Shanghai, China. SOME NEW FIXED ... *Projects supported by Provincial Natural Science Foundation of Jiangxi, China ..... Deg (I-T, D,O) ---Dog (I,D,O) --- 1~ o.
Applied Mathematics and Mechanics (English Edition, Vol. 16,'No. 2, Feb. 1995)
Published by SU, Shanghai, China
SOME NEW FIXED POINT THEOREMS
IN PROBABILISTIC
METRIC SPACES* Zhu Chuan-xi ( ~ t " ~ ) (Mathematics Division. Nanchang University. Nanchang)
(Received July 25, 1993; Communicated by Zhang Shi-sheng) Abstract hi this paper, we introduce the concept o/ the Z-M-PN .vmce. and obtain some new./i'xed pohlt theorems in probabilistic metric spaces. Meamvhile. some.[amous./~.ved poiot theorems are generalized in Abstract probabilistic metric spaces, stwh as ./ixed point theorem o[ Schauder. Guo's theorem attd Jixed point theorem of Petrvshyn are
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has generalized in Menger PN-.v~gwe. And ./i.ved po#tt theorem of AItman is ktlso an analytic solution only when the polytropic index of detonation products equals to three. In in the Z-M-PN space. In this paper, however, by utilizing the "weak" shock general, a genertdized numerical analysis is required. behavior of theKey reflection in the explosive and applying the small parameter purwordsshock topological degree, products, probabilistic metric spaces, operator. terbation method, an analytic, compact first-ordercontinuous approximate solution is obtained for the problem of flying operator, Z-M-PN space plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final Let velocities plate obtained agree very well numerical results by Thus R be of theflying set denoting all real numbers, R §with be the set denoting all computers. nonnegative real an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic numbers. A mapping f z R . - - ~ R § is called a distribution function, if it is a nondecreasing index) for estimation of the velocity of flying plate is established. function and left continuation satisfying the following conditions: i n f f1.( t ) =Introduction 0, sup/(t)=l IER
#~R
the set driven of all distribution Explosive flying-plate functions. technique ffmds its important use in the study of behavior of In this paper, let supposeloading, that t-norm is continuous. materials under intense us impulsive shock Asynthesis of diamonds, and explosive welding and D e fof i nmetals. i t i o n 1The (Zhang Shi-sheng r~3) Aof Menger probabilistic linear normed space (M-PN cladding method of estimation flyor velocity and the way of raising it are questions for Short) is a mapofSpace common interest.is a triplet (E', F , A) , where E is a real linear space, F ' F - , ~ of one-dimensional plane detonation flying plate, normal ping Under (we sthe h a lassumptions l denote the distribution function F ( x ) byand]',rigid ) satisfying thethefollowing approach of solving the problem of motion of flyor is to solve the following system of equations conditions: governing the ]flow (PN-I) ' , ( 0 )field , = 0jof detonation products behind the flyor (Fig. I): (PN-2)f,(t)=H(t), VtER, ifandonlyifx=0; (PN-3) for any real number a ~--ff 0ap, +u_~_xp f o , ( t ) - +- f , ( au t / l a =o, l)l (PN-4) for any x, u E E , tl, t ~ E R , if f , ( t l ) = l , f,(t~)---l, ther~f,§ au au 1 =0, I , § (PN-5) for any x, ,gEE', and all t l , t~ER §y , we have /w(tz)). (i.0 in the M-PN space, we prove some new fixed point theorems as following: aS as L e m m a 1 (Zhang Shi-shenga--T and Chen =o, Yu-qing '~21) Let D be an open subset of an M-PN space ( E , F , l k ) with A ( t , t ) ~ t , V t E [ O , 1] . Suppose E is an infinite dimension p =p(p, s), space, and T~ D ~ E i s a compact continuous operator satisfying the following conditions: where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, the trajectory R ofNatural reflected shockFoundation of detonation wave D China as a boundary and the *Projects with supported by Provincial Science of Jiangxi, trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraFirst Received May 31, 1993. meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave 179 D and by initial stage of motion of flyor also; the position of F and the state parameters of products 293
Zhu Chuan-xi
180
05 ~ ( i i ) Tx~-a2x, ' d a E ( 0 , 1 ] , xEOD. Then D e g ( I - - T , D , 0 ) - - 0 . T h e o r e m 1 Let D be an open subset of an M-PN space ( E , F , A ) w i t h A (t, t) ~ t , V t E [ 0 , 1]," and let E be an infinite dimension space. Suppose that T'D->E is a compact continuous operator satisfying the following conditions: (i)
OED~ k/xEOD, x~Tx,f~,(t')0,aE(0,
1]. Then D e g ( I - T ,
D , 0) = c .
By virtue of Riven condition: VxEOD, f ~ , ( t ) ~ f ~ Vt>O , we know that Oq~T(o~D) and Tx~aZx, VaC: ( 0 , 1 J . In fact, 1~ suppose OET(.~D) , then there exist x,,~dDsuch that T(x.). 5r>O, tha,t is r e > 0 , 3,>0, there exists an I V > 0 , when n > N , we have ]Tx. ( e ) > l - - 3 . . Since fax. (e)>~fTx. ( e ) > l - - 2 , hence ax., "Y->0, i.e. x,. ~ > 0 . Since ODis closed set (for ,9"), hence OEOD, this is in contradiction Abstractwith OED. By this contradiction it follows that O$T(OD). The of thexoEOD motion, ofsuch a rigid plate underhence explosive attack ahas 2 ~one-dimensional Suppose there problem exists an thatflying Txo=aZxo~ we have ~,l an analytic solution only when the polytropic index of detonation products equals to three. In (otherwise a = I, Txo=xo, xoEOD , this is in contradiction with condition Tx~-x, VxE(YD), general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock thus a6: (0, 1) . And so we have Proof
behavior of the reflection shock in the explosive products, and applying the small parameter purt is obtained z t for the problem of flying terbation method, an analytic, first-order approximate solution plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained results by computers. Thus t agree very t well with ~ numerical t an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.
,o...(§ 1.
,o.o
,,.0
Introduction
Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest. Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal Because of 1 / athe > l , problem let fTxoof(t)~.fTxo the limit in the two sides,of(let n-~oo) approach of solving motion of(t/an). flyor isget to solve the following system equations obtaining the fTxo (t)field = H of(t)detonation . governing flow products behind the flyor (Fig. I): By (PN-2), we know that Txo----O, that is OET(OD)~T(OD) , this is in contradiction with 1~ Thus T x # aZx,aE (o, 1], V xEOD. --ff =o, ap +u_~_xp + auall the conditions of Lemma 1, hence, by Since 1 ~ 2 ~ and the given condition satisfy au Lemma 1 we know that Deg ( I - T , au D, 0)----0. y1 =0, T h e o r e m 9. Let T be a compact continuous operator. Suppose D is an open subset (i.0 of an M-PN space ( E , F , A ) , T'D-->E,aS OED,a s and A(t,t)>~t, ' v ' t E [ 0 , 1 ] . If it satisfies the a--T =o, following conditions: Oq~T(OD) and f~,(t)>~fr,_,(t), Vt>O,xEOD , then Deg (I--T, D , 0 ) = 1 , i.e. T has fixed point inp D. =p(p, s), P r o o f Let ha (x) = x - - ( 1 -- ~,) Tx, 3.E [ 0, I ] , V x E D . We prove that 0 $ ha (OD'), V'~E where p, fact, S, u are pressure, specific i.e. entropy particle velocity ofand detonation products [ 0 , 1 ] .p, I~a suppose thatdensity, OEha(OD) there and exists a ~,0E[0,1], an xoEaD such respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the that O=hxo(xo) ; i.e. trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field wave behind the detonation wave X o - I (of1 -central A o ) T x rarefaction o=O ( 1) D and by initial stage of motion of flyor also; the position of F and the state parameters of products 293
Some New Fixed PointTheorems in Probabilistic Metric Spaces
181
For xo#Txo, x,EOD, It is proved easily. (In fact, suppose that x~EOD~ xo=Txo, we have Txo-- xo ==O.By fTxo (t) ~ fTxo-xo (t), ~ t>O, we obtain that fTxo (t) ~]Txo-xo (t) =
H(t). i.e. ]Txo ( t ) = H ( t ) , k i t > 0 , i.e. Txo=O, i.e. OE'_I'(OI3) , this is in contradiction with O~7"(OD)). Thus, in (1)A,o4=0,2o.--V-1 (otherwise xo=OE~gD, tins is m contradiction with oED) , therefore A.0E(0,1). By virtue of(1), it gets Txo~Xo=aoTxo , and so we have fTxo (t) ~fT'xo-x, (t) = f 2oxo(t) =jTx, (t/ ao) ~frxo-xo (t/ ~,~ =faor~o(t/ao) =trio (~/ag) >... >.>... =]Txo(t/,,l"o), n= 1,2,-.., V t.~0, i.e.
]T:,o(t)>~fr,,o(t/,~"o), Let .fTxo(t)~fTxo(t//~"o) distribution, we know that
kit>a, 0 < ; t 0 K t ,
n=l,2,...
(g)
get the limit in the two sides. By the definition of
Abstract fr:,o(t)=H(t),
kit>0, The one-dimensional problem of the motion of a rigid flying plate under explosive attack has By (PN-2), it gets Txo=O, i.e. OET(OD) , this is in contradiction with 0 q3T(OD).Thus an analytic solution only when the polytropic index of detonation products equals to three. In Oq~h,(&D), V a E Eanalysis o , 1]. is required. In this paper, however, by utilizing the "weak" shock general, a numerical By the honaotopy degree in and [2], applying we obtain behavior of the reflectioninvariance shock in of thetopological explosive products, the that small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying Deg(I-T, D, 0)=Deg(/, D, 0)=14=0. plate driven by various high explosives with polytropic indices other than but nearly equal to three. And by the solution property of topological degree in [2~, we know that T has fixed Final velocities of flying plate obtained agree very well with numerical results by computers. Thus point in D.formula with two parameters of high explosive (i.e. detonation velocity and polytropic an analytic h e oestimation r e m 3 of LettheD~. Dz be of two openplate subset o f a n infinite dimension M-PN space ( E , F , index)Tfor velocity flying is established. A ) , OED~, 131~D2, where A is a t-norm, A(t, t)>/t, k i t E [ 0 , 1 ] . Suppose that T " 0 ~ E ' is a compact continuous operator, tbllowing inequalities hold: 1. Introduction ( H I ) f~,,_,(t)~f~,(t), kixEOD~, t>O. OET(OD,) Explosive driven flying-plate technique ffmds its important use in the study of behavior of (Hz) f , , ( t ) ~ f T , ( t ) , kixEODz, t>O, aE(O,1],x-~Tx. materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and Then T has atThe least a fixed in Dz\DI cladding of metals. method of point estimation of flyor velocity and the way of raising it are questions of common P r o o finterest. Since condition (H0, by virtue of Theorem 2, we obtain that Under one-dimensional planebydetonation rigid flying normal Deg ( Ithe - T . assumptions D~. 0 ) = l . of Since condition (Ha), virtue of and Theorem 1, weplate, havetheDeg(I--T. approach of solving the problem of motion of flyor is to solve the following system of equations D2, 0)=0. By the additivity of topological degree in [2], It gets governing the flow field of detonation products behind the flyor (Fig. I): Deg(I-T, Dz\,O~, O ) = D e g ( I - T , D2, O ) - D e g ( I - T , D., 0 ) = 0 - 1 # 0 . By the solution property of topological degree in=o, [23, we know that T has at least a fixed --ff ap +u_~_xp + au point in D~\DI. au openau T h e o r e m 4 Let D~, D2 be two subsety1 of an=0, M-PN space (E, F , tx) , OEDj, Ol~D= , where A is a t-norm. A(t, t)~r k i t E [ 0 , 1 ] Suppose that T'Dz-->E(i.0 is a s compact continuous operator, and aS E is an ainfinite dimension space. In the meantime, if one of a--T =o, the following conditions holds: p =p(p, s), (H3) /~,(t)~f~_,~,(t), Vt>o, o~f,(t), V t > 0 , xEODt (5) trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraV t > 0I ,of central 0 ~ < e l and hEN. Meanwhile there exists at least one t~>0 such that f(Tx+x)"(t~)~ f(Tx)"+x'(tO, VxgzOD, where n > l .and nC:N. Then Deg (I--T. D, 0)=Deg (I, D, 0)=1, i.e. T has a fixed point in D. P r o o f Letting h ~ ( x ) = x - 2 T x , 2 E [ 0 , 1 ] , ~/xED, we prove that O~h~(OD), h E [ 0 , 1 ] . In fact, suppose that OEha(OD), i.e. there exists a 2r,E [ 0 , 1 ] , xoEOD, such that Abstract O=xo-- ?..~Txo, then 20ev 0 (otherwise xo =Og:o~D , this contradicts OED), and 20::k= 1 (otherwise The one-dimensional of a rigid flying plate under explosive xe=Txo, xoEOD, this c oproblem n t r a d i c of t s xthe : ~ Tmotion x , VxEOD). Thus h0E(0,1). Because of attack 0 = x 0 -has 20 an analytic solution only when the polytropic index of detonation products equals to three. In Txo, obtaining Txo=xo/2o . By the given condition, we have general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock f(Txo+xo)'(t)~f(Txo)'+x~ n > l ,thehEN (lt) behavior of the reflection shock in the explosive (t), products, ~'t>O, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying In (I I), let t be the fixed number tt>O in the given condition, it gets plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very with numerical (12) f('rx~+xo)" (t~) well ~>f(Vxo) +~; (t~) results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic Let Txo-~Xo/2o in (12). it gets index) for estimation of the velocity of flying plate is established.
fC;,o/~o+~,o) o (t~)
>fCtl;.~+t)~,"o(tt),
i.e.
1. Introduction .fCl/;.o+t)"~,; (t~) > / ( t / ~ ; + l ) ~ , ; (tj) ,,
i.e
Explosive driven flying-plate technique ffmds its important use in the study of behavior of (.13) .f o~[loading, 2"ot,/ (1shock + &).]synthesis > ?~ [of2"ot,/ (1 -J--A.j) ], explosive welding and materials under intense impulsive diamonds, and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions where xoEOD,2oE(o,l) , t j > 0 ) of common interest. By the nondecreasing of f x ~ E . ~ plane , it gets Under the assumptionsproperty of one-dimensional detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations 2~tl ,.,~ ,,].~t i 1 l governing the flow field of- (detonation products behind the flyor (Fig. I): 1 - F 2 o ) " ~ 1--}-A~ ' i.e. ( 1 . . } _ 2 o ) n ~ i + 2 ~ i.e. 1-F2~>(1-{-2o)", i.e.
--ff ap +u_~_xp+ a
=o,
au
"n-l.L]n l+2"o>l+n2o+C, 2o+'"+C, 2o+'"+C~-Xao --,~o au au 1
i.e.
2
k
y
k
=0,
(i.0 aS~,4'0 + a"'"s +C: -'2~ - ' < 0 . n 2 o + C a--T =o,
This is in contradiction with g0>0 and n2o-FCS.2~o+.,. +C"-~2"o-~o. p =p(p, s), Thus0q~hx(OD), V 2 E [ 0 , 1 ] . homotopy invariance of topological degree [2], wevelocity have of detonation products whereByp,the p, S, u are pressure, density, specific entropy andinparticle respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the D e g boundary. ( I - T , D,O) (I,D,O)the --- 1~ o of R and the state paratrajectory F of flyor as another Both---Dog are unknown; position meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products 293
Some New Fixed Point Theorems in Probabilistic Metric Spaces
185
And by the solution property of topological degree in I-2], we know that T has a fixed point in D. R e m a r k 1 Because of the limited space, we have not listed other patterns of fixed point theorem in Z-M-PN spaces. R e m a r k 2 In the M-PN space, Theorem 2 is a generalization of the fixed point theorem of Petryshyn; Theorem 4 is a generalization of the Guo's theorem; Theorem 5 is a generalization of the fixed point theorem of Schauder. In the Z-M-PN space, Theorem 6 is a generalization of the fixed point theorem of Altman. References Zhang Shi-sheng, Fixed Point T/woo' and Application. Chongqing Press, Chongqing (1984), (in Chinese) [ 2 ] Zhang Shi-sheng and Chen Yu-qing, Topological degree theory and fixed point theorems in probabilistic metric spaces, AbstractApplied Mathematics and Mechanics (English Ed.), 10, 6 (1989), 495.--505. [ 3 ] TheCao Jue-sheng andproblem Lin Yi-qi, Themotion extension theorem and topological degree attack of compact one-dimensional of the of a rigid flying plate under explosive has an analytic solution operators only whenonthe polytropic index of detonation products to three. In continuous a probabilistic normed linear space, Journalequals of NanjhTg Normal general, University. a numerical 14, analysis is required. In this paper, however, by utilizing the "weak" shock 1 (1991), 1-- 8. (in Chinese) behavior of theHuai-yun, reflection The shockachievement in the explosive andtheapplying theofsmall parameter metric pur[ 4 ] Gong and products, prospect in research probabilistic terbation method, an analytic, first-order approximate solution is obtained for the problem of flying spaces in China. The Literature Contributed by Experts at the Academic Symposium of plate driven by various high explosives with polytropic indices other than but nearly equal to three. the Chh~a~ Fifthplate Session q/" Fixed Probabilistic Metric Space Variational Final velocities of flying obtained agree Pohtt, very well with numerical results by and computers. Thus h7 Equalities (5, 1991). (in Chinese) an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic [ 5 3 for Xia Dao-xing et al., RealofVariable Function Theory and Functional Analysis (Vol. II), index) estimation of the velocity flying plate is established. The Publishing House of People's Education, Beijing (1979). (in Chinese) Guo's theorem, Journal o[" Jiangxi Normal UniversiE63 Li Guo-zhen, On generalization 1. ofIntroduction ty, 9+(t984), 10--12. (in Chinese) [ 7 ] Explosive Ding Guang-gui, An bm'oduetion Banach Space Theoo', Science (1984). of(in driven flying-plate technique toffmds its important use in the study Press of behavior materialsChinese) under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. TheSome method of estimation of flyorand velocity and the way of raisingAnal., it are questions Da-jun, fixed point theorems applications, Nonlinear 10 (1986), [ 8 3 Guo of common interest. 1293 -- 1302. Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal E 9 ] Sun Jing-xian, A generalization of Guo's theorem and applications, J. Math. Anal. approach of solving the problem of motion of flyor is to solve the following system of equations 126field (1987). governingAppl., the flow of detonation products behind the flyor (Fig. I): [lO] Zhu Chuan-xi, Several new mappings of expansion and fixed point theorems, Journal of Jiangxi Normal University, 3 (1991), 244--248. (in Chinese) --ff +u_~_xp+ au =o, Ell] Li Guo-zhen, Zhu Chuan-xi,ap lterative technique for quasiweakly continuous nonlinear operator equations, The Literature Contributed b.l" Experts at the Academic" Symposium au au y1 =0, ~[" the China's F(['th Session of Fixed PohTt, Pro.babilistie Metric Space and Variational (i.0 hwqualities (5, 1991), 8--14. aS (in Chinese) as [ 1]
a--T =o, p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products 293
