.S.Jug [5], vesgaed he soluo ad sably of he fucoal equao f (x ) f (x ) f (x ) f (x ) (.7) where r, r R {0}, ( 0,, ) ad s a posve eger Radom ormed spaces. I Seco, he geeral soluo of he fucoal equadervg from cubc ad quadrac fucos. Recely he o (.7) s gve, I Seco, basc defo ad prelmfuzzy sably of (.) was dscussed by Z.H.Wag ad W.X.Zhagares of Radom ormed space s prese, I Seco, he []. Yeol Je Cho e.al., [7] esablshed he geeral soluo geeralzed Ulam - Hyers sably of he fucoal equao ad sably of geeralzed mxed ype quadrac-cubc fuc- (.7) s proved va Hyers mehod. f (x y) 9 f (x) (.) 6

3 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 6/76. Geeral Soluo I hs seco, we prese he geeral soluo of he fucoal equao (.7). Through ou hs seco le X ad Y be real vecor spaces. eg x y 0 (.), we ge f (0) 0. Replacg y by x (.), we ge f (x) f (x) (.8) emma.. A eve fuco f : X Y sasfes he quadrac fucoal equao (.) f ad oly f f : X Y sasfes he fucoal equao (.7) for all x0, x,..., x, x X. for all x X. I geeral for ay posve eger a, we have Proof. Assume f : X Y sasfes he fucoal equao (.7). Usg eveess of f (.7), we arrve for all x X. By [[, 9]], here exss a uque symmerc b-addve mappg B : X X Y such ha f (x) B(x, x) for all x X ad f (ax) a f (x) [ f (r x r x )] B(x, y) [ f (x y) f (x y)] r f (x ) r f (x ) (.) f (r x r x ) for all x0, x,..., x, x X. Subsug (x0, x,..., x, x ) by (0, 0..., 0, 0) (.), we ge f (0) 0. Replacg (x0, x,..., x, x ) by (x, y, 0,..., 0, 0) (.), we have f (r0 x r y) r0 f (x) r f (y) r0 r [ f (x y) f (x y)] (.) B(r x, r x r x ) B(r x, r x ) B(r x, r x ) r r B(x, x ) r r B(x, x ) r B(x, x ) r r B(x, x ) r r B(x, x ) r B(x, x ) (.) f (r y) r f (y) (.) for all y X. Seg y by y ad usg eveess of f (.), we reach f (r0 x r y) r0 f (x) r f (y) r0 r B(r x, r x r x ) r r B(x, x ) r r B(x, x ) for all x X. Aga, f we pu x by 0 (.) ad usg eveess of f, we ge B(r x r x, r x r x ) B(r x, r x ) B(r x, r x ) for all x, y X. If we pu y by 0 (.), we oba f (r0 x) r0 f (x) [ f (x y) f (x y)] (.0) for all x, y X. Hece, for ( 0,, ), we have r r ( f (x x ) f (x x )) (.9) (.5) r B(x, x ) r r B(x, x ) r r B(x, x ) r B(x, x ) r B(x, x ) r r B(x, x ) r B(x, x ) r r ( f (x x ) r f (x ) f (x x )) r f (x ) (.) for all x, x X. Thus, from (.) ha [ f (r x r x )] for all x, y X. Addg (.) ad (.5), we arrve f (r0 x r y) f (r0 x r y) r0 f (x) r f (y) (.6) for all x, y X. Usg (.) ad (.) (.6), we have f (r0 x r y) f (r0 x r y) f (r0 x) f (r y) f (x ) r f (x ) r (.7) for all x, y X. Fally, replacg (x, y) by rx0, ry (.7), we arrve (.). Coversely, e f : X Y sasfes (.). 6 r r ( f (x x ) f (x x )) (.) for all x0, x,..., x, x X. Sce f s a eve fuco, ca be wre as f (x) ( f (x) f (x)) (.)

4 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 65/76 for all x X. Wh he help of (.), (.) ca be wre as (x0, x,..., x, x ) by (x0, x, 0,..., 0, 0) (.6),we have f (r0 x0 r x ) [ f (r x r x )] r r [ f (x ) f (x )] [ f (x ) f (x )] r0 r [ f (x0 x ) f (x0 x )] r0 r [ f (x0 x ) f (x0 x )] (r0 r r0 ) f (x0 ) (r0 r r ) f (x ) (.7) for all x0, x X. If we pu x by 0 (.7), we oba r r [ f (x x ) f (x x ) f ((x x )) f ((x x ))]] f (r0 x0 ) r0 f (x0 ) (.) for all x0, x,..., x, x X. Addg he followg erms (.8) for all x0 X. Aga, f we pu x0 by 0 (.7) ad usg oddess of f, we ge f (r x ) r f (x ) (.9) r ) f (x ), f (r r for all x X. Seg x by x ad usg oddess of f (.7), we reach ) f (x ), f (r r r f (r0 x0 r x ) r r [ f (x x ) f (x x )], r r [ f (x x ) f (x x )] (.5) o boh sdes of (.), ad usg eveess of f, we reach (.7) for all x0, x,..., x, x X. Hece he proof s complee. r0 r [ f (x0 x ) f (x0 x )] r0 r [ f (x0 x ) f (x0 x )] (r0 r r0 ) f (x0 ) (r0 r r ) f (x ) (.0) for all x0, x X. Addg (.7) ad (.0), we arrve (.) he form of f (r0 x0 r x ) f (r0 x0 r x ) r0 r f (x0 x ) r0 r f (x0 x ) r0 (r0 r ) f (x0 ) (.) emma.. A odd fuco f : X Y sasfes he cubc fucoal equao (.) where r 6 ±, 0, s are real umbers f ad oly f f : X Y sasfes he fucoal equao (.7) for all x0, x,..., x, x X. Proof. Assume f : X Y sasfes he fucoal equao (.7). Usg oddess of f (.7), we arrve for all x0, x X. where r0 6 ±, 0, r are real umbers. Hece f Sasfes (.). Aga, replacg (x0, x, x, x..., x, x ) by (0, 0, x, x,..., 0, 0) (.6), we have f (r x r x ) [ f (r x r x )] r r [ f (x x ) f (x x )] r r [ f (x x ) f (x x )] (r r r ) f (x ) (r r r ) f (x ) (.) " r r ( f (x x ) f (x x )) r r ( f (x x ) f (x x )) (r r r ) f (x ) (r r r ) f (x ) (.6) for all x0, x,..., x, x X. Subsug (x0, x,..., x, x ) by (0,..., 0, 0) (.6), we ge f (0) 0. Replacg 65 for all x, x X. I follows from he seps (.8) o (.), we have (.) he form of f (r x r x ) f (r x r x ) r r f (x x ) r r f (x x ) r (r r ) f (x ) (.) for all x, x X. where r 6 ±, 0 ad r are real umbers. By coug hs maer, fally replacg (x0, x,..., x, x ) by

5 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 66/76 for all x, x X, Fally ( 0,..., 0, x, x ) mes f (r x r x ) (.6), we have f (r x r x ) r r [ f (x x ) f (x x )] r r [ f (x x ) f (x x )] (r r r ) f (x ) r r [ f (x x ) f (x x )] r r [ f (x x ) f (x x )] (r r r ) f (x ) (r r r ) f (x ) (r r r ) f (x ) (.) for all x, x X. Aga, I follows from he seps (.8) o (.), we have (.) he mold of (.5) (.6) for all x0 X. Replacg x0 by r x ad x by r0 x0 (.), ad dvdg he resula by r0 r, we oba f (r x r0 x0 ) f (r x r0 x0 ) r0 r [ f (x x0 ) f (x x0 )] r r ( f (x x ) f (x x )) r r ( f (x x ) f (x x )) (r r r ) f (x ) r r r ) f (x ) (.) for all x0,..., x, x X. Sce f s a odd fuco, ca be wre as f (x) ( f (x) f (x)) (.) for all x X. Wh he help of (.), (.) ca be remodfy as, (.7) for all x0, x X. Aga, replacg x0 by x ad x by x0 ad usg oddess of f (.7), we ge [ f (r x r x )] " f (r0 x0 r x ) f (r0 x0 r x ) r0 r [ f (x0 x ) f (x0 x )] r (r0 r ) f (x ) " for all x, x X. where r 6 ±, 0 ad r are real umbers. Coversely, assume f : X Y sasfes he fucoal equao (.). Subsug (x0, x ) by (0, 0) (.), we ge f (0) 0. Replacg (x0, x ) by (x0, 0) (.), we have r (r0 r ) f (x ) [ f (r r r x )] r r f (x y) r r f (x y) f (r0 x0 ) r0 f (x0 ) for all x, x X. Addg (.9), (.0) ad (.), we reach f (r x r y) f (r x r y) r (r r ) f (x) (.) r r [[( f (x x ) f (x x ))] [ f ((x x )) f ((x x )))]] r r [[( f (x x ) f (x x ))] [( f ((x x )) f ((x x )))]] (.8) for all x0, x X. Subsug (.8) (.7), we arrve r r r [ f (x ) f (x )] # r (r r ) [ f (x ) f (x )] r0 r f (r0 x0 r x ) [ f (x0 x ) f (x0 x )] r r 0 [ f (x0 x ) f (x0 x )] (r0 r r0 ) f (x0 ) (r0 r r ) f (x ) (.) for all x0,..., x, x X. Addg he followgs erms (.9) for all x0, x X. By applyg he procedure from (.6) o (.9), (.) ad (.5), we have he followg equaos r r f (r x r x ) [ f (x x ) f (x x )] r r [ f (x x ) f (x x )] (r r ) f (x ) (r r r ) f (x ) (.0) 66 r f (x ), r f (x ), r r ( f (x x ) f (x x )) (.5)

6 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 67/76 o boh sdes of (.), ad usg oddess of f, we reach (.7) (RN) µxy ( s) τ (µx (), µy (s)) for all x, y X ad, s for all x0,..., x, x X. Hece he proof s compleed. 0. Example.. Every ormed spaces (X, ) defes a radom ormed space (X, µ, τm ), where. Prelmares of Radom Normed Spaces µx () I he sequel, we adop he usual ermology, oaos ad coveos of he heory of radom ormed spaces as [6, 8, 9]. Throughou hs paper, s he space of dsrbuo fucos, ha s, he space of all mappgs F : R {, } [0, ], such ha F s lefcouous ad odecreasg o R, F(0) 0 ad F( ). D s a subse of cossg of all fucos F for whch l F( ), where l f (x) deoes he lef lm of he fuco f a he po x, ha s, l f (x) lm f (). The space s parally ordered x by he usual powse orderg of fucos, ha s, F G f ad oly f F() G() for all R. The maxmal eleme for hs order s he dsrbuo fuco ε0 gve by 0, f 0, ε0 () (.), f > 0. Defo.. [8] A mappg τ : [0, ] [0, ] [0, ] s called a couous ragular orm (brefly, a couous orm) f τ sasfes he followg codos: (a) τ s commuave ad assocave; (b) τ s couous; Defo.. e (X, µ, τ) be a RN-space. () A sequece {x } X s sad o be coverge o a po x X f, for ay ε > 0 ad λ > 0, here exss a posve eger N such ha µx x (ε) > λ for all N. () A sequece {x } X s called a Cauchy sequece f, for ay ε > 0 ad λ > 0, here exss a posve eger N such ha µx xm (ε) > λ for all m N. () A RN-space (X, µ, τ) s sad o be complee f every Cauchy sequece X s coverge o a po X. Theorem.5. If (X, µ, τ) s a RN-space ad {x} s a sequece X such ha x x, he lm µx () µx () almos everywhere. Hereafer, hroughou hs paper, le us cosder U be a lear space ad (V, µ, τ) s a complee RN-space. Defe a mappg f : U V by (d) τ(a, b) τ(c, d) wheever a c ad b d for all a, b, c, d [0, ]. Typcal examples of couous orms are τp (a, b) ab, τm (a, b) m(a, b) ad τ (a, b) max(a b, 0) (he ukasewcz orm). Recall (see [0, ]) ha f τ s a orm ad x s a gve sequece of umbers [0, ], he x τ s defed recurrely by τ x x ad τ x τ τ x, x f or. x s defed as τ x τ. I s kow [] ha, for he ukasewcz orm, he followg mplcao holds: ad τm s he mmum orm. Ths space s called he duced radom ormed space. f (x0, x,, x, x ) (c) τ(a, ) a for all a [0, ]; lm (τ ) x x ( x ) < (.) [ f (r x r x )] r r ()uv ()u r r u0 v0 r r ()v ( f (()u x ()v x )) r r r r r r r r r r r r r r r r Defo.. [9] A radom ormed space (brefly, RNspace) s a rple (X, µ, τ), where X s a vecor space, τ s a couous orm ad µ s a mappg from X o D sasfyg he followg codos: f (x ) f (x ) f (x ) f (x ) for all x0, x,..., x, x U.. Sably Resuls : Drec Mehod (RN) µx () ε0 () for all > 0 f ad oly f x 0; (RN) µα x () µx (/ α ) for all x X, ad α R wh α 6 0; 67 I hs seco, he geeralzed Ulam Hyers sably of he fucoal equao (.7) usg drec mehod s provded.

7 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 68/76 Theorem.. e j ±. e f : U V be a eve mappg for whch here exs a fuco : U D wh he codo lm τ (0,, 0,Ω j x,ω j x) Ω() j mes (.) lm Ω j x0,ω j x,,ω j x,ω j x Ω j such ha he fucoal equaly such ha Λ f (x0,x,,x,x ) () x0,x,,x,x () for all x U ad all > 0. I follows from (.0) ha Λ f (Ωk x) f (Ωk x) (0,, 0,Ωk x,ωk x) () (.) k Ω(k) Ω Ω(k) mes mes for all x U ad all > 0. The mappg Q(x) s defed by mes (.) for all x0, x,..., x, x U ad all > 0. The here exss a uque quadrac mappg Q : U V sasfyg he fucoal equao (.7) ad ΛQ(x) f (x) () τ (0,, 0,Ω j x,ω j x) Ω() j (.) ΛQ(x) () lm Λ f (Ω j x) () for all x U ad all > 0. Replacg x by Ωk x (.9), we arrve (0,, 0,Ωk x,ωk x) () (.0) Λ f (Ωk x) f (Ωk x) Ω Ω (.) Ω j for all x U ad all > 0. Subsue by Ω(k) (.), we arrve Λ f (Ωk x) f (Ωk x) () (0,, 0,Ωk x,ωk x) Ω(k) (.) k Ω Ω(k) mes for all x U ad all > 0. I s easy o see ha f (Ωk x) f (Ω x) f (Ω x) f (x) () Ω Ωk Ω (.) for all x U. From equaos (.) ad (.), we have for all x U ad all > 0. Λ f (Ωk x) Proof. Assume j. Seg (x0, x,..., x, x ) by ( 0,, 0, x, x) ad usg eveess Ωk f (x) Λ k mes of f (.), we ge Λ f [(r () f (Ω x) f (Ω x) Ω Ω() k τ Λ f (Ω x) (0,, 0,x,x) () (.5) f (Ω x) Ω Ω() k (0,, 0,Ωk j x,ωk j x) Ω() j τ Ω() r r () f (x) r )x]r f (x)r f (x) () (.) mes mes for all x U adall >o0. I order o prove he covergece k x) of he sequece f (Ω, we replace x by Ωm x (.), we Ωk arrve for all x U ad all > 0. Usg (.8) (.5), we have Λ f [(r r )x]r f (x)r f (x)r r f (x) (0,, 0,x,x) () () (.6) Λ f (Ωkm x) Ω(km) mes for all x U ad all > 0. The above equaly ca be wre as Λ f [(r r )x](r r ) f (x) () (0,, 0,x,x) () (.7) mes for all x U ad all > 0. Defe Ω r r (.7), ca be wre as Λ f (Ωx)(Ω) f (x) () (0,, 0,x,x) () (.8) mes for all x U ad all > 0. I follows from (.8) ad (RN), we have Λ f (Ωx) f (x) (0,, 0,x,x) () (.9) Ω Ω mes 68 f (x) () k τ (0,, 0,Ωm x,ωm x) Ω(m) j mes mk τm (0,, 0,Ω x,ω x) Ω() j mes as m (.5) f (Ωk x) for all x U ad all > 0. Thus s a Cauchy Ωk sequece. Sce V s complee here exss a mappg Q : U V, we defe ΛQ(x) () lm Λ f (Ωk x) () Ωk for all x U ad all > 0. eg m 0 ad (.), we arrve (.) for all x U ad all > 0. Now, we have o

8 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 69/76 show ha Q sasfes (.7), replacg (x0, x,..., x, x ) by (Ωk x0, Ωk x,..., Ωk x, Ωk x ), we have Λ f (Ωk x0,ωk x,,ωk x,ωk x ) () Ωk x0,ωk x,,ωk x,ω x () (.6) for all x U ad all > 0. Takg k boh sdes, we fd ha Q sasfes (.7) for all x0, x,..., x, x U. Therefore he mappg Q : X Y s Quadrac. Fally, o prove he uqueess of he quadrac fuco Q subjec o (.), le us assume ha here exs a quadrac fuco Q0 whch sasfes (.) ad (.). Sce Q(Ωk x) Ωk Q(x) ad Q0 (Ωk x) Ωk Q0 (x) for all x U ad all N, follows from (.) ha ΛQ(x)Q0 (x) () ΛQ(Ωk x)q0 (Ωk x) (Ωk ) Theorem.. e j ±. e f : U V be a odd mappg for whch here exs a fuco : U D wh he codo lm τ (0,, 0,Ω j x,ω j x) Ω() j mes lm Ω j x0,ω j x,,ω j x,ω j x Ω j (.9) such ha he fucoal equaly such ha Λ f (x0,x,,x,x ) () x0,x,,x,x () (.0) for all x0, x,..., x, x U ad all > 0. The here exss a uque cubc mappg C : U V sasfyg he fucoal equao (.7) ad ΛC(x) f (x) () τ (0,, 0,Ω j x,ω j x) Ω() j (.) mes ΛQ(Ωk x) f (Ωk x) f (Ωk x)q0 (Ωk x) (Ωk ) τ ΛQ(Ωk x) f (Ωk x) (Ωk ), Λ f (Ωk x)q0 (Ωk x) (Ωk ) τ τ (0,, 0,Ω x,ω x) Ω() j, mes τ (0,, 0,Ω x,ω x) Ω() j for all x U ad all > 0. The mappg Q(x) s defed by ΛC(x) () lm Λ f (Ω j x) () (.) Ω j for all x U ad all > 0. Proof. Assume j. Seg (x0, x,..., x, x ) by ( 0,, 0, x, x) ad usg oddess of f (.0), we ge mes mes Λ f [(r r )x] as r r f (x) r r (r r r ) f (x) ) ( r for all x U ad all > 0. Hece Q s uque. For j, we ca prove a smlar sably resul. Ths complees he proof of he heorem. (.) r r f (x) f (x) (0,, 0,x,x) () (.) mes for all x U ad all > 0. Usg (.8) (.), we have Corollary.. e Φ ad s be oegave real umbers. e a eve fuco f : U V sasfes he equaly Λ f [(r r )x]r f (x)r f (x)r r f (x)r r f (x) (0,, 0,x,x) () (.5) Λ f (x0,x,,x,x ) () mes Φ (), (), s 6 ; s Φ x for all x U ad all > 0. The above equaly ca be wre (), s 6 ; as ()s s Φ( x x () ) (.7) for all x0, x,, x, x U ad all > 0. The here exss a uque quadrac fuco Q : U V such ha Φ (), Ω s Φ x (), Λ f (x)q(x) () (.8) Ω Ωs Φ x ()s (), Λ f [(r r )x](r r ) f (x) () (0,, 0,x,x) () (.6) mes for all x U ad all > 0. Defe Ω r r (.6), we ge Λ f (Ωx)(Ω) f (x) () (0,, 0,x,x) () (.7) mes for all x U ad all > 0. Ω Ω()s The res of he proof s smlar o ha of Theorem.. Hece he deals of he proof are omed. for all x U ad all > ()

9 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 70/76 Corollary.. e Φ ad s be oegave real umbers. e a odd fuco f : U V sasfes he equaly Λ f (x0,x,,x,x ) () (.8) Φ (), s 6 ; Φ x s (), 6 () ; Φ( x s x ()s ) (), s (.9) for all x0, x,, x, x U ad all > 0. The here exss a uque cubc fuco C : U V such ha Φ (), Ω s Φ x (), Λ f (x)c(x) () (.0) Ω Ωs Φ x ()s (), all x0, x,, x, x U ad all > 0. By Theorem., we have ΛQ(x) fe (x) () () j, τ τ (0,, 0,Ω j x,ω j x) Ω mes, (0,, 0,Ω j x,ω j x) Ω() j mes (.5) Ω Ω()s for all x U ad all > 0. Theorem.5. e j ±. e f : U V be a mappg for whch here exs a fuco : U D sasfyg he codos (.) ad (.9) ad he fucoal equaly such ha Λ f (x0,x,,x,x ) () x0,x,,x,x () for all x U. Also, e fo (x) { f (x) f (x)} for all x U. The fo (0) 0, fo (x) fo (x) for all x U. Hece (.) for all x0, x,, x, x U ad all > 0. The here exss a uque quadrac fuco Q : U V ad a uque cubc fuco C : U V sasfyg he fucoal equao (.7) such ha µq(x)c(x) f (x) () (.) () j τ τ, τ (0,, 0,Ω j x,ω j x) Ω mes, (0,, 0,Ω j x,ω j x) Ω() j mes () j, τ, τ (0,, 0,Ω j x,ω j x) Ω mes, (0,, 0,Ω j x,ω j x) Ω() j mes (.) Λ fo (x0,x,,x,x ) () τ x0,x,,x,x (),, x0,x,,x,x () for all x0, x,, x, x U ad all > 0. By Theorem., we have ΛC(x) fo (x) () () j τ, τ (0,, 0,Ω j x,ω j x) Ω mes, (0,, 0,Ω j x,ω j x) Ω() j mes (.6) for all x U ad all > 0. The mappg ΛQ(x) ad ΛC(x) are defed (.) ad (.) respecvely for all x U ad all > 0. Proof. e fe (x) { f (x) f (x)} for all x X. The fe (0) 0, fe (x) fe (x) for all x U. Hece for all x U ad all > 0.Defe Λ fe (x0,x,,x,x ) () τ x0,x,,x,x (),, x0,x,,x,x () (.) 70 f (x) fe (x) fo (x) (.7)

10 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 7/76 for all x U. From.5,.6 ad.7, we arrve Theorem 5.. [](The alerave of fxed po) Suppose ha for a complee geeralzed merc space (X, d) ad a srcly coracve mappg T : X X wh pschz cosa. The, for each gve eleme x X, eher (B) d(t x, T x) 0, or (B) here exss a aural umber 0 such ha: () d(t x, T x) < for all 0 ; ()The sequece (T x) s coverge o a fxed po y of T () y s he uque fxed po of T he se Y {y X : d(t 0 x, y) < }; (v) d(y, y) d(y, Ty) for all y Y. ΛQ(x)C(x) f (x) () ΛQ(x)C(x) fo (x) fe (x) () τ ΛQ(x) fe (x) (), ΛC(x) fo (x) () () j, τ τ τ (0,, 0,Ω j x,ω j x) Ω mes, (0,, 0,Ω j x,ω j x) Ω() j mes () j, τ, τ (0,, 0,Ω j x,ω j x) Ω mes () j, (0,, 0,Ω j x,ω j x) Ω For o prove he sably resul we defe he followg: δ s a cosa such ha Ω f 0, δ f Ω ad Γ s he se such ha Γ {g g : X Y, g(0) 0}. mes for all x U ad all > 0. Hece he heorem s proved. Corollary.6. e Φ ad s be oegave real umbers. e a fuco f : U V sasfes he equaly Theorem 5.. e f : U V be a eve mappg for whch here exs a fuco : U D wh he codo lm δ k x0,δ k x,,δ k x,δ k x δk (5.) k Λ f (x0,x,,x,x ) () for all x0, x,, x, x U ad all > 0 ad sasfyg Φ (), he fucoal equaly Φ x s (), s 6, ; 6 (), () ; Λ () () (5.) Φ( x s x ()s ) (), s D f (x0,x,,x,x ) x0,x,,x,x (.8) for all x0, x,, x, x U ad all > 0. The here exss a uque quadrac fuco Q : U V ad a uque cubc fuco C : U V such ha for all x0, x,, x, x U ad all > 0. If here exss () such ha he fuco D (x,) (0,, 0, x, x ) () Ω Ω (5.) mes Φ Φ (), Ω s Ω s Φ x Φ x (), Λ f (x)q(x)c(x) () Ω Ωs Ω Ωs Φ x ()s Φ x ()s (), has he propery D (x,) Ω Ω()s Ω Ω()s (.9) for all x U ad all > 0. I hs seco, he auhors prese he geeralzed Ulam Hyers sably of he fucoal equao (.7) Radom ormed space usg fxed po mehod. Through ou hs seco, e us cosder U be a vecor space ad (V, Λ, T ) s a complee RN-space. Now we wll recall he fudameal resuls fxed po heory. 7 (5.) The here exss a uque quadrac fuco Q : U V sasfyg he fucoal equao (.7) ad ΛQ(x) f (x) 5. Sably Resuls: Fxed po Mehod D (δ x,), x X, > 0. δ D (x,), x X, > 0. (5.5) Proof. e d be a geeral merc o Γ, such ha d(g, h) f K (0, ) Λg(x)h(x) (K) D (x,), x X, > 0. I s easy o see ha (Γ, d) s complee. Defe T : Γ Γ by T g(x) g(δ x), for all x X. Now for g, h Γ, we have δ

11 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 7/76 d(g, h) K herefore Q s a uqe fuco such ha Λg(x)h(x) (K) D (x,) K Λ g(δ x) h(δ x) D (δ x,) δ δ δ K ΛT g(x)t h(x) D (x,) δ Λ f (x)q(x) (K) D (x,) for all x U, > 0 ad K > 0. Aga usg he fxed po alerave, we oba d (T g(x), T h(x)) K d (T g, T h) d(g, h) (5.6) mes for all x U, > 0, Usg (5.) for he case 0 reduces o Λ f (Ωx) f (x) D (x,) Ω Ω Aga replacg x by x Ω From Theorem 5., we oba he followg corollary cocerg he sably for he fucoal equao (.7). Corollary 5.. e Φ ad s be oegave real umbers. e a eve fuco f : U V sasfes he equaly (5.9) ΛD f (x0,x,,x,x ) () Φ (), Φ x s (), s 6 ; 6 () ; Φ( x s x ()s ) (), s (5.7), we oba Λ f (x)ω f ( x ) () (0,, 0, x, x ) () Ω Ω Ω (5.0) mes (5.5) for all x U, > 0 wh he help of (5.) whe, follows from (5.0), we ge for all x0, x,, x, x U ad all > 0. The here exss a uque quadrac fuco Q : U V such ha Λ f (x)ω f ( x ) () D (x,) Ω d( f, T f ) 0 (5.) for all x U ad > 0. Ths complees he proof of he heorem. for all x U ad all > 0. I follows from (5.7), we have Λ f (Ωx) f (x) (0,, 0,x,x) () (5.8) Ω Ω d(t f, f ) 0 <. (5.7) mes d( f, T f ) d( f, Q) Λ f (x)q(x) D (x,) d( f, Q) for all g, h Γ. There fore T s srcly coracve mappg o Γ wh pschz cosa. From (.8) ha Λ f (Ωx)(Ω) f (x) () (0,, 0,x,x) () (5.) (5.) Φ (), Ω s Φ x (), Λ f (x)q(x) () Ω Ωs (), Φ x ()s The from (5.9) ad (5.), we ca coclude, d( f, T f ) 0 < (5.6) Ω Ω()s Now from he fxed po alerave boh cases, follows ha here exss a fxed po Q of T Γ such ha for all x U ad all > 0. ΛQ(x) () lm k Λ f (δ k x) δk (), x U, > 0. (5.) Proof. Seg Replacg (x0, x,, x, x ) by (δk x0, δk x,, δk x, δk x ) (5.) ad dvdg by δk, follows from (5.) ad (5.), Q sasfes (.7) for all x0, x,, x, x U ad all > 0. By fxed po alerave, sce Q s uque fxed po of T he se { f Γ d( f, Q) < }, 7 x0,x,,x,x () Φ (), Φ x s (), (), Φ x s x ()s

12 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 7/76 Case: Ωs for s < f, D (x,) ΛQ(x) f (x) Ωs Ωs ΛQ(x) f (x). Ωs Ω for all x0, x,, x, x U ad all > 0. The, (δk x0,δk x,,δk x,δk x ) () Φ (), δk (), δk Φ δ k δk x s δk Φ δk x s (), Φδk k(s) Φδ x s δk x ()s Case:5 Ω()s for s > (), (), ΛQ(x) f (x) Φδ Ω()s Ω()s. Ω Ω()s Case:6 Ω()s for s < () f, D (x,) ΛQ(x) f (x) Ω()s Ω()s ΛQ(x) f (x). Ω()s Ω Thus, (5.) s holds. From (5.), we have x U, > 0. Also From (5.), we arrve β (δ x,) δ (), Φδk s s (), Φδ x Ωs ()s f 0, Ω()s Ω()s D (x,) ΛQ(x) f (x) (), k(()s) Φδ x s x ()s as k, as k, as k. () Hece he proof s complee. x ()s Theorem 5.. e f : U V be a odd mappg for whch here exs a fuco : U D wh he codo lm δ k x0,δ k x,,δ k x,δ k x δk (5.7) (). k for all x0, x,, x, x U ad all > 0 ad sasfyg he fucoal equaly k δ β (x,), δ s β (x,), ()s δ β (x,). ΛD f (x0,x,,x,x ) () x0,x,,x,x () (5.8) Now from (5.5), we prove he followg cases for codos () ad (). Case: Ω for s 0 f 0, Ω(0) D (x,) ΛQ(x) f (x) Ω ΛQ(x) f (x). Ω for all x0, x,, x, x U ad all > 0. If here exss () such ha he fuco Case: Ω for s 0 f, The here exss a uque cubc fuco C : X Y sasfyg he fucoal equao (.7) ad ΛC(x) f (x) D (x,), x U, > 0. (5.0) Ω() Ω ΛQ(x) f (x). Ω D (x,) ΛQ(x) f (x) D (x,) (0,, 0, x, x ) () Ω Ω mes has he propery D (x,) D (δ x,), x U, > 0. δ (5.9) Corollary 5.5. e Φ ad s be oegave real umbers. e a odd fuco f : U V sasfes he equaly Case: Ωs for s > f 0, Ωs D (x,) ΛQ(x) f (x) Ωs Ωs ΛQ(x) f (x). Ω Ωs ΛD f (x0, x,, x, x )() Φ (), Φ x s (), s 6 ; 6 () ; Φ( x s x ()s ) (), s (5.) 7

13 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 7/76 for all x0, x,, x, x U ad all > 0. The here exss a uque cubc fuco C : U V such ha Φ (), Ω s Φ x (), Λ f (x)c(x) () Ω Ωs Φ x ()s (), for all x0, x,, x, x U ad all > 0. By Theorem 5., we have ΛC(x) fo (x) (5.) (5.8) f (x) fe (x) fo (x) (5.9) Defe Ω Ω()s for all x U ad all > 0. Theorem 5.6. e f : U V be a mappg for whch here exs a fuco : U D wh he codo (5.) ad (5.7) sasfyg he fucoal equaly wh ΛD f (x0,x,,x,x ) () x0,x,,x,x () τ [D (x,), D (x,)], x X, > 0. (5.) for all x0, x,, x, x U ad all > 0. If here exss () such ha he fuco D (x,) (0,, 0, x, x ) () Ω Ω mes has he propery (5.) ad (5.9) for all x U. The here exss a uque quadrac fuco Q : U V ad a uque cubc fuco C : U V sasfyg he fucoal equao (.7) ad Λ f (x)q(x)c(x) h h τ τ D x,, D x, h, τ D x,, D x, (5.) for all x U. From (5.6),(5.8) ad (5.9), we arrve ΛQ(x)C(x) f (x) ΛQ(x)C(x) fo (x) fe (x), ΛC(x) fo (x) τ ΛQ(x) fe (x) h h τ τ D x,, D x,,, τ [D (x,), D (x,)]] for all x U ad all > 0. Hece he heorem s proved. Corollary 5.7. e Φ ad s be oegave real umbers. If a fuco f : U V sasfes he equaly Λ f (x0,x,,x,x ) () Φ (), s 6, ; s (), Φ x s Φ x, () ; 6 () x ()s (), s forall x U ad all > 0. (5.0) Proof. e fe (x) { f (x) f (x)} for all x X. The fe (0) 0, fe (x) fe (x) for all x U. Hece ΛD fe (x0,x,,x,x ) () τ x0,x,,x,x (), x0,x,,x,x () (5.5) for all x0, x,, x, x U ad all > 0. The here exss a uque quadrac fuco Q : U V ad a uque cubc fuco C : U V such ha Φ Φ (), Ω s Ω s Φ x Φ x (), Λ f (x)q(x)c(x) () Ω Ωs Ω Ωs (), Φ x ()s Φ x ()s forall x0, x,, x, x U ad all > 0. By Theorem 5., we have ΛQ(x) fe (x) τ [D (x,), D (x,)], x X, > 0. (5.6) Ω Ω()s (5.) for all x U ad all > 0. Also,e fo (x) { f (x) f (x)} for all x X. The fo (0) 0, fo (x) fo (x) for all x U. Hece ΛD fo (x0,x,,x,x ) () τ x0,x,,x,x (), x0,x,,x,x () (5.7) 7 Ω Ω()s Refereces [] [] J. Aczel ad J. Dhombres, Fucoal Equaos Several Varables, Cambrdge Uv, Press, 989. T. Aok, O he sably of he lear rasformao Baach spaces, J. Mah. Soc. Japa, (950), 6-66.

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15 Geeral soluo ad geeralzed Ulam - Hyers sably of r ype dmesoal quadrac-cubc fucoal equao radom ormed spaces: Drec ad fxed po mehods 76/76 [6] [7] [8] [9] [0] [] mappgs, A. Uv. Tm s oara Ser. Ma.-Iform., 8 No. (000),. J.M. Rassas, H.M. Km, Geeralzed Hyers-Ulam sably for geeral addve fucoal equaos quas β ormed spaces J. Mah. Aal. Appl. 56 (009), o., Joh M. Rassas, M. Arukumar, E. Sahya, N. MaheshKumar, Soluo Ad Sably Of A ACQ Fucoal Equao I Geeralzed -Normed Spaces, Ier. J.Fuzzy Mahemacal Archve, Vol. 7, No., (05), -. T.Z. Xu, J.M. Rassas, W.X Xu,Geeralzed Ulam-Hyers sably of a geeral mxed AQCQ-fucoal equao mul-baach spaces: a fxed po approach, Eur. J. Pure Appl. Mah., (00), T.Z. Xu, J.M. Rassas, M.J. Rassas, W.X. Xu, A fxed po approach o he sably of quc ad sexc fucoal equaos quas β -ormed spaces, J. Iequal.Appl. 00, Ar. ID, pp. T.Z. Xu, J. M. Rassas, W.X. Xu, A fxed po approach o he sably of a geeral mxed AQCQ-fucoal equao o-archmedea ormed spaces, Dscree Dy.Na. Soc. 00, Ar. ID 855, pages. T.Z. Xu, J.M. Rassas, Approxmae Sepc ad Occ mappgs quas β ormed spaces, J. Compu. Aal. Appl., 5, No. 6 (0), 0 9.????????? ISSN(P):9 786 Malaya Joural of Maemak ISSN(O): 5666????????? 76