(Φ,△)-TYPE PROBABILISTIC CONTRACTOR AND SOLUTIONS FOR A NONLINEAR OPERATOR EQUATIONS IN MENGER PN-SPACES

The purpose of this paper is to introduce the coneept of (Φ,△)-type probabilistic contractor in Menger PN-spaces and to study the existence and uniqueness of solutions for the nonlinear operator equations with such probabilistic contractor in Menger PN-spaces.The results presented in this paper improve and extend the corresponding results in [1] and [4-8].     

On the isometric isomorphism of probabilistic metric spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｒｅａｒｅｔｗｏｋｉｎｄｓｏｆｉｓｏｍｅｔｒｉｃｉｓｏｍｏｒｐｈｉｓｍｉｎｐｒｏｂａｂｉｌｉｓｔｉｃｍｅｔｒｉｃｓｐａｃｅｔｈｅｏｒｙ (ｓｅｅ[1 -9] ) .ＴｈｅｆｉｒｓｔｉｓｔｈａｔａＰＭｓｐａｃｅ (Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｎｏｔｈｅｒＰＭｓｐａｃｅ(Ｅ′,Ｆ′) ,ａｎｄｔｈｅｓｅｃｏｎｄｉｓｔｈａｔａＰＭｓｐａｃｅｓ(Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｇｅｎｅｒａｔｉｎｇｓｐａｃｅｏｆｑｕａｓｉ＿ｍｅｔ…     

Probabilistic norm of operators and resonance theorems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｓｔ,ａｓｔｈｅｄｅｆｉｎｉｔｉｏｎｓａｂｏｕｔｓｏｍｅｃｏｎｃｅｐｔｓｉｎｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｌｉｎｅａｒｓｐａｃｅ（ｂｒｉｅｆｌｙ,ＰＮｓｐａｃｅ）ｃｏｐｉｅｄｔｈｅｃｏｒｅｓｐｏｎｄｉｎｇ...     

A note on probabilistic contractor couple and solutions for a system of nonlinear equations in n a menger pn-spaces

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＤｅｆｉｎｉｔｉｏｎｓＩｎ [1 ] ,ＣｈａｎｇＳｈｉｓｈｅｎｇａｎｄＰｅｎｇＹｏｎｇｃｈｅｎｇｉｎｔｒｏｄｕｃｅｄｔｈｅｃｏｎｃｅｐｔｏｆｐｒｏｂａｂｉｌｉｓｔｉｃｃｏｎｔｒａｃｔｏｒｃｏｕｐｌｅｉｎＮｏｎ＿ＡｒｃｈｉｍｅｄｅａｎＰＮ＿ｓｐａｃｅｓ (ｆｏｒｓｈｏｒｔ,Ｎ .Ａ .ＭｅｎｇｅｒＰＮ＿ｓｐａｃｅｓ) .Ｕｓｉｎｇｔｈｉｓｃｏｎｃｅｐｔ,ｕｎｄｅｒｔｈｅｃｏｎｄｉｔｉｏｎｔ＿ｎｏｒｍΔ =ｍｉｎ ,ｔｈｅｙｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏ…     

NEW VERSION OF KKMTHEOREM IN PROBABILISTIC METRIC SPACES WITH APPLICATIONS

ＮＥＷＶＥＲＳＩＯＮＯＦＫＫＭＴＨＥＯＲＥＭＩＮＰＲＯＢＡＢＩＬＩＳＴＩＣＭＥＴＲＩＣＳＰＡＣＥＳＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＺｈａｎｇＳｈｉｓｈｅｎｇ（张石生）ＹｅｏｌＪｅＣｈｏ（赵烈济）ＷｕＸｉａｎｇ（吴鲜）（ＲｅｃｅｉｖｅｄＭａｙ７，１９９...     

Probabilistic norm of linear operators on PN space

I.IntroductionInl4],adefinitionofprobabilisticnormoflinearoperatorsonMengerprobabiIisticnormedlinearspace(brieflyM-PNspace)wasgiven.Itissuperiortoeachdefinitionin[l],I5J,I6],I7j,becausetheboundednessoflinear0peratorscanbedescribedwellbymeansofit.Thispaper…     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

SET-VALUED EXTENSION OF OPERATORS VIA STEINER SELECTIONS (Ⅰ)-THEORETICAL RESULTS

A way to extend operators in spaces of continuous functions to spaces of continuous set-valued functions is proposed. This extension is developed through the Steiner selections of the set-valued functions. Their properties and characteristics of the convergence of sequences of operators of this class are studied. In Part Ⅱ of this series some applications to approximation theory will be shown.     

Mikusinski’s operators solution of three-order linear difference equation with variable coefficients

This paper proceeds the papers [3] [4], we make use of the idea of the variable number operators and some concepts and conclusions of the shifting operators series with variable coefficients in the operational field of Mikusinski, it is devoted to the solution of the general three-order linear difference equation with variable coefficients, and it is also devoted to the better solution formula for the some special three-order linear difference equations with variable coefficients: in addition, we try to provide the idea and method for realizing solution of the more than three-order linear difference equation with variable coefficients. Project Supported by the Science Foundation of Anhui Province     

SET-VALUED EXTENSION OF OPERATORS VIA STEINER SELECTIONS(Ⅱ)-APPLICATIONS TO APPROXIMATION

On the basis of Part (Ⅰ) of this series some applications to the approximation of set-valued functions are obtained: Korovkin type theorems, a method to extend classical approximation operators to the set-valued setting and a Jackson type estimate.     

PROBABILISTIC MODELS FOR LONG FATIGUE CRACK GROWTH RATES OF LZ50 AXLE STEEL

IntroductionStudy on the model of fatigue crack growth rate (da/dN) and measurement of theprobabilistic curves of the model are a basis of the fatigue damage tolerate design and theprobabilistic safety assessment. Some benefit efforts have been made in th…     

Topological degree theory and fixed point theorems in probabilistic metric spaces

The Leray-Schauder topological degree theory is established in the probabilistic linear normed spaces. Based on this theory, some fixed point theorems for mappings in the probabilistic linear normed spaces are shown.The projects supported by National Natural Science Foundation of China.     

Some new fixed point theorems in probabilistic metric spaces

ＳＯＭＥＮＥＷＦＩＸＥＤＰＯＩＮＴＴＨＥＯＲＥＭＳＩＮＰＲＯＢＡＢＩＬＩＳＴＩＣＭＥＴＲＩＣＳＰＡＣＥＳｚｈｕＣｈｕａｎ－ｘｉ（朱传喜）（ＭａｔｈｅｍａｔｉｅｓＤｉｖｉｓｉｏｎ,Ｎａｎｃｈａｎｇ．Ｕｎｉｖｅｒｓｉｔｙ,Ｎａｎｃｈａｎｇ）（Ｒｅｃｅｉｖ...     

Some problems in the Z-C-X space

The new concepts of the Z-C-X space and excellent cone are introduced. Some problems of random semiclosed 1-set-contractive operator are investigated in the Z-C-X space. At first, an important inequality is proved. Secondly, several new conclusions are proved by means of random fixed point index in the theory of random topological degree. A random solution of a class of random operator equations under conditions of imitating the parallelogram law is obtained, famous Altman’s theorem is generalized in partially ordered Z-C-X space. Therefore, some new results are obtained.     

Some bounded results of θ(t)-type singular integral operators

Let B be a Banach space in UMD with an unconditional basis. The boundedness of the θ(t)-type singular integral operators in L _B ^p (R ⁿ), (1≤p<+∞) and H _B ¹ (R ⁿ) spaces are discussed. Foundation item: the Education Commission of Shandong Province (J98P51) Biography: Zhao Kai (1960-)     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

Removable Singularities for Fully Nonlinear Elliptic Equations

We obtain a removability result for the fully nonlinear uniformly elliptic equations F(D ² u)+f(u)=0. The main theorem states that every solution to the equation in a punctured ball (without any restrictions on the behaviour near the centre of the ball) is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on F. Previously such results were known for linear and quasilinear operators F. In comparison with the semi- or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions. Accepted May 3, 2000?Published online November 16, 2000     

Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

IntroductionLetN ∈Rm0 ×n(m0 ≥n)beaverticalblockmatrixoftype(m1,… ,mn) ,q∈Rm0 beaconstantvectorpartitionedconformablywithN ,thatareN =N1Nn,　q =q1qn,　m0 = ni=1mi,whereNi ∈Rmi×n,qi ∈Rmi,i ∈I =1 ,… ,n .WeconsidertheverticallinearomplementarityproblemVLCP(N ,q)associatedwith (N ,q)offindingavectorx∈Rnsuchthatx≥ 0 ,　si(x) :=Nix+qi ≥ 0 ,　xi∏mij=1sij(x) =0　　 (i∈I) ,wherexi,sij(x)denotetheithcomponentofxandthejthcomponentofsi(x) ,respectively ;∏ mj=1ajdenotesa1…am.Iti…     

Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

Eigensystem analysis techniques are applied to finite difference formulations of the Navier-Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix. On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. This procedure has improved the rates of convergence of CFD codes (developed at NASA Ames Research Center) by 20%–50%. The sensitivity of the computed solution with respect to the shift is also studied. Finally, an adaptive shifting of the spectrum together with Wynn's acceleration algorithm are discussed. It turns out that the shifting process is a preconditioner for Wynn's method.