THE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION

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1 Joural of Scece ad Arts Year 12, No. 3(2), pp , 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: ; Accepted paper: ; ublshed ole: Abstract. We obta a stablty result for the Baker fuctoal equato, the settg of probablstc quas-metrc spaces. As a partcular case, we dscuss the probablstc stablty of the Gamma fuctoal equato. Keywords: Hyers - Ulam stablty, probablstc quas-metrc space, probablstc cotracto. 2 Mathematcs Subject Classfcato: 54E7; 39B52; 47H1. 1. INTRODUCTION By usg a fxed pot techque, J. A. Baker [1] establshed the followg Ulam - Hyers stablty result for the olear fuctoal equato f x x, f x. (1.1) Theorem 1.1 ([1], Theorem 2) Suppose S s a oempty set, (X,d) s a complete metrc space, : S S, : S X X,,1, ad,,,, d u x u y d x y, for all u S, x, y X Also, suppose that f:s X, δ>, ad d f u, u, f u for all u S. The there exsts a uque mappg g:s X such that g u u, g u, for all u S, ad d f u, gu, for all u S. 1 The am of ths paper s to obta a smlar result the settg of probablstc quasmetrc spaces edowed wth the łukasewcz t-orm. For the reader s coveece, we recall some useful termology from the theory of probablstc metrc spaces. For more detals, see the books [2] ad [3]. A tragular orm (or t-orm) s a bary operato T :[,1 [,1 [,1] whch s commutatve, assocatve, mootoe each varable ad has 1 as the ut elemet. Some basc examples are 1 West Uversty of Tmsoara, Departmet of Mathematcs, 3223 Tmsoara, Romaa. E-mal: mhet@math.uvt.ro; czahara@math.uvt.ro. ISSN: Mathematcs Secto

2 298 The probablstc stablty for Dorel Mhet, Clauda Zahara ad L, max 1, T a, b a b T a b ab (the Lukasewcz t-orm) T a, b m{ a, b} (the mmum t-orm). M We deote by Δ + the space of all fuctos f ad oly f x y; (the product t-orm) F : [,1], such that F s leftcotuous ad o-decreasg o, F()=, ad F( )=1, ad let D + be the subspace of Δ + of fuctos F wth lm t F(t)=1. Defto 1.1 A probablstc quas-metrc space s a trple (X,,T), where X s a oempty set, T s a t-orm, ad :X X D + s a mappg satsfyg () xy yx () xy t s T xz t, zy s, x, y, zx, t, s. If has the addtoal symmetry property xy = yx for all x, y X, the (X,,T) s called a Meger space. If the mappg Defto? has values Δ + stead of D +, the (X,,T) s sad to be a geeralzed probablstc quas-metrc space. 2 The mappg Q: X D defed by Q xy = yx for all x, y X s called the cojugate probablstc quas-metrc of. Defto 1.2 Let (X,,T) be a probablstc quas-metrc space. A sequece (x ) X s sad to be: () rght K-Cauchy (left K-Cauchy) f, for each ad,1, there exsts k so that, for all m k, ( ) 1 exsts x xm Q 1 respectvely ; () -coverget (Q-coverget) to x X f, for each ad,1, there k so that xx 1Q 1 xx, for all k. xx m Defto 1.3 Let Arght K, left Kad B, Q complete f every A-Cauchy sequece s B coverget.. The space (X,,T) s (A B) Defto 1.4 The probablstc quas-metrc space (X,,T) has the L-US (R-US) property f every - (Q-) coverget sequece has a uque lmt. 2. RESULTS The proof of our ma result s based o a fxed pot theorem for - cotractve mappgs probablstc quas-metrc spaces (Lemma 2.1), whch exteds a result from [4]. Recall that a - cotracto s a mappg f from a Meger space (X,F,T) to tself havg the property that there exsts k,1 such that Mathematcs Secto

3 The probablstc stablty for Dorel Mhet, Clauda Zahara 299 xy f x f y - cotracto satsfes,,1 : F 1 F k 1k. Note that every F k F t x yx, xy,, y f x f that s, t s a Sehgal cotracto o (X,F,T). Lemma 1 Let (X,, T L ) be a (rght K Q)-complete geeralzed probablstc quasmetrc space wth the R US property, ad let f : X X be a mappg for whch there exsts k,1 such that, for all Suppose there exst mappg f has a fxed pot x*, ad ad,1, 1 k 1k (2.1) xy f x f y,,1 ad x X. The the wth 1 xf x xx * max1, 1k 1k (2.2) obta that 1 x* roof: Let,,1ad x X f k 1 x f 1 x k, for all. be such that 1 Let t ad,1 be gve. Sce the seres such that k 1 t ad m1 TL 1 k f x f x m1 T 1 k L m1 max 1 k,1 Cosequetly, k 1. Iductvely, we xf x k s coverget, there exsts. The, for all 1 ad m *, m1 m t m k f x f x f x f x (2.3) Error! Bookmark ot defed. f x s rght K- Cauchy X, thus t s Q-coverget to some t 1 t f x x* X, that s, whe, for all. From hypothess (2.1), we derve that f s a Sehgal cotracto, wth cotracto costat k. Therefore 1 kt t 1, t, f x f x* f x x* ISSN: Mathematcs Secto

4 3 The probablstc stablty for Dorel Mhet, Clauda Zahara meag that f x s Q-coverget to * f x. By the R US property of the space X, we coclude that x * s a fxed pot of f. Addtoally, for all 1, relato (2.3.) mples 1 1 k max 1 k, xf x so 1 1 k k max 1, xf x xf 1 x k 1k max 1, 1 k For a arbtrary, xx* TL, xf x f x x* 1k 1k. But whe. As a cosequece, 1 f x x* xx* TL max 1,,1 max 1, 1 k 1 k 1 k By lettg, we obta the estmato (2.2). Theorem 2.2 Let S be a oempty set, ad (X,, T L ) be a (rght K Q) - complete geeralzed probablstc quas-metrc space wth the R US property. Suppose that : S X X s a mappg for whch there exsts k,1 so that, for all ad,1, 1 u, x u, y k 1k, u S (2.4) xy The, for every f:s X havg the property that, for some ad,1, f u u, f u 1, u S (2.5) there exsts a mappg a:s X satsfyg the equato (1.1), wth max1, 1k 1k f u a u, u S (2.6) by roof: We cosder the space Y g: S X ad Baker s operator J : Y Y gve Jgu u, g u, for all g Y ad all u S. We defe the mappg F : Y Y D by F t supf s, gh s us for all gh, Y. From the hypotheses o (X,, T L ), we fer that (Y, F, T L ) s a (rght K Q) - complete geeralzed quas-metrc space wth the R US property. Mathematcs Secto

5 The probablstc stablty for Dorel Mhet, Clauda Zahara 31 Next, we show that, f gh, Y, ad ad,1 are such that gh 1 the F k 1 k. To ths ed, frst ote that, f F 1, there exsts ' J g J h (,1) for whch F 1 ' 1. Ths mples gh whece there exsts s So It follows that The, va (2.4), Therefore gh supf s 1 ' s us wth the property f s 1 ' us. s 1 ', u S s 1 ', u S ks 1 k ', u S J g u J h u F k sup f ks 1 k' 1k J g J h u S J g u J h u ksk Now, let f be a mappg satsfyg (2.5), for some gve ad clam that F 1. for all u fj f Ideed, from (2.5) t follows that there exsts ' wth 1 f u J f u S. By the left cotuty of, there exsts s all u S. We ca deduce that f s 1 ', so us f u J f u F fj 1 ' 1 f F,,1. We ' (2.7) wth s ', for f u J f u 1 Oe ca ow apply Lemma 2.1 to obta that the operator J has a fxed pot a Y, meag that the mappg a: S X s a exact soluto of (1.1). Moreover, F fa max1, 1k 1k, provdg the estmato (2.6). S, X, u, x u1 x ad u u1 the above theorem, By settg we obta the followg probablstc stablty result for the Gamma fuctoal equato: Theorem 2.3 Let (R,, T L ) be a (rght K Q) complete geeralzed probablstc quasmetrc space wth the R US property. If there exsts k,1 so that, for all ad,1, 1 k 1k, u, ad f : xy u1 x, u1 y s a mappg satsfyg 1, u f u, u1 f u1 ISSN: Mathematcs Secto

6 32 The probablstc stablty for Dorel Mhet, Clauda Zahara for some ad,1, the there exsts a : wth ad 1 1 au u au, u max1, 1k 1k, u. f u a u Ackowledgemets: The work of the frst author was supported by a grat of the Romaa Natoal Authorty for Scetfc Research, CNCS- UEFISCDI, project umber N-II-ID-CE The work of the secod author was supported by the strategc grat OSDRU/C17 /DMI1.5 /S /78421, roject ID (21), co-faced by the Europea Socal Fud - Ivestg eople, wth the Sectoral Operatoal rogramme Huma Resources Developmet REFERENCES [1] Baker, J.A., roc. Amer. Math. Soc, 112, 729, [2] Cho, Y. J., Grabec, M., Radu, V., O Nosymmetrc Topologcal ad robablstc Structures, Nova Scece ublshers, 26. [3] Hadžć, O., ap, E., Fxed pot theory probablstc metrc spaces, Kluwer Academc ublshers, 21. [4] Mheţ, D., Aals of the West Uversty of Tmsoara, Mathematcs ad Computer Scece seres, 37, 15, [5] Mheţ, D., The Semar of robablty Theory ad Applcatos (STA), 14, Mathematcs Secto