Sentence examples for methods for variational from inspiring English sources

We refer to some results of sub-supersolution methods for variational inequalities and the existence of solutions for differential equations studied in variable exponent Sobolev or Orlicz-Sobolev spaces (see, e.g., [4 11]).

So, [23] proposed several projection and extragradient methods rather than methods based on gap functions, which generalized the double-projection methods for variational inequality problem to equilibrium problems with a moving constraint set (K x)).

However, most of the results related to the existence of solutions and iterative methods for variational inequality problems have been investigated and considered so far to the case where the underlying set is convex.

This paper presents an adaptive method for variational curve smoothing based on level set implementation.

So, a high efficiency numerical method for variational inequalities is often beneficial to the relevant subject.

We provide different conditions and establish a strong convergence theorem for the MRHSD method for variational inequalities under these conditions.

Le [1] established a sub-supersolution method for variational inequalities with Leray-Lions operators in Sobolev spaces with variable exponents.

We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies.

In 2010, Zhang et al. (see [10]) proposed the following iteration method for variational inclusion problem (1.5) and equilibrium problem (1.6) in a Hilbert space : (1.9).

Ding et al. provided a three-step relaxed hybrid steepest-descent method for variational inequalities [11] and Yao et al. [19] provided a simple proof of the method under different conditions.

It is well known that the extragradient projection method is an efficient solution method for variational inequalities due to its low memory and low cost of computing [29, 30].