Interative approximation of fixed points for almost asymptotically nonexpansive type mappings in Banach spaces

IntroductionThroughoutthispaperweassumethatEisarealBanachspace ,E isthedualspaceofE ,DisanonemptysubsetofEandJ:E →2 E isthenormalizeddualitymappingdefinedbyJ(x) =f∈E :〈x ,f〉=‖x‖·‖f‖,‖f‖=‖x‖,　　 x∈E .　　Definition 1　LetT :D →Dbeamapping .1 )Tissaidtobeasymptoticallynonexpansive[1],ifthereexistsasequence kn [1 ,∞)withlimn→∞kn =1suchthat　　　　　‖Tnx-Tny‖≤kn‖x-y‖forall　　x ,y∈D ,n≥0 ;(1 )　　2 )Tissaidtobeofasymptoticallynonexpansivetype[2 ],if　　　　　lims…     

ON REICH''''S OPEN QUESTION

1　IntroductionandPreliminariesThroughoutthispaperweassumethatEisarealBanachspace ,E isthedualspaceofE ,DisanonemptysubsetofEandJ:E → 2 E isthenormalizeddualitymappingdefinedbyJ(x) =f∈E ,　〈x ,f〉 =‖x‖‖f‖ ,　‖x‖ =‖f‖ 　 (x∈E) . (1 )　　Definition 1　LetT :D →Dbeamapping .1 .Tissaidtobeasymptoticallynonexpansive[1],ifthereexistsasequence kn [1 ,∞ )withlimn→∞kn =1suchthat‖Tnx-Tny‖ ≤kn‖x-y‖　forall　x ,y∈D ,n≥ 1 ;　　 2 .Tissaidtobenonexpansive,ifthesequence kn app…     

Singular perturbation of initial value problem for a nonlinear second order systems

In this paper,the singular perturbation of initial value problem for nonlinearsecond order vector differential equationsε~rx″=f(t,x,x′,ε)x(0,ε)=a,x′(0,ε)=βis discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,aandβ∈R~n.Under suitable assumptions,by using the method of many-parameterexpansion and the technique of diagonalization,the existence of the solution of pertur-bation problem is proved and its uniformly valid asymptotic expansion of higher order isderived.     

Generalized derivatives of a region function and its applications

This paper is based on some fundamental concepts im [7], Clarke’s generalizedderivatives,as well as Lasotra’s and Strauss’s definitions of differential D(x) of amultivalued function f(x).Thereby,the generalized derivatives of a region function F(x) isdefined asD_F(x)=U∩{G(x)(?)B(R), (?)x∈B(R); G(x)=F’_x=F’(x)}The existence of the generalized derivatives of a region function F(x) is discussed:thenecessary and sufficient conditions of existence of the Fréchet generalized derivatives ofsuch a function is established.     

Applications of the signs of Melnikov's function

The existence and stability of peridic solutions for the two-dimensional system x′=f(x)+g(x, a), O<ε<1,a∈R whoseunperturbed system is Hamlitonan be decided by using the signs of Melnikov's function. Theresults can be applied to the comstuction of phase portralts in the bifiacation set of codimension two bifurcations of flows with double aero eignvahues.     

Singular perturbation of nonlinear vector boundary value problem

In this paper we study the perturbed boundary value problem of the form dx/dt=f(x,y,t;ε), εdy/dt=g(x,y,t;ε), a_1(ε)x(0,ε)+a_2(ε)y(0,ε)=a(ε) b_1(ε)x(1,ε)+εb_2(ε)y(1,ε)=β(ε)in whichx,f,β∈E~m, y,g,a∈E~n, 0<ε(?)1and a_1(ε), a_2(ε), b_2(ε)and b_2(ε) are matrices of the appropriate size. Under the condition that g_y(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.     

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

IntroductionLetX Rn,F:X→Rnbe given, the variational inequality, denoted byVI(X,F),is tofind a vectorx∈ Xsuch thatF(x)T(y-x)≥0, y∈ X. (1)DenoteN∶={1,2,…,n},whenX =[a,b]∶={x∈Rn|ai≤xi≤bi,i∈N},theVI(X,F)is called the box constrained variational ine     

Extended airy function and differential equations with n-turning points

This paper studies a second order linear ordinary differential equation with n-turningpoints(d~2y)/(dx~2) [λ~2q_1(x) q_2(x)]y=0Where q_1(x)=(x-μ_1)(x-μ_2)...(x-μ_n)f(x),f(x)≠0 ,and λis a largeparameter.The formal uniformly valid asymptotic solution of the equation is obtained based onthe analysis of the three points by means of the matched method.By the work a method isdeveloped and the applicability of this method to the n-turning points is demonstrated.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

A SIGNIFICANT IMPROVEMENT ON NEWTON’S ITERATIVE METHOD

For solving nonlinear and transcendental equation f(x)=0, a singnificant improvement on Newton’s method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov’s methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton’s method, and remove the monotoneity condition imposed on f(x):f′(x)≠0.     

THE WEAK DAMPED OSCILLATION IN A RANDOM MEDIUM

The paper gives a solution of the random differential equation with random coefficient,that is (?)(t) K~2(t)x(t)=-εf(x,t),where K(t)is a random process and ε is adamp coefficient,it is a little parameter.     

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

Extended graphical representation of rational fractions with applications to cybernetics

In this paper,we discuss the extended graphical representation ofthe fraction of a complex variable sWhere K is confined to be real.Three figures of the above fraction canbe used in feedback systems as well as to study the properties of figuresfor any one coefficient of a characteristic equation as a real parameter.It is easy to prove the following theorem:K_1=f(n)(s)/F(d)(s)and K_2=F(d)(s)/f(n)(s)have the same root locus.By this graphical theory,we find out that if the zeros and polesof a fraction are alternatively placed on the axis x,then there is nocomplex root locus of this fraction,therefore the state of such a sys-tem is always non-oscillatory.Using these figures of this,fraction,wecan discuss its stable interval systematically.     

Experimental study of vortex-induced vibrations of a cylinder near a rigid plane boundary in steady flow

In this study, the vortex-induced vibrations of a cylinder near a rigid plane boundary in a steady flow are studied experimentally. The phenomenon of vortex-induced vibrations of the cylinder near the rigid plane boundary is reproduced in the flume. The vortex shedding frequency and mode are also measured by the methods of hot film velocimeter and hydrogen bubbles. A parametric study is carded out to investigate the influences of reduced velocity, gap-to-diameter ratio, stability parameter and mass ratio on the amplitude and frequency responses of the cylinder. Experimental results indicate： （1） the Strouhal number （St） is around 0.2 for the stationary cylinder near a plane boundary in the sub-criti- cal flow regime; （2） with increasing gap-to-diameter ratio （eo/D）, the amplitude ratio （A/D） gets larger but frequency ratio （f/fn） has a slight variation for the case of larger values of eo/D（eo/D 〉 0.66 in this study）; （3） there is a clear difference of amplitude and frequency responses of the cylin- derbetween the larger gap-to-diameter ratios （e0/D 〉 0.66） and the smaller ones （e0/D 〈 0.3）; （4） the vibration of the cylinder is easier to occur and the range of vibration in terms of Vr number becomes more extensive with decrease of the stability parameter, but the frequency response is affected slightly by the stability parameter; （5） with decreasing mass ratio, the width of the lock-in ranges in terms of Vr and the frequency ratio （f/fn） become larger.     

THE TRANSFORMATION FUNCTION Φ AND THE CONDITION NEEDED FOR KUR SPACE HAVING THE FIXED POINT

In the last several years some progress has been made in the study of the properties ofthe extent of Banach space:In 1979 for example.when Suillivan discussed a relatedcharacterization of real L~P(X)space.he used uniform behavior of all two-dimensionalsubspace and defined this concept of a KUR space.In 1980 Huff used the concept of a NUCspace when he discussed the property of generalizing uniform convexity which was definedin terms of sequence.And in 1980 Yu Xin-tai stated,certainly and proved that the KURspace is equal to the NUC space.However,the following quite interesting questions raised respectively by Suillivan andHuff merit attention:Does every super-reflexive space have the fixed point property?The purpose of this paper is to study the characterization of transformationfunction and relationships between transformation function and the two questions above.     

THE RELATIONSHIP BETWEEN THE STABILITY AND THE OPTIMALITY OF LINEAR SYSTEMS——Another kind of the inverse problem of linear optimal control

Different from the inverse problem put forward by R. E. Kalman, another kind ofinverse problem of linear optimal control is proposed and dicussed in this paper.which is asfollows: If an asymptotically stable system x=Ax and a performance indexJ=integral from n=0 to ∞(x(?)Qx u(?)u)at are given, when can A be decomposedinto (?)A BK(?)so thatthe control lawu=K(?)x is optimal for the systeem x=Ax Bu and the index J? This papergives the solution for the problem and presents certain correspondence between theasymptotically stable system and the optimal system.     

A class of incompletely exponentially fitted difference schemes for a singular perturbation problem

Some sufficient conditions are considered, under which the solutions of a class of incompletely exponentially fitted difference schemes converge uniformly in e, with orders one and two, to the solution of the singular perturbation problem: eu"+a(x)u’-b(x)u=f(x), for 0a>0, b(x)≥0. From these conditions.an incompletely exponentially fitted second-order scheme is derived. Finally, the results of some numerical experiments are given.     

Effect of irregularity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer

The paper studies the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer over a semi-infinite heterogeneous half space with linearly varying rigidity and density due to irregularity at the interface. The irregularity is taken in the half space in the form of a rectangle. It is observed that torsional surface waves propagate in this assumed medium. In the absence of the irregularity, the velocity equation of the torsional surface wave is also obtained. For a layer over a homogeneous half space, the velocity of torsional surface waves coincides with that of the Love waves.     

On resolution to Wu＇s conjecture on Cauchy function＇s exterior singularities

This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral J[f(z) ] ≡(2πi) -1 C f(t)(t z) -1dt taken along the unit circle as contour C,inside which(the open domain D+) f(z) is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C(|t| = 1) ,as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.     

FIXED POINT THEOREM OF NONEXPANSIVE MAPPINGS IN CONVEX METRIC SPACES

Let X be a convex metric space with the property that every decreasing sequence ofnonempty closed subsets of X with diameters tending to(?)has nonempty intersection.This paper proved that if T is a mapping of a closed convex nonempty subset K of X intoitself satisfying the inequality:d(Tx,ty)≤ad(x.y) b{d(x,Tx) d(y,Ty} c{d(x,Ty) d(y,Tx)}for allλ.yin K,where0≤a<1,b≥0,c≥0,a c≠0 and a 2b 3c≤1,then T has auniquc fixed point in K.