SOME REMARKS ON TARDIFF S FIXED POINT THEOREM ON MENGER SPACES

Transcription

1 PORTUGALIAE MATHEMATICA Vol. 54 Fac SOME REMARKS ON TARDIFF S FIXED POINT THEOREM ON MENGER SPACES E. Pără and V. Rad Inrodcion Le D + be he family of all diribion fncion F : R [0, ] ch ha F (0) = 0, and H 0 be he elemen of D + which i defined by { 0, if x 0, H 0 =, if x > 0. A -norm T i a binary operaion on [0, ] which i aociaive, commaive, ha a ideniy, and i non-decreaing in each place. We ay ha T i ronger han T and we wrie T T if T (a, b) T (a, b), a, b [0, ]. Definiion.. Le X be a e, F : X 2 D + a mapping (F(x, y) will be denoed F xy ) and T : [0, ] [0, ] [0, ] a -norm. The riple (X, F, T ) i called a Menger pace iff i aifie he following properie: (PM0) If x y hen F xy H 0 ; (PM) If x = y hen F xy = H 0 ; (PM2) F xy = F yx, x, y X ; (M) F xy ( + v) T (F xz (), F zy (v)), x, y, z X,, v R. Le f : [0, ] [0, ] be a conino fncion which i ricly decreaing and vanihe a. Received: March, 996; Revied: Ocober 3, 996. AMS Sbjec Claificaion: 60E54. Keyword and Phrae: Menger pace, Fixed poin heorem.

2 432 E. PĂRĂU and V. RADU Definiion.2 ([6]). The pair (X, F) which ha he properie (PM0) (PM2) i called a probabiliic f-meric rcre iff > 0 > 0 ch ha [ ] f F xz () <, f F zy () < f F xy () <. Remark.3. If (X, F) i a probabiliic f-meric rcre hen he family W f F :={Wɛ f } ɛ (0,f(0)), where Wɛ f :={(x, y) F xy(ɛ) > f (ɛ)}, i a niformiy bae which generae a niformiy on X called U F [6, p.46, Th..3.39]. We define he -norm generaed by f by: T f (a, b) = f ( )( ) f(a) + f(b) where f ( ) i he qai-invere of f, namely f ( ) (x) = { f (x), x f(0), 0, x > f(0). I i well known and eay o ee ha f f ( ) (x) x, x [0, ] and f ( ) f(a) = a, a [0, ]. In he nex ecion of hi noe we ll conrc generalized meric on Menger pace, relaed o ome idea which have appeared in [] and [4], and ing ome properie of he probabiliic f-meric rcre. In he la ecion, ing hi generalized meric, we ll obain a fixed poin heorem on complee Menger pace and we ll give ome coneqence. We ll give alo, a fixed poin alernaive in complee Menger pace. The noaion and he noion no given here are andard and follow [], [8]. 2 A generalized meric on probabiliic f-meric rcre Le f : [0, ] [0, ] a conino and ricly decreaing fncion, ch ha f() = 0. Lemma 2.. We conider a Menger pace (X, F, T ), where T T f. For each k > 0 le define d k (x, y) := p >0 k f F xy () d

3 TARDIFF S FIXED POINT THEOREM ON MENGER SPACES 433 and ( ) ρ k (x, y) := d k (x, y) k+. Then ρ k i a generalized meric on X. I i clear ha ρ k i ymmeric and ρ k (x, x) = 0. If ρ k (x, y) = 0, hen for each > 0, f F xy () d = 0, > > 0. Since f F xy() f F xy() f F xy () 0 for each (, ), hen d = 0 which implie 0 = f F xy () d f F xy() ( ), > > 0, which implie f F xy () = 0, > 0. Since f i a icly decreaing fncion and f() = 0 hen F xy () =, > 0, ha i x = y. Becae (X, F, T ) i a Menger pace and T T f, we have ( ) ( ) F xy (+v) T F xz (), F zy (v) T f F xz (), F zy (v), x, y, z X,, v R. Le ake = α and v = β, where α, β (0, ), α + β =. Then F xy () f ( )( ) f F xz (α) + f F zy (β), and o x, y, z X, > 0, α, β (0, ), α + β =, ( ) f F xy () (f f ( ) ) f F xz (α) + f F zy (β) f F xz (α) + f F zy (β), x, y, z X, > 0, α, β (0, ), α + β =. We divide he boh member of ineqaliy by, inegrae from o m and mliply wih k, where > 0, m >, k > 0. We obain m k f F xy () m d k f F xz (α) m d + k f F zy (β) d, m>, >0.

4 434 E. PĂRĂU and V. RADU We ake α =, repecively β = v in he fir, repecively, he econd erm of he righ ide of he previo ineqaliy and i follow ha: m k f F xy () d mα α k (α)k α α k (α)k α k p >0 α (α) k f F xz () f F xz () α f F xz () m >, > 0. d + β k (β)k d + β k (β)k mβ β β f F zy (v) v f F zy (v) v d + β k p (β) k >0 β dv dv f F zy (v) v By making m and aking p in he lef ide of he previo ineqaliy and by oberving ha and we obain ha (2.) p >0 p >0 p >0 Le denoe (α) k α (β) k k β >0 f F xz () f F zy (v) v f F xy () a = p >0 d = p >0 dv = p >0 d α k p >0 k b = α k p >0 c = β k p >0 + β k p k k k k >0 f F xy () k k f F xz () f F zy () f F xz () d, f F xz () f F zy () f F zy (v) v d, d. d d, d dv. dv,

5 TARDIFF S FIXED POINT THEOREM ON MENGER SPACES 435 If b = or/and c = i follow ha ρ k (x, z) = b k+ = or/and ρk (z, y) = c k+ = and i i obvio ha ρk (x, y) = ρ k (x, z) + ρ k (z, y). We ppoe ha b < and c <. The ineqaliy (2.) become: a b α k + c β k = b α k + c, α (0, ), ( α) k ( b which implie a inf 0<α< α k + c ) ( α) k, α (0, ). We define he fncion g : (0, ) R +, g(α) = b α k + ha g ha a minimm in α 0 = Therefore and i i clear ha a b α k 0 + b k+ b k+ + c k+ (g (α 0 ) = 0). c ( α 0 ) k = (b k+ + c k+ ) k+ ρ k (x, y) = a k+ b k+ + c k+ = ρk (x, z) + ρ k (z, y). c. We oberve ( α) k Lemma 2.2. Le (X, F, T ) be a Menger pace wih T T f. Then U F U ρk. I can be hown ha p a< T (a, a) p T f (a, a) = and, ing a< [6, p.4, Th..3.22] we obain ha (X, F) i a probabiliic f-meric rcre. By ing Remark.3 i ffice o how ha (2.2) ɛ (0, f(0)), δ(ɛ): ρ k (x, y) < δ F xy (ɛ) > f (ɛ). We oberve ha ρ k (x, y) < δ p >0 We ake fixed, = ɛ 2 k > 0, f F xy () k d < δ k+ f F xy () d < δ k+ m = m >, > 0, k f F xy () and m = 2. I follow d < δ k+. ( ɛ k ɛ 2) ɛ 2 f F xy () d < δ k+.

6 436 E. PĂRĂU and V. RADU B ɛ F xy () F xy (ɛ) f F xy () f F xy (ɛ) f F xy() Therefore f F xy(ɛ). ɛ ( ɛ k ɛ 2) ɛ 2 f F xy (ɛ) ɛ d ( ɛ k ɛ 2) ɛ 2 f F xy () d < δ k+, ( ɛ k+ f F xy (ɛ) which implie <δ 2) k+. If we chooe δ = ɛ ɛ 2 we have f F xy(ɛ)<ɛ, which how ha he relaion (2.2) i aified for δ(ɛ) = ɛ 2. Lemma 2.3. If (X, F, T ) i a complee Menger pace nder T T f, hen (X, ρ k ) i complee. We ppoe ha (x n ) i a ρ k -Cachy eqence, ha i, (2.3) ɛ > 0, n 0 (ɛ): n n 0 (ɛ), p 0 ρ(x n, x n+p ) < ɛ. From Lemma 2.2 we have ha (x n ) i a U F -Cachy eqence. Since (X, F, T ) i a complee Menger pace, we obain ha (x n ) i a U F -convergen eqence, ha i x 0 X ch ha ɛ > 0, n (ɛ): n n (ɛ) F xnx 0 (ɛ) > f (ɛ). I remain o how ha (x n ) i a ρ k -convergen eqence. From (2.3) we obain ha ɛ lim ρ(x n, x n+p ) = lim p >0 lim k k f F xnx n+p () f F xnx n+p () d d, n n 0 (ɛ), > 0. By ing he Fao lemma and he coniniy of f we obain: ɛ lim k f F xnx n+p () = k d k lim f F xnx n+p () d = ( ) f lim F x nx n+p () d, n n 0 (ɛ), > 0.

7 TARDIFF S FIXED POINT THEOREM ON MENGER SPACES 437 I can be proved ha lim F xnx n+p () = F xnx 0 () (acally we ll e only he fac ha lim F xnx n+p () F xnx 0 ()) and he previo relaion become which implie ɛ k ρ k (x n, x 0 ) = p >0 Th he lemma i proved. f F xnx 0 () k d, n n 0 (ɛ), > 0, f F xnx 0 () d ɛ, n n 0 (ɛ). 3 A fixed poin heorem and ome coneqence I i well-known ha a mapping A : X X (where (X, F) i a PM-pace) i called -conracion if here exi L (0, ) ch ha F AxAy (L) F xy () for all R, for all x, y X. Lemma 3.. If (X, F) i a probabiliic f-meric rcre and A i an -conracion hen A i, for each k > 0, a ric conracion in (X, ρ k ). we have Since F AxAy (L) F xy () for ome L (0, ), and every real hen k f F AxAy (L) d k If we make L = in he lef ide, hen we obain L k (L)k L f F AxAy ()) Therefore, if we ake p obain ha >0 d k p >0 f F xy ()) k f F xy () d f F xy ()) d. d = d k (x, y), > 0. in he fir member of he above ineqaliy, hen we L k d k(ax, Ay) d k (x, y) and i i clear ha (3.) ρ k (Ax, Ay) L ρ k (x, y) where L = L k k+ (0, ) and he lemma i proved. Now, we can prove or main rel:

8 438 E. PĂRĂU and V. RADU Theorem 3.2. Le (X, F, T ) be a complee Menger pace wih T T f. If here exi ome k > 0 ch ha for every pair (x, y) X one ha (3.2) p >0 k f F xy () d <, hen every -conracion on X ha a niqe fixed poin. The relaion (3.2) how ha ρ k i a meric. From Lemma 3. we obain ha A i a ric conracion in (X, ρ k ). Le x X be an arbirary poin. From (3.) we have ha (A i x) i a U ρk -Cachy eqence. By ing he Lemma 2.3, we oberve ha (A i x) i a ρ k -convergen eqence o x 0. I i eay o ee ha x 0 i he niqe fixed poin of A. Corollary 3.3 (cf. [0]). Le (X, F, T ) be a complee Menger pace nder T T f, where f(0) < and ppoe ha for each pair (x, y) X 2 here exi xy for which F xy ( xy ) =. Then every -conracion on X ha a niqe fixed poin. Since for xy we have 0 k f F xy () k and for > xy we have k he heorem. d = k xy xy f F xy () f F xy () f F xy () d ( ) d k f(0) ln( xy ) ln() d = 0 hen (3.2) hold and we can apply Corollary 3.4 ([6]). Le (X, F, T ) be a complee Menger pace wih T T ch ha for ome k > 0 and every pair (x, y) X one ha (3.3) p >0 k F xy () d <. Then every -conracion on X ha a niqe fixed poin. We ake f() = f () = and we apply he heorem.

9 TARDIFF S FIXED POINT THEOREM ON MENGER SPACES 439 Corollary 3.5. Le (X, F, T ) be a complee Menger pace nder T T and ppoe ha here exi k > 0 ch ha every F xy ha a finie k-momen. Then every -conracion on X ha a niqe fixed poin. I i well-known ha (µ k ) k xy = k F xy () d and he corollary follow. k F xy() d = 0 k ( F xy ()) d <. Therefore k ( F xy ()) d (µ k ) k xy < Remark 3.6. For k = i can be obained a known rel (ee [, Corollary 2.2]). Generally from he fixed poin alernaive ([3]) we obain he following Theorem 3.7. Le (X, F, T ) be a complee Menger pace nder T T f and A an -conracion. Then for each x X eiher, i) here i ome k > 0 ch ha (A i x) i ρ k -convergen o he niqe fixed poin of A, or ii) for all k > 0, for all n N and for all M > 0 here exi := (k, n, M) ch ha k f F A n xa n+ x() d > M. We ppoe ha ii) i no re: k >0, n 0 >0, M >0, >0 ch ha k f F A n 0 xa n 0 + x () So, we have for ome k > 0, ρ k (A n 0 x, A n 0+ x) <. I follow ha p > 0, p ρ k (A n 0 x, A n0+p x) ρ k (A n0+p x, A n0+p+ x) i=0 d M. ( + L + L L p ) ρ k (A n 0 x, A n 0+ x) = Lp L ρ k (A n 0 x, A n 0+ x) ρ k(a n 0 x, A n 0+ x) L <,

10 440 E. PĂRĂU and V. RADU where L := L k k+ <. Therefore, he eqence of cceive approximaion, (A i x) i a ρ k -Cachy eqence. From Lemma 2.3 we obain ha (A i x) i ρ k -convergen and i i eay o ee ha he limi of he eqence (A i x) i he niqe fixed poin of A. REFERENCES [] Conanin, Gh. and Irăţec, I. Elemen of Probabiliic Analyi wih Applicaion, Ed. Acad. and Klwer Academic Pbliher, 989. [2] Had zić, O. Fixed Poin Theory in Probabiliic Meric Space, Univeriy of Novi Sad, 995. [3] Margoli, B. and Diaz, J.B. A fixed poin heorem of he alernaive for conracion on a generalized complee meric pace, Bll. Amer. Mah. Soc., 74 (968), [4] Rad, V. A Remark on Conracion in Menger Space, Seminarl de Teoria Probabiliăţilorşi Aplicaţii(=STPA), Univeriaea din Timişoara, Nr.64, 983. [5] Rad, V. Some fixed Poin Theorem in Probabiliic Meric Space, in Lecre Noe in Mah., Vol.223, pp.25 33, Springer-Verlag, New York, 987. [6] Rad, V. Lecre on probabiliic analyi, in Srvey, Lecre Noe and Monograph, Serie on Probabiliy, Saiic and Applied Mahemaic, 2, Univeriaea de Ve din Timişoara, (Seminarl de Teoria Probabiliăţilorşi Aplicaţii), 994. [7] Schweizer, B., Sherwood, H. and Tardiff, R. Comparing wo conracion noion on probabiliic meric pace, Sochaica, 2 (988). [8] Schweizer, B. and Sklar, A. Probabiliic Meric Space, Norh-Holland, New York, 983. [9] Sehgal, V.M. and Bharcha-Reid, A.T. Fixed poin of conracion mapping on probabiliic meric pace, Mah. Syem Theory, 6 (972), [0] Sherwood, H. Complee probabiliic meric pace, Z. Wahrch. Verw. Geb., 20 (97), [] Tardiff, R.M. Conracion map on probabiliic meric pace, Jornal Mah. Anal. Appl., (990), Emilian Pără and Viorel Rad, We Univeriy of Timişoara, Facly of Mahemaic, Bv. V. Pârvan, No.4, 900 TIMIŞOARA ROMANIA rad@im.v.ro