Variational statements on KST-metric structures

Transcription

1 A. Şt. Uiv. Ovidius Costaţa Vol. 17(1), 2009, Variatioal statmts o KST-mtric structurs M. Turiici Abstract Structural rfimts of th variatioal rsult i Kada, Suzuki ad Takahashi [Math. Japoica, 44 (1996), ] ar giv. Applicatios to quilibrium poits thory (ovr th cosidrd sttig) ar also providd. 1 Itroductio Lt (M, d) b a complt mtric spac; ad ϕ : M R { }, som fuctio with (1a) ϕ is if-propr (Dom(ϕ) ad ϕ := if[ϕ(m)] > ) d (1b) ϕ is d-lsc (limif ϕ(x ) ϕ(x), whvr x x). Th followig 1979 statmt i Eklad [12] (rfrrd to as Eklad s variatioal pricipl; i short: EVP) is wll kow. Thorm 1 Lt th prcis coditios hold. Th, a) for ach u Dom(ϕ) thr xists v = v(u) Dom(ϕ) with d(u,v) ϕ(u) ϕ(v) (hc ϕ(u) ϕ(v)) (1) Ky Words: Complt mtric spac; If-propr lsc fuctio; Variatioal pricipl; w- distac; Psudomtric; Cauchy/asymptotic squc; Trasitiv rlatio; (strog) KSTmtric; τ-distac/fuctio; quilibrium poit. Mathmatics Subjct Classificatio: Primary 49J53. Scodary 47J30. Rcivd: Dcmbr, 2008 Accptd: April, 2009 This rsarch was supportd by Grat PN II PCE ID 387, from th Natioal Authority for Scitific Rsarch, Romaia. 231

2 232 M. Turiici v is E-variatioal (modulo (d,ϕ)): d(v,x) > ϕ(v) ϕ(x), for all x M \ {v} (2) aa) if u Dom(ϕ), ρ > 0 fulfill ϕ(u) ϕ ρ, th (1) givs (ϕ(u) ϕ(v) ad) d(u,v) ρ. (3) This pricipl foud som basic applicatios to cotrol ad optimizatio, gralizd diffrtial calculus, critical poit thory ad global aalysis; w rfr to th quotd papr for dtails. So, it caot b surprisig that, soo aftr, may xtsios of EVP wr proposd. For xampl, th abstract (ordr) o starts from th fact that, with rspct to th (quasi-)ordr (a1) (x,y M) x y iff d(x,y) + ϕ(y) ϕ(x) th poit v M apparig i (2) is maximal; so that, EVP is othig but a variat of th Zor-Bourbaki maximality pricipl [7]. Th dimsioal way of xtsio rfrs to th support spac (R) of Codom(ϕ) big substitutd by a (topological or ot) vctor spac. A accout of th rsults i this ara is to b foud i th 2003 moograph by Gopfrt, Riahi, Tammr ad Zăliscu [13, Ch 3]; s also Rozovau [23] ad Turiici [28]. Fially, th mtrical o cosists i th coditios imposd to th ambit mtric ovr M big rlaxd. Th basic 1996 rsult i this dirctio obtaid by Kada, Suzuki ad Takahashi [16] may b statd as follows. By a psudomtric ovr M w shall ma ay map (x,y) (x,y) from M M to R + := [0, [. Suppos that w fixd such a objct; which i additio, is triagular [(x,z) (x,y)+(y,z), x,y,z M]. W say that it is a w-distac (modulo d) ovr M providd (b1) is strogly d-sufficit: for ach ε > 0, thr xists δ > 0 such that: (z,x),(z,y) δ = d(x,y) ε. (c1) y (x,y) is d-lsc o M (s abov), x M. Thorm 2 Lt th coditios i Thorm 1 b admittd; ad b som w-distac (modulo d) ovr M. Th b) For ach u Dom(ϕ), thr xists a E-variatioal (modulo (,ϕ)) v = v(u) Dom(ϕ) with ϕ(u) ϕ(v) bb) For ach ρ > 0 ad ach u Dom(ϕ) with (u,u) = 0, ϕ(u) ϕ + ρ thr xists a E-variatioal (modulo (,ϕ)) v = v(u,ρ) Dom(ϕ) with ϕ(u) ϕ(v) ad (u,v) ρ. I particular, wh = d, ths rgularity coditios hold; ad Thorm 2 icluds th local vrsio of Thorm 1 basd upo (3). Th rlativ form of th sam, basd upo (1) also holds, but idirctly; s Bao ad Khah [5]

3 Variatioal statmts 233 for dtails. Not that th (rathr ivolvd) authors argumt rlis o th ocovx miimizatio thorm i Takahashi [26]. It is our aim i th prst xpositio to show (i Sctio 3) that a simplificatio of this is possibl. Th basic tool of our ivstigatios is (cf. Sctio 2) a psudomtric variatioal pricipl (xtdig th o i Tataru [27]) dductibl, i fact, by th origial Eklad s argumt. Fially, i Sctio 4, a applicatio of th obtaid facts is giv to quilibrium poits thory ovr such structurs. 2 Psudomtric variatioal statmts Lt M b som ompty st. Rmmbr that, by a psudomtric ovr it w shall ma ay map : M M R +. Suppos that w fixd such a objct; which i additio, is triagular (cf. Sctio 1). Lt also ϕ : M R { } b som if-propr fuctio (cf. (1a)). For a asy rfrc, w shall formulat th basic rgularity coditios ivolvig our data. Call th squc (x ) i M, strogly -asymptotic wh th sris (x,x +1 ) covrgs (i R). Furthr, lt th -Cauchy proprty of this objct b th usual o: δ > 0, (δ), such that (δ) p < q = (x p,x q ) δ. By th triagular proprty of w hav (for ach squc): strogly -asymptotic = -Cauchy; but th covrs is ot i gral tru. Nvrthlss, i may coditios ivolvig all such objcts, this is rtaiabl. A cocrt xampl is to b costructd udr th lis blow. Lt us itroduc a -covrgc structur ovr M by: x x iff (x,x) 0 as. W cosidr th rgularity coditio (2a) (,ϕ) is wakly dscdig complt: for ach strogly -asymptotic squc (x ) i Dom(ϕ) with (ϕ(x )) dscdig thr xists x M with x x ad lim ϕ(x ) ϕ(x). By th rmark abov, it is implid by its (strogr) coutrpart (2b) (,ϕ) is dscdig complt: for ach -Cauchy squc (x ) i Dom(ϕ) with (ϕ(x )) dscdig thr xists x M with x x ad lim ϕ(x ) ϕ(x). A rmarkabl fact to b addd is that th rciprocal iclusio also holds: Lmma 1 W hav (2a) = (2b); hc (2a) (2b). Proof Assum that (2a) holds; ad lt (x ) b a -Cauchy squc i Dom(ϕ) with (ϕ(x )) dscdig. By th vry dfiitio of this proprty, thr must b a strogly -asymptotic subsquc (y = x i() ) of (x ) with (ϕ(y )) dscdig. This, alog with (2a), yilds a lmt y M fulfillig

4 234 M. Turiici y y ad lim ϕ(y ) ϕ(y). It is ow clar (by th choic of (x )) that th poit y has all dsird i (2b) proprtis. Now, lt = (,ϕ) stad for th rlatio (ovr M) (a2) (x,y M) x y iff (x,y) + ϕ(y) ϕ(x). This is clarly trasitiv (x y, y z = x z); but ot i gral rflxiv (x x may b fals for crtai x M). So, for u M, it is possibl that M(u, ) := {x M;u x} b mpty. If this dos ot hold (M(u, ) ), w say that u is -startig; i.., (b2) (u,x) + ϕ(x) ϕ(u), for at last o x M; also rfrrd to as: u is startig (modulo (,ϕ)). Fially, call v Dom(ϕ), BB-variatioal (modulo (,ϕ)) providd (c2) (v,x) ϕ(v) ϕ(x) = ϕ(v) = ϕ(x) [= (v,x) = 0]. (This trmiology is rlatd to th mthods itroducd by Brzis ad Browdr [8] ad dvlopd by Kag ad Park [17]; s also Altma [2], Aisiu [3] ad Szaz [25]). For a o-trivial cocpt, w must tak v as startig (modulo (,ϕ)) (cf. (b2)); for, othrwis, (c2) is vacuously fulfilld. Som basic proprtis of such poits ar giv i Lmma 2 Assum v Dom(ϕ) is BB-variatioal (modulo (, ϕ)). Th (v,x) ϕ(v) ϕ(x), for all x M (1) (v,x) > ϕ(v) ϕ(x), for ach x M with (v,x) > 0. (2) Proof Th lattr part is clar, by dfiitio; so, it rmais to stablish th formr o. Assum this would b fals: (v,x) < ϕ(v) ϕ(x), for som x M. This, alog with (c2), yilds ϕ(v) = ϕ(x); whrfrom 0 (v, x) < 0, cotradictio; hc th claim. W may ow stat a usful psudomtric variatioal pricipl. Thorm 3 Lt th gral coditios upo (,ϕ) b accptd; as wll as (2a)/(2b). Th, for ach startig (modulo (, ϕ)) u Dom(ϕ) thr xists a BB-variatioal (modulo (,ϕ)) v = v(u) Dom(ϕ) with th proprty (1) (modulo ). Proof Lt ( ) stad for th trasitiv rlatio (a2); ad u Dom(ϕ) b as i th statmt. By dfiitio, M(u, ) ; hc u u 0, for som u 0 Dom(ϕ). Put = 0. If th altrativ blow holds

5 Variatioal statmts 235 (2c) u is BB-variatioal (modulo (,ϕ)) [s abov] w ar do, with v = u. Othrwis, o has (for = 0) (2d) u is ot BB-variatioal (modulo (,ϕ)). I this cas, th rlatios blow hold (for = 0) S := M(u, ), if[ϕ(s )] < ϕ(u ); (3) whrfrom, by dfiitio (agai for = 0) u +1 S : ϕ(u +1 ) (1/2)(ϕ(u ) + if[ϕ(s )]) < ϕ(u ). (4) Now, lik bfor, th coupl (2c)+(2d) (with + 1 i plac of ) coms ito discussio, tc. Th oly poit to b clarifid is that of (2d) takig plac for all. Th, a squc (u ) i M(u, ) may b foud so as (cf. (4)) (u,u m ) ϕ(u ) ϕ(u m ), whvr < m. (5) Th squc (ϕ(u )) is (strictly) dscdig ad boudd blow i R; hc a Cauchy o. This, alog with (5), tlls us that (u ) is a -Cauchy squc i Dom(ϕ). Puttig ths togthr, it follows via (2b) that thr must b som v M with u v ad λ := lim ϕ(u ) ϕ(v). (6) Th scod half of this givs v Dom(ϕ); sic (u ) Dom(ϕ). O th othr had, fix som rak. By (5) (ad th triagular proprty of ) (u,v) (u,u m ) + (u m,v) ϕ(u ) ϕ(u m ) + (u m,v), m >. This, alog with (6), yilds by a limit procss (rlativ to m) (u,v) ϕ(u ) lim m ϕ(u m ) ϕ(u ) ϕ(v) (i..: u v). (7) Lt x M(v, ) b arbitrary fixd. By (7) (ad th dfiitio of ( )) ϕ(u ) ϕ(v) ϕ(x), ; hc λ ϕ(v) ϕ(x). O th othr had, (4) yilds ϕ(u +1 ) (1/2)[ϕ(u ) + ϕ(x)], ; hc λ ϕ(x) ϕ(v). This givs ϕ(v) = ϕ(x); which (by th arbitrariss of x) tlls us that v is BB-variatioal (modulo (,ϕ)). Th proof is complt. Now, th rgularity coditio (2a) holds udr

6 236 M. Turiici (2) (,ϕ) is wakly complt: for ach strogly -asymptotic squc (x ) i Dom(ϕ) thr xists x M with x x ad limif ϕ(x ) ϕ(x). For xampl, this is rtaiabl whvr (2f) is wakly complt: ach strogly -asymptotic squc is -covrgt (2g) ϕ is wakly -lsc: limif ϕ(x ) ϕ(x) whvr th strogly -asymptotic squc (x ) fulfills x x. I particular, wh is (i additio) rflxiv [(x,x) = 0, x M], Thorm 3 icluds th variatioal pricipl i Tataru [27]. Th qustio of th covrs iclusio big also tru rmais op; w cojctur that th aswr is positiv. Not fially that, by th proposd proof, Thorm 3 is logically rducibl to th Pricipl of Dpdt Choics (cf. Wolk [31]). I fact, a similar coclusio is tru for th Brzis-Browdr ordrig pricipl [8]; s, for istac Cârjă, Ncula ad Vrabi [9, Ch 2, Sct 2.1]. So, it is atural askig of Thorm 3 big dductibl from th quotd pricipl. A positiv aswr for this is availabl; w shall dvlop such facts lswhr. Furthr aspcts wr dliatd i Hyrs, Isac ad Rassias [15, Ch 5]; s also Ba, Cho ad Yom [4]. Lt us ow rtur to our iitial sttig. A itrstig compltio of Thorm 3 is th followig. Lt th cocpt of E-variatioal (modulo (, ϕ)) poit b that of (2) (with i plac of d). This is strogr tha th cocpt of BB-variatioal (modulo (, ϕ)) lmt itroducd via (c2). To gt a corrspodig form of Thorm 3 ivolvig such poits w hav to impos (i additio to (2a)/(2b)) (2h) is trasitivly sufficit ((z,x) = (z,y) = 0 = x = y). Thorm 4 Lt th prcis coditios b i forc. Th, for ach startig (modulo (, ϕ)) u Dom(ϕ) thr xists a E-variatioal (modulo (, ϕ)) w = w(u) Dom(ϕ) with th proprty (1) (modulo (, w)). Proof Lt u Dom(ϕ) b tak as i th statmt. By Thorm 3, w hav promisd a BB-variatioal (modulo (, ϕ)) v = v(u) Dom(ϕ) with th proprty (1) (modulo ). If v is E-variatioal (modulo (,ϕ)) w ar do (with w = v); so, it rmais th altrativ of v fulfillig th opposit proprty: v w (hc ϕ(v) = ϕ(w)), for som w M \ {v}. I this cas, w is our dsird lmt. Assum ot: w y, for som y M, y w. By th prcdig rlatio, v y (hc ϕ(v) = ϕ(y)). Summig up, v w, v y ad

7 Variatioal statmts 237 ϕ(v) = ϕ(w) = ϕ(y); whrfrom (by (2h) ad th dfiitio of ( )) w = y, cotradictio. This ds th argumt. As bfor, th rgularity coditio (2b) holds udr (2i) (,ϕ) is complt: for ach -Cauchy squc (x ) i Dom(ϕ) thr xists x M with x x ad limif ϕ(x ) ϕ(x). For xampl, this is rtaiabl whvr (2j) is complt: ach -Cauchy squc is -covrgt (2k) ϕ is Cauchy -lsc: limif ϕ(x ) ϕ(x), whvr th -Cauchy squc (x ) fulfills x x. O th othr had, (2h) holds whvr is sufficit [(x,y) = 0 = x = y]. Not that, i such a cas, Thorm 4 bcoms th variatioal pricipl i Turiici [29]; s also Cosrva ad Rizzo [10]. I particular, wh is (i additio) rflxiv ad symmtric [(x,y) = (y,x), x,y M] (hc, a (stadard) mtric o M) Thorm 4 is othig but Eklad s variatioal pricipl (EVP). Furthr aspcts wr dliatd i Dacs, Hgdus ad Mdvgyv [11]; s also Haml [14, Ch 4]. 3 Extdd KST pricipls W ar ow i positio to gt a appropriat aswr to th qustios i Sctio 1. Lt M b a ompty st; ad d : M M R + b a psudomtric ovr it; supposd to b triagular (cf. Sctio 1) ad rflxiv sufficit (d(x, y) = 0 x = y). This map has all proprtis of a mtric, xcpt symmtry; w shall trm it, a almost mtric o M. Giv such a objct, th d-cauchy ad d-covrgc proprtis for a squc i M ar thos i Sctio 2. Lt us cosidr th coditio (a3) d is complt: ach d-cauchy squc is d-covrgt. Not that, by th lack of symmtry, a d-covrgt squc d ot b d-cauchy; hc, this cocpt has a tchical motivatio oly. Lt : M M R + b a triagular psudomtric ovr M. W shall say that this objct is a KST-mtric (modulo d) providd (b3) is Cauchy subordiatd to d: ach -Cauchy squc is d-cauchy (hc d-covrgt) (c3) is Cauchy d-lsc i th scod variabl: (y ) is -Cauchy ad y d y imply limif (x,y ) (x,y), x M.

8 238 M. Turiici Furthr, lt ϕ : M R { } b som if-propr, d-lsc fuctio (cf. (1a)+(1b)). Th followig auxiliary fact will b usful for us. Lmma 3 Assum that is som KST-mtric (modulo d) ovr M. Th, (, ϕ) is complt (i th ss of (2i)); hc (a fortiori), dscdig complt (i th ss of (2b)). Proof Lt (x ) b som -Cauchy squc i Dom(ϕ). From (b3), (x ) is d d-cauchy; so, by compltss, x y as, for som y M. W claim that this is our dsird poit for (2i). I fact, lt γ > 0 b arbitrary fixd. By th choic of (x ), thr xists k = k(γ) so that (x p,x m ) γ, for ach p k ad ach m > p. Passig to limit upo m o gts (via (c3)) (x p,y) γ, for ach p k; ad sic γ > 0 was arbitrarily chos, x y. This, alog with (1b), yilds th dd coclusio. Now, combiig ths with Thorm 3 (ad Lmma 2), w driv Thorm 5 Lt b som KST-mtric (modulo d) ad ϕ b if-propr, d-lsc. Th, for ach startig (modulo (,ϕ)) u Dom(ϕ) thr xists v = v(u) Dom(ϕ) with th proprtis (1) (modulo ) ad (1)-(2). A itrstig problm to b posd is that of gttig a corrspodig form of this rsult ivolvig E-variatioal (modulo (,ϕ)) poits. Th appropriat aswr to this is obtaiabl via Thorm 4. Call th triagular psudomtric : M M R +, a strog KST-mtric (modulo d) wh it is a KST-mtric (modulo d) ad fulfills (2h). Thorm 6 Assum that is som strog KST-mtric (modulo d) ad ϕ is if-propr, d-lsc. Th, for ach startig (modulo (, ϕ)) u Dom(ϕ) thr xists w = w(u) Dom(ϕ) fulfillig (1)+(2) (with (, w) i plac of (d, v)). I th followig, w shall giv som particular cass of our statmts. I) Lt d b a mtric (i..: symmtric almost mtric) o M; supposd to b complt (cf. (a3)). Furthr, lt : M M R + b som w-distac (modulo d). By (c1), o gts at oc (c3) (s abov). O th othr had, (b1) yilds (b3) as wll as th xtra coditio (2h); bcaus (as d=mtric) (z,x) = (z,y) = 0 = [d(x,y) ε, ε > 0] = x = y. Summig up, ay w-distac is a strog KST-mtric (modulo d). Hc th variatioal statmt i Kada, Suzuki ad Takahashi [16] (subsumd to Thorm 2) is a particular cas of Thorm 6. But, as xplicitly statd by ths authors, thir cotributio xtds th o du to T. H. Kim, E. S. Kim ad S. S. Shi [18]; hc so dos our statmt. This also shows that ay rcursio to th ocovx miimizatio thorm i Takahashi [26] is avoidabl i such

9 Variatioal statmts 239 approachs. Som rlatd facts may b foud i Um [30]; s also Zhu, Zhog ad Wag [33]. II) Lt (M,d) b a complt mtric spac; ad : M M R + b a triagular psudomtric ovr M. Accordig to Suzuki [24], w say that this objct is a τ-distac (modulo d) ovr M wh thr xists a fuctio η = η() from M R + to R + with th proprtis (3a) t η(x, t) is icrasig ad lim t 0 η(x,t) = 0 = η(x,0), x M (3b) lim [sup{η(z,(z,y m ));m }] = 0 ad y lim if (x,y ) (x,y), for ach x M d y imply (3c) lim [sup{(x,y m );m }] = 0 ad lim η(x,t ) = 0 imply lim η(y,t ) = 0 (3d) lim η(z,(z,x )) = 0 ad lim η(z,(z,y )) = 0 imply lim d(x,y ) = 0. Clarly, ay w-distac is a τ-distac too; just tak η(x,t) = t, x M, t R +. O th othr had (as w alrady rmarkd), ay w-distac is a strog KST-mtric. So, it is atural askig of what ca b said about th rlatioships btw ths largmts of our iitial cocpt. Th aswr to this is giv i Lmma 4 Lt th prcis covtios b i us. Th, ach τ-distac (modulo d) is cssarily a strog KST-mtric (modulo d); so that, w-distac = τ-distac = strog KST-mtric (modulo d). (1) Proof Lt : M M R + b a τ-distac; ad η = η() stad for som associatd map fulfillig (3a)-(3d). i) Lt x,y,z M b such that (z,x) = (z,y) = 0. By (3a), w hav η(z,(z,x)) = η(z,(z,y)) = 0; ad this, addd to (3d), givs d(x,y) = 0 (hc x = y); whrfrom (2h) holds. ii) Call th squc (x ) i M, (η,)-cauchy providd (d3) lim [sup{η(z,(z,x m ));m }] = 0, for som (z ) M. By [24, Lmma 3], th gric iclusio holds [for ach squc] -Cauchy = (η,)-cauchy. (2) O th othr had, ot that (3b) may b writt as

10 240 M. Turiici (3) (y ) is (η,)-cauchy ad y y imply lim if (x,y ) (x,y), for ach x M. Combiig ths givs (c3). iii) Fially, ot that by [24, Lmma 1] d [for ach squc] (η,)-cauchy = d-cauchy; (3) This, via (2), yilds (b3). Th proof is thrby complt. Now, by simply addig this to Thorm 6, o gts Thorm 7 Assum that is som τ-distac (modulo d), ad ϕ is ifpropr, d-lsc. Th, for ach startig (modulo (, ϕ)) u Dom(ϕ) thr xists a E-variatioal (modulo (,ϕ)) w = w(u) Dom(ϕ) with th proprty (1) (modulo ). Th origial proof of this rsult was providd i Suzuki [24]; but, it is rathr ivolvd. Th proposd rasoig (basd o th dvlopmts i Sctio 2) may b viwd as a rfimt of this o; howvr, it still dpds o Suzuki s argumts cocrig th iclusios (2)+(3). It would b itrstig to hav altrat proofs of ths, so as to avoid Lmma 4 abov. Furthr structural aspcts may b foud i G. M. L, B. S. L, J. S. Jug ad S. S. Chag [19]; s also Zhag, Ch ad Guo [32]. III) Lt agai d b a mtric ovr M; supposd to b complt (cf. (a3)); ad : M M R + b a triagular psudomtric. Accordig to Li ad Du [20] w say that it is a τ-fuctio (modulo d) providd (2h) holds ad (3f) x M, y y ad (x,y ) M, (for som M = M(x) > 0) imply (x,y) M (3g) lim [sup{(x,x m );m > }] = 0 ad lim (x,y ) = 0 imply lim d(x,y ) = 0. By [20, Rmark 1] ach w-distac (modulo d) is a τ-fuctio (modulo d). So, it is atural askig of th rlatioships btw this last cocpt ad that of strog KST-mtric (modulo d). Th aswr is cotaid i Lmma 5 Lt th prcis covtios hold. Th, ach τ-fuctio (modulo d) is a strog KST-mtric (modulo d); so (combiig with th abov) w-distac = τ-fuctio = strog KST-mtric (modulo d). (4)

11 Variatioal statmts 241 Proof Lt : M M R + b som τ-fuctio (modulo d). By dfiitio, it fulfills (2h); so, it rmais to prov that (b3)+(c3) hold. Th lattr of ths is just (3f). To vrify th formr, call th squc (x ), almost -Cauchy wh lim [sup{(x,x m );m > }] = 0. By dfiitio, [for ach squc] -Cauchy = almost -Cauchy. O th othr had, [20, Lmma 2.1] tlls us that [for ach squc] almost -Cauchy = d-cauchy. (5) Combiig with th abov givs (b3); ad th claim follows. As a cosquc of this, th variatioal statmt (ivolvig τ-fuctios (modulo d)) obtaid by th quotd authors is dductibl from Thorm 6 abov. Not that (as bfor) th proposd proofs ar still dpdig o Li- Du s rasoig ivolvig th rlatio (5); so, it would b itrstig to hav altrat proofs of it. Furthr aspcts may b foud i Li ad Du [21]. IV) Lt (M,d) b a complt almost mtric spac (s abov); ad : M M R + b a triagular psudomtric ovr M. Accordig to Al-Homida Asari ad Yao [1], w say that it is a Q-fuctio (modulo d) providd (b1) ad (3f) hold. Clarly, th lattr coditio is just (c1); ad from this, (c3) is fulfilld. O th othr had, (b1) yilds (b3) ad (2h). [Th argumt is similar to th mtrical o i I)]. W thrfor provd Q-fuctio = strog KST-mtric (modulo d). As a dirct cosquc, th variatioal statmt (ivolvig Q-fuctios (modulo d)) obtaid by th quotd authors [1, Thorm 3.1] is dductibl from Thorm 6. Th covrs qustio is still op; w cojctur that a positiv aswr is ultimatly availabl. A itrstig particular cas rfrs to d big (i additio) symmtric; hc, a mtric o M. By th prvious rmark ivolvig (c1), it follows that j) ach τ-fuctio is a Q-fuctio (modulo d) (cf. [1, Rmark 2.1]) jj) th cocpts of Q-fuctio ad w-distac ar idtical. This, alog with Lmma 5, givs us th iclusios w-distac = τ-fuctio = w-distac (modulo d); i.., th cocpts of w-distac ad τ-fuctio ar idtical (withi th mtric framwork). This, howvr, sms to b i cotradictio with [1, Exampl 2.1]; furthr aspcts will b dliatd lswhr.

12 242 M. Turiici 4 Applicatio (quilibrium poits) Roughly spakig, th xtsio of Thorm 2 giv i Sctio 3 cosists i takig : M M R + as a strog KST-mtric (modulo d) i plac of a w-distac (modulo d) at th lvl of th comparativ rlatio (1) ad th variatioal proprty (2). It is thrfor atural to ask whthr th right mmbr of ths could b also xtdd i som way. A appropriat aswr to this may b giv alog th followig lis. Lt M b som ompty st. Tak a psudomtric : M M R + ; which i additio, is triagular (cf. Sctio 1) ad trasitivly sufficit (cf. (2h)). By a rlativ psudomtric ovr M w shall ma ay map (x,y) F(x,y) from M M to R = R {, }. Fix such a objct; supposd to b triagular [F(x,z) F(x,y) + F(y,z), whvr th right mmbr xists] ad rflxiv [F(x,x) = 0, x M]. Lt µ = µ F stad for th associatd fuctio µ(x) = sup{ F(x,y);y M}, x M. By rflxivity, µ(x) 0, for ach x M; hc its valus ar i R + { } = [0, ]. Not that th altrativ µ(m) = { } caot b xcludd; so, to avoid this, w must assum (4a) µ is propr (Dom(µ) := {x M;µ(x) < } = ); rfrrd to as: F is smi-propr. For th arbitrary fixd u Dom(µ) put F u := F(u,.). W hav by dfiitio F u (u) = 0; F u (x) if{f(u,x);x M} = µ(u) > ; (1) whrfrom, F u is if-propr (cf. (1a)) for all such u; w shall rfr to such a proprty as: F is smi if-propr. I this cotxt, u is startig (modulo (,F u )) whvr (a4) (u,x) F u (x) = F(u,x), for som x M; this will b trmd as: u is startig (modulo (,F)). [I particular, (a4) holds whvr (u,u) = 0, if o taks th rflxiv proprty of F ito accout; othrwis, it may b fals]. Fially, lt us say that F is -complt if (b4) (,F u ) is complt (cf. (2i)), for all u Dom(µ). Th followig variatioal statmt is availabl. Thorm 8 Lt th triagular trasitivly sufficit psudomtric ad th triagular rflxiv rlativ psudomtric F b such that (4a)+(b4) hold. Th,

13 Variatioal statmts 243 for ach startig (modulo (,F)) u Dom(µ) thr xists v = v(u) Dom(F u ) with (u,v) F(u,v) µ(u)(< ) (2) (v,x) > F(v,x), for all x M \ {v}. (3) Proof Lt u b tak as bfor. By th imposd coditios, Thorm 4 is applicabl to (,F u ); morovr (s abov) u Dom(F u ) ad (by its vry choic) u is startig (modulo (,F u )). It th follows that, thr must b som v = v(u) Dom(F u ) with th proprtis (u,v) F u (u) F u (v)(= F(u,u) F(u,v)) (4) (v,x) > F u (v) F u (x)(= F(u,v) F(u,x)), x M \ {v}. (5) Th formr of ths is just (2), by th rflxivity of F. Ad th lattr o givs (3); bcaus (from th triagular proprty of F ad (2)) F(u,v) F(u,x) F(u,v) (F(u,v) + F(v,x)) = F(v,x). A basic particular cas of this is th o prcis by th costructios i Sctio 3. Prcisly, lt (M,d) b a complt almost mtric spac; ad : M M R + b a strog KST-mtric (modulo d) ovr it. Furthr, lt F : M M R b a triagular ad rflxiv rlativ psudomtric ovr M, fulfillig (4a). Rmmbr that F is th smi if-propr, i th ss: F u is if-propr, for ach u Dom(µ). Suppos i additio that (4b) F is smi d-lsc: F u is d-lsc, for ach u Dom(µ). Not that, as a cosquc of this, (b4) holds, if o taks Lmma 3 ito accout. By Thorm 8, w th hav Thorm 9 Lt th coupl (d,) b tak as bfor; ad th (triagular rflxiv) rlativ psudomtric F b such that (4a)+(4b) hold. Th, for ach startig (modulo (,F)) u Dom(µ) thr xists v = v(u) Dom(F u ) with th proprtis (2)+(3). Som rmarks ar i ordr. Lt ϕ : M R { } b som if-propr fuctio. Th rlativ psudomtric (c4) F(x,y) = ϕ(y) ϕ(x), x,y M (whr = 0) is triagular ad rflxiv, as it ca b dirctly s. I additio, (4a) holds, bcaus µ(x) = ϕ(x) ϕ,x M (hc Dom(µ) = Dom(ϕ)). This, alog with F u (.) = ϕ(.) ϕ(u), u Dom(µ), tlls us that i) F is -complt iff (,ϕ) is complt (cf. (2i))

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16 246 M. Turiici [32] S. Zhag, Y. Ch ad J. Guo, Eklad s variatioal pricipl ad Caristi s fixd poit thorm i probabilistic mtric spacs, Acta Math. Appl. Siica, 7 (1991), [33] J. Zhu, C. K. Zhog ad G. P. Wag, A xtsio of Eklad s variatioal pricipl i fuzzy mtric spac ad its applicatios, Fuzzy Sts Syst., 108 (1999), A. Myllr Mathmatical Smiar, A. I. Cuza Uivrsity 11, Copou Boulvard, Iaşi, Romaia -mail: mturi@uaic.ro