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In this paper, we propose a new inertial iterative method to solve classical variational inequalities with pseudomonotone and Lipschitz continuous operators in the setting of a real Hilbert space. The proposed iterative scheme is basically analogous to the extragradient method used to solve the problems of variational inequalities in real Hilbert spaces. The strong convergence of the proposed algorithm is set up with the prior knowledge of Lipschitz's constant of an operator. Finally, several computational experiments are listed to show the applicability and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a new method to solve the pseudomonotone equilibrium problem. This method can be seen as an extension and improvement of the Popov's extragradient method. We replace the fixed stepsize with a self-adapting stepsize formula that is revised on each iteration depends on previous iterations. A weak convergence theorem of the method is well established based on typical bifunctional cost assumptions. We also provide the application of our results to solve two kinds of variational inequality problems. Various numerical examples are provided to support our well-established convergence results, and we can see that the new approach provides a significant improvement in the number of iterations and the execution time. [ABSTRACT FROM AUTHOR]

The aim of this paper is to improve conditions proposed in recently published fixed point results for complete and compact fuzzy metric spaces (Ćirić in Chaos Solitons Fractals 42:146-154, 2009; Shen et al. in Appl. Math. Lett. 25:138-141, 2012). For this purpose, the altering distance functions are used. Moreover, in some of the results presented the class of t-norms is extended by using the theory of countable extensions of t-norms. The mentioned generalizations are illustrated by examples. [ABSTRACT FROM AUTHOR]

This paper deals with new methods for approximating a solution to the fixed point problem; find x ∈ F(T), where H is a Hilbert space, C is a closed convex subset of H, f is a ρ-contraction from C into H, 0 < ρ < 1, A is a strongly positive linear-bounded operator with coefficient γ > 0, 0 < γ < γ/ρ, T is a nonexpansive mapping on C, and PF(T) denotes the metric projection on the set of fixed point of T.Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality 〈(A-γf)x+τ(I-S)x, x- x) = 0 for x ∈ F(T), where τ ∈ [0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part. [ABSTRACT FROM AUTHOR]

Symmetries play an important role in the dynamics of physical systems. As an example, quantum physics and microworld are the basis of symmetry principles. These problems are reduced to solving inequalities in general. That is why in this article, we study the numerical approximation of solutions to variational inequality problems involving quasimonotone operators in an infinite-dimensional real Hilbert space. We prove that the iterative sequences generated by the proposed iterative schemes for solving variational inequalities with quasimonotone mapping converge strongly to some solution. The main advantage of the proposed iterative schemes is that they use a monotone and non-monotone step size rule based on operator knowledge rather than a Lipschitz constant or some line search method. We present a number of numerical experiments for the proposed algorithms. [ABSTRACT FROM AUTHOR]

Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator's Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems. [ABSTRACT FROM AUTHOR]

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method. [ABSTRACT FROM AUTHOR]

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones. [ABSTRACT FROM AUTHOR]

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others. [ABSTRACT FROM AUTHOR]

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method's efficiency and to compare it to other methods in the literature. [ABSTRACT FROM AUTHOR]