A study of Variational Inequality Problems and Numerical Methods

Abstract

The theory of variational inequality problem and#136;V IP is one of the best tools to newlinesolve equilibrium and non equilibrium problems arise in the branches of Applied newlineMathematics, Applied Physics, Engineering, Medical Science, and Economics. The newlineconcept of variational inequality problem was introduced and developed by Stampacchia newlinein 1964. Since then, many researchers have defined various types of V IPs newlineand investigated the existence of solutions of the problems in various spaces under newlinecertain conditions. Complementarity Problems and fixed point problems are newlinethe particular cases of V IP. Gradient methods, extragradient methods, proximal newlinemethods, variable step iterative methods and projection methods are some important newlinenumerical methods to solve the V IPs. In the thesis, the concept of and#961;-projective newlineand and#963;-involutory operators are defined and the relation between them is studied. newlineThe concept of and#961;-projective V IP is introduced and its existence theorem is established. newlineThe concept of golden ratio variational inequality problem is introduced newlineand its existence theorems are studied under certain conditions. The existence of newlinegolden ratio variational inequality problem is studied in H-space. The concept of newlinegeometric H-space is introduced and geometric variational inequality problem is newlinedefined in it. The concept of geometric convexity of the join function is defined newlineand some basic results in geometric H-space are discussed. The existence of the newlinesolution of the geometric variational inequality problem is studied. The equivalence newlinetheorem between the geometric variational inequality problem and its dual problem newlineis also discussed. The concept of f-approximation problem and#136;AP is defined and newlineits existence theorem is studied with the help of homotopy theory. The concept of newlinegeneralized AP, Functional AP, Graph AP are also defined to study the existence newlineof V IP. The existence of the V IP is studied with the help of Cauchy sequence newlinevi newlineconvergence. The existence of f-variational inequality problem is studied with the newlinehelp of golden ratio algorithm and#136;GRA , conjugate golden ratio algorithm and generalized newlinegolden ratio algorithm. The concept of hemi-quasidomonotone and potential newlineoperator, hemi-pseudomonotone and potential operator, hemi-quasimonotone and newlinepotential operator, asymptotically hybrid and#136;and#951;; t, s -uniformly and strongly monotone newlineoperator, asymptotically hybrid and#136;and#951;; t, s -uniformly and strongly dissipative operators newlineare defined and studied their existence theorems under certain conditions. The existence newlinetheorem of non-smooth generalized vector F-variational inequality problem newlineis studied using pre and post affine mapping in the affine sets. newline