Existence and algorithm of solutions for general multivalued mixed implicit quasi-variational inequalities

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case , introduced and studied by Ding Xie-ping . The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex , lower semicontinuous , binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality . Secondly , this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities . Here , the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the e     

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle technique. The convergence of three-step iterative sequences generated by the algorithm is also proved.     

NEW COLLECTIVELY FIXED POINT THEOREMS AND APPLICATIONS IN G-CONVEX SPACES

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ 1 991 ,Ｔａｒａｆｄａｒ[1]ｆｉｒｓｔｅｓｔａｂｌｉｓｈｅｄｔｈｅｆｏｌｌｏｗｉｎｇｃｏｌｌｅｃｔｉｖｅｌｙｆｉｘｅｄｐｏｉｎｔｔｈｅｏｒｅｍｏｎｔｈｅｐｒｏｄｕｃｔｓｐａｃｅｏｆｎｏｎｅｍｐｔｙｃｏｍｐａｃｔｃｏｎｖｅｘｓｕｂｓｅｔｓｏｆｔｏｐｏｌｏｇｉｃａｌｖｅｃｔｏｒｓｐａｃｅｓａｎｄｇａｖｅｉｔｓａｐｐｌｉｃａｔｉｏｎｓｔｏｔｈｅｅｘｉｓｔｅｎｃｅｏｆｅｑｕｉｌｉｂｒｉｕｍｐｏｉｎｔｓｆｏｒａｂｓｔｒａｃｔｅｃｏｎｏｍｉｅｓ.Ｔｈｅｏｒｅｍ 1　Ｌｅｔ Ｘｉ ｉ∈Ｉ…     

Iterative algorithms for finding approximate solutions of completely generalized strongly nonlinear quasivariational inequalities

ＩＴＥＲＡＴＩＶＥＡＬＧＯＲＩＴＨＭＳＦＯＲＦＩＮＤＩＮＧＡＰＰＲＯＸＩＭＡＴＥＳＯＬＵＴＩＯＮＳＯＦＣＯＭＰＬＥＴＥＬＹＧＥＮＥＲＡＬＩＺＥＤＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＱＵＡＳＩＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＩＥＳＺｅｎｇＬｕ－ｃ...     

Generalized variational inequalities and generalized quasi-variational inequalities

Under much weaker hypotheses and in a more general setting some existence theorems of solutions to generalized variational inequalities,generalized quasi-varia-tional inequalities and minimax inequalities are established.The results presented in this paper generalize the corresponding results of[3—13]to the noncompact case,thus improving these results.     

Boundary and interior layer behavior for singularly perturbed vector problem

In this paper,we consider the vector nonlinear boundary value problem:εy~v=f(x,y,z,y′,ε),y(0)=A_1,y(1)=B_1εz~v=g(x,y,z,z′,ε),z(0)=A_2,z(1)=B_2whereε>0 is a small parameter,0≤x≤1 ,f and g are continuous functions in R~4,Under appropriate assumptions,by means of the differential inequalities,we demonstratethe existence and estimation,involving boundary and interior layers,of the solutions to theabove problem.     

关于变分不等式问题近似解的数值证明

基于文献[1]给出了一种数值证明变分不等式解的存在性方法。通过Hilbert空间中的Riesz表示定理,首先将变分不等式问题的迭代过程转化为一种不动点形式,再利用Schauder不动点定理构造了一个高效率的数值证明过程,即通过数值计算产生一个包含近似解的有界闭凸子集。非线性Helmholtz方程的算例说明这一方法的可行性和高效性。     

New collectively fixed point theorems and applications inG-convex spaces

By applying the technique of continuous partition of unity and Tychonoff’s fixed point theorem, some new collectively fixed point theorems for a family of set-valued mappings defined on the product space of noncompact G-convex spaces are proved. As applications, some nonempty intersetion theorems of Ky Fan type for a family of subsets of the product space of G-convex spaces are proved; An existence theorem of solutions for a system of nonlinear inequalities is given in G-convex spaces and some equilibrium existence theorems of abstract economies are also obtained in G-convex spaces. Our theorems improve, unify and generalized many important known results in recent literature.     

Some generalizations of generalized bi-quasivariational inequalities

The purpose of this paper is to introduce and study the existenee problems of solutions for a class of new bi-quasi-variational inequalities. The resuhs presented in this paper unify, sharpen and extend many recent results.     

On the existence and uniqueness of solutions for a class of variational inequalities with applications to the signorini problem in mechanics

In this paper,we introduce a new unified and general class of variational inequalities,and show some existence and uniqueness results of solutions for this kind of variationalinequalities.As an application,we utilize the results presented in this paper to study theSignorini problem in mechanics.     

Section theorems,coincidence theorems and intersection theorems on H-spaces with applications

The purpose of this paper is to study the section theorems, coincidence theorems and intersection theorems on H-spaces. As a way of application, we use these results to study the existence problems of solutions for minimax inequalities and variational inequalities. The results presented in this paper improve and extend the corresponding results in [1, 3, 5, 6, 8, 9, 12, 14,15, 17]     

第二类抛物型变分不等式中的MRM方法

本文讨论了第二类抛物型变分不等式中的MRM(多重互易方法)方法。首先采用时间项半离散和隐格式方法将抛物型变分不等式化解为一个椭圆变分不等式，然后利用MRM-边界积分方程，将其化解为MRM-边界混合变分不等式，并给出了MRM-边界混合变分不等式解的存在唯一性。说明了该MRM-边界混合变分不等式与常规边界积分方程得到的边界混合变分不等式是一致的，并且具有更容易编程实现。这为使用MRM边界元方法数值求解抛物型变分不等式提供了方法和理论依据。文末给出了数值算例。     

On the L_p intersection body

In this paper,by using the Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory,associated with L_p intersection body and dual mixed volume,some dual Brunn-Minkowski inequalities and their isolate forms are established for L_p intersection body about the normalized L_p radial addition and L_p radial linear combination.Some properties of operator Lp are given.     

On Hardy-type integral inequalities

The Hardy integral inequality is one of the most important inequalities in analysis. The present paper establishes some new Copson-Pachpatte(C-P) type inequalities, which are the generalizations of the Hardy integral inequalities on binary functions.     

Existence and iterative algorithms of solutions for generalized mixed quasi-variational inequalities

１　ＰｒｏｂｌｅｍｓＬｅｔＨｂｅａｒｅａｌＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｎｏｒｍａｎｄｉｎｎｅｒｐｒｏｄｕｃｔｓｄｅｎｏｔｅｄｂｙ‖·‖ａｎｄ〈·,·〉ｒｅｓｐｅｃｔｉｖｅｌｙ．ＬｅｔＫｂｅａｃｌｏｓｅｄｃｏｎｖｅｘｓｕｂｓｅｔｉｎＨａｎｄｆ：Ｈ→Ｈｂｅａｓｉｎｇｌｅｖａｌｕｅｄｍａｐｐｉｎｇ．Ｔｈｅｃｌａｓｓｉｃａｌｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｙｐｒｏｂｌｅｍ（ＶＩＰ（ｆ,Ｋ））ｉｓｔｏｆｉｎｄ＾ｘ∈Ｋ（＾ｘ）,ｓｕｃｈｔｈａｔ〈ｆ（＾ｘ）,ｙ－＾ｘ〉≥０　　（ｙ∈Ｋ）．（１）…     

Singular integral inequalities with several nonlinearities and integral equations with singular kernels

We deal with an integral inequality with a power nonlinearity on its left-hand side, n nonlinearities on its right-hand side, and weakly singular kernels. The obtained result is an extension of the Bihari, Henry, Pachpatte, and Pinto inequalities and results obtained by the author. Using these results, we prove sufficient conditions for the existence of global solutions of some nonlinear Volterra integral equations with singular kernels and n nonlinearities. Published in Neliniini Kolyvannya, Vol. 11, No. 1, pp. 71–80, January–March, 2007.     

Optimal control of parabolic variational inequalities with state constraint

1　IntroductionandProblemWeshallstudytheoptimalcontrolproblemsgovernedbynonlinearparabolicvariationalinequalitiesoftheformy′+Ay +β(y) ∈Bu+f(a.e .(x,t)∈Q =Ω× [0 ,t]) ,y(0 ) =y0 , ( )withthestateconstraintF(y) S ,andthecostfunctionalI(y,u) .Whereβisadiscontinuous,nonlinearandnonmonotonemulti_valuedmapping .Theoptimalcontrolproblemsofthedifferentialsystemshavebeenstudiedforalongtime.Manyscholars,suchasJ.L .Lions ,V .Barbu ,D .Tiba,andF .Mignotetal.,haveresearchedtheoptimalcontrolpr…     

The perturbation finite element method for solving problems with nonlinear materials

The perturbation method is one of the effective methods for so-lving problems in nonlinear continuum mechanics.It has been de-veloped on the basis of the linear analytical solutions for the o-riginal problems.If a simple analytical solution cannot be ob-tained.we would encounter difficulties in applying this method tosolving certain complicated nonlinear problems.The finite ele-ment method appears to be in its turn a very useful means for sol-ving nonlinear problems,but generally it takes too much time incomputation.In the present paper a mixed approach,namely,theperturbation finite element method,is introduced,which incorpo-rates the advantages of the two above-mentioned methods and enablesus to solve more complicated nonlinear problems with great savingin computing time.Problems in the elastoplastic region have been discussed anda numerical solution for a plate with a central hole under tensionis given in this paper.     

Optimal control of hyperbolic H-hemivariational inequalities with state constraints

The optimal control problems of hyperbolic H-hemivariational inequalities with the state constraints and nonnornotone multivalued mapping term are considered. The optimal solutions are obtained. In addition, their approximating problems are also studied.     

On the boundedness of motions of mechanical systems described by fuzzy ordinary differential equations

A new approach is proposed for analyzing the dynamics of fuzzy systems. Within the framework of the approach, boundedness conditions for the solutions of fuzzy systems are established on the basis of the classical inequalities of convex geometry __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 93–99, December 2005.