Sentence examples for equilibria problem from inspiring English sources

The system of mixed type equilibria problem (MSSMVIP-4) is defined as follows.

If F i = F 1, ∀ i = 1, 2, …, N, then the combination of equilibria problem (1.3) reduces to the equilibrium problem (1.1).

For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem.

In this section, we suggest and analyze our method for finding common solutions of the combination of equilibria problem (1.3), the combination of variational inequality problems (1.4), and the hierarchical fixed point problem (1.8).

Define the mapping ∑ i = 1 N a i F i : C × C → R. The combination of equilibria problem is to find x ∈ C such that ∑ i = 1 N a i F i ( x, y ) ≥ 0, ∀ y ∈ C, (1.3).

This high CPU time demand becomes particularly acute during the execution of compositional reservoir simulations as the phase equilibria problem needs to be solved repeatedly for each discretization block and at each iteration of the non-linear solver.

Equilibrium problems include fixed point problems, optimization problems, variational inequalities, Nash equilibria problems, and complementary problems as special cases.

Moreover, vector variational inequality problems have many important applications in vector optimization problems [7 9], vector equilibria problems [10, 11], and variational relation problems [12, 13].

On the other hand, the equilibrium problem provides a general mathematical model for a wide range of practical problems, such as optimization problems, Nash equilibria problems, fixed point problems, variational inequality problems, and complementarity problems, and has been investigated intensively.

Recently, a great deal of research on the solvability of inclusion problems is carried out using resolvent operator techniques, that have applications to other problems such as equilibria problems in economics, optimization and control theory, operations research, and mathematical programming.

In this section, we will recall some concepts of generalized and extended Nash equilibria of non-monetized, non-cooperative game which are defined in [3, 4] and we point out that our main result can be applied to prove some existence theorems for generalized and extended Nash equilibria problems.