GENERAL RANDOM FIXED POINT THEOREM AND ITS APPLICATIONS

摘要/Abstract

摘要： In this paper,we obtain a general random fixed point theorem which generalizes the main results of Engle[4,7] and Bocsan[8]. The usefulness of this theorem seems to lie in the fact that unlike many random fixed point theorems obtained by using special methods[1,4,5-13] can be proved by using out general theorem (Theorem 1 and Corollaries 1 and 2). Finally,we indicate some possible applications of our results to nonlinear random integral and differential equations.

关键词: Euler equation, finite difference method, numerical simulation, Crank-Nicolson scheme

Abstract: In this paper,we obtain a general random fixed point theorem which generalizes the main results of Engle[4,7] and Bocsan[8]. The usefulness of this theorem seems to lie in the fact that unlike many random fixed point theorems obtained by using special methods[1,4,5-13] can be proved by using out general theorem (Theorem 1 and Corollaries 1 and 2). Finally,we indicate some possible applications of our results to nonlinear random integral and differential equations.

Key words: Euler equation, finite difference method, numerical simulation, Crank-Nicolson scheme

丁协平. GENERAL RANDOM FIXED POINT THEOREM AND ITS APPLICATIONS[J]. Applied Mathematics and Mechanics (English Edition), 1984, 5(5): 1667-1678.

Ding Xie-ping . GENERAL RANDOM FIXED POINT THEOREM AND ITS APPLICATIONS[J]. Applied Mathematics and Mechanics (English Edition), 1984, 5(5): 1667-1678.

参考文献

[1] Bharucha-Reid,A.T.,Random Integral Equations,Acad.Press,New York,(1972).
[2] Tsokos,C.P.and W.J.Padgett,Random Integral Equations,with Applications to Life Sciences and Engineering,Acad.Press,New York,(1974).
[3] Morse,W.L.The dishonest method in stream temperature modeling,Water Resour.Res.,14(1978)h45-51.
[4] Engl.H.W.,A general stochastic fixed point theorem for continuous random operators on stochastic domains,J.Math.Anal.Appl.,66(1978),220-231.
[5] Bharugha-Reid,A.T.,Fixed point theorems in probabilistic analysis,Bull.Amer.Math.Soc.,82(1976),641-657.
[6] Wang Zi-kun,A introduction of random functional analysis,Advances Math.,5(1962),45-71.(in Chinese).
[7] Engl,H.W.,Random fixed point theorems,in Nonlinear Equations in Abstract Spaces,Acad.Press,New York,(1978),67-80.
[8] Bocsan,Gh.,A general random fixed point theorem and applications to random equation,Rev.Roum.Math.Pures et Appl.,26(1981),381-384.
[9] Iton,S.,A random fixed point theorem for a multivalued contraction mapping,Pacific J.Math.,68(1977),85-90.
[10] Ding Xie-ping,Fixed point theorems for continuous random operators,Advances Math.,12(1983),294-298.(in Chinese)..
[11] Liu Zuo-shu and Chen Shao-zhong,On fixed-point theorems of random set-valued maps,Kexue Tongbao,27(1982),1161-1162.(in Chinese).
[12] Chang,S.S.and S.Z.Chen,Random fixed point theorems in probabilistic analysis and their applications to random approximation theory,Acta.Math.Appl.Sinica,5(1982),300-309.(in Chinese).
[13] Chang,S.S.,Random fixed point theorem in probabilistic analysis,Nonlinear Anal.,5(1981),113-122.
[14] Mimmeberg,C.J.,Measurable relations,Fund.Math.,87 (1975),53-72.
[15] Castaing,C.and M.Valadier,Convex Analysis and Measurable Multifunctions,Springer-Verlag,(1977),580.
[16] Himmeberg,C.J.and F.S.Van Vleck,Multifunctions on abstract measurable spaces and application to stochastic decision theory,Annali di Mat.Pura ed Appl.,101(1974),229-236.
[17] Ding Xie-ping,Common fixed point theorems for set-valued contractive type mappings,Sichuan Shiyuan Xuebao,A special issue in Math.,(1981),26-34.(in Chinese).
[18] Ding Xie-ping,Some fixed point theorems of nonexpansive type mappings,Advances Math.,12(1983),133-140.(in Chinese).
[19] Ding Xie-ping,Some fixed point theorems of set-valued non-self-maps,J.Math.Res.Exp.,(to be published).(in Chinese).
[20] Assad,N.A.and W.A.Kirk,Fixed point theorems for set-valued mappings of contractive type,Pacific J.Math.,43(1972),553-562.
[21] Khan,M.S.,Common fixed point theorems for multivalued mappings,Pacific J.Math.,95(1981),337-347.
[22] Ding-Xie-Ping,Fixed point theorems of some contractive type mappings and relations between them,Acta Math.Sinica,(to be published).(in Chinese).
[23] Ding Xie-ping,Fixed point theorems of orbitally contraction mappings,Chin.Ann.of Math.,2(1981),511-517.(in Chinese).
[24] Ding Xie-ping,Fixed point theorems of gneeralized contractive type mappings (Ⅱ),Chin.Ann.of Math.,4B(1983),153-163.
[25] Roseau,M.,Equations Differentielles,Masson,(1976).
[26] Ding Xie-ping,Fixed point theorems of random set-valued mappings and their applications,Appl.Math.and Mech.,Vol.5,No.4,(1984).
[27] Ding Xie-ping,On existence and uniqueness theorems of solutions for abstract nonlinear volterra integral equations,Trans.Math.,3(1982),56-61.(in Chinese)