Common solution of variational inequalities, fixed point, and equilibrium problems using the iterative method

Abstract

In this thesis, two novel iterative algorithms are proposed and analyzed to obtain a common solution to generalized nonlinear variational inequality, equilibrium, and fixed-point problems for nonexpansive mappings in real Hilbert spaces. The first algorithm is developed to approximate a common solution of these three classes of problems. It is proved that the sequence generated by the algorithm converges strongly under mild and standard assumptions in Hilbert spaces. The second algorithm is developed within the framework of the D-plus iterative algorithm. Unlike extragradient-type methods, the proposed algorithm employs a projection operator for the generalized nonlinear variational inequality (GNVI) problem, a resolvent operator for the equilibrium problem (EP), and a nonexpansive mapping derived from the fixed-point structure. It is shown that the sequence generated by the algorithm converges strongly under standard assumptions. Furthermore, the stability and perturbation properties of the algorithm are investigated, and explicit error estimates are derived. In addition, numerical examples and various applications are presented to evaluate the performance of the proposed algorithms and to demonstrate their advantages over existing iterative algorithms.