Sentence examples for been shown that variational from inspiring English sources

It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of abstract equilibrium problems (e.g., [1, 2]).

It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems.

It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems (e.g., [20 23]).

It has been shown that variational inequalities provide the most natural, direct, simple and efficient framework for a general treatment of some unrelated problems arising in various fields of pure and applied sciences.

It has been shown that variational inequality theory provides the unified and efficient framework for a general treatment of a wide class of problems; for details, see Baiocchi and Capelo [1], Fukushima [2], Giannessi and Maugeri [3], Glowinski and Tallec [4], Noor et al. [5], Patriksson [6], Kinderlehrer and Stampacchia [7] and references therein.

Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems.

In fact, it has been shown that the variational inequalities theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of problems.

Moreover, it has been shown that the existing variational models are special cases of the current formulation.

It has been shown that equilibrium-like problems include equilibrium problems, variational-like inequalities, variational inequalities and their invariant forms as special cases.

It has been shown that equilibrium problems cover fixed point problems, variational inequality problems, inclusion problems, saddle point problems, complementarity problem, minimization problem, and Nash equilibrium problem.

It has been shown that equilibrium problems cover fixed point problems, variational inequality problems, inclusion problems, saddle problems, complementarity problem, minimization problem, and Nash equilibrium problem; see [1 20] and the references therein.