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Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities  [cached]
Liu Zeqing,Chen Lin,Ume JeongSheok,Kang ShinMin
Journal of Inequalities and Applications , 2009,
Abstract: We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.
Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities
Zeqing Liu,Lin Chen,Jeong Sheok Ume,Shin Min Kang
Journal of Inequalities and Applications , 2009, DOI: 10.1155/2009/758786
Abstract: We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.
A study of variational inequalities for set-valued mappings  [cached]
Tan Kok-Keong,Tarafdar Enayet,Yuan George Xian-Zhi
Journal of Inequalities and Applications , 1999,
Abstract: In this paper, Ky Fan's KKM mapping principle is used to establish the existence of solutions for simultaneous variational inequalities. By applying our earlier results together with Fan–Glicksberg fixed point theorem, we prove some existence results for implicit variational inequalities and implicit quasi-variational inequalities for set-valued mappings which are either monotone or upper semi-continuous.
On solvability of general nonlinear variational-like inequalities in reflexive Banach spaces
Zeqing Liu,Juhe Sun,Soo Hak Shim,Shin Min Kang
International Journal of Mathematics and Mathematical Sciences , 2005, DOI: 10.1155/ijmms.2005.1415
Abstract: We introduce and study a new class of general nonlinear variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality, we establish two existence and uniqueness theorems of solutions for the general nonlinear variational-like inequality.
Variational inequalities  [PDF]
Nikolaos E. Sofronidis
Mathematics , 2015,
Abstract: If $- \infty < \alpha < \beta < \infty $ and $f \in C^{3} \left( [ \alpha , \beta ] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${\phi } \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ for which ${\phi }^{(k)}( \alpha ) = {\phi }^{(k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {\phi }^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {\phi }^{3} \right) dx$ $\geq \int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {\phi }^{2} \phi ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } \phi \phi " + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } \phi ' {\phi }^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } \phi {\phi '}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$.
Applications of W. A. Kirk's fixed-point theorem to generalized nonlinear variational-like inequalities in reflexive Banach spaces  [cached]
Liu Zeqing,Ume Jeong Sheok,Kang Shin Min
Journal of Inequalities and Applications , 2005,
Abstract: We introduce and study a new class of generalized nonlinear variational-like inequalities, which includes these variational inequalities and variational-like inequalities due to Bose, Cubiotti, Dien, Ding, Ding and Tarafdar, Noor, Parida, Sahoo, and Kumar, and Yao, and others as special cases. By applying Kirk's fixed-point theorem and Ding-Tan minimax inequality, we establish the existence theorems of solutions for the generalized nonlinear variational-like inequalities in reflexive Banach spaces.
The Solvability of a Class of General Nonlinear Implicit Variational Inequalities Based on Perturbed Three-Step Iterative Processes with Errors  [cached]
Liu Zeqing,Kang ShinMin,Ume JeongSheok
Fixed Point Theory and Applications , 2008,
Abstract: We introduce and study a new class of general nonlinear implicit variational inequalities, which includes several classes of variational inequalities and variational inclusions as special cases. By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative algorithm with errors for solving the class of variational inequalities. Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities, and convergence and stability results of the sequence generated by the algorithm are obtained. The results presented in this paper extend, improve, and unify a host of results in recent literatures.
The Solvability of a Class of General Nonlinear Implicit Variational Inequalities Based on Perturbed Three-Step Iterative Processes with Errors  [cached]
Zeqing Liu,Shin Min Kang,Jeong Sheok Ume
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/634921
Abstract: We introduce and study a new class of general nonlinear implicit variational inequalities, which includes several classes of variational inequalities and variational inclusions as special cases. By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative algorithm with errors for solving the class of variational inequalities. Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities, and convergence and stability results of the sequence generated by the algorithm are obtained. The results presented in this paper extend, improve, and unify a host of results in recent literatures.
On nonlinear variational inequalities  [cached]
Muhammed Aslam Noor
International Journal of Mathematics and Mathematical Sciences , 1991, DOI: 10.1155/s0161171291000479
Abstract: The fixed point technique is used to prove the existence of a solution for a class of nonlinear variational inequalities related with odd order constrained boundary value problems and to suggest an iterative algorithm to compute the approximate solution.
An approximation of solutions of variational inequalities
Jinlu Li,B. E. Rhoades
Fixed Point Theory and Applications , 2005, DOI: 10.1155/fpta.2005.377
Abstract: We use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.
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