Algorithm of solutions for mixed-nonlinear variational-like inequalities in reflexive banach space

I.IntroductionLetDbeanonemptyconvexsubsetofaBanachspaceB,BbethetopologicaldualspaceofBand(u,v)bethepairingbetweenneBandr6B.LetT,A:D~B'and9:DxD-Bbethreemappingsandf:D~(-ac, ac1beareal-valuedfunction.Inthispaper,weshallinvestigatethemixednonlinearvariationalinequalityproblemMNna(T,A,9,f,D):FinduEDsuchthatSpecialCases.(l)f-0,thentheproblem(l.l)isequivalenttofindingueDsuchthatTheproblem(l.2)isthestronglynonlinearvariational-likeinequalityproblemstudiedbyNoorllJwhicharisesnaturallyincon…     

Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces

A new system of generalized mixed equilibrium problems (SGMEPs) involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalized mixed equilibrium problems (SAGMEPs) for solving the SGMEPs is first introduced. Existence and uniqueness of the solutions to the SAGMEPs is proved under quite mild assumptions without any coercive conditions in reflexive Banach spaces. Using the auxiliary principle technique, a new iterative algorithm for solving the SGMEPs is proposed and analyzed. Strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild assumptions without any coercive conditions. These results improve, unify, and generalize some recent results in this field.     

Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces

A new system of generalized mixed equilibrium problems (SGMEPs) involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalized mixed equilibrium problems (SAGMEPs) for solving the SGMEPs is first introduced. Existence and uniqueness of the solutions to the SAGMEPs is proved under quite mild assumptions without any coercive conditions in reflexive Banach spaces. Using the auxiliary principle technique, a new iterative algorithm for solving the SGMEPs is proposed and analyzed. Strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild assumptions without any coercive conditions. These results improve, unify, and generalize some recent results in this field.     

Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces

A new bilevel generalized mixed equilibrium problem(BGMEP) involving generalized mixed variational-like inequality problems(GMVLIPs) is introduced and studied in the reflexive Banach spaces.First,an auxiliary generalized mixed equilibrium problem(AGMEP) is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs.By using a minimax inequality,the existence and the uniqueness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions.By using an au...     

Iterative algorithm for solutions to new system of generalized mixed implicit equilibrium

A new system of generalized mixed implicit equilibrium problems(SGMIEP) involving nonmonotone set-valued mappings is introduced and studied in real reflexive Banach spaces.First,an auxiliary mixed equilibrium problem(AMEP) is introduced.The existence and the uniqueness of the solutions to the AMEP are proved under quite mild assumptions without any coercive conditions.Next,by using the solution mapping of the AMEP,a system of generalized equation problems(SGEP) is considered,and its equivalence with the SGMIEP is shown.By using the SGEP,a new iterative algorithm for solving the SGMIEP is proposed and analyzed.The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions.These results are new,which unify and generalize some recent results in this field.     

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     

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）…     

Iterative algorithm of solutions for multivalued general mixed implicit equilibrium-like problems

The multivalued general mixed implicit equilibrium-like problems are introduced and studied. To solve these problems, a new predictor-corrector iterative algorithm is proposed and analyzed using the auxiliary principle technique. The convergence of the suggested algorithm is also proved in weaker conditions.     

System of set-valued mixed quasi-variational-like inclusions involving H-η-monotone operators in Banach spaces

A new system of the set-valued mixed quasi-variational-like inclusions (SS-MQVLI) involving H-η-monotone operators is studied in general Banach spaces without uniform smoothness. By using the resolvent operator technique of H-η-monotone opera-tors, a new iterative algorithm for finding approximate solutions to SSMQVLI is proposed. It is shown that the iterative sequences generated by the algorithm converge strongly to the exact solution of SSMQVLI under appropriate assumptions. These obtained new re-sults have extended and improved previous results.     

g-η-Monotone mapping and resolvent operator technique for solving generalized implicit variational-like inclusions

A new class of g-η-monotone mappings and a class of generalized implicit variational-like inclusions involving g-η-monotone mappings are introduced. The resolvent operator of g-η-monotone mappings is defined and its Lipschitz continuity is presented.An iterative algorithm for approximating the solutions of generalized implicit variationallike inclusions is suggested and analyzed. The convergence of iterative sequences generated by the algorithm is also proved.     

Generalized Strongly Nonlinear Quasi-Complementarity Problems

ＧＥＮＥＲＡＬＩＺＥＤＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＱＵＡＳＩ－ＣＯＭＰＬＥＭＥＮＴＡＲＩＴＹＰＲＯＢＬＥＭＳＬｉＨｏｎｇ－ｍｅｉ（李红梅）ＤｉｎｇＸｉｅ－ｐｉｎｇ（丁协平）（ＳｉｃｈｕａｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ），Ｃｈｅｎｇｄｕ（Ｒｅ...     

PREDICTOR-CORRECTOR ALGORITHMS FOR SOLVING GENERALIZED MIXED IMPLICIT QUASI-EQUILIBRIUM PROBLEMS

A new class of generalized mixed implicit quasi-equilibrium problems （GMIQEP） with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems and generalized mixed implicit quasi-variational inequality problems as many special cases. By employing the auxiliary principle technique, some predictor-corrector iterative algorithms for solving the GMIQEP are suggested and analyzed. The convergence of the suggested algorithm only requires the continuity and the partially relaxed implicit strong monotonicity of the mappings     

g-η-Monotone mapping and resolvent operator technique for solving generalized implicit variational-like inclusions

A new class of g-η-monotone mappings and a class of generalized implicit variational-like inclusions involving g-η-monotone mappings are introduced. The resolvent operator of g-η-monotone mappings is defined and its Lipschitz continuity is presented. An iterative algorithm for approximating the solutions of generalized implicit variational-like inclusions is suggested and analyzed. The convergence of iterative sequences generated by the algorithm is also proved.     

On a class of generalized nonlinear implicit quasivariational inclusions

ＩｎｔｒｏｄｕｃｔｉｏｎＶａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｙｔｈｅｏｒｙａｎｄｃｏｍｐｌｅｍｅｎｔａｒｉｔｙｐｒｏｂｌｅｍｔｈｅｏｒｙｈａｖｅｂｅｃｏｍｅｖｅｒｙｅｆｆｅｃｔｉｖｅａｎｄｐｏｗｅｒｆｕｌｔｏｏｌｓｆｏｒｓｔｕｄｙｉｎｇａｗｉｄｅｒａｎｇｅｏｆｐｒｏｂｌｅｍｓａｒｉｓｉｎｇｉｎｍｅｃｈａｎｉｃｓ,ｍａｔｈｅｍａｔｉｃａｌｐｒｏｇｒａｍｍｉｎｇ,ｏｐｔｉｍｉｚａｔｉｏｎａｎｄｃｏｎｔｒｏｌ,ｅｑｕｉｌｉｂｒｉｕｍｔｈｅｏｒｙｏｆｅｃｏｎｏｍｉｃｓ,ｍａｎａｇｅｍｅｎｔｓｃｉｅｎ…     

System of set-valued mixed quasi-variational-like inclusionsinvolving H-η-monotone operators in Banach spaces

A new system of the set-valued mixed quasi-variational-like inclusions （SSMQVLI） involving H-η-monotone operators is studied in general Banach spaces without uniform smoothness. By using the resolvent operator technique of H-η-monotone operators, a new iterative algorithm for finding approximate solutions to SSMQVLI is proposed. It is shown that the iterative sequences generated by the algorithm converge strongly to the exact solution of SSMQVLI under appropriate assumptions. These obtained new results have extended and improved previous results.     

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.     

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

A projected subgradient method for solving a class of set-valued mixed variational inequalities (SMVIs) is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.     

Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions

I.IntroductionLetHbeaHibertspaceendowedwithanorm11'11andaninnerproduct',.)andC(H)bethefamilyofallnonemptycompactsubsetsofH.LetT,A:H~C(H)besetvaluedmappings,g:H~Hbeasingle-valuedmappingand9:HxH~RU{ ac}besuchthatforeachfixedyeH,9.,y):H~RUt co}isaproperconvexlowersemicontinuousfunctiononHandg(H)ndomag'Y)#oforeachyeH,ThentheproblemoffindingxeH,ueT(x)andveA(x)suchthatg(x)edomag',x)and(u~v,y~g(x))39(g(x)'x)~9(y'x)(VYeH)(l.l)iscalledthegeneralizedstronglynonlinearquasivariationalinclusio…     

弹性力学Hamilton方法广义解的适定性

首先引入了Hamilton体系中平面应力弹性力学问题正则方程的Galerkin变分方程,证明了Galerkin变分方程和目前文献中所用的Ritz变分方程的等价性,以及相应广义解的适定性,从而为目前的数值方法提供了理论基础。从证明过程中可以看到广义解实际上是Ritz变分泛函的一个鞍点。     

Existence and algorithm of solutions for a system of generalized mixed implicit equilibrium problems in Banach spaces

A new system of generalized mixed implicit equilibrium problems is introduced and studied in Banach spaces. First, the notion of the Yosida proximal mapping for generalized mixed implicit equilibrium problems is introduced. By using the notion, a system of generalized equation problems is considered, and its equivalence with the system of generalized mixed implicit equilibrium problems is also proved. Next, by applying the system of generalized equation problems, we suggest and analyze an iterative algorithm to compute the approximate solutions of the system of generalized mixed implicit equilibrium problems. The strong convergence of the iterative sequences generated by the algorithm is proved under quite mild conditions. The results are new and unify and generalize some recent results in this field.