Second, the excitations within the solute were projected out by using the projection operator, where represents the excitations within the solute's MO space.
Now, by using the projection operator technique, we establish the equivalence between problem (11) and fixed point problem (8).
By using the projection operator technique, one can easily establish the equivalence between variational inequalities and fixed point problems.
By using the projection operator technique, we have verified the equivalence between RNVI (11) and the fixed point problem (8) as well as NWHE (9).
One can establish the equivalence between problems (9) and (10) by using the projection operator technique; see Noor [17, 18, 22].
Verma [6], Chang et al. [7] and Huang and Noor [8] introduced and studied systems of nonlinear variational inequalities, and by using the projection operator technique, they proposed some projection iterative algorithms for solving these systems of variational inequalities.
By using the projection operator technique and the system of Wiener-Hopfequations technique, we suggest several new iterative algorithms to find the approximatesolutions to the problems and prove the convergence of the different types of iterativesequences.
For this end, we need to the following lemma in which by using the projection operator technique, we verify the equivalence between the problem (3.1) and the fixed point problem.
For this end, we need the following lemma in which by using the projection operator technique, we verify the equivalence between the system of general nonlinear regularized nonconvex variational inequalities (3.1) and a fixed point problem.
Each iteration of the above methods contain a prediction and a correction, the predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original LQP method for [12], while the new iterate is computed by using the projection operator for [14, 20, 21].
