تعادل عمومی تابع دو طرفه نابرابری تغییرات

In this paper, we introduce and consider a new class of equilibrium variational inequalities, called the mixed general equilibrium bifunction variational inequalities. We suggest and analyze some proximal methods for solving mixed general equilibrium bifunction variational inequalities using the auxiliary principle technique. Convergence of these methods is considered under some mild suitable conditions. Several cases are also discussed. Results in this paper include some new and known results as special cases.

Variational inequalities, which were introduced and first studied in the sixties, have been seen to be an important and interesting branch of mathematical sciences with applications in industry, regional sciences, and pure and applied sciences. Variational inequalities can be viewed as natural extensions of the variational principles. It is a well-known fact that the optimality conditions for the minimum of a differentiable convex function on a convex set can be characterized by means of the variational inequalities. Noor [1] has shown that the optimality conditions for differentiable nonconvex functions on the nonconvex set can by characterized by means of a class of variational inequalities, called the general variational inequalities. It has been shown that a wide class of odd-order and nonsymmetric boundary value problems can be studied in the unified and general framework of the general variational inequalities. Related to the variational inequalities, there is an equilibrium problem, which is mainly due to Blum and Oettli [2] and Noor and Oettli [3]. Such problems have been studied extensively in recent years due to their importance in pure and applied sciences. These problems have been extended and generalized in several directions using novel and innovative ideas and techniques. See [2], [4], [5], [6], [7], [8], [9], [10], [1], [11], [12], [13], [14], [15], [16], [17], [18], [3], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and [32] and the references therein for the applications, formulation, numerical methods and other aspects for the variational inequalities and equilibrium problems. Motivated and inspired by the research going on in this dynamic and interesting field, we introduce and study a new class of equilibrium problems and bifunction variational inequalities, called the mixed general equilibrium bifunction variational inequalities, in a unified manner. This class includes the mixed general equilibrium problems and the mixed general bifunction variational inequalities as special cases. We have shown that the minimum of a sum of differentiable nonconvex and directionally (Gateaux) differentiable nonconvex functions can be characterized by means of this new class of general equilibrium bifunction variational inequalities. We note that the projection method and its variant form, the resolvent method, cannot be used to suggest some iterative methods for solving the mixed general equilibrium bifunction variational inequalities. This fact motivated us to use the technique of the auxiliary principle of Glowinski et al. [9] to suggest and analyze some implicit iterative methods for solving the mixed general equilibrium bifunction variational inequalities; see Algorithm 3.1, Algorithm 3.2 and Algorithm 3.3. We also consider the convergence criterion for the proposed method (Algorithm 3.1) under suitable mild conditions, thus obtaining the main results (Theorem 3.1 and Theorem 3.2) of this paper. Several special cases of our main results are also considered. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results. These may be extended to other classes of variational inequalities and equilibrium problems. Comparison of these methods with other methods is an interesting problem for future research. Readers are encouraged to find novel applications of the general equilibrium bifunction variational inequalities in pure and applied sciences.